Finite Group Databasebeta

← #37584057

Subgroups — #37584057

Order 24448.

510 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
1C2×#20259346C211yesmaximal, characteristic, complemented
2C2×#20259344C211yesmaximal, characteristic, complemented
3#27133331C211yesmaximal, characteristic, complemented
4#27133310C211yesmaximal, characteristic, complemented
5#27133051C211yesmaximal, characteristic, complemented
6#27132910C211yesmaximal, characteristic, complemented
7#27132884C211yesmaximal, characteristic, complemented
8#27132857C211yesmaximal, characteristic, complemented
9#27132826C211yesmaximal, characteristic, complemented
10#27132806C211yesmaximal, characteristic, complemented
11#27132806C211yesmaximal, characteristic, complemented
12#27132798C211yesmaximal, characteristic, complemented
13#27132762C211yesmaximal, characteristic, complemented
14#27132734C211yesmaximal, characteristic, complemented
15#466×C191C211yesmaximal, characteristic, complemented
16C2×#15690287C2211yescharacteristic
17C2×#15690287C2211yescharacteristic, complemented
18C2×#15690286C2211yescharacteristic
19C2×D4×D191C2211yescharacteristic
20#20259346—24no—
21#20259344—24no—
22C2×Dic764C2211yescharacteristic, complemented
23C2×#15690276C2211yescharacteristic
24C2×#15690276C2211yescharacteristic, complemented
25C2×#15690275C2211yescharacteristic
26C2×#15690275C2211yescharacteristic, complemented
27C2×D1528C2211yescharacteristic
28C2×#15690272C2211yescharacteristic
29C2×#15690272C2211yescharacteristic
30C2×#15690270C2211yescharacteristic
31#20259318—21no—
32#20259318C2211yescharacteristic, complemented
33#20259309C2211yescharacteristic
34#20259304—21no—
35#20259302C2211yescharacteristic
36#20259297C2211yescharacteristic
37#20259295—21no—
38#20259294—21no—
39#20259293—21no—
40#20259290—21no—
41#20259287C2211yescharacteristic, complemented
42#20259284C2211yescharacteristic
43C4×C22⋊D191C2211yescharacteristic
44(C22⋊D191)⋊C4—21no—
45C22⋊C4×D191—21no—
46Dic191⋊C8C2211yescharacteristic, complemented
47#20259245C2211yescharacteristic, complemented
48#20259245C2211yescharacteristic, complemented
49#20259242C2211yescharacteristic
50#20259242C2211yescharacteristic, complemented
51C22⋊(C191⋊C8)C2211yescharacteristic, complemented
52(C2×D191)⋊C8C2211yescharacteristic
53#20259232C2211yescharacteristic, complemented
54C2×D8×C191C2211yescharacteristic
55C22×C8×C191C2211yescharacteristic, complemented
56#121×C191C2211yescharacteristic
57#121×C191C2211yescharacteristic, complemented
58#107×C191C2211yescharacteristic, complemented
59#102×C191C2211yescharacteristic
60#102×C191C2211yescharacteristic, complemented
61C23×D191—21no—
62#15690287—22no—
63#15690287—22no—
64#15690287—41no—
65#15690287—41no—
66#15690286—22no—
67#15690286—41no—
68C2×Dic382C2311yescharacteristic
69C2×Dic382C2311yescharacteristic
