Finite Group Databasebeta

← #29737210

Subgroups — #29737210

Order 15616.

966 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840058C211yesmaximal, characteristic, complemented
2C2×#16839759C211yesmaximal, characteristic
3#21449510C211yesmaximal, characteristic, complemented
4#21448659C211yesmaximal, characteristic
5#21447339C211yesmaximal, characteristic, complemented
6#21447206C211yesmaximal, characteristic, complemented
7#21447206C211yesmaximal, characteristic, complemented
8#21446956C211yesmaximal, characteristic, complemented
9#21446090C211yesmaximal, characteristic, complemented
10#21446076C211yesmaximal, characteristic, complemented
11#21446020C211yesmaximal, characteristic, complemented
12#21445848C211yesmaximal, characteristic, complemented
13#21445835C211yesmaximal, characteristic, complemented
14#21445760C211yesmaximal, characteristic
15C61×#1850C211yesmaximal, characteristic, complemented
16C22×#11758222C2211yescharacteristic
17C2×#14519777C2211yescharacteristic
18C2×C4×Dic122C2211yescharacteristic
19C2×C4×Dic122C2211yescharacteristic
20C2×#14519736C2211yescharacteristic
21#16840058—24no—
22C2×#14519711C2211yescharacteristic
23C2×#14519711C2211yescharacteristic
24C2×#14519711C2211yescharacteristic
25C2×#14519709C2211yescharacteristic
26#16839759—24no—
27C2×#14519702C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519702C2211yescharacteristic
30C2×#14519702C2211yescharacteristic
31C2×#14519699C2211yescharacteristic
32C2×#14519699C2211yescharacteristic
33C2×#14519699C2211yescharacteristic
34C2×#14519699C2211yescharacteristic
35#16839685—21no—
36#16839674—21no—
37#16839666—21no—
38#16839664C2211yescharacteristic
39#16839664C2211yescharacteristic
40#16839662—21no—
41#16839662—21no—
42#16839660C2211yescharacteristic
43#16839660C2211yescharacteristic
44#16839649—21no—
45#16839628C2211yescharacteristic
46#16839622C2211yescharacteristic
47#16839622C2211yescharacteristic
48#16839621—21no—
49#16839609—21no—
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839525C2211yescharacteristic
53#16839525C2211yescharacteristic
54C23×C8×C61C2211yescharacteristic
55C2×C4⋊C8×C61C2211yescharacteristic
56C2×C4⋊C8×C61C2211yescharacteristic
57C2×C22⋊C8×C61C2211yescharacteristic
58C4×C22⋊C4×C61C2211yescharacteristic
59C61×#336C2211yescharacteristic
60C61×#336C2211yescharacteristic
61C2×#11758222—22no—
62C2×#11758222—22no—
63C2×#11758222—22no—
64C2×#11758222—41no—
65C2×#11758222—41no—
66C2×#11758222Q814yescomplemented
67C22×Dic122—21no—
68C22×Dic122C2311yescharacteristic
69C22×D244C2311yescharacteristic
70C22×C22⋊D61—21no—
71C22×C4×D61—21no—
72#14519777—22no—
73#14519777—41no—
74C4×Dic122—22no—
75C4×Dic122—22no—
76C4×Dic122—41no—
77C4×Dic122—41no—
78#14519736—22no—
79#14519736—41no—
80C2×C4⋊Dic61—21no—
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×C4⋊Dic61C2311yescharacteristic
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×#11758196—21no—
90C2×#11758196—21no—
91C2×C22⋊Dic61—21no—
92C2×C22⋊Dic61—21no—
93C2×C22⋊Dic61—21no—
94C2×#11758198—21no—
95C2×#11758198—21no—
