Finite Group Databasebeta

← #29712452

Subgroups — #29712452

Order 15616.

1228 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840452C211yesmaximal, characteristic
2C2×#16839918C211yesmaximal, characteristic, complemented
3C2×#16839771C211yesmaximal, characteristic, complemented
4#21446970C212yesmaximal, complemented
5#21446827C212yesmaximal, complemented
6#21446250C211yesmaximal, characteristic, complemented
7#21446163C212yesmaximal, complemented
8#21446127C212yesmaximal, complemented
9#21446117C211yesmaximal, characteristic
10#21445911C211yesmaximal, characteristic, complemented
11C61×#2409C211yesmaximal, characteristic
12C22×#11758202C2212yes—
13C22×#11758194C2211yescharacteristic
14#16840452—24no—
15#16840452—24no—
16C2×#14519770—21no—
17C2×#14519769C2211yescharacteristic
18C2×#14519767—21no—
19C2×#14519758—21no—
20C2×#14519756C2211yescharacteristic
21C2×#14519756C2211yescharacteristic
22C2×C4×Dic122C2211yescharacteristic
23C2×#14519736C2211yescharacteristic
24C2×#14519733—21no—
25C22×C4⋊Dic61C2211yescharacteristic
26#16839918—24no—
27#16839771—24no—
28C2×#14519701C2212yes—
29C2×#14519701C2212yes—
30C2×#14519701C2212yes—
31C2×#14519700C2212yes—
32C2×C22⋊#92418C2211yescharacteristic
33C2×C22⋊#92418C2212yes—
34#16839524C2212yes—
35#16839524C2212yes—
36#16839524C2212yes—
37#16839517C2212yes—
38C23×Q8×C61C2211yescharacteristic
39C2×C22⋊Q8×C61C2212yes—
40C2×C4×D4×C61C2211yescharacteristic
41C61×#395—21no—
42C61×#393—21no—
43C61×#391C2212yes—
44C61×#386—21no—
45C61×#385—21no—
46C61×#380C2211yescharacteristic
47C22×Dic122C2311yescharacteristic
48C23×Dic61—21no—
49C2×#11758202—24no—
50C2×#11758202—24no—
51C2×#11758202—42no—
52C2×#11758194—22no—
53C2×#11758194—42no—
54C22×#92418C2312yes—
55C22×#92418C2312yes—
56#14519770—28no—
57#14519769—22no—
58#14519769—22no—
59#14519767—28no—
60#14519758—28no—
61#14519756—22no—
62#14519756—22no—
63#14519756—22no—
64#14519756D414yes—
65C4×Dic122—22no—
66C4×Dic122—22no—
67#14519736—22no—
68#14519736D414yes—
69#14519733—28no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—22no—
72C2×C4⋊Dic61—22no—
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×C22⋊Dic61—21no—
82C2×C22⋊Dic61—21no—
83C2×C22⋊Dic61—21no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61—21no—
86#14519701—24no—
87#14519701—24no—
88#14519701—24no—
89#14519700—42no—
90C22⋊#92418—22no—
91C22⋊#92418—42no—
92C22×Q8×C61—21no—
93C22×Q8×C61—24no—
94C22×Q8×C61—22no—
95C22×Q8×C61—22no—
96C22×Q8×C61—42no—
97C22×Q8×C61C2312yes—
98C22×D4×C61D411yescharacteristic
99C23×C4×C61—21no—
100C23×C4×C61C2311yescharacteristic
101C23×C4×C61C2311yescharacteristic
102C22⋊Q8×C61—24no—
103C22⋊Q8×C61—42no—
104C4×D4×C61—22no—
105C4×D4×C61—22no—
106C2×C4⋊C4×C61—22no—
107C2×C4⋊C4×C61D411yescharacteristic
108C2×C4⋊C4×C61C2312yes—
109C2×C22⋊C4×C61—21no—
110C2×C22⋊C4×C61—22no—
111C2×C22⋊C4×C61D411yescharacteristic
112C2×C22⋊C4×C61D411yescharacteristic
113C2×C42×C61C2311yescharacteristic
114#95×C61—21no—
115#95×C61—22no—
116#95×C61—22no—
117C24×C61C2×D411yescharacteristic
118C24×C61C2×D411yescharacteristic
