Finite Group Databasebeta

← #29663957

Subgroups — #29663957

Order 15616.

608 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456330C211yesmaximal, characteristic, complemented
2#21455980C212yesmaximal
3C8×#11758226C211yesmaximal, characteristic
4#21445223C211yesmaximal, characteristic, complemented
5#21445219C211yesmaximal, characteristic, complemented
6#21445205C211yesmaximal, characteristic, complemented
7#21445199C212yesmaximal, complemented
8#21445193C212yesmaximal, complemented
9#21444581C211yesmaximal, characteristic, complemented
10#21444526C212yesmaximal, complemented
11C61×#1674C211yesmaximal, characteristic, complemented
12#16840317C2212yes—
13C4×#11758226C2211yescharacteristic
14#16840167C2211yescharacteristic
15#16840163—21no—
16#16840163—21no—
17#16840158C2212yes—
18#16840101—22no—
19#16840057C2212yes—
20#16840053C2212yes—
21#16839964C2212yes—
22#16839923—22no—
23Q8×#92418C2211yescharacteristic
24#16839869—22no—
25#16839865—22no—
26C8×D244—21no—
27C8×D244C2211yescharacteristic
28#16839827—21no—
29#16839827C2211yescharacteristic
30#16839818C2212yes—
31#16839814—21no—
32#16839814C2211yescharacteristic
33#16839799—21no—
34#16839799—21no—
35C4×C8×D61—21no—
36C4×C8×D61C2211yescharacteristic
37#16839515C2211yescharacteristic
38#16839512C2212yes—
39#16839510C2212yes—
40#16839510C2212yes—
41#16839508C2212yes—
42C61×#500C2211yescharacteristic
43C61×#462C2212yes—
44C8×Q8×C61C2211yescharacteristic
45C61×#335C2211yescharacteristic
46C61×#332C2212yes—
47C2×#11758226D411yescharacteristic
48C2×#11758201—22no—
49C2×C8×D61—21no—
50C2×C8×D61—21no—
51C2×C8×D61D411yescharacteristic
52C2×C8×D61D411yescharacteristic
53#14519777C2312yes—
54#14519768—22no—
55#14519757—22no—
56#14519753—22no—
57Q8×Dic61—22no—
58Q8×Dic61D411yescharacteristic
59#14519742—22no—
60C4×D244—21no—
61C4×D244C2311yescharacteristic
62#14519739—21no—
63#14519739D411yescharacteristic
64#14519735—21no—
65#14519735—22no—
66#14519735D411yescharacteristic
67C42×D61—21no—
68C42×D61C2311yescharacteristic
69#14519711—21no—
70#14519711—21no—
71#14519710—22no—
72#14519709—22no—
73#14519708—22no—
74#14519708—22no—
75C4⋊#92418—21no—
76C4⋊#92418C2312yes—
77C4⋊#92418C2311yescharacteristic
78#14519701—22no—
79#14519701—22no—
80#14519699—22no—
81#14519689—21no—
82#14519689—22no—
83#14519689—21no—
84#14519689—21no—
85#14519687—22no—
86C4×#92418—21no—
87C4×#92418C2311yescharacteristic
88C8×Dic61—21no—
89C8×Dic61D411yescharacteristic
90C8×Dic61D411yescharacteristic
91C2×Q16×C61—22no—
92C4⋊Q8×C61C2312yes—
93C4×Q8×C61C2311yescharacteristic
94#107×C61—21no—
95#107×C61—21no—
96C4⋊C8×C61—21no—
97C4⋊C8×C61C2311yescharacteristic
98C4⋊C8×C61C2312yes—
99#103×C61—22no—
100C4×C8×C61—21no—
101C4×C8×C61C2311yescharacteristic
102C2×Q8×C61—22no—
103C2×Q8×C61C2×D411yescharacteristic
104#11758226—22no—
105#11758226—22no—
106C2×D244—21no—
107C2×D244C2×D411yescharacteristic
108C2×C4×D61—21no—
109C2×C4×D61—21no—
110C2×C4×D61—21no—
111C2×C4×D61C2×D411yescharacteristic
112C2×C4×D61C2×D411yescharacteristic
113C2×C4×D61#4111yescharacteristic
114C2×Dic122—22no—
115Q16×C61—44no—
116C2×C8×C61—21no—
117C2×C8×C61—22no—
118C2×C8×C61C2×D411yescharacteristic
119C2×C8×C61C2×D411yescharacteristic
120C4⋊C4×C61—21no—
121C4⋊C4×C61—22no—
122C4⋊C4×C61—22no—
123C4⋊C4×C61C2×D411yescharacteristic
124C42×C61—21no—