70C2×D764—21no—
71C2×D764C2311yescharacteristic
72C2×D764C2311yescharacteristic
73C2×C22⋊D191—21no—
74C2×C22⋊D191—21no—
75C2×C22⋊D191—21no—
76C2×C22⋊D191—21no—
77C2×C22⋊D191—21no—
78C2×C22⋊D191—21no—
79C2×C22⋊D191D411yescharacteristic
80D4×D191—22no—
81D4×D191—41no—
82C22×Dic191—21no—
83C2×C4×D191—21no—
84C2×C4×D191—21no—
85C2×C4×D191—21no—
86C2×C4×D191C2311yescharacteristic
87Dic764—22no—
88#15690276—22no—
89#15690276D414yescomplemented
90#15690275—22no—
91#15690275—41no—
92D1528—22no—
93#15690272—22no—
94#15690272—41no—
95#15690270—22no—
96C2×C191⋊C8C2311yescharacteristic
97C2×C191⋊C8C2311yescharacteristic
98C4⋊Dic191—21no—
99Dic191⋊C4—21no—
100Dic191⋊C4D411yescharacteristic
101C22⋊Dic191—21no—
102C22⋊Dic191—21no—
103C22⋊Dic191D411yescharacteristic
104(C2×D191)⋊C4—21no—
105(C2×D191)⋊C4—21no—
106(C2×D191)⋊C4—21no—
107(C2×D191)⋊C4D411yescharacteristic
108C4×Dic191—21no—
109C4×Dic191C2311yescharacteristic
110C2×D4×C191—21no—
111C2×D4×C191—21no—
112C2×D4×C191C2311yescharacteristic
113C2×D4×C191C2311yescharacteristic
114C22×C4×C191C2311yescharacteristic
115D8×C191—22no—
116C2×C8×C191—22no—
117C2×C8×C191C2311yescharacteristic
118C2×C8×C191C2311yescharacteristic
119C4⋊C4×C191C2311yescharacteristic
120C4⋊C4×C191C2311yescharacteristic
121C22⋊C4×C191—21no—
122C22⋊C4×C191—21no—
123C22×D191—21no—
124C22×D191—21no—
125C22×D191—42no—
126C22×D191—42no—
127C22×D191—81no—
128C22×D191C2×D411yescharacteristic
129Dic382—21no—
130Dic382—21no—
131Dic382—21no—
132Dic382C2×D412yes—
133D764—21no—
134D764—21no—
135D764—21no—
136D764—81no—
137D764C2×D412yes—
138C22⋊D191—41no—
139C22⋊D191—42no—
140C22⋊D191—42no—
141C22⋊D191—42no—
142C22⋊D191—42no—
143C22⋊D191—42no—
144C22⋊D191—81no—
145C2×Dic191—21no—
146C2×Dic191—21no—
147C2×Dic191—42no—
148C2×Dic191C2×D411yescharacteristic
149C2×Dic191C2×D411yescharacteristic
150C2×Dic191C2×D411yescharacteristic
151C4×D191—21no—
152C4×D191—21no—
153C4×D191—42no—
154C4×D191—42no—
155C4×D191—81no—
156C191⋊C8—21no—
157C191⋊C8—21no—
158C23×C191—21no—
159C23×C191—21no—
160C23×C191C2×D411yescharacteristic
161D4×C191—21no—
162D4×C191—21no—
163D4×C191—21no—
164D4×C191—21no—
165D4×C191—81no—
166D4×C191—81no—
167C2×C4×C191—21no—
168C2×C4×C191—21no—
169C2×C4×C191—21no—
170C2×C4×C191—21no—
171C2×C4×C191C2×D411yescharacteristic
172C2×C4×C191C2411yescharacteristic
173C8×C191—21no—
174C8×C191C2×D412yes—
175C22×C191—21no—
176C22×C191—21no—
177C22×C191—21no—
178C22×C191—41no—
179C22×C191—41no—
180C22×C191—41no—
181C22×C191—41no—
182C22×C191—41no—
183C22×C191—41no—
184C22×C191C22×D411yescharacteristic
185C2×D191—21no—
186C2×D191—22no—
187C2×D191—21no—
188C2×D191—21no—
189C2×D191—41no—
190C2×D191—41no—
191C2×D191—41no—
192C2×D191—41no—
193C2×D191—41no—
194C2×D191—84no—
195C4×C191—21no—
196C4×C191—41no—
197C4×C191—41no—
198C4×C191C22×D411yescharacteristic
