Finite Group Databasebeta

← #29737201

Subgroups — #29737201

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×#16839757C211yesmaximal, characteristic
2#21449506C211yesmaximal, characteristic, complemented
3#21448657C211yesmaximal, characteristic
4#21447337C211yesmaximal, characteristic, complemented
5#21447206C211yesmaximal, characteristic, complemented
6#21447202C211yesmaximal, characteristic, complemented
7#21446952C211yesmaximal, characteristic, complemented
8#21446087C211yesmaximal, characteristic, complemented
9#21446075C211yesmaximal, characteristic, complemented
10#21446015C211yesmaximal, characteristic, complemented
11#21445995C211yesmaximal, characteristic, complemented
12#21445845C211yesmaximal, characteristic, complemented
13#21445832C211yesmaximal, characteristic, complemented
14#21445758C211yesmaximal, characteristic
15C2×C61×#477C211yesmaximal, characteristic, complemented
16C22×#11758226C2211yescharacteristic
17C22×C8×D61C2211yescharacteristic
18C2×Q8×Dic61C2211yescharacteristic
19C2×#14519735C2211yescharacteristic
20C2×#14519711C2211yescharacteristic
21C2×#14519708C2211yescharacteristic
22C2×#14519706C2211yescharacteristic
23#16839757—24no—
24C2×#14519701C2211yescharacteristic
25C2×#14519701C2211yescharacteristic
26C2×#14519699C2211yescharacteristic
27C2×#14519699C2211yescharacteristic
28C2×#14519697C2211yescharacteristic
29C2×#14519697C2211yescharacteristic
30C2×#14519689C2211yescharacteristic
31#16839684—21no—
32#16839673—21no—
33#16839669—21no—
34#16839664—21no—
35#16839664C2211yescharacteristic
36#16839663—21no—
37#16839663C2211yescharacteristic
38#16839661C2211yescharacteristic
39#16839660C2211yescharacteristic
40#16839643—21no—
41#16839627C2211yescharacteristic
42#16839619—21no—
43#16839619C2211yescharacteristic
44#16839619C2211yescharacteristic
45C4×#11758198—21no—
46C4×#11758198C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839517C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×C4⋊Q8×C61C2211yescharacteristic
54C2×C4×Q8×C61C2211yescharacteristic
55C61×#477—24no—
56C2×#107×C61C2211yescharacteristic
57C2×#107×C61C2211yescharacteristic
58C2×C4⋊C8×C61C2211yescharacteristic
59C2×#103×C61C2211yescharacteristic
60C2×#103×C61C2211yescharacteristic
61C2×#11758226—22no—
62C2×#11758226—22no—
63C2×#11758226—22no—
64C2×#11758226—41no—
65C2×#11758226—41no—
66C2×#11758226Q814yescomplemented
67C22×D244—21no—
68C22×D244C2311yescharacteristic
69C22×C4×D61—21no—
70C22×C4×D61C2311yescharacteristic
71C22×#92418C2311yescharacteristic
72C22×#92418C2311yescharacteristic
73C2×C8×D61—22no—
74C2×C8×D61—22no—
75C2×C8×D61—41no—
76Q8×Dic61—22no—
77Q8×Dic61—41no—
78#14519735—22no—
79#14519735—41no—
80C2×C4⋊Dic61—21no—
81C2×C4⋊Dic61—21no—
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×#11758196—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×#11758198—21no—
91C2×#11758198—21no—
92C2×#11758198—21no—
93C2×#11758198D411yescharacteristic
94C2×#11758198D411yescharacteristic