125C42×C61C2411yescharacteristic
126#11758201—44no—
127#11758198—21no—
128#11758198—21no—
129#11758198—21no—
130#11758198—42no—
131C4⋊Dic61—21no—
132C4⋊Dic61—22no—
133C4⋊Dic61—22no—
134C4⋊Dic61—42no—
135C4⋊Dic61C2×D411yescharacteristic
136#11758196—42no—
137C4×Dic61—21no—
138C4×Dic61—21no—
139C4×Dic61—42no—
140C4×Dic61C2×D411yescharacteristic
141C4×Dic61C2×D411yescharacteristic
142C4×Dic61#4111yescharacteristic
143C2×#92418—21no—
144C2×#92418—22no—
145C2×#92418C2×D411yescharacteristic
146C2×#92418C2×D411yescharacteristic
147C8×D61—22no—
148C8×D61—22no—
149C8×D61—42no—
150C8×D61—42no—
151C22×D61—21no—
152C22×D61#12111yescharacteristic
153Q8×C61—21no—
154Q8×C61—42no—
155Q8×C61—42no—
156Q8×C61#13512yes—
157C2×C4×C61—21no—
158C2×C4×C61—21no—
159C2×C4×C61—42no—
160C2×C4×C61C22×D411yescharacteristic
161C2×C4×C61C22×D411yescharacteristic
162C2×C4×C61C2×#4111yescharacteristic
163C2×Dic61—21no—
164C2×Dic61—42no—
165C2×Dic61#12111yescharacteristic
166C2×Dic61#12111yescharacteristic
167C2×Dic61#12111yescharacteristic
168D244—24no—
169D244—21no—
170D244#13512yes—
171C4×D61—24no—
172C4×D61—21no—
173C4×D61—21no—
174C4×D61—21no—
175C4×D61—21no—
176C4×D61—41no—
177C4×D61—42no—
178C4×D61—41no—
179C4×D61—41no—
180Dic122—82no—
181C8×C61—21no—
182C8×C61—42no—
183C8×C61—41no—
184C8×C61#13712yes—
185#92418—21no—
186#92418—42no—
187#92418—41no—
188#92418#13712yes—
189#19273—611nomaximal, Sylow
190C22×C61C2×#12111yescharacteristic
191C2×D61—21no—
192C2×D61—21no—
193C2×D61—22no—
194C2×D61—21no—
195C2×D61—41no—
196C2×D61—41no—
197C4×C61—22no—
198C4×C61—21no—
199C4×C61—41no—
200C4×C61—82no—
201C4×C61C2×#13511yescharacteristic
202C4×C61C2×#13511yescharacteristic
203C4×C61C2×#13711yescharacteristic
204C4×C61C2×#13711yescharacteristic
205Dic61—21no—
206Dic61—21no—
207Dic61—21no—
208Dic61—41no—
209Dic61—82no—
210#3160—611no—
211#3116—612no—
212C8×#41—611no—
213#1676—611no—
214#1675—611no—
215#1674D6111yescharacteristic, complemented
216#1673—611no—
217#1672—612no—
218#1671—612no—
219#1568—611no—
220#1554—612no—
221C2×C61#303911yescharacteristic
222C2×C61#305011yescharacteristic
223C2×C61#306711yescharacteristic
224D61—41no—
225D61—42no—
226#533—612no—
227C4×#41—611no—
228#500—611no—
229#500C2×D6111yescharacteristic
230#499—1221no—
231#499—1221no—
232#498—612no—
233#485—1222no—
234#477—612no—
235#476—612no—
236#462C2×D6112yes—
237#461—612no—
238#452—1222no—
239C8×Q8—611no—
240C8×Q8C2×D6111yescharacteristic
241#441—1222no—
242#440—1222no—
243C8×D4—611no—
244C8×D4—611no—
245C8×D4—1221no—
246C8×D4—1221no—
247#432—612no—
248#431—611no—
249#431—1221no—
250#427—1221no—
251#427—1221no—
252C2×C4×C8—611no—
253C2×C4×C8—1221no—
254#335—611no—
255#335C2×D6111yescharacteristic
256#334—612no—
257#333—612no—
258#333—612no—
259#332—612no—
260#332C2×D6112yes—
261C61#1927311yescharacteristic, complemented, Sylow
262C2×#41—611no—
263C2×Q16—22no—
264C2×SD16—1222no—
265C22×C8—611no—
266C22×C8—611no—
267C22×C8—1221no—
268C22×C8—1221no—
269C4⋊Q8—612no—
270C4⋊Q8C22×D6112yes—
271#125—1222no—
272C22⋊Q8—1222no—
273C22⋊Q8—1222no—
274C4×Q8—611no—
275C4×Q8—1222no—
276C4×Q8—1222no—
277C4×Q8C22×D6111yescharacteristic
278C4×D4—611no—
279C4×D4—611no—
280C4×D4—1221no—
281C4×D4—1221no—
282#117—611no—
283#117—1221no—
284#117—1222no—
285C2×C42—611no—
286C2×C42—1221no—
287#107—21no—
288#107—21no—
289#107—1221no—
290#107—1221no—