96C2×#11758198—21no—
97C2×#11758198—21no—
98C2×C4×Dic61—21no—
99C2×C4×Dic61—21no—
100C2×C4×Dic61—21no—
101#14519711—22no—
102#14519711—22no—
103#14519711—41no—
104#14519709—41no—
105#14519702—22no—
106#14519702—22no—
107#14519702—22no—
108#14519702D414yes—
109#14519699—22no—
110#14519699—22no—
111#14519699—41no—
112#14519699—41no—
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118#14519684—21no—
119C23×C4×C61C2311yescharacteristic
120C22×C8×C61—22no—
121C22×C8×C61—22no—
122C22×C8×C61—41no—
123C22×C8×C61C2311yescharacteristic
124C22×C8×C61C2311yescharacteristic
125C22×C8×C61C2311yescharacteristic
126C22×C8×C61C2311yescharacteristic
127C2×C22⋊C4×C61D411yescharacteristic
128C2×C22⋊C4×C61D411yescharacteristic
129C2×C42×C61C2311yescharacteristic
130C2×C42×C61C2311yescharacteristic
131C4⋊C8×C61—22no—
132C4⋊C8×C61—41no—
133C22⋊C8×C61—22no—
134#95×C61D411yescharacteristic
135#95×C61D411yescharacteristic
136C24×C61C2×D411yescharacteristic
137C23×D61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×Q812yes—
148C22×C4×C61#4112yes—
149C22×C4×C61C2411yescharacteristic
150C2×C22⋊D61—24no—
151C2×C22⋊D61—42no—
152C2×C22⋊D61—42no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160#11758222—28no—
161#11758222—44no—
162#11758222—44no—
163#11758222—44no—
164C2×D244—21no—
165C2×D244—21no—
166C2×D244—21no—
167C2×D244—21no—
168C2×D244—21no—
169C2×D244C2×Q812yes—
170C2×C4×D61—24no—
171C2×C4×D61—42no—
172C2×C4×D61—42no—
173C2×Dic122—21no—
174C2×Dic122—21no—
175C2×Dic122—21no—
176C2×Dic122—42no—
177C2×Dic122—42no—
178C2×Dic122—81no—
179C2×Dic122C2×D412yes—
180C2×Dic122C2×Q812yes—
181C2×Dic122#4112yes—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C2×C8×C61—21no—
185C2×C8×C61—21no—
186C2×C8×C61—21no—
187C2×C8×C61—21no—
188C2×C8×C61—21no—
189C2×C8×C61—21no—
190C2×C8×C61—21no—
191C2×C8×C61—21no—
192C2×C8×C61—21no—
193C2×C8×C61—44no—
194C2×C8×C61C2×D412yes—
195C22⋊C4×C61—22no—
196C22⋊C4×C61—41no—
197C42×C61—21no—
198C42×C61—21no—
199C42×C61—21no—
200C42×C61—21no—
201C22⋊Dic61—42no—
202C22⋊Dic61—81no—
203C22⋊Dic61—81no—
204#11758198—42no—
205#11758198—42no—
206#11758198—81no—
207#11758198—81no—
208C4⋊Dic61—21no—
209C4⋊Dic61—21no—
210C4⋊Dic61—21no—
211C4⋊Dic61—21no—
212C4⋊Dic61—21no—
213C4⋊Dic61—21no—
214C4⋊Dic61—21no—
215C4⋊Dic61—21no—
216C4⋊Dic61—21no—
217C4⋊Dic61—21no—
218C4⋊Dic61—21no—
219C4⋊Dic61—42no—
220C4⋊Dic61C2×D412yes—
221#11758196—42no—
222#11758196—42no—
223#11758196—42no—
224#11758196—81no—
225C4×Dic61—42no—
226C4×Dic61—42no—
227C4×Dic61—42no—
228C23×C61—21no—
229C23×C61—21no—
230C23×C61—21no—
231C23×C61—21no—
232C23×C61—21no—
233C23×C61—21no—
234C23×C61C22⋊Q812yes—
235C23×C61C22×D411yescharacteristic
236C22×D61—22no—
237C22×D61—41no—
238C22×D61—41no—
239C22×D61—41no—
240C22×D61—41no—
241C22×D61—41no—
242C22×D61—41no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—22no—
248C2×C4×C61—22no—
249C2×C4×C61—22no—
250C2×C4×C61—22no—
251C2×C4×C61—21no—
252C2×C4×C61—21no—
253C2×C4×C61—21no—
254C2×C4×C61—21no—
255C2×C4×C61—21no—
256C2×C4×C61—21no—
257C2×C4×C61—21no—
258C2×C4×C61—21no—
259C2×C4×C61—21no—