199C4×C191H(5,𝔽2)11yescharacteristic
200Dic191—21no—
201Dic191—22no—
202Dic191—21no—
203Dic191—21no—
204Dic191—41no—
205C2×C191—41no—
206C2×C191—81no—
207C2×C191—81no—
208C2×C191D4211yescharacteristic
209C2×C191#57511yescharacteristic
210C2×C191#57611yescharacteristic
211D191—41no—
212D191—42no—
213D191—81no—
214C191#326811yescharacteristic, complemented, Sylow
215#3268—1911nomaximal, Sylow
216C2×AGL(1,ℤ/8)—1911no—
217C2×#135—1911no—
218#550—1911no—
219#546—1911no—
220#496—1911no—
221#468—1911no—
222#466D19111yescharacteristic, complemented
223#464—1911no—
224#459—1911no—
225#452—1911no—
226#449—1911no—
227#449—1911no—
228#447—1911no—
229#440—1911no—
230#435—1911no—
231C2×#41—1911no—
232C2×#41—1911no—
233C2×#41—1911no—
234C22×D4—1911no—
235AGL(1,ℤ/8)—3824no—
236#135—3824no—
237C2×Q16—1911no—
238C2×SD16—1911no—
239C2×SD16—1911no—
240C2×SD16—1911no—
241C2×SD16—1911no—
242C2×D8—1911no—
243C2×D8—1911no—
244C2×D8—1911no—
245C2×D8C2×D19111yescharacteristic
246C2×M16—1911no—
247C22×C8C2×D19111yescharacteristic, complemented
248#127—3821no—
249#127—1911no—
250#124—1911no—
251#123—3821no—
252C22⋊Q8—1911no—
253#121—3821no—
254#121—3821no—
255#121—3821no—
256#121—1911no—
257#121C2×D19111yescharacteristic
258#121C2×D19111yescharacteristic, complemented
259#120—3821no—
260C4×Q8—1911no—
261C4×D4—3821no—
262C4×D4—1911no—
263C4×D4—1911no—
264C2×C22⋊C4—3821no—
265#107C2×D19111yescharacteristic, complemented
266C4⋊C8—1911no—
267#103—1911no—
268#103—1911no—
269#102—1911no—
270#102—1911no—
271#102C2×D19111yescharacteristic
272#102C2×D19111yescharacteristic, complemented
273C22⋊C8—1911no—
274C22⋊C8—1911no—
275#97—1911no—
276C24—3821no—
277#41—3822no—
278#41—3822no—
279#41—3822no—
280#41—7641no—
281#41—7641no—
282#41—7641no—
283C2×Q8—1911no—
284C2×Q8—1911no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—3821no—
288C2×D4—3822no—
289C2×D4—3821no—
290C2×D4—3821no—
291C2×D4—3821no—
292C2×D4—3821no—
293C2×D4—3821no—
294C2×D4—3821no—
295C2×D4—1911no—
296C2×D4—1911no—
297C2×D4—1911no—
298C2×D4—7641no—
299C2×D4C22×D19111yescharacteristic
300C2×D4C22×D19111yescharacteristic
301C22×C4—3821no—
302C22×C4—3821no—
303C22×C4—3821no—
304C22×C4—3821no—
305C22×C4—1911no—
306C22×C4C22×D19111yescharacteristic
307Q16—3822no—
308SD16—3822no—
309SD16—3822no—
310SD16—1914no—
311SD16—7641no—
312D8—22no—
313D8—3822no—
314D8—3822no—
315D8—7641no—
316M16—3822no—
317C2×C8—22no—
318C2×C8—1911no—
319C2×C8—1911no—
320C2×C8C22×D19111yescharacteristic
321C2×C8C22×D19111yescharacteristic
322C4⋊C4—3821no—
323C4⋊C4—3821no—
324C4⋊C4—1911no—
325C4⋊C4C22×D19111yescharacteristic
326C4⋊C4C22×D19111yescharacteristic
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—3821no—
330C22⋊C4—3821no—
331C22⋊C4—3821no—
332C22⋊C4—3821no—
333C22⋊C4—3821no—
334C22⋊C4—1911no—
335C22⋊C4—1911no—
336C42—3821no—
