Finite Group Databasebeta

← #29835185

Subgroups — #29835185

Order 15616.

556 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456471C211yesmaximal, characteristic, complemented
2#21454670C211yesmaximal, characteristic
3#21448027C211yesmaximal, characteristic, complemented
4#21447979C211yesmaximal, characteristic, complemented
5#21447975C211yesmaximal, characteristic, complemented
6#21447863C211yesmaximal, characteristic, complemented
7#21447863C211yesmaximal, characteristic, complemented
8#21447750C211yesmaximal, characteristic
9#21447747C211yesmaximal, characteristic
10#21447686C211yesmaximal, characteristic, complemented
11#21447682C211yesmaximal, characteristic, complemented
12#21447670C211yesmaximal, characteristic, complemented
13#21447668C211yesmaximal, characteristic, complemented
14#21447623C211yesmaximal, characteristic, complemented
15C61×#2237C211yesmaximal, characteristic, complemented
16C22×D488C2211yescharacteristic
17#16840372—21no—
18#16840236—21no—
19C2×#14519722C2211yescharacteristic
20C2×#14519722C2211yescharacteristic
21C2×#14519720C2211yescharacteristic
22C2×#14519720C2211yescharacteristic
23#16840163C2211yescharacteristic, complemented
24#16840140C2211yescharacteristic, complemented
25#16840129—21no—
26#16840018—21no—
27#16840003C2211yescharacteristic
28#16839873—21no—
29#16839873—21no—
30#16839866—21no—
31Q16×Dic61C2211yescharacteristic, complemented
32#16839856C2211yescharacteristic, complemented
33#16839844C2211yescharacteristic
34#16839844C2211yescharacteristic
35#16839844C2211yescharacteristic
36#16839844C2211yescharacteristic
37#16839766—21no—
38#16839717C2211yescharacteristic
39#16839589C2211yescharacteristic, complemented
40#16839587C2211yescharacteristic, complemented
41#16839580C2211yescharacteristic, complemented
42#16839580C2211yescharacteristic, complemented
43#16839565C2211yescharacteristic, complemented
44#16839565C2211yescharacteristic, complemented
45#16839564C2211yescharacteristic
46#16839564C2211yescharacteristic
47#16839562C2211yescharacteristic
48#16839562C2211yescharacteristic
49C22⋊#11758185C2211yescharacteristic
50#16839537C2211yescharacteristic, complemented
51C16×Dic61C2211yescharacteristic, complemented
52C61×#467C2211yescharacteristic
53C61×#427C2211yescharacteristic
54C61×#367C2211yescharacteristic, complemented
55C61×#366C2211yescharacteristic, complemented
56C61×#358C2211yescharacteristic, complemented
57C61×#358C2211yescharacteristic, complemented
58C61×C22⋊C16C2211yescharacteristic
59C22×D244—21no—
60C2×Dic244C2311yescharacteristic
61C2×#11758202—21no—
62C2×D488—21no—
63C2×D488—22no—
64C2×D488—22no—
65C2×D488C2311yescharacteristic
66C2×D488C2311yescharacteristic
67C2×#11758194D411yescharacteristic
68#14519775—21no—
69#14519770—41no—
70#14519770—41no—
71#14519768—21no—
72#14519768—41no—
73#14519767—41no—
74#14519756—21no—
75Q8×Dic61—21no—
76#14519742—21no—
77C4×D244—41no—
78C4×D244—41no—
79#14519739—21no—
80#14519739—21no—
81#14519739—21no—
82#14519739—21no—
83#14519736D411yescharacteristic
84C22⋊C4×D61—41no—
85C22⋊C4×D61—41no—
86#14519722—22no—
87#14519722—22no—
88#14519720—22no—
89#14519720—22no—
90C2×#11758185C2311yescharacteristic
91C2×#11758185C2311yescharacteristic
92#14519711C2311yescharacteristic
93#14519708—21no—
94#14519702—21no—
95#14519701—21no—
96#14519700—21no—
97#14519697—21no—
98#14519697—21no—
99#14519697—21no—
100#14519697—21no—
101#14519687D411yescharacteristic
102#14519687D411yescharacteristic
103C8×Dic61C2311yescharacteristic
104C8×Dic61C2311yescharacteristic
105C2×Q16×C61C2311yescharacteristic
106C22×C8×C61C2311yescharacteristic
107C22⋊Q8×C61—21no—
108#117×C61—21no—
109C2×C16×C61C2311yescharacteristic
110C2×C16×C61C2311yescharacteristic
111#107×C61C2311yescharacteristic