119C2×Q8×C61—22no—
120C2×Q8×C61—28no—
121C2×Q8×C61—22no—
122C2×Q8×C61—22no—
123C2×Q8×C61—22no—
124C2×Q8×C61—22no—
125C2×Q8×C61—44no—
126C2×Q8×C61—48no—
127C2×Q8×C61—44no—
128C2×Q8×C61—42no—
129C2×Q8×C61#4114yes—
130C2×D4×C61—22no—
131C2×D4×C61—22no—
132C2×D4×C61—41no—
133C22×C4×C61—22no—
134C22×C4×C61—22no—
135C22×C4×C61—22no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—22no—
139C22×C4×C61—22no—
140C22×C4×C61—22no—
141C22×C4×C61—44no—
142C22×C4×C61—42no—
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D412yes—
148C22×C4×C61C2×D412yes—
149C22×C4×C61C2411yescharacteristic
150C22×Dic61—24no—
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—42no—
156C22×Dic61—42no—
157C2×Dic122—22no—
158C2×Dic122—22no—
159C2×Dic122—21no—
160C2×Dic122C2×D412yes—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—22no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61—44no—
165C22⋊C4×C61—22no—
166C22⋊C4×C61—22no—
167C22⋊C4×C61—44no—
168C22⋊C4×C61—42no—
169C42×C61—21no—
170C42×C61—21no—
171C22⋊Dic61—24no—
172C22⋊Dic61—24no—
173C22⋊Dic61—24no—
174#11758202—48no—
175C4⋊Dic61—24no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61C2×D412yes—
180C4⋊Dic61C2×D412yes—
181C4⋊Dic61C2×D412yes—
182#11758196—24no—
183#11758196—24no—
184#11758196—24no—
185#11758196—24no—
186#11758196—24no—
187C4×Dic61—24no—
188C4×Dic61—24no—
189#11758194—44no—
190C2×#92418—22no—
191C2×#92418—22no—
192C2×#92418—22no—
193C2×#92418—22no—
194C2×#92418—22no—
195C2×#92418—22no—
196C23×C61—21no—
197C23×C61—22no—
198C23×C61—22no—
199C23×C61—21no—
200C23×C61—21no—
201C23×C61—21no—
202C23×C61—22no—
203C23×C61—22no—
204C23×C61#12012yes—
205C23×C61#13712yes—
206C23×C61C22×D411yescharacteristic
207Q8×C61—24no—
208Q8×C61—24no—
209Q8×C61—48no—
210Q8×C61—42no—
211Q8×C61—42no—
212D4×C61—44no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—24no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—24no—
221C2×C4×C61—22no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—22no—
225C2×C4×C61—48no—
226C2×C4×C61—42no—
227C2×C4×C61—42no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—42no—
231C2×C4×C61—42no—
232C2×C4×C61—42no—
233C2×C4×C61—42no—
234C2×C4×C61#12012yes—
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C22×D411yescharacteristic
237C2×C4×C61C2×#4112yes—
238C2×C4×C61H(5,𝔽2)12yes—
239C2×Dic61—22no—
240C2×Dic61—22no—
241C2×Dic61—22no—
242C2×Dic61—22no—
243C2×Dic61—48no—
244C2×Dic61—42no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—42no—
250Dic122—24no—
251Dic122—41no—
252Dic122—41no—
253#92418—42no—
254#92418—42no—
255#23983—611nomaximal, Sylow
256C22×C61—22no—
257C22×C61—22no—
258C22×C61—22no—
259C22×C61—22no—
260C22×C61—24no—
261C22×C61—41no—
262C22×C61—41no—
263C22×C61—42no—
264C22×C61—42no—
265C22×C61—42no—
266C22×C61C2×#12011yescharacteristic
267C22×C61#54612yes—
268C22×C61C2×#13711yescharacteristic
269C22×C61#57511yescharacteristic
270C22×C61#57812yes—
271C4×C61—22no—
272C4×C61—41no—
273C4×C61—44no—
274C4×C61—41no—
275C4×C61—41no—
276C4×C61—82no—
277C4×C61#52511yescharacteristic
278C4×C61#53511yescharacteristic
