Finite Group Databasebeta

← #29737166

Subgroups — #29737166

Order 15616.

986 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840064C211yesmaximal, characteristic, complemented
2C2×#16839755C211yesmaximal, characteristic
3#21449535C211yesmaximal, characteristic, complemented
4#21448644C211yesmaximal, characteristic
5#21447358C211yesmaximal, characteristic, complemented
6#21447215C211yesmaximal, characteristic, complemented
7#21447199C211yesmaximal, characteristic, complemented
8#21446969C211yesmaximal, characteristic, complemented
9#21446066C211yesmaximal, characteristic, complemented
10#21446064C211yesmaximal, characteristic, complemented
11#21446019C211yesmaximal, characteristic, complemented
12#21445995C211yesmaximal, characteristic, complemented
13#21445847C211yesmaximal, characteristic, complemented
14#21445756C211yesmaximal, characteristic
15C61×#1802C211yesmaximal, characteristic, complemented
16C22×#11758224C2211yescharacteristic
17C22×C8×D61C2211yescharacteristic
18C2×#14519768C2211yescharacteristic
19C2×D4×Dic61C2211yescharacteristic
20C2×#14519735C2211yescharacteristic
21#16840064—24no—
22C2×#14519710C2211yescharacteristic
23C2×#14519710C2211yescharacteristic
24C2×#14519710C2211yescharacteristic
25C2×#14519709C2211yescharacteristic
26C2×#14519706C2211yescharacteristic
27#16839755—24no—
28C2×#14519702C2211yescharacteristic
29C2×#14519702C2211yescharacteristic
30C2×#14519700C2211yescharacteristic
31C2×#14519700C2211yescharacteristic
32C2×#14519698C2211yescharacteristic
33C2×#14519698C2211yescharacteristic
34C2×#14519689C2211yescharacteristic
35#16839697—21no—
36#16839684—21no—
37#16839668C2211yescharacteristic
38#16839667C2211yescharacteristic
39#16839660C2211yescharacteristic
40#16839658C2211yescharacteristic
41#16839657—21no—
42#16839648—21no—
43#16839641—21no—
44#16839639—21no—
45#16839623—21no—
46#16839619C2211yescharacteristic
47#16839619C2211yescharacteristic
48C4×#11758198C2211yescharacteristic
49C2×#14519684—21no—
50#16839525C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839517C2211yescharacteristic
53#16839516C2211yescharacteristic
54C2×C4×D4×C61C2211yescharacteristic
55C2×C4⋊C8×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C2×#102×C61C2211yescharacteristic
58C61×#384C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C2×#11758224—22no—
62C2×#11758224—22no—
63C2×#11758224—22no—
64C2×#11758224—41no—
65C2×#11758224—41no—
66C2×#11758224Q814yes—
67C22×Dic122C2311yescharacteristic
68C22×C22⋊D61—21no—
69C23×Dic61—21no—
70C23×Dic61—21no—
71C22×C4×D61C2311yescharacteristic
72C22×#92418C2311yescharacteristic
73C22×#92418C2311yescharacteristic
74C2×C8×D61—22no—
75C2×C8×D61—22no—
76C2×C8×D61—41no—
77#14519768—22no—
78#14519768—41no—
79D4×Dic61—22no—
80D4×Dic61—41no—
81#14519735—22no—
82#14519735—41no—
83C2×C4⋊Dic61—21no—
84C2×C4⋊Dic61C2311yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×C4⋊Dic61C2311yescharacteristic
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×#11758196—21no—
90C2×C22⋊Dic61—21no—