112#107×C61C2311yescharacteristic
113#107×C61C2311yescharacteristic
114#106×C61—21no—
115#103×C61—21no—
116C23×D61—41no—
117C2×Q8×C61—21no—
118C22×C4×C61C2×D411yescharacteristic
119C2×D244—24no—
120C2×D244—21no—
121C2×D244—21no—
122C2×D244—42no—
123C2×D244—41no—
124C2×C4×D61—41no—
125C2×C4×D61—41no—
126C2×C4×D61—41no—
127C2×C4×D61—41no—
128C2×Dic122—21no—
129Q16×C61—21no—
130Q16×C61—21no—
131C2×C8×C61—21no—
132C2×C8×C61—21no—
133C2×C8×C61C2×D411yescharacteristic
134C2×C8×C61C2411yescharacteristic
135C4⋊C4×C61—21no—
136C4⋊C4×C61—21no—
137C4⋊C4×C61—21no—
138C4⋊C4×C61—41no—
139C22⋊C4×C61—41no—
140C22⋊C4×C61—41no—
141C42×C61—41no—
142C22⋊Dic61—41no—
143C22⋊Dic61D811yescharacteristic
144C22⋊Dic61D811yescharacteristic
145#11758202—42no—
146#11758198—41no—
147#11758198—41no—
148#11758198—41no—
149#11758198—41no—
150C4⋊Dic61—21no—
151C4⋊Dic61—41no—
152C4⋊Dic61—41no—
153C4⋊Dic61D811yescharacteristic
154C4⋊Dic61D811yescharacteristic
155#11758196—41no—
156#11758196—41no—
157#11758196—41no—
158#11758196—41no—
159#11758196—41no—
160C4×Dic61—41no—
161C4×Dic61—41no—
162C4×Dic61C2×D411yescharacteristic
163C4×Dic61C2×D411yescharacteristic
164#11758194—41no—
165C2×#92418C2×D411yescharacteristic
166C2×#92418C2×D411yescharacteristic
167Dic244—21no—
168Dic244—21no—
169D488—24no—
170D488—21no—
171D488—21no—
172D488#4112yes—
173D488#4112yes—
174C16×C61—21no—
175C16×C61—21no—
176#11758185—21no—
177#11758185—21no—
178C23×C61C2×D811yescharacteristic
179C22×D61—41no—
180C22×D61—41no—
181C22×D61—44no—
182C22×D61—161no—
183C22×D61—161no—
184Q8×C61—41no—
185Q8×C61—41no—
186C2×C4×C61—21no—
187C2×C4×C61—21no—
188C2×C4×C61—41no—
189C2×C4×C61—41no—
190C2×C4×C61—41no—
191C2×C4×C61—41no—
192C2×C4×C61C2×D811yescharacteristic
193C2×C4×C61C22×D411yescharacteristic
194C2×Dic61—41no—
195C2×Dic61—41no—
196C2×Dic61C2×D811yescharacteristic
197C2×Dic61C2×D811yescharacteristic
198C2×Dic61C2×D811yescharacteristic
199C2×Dic61C2×D811yescharacteristic
200D244—22no—
201D244—22no—
202D244—44no—
203D244—41no—
204D244—41no—
205C4×D61—161no—
206C4×D61—161no—
207C4×D61—161no—
208C4×D61—161no—
209Dic122—41no—
210Dic122—41no—
211C8×C61—21no—
212C8×C61C2×#4111yescharacteristic
213C8×C61C2×#4111yescharacteristic
214#92418—21no—
215#92418—21no—
216#33683—611nomaximal, Sylow
217C22×C61—21no—
218C22×C61—21no—
219C22×C61—21no—
220C22×C61C22×D811yescharacteristic
221C2×D61—42no—
222C2×D61—42no—
223C2×D61—81no—
224C2×D61—81no—
225C2×D61—81no—
226C2×D61—81no—
227C2×D61—164no—
228C4×C61—21no—
229C4×C61—81no—
230C4×C61—81no—
231C4×C61—81no—
232C4×C61—81no—
233C4×C61#52511yescharacteristic
234C4×C61#57511yescharacteristic
235Dic61—21no—
236Dic61—21no—
237Dic61—21no—
238Dic61—21no—
239Dic61—81no—
240Dic61—81no—
241#3177—611no—
242#2948—611no—
243#2237—611no—
244#2237D6111yescharacteristic, complemented
245#2228—611no—
246#2228—611no—
247#2208—611no—
248#2208—611no—
249#2187—611no—
250#2187—611no—
251#2175—611no—
252#2175—611no—
253C4×SD32—611no—
254C4×SD32—611no—
255#2164—611no—
256C2×C61—41no—
257C2×C61#310011yescharacteristic
258C2×C61#341611yescharacteristic
259C2×C61#341611yescharacteristic
260D61—82no—
261D61—82no—
262C22×D8—611no—
263#541—1221no—
264#518—1221no—
265C2×SD32—611no—
266C2×SD32—611no—
267C2×SD32—611no—
268C2×SD32—611no—
269#499—611no—
270#494—611no—
271#491—1221no—
272#470—1221no—
273#467—611no—
274#467C2×D6111yescharacteristic
275#442—1221no—
276#442—1221no—
277#441—1221no—
278C4×Q16—611no—
279C4×Q16—611no—
280C4×D8—611no—
281C4×D8—611no—
282C4×D8—611no—
283C4×D8—611no—
284#427C2×D6111yescharacteristic
285#418—1221no—
286#405—611no—
287#367—611no—
288#367C2×D6111yescharacteristic, complemented
289#366—611no—
290#366C2×D6111yescharacteristic, complemented
291C4⋊C16—611no—
292C4⋊C16—611no—
293#358—611no—
294#358—611no—