95C2×C4×Dic61—21no—
96C2×C4×Dic61—21no—
97C2×C4×Dic61C2311yescharacteristic
98#14519711—41no—
99#14519708—41no—
100#14519706—41no—
101#14519701—22no—
102#14519701—22no—
103#14519699—22no—
104#14519699—41no—
105#14519697—22no—
106#14519697—41no—
107#14519689—22no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684D411yescharacteristic
115#14519684D411yescharacteristic
116C22×Q8×C61—21no—
117C22×Q8×C61C2311yescharacteristic
118C22×C8×C61C2311yescharacteristic
119C22×C8×C61C2311yescharacteristic
120C4⋊Q8×C61—22no—
121C4⋊Q8×C61—41no—
122C4×Q8×C61—22no—
123C4×Q8×C61—41no—
124C2×C4⋊C4×C61—21no—
125C2×C4⋊C4×C61—21no—
126C2×C4⋊C4×C61C2311yescharacteristic
127C2×C4⋊C4×C61C2311yescharacteristic
128C2×C4⋊C4×C61C2311yescharacteristic
129C2×C42×C61—21no—
130C2×C42×C61C2311yescharacteristic
131#107×C61—22no—
132#107×C61—22no—
133C4⋊C8×C61—22no—
134#103×C61—22no—
135#103×C61D414yes—
136C23×D61—21no—
137C23×D61C2×D411yescharacteristic
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—42no—
142C2×Q8×C61—42no—
143C2×Q8×C61—81no—
144C2×Q8×C61C2×D412yes—
145C2×Q8×C61C2×Q812yes—
146C2×Q8×C61#4112yes—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2411yescharacteristic
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61C2×D411yescharacteristic
158C22×Dic61C2×D411yescharacteristic
159C22×Dic61C2×D411yescharacteristic
160#11758226—28no—
161#11758226—44no—
162#11758226—44no—
163#11758226—44no—
164C2×D244—21no—
165C2×D244—21no—
166C2×D244—21no—
167C2×D244—21no—
168C2×D244—24no—
169C2×D244—21no—
170C2×D244—42no—
171C2×D244—42no—
172C2×D244C2×Q812yes—
173C2×C4×D61—21no—
174C2×C4×D61—21no—
175C2×C4×D61—21no—
176C2×C4×D61—21no—
177C2×C4×D61—24no—
178C2×C4×D61—42no—
179C2×C4×D61—42no—
180C2×C4×D61C2×Q812yes—
181C2×C4×D61#4112yes—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C2×C8×C61—21no—
185C2×C8×C61—21no—
186C2×C8×C61—21no—
187C2×C8×C61C2×D412yes—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—21no—
193C4⋊C4×C61—42no—
194C4⋊C4×C61—42no—
195C4⋊C4×C61C2×D412yes—
196C42×C61—21no—
197C42×C61—21no—
198C42×C61—42no—
199#11758198—22no—
200#11758198—41no—
201#11758198—42no—
202#11758198—42no—
203#11758198—42no—
204#11758198—81no—
205#11758198—81no—
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—42no—
215C4⋊Dic61—81no—
216#11758196—42no—
217C4×Dic61—21no—
218C4×Dic61—21no—
219C4×Dic61—42no—
220C4×Dic61—42no—
221C2×#92418—21no—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418—21no—
225C2×#92418—21no—
226C2×#92418—21no—
227C8×D61—44no—
228C23×C61C22×D411yescharacteristic
229C22×D61—21no—
230C22×D61—21no—
231C22×D61—21no—
232C22×D61—21no—
233C22×D61—21no—
234C22×D61—21no—
235C22×D61—22no—
236C22×D61—41no—
237C22×D61—41no—
238C22×D61—41no—
239C22×D61—41no—
240C22×D61—41no—
241C22×D61—41no—
242C22×D61C22⋊Q812yes—
243Q8×C61—22no—
244Q8×C61—22no—
245Q8×C61—22no—
246Q8×C61—84no—
247Q8×C61C22⋊Q814yes—
248C2×C4×C61—22no—
249C2×C4×C61—21no—
250C2×C4×C61—22no—
251C2×C4×C61—22no—