260C2×C4×C61C22⋊Q812yes—
261C2×C4×C61#13512yes—
262C2×C4×C61C22×D411yescharacteristic
263C2×C4×C61C22×Q811yescharacteristic
264C2×C4×C61C2×#4111yescharacteristic
265C2×C4×C61C2×#4111yescharacteristic
266C2×C4×C61H(5,𝔽2)11yescharacteristic
267C2×C4×C61H(5,𝔽2)11yescharacteristic
268C22⋊D61—48no—
269C2×Dic61—22no—
270C2×Dic61—22no—
271C2×Dic61—22no—
272C2×Dic61—22no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61—41no—
288C2×Dic61—41no—
289C2×Dic61—41no—
290D244—22no—
291D244—22no—
292D244—41no—
293D244—41no—
294C4×D61—48no—
295Dic122—22no—
296Dic122—22no—
297Dic122—22no—
298Dic122—84no—
299Dic122C22⋊Q814yes—
300C8×C61—22no—
301C8×C61—41no—
302C8×C61—41no—
303C8×C61—41no—
304#25959—611nomaximal, Sylow
305C22×C61—22no—
306C22×C61—22no—
307C22×C61—22no—
308C22×C61—41no—
309C22×C61—41no—
310C22×C61—41no—
311C22×C61—41no—
312C22×C61C2×C22⋊Q811yescharacteristic
313C22×C61D4211yescharacteristic
314C22×C61#54611yescharacteristic
315C22×C61C2×#13511yescharacteristic
316C22×C61#57511yescharacteristic
317C22×C61#57611yescharacteristic
318C22×C61#57811yescharacteristic
319C2×D61—42no—
320C2×D61—42no—
321C2×D61—42no—
322C2×D61—81no—
323C2×D61—81no—
324C2×D61—81no—
325C2×D61—81no—
326C4×C61—22no—
327C4×C61—22no—
328C4×C61—41no—
329C4×C61—41no—
330C4×C61—41no—
331C4×C61C2×C22⋊Q811yescharacteristic
332C4×C61#54311yescharacteristic
333C4×C61#54611yescharacteristic
334C4×C61#55111yescharacteristic
335Dic61—24no—
336Dic61—42no—
337Dic61—81no—
338Dic61—81no—
339Dic61—81no—
340Dic61—81no—
341Dic61—81no—
342C2×#477—611no—
343C2×#416—611no—
344#2439—611no—
345#2341—611no—
346#2092—611no—
347#2067—611no—
348#2067—611no—
349#2023—611no—
350#1865—611no—
351#1863—611no—
352#1854—611no—
353#1850D6111yescharacteristic, complemented
354#1808—611no—
355#1805—611no—
356#1788—611no—
357C2×C61—42no—
358C2×C61#243711yescharacteristic
359C2×C61#307611yescharacteristic
360C2×C61#308311yescharacteristic
361C2×C61#326811yescharacteristic
362C2×C61#328511yescharacteristic
363C2×C61#330511yescharacteristic
364C2×C61#331311yescharacteristic
365D61—82no—
366C22×#41—611no—
367C23×C8C2×D6111yescharacteristic
368C2×C4⋊Q8—611no—
369C2×C4×Q8—611no—
370C2×C4×Q8—611no—
371C2×#117—611no—
372#477—1224no—
373C2×#107—611no—
374C2×#107—611no—
375C2×#107—611no—
376C2×#106—611no—
377C2×C4⋊C8C2×D6111yescharacteristic
378C2×C4⋊C8C2×D6111yescharacteristic
379#416—1224no—
380C2×#103—611no—
381C2×#103—611no—
382C2×#103—611no—
383C2×#103—611no—
384C2×#102—611no—
385C2×#102—611no—
386C2×#102—611no—
387C2×#102—611no—
388C2×C22⋊C8C2×D6111yescharacteristic
389#396—1221no—
390#393—1221no—
391#391—1221no—
392#390—611no—
393#390—611no—
394#390—1221no—
395#390—1221no—
396#389—611no—
397#389—611no—
398#387—1221no—
399#382—611no—
400#380—611no—
401#380—611no—
402#380—1221no—
403C4×C22⋊C4—1221no—
404C4×C22⋊C4C2×D6111yescharacteristic
405#340—611no—
406#340—611no—
407#340—611no—
408#340—611no—
409#336C2×D6111yescharacteristic
410#336C2×D6111yescharacteristic
411C61#2595911yescharacteristic, complemented, Sylow
412C2×#41—614no—
413C2×#41—1222no—