295#358C2×D6111yescharacteristic, complemented
296#358C2×D6111yescharacteristic, complemented
297#357—611no—
298#357—611no—
299#357—611no—
300#357—611no—
301C22⋊C16—611no—
302C22⋊C16C2×D6111yescharacteristic
303#346—611no—
304C4×C16—611no—
305C61#3368311yescharacteristic, complemented, Sylow
306C22×D4—1221no—
307C2×Q16—611no—
308C2×Q16—1221no—
309C2×Q16C22×D6111yescharacteristic
310C2×D8—611no—
311C2×D8—611no—
312C2×D8—1222no—
313C2×D8—1222no—
314C2×D8—1221no—
315C2×M16—611no—
316C22×C8C22×D6111yescharacteristic
317C4⋊Q8—1221no—
318#126—2441no—
319#126—2441no—
320#125—1221no—
321#125—2441no—
322#125—2441no—
323C22⋊Q8—21no—
324C22⋊Q8—1221no—
325C4×Q8—1221no—
326C4×Q8—1221no—
327C4×D4—1221no—
328C4×D4—1221no—
329C4×D4—1221no—
330C4×D4—1221no—
331C4×D4—2441no—
332C4×D4—2441no—
333#117—21no—
334#117—611no—
335C2×C22⋊C4—2441no—
336C2×C22⋊C4—2441no—
337SD32—1222no—
338SD32—1222no—
339SD32—1222no—
340SD32—1222no—
341C2×C16—611no—
342C2×C16—611no—
343C2×C16C22×D6111yescharacteristic
344C2×C16C22×D6111yescharacteristic
345#107—611no—
346#107C22×D6111yescharacteristic
347#107C22×D6111yescharacteristic
348#107C22×D6111yescharacteristic
349#106—21no—
350#106—1221no—
351#103—21no—
352#103—1221no—
353#103—1221no—
354#103—1221no—
355#102—1221no—
356#102—1221no—
357#102—1221no—
358#102—1221no—
359#97—611no—
360#97—611no—
361C4×C8—611no—
362C4×C8—611no—
363C24—2441no—
364C2×Q8—21no—
365C2×Q8—1221no—
366C2×D4—1224no—
367C2×D4—1221no—
368C2×D4—1221no—
369C2×D4—2442no—
370C2×D4—2441no—
371C22×C4—2441no—
372C22×C4—2441no—
373C22×C4—2441no—
374C22×C4—2441no—
375C22×C4D4×D6111yescharacteristic
376Q16—21no—
377Q16—21no—
378Q16—1221no—
379Q16—1221no—
380Q16—2442no—
381D8—612no—
382D8—612no—
383D8—1224no—
384D8—1221no—
385D8—1221no—
386M16—2441no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—611no—
390C2×C8—611no—
391C2×C8D4×D6111yescharacteristic
392C2×C8C23×D6111yescharacteristic
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—41no—
397C4⋊C4—611no—
398C4⋊C4—611no—
399C4⋊C4—1221no—
400C4⋊C4—2441no—
401C4⋊C4—2441no—
402C4⋊C4—2441no—
403C4⋊C4—2441no—
404C4⋊C4—2441no—
405C4⋊C4—2441no—
406C4⋊C4—2441no—
407C22⋊C4—41no—
408C22⋊C4—41no—
409C22⋊C4—611no—
410C22⋊C4—611no—
411C22⋊C4—2441no—
412C22⋊C4—2441no—
413C22⋊C4—2441no—
414C22⋊C4—2441no—
415C22⋊C4—2441no—
416C42—41no—
417C42—611no—
418C42—611no—
419C42—2441no—
420C42—2441no—
421C16—21no—
422C16—21no—
423C16—1221no—
424C16—1221no—
425C23—9761no—
426C23—9761no—
427C23—2444no—
428C23—2441no—
429C23—2441no—
430C23D8×D6111yescharacteristic
431Q8—41no—
432Q8—41no—
433Q8—2441no—
434Q8—2441no—
435D4—1222no—
436D4—1222no—
437D4—2444no—
438D4—2441no—
439D4—2441no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—611no—
447C2×C4—611no—
448C2×C4—611no—
449C2×C4—611no—
450C2×C4—9761no—
451C2×C4—9761no—
452C2×C4—9761no—
453C2×C4—9761no—
454C2×C4—2441no—
455C2×C4—2441no—
456C2×C4D8×D6111yescharacteristic
457C2×C4C2×D4×D6111yescharacteristic
458C8—21no—
459C8—1221no—
460C8—1221no—
461C8#41×D6111yescharacteristic
462C8#41×D6111yescharacteristic
463C22—21no—
464C22—21no—
465C22—21no—
466C22—9764no—
467C22—4881no—
468C22—4881no—
469C22—4881no—
470C22—4881no—
471C22—2442no—
472C22—2442no—
473C22C2×D8×D6111yescharacteristic
474C4—21no—
475C4—81no—
476C4—81no—
477C4—81no—
478C4—81no—
479C4—1221no—
480C4—1221no—
481C4—1221no—
482C4—1221no—
483C4—4881no—
484C4—4881no—
485C4#1684027011yescharacteristic
486C4#1684056111yescharacteristic
487C2—41no—
488C2—4882no—
489C2—4882no—
490C2#2145581611yescharacteristic
491C2#2145896211yescharacteristic
492C2#2145896411yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2237C2
socleC22×C61C22×D8
prime radicalC22×D488C22
hypercenter#2237D61
hypocenterC61#33683
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2237