252C2×C4×C61—21no—
253C2×C4×C61—22no—
254C2×C4×C61—21no—
255C2×C4×C61—21no—
256C2×C4×C61—21no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—41no—
262C2×C4×C61—41no—
263C2×C4×C61—41no—
264C2×C4×C61—41no—
265C2×C4×C61#13512yes—
266C2×C4×C61C22×D411yescharacteristic
267C2×C4×C61C22×Q811yescharacteristic
268C2×C4×C61C2×#4111yescharacteristic
269C2×C4×C61C2×#4111yescharacteristic
270C2×C4×C61H(5,𝔽2)11yescharacteristic
271C2×C4×C61H(5,𝔽2)11yescharacteristic
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—21no—
275C2×Dic61—21no—
276C2×Dic61—21no—
277C2×Dic61—21no—
278C2×Dic61—21no—
279C2×Dic61—21no—
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×Dic61C22⋊Q812yes—
290D244—22no—
291D244—22no—
292D244—48no—
293D244—41no—
294D244—41no—
295C4×D61—22no—
296C4×D61—22no—
297C4×D61—22no—
298C4×D61—22no—
299C4×D61—48no—
300C8×C61—22no—
301C8×C61—41no—
302#92418—41no—
303#92418—41no—
304#25959—611nomaximal, Sylow
305C22×C61C2×C22⋊Q811yescharacteristic
306C22×C61D4211yescharacteristic
307C22×C61#54611yescharacteristic
308C22×C61C2×#13511yescharacteristic
309C22×C61#57511yescharacteristic
310C22×C61#57611yescharacteristic
311C22×C61#57811yescharacteristic
312C2×D61—22no—
313C2×D61—22no—
314C2×D61—22no—
315C2×D61—41no—
316C2×D61—41no—
317C2×D61—41no—
318C2×D61—41no—
319C2×D61—42no—
320C2×D61—42no—
321C2×D61—42no—
322C2×D61—81no—
323C2×D61—81no—
324C2×D61—81no—
325C2×D61—81no—
326C4×C61—24no—
327C4×C61—22no—
328C4×C61—41no—
329C4×C61—42no—
330C4×C61—81no—
331C4×C61—81no—
332C4×C61C2×C22⋊Q811yescharacteristic
333C4×C61#54311yescharacteristic
334C4×C61#54611yescharacteristic
335C4×C61#55111yescharacteristic
336Dic61—22no—
337Dic61—41no—
338Dic61—41no—
339Dic61—81no—
340Dic61—81no—
341Dic61—81no—
342C2×#477D6111yescharacteristic, complemented
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#1850—611no—
354#1808—611no—
355#1805—611no—
356#1788—611no—
357C2×C61#243711yescharacteristic
358C2×C61#307611yescharacteristic
359C2×C61#308311yescharacteristic
360C2×C61#326811yescharacteristic
361C2×C61#328511yescharacteristic
362C2×C61#330511yescharacteristic
363C2×C61#331311yescharacteristic
364D61—42no—
365D61—82no—
366C22×#41—611no—
367C23×C8—611no—
368C2×C4⋊Q8C2×D6111yescharacteristic
369C2×C4×Q8—611no—
370C2×C4×Q8C2×D6111yescharacteristic
371C2×#117—611no—
372#477—24no—
373C2×#107—611no—
374C2×#107C2×D6111yescharacteristic
375C2×#107C2×D6111yescharacteristic
376C2×#106—611no—
377C2×C4⋊C8—611no—
378C2×C4⋊C8C2×D6111yescharacteristic
379#416—1224no—
380C2×#103—611no—
381C2×#103—611no—
382C2×#103C2×D6111yescharacteristic
383C2×#103C2×D6111yescharacteristic
384C2×#102—611no—
385C2×#102—611no—
386C2×#102—611no—
387C2×#102—611no—
388C2×C22⋊C8—611no—
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—611no—
404C4×C22⋊C4—1221no—
405#340—611no—
406#340—611no—
407#340—611no—
408#340—611no—
409#336—611no—
410#336—611no—
411C61#2595911yescharacteristic, complemented, Sylow
412C2×#41—614no—
413C2×#41—1222no—