279C4×C61#54612yes—
280Dic61—42no—
281Dic61—42no—
282Dic61—42no—
283Dic61—42no—
284C2×#552—611no—
285C2×#451—611no—
286C2×#419—611no—
287#2409D6111yescharacteristic
288#2026—612no—
289#2001—612no—
290#1894—611no—
291#1878—612no—
292#1872—612no—
293#1870—611no—
294#1825—611no—
295C2×C61—42no—
296C2×C61—42no—
297C2×C61#240411yescharacteristic
298C2×C61#300711yescharacteristic
299C2×C61#301111yescharacteristic
300C2×C61#331212yes—
301C2×C61#332712yes—
302C23×Q8C2×D6111yescharacteristic
303C22×Q16—612no—
304C22×M16—611no—
305#552—1224no—
306#552—1224no—
307C2×#126—1221no—
308C2×#125—611no—
309C2×#125—1221no—
310C2×#123—1221no—
311C2×C22⋊Q8—611no—
312C2×C22⋊Q8—611no—
313C2×C22⋊Q8C2×D6112yes—
314C2×C4×Q8—611no—
315C2×C4×D4C2×D6111yescharacteristic
316C2×#117—611no—
317C2×#117—1221no—
318C22×C4⋊C4—611no—
319#451—1224no—
320#419—1224no—
321C2×#103—612no—
322C2×#103—612no—
323C2×#103—612no—
324C2×#103—612no—
325C2×C22⋊C8—611no—
326C2×C22⋊C8—612no—
327#395—21no—
328#393—21no—
329#391C2×D6112yes—
330#386—21no—
331#385—21no—
332#380C2×D6111yescharacteristic
333#340—612no—
334#340—612no—
335#340—612no—
336#336—612no—
337C61#2398311yescharacteristic, complemented, Sylow
338C22×Q8—21no—
339C22×Q8—24no—
340C22×Q8—22no—
341C22×Q8—22no—
342C22×Q8—42no—
343C22×Q8—611no—
344C22×Q8C22×D6112yes—
345C22×D4C22⋊D6111yescharacteristic
346C23×C4—21no—
347C23×C4—1221no—
348C23×C4C22×D6111yescharacteristic
349C23×C4C22×D6111yescharacteristic
350C2×Q16—1224no—
351C2×Q16—1224no—
352C2×Q16—2442no—
353C2×M16—1222no—
354C2×M16—2442no—
355C22×C8—612no—
356C22×C8—612no—
357#126—1228no—
358#125—1222no—
359#125—1222no—
360#125—1228no—
361#123—1228no—
362C22⋊Q8—24no—
363C22⋊Q8—42no—
364C22⋊Q8—614no—
365C22⋊Q8—1222no—
366C22⋊Q8—1222no—
367C22⋊Q8—1222no—
368C4×Q8—1222no—
369C4×Q8—1222no—
370C4×D4—22no—
371C4×D4—22no—
372#117—614no—
373#117—1228no—
374#117—1222no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—611no—
377C2×C4⋊C4—611no—
378C2×C4⋊C4—611no—
379C2×C4⋊C4—1222no—
380C2×C4⋊C4—1222no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4—1221no—
383C2×C4⋊C4—1221no—
384C2×C4⋊C4—1221no—
385C2×C4⋊C4—1221no—
386C2×C4⋊C4—1221no—
387C2×C4⋊C4C22⋊D6111yescharacteristic
388C2×C4⋊C4C22×D6112yes—
389C2×C22⋊C4—22no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—1221no—
392C2×C22⋊C4—1221no—
393C2×C22⋊C4—1221no—
394C2×C22⋊C4C22⋊D6111yescharacteristic
395C2×C22⋊C4C22⋊D6111yescharacteristic
396C2×C42—1221no—
397C2×C42—1221no—
398C2×C42C22×D6111yescharacteristic
399#103—1224no—
400#103—1224no—
401#103—1224no—
402#103—2442no—
403C22⋊C8—1222no—
404C22⋊C8—2442no—
405#95—21no—
406#95—22no—
407#95—22no—
408C24C2×C22⋊D6111yescharacteristic
409C24C2×C22⋊D6111yescharacteristic
410C2×Q8—22no—
411C2×Q8—28no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—44no—
417C2×Q8—48no—
418C2×Q8—44no—
419C2×Q8—42no—
420C2×Q8—612no—
421C2×Q8—1221no—
422C2×Q8—1222no—
423C2×Q8—1222no—
424C2×Q8#1175822414yes—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—41no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—44no—
437C22×C4—42no—