91C2×#11758198—21no—
92C2×#11758198—21no—
93C2×#11758198—21no—
94C2×#11758198D411yescharacteristic
95C2×#11758198D411yescharacteristic
96C2×C4×Dic61—21no—
97C2×C4×Dic61C2311yescharacteristic
98#14519710—22no—
99#14519710—41no—
100#14519710—41no—
101#14519709—22no—
102#14519706—22no—
103#14519702—22no—
104#14519702—41no—
105#14519700—22no—
106#14519700—41no—
107#14519698—22no—
108#14519698D414yes—
109#14519689—22no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118#14519684D411yescharacteristic
119#14519684D411yescharacteristic
120C22×D4×C61C2311yescharacteristic
121C23×C4×C61—21no—
122C22×C8×C61C2311yescharacteristic
123C22×C8×C61C2311yescharacteristic
124C4×D4×C61—22no—
125C4×D4×C61—41no—
126C2×C4⋊C4×C61—21no—
127C2×C4⋊C4×C61—21no—
128C2×C4⋊C4×C61C2311yescharacteristic
129C2×C4⋊C4×C61C2311yescharacteristic
130C2×C4⋊C4×C61C2311yescharacteristic
131C2×C22⋊C4×C61—21no—
132C2×C42×C61C2311yescharacteristic
133C4⋊C8×C61—41no—
134#102×C61—22no—
135#102×C61—22no—
136C24×C61—21no—
137C23×D61C2×D411yescharacteristic
138C2×D4×C61—21no—
139C2×D4×C61—21no—
140C2×D4×C61—21no—
141C2×D4×C61C2×D412yes—
142C2×D4×C61C2×Q812yes—
143C2×D4×C61#4112yes—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—42no—
148C22×C4×C61—42no—
149C22×C4×C61—81no—
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2411yescharacteristic
153C2×C22⋊D61—24no—
154C2×C22⋊D61—42no—
155C2×C22⋊D61—42no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—24no—
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61—42no—
162C22×Dic61—42no—
163C22×Dic61—42no—
164C22×Dic61—42no—
165C22×Dic61—81no—
166C22×Dic61C2×D411yescharacteristic
167C22×Dic61C2×D411yescharacteristic
168C22×Dic61C2×D411yescharacteristic
169#11758224—28no—
170#11758224—44no—
171#11758224—44no—
172#11758224—44no—
173C2×C4×D61—21no—
174C2×C4×D61—21no—
175C2×C4×D61—21no—
176C2×C4×D61—21no—
177C2×C4×D61C2×Q812yes—
178C2×C4×D61#4112yes—
179C2×Dic122—21no—
180C2×Dic122—21no—
181C2×Dic122—21no—
182C2×Dic122—21no—
183C2×Dic122—21no—
184C2×Dic122C2×Q812yes—
185C2×C8×C61—21no—
186C2×C8×C61—21no—
187C2×C8×C61—21no—
188C2×C8×C61—21no—
189C2×C8×C61—21no—
190C2×C8×C61—21no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—21no—
193C4⋊C4×C61—21no—
194C4⋊C4×C61—21no—
195C4⋊C4×C61—21no—
196C4⋊C4×C61—21no—
197C4⋊C4×C61—42no—
198C4⋊C4×C61—81no—
199C22⋊C4×C61—42no—
200C42×C61—21no—
201C42×C61—21no—
202C22⋊Dic61—42no—
203#11758198—22no—
204#11758198—41no—
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—42no—
214C4⋊Dic61C2×D412yes—
215#11758196—42no—
216#11758196—42no—
217#11758196—81no—
218C4×Dic61—21no—
219C4×Dic61—21no—
220C4×Dic61—42no—
221C2×#92418—21no—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418—21no—
225C2×#92418—21no—
226C2×#92418C2×D412yes—
227C8×D61—44no—
228C23×C61—22no—
229C23×C61—22no—
230C23×C61—22no—
231C23×C61—41no—
232C23×C61—41no—
233C23×C61—41no—
234C23×C61—41no—
235C23×C61C22×D411yescharacteristic
236C22×D61—21no—
237C22×D61—21no—
238C22×D61—21no—
239C22×D61—21no—
240C22×D61—21no—