291#107—1222no—
292#106—1222no—
293#106—1222no—
294#106—1222no—
295C4⋊C8—21no—
296C4⋊C8—611no—
297C4⋊C8—612no—
298C4⋊C8—1221no—
299C4⋊C8C22×D6111yescharacteristic
300C4⋊C8C22×D6112yes—
301#103—22no—
302#103—1222no—
303#103—1222no—
304#102—1222no—
305C22⋊C8—1221no—
306C22⋊C8—1221no—
307C22⋊C8—1222no—
308C22⋊C8—1221no—
309#97—1222no—
310C4×C8—21no—
311C4×C8—611no—
312C4×C8—611no—
313C4×C8—611no—
314C4×C8—1221no—
315C4×C8—1221no—
316C4×C8C22×D6111yescharacteristic
317#41—1222no—
318#41—1222no—
319C2×Q8—22no—
320C2×Q8—1222no—
321C2×Q8D4×D6111yescharacteristic
322C2×D4—611no—
323C2×D4—1221no—
324C22×C4—611no—
325C22×C4—611no—
326C22×C4—611no—
327C22×C4—1221no—
328C22×C4—1221no—
329C22×C4—1221no—
330Q16—44no—
331SD16—2444no—
332C2×C8—21no—
333C2×C8—22no—
334C2×C8—611no—
335C2×C8—611no—
336C2×C8—1222no—
337C2×C8—1222no—
338C2×C8—1221no—
339C2×C8—1222no—
340C2×C8—2442no—
341C2×C8—2442no—
342C2×C8D4×D6111yescharacteristic
343C2×C8D4×D6111yescharacteristic
344C4⋊C4—21no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—611no—
348C4⋊C4—1221no—
349C4⋊C4—1222no—
350C4⋊C4—1222no—
351C4⋊C4—2442no—
352C4⋊C4—2442no—
353C4⋊C4D4×D6111yescharacteristic
354C22⋊C4—1221no—
355C22⋊C4—1221no—
356C22⋊C4—1221no—
357C22⋊C4—2442no—
358C42—21no—
359C42—611no—
360C42—611no—
361C42—611no—
362C42—1221no—
363C42—1221no—
364C42—2442no—
365C42C23×D6111yescharacteristic
366C23—611no—
367C23—1221no—
368Q8—21no—
369Q8—42no—
370Q8—42no—
371Q8—4882no—
372Q8#1451979612yes—
373D4—612no—
374D4—1224no—
375D4—1221no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—42no—
379C2×C4—611no—
380C2×C4—611no—
381C2×C4—611no—
382C2×C4—1224no—
383C2×C4—1221no—
384C2×C4—1221no—
385C2×C4—1221no—
386C2×C4—1221no—
387C2×C4—1221no—
388C2×C4—2441no—
389C2×C4—2442no—
390C2×C4—2442no—
391C2×C4—2441no—
392C2×C4—2441no—
393C2×C4C2×D4×D6111yescharacteristic
394C2×C4C2×D4×D6111yescharacteristic
395C2×C4#41×D6111yescharacteristic
396C8—21no—
397C8—42no—
398C8—41no—
399C8—612no—
400C8—1221no—
401C8—2442no—
402C8—2441no—
403C8#1451980912yes—
404C22—1222no—
405C22—1221no—
406C22—1221no—
407C22—1221no—
408C22—2441no—
409C22—2441no—
410C22#121×D6111yescharacteristic
411C4—22no—
412C4—21no—
413C4—41no—
414C4—82no—
415C4—1221no—
416C4—1221no—
417C4—1221no—
418C4—4882no—
419C4—2441no—
420C4#135×D6111yescharacteristic
421C4C2×#1451979611yescharacteristic
422C4#137×D6111yescharacteristic
423C4C2×#1451980911yescharacteristic
424C2—2441no—
425C2—2442no—
426C2#2145530311yescharacteristic
427C2#2145539811yescharacteristic
428C2#2145554111yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1674C2
socleC22×C61C2×#121
prime radicalC2×#11758226D4
hypercenter#1674D61
hypocenterC61#19273
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1674

Complements

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

NH ≅ G/Ntype
1. #21456330C2semidirect
4. #21445223C2semidirect
5. #21445219C2semidirect
6. #21445205C2semidirect
7. #21445199C2semidirect
8. #21445193C2semidirect
9. #21444581C2semidirect
10. #21444526C2semidirect
11. C61×#1674C2semidirect
215. #1674D61semidirect
261. C61#19273semidirect

Maximal subgroups

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

subgroupindex
#214563302
#214559802
C8×#117582262
#214452232
#214452192
#214452052
#214451992
#214451932
#214445812
#214445262
C61×#16742
#1927361