414C2×#41—1222no—
415C2×#41—1222no—
416C2×#41—2441no—
417C2×#41—2441no—
418C22×Q8—21no—
419C22×Q8C22×D6111yescharacteristic
420C22×D4—611no—
421C22×D4—1221no—
422C23×C4—611no—
423C23×C4—1221no—
424C22×C8—611no—
425C22×C8—611no—
426C22×C8—1222no—
427C22×C8—1222no—
428C22×C8—2441no—
429C22×C8C22×D6111yescharacteristic
430C22×C8C22×D6111yescharacteristic
431C4⋊Q8—22no—
432C4⋊Q8—41no—
433C4×Q8—22no—
434C4×Q8—41no—
435C4×Q8—1222no—
436C4×Q8—2441no—
437#117—1222no—
438#117—2441no—
439C2×C4⋊C4—21no—
440C2×C4⋊C4—21no—
441C2×C4⋊C4—611no—
442C2×C4⋊C4—611no—
443C2×C4⋊C4—611no—
444C2×C4⋊C4—1221no—
445C2×C4⋊C4—1221no—
446C2×C4⋊C4—1221no—
447C2×C4⋊C4C22×D6111yescharacteristic
448C2×C4⋊C4C22×D6111yescharacteristic
449C2×C4⋊C4C22×D6111yescharacteristic
450C2×C22⋊C4—611no—
451C2×C22⋊C4—611no—
452C2×C22⋊C4—1221no—
453C2×C22⋊C4—1221no—
454C2×C22⋊C4—1221no—
455C2×C22⋊C4—1221no—
456C2×C22⋊C4—1221no—
457C2×C22⋊C4—1221no—
458C2×C22⋊C4—1221no—
459C2×C42—21no—
460C2×C42—611no—
461C2×C42—1221no—
462C2×C42—1221no—
463C2×C42C22×D6111yescharacteristic
464#107—22no—
465#107—22no—
466#107—2441no—
467#106—2441no—
468C4⋊C8—22no—
469C4⋊C8—2441no—
470#103—22no—
471#103—1222no—
472#103—1222no—
473#103C22⋊D6114yes—
474#102—1222no—
475#102—1222no—
476#102—2441no—
477#102—2441no—
478C22⋊C8—1222no—
479#95—611no—
480#95—611no—
481#95—1221no—
482#95—1221no—
483#95—1221no—
484#95—1221no—
485#95—1221no—
486#95—1221no—
487C24—611no—
488C24—1221no—
489#41—1228no—
490#41—2444no—
491#41—2444no—
492#41—2444no—
493C2×Q8—21no—
494C2×Q8—21no—
495C2×Q8—21no—
496C2×Q8—42no—
497C2×Q8—42no—
498C2×Q8—81no—
499C2×Q8Q8×D6112yes—
500C2×Q8#1175822612yes—
501C2×Q8C2×C22⋊D6112yes—
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—611no—
516C22×C4—612no—
517C22×C4—612no—
518C22×C4—611no—
519C22×C4—611no—
520C22×C4—1224no—
521C22×C4—1221no—
522C22×C4—1221no—
523C22×C4—1221no—
524C22×C4—1221no—
525C22×C4—1221no—
526C22×C4—1221no—
527C22×C4—1221no—
528C22×C4—2442no—
529C22×C4—2442no—
530C22×C4D4×D6111yescharacteristic
531C22×C4D4×D6111yescharacteristic
532C22×C4C23×D6111yescharacteristic
533C2×C8—21no—
534C2×C8—21no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—21no—
538C2×C8—1221no—
539C2×C8—1221no—
540C2×C8—1221no—
541C2×C8—1221no—
542C2×C8—1221no—
543C2×C8—1221no—
544C2×C8—2444no—
545C2×C8C2×C22⋊D6112yes—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—21no—
549C4⋊C4—21no—
550C4⋊C4—21no—
551C4⋊C4—42no—
552C4⋊C4—42no—
553C4⋊C4—1221no—
554C4⋊C4—1221no—
555C4⋊C4—1221no—
556C4⋊C4—1221no—
557C4⋊C4—1221no—
558C4⋊C4—1221no—
559C4⋊C4—4881no—
560C4⋊C4—2442no—
561C4⋊C4—2442no—
562C4⋊C4C2×C22⋊D6112yes—
563C22⋊C4—1222no—
564C22⋊C4—4881no—
565C22⋊C4—4881no—
566C22⋊C4—4881no—
567C22⋊C4—4881no—
568C22⋊C4—2441no—
569C22⋊C4—2442no—
570C22⋊C4—2442no—
571C22⋊C4—2442no—
572C42—21no—
573C42—21no—
574C42—42no—
575C42—1221no—
576C42—1221no—
577C42—2442no—
578C42—2442no—
579C23—612no—
580C23—1221no—
581C23—1221no—
582C23—1221no—
583C23—1221no—
584C23—1221no—