241C22×D61—21no—
242C22×D61C22⋊Q812yes—
243D4×C61—22no—
244D4×C61—22no—
245D4×C61—22no—
246D4×C61C22⋊Q814yes—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—21no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—84no—
262C2×C4×C61#13512yes—
263C2×C4×C61C22×D411yescharacteristic
264C2×C4×C61C22×Q811yescharacteristic
265C2×C4×C61C2×#4111yescharacteristic
266C2×C4×C61C2×#4111yescharacteristic
267C2×C4×C61H(5,𝔽2)11yescharacteristic
268C2×C4×C61H(5,𝔽2)11yescharacteristic
269C22⋊D61—48no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—22no—
273C2×Dic61—21no—
274C2×Dic61—21no—
275C2×Dic61—22no—
276C2×Dic61—21no—
277C2×Dic61—21no—
278C2×Dic61—21no—
279C2×Dic61—21no—
280C2×Dic61—48no—
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—
290C2×Dic61—41no—
291C2×Dic61—84no—
292C2×Dic61C22⋊Q812yes—
293C4×D61—22no—
294C4×D61—22no—
295C4×D61—22no—
296C4×D61—22no—
297Dic122—22no—
298Dic122—22no—
299Dic122—41no—
300Dic122—41no—
301C8×C61—41no—
302C8×C61—41no—
303#92418—22no—
304#92418—41no—
305#25956—611nomaximal, Sylow
306C22×C61—24no—
307C22×C61—42no—
308C22×C61—42no—
309C22×C61—42no—
310C22×C61—42no—
311C22×C61—42no—
312C22×C61—42no—
313C22×C61C2×C22⋊Q811yescharacteristic
314C22×C61D4211yescharacteristic
315C22×C61#54611yescharacteristic
316C22×C61C2×#13511yescharacteristic
317C22×C61#57511yescharacteristic
318C22×C61#57611yescharacteristic
319C22×C61#57811yescharacteristic
320C2×D61—22no—
321C2×D61—22no—
322C2×D61—22no—
323C2×D61—41no—
324C2×D61—41no—
325C2×D61—41no—
326C2×D61—41no—
327C4×C61—22no—
328C4×C61—41no—
329C4×C61—81no—
330C4×C61—81no—
331C4×C61—81no—
332C4×C61C2×C22⋊Q811yescharacteristic
333C4×C61#54311yescharacteristic
334C4×C61#54611yescharacteristic
335C4×C61#55111yescharacteristic
336Dic61—22no—
337Dic61—42no—
338Dic61—41no—
339Dic61—41no—
340Dic61—42no—
341Dic61—81no—
342Dic61—81no—
343C2×#478—611no—
344C2×#416—611no—
345#2441—611no—
346#2339—611no—
347#2095—611no—
348#2068—611no—
349#2066—611no—
350#2026—611no—
351#1862—611no—
352#1862—611no—
353#1854—611no—
354#1850—611no—
355#1808—611no—
356#1802D6111yescharacteristic, complemented
357#1788—611no—
358C2×C61—44no—
359C2×C61#243711yescharacteristic
360C2×C61#307611yescharacteristic
361C2×C61#308311yescharacteristic
362C2×C61#326811yescharacteristic
363C2×C61#328311yescharacteristic
364C2×C61#329711yescharacteristic
365C2×C61#331511yescharacteristic
366D61—42no—
367C22×#41—611no—
368C23×C8—611no—
369C2×#125—611no—
370C2×C4×D4—611no—
371C2×C4×D4C2×D6111yescharacteristic
372C2×#117—611no—
373#478—1224no—
374C2×#107—611no—
375C2×#107—611no—
376C2×#107—611no—
377C2×#106—611no—
378C2×C4⋊C8—611no—
379C2×C4⋊C8C2×D6111yescharacteristic
380#416—1224no—
381C2×#103—611no—
382C2×#103—611no—
383C2×#103—611no—
384C2×#103—611no—
385C2×#102—611no—
386C2×#102—611no—
387C2×#102C2×D6111yescharacteristic
388C2×#102C2×D6111yescharacteristic
389C2×C22⋊C8—611no—
390#399—1221no—
391#396—1221no—
392#391—611no—
393#391—611no—
394#389—611no—
395#389—611no—
396#389—1221no—
397#387—1221no—
398#386—1221no—