414C2×#41—1222no—
415C2×#41—1222no—
416C2×#41—2441no—
417C2×#41—2441no—
418C22×Q8—611no—
419C22×Q8—1221no—
420C22×D4—611no—
421C22×D4—1221no—
422C23×C4—1221no—
423C23×C4C22×D6111yescharacteristic
424C22×C8—22no—
425C22×C8—22no—
426C22×C8—41no—
427C22×C8C22×D6111yescharacteristic
428C22×C8C22×D6111yescharacteristic
429C22×C8C22×D6111yescharacteristic
430C22×C8C22×D6111yescharacteristic
431C4⋊Q8—1222no—
432C4⋊Q8—2441no—
433C4×Q8—1222no—
434C4×Q8—1222no—
435C4×Q8—2441no—
436C4×Q8—2441no—
437#117—1222no—
438#117—2441no—
439C2×C4⋊C4—611no—
440C2×C4⋊C4—611no—
441C2×C4⋊C4—611no—
442C2×C4⋊C4—611no—
443C2×C4⋊C4—611no—
444C2×C4⋊C4—611no—
445C2×C4⋊C4—1221no—
446C2×C4⋊C4—1221no—
447C2×C4⋊C4—1221no—
448C2×C4⋊C4—1221no—
449C2×C4⋊C4—1221no—
450C2×C22⋊C4—1221no—
451C2×C22⋊C4—1221no—
452C2×C22⋊C4—1221no—
453C2×C22⋊C4—1221no—
454C2×C22⋊C4—1221no—
455C2×C22⋊C4—1221no—
456C2×C22⋊C4—1221no—
457C2×C22⋊C4D24411yescharacteristic
458C2×C22⋊C4D24411yescharacteristic
459C2×C42—1221no—
460C2×C42—1221no—
461C2×C42—1221no—
462C2×C42C22×D6111yescharacteristic
463C2×C42C22×D6111yescharacteristic
464#107—1222no—
465#107—1222no—
466#107—2441no—
467#106—2441no—
468C4⋊C8—22no—
469C4⋊C8—41no—
470#103—614no—
471#103—1222no—
472#103—1222no—
473#103—1222no—
474#102—1222no—
475#102—1222no—
476#102—2441no—
477#102—2441no—
478C22⋊C8—22no—
479#95—1221no—
480#95—1221no—
481#95—1221no—
482#95—1221no—
483#95—1221no—
484#95—1221no—
485#95D24411yescharacteristic
486#95D24411yescharacteristic
487C24—1221no—
488C24C2×D24411yescharacteristic
489#41—1228no—
490#41—2444no—
491#41—2444no—
492#41—2444no—
493C2×Q8—612no—
494C2×Q8—612no—
495C2×Q8—612no—
496C2×Q8—1221no—
497C2×Q8—1221no—
498C2×Q8—1221no—
499C2×Q8—4881no—
500C2×Q8—2442no—
501C2×Q8—2442no—
502C2×D4—612no—
503C2×D4—1224no—
504C2×D4—1221no—
505C2×D4—1221no—
506C2×D4—1221no—
507C2×D4—1221no—
508C2×D4—1221no—
509C2×D4—2442no—
510C2×D4—2442no—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—21no—
514C22×C4—21no—
515C22×C4—1221no—
516C22×C4—1221no—
517C22×C4—1224no—
518C22×C4—1221no—
519C22×C4—1221no—
520C22×C4—1221no—
521C22×C4—1221no—
522C22×C4—1221no—
523C22×C4—2442no—
524C22×C4—2442no—
525C22×C4C2×D24411yescharacteristic
526C22×C4C2×D24411yescharacteristic
527C22×C4C2×D24411yescharacteristic
528C22×C4C2×D24411yescharacteristic
529C22×C4C2×D24411yescharacteristic
530C22×C4#1175822412yes—
531C22×C4Q8×D6112yes—
532C22×C4C23×D6111yescharacteristic
533C2×C8—21no—
534C2×C8—21no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—21no—
538C2×C8—21no—
539C2×C8—21no—
540C2×C8—21no—
541C2×C8—21no—
542C2×C8—21no—
543C2×C8—21no—
544C2×C8—44no—
545C2×C8D4×D6112yes—
546C4⋊C4—612no—
547C4⋊C4—1221no—
548C4⋊C4—1221no—
549C4⋊C4—1221no—
550C4⋊C4—1221no—
551C4⋊C4—1221no—
552C4⋊C4—1221no—
553C4⋊C4—1221no—
554C4⋊C4—1221no—
555C4⋊C4—1221no—
556C4⋊C4—1221no—
557C4⋊C4—1221no—
558C4⋊C4—4881no—
559C4⋊C4—2442no—
560C4⋊C4—2442no—
561C4⋊C4—2442no—
562C4⋊C4—2442no—
563C22⋊C4—22no—
564C22⋊C4—41no—
565C22⋊C4—4881no—
566C22⋊C4—4881no—
567C22⋊C4—4881no—
568C22⋊C4—4881no—
569C22⋊C4—2442no—
570C22⋊C4—2442no—
571C22⋊C4—2442no—
