Finite Group Databasebeta

← #28402178

Subgroups — #28402178

Order 14464.

520 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16813649C211yesmaximal, characteristic, complemented
2C2×#16813647C211yesmaximal, characteristic, complemented
3#20299239C211yesmaximal, characteristic, complemented
4#20299236C211yesmaximal, characteristic, complemented
5#20298983C211yesmaximal, characteristic, complemented
6#20298847C211yesmaximal, characteristic, complemented
7#20298846C211yesmaximal, characteristic, complemented
8#20298816C211yesmaximal, characteristic, complemented
9#20298815C211yesmaximal, characteristic, complemented
10#20298759C211yesmaximal, characteristic, complemented
11#20298734C211yesmaximal, characteristic, complemented
12#20298733C211yesmaximal, characteristic, complemented
13#20298678C211yesmaximal, characteristic, complemented
14#20298673C211yesmaximal, characteristic, complemented
15#452×C113C211yesmaximal, characteristic, complemented
16C2×#14268648C2211yescharacteristic
17C22×D452C2211yescharacteristic, complemented
18C2×D4×D113C2211yescharacteristic, complemented
19#16813649—24no—
20#16813647—24no—
21C2×#14268634C2211yescharacteristic, complemented
22C2×#14268634C2211yescharacteristic, complemented
23C2×#14268634C2211yescharacteristic, complemented
24C2×D904C2211yescharacteristic
25C2×D904C2211yescharacteristic
26C2×#14268632C2211yescharacteristic
27C2×#14268632C2211yescharacteristic
28C2×#14268631C2211yescharacteristic, complemented
29C2×#14268630C2211yescharacteristic, complemented
30#16813611C2211yescharacteristic, complemented
31#16813611C2211yescharacteristic, complemented
32#16813607—21no—
33#16813605—21no—
34#16813598C2211yescharacteristic, complemented
35#16813597—21no—
36#16813597C2211yescharacteristic
37#16813596C2211yescharacteristic
38#16813594—21no—
39#16813592—21no—
40#16813590—21no—
41D4×Dic113C2211yescharacteristic, complemented
42#16813584C2211yescharacteristic, complemented
43C4×C22⋊D113C2211yescharacteristic
44C2×#14268624—21no—
45C22⋊C4×D113—21no—
46#16813555C2211yescharacteristic, complemented
47#16813551C2211yescharacteristic, complemented
48#16813546C2211yescharacteristic, complemented
49#16813544C2211yescharacteristic, complemented
50#16813544C2211yescharacteristic, complemented
51#16813542C2211yescharacteristic, complemented
52#16813534C2211yescharacteristic, complemented
53C8×Dic113C2211yescharacteristic, complemented
54C2×#41×C113C2211yescharacteristic
55C2×Q16×C113C2211yescharacteristic, complemented
56C2×SD16×C113C2211yescharacteristic, complemented
57C22⋊Q8×C113C2211yescharacteristic, complemented
58#103×C113C2211yescharacteristic, complemented
59#102×C113C2211yescharacteristic, complemented
60C22⋊C8×C113C2211yescharacteristic, complemented
61C23×D113—21no—
62C23×D113—21no—
63#14268648—22no—
64#14268648—41no—
65C2×D452—22no—
66C2×D452—21no—
67C2×D452—21no—
68C2×D452—21no—
69C2×D452—41no—
70C2×D452C2311yescharacteristic
71C2×D452C2311yescharacteristic