399#386—1221no—
400#384C2×D6111yescharacteristic
401#381—1221no—
402#380—611no—
403#380—611no—
404C4×C22⋊C4—611no—
405C2×#95—1221no—
406#340—611no—
407#340—611no—
408#340C2×D6111yescharacteristic
409#340C2×D6111yescharacteristic
410#336—611no—
411#336—611no—
412C61#2595611yescharacteristic, complemented, Sylow
413C2×#41—614no—
414C2×#41—1222no—
415C2×#41—1222no—
416C2×#41—1222no—
417C2×#41—2441no—
418C2×#41—2441no—
419C22×Q8—611no—
420C22×D4—1221no—
421C22×D4C22×D6111yescharacteristic
422C23×C4—21no—
423C23×C4—611no—
424C23×C4—1221no—
425C23×C4—1221no—
426C22×C8—611no—
427C22×C8—611no—
428C22×C8—1222no—
429C22×C8—1222no—
430C22×C8—2441no—
431C22×C8C22×D6111yescharacteristic
432C22×C8C22×D6111yescharacteristic
433#125—1222no—
434#125—2441no—
435C4×D4—22no—
436C4×D4—41no—
437C4×D4—1222no—
438C4×D4—2441no—
439#117—1222no—
440#117—2441no—
441C2×C4⋊C4—21no—
442C2×C4⋊C4—21no—
443C2×C4⋊C4—611no—
444C2×C4⋊C4—611no—
445C2×C4⋊C4—611no—
446C2×C4⋊C4—1221no—
447C2×C4⋊C4—1221no—
448C2×C4⋊C4—1221no—
449C2×C4⋊C4—1221no—
450C2×C4⋊C4C22×D6111yescharacteristic
451C2×C4⋊C4C22×D6111yescharacteristic
452C2×C4⋊C4C22×D6111yescharacteristic
453C2×C22⋊C4—21no—
454C2×C22⋊C4—611no—
455C2×C22⋊C4—611no—
456C2×C22⋊C4—1221no—
457C2×C22⋊C4—1221no—
458C2×C22⋊C4—1221no—
459C2×C22⋊C4—1221no—
460C2×C42—611no—
461C2×C42—1221no—
462C2×C42C22×D6111yescharacteristic
463#107—1222no—
464#107—2441no—
465#107—2441no—
466#106—1222no—
467C4⋊C8—41no—
468C4⋊C8—1222no—
469#103—1222no—
470#103—1222no—
471#103—2441no—
472#103—2441no—
473#102—22no—
474#102—22no—
475#102—614no—
476#102—1222no—
477C22⋊C8—1222no—
478#95—611no—
479#95—611no—
480#95—1221no—
481#95—1221no—
482#95—1221no—
483#95—1221no—
484#95—1221no—
485#95—1221no—
486#95—1221no—
487#95—1221no—
488C24—21no—
489C24—611no—
490#41—1228no—
491#41—2444no—
492#41—2444no—
493#41—2444no—
494C2×Q8—612no—
495C2×Q8—1221no—
496C2×Q8—1221no—
497C2×Q8—1221no—
498C2×Q8—1221no—
499C2×Q8—1221no—
500C2×D4—21no—
501C2×D4—21no—
502C2×D4—21no—
503C2×D4—1224no—
504C2×D4—2442no—
505C2×D4—2442no—
506C2×D4#1175822212yes—
507C2×D4D4×D6112yes—
508C2×D4Q8×D6112yes—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—42no—
513C22×C4—42no—
514C22×C4—81no—
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—1221no—
529C22×C4—4881no—
530C22×C4—2442no—
531C22×C4—2442no—
532C22×C4—2442no—
533C22×C4—2442no—
534C22×C4D4×D6111yescharacteristic
535C22×C4D4×D6111yescharacteristic
536C22×C4C23×D6111yescharacteristic
537C2×C8—21no—
538C2×C8—21no—
539C2×C8—21no—
540C2×C8—21no—
541C2×C8—21no—
542C2×C8—21no—
543C2×C8—612no—
544C2×C8—1221no—
545C2×C8—1221no—
546C2×C8—1221no—
547C2×C8—1221no—
548C2×C8—1221no—
549C2×C8—2444no—
550C4⋊C4—21no—
551C4⋊C4—21no—
552C4⋊C4—21no—
553C4⋊C4—21no—
554C4⋊C4—21no—
555C4⋊C4—21no—
556C4⋊C4—42no—
557C4⋊C4—81no—
558C4⋊C4—612no—
559C4⋊C4—1221no—
560C4⋊C4—1221no—
561C4⋊C4—1221no—
562C4⋊C4—1221no—
563C4⋊C4—1221no—
564C4⋊C4—4881no—
565C4⋊C4—2442no—
566C4⋊C4—2442no—
567C4⋊C4—2442no—
568C22⋊C4—42no—
569C22⋊C4—1222no—
570C22⋊C4—4881no—
571C22⋊C4—4881no—