585C23—1221no—
586C23—1222no—
587C23—2441no—
588C23—2441no—
589C23—2441no—
590C23—2441no—
591C23—2441no—
592C23—2441no—
593C23C2×D4×D6111yescharacteristic
594Q8—22no—
595Q8—22no—
596Q8—22no—
597Q8—84no—
598Q8#1451975314yes—
599D4—1222no—
600D4—1222no—
601D4—2448no—
602D4—2441no—
603D4—2441no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—22no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—612no—
622C2×C4—1221no—
623C2×C4—1221no—
624C2×C4—1222no—
625C2×C4—1222no—
626C2×C4—1222no—
627C2×C4—1222no—
628C2×C4—1221no—
629C2×C4—1221no—
630C2×C4—1221no—
631C2×C4—1221no—
632C2×C4—1221no—
633C2×C4—1221no—
634C2×C4—2448no—
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—2441no—
644C2×C4#1451980012yes—
645C2×C4C22×C22⋊D6111yescharacteristic
646C2×C4C2×Q8×D6111yescharacteristic
647C2×C4#41×D6111yescharacteristic
648C2×C4C2×#1175822611yescharacteristic
649C2×C4#1451982211yescharacteristic
650C2×C4#1451982311yescharacteristic
651C8—22no—
652C8—41no—
653C8—2441no—
654C8—2441no—
655C22—1222no—
656C22—1222no—
657C22—1222no—
658C22—4881no—
659C22—4881no—
660C22—4881no—
661C22—4881no—
662C22—2442no—
663C22—2442no—
664C22—2442no—
665C22—2441no—
666C22—2441no—
667C22—2441no—
668C22—2441no—
669C22C22⋊Q8×D6111yescharacteristic
670C22D4×C22⋊D6111yescharacteristic
671C22#1684040311yescharacteristic
672C22C2×#1451980011yescharacteristic
673C22#1684055811yescharacteristic
674C22#1684056611yescharacteristic
675C22#1684057511yescharacteristic
676C4—24no—
677C4—22no—
678C4—41no—
679C4—42no—
680C4—81no—
681C4—81no—
682C4—1222no—
683C4—4881no—
684C4—4881no—
685C4—4881no—
686C4—2441no—
687C4—2441no—
688C4C2×#1451975311yescharacteristic
689C4#1684038711yescharacteristic
690C4#1684040011yescharacteristic
691C4#1684043911yescharacteristic
692C2—4882no—
693C2—2442no—
694C2#2144948211yescharacteristic
695C2#2145561011yescharacteristic
696C2#2145566711yescharacteristic
697C2#2145735311yescharacteristic
698C2#2145756011yescharacteristic
699C2#2145776611yescharacteristic
700C2#2145788211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#477C2
socleC23×C61C22×D4
prime radicalC22×#11758226C22
hypercenterC2×#477D61
hypocenterC61#25959
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#477

Complements

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

NH ≅ G/Ntype
2. #21449506C2semidirect
4. #21447337C2semidirect
5. #21447206C2semidirect
6. #21447202C2semidirect
7. #21446952C2semidirect
8. #21446087C2semidirect
9. #21446075C2semidirect
10. #21446015C2semidirect
11. #21445995C2semidirect
12. #21445845C2semidirect
13. #21445832C2semidirect
15. C2×C61×#477C2semidirect
66. C2×#11758226Q8semidirect
342. C2×#477D61semidirect
411. C61#25959semidirect

Maximal subgroups

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

subgroupindex
C2×#168397572
#214495062
#214486572
#214473372
#214472062
#214472022
#214469522
#214460872
#214460752
#214460152
#214459952
#214458452
#214458322
#214457582
C2×C61×#4772
#2595961