72C2×D452C2311yescharacteristic
73C2×C22⋊D113—21no—
74C2×C22⋊D113—21no—
75C2×C22⋊D113D411yescharacteristic
76D4×D113—22no—
77D4×D113—41no—
78C22×Dic113—21no—
79C2×C4×D113—21no—
80C2×C4×D113—21no—
81C2×C4×D113C2311yescharacteristic
82#14268634—22no—
83#14268634—22no—
84#14268634—22no—
85D904—22no—
86D904—41no—
87#14268632—22no—
88#14268632D414yescomplemented
89#14268631—22no—
90#14268630—22no—
91C2×#11744751C2311yescharacteristic
92C2×#11744751C2311yescharacteristic
93C4⋊Dic113C2311yescharacteristic
94#14268626—21no—
95#14268626D411yescharacteristic
96C22⋊Dic113—21no—
97C22⋊Dic113D411yescharacteristic
98#14268624—21no—
99#14268624—21no—
100#14268624—21no—
101#14268624—21no—
102#14268624—21no—
103#14268624—21no—
104#14268624—21no—
105#14268624—21no—
106#14268624D411yescharacteristic
107C4×Dic113C2311yescharacteristic
108#41×C113—22no—
109#41×C113—41no—
110C2×Q8×C113C2311yescharacteristic
111C2×Q8×C113C2311yescharacteristic
112C2×D4×C113—21no—
113C2×D4×C113C2311yescharacteristic
114C22×C4×C113—21no—
115C22×C4×C113C2311yescharacteristic
116Q16×C113—41no—
117SD16×C113—22no—
118C2×C8×C113C2311yescharacteristic
119C2×C8×C113C2311yescharacteristic
120C4⋊C4×C113—21no—
121C4⋊C4×C113C2311yescharacteristic
122C22⋊C4×C113—21no—
123C23×C113—21no—
124C23×C113C2×D411yescharacteristic
125C22×D113—21no—
126C22×D113—21no—
127C22×D113—21no—
128C22×D113—42no—
129C22×D113—42no—
130C22×D113—42no—
131C22×D113—42no—
132C22×D113—81no—
133C22×D113—81no—
134C22×D113C2×D411yescharacteristic
135Q8×C113—21no—
136Q8×C113—21no—
137Q8×C113—21no—
138Q8×C113—21no—
139D4×C113—21no—
140D4×C113—42no—
141D4×C113C2×D412yes—
142C2×C4×C113—21no—
143C2×C4×C113—21no—
144C2×C4×C113—21no—
145C2×C4×C113—21no—
146C2×C4×C113—21no—
147C2×C4×C113—42no—
148C2×C4×C113C2×D411yescharacteristic
149C2×C4×C113C2411yescharacteristic
150C22⋊D113—41no—
151C22⋊D113—42no—
152C22⋊D113—81no—
153C2×Dic113—21no—
154C2×Dic113—81no—
155C2×Dic113C2×D411yescharacteristic
156C2×Dic113C2×D411yescharacteristic
157C2×Dic113C2×D411yescharacteristic
158D452—21no—
159D452—21no—
160D452—21no—
161D452—21no—
162D452—21no—
163D452—42no—
164D452—42no—
165D452—81no—
166D452C2×D412yes—
167C4×D113—21no—
168C4×D113—21no—
169C4×D113—42no—
170C4×D113—81no—
171C8×C113—21no—
172C8×C113—21no—
173#11744751—21no—
174#11744751C2×D412yes—
175C22×C113—22no—
176C22×C113—21no—
177C22×C113—21no—
178C22×C113—21no—
179C22×C113—41no—
180C22×C113—41no—
181C22×C113C22×D411yescharacteristic
182C2×D113—21no—
183C2×D113—22no—
184C2×D113—21no—
185C2×D113—21no—
186C2×D113—41no—
187C2×D113—41no—
188C2×D113—41no—
189C2×D113—41no—
190C2×D113—41no—
191C2×D113—41no—
192C2×D113—41no—
193C2×D113—41no—
194C2×D113—84no—
195C2×D113—84no—
196C4×C113—21no—
197C4×C113—41no—
198C4×C113—41no—
199C4×C113—41no—
200C4×C113C22×D411yescharacteristic
201C4×C113H(5,𝔽2)11yescharacteristic
202Dic113—21no—
203Dic113—21no—
204Dic113—21no—
205Dic113—41no—
206C2×C113—41no—
207C2×C113—42no—
208C2×C113D4211yescharacteristic
209C2×C113#57511yescharacteristic