572C22⋊C4—2441no—
573C22⋊C4—2442no—
574C22⋊C4—2442no—
575C42—21no—
576C42—21no—
577C42—1221no—
578C42—1221no—
579C42—2442no—
580C23—22no—
581C23—22no—
582C23—22no—
583C23—41no—
584C23—41no—
585C23—41no—
586C23—41no—
587C23—612no—
588C23—1221no—
589C23—1221no—
590C23—1221no—
591C23—1221no—
592C23—1221no—
593C23—1221no—
594C23C2×D4×D6111yescharacteristic
595Q8—1222no—
596Q8—1222no—
597Q8—2441no—
598Q8—2441no—
599D4—22no—
600D4—22no—
601D4—22no—
602D4—2448no—
603D4#1451975514yes—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—84no—
619C2×C4—612no—
620C2×C4—1221no—
621C2×C4—1221no—
622C2×C4—1222no—
623C2×C4—1222no—
624C2×C4—1222no—
625C2×C4—1222no—
626C2×C4—1222no—
627C2×C4—1221no—
628C2×C4—1221no—
629C2×C4—1221no—
630C2×C4—1221no—
631C2×C4—1221no—
632C2×C4—1221no—
633C2×C4—1222no—
634C2×C4—4884no—
635C2×C4—2448no—
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—2441no—
645C2×C4—2441no—
646C2×C4#1451979812yes—
647C2×C4C2×D4×D6111yescharacteristic
648C2×C4C2×Q8×D6111yescharacteristic
649C2×C4#41×D6111yescharacteristic
650C2×C4C2×#1175822211yescharacteristic
651C2×C4#1451982211yescharacteristic
652C2×C4#1451982311yescharacteristic
653C8—41no—
654C8—41no—
655C8—1222no—
656C8—2441no—
657C22—24no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—1222no—
665C22—1222no—
666C22—1222no—
667C22—2441no—
668C22—2441no—
669C22—2441no—
670C22—2441no—
671C22C22⋊Q8×D6111yescharacteristic
672C22#1684039611yescharacteristic
673C22#1684040211yescharacteristic
674C22C2×#1451979811yescharacteristic
675C22#1684055811yescharacteristic
676C22#1684056311yescharacteristic
677C22#1684057811yescharacteristic
678C4—22no—
679C4—41no—
680C4—81no—
681C4—81no—
682C4—81no—
683C4—1222no—
684C4—4881no—
685C4—4881no—
686C4—2442no—
687C4—2441no—
688C4—2442no—
689C4—2441no—
690C4C2×#1451975511yescharacteristic
691C4#1684038711yescharacteristic
692C4#1684040111yescharacteristic
693C4#1684043611yescharacteristic
694C2—44no—
695C2—2442no—
696C2#2144948711yescharacteristic
697C2#2145560811yescharacteristic
698C2#2145566511yescharacteristic
699C2#2145736511yescharacteristic
700C2#2145753611yescharacteristic
701C2#2145766511yescharacteristic
702C2#2145791611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1802C2
socleC23×C61C22×D4
prime radicalC22×#11758224C22
hypercenter#1802D61
hypocenterC61#25956
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
1. C2×#16840064C2semidirect
3. #21449535C2semidirect
5. #21447358C2semidirect
6. #21447215C2semidirect
7. #21447199C2semidirect
8. #21446969C2semidirect
9. #21446066C2semidirect
10. #21446064C2semidirect
11. #21446019C2semidirect
12. #21445995C2semidirect
13. #21445847C2semidirect
15. C61×#1802C2semidirect
356. #1802D61semidirect
412. C61#25956semidirect

Maximal subgroups

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

subgroupindex
C2×#168400642
C2×#168397552
#214495352
#214486442
#214473582
#214472152
#214471992
#214469692
#214460662
#214460642
#214460192
#214459952
#214458472
#214457562
C61×#18022
#2595661