337C42—1911no—
338C23—21no—
339C23—21no—
340C23—3821no—
341C23—3821no—
342C23—1911no—
343C23—15281no—
344C23—7642no—
345C23—7642no—
346C23D4×D19111yescharacteristic
347Q8—3821no—
348Q8—3821no—
349Q8—3821no—
350Q8—1912no—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—81no—
356D4—81no—
357D4—3821no—
358D4—3821no—
359D4—3821no—
360D4—1912no—
361D4—15281no—
362D4—15281no—
363D4—7641no—
364D4—7642no—
365D4—7642no—
366D4—7642no—
367D4—7642no—
368D4—7642no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—3821no—
374C2×C4—3821no—
375C2×C4—3821no—
376C2×C4—3821no—
377C2×C4—1911no—
378C2×C4—1911no—
379C2×C4—1911no—
380C2×C4—15281no—
381C2×C4—7642no—
382C2×C4—7642no—
383C2×C4—7642no—
384C2×C4D4×D19111yescharacteristic
385C2×C4C23×D19111yescharacteristic
386C8—21no—
387C8—3821no—
388C8—3821no—
389C8D4×D19112yes—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—3821no—
400C22—3821no—
401C22—3821no—
402C22—3822no—
403C22—15284no—
404C22—7641no—
405C22—7641no—
406C22—7641no—
407C22—7641no—
408C22—7641no—
409C22C2×D4×D19111yescharacteristic
410C4—21no—
411C4—41no—
412C4—41no—
413C4—3821no—
414C4—3821no—
415C4—3821no—
416C4—3822no—
417C4—7641no—
418C4C2×D4×D19111yescharacteristic
419C4#2025936811yescharacteristic
420C2—41no—
421C2—81no—
422C2—81no—
423C2—15281no—
424C2—7641no—
425C2—7642no—
426C2#2713329711yescharacteristic
427C2#2713346011yescharacteristic
428C2#2713346411yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#466×C191C2
socleC22×C191C22×D4
hypercenter#466D191
hypocenterC191#3268
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. C2×#20259346C2semidirect
2. C2×#20259344C2semidirect
3. #27133331C2semidirect
4. #27133310C2semidirect
5. #27133051C2semidirect
6. #27132910C2semidirect
7. #27132884C2semidirect
8. #27132857C2semidirect
9. #27132826C2semidirect
10. #27132806C2semidirect
11. #27132806C2semidirect
12. #27132798C2semidirect
13. #27132762C2semidirect
14. #27132734C2semidirect
15. #466×C191C2semidirect
17. C2×#15690287C22semidirect
22. C2×Dic764C22semidirect
24. C2×#15690276C22semidirect
26. C2×#15690275C22semidirect
32. #20259318C22semidirect
41. #20259287C22semidirect
46. Dic191⋊C8C22semidirect
47. #20259245C22semidirect
48. #20259245C22semidirect
50. #20259242C22semidirect
51. C22⋊(C191⋊C8)C22semidirect
53. #20259232C22semidirect
55. C22×C8×C191C22semidirect
57. #121×C191C22semidirect
58. #107×C191C22semidirect
60. #102×C191C22semidirect
89. #15690276D4semidirect
214. C191#3268semidirect
222. #466D191semidirect
247. C22×C8C2×D191semidirect
258. #121C2×D191semidirect
265. #107C2×D191semidirect
272. #102C2×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593462
C2×#202593442
#271333312
#271333102
#271330512
#271329102
#271328842
#271328572
#271328262
#271328062
#271327982
#271327622
#271327342
#466×C1912
#3268191