210C2×C113#57611yescharacteristic
211D113—41no—
212D113—42no—
213D113—81no—
214D113—81no—
215#3273—1131nomaximal, Sylow
216C113#327311yescharacteristic, complemented, Sylow
217C2×AGL(1,ℤ/8)—1131no—
218C2×AGL(1,ℤ/8)—1131no—
219#546—1131no—
220#546—1131no—
221#495—1131no—
222#468—1131no—
223#468—1131no—
224#463—1131no—
225#463—1131no—
226#452—1131no—
227#452D11311yescharacteristic, complemented
228#447—1131no—
229#447—1131no—
230C4×D8—1131no—
231#436—1131no—
232C2×#41—1131no—
233C2×#41C2×D11311yescharacteristic
234C22×D4—1131no—
235C22×D4—1131no—
236AGL(1,ℤ/8)—2264no—
237AGL(1,ℤ/8)—2264no—
238C2×Q16C2×D11311yescharacteristic, complemented
239C2×SD16—1131no—
240C2×SD16—1131no—
241C2×SD16—1131no—
242C2×SD16C2×D11311yescharacteristic, complemented
243C2×D8—1131no—
244C2×D8—1131no—
245C2×D8—1131no—
246C2×D8—1131no—
247C2×M16—1131no—
248C2×M16—1131no—
249#124—1131no—
250#124—1131no—
251#123—2261no—
252#123—2261no—
253C22⋊Q8—1131no—
254C22⋊Q8C2×D11311yescharacteristic, complemented
255#121—1131no—
256#121—1131no—
257#121—2261no—
258#121—2261no—
259#120—2261no—
260#120—2261no—
261C4×D4—1131no—
262C4×D4—1131no—
263C4×D4—1131no—
264C2×C22⋊C4—2261no—
265C2×C22⋊C4—2261no—
266#107—1131no—
267C4⋊C8—1131no—
268#103—1131no—
269#103C2×D11311yescharacteristic, complemented
270#102—1131no—
271#102—1131no—
272#102—1131no—
273#102C2×D11311yescharacteristic, complemented
274C22⋊C8—1131no—
275C22⋊C8C2×D11311yescharacteristic, complemented
276C4×C8—1131no—
277C24—2261no—
278C24—2261no—
279#41—22no—
280#41—41no—
281#41—4521no—
282#41—2262no—
283C2×Q8C22×D11311yescharacteristic
284C2×Q8C22×D11311yescharacteristic
285C2×D4—21no—
286C2×D4—1131no—
287C2×D4—1131no—
288C2×D4—1131no—
289C2×D4—1131no—
290C2×D4—4521no—
291C2×D4—4521no—
292C2×D4—2262no—
293C2×D4—2261no—
294C2×D4—2261no—
295C2×D4—2261no—
296C2×D4—2261no—
297C2×D4—2262no—
298C2×D4—2261no—
299C2×D4C22×D11311yescharacteristic
300C22×C4—21no—
301C22×C4—1131no—
302C22×C4—2261no—
303C22×C4—2261no—
304C22×C4—2261no—
305C22×C4C22×D11311yescharacteristic
306Q16—41no—
307SD16—22no—
308SD16—2262no—
309SD16—2262no—
310SD16—2262no—
311D8—1134no—
312D8—4521no—
313D8—2262no—
314D8—2262no—
315M16—2262no—
316M16—2262no—
317C2×C8—1131no—
318C2×C8—1131no—
319C2×C8C22×D11311yescharacteristic
320C2×C8C22×D11311yescharacteristic
321C4⋊C4—21no—
322C4⋊C4—1131no—
323C4⋊C4—1131no—
324C4⋊C4—2261no—
325C4⋊C4C22×D11311yescharacteristic
326C22⋊C4—21no—
327C22⋊C4—1131no—
328C22⋊C4—1131no—
329C22⋊C4—2261no—
330C22⋊C4—2261no—
331C22⋊C4—2261no—
332C22⋊C4—2261no—
333C22⋊C4—2261no—
334C22⋊C4—2261no—
335C22⋊C4—2261no—
336C22⋊C4—2261no—
337C22⋊C4—2261no—
338C42—1131no—
339C23—21no—
340C23—1131no—
341C23—9041no—
342C23—9041no—
343C23—4522no—
344C23—4522no—
345C23—4522no—
346C23—4522no—
347C23—2261no—
348C23—2261no—
349C23—2261no—
350C23D4×D11311yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—21no—
354Q8—21no—
355D4—21no—
356D4—42no—
357D4—1132no—
358D4—9041no—
359D4—9041no—
360D4—4522no—
361D4—4522no—