438C22×C4—1224no—
439C22×C4—1221no—
440C22×C4—1221no—
441C22×C4—1221no—
442C22×C4—1221no—
443C22×C4—2442no—
444C22×C4—2442no—
445C22×C4D4×D6112yes—
446C22×C4D4×D6112yes—
447C22×C4C2×C22⋊D6111yescharacteristic
448C22×C4C2×C22⋊D6111yescharacteristic
449C22×C4C2×C22⋊D6111yescharacteristic
450C22×C4C2×C22⋊D6111yescharacteristic
451C22×C4C23×D6111yescharacteristic
452Q16—2448no—
453M16—2444no—
454C2×C8—1222no—
455C2×C8—1222no—
456C2×C8—1222no—
457C2×C8—1222no—
458C2×C8—1222no—
459C2×C8—1222no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—44no—
464C4⋊C4—612no—
465C4⋊C4—612no—
466C4⋊C4—612no—
467C4⋊C4—1224no—
468C4⋊C4—1224no—
469C4⋊C4—1224no—
470C4⋊C4—1224no—
471C4⋊C4—1224no—
472C4⋊C4—1221no—
473C4⋊C4—1221no—
474C4⋊C4—1221no—
475C4⋊C4—1224no—
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—44no—
479C22⋊C4—42no—
480C22⋊C4—1224no—
481C22⋊C4—1224no—
482C22⋊C4—1224no—
483C42—21no—
484C42—21no—
485C42—1224no—
486C42—1224no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23#1451974512yes—
496C23#1451980812yes—
497C23C22×C22⋊D6111yescharacteristic
498Q8—24no—
499Q8—24no—
500Q8—48no—
501Q8—42no—
502Q8—42no—
503Q8—1224no—
504Q8—2441no—
505Q8—2441no—
506D4—44no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—48no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—1222no—
529C2×C4—1222no—
530C2×C4—1222no—
531C2×C4—1222no—
532C2×C4—2448no—
533C2×C4—2441no—
534C2×C4—2441no—
535C2×C4—2441no—
536C2×C4—2441no—
537C2×C4—2442no—
538C2×C4—2442no—
539C2×C4#1451974512yes—
540C2×C4C2×D4×D6111yescharacteristic
541C2×C4C2×D4×D6111yescharacteristic
542C2×C4C2×#1175822412yes—
543C2×C4#1451982212yes—
544C8—2442no—
545C8—2442no—
546C22—22no—
547C22—22no—
548C22—24no—
549C22—22no—
550C22—22no—
551C22—41no—
552C22—42no—
553C22—42no—
554C22—41no—
555C22—42no—
556C22C2×#1451974511yescharacteristic
557C22#1684040612yes—
558C22C2×#1451980811yescharacteristic
559C22#1684056011yescharacteristic
560C22#1684057712yes—
561C4—22no—
562C4—41no—
563C4—41no—
564C4—44no—
565C4—41no—
566C4—82no—
567C4—2442no—
568C4—2442no—
569C4—2442no—
570C4—2442no—
571C4#1684026811yescharacteristic
572C4#1684032711yescharacteristic
573C4#1684039912yes—
574C2—42no—
575C2—42no—
576C2#2144913511yescharacteristic
577C2#2145507711yescharacteristic
578C2#2145511211yescharacteristic
579C2#2145787012yes—
580C2#2145808712yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2409C2
socleC23×C61C22×D4
prime radicalC22×D4×C61D4
hypercenter#2409D61
hypocenterC61#23983
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2409

Complements

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

NH ≅ G/Ntype
2. C2×#16839918C2semidirect
3. C2×#16839771C2semidirect
4. #21446970C2semidirect
5. #21446827C2semidirect
6. #21446250C2semidirect
7. #21446163C2semidirect
8. #21446127C2semidirect
10. #21445911C2semidirect
337. C61#23983semidirect

Maximal subgroups

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

subgroupindex
C2×#168404522
C2×#168399182
C2×#168397712
#214469702
#214468272
#214462502
#214461632
#214461272
#214461172
#214459112
C61×#24092
#2398361