572C42—21no—
573C42—21no—
574C42—21no—
575C42—21no—
576C42—2442no—
577C42—2442no—
578C42—2442no—
579C23—21no—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—1222no—
586C23—2441no—
587C23—2441no—
588C23—2441no—
589C23—2441no—
590C23—2441no—
591C23—2441no—
592C23#1451975712yes—
593C23C22×D24411yescharacteristic
594Q8—614no—
595Q8—1222no—
596Q8—1222no—
597Q8—1222no—
598Q8—4884no—
599D4—1222no—
600D4—1222no—
601D4—2448no—
602D4—2441no—
603D4—2441no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—1222no—
622C2×C4—1222no—
623C2×C4—1222no—
624C2×C4—1222no—
625C2×C4—2448no—
626C2×C4—2441no—
627C2×C4—2441no—
628C2×C4—2441no—
629C2×C4—2441no—
630C2×C4—2441no—
631C2×C4—2441no—
632C2×C4—2441no—
633C2×C4—2441no—
634C2×C4—2441no—
635C2×C4—2441no—
636C2×C4—2441no—
637C2×C4—2441no—
638C2×C4—2441no—
639C2×C4—2441no—
640C2×C4—2441no—
641C2×C4—2441no—
642C2×C4—2441no—
643C2×C4#1451975712yes—
644C2×C4#1451979912yes—
645C2×C4C2×D4×D6111yescharacteristic
646C2×C4C2×Q8×D6111yescharacteristic
647C2×C4#41×D6111yescharacteristic
648C2×C4C2×#1175822411yescharacteristic
649C2×C4#1451982211yescharacteristic
650C2×C4#1451982311yescharacteristic
651C8—22no—
652C8—41no—
653C8—41no—
654C8—41no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—41no—
659C22—41no—
660C22—41no—
661C22—41no—
662C22—4881no—
663C22—4881no—
664C22—4881no—
665C22—4881no—
666C22—2442no—
667C22—2442no—
668C22—2442no—
669C22C2×#1451975711yescharacteristic
670C22D4×D24411yescharacteristic
671C22#1684040711yescharacteristic
672C22C2×#1451979911yescharacteristic
673C22#1684056211yescharacteristic
674C22#1684056711yescharacteristic
675C22#1684057911yescharacteristic
676C4—22no—
677C4—22no—
678C4—41no—
679C4—41no—
680C4—41no—
681C4—1224no—
682C4—4881no—
683C4—4881no—
684C4—4881no—
685C4—4881no—
686C4—4881no—
687C4—2442no—
688C4C22⋊Q8×D6111yescharacteristic
689C4#1684038711yescharacteristic
690C4#1684039911yescharacteristic
691C4#1684043511yescharacteristic
692C2—42no—
693C2—4882no—
694C2#2144948311yescharacteristic
695C2#2145560911yescharacteristic
696C2#2145567211yescharacteristic
697C2#2145736411yescharacteristic
698C2#2145756211yescharacteristic
699C2#2145777011yescharacteristic
700C2#2145788711yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1850C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
hypercenter#1850D61
hypocenterC61#25959
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1850

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#16840058C2semidirect
3. #21449510C2semidirect
5. #21447339C2semidirect
6. #21447206C2semidirect
7. #21447206C2semidirect
8. #21446956C2semidirect
9. #21446090C2semidirect
10. #21446076C2semidirect
11. #21446020C2semidirect
12. #21445848C2semidirect
13. #21445835C2semidirect
15. C61×#1850C2semidirect
66. C2×#11758222Q8semidirect
353. #1850D61semidirect
411. C61#25959semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168400582
C2×#168397592
#214495102
#214486592
#214473392
#214472062
#214469562
#214460902
#214460762
#214460202
#214458482
#214458352
#214457602
C61×#18502
#2595961