362D4—4522no—
363D4—4521no—
364D4—2261no—
365D4—2261no—
366D4—2261no—
367D4—2261no—
368D4—2261no—
369D4D4×D11312yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—42no—
376C2×C4—1131no—
377C2×C4—1131no—
378C2×C4—1131no—
379C2×C4—9041no—
380C2×C4—9041no—
381C2×C4—4522no—
382C2×C4—2261no—
383C2×C4—2261no—
384C2×C4—2261no—
385C2×C4D4×D11311yescharacteristic
386C2×C4C23×D11311yescharacteristic
387C8—21no—
388C8—21no—
389C8—1132no—
390C8—2261no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—22no—
395C22—41no—
396C22—41no—
397C22—9044no—
398C22—9044no—
399C22—4521no—
400C22—4521no—
401C22—4521no—
402C22—4521no—
403C22—4521no—
404C22—4521no—
405C22—4521no—
406C22—4521no—
407C22—2261no—
408C22—2261no—
409C22—2262no—
410C22—2261no—
411C22C2×D4×D11311yescharacteristic
412C4—21no—
413C4—41no—
414C4—41no—
415C4—41no—
416C4—4521no—
417C4—2261no—
418C4—2261no—
419C4—2261no—
420C4C2×D4×D11311yescharacteristic
421C4#1681366711yescharacteristic
422C2—41no—
423C2—42no—
424C2—9041no—
425C2—9041no—
426C2—4521no—
427C2—4522no—
428C2#2029923011yescharacteristic
429C2#2029939711yescharacteristic
430C2#2029940011yescharacteristic
subgroupG/H
centerC22C2×D4×D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#452×C113C2
socleC22×C113C22×D4
hypercenter#452D113
hypocenterC113#3273
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. C2×#16813649C2semidirect
2. C2×#16813647C2semidirect
3. #20299239C2semidirect
4. #20299236C2semidirect
5. #20298983C2semidirect
6. #20298847C2semidirect
7. #20298846C2semidirect
8. #20298816C2semidirect
9. #20298815C2semidirect
10. #20298759C2semidirect
11. #20298734C2semidirect
12. #20298733C2semidirect
13. #20298678C2semidirect
14. #20298673C2semidirect
15. #452×C113C2semidirect
17. C22×D452C22semidirect
18. C2×D4×D113C22semidirect
21. C2×#14268634C22semidirect
22. C2×#14268634C22semidirect
23. C2×#14268634C22semidirect
28. C2×#14268631C22semidirect
29. C2×#14268630C22semidirect
30. #16813611C22semidirect
31. #16813611C22semidirect
34. #16813598C22semidirect
41. D4×Dic113C22semidirect
42. #16813584C22semidirect
46. #16813555C22semidirect
47. #16813551C22semidirect
48. #16813546C22semidirect
49. #16813544C22semidirect
50. #16813544C22semidirect
51. #16813542C22semidirect
52. #16813534C22semidirect
53. C8×Dic113C22semidirect
55. C2×Q16×C113C22semidirect
56. C2×SD16×C113C22semidirect
57. C22⋊Q8×C113C22semidirect
58. #103×C113C22semidirect
59. #102×C113C22semidirect
60. C22⋊C8×C113C22semidirect
88. #14268632D4semidirect
216. C113#3273semidirect
227. #452D113semidirect
238. C2×Q16C2×D113semidirect
242. C2×SD16C2×D113semidirect
254. C22⋊Q8C2×D113semidirect
269. #103C2×D113semidirect
273. #102C2×D113semidirect
275. C22⋊C8C2×D113semidirect

Maximal subgroups

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

subgroupindex
C2×#168136492
C2×#168136472
#202992392
#202992362
#202989832
#202988472
#202988462
#202988162
#202988152
#202987592
#202987342
#202987332
#202986782
#202986732
#452×C1132
#3273113