Complements

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

NH ≅ G/Ntype
1. #21456471C2semidirect
3. #21448027C2semidirect
4. #21447979C2semidirect
5. #21447975C2semidirect
6. #21447863C2semidirect
7. #21447863C2semidirect
10. #21447686C2semidirect
11. #21447682C2semidirect
12. #21447670C2semidirect
13. #21447668C2semidirect
14. #21447623C2semidirect
15. C61×#2237C2semidirect
23. #16840163C22semidirect
24. #16840140C22semidirect
31. Q16×Dic61C22semidirect
32. #16839856C22semidirect
39. #16839589C22semidirect
40. #16839587C22semidirect
41. #16839580C22semidirect
42. #16839580C22semidirect
43. #16839565C22semidirect
44. #16839565C22semidirect
50. #16839537C22semidirect
51. C16×Dic61C22semidirect
54. C61×#367C22semidirect
55. C61×#366C22semidirect
56. C61×#358C22semidirect
57. C61×#358C22semidirect
244. #2237D61semidirect
288. #367C2×D61semidirect
290. #366C2×D61semidirect
295. #358C2×D61semidirect
296. #358C2×D61semidirect
305. C61#33683semidirect

Maximal subgroups

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

subgroupindex
#214564712
#214546702
#214480272
#214479792
#214479752
#214478632
#214477502
#214477472
#214476862
#214476822
#214476702
#214476682
#214476232
C61×#22372
#3368361