Finite Group Databasebeta

← #37584029

Subgroups — #37584029

Order 24448.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20259354C211yesmaximal, characteristic, complemented
2C2×SD16×D191C211yesmaximal, characteristic, complemented
3#27133328C211yesmaximal, characteristic, complemented
4#27133306C211yesmaximal, characteristic
5#27132906C211yesmaximal, characteristic, complemented
6#27132901C211yesmaximal, characteristic, complemented
7#27132882C211yesmaximal, characteristic, complemented
8#27132865C211yesmaximal, characteristic, complemented
9#27132824C211yesmaximal, characteristic, complemented
10#27132817C211yesmaximal, characteristic, complemented
11#27132812C211yesmaximal, characteristic, complemented
12#27132800C211yesmaximal, characteristic, complemented
13#27132765C211yesmaximal, characteristic, complemented
14#27132736C211yesmaximal, characteristic, complemented
15#497×C191C211yesmaximal, characteristic, complemented
16C2×#15690288C2211yescharacteristic
17C2×Q8×D191C2211yescharacteristic
18C2×Q8×D191C2211yescharacteristic, complemented
19C2×D4×D191C2211yescharacteristic
20#20259354—24no—
21C2×#15690277C2211yescharacteristic, complemented
22C2×#15690276C2211yescharacteristic
23C2×#15690275C2211yescharacteristic
24C2×#15690274C2211yescharacteristic
25C2×#15690274C2211yescharacteristic
26SD16×D191—24no—
27C2×#15690270C2211yescharacteristic, complemented
28C2×C8×D191C2211yescharacteristic
29#20259321—21no—
30#20259304—21no—
31#20259300—21no—
32#20259300—21no—
33#20259298C2211yescharacteristic
34#20259298C2211yescharacteristic
35#20259298C2211yescharacteristic, complemented
36#20259295—21no—
37#20259293C2211yescharacteristic
38#20259292—21no—
39C4×Dic382C2211yescharacteristic, complemented
40C4×D764C2211yescharacteristic
41C4×D764C2211yescharacteristic
42#20259284—21no—
43C22⋊C4×D191—21no—
44#20259256C2211yescharacteristic, complemented
45C4⋊(C191⋊C8)C2211yescharacteristic, complemented
46#20259247C2211yescharacteristic, complemented
47#20259246C2211yescharacteristic, complemented
48#20259246C2211yescharacteristic, complemented
49#20259246C2211yescharacteristic, complemented
50#20259244C2211yescharacteristic
51#20259243C2211yescharacteristic
52(C2×D191)⋊C8C2211yescharacteristic
53(C2×D191)⋊C8C2211yescharacteristic, complemented
54C2×Q16×C191C2211yescharacteristic, complemented
55C2×Q16×C191C2211yescharacteristic, complemented
56C2×SD16×C191C2211yescharacteristic
57C2×SD16×C191C2211yescharacteristic
58C4⋊Q8×C191C2211yescharacteristic, complemented
59#124×C191C2211yescharacteristic
60#97×C191C2211yescharacteristic, complemented
61C23×D191—21no—
62#15690288—22no—
63#15690288—41no—
64C2×Dic382—21no—
65C2×Dic382—21no—
66C2×Dic382C2311yescharacteristic
67Q8×D191—22no—
68Q8×D191—22no—
69Q8×D191—41no—
70Q8×D191—41no—
71C2×D764—21no—
72C2×D764D411yescharacteristic
73C2×D764C2311yescharacteristic
74C2×C22⋊D191—21no—
75C2×C22⋊D191—21no—
76D4×D191—22no—
77D4×D191—41no—
78C2×C4×D191—21no—
79C2×C4×D191—21no—
80C2×C4×D191—21no—
81C2×C4×D191—21no—
82C2×C4×D191C2311yescharacteristic
83C2×C4×D191C2311yescharacteristic
84#15690277—22no—
85#15690276D414yescomplemented
86#15690275—22no—
87#15690274—22no—
88#15690274—22no—
89#15690270—22no—
90C2×C191⋊C8C2311yescharacteristic
91C2×C191⋊C8C2311yescharacteristic
92C8×D191—22no—
93C4⋊Dic191D411yescharacteristic
94C4⋊Dic191C2311yescharacteristic
95C4⋊Dic191C2311yescharacteristic
96Dic191⋊C4—21no—
97Dic191⋊C4—21no—
98Dic191⋊C4—21no—
99Dic191⋊C4—21no—
100C22⋊Dic191—21no—
101(C2×D191)⋊C4—21no—
102(C2×D191)⋊C4—21no—
103(C2×D191)⋊C4—21no—
104(C2×D191)⋊C4—21no—
105(C2×D191)⋊C4D411yescharacteristic
106(C2×D191)⋊C4D411yescharacteristic
107C4×Dic191—21no—
108C2×Q8×C191C2311yescharacteristic
109C2×Q8×C191C2311yescharacteristic
110C2×Q8×C191C2311yescharacteristic
111C2×D4×C191C2311yescharacteristic
112Q16×C191—22no—
113Q16×C191—41no—
114SD16×C191—22no—
115SD16×C191—41no—
116C2×C8×C191C2311yescharacteristic
117C2×C8×C191C2311yescharacteristic
118C4⋊C4×C191—21no—
119C4⋊C4×C191—21no—
120C22⋊C4×C191—21no—
121C22⋊C4×C191—21no—
122C42×C191C2311yescharacteristic
123C22×D191—21no—
124C22×D191—42no—
125C22×D191—42no—
126C22×D191—81no—
127C22×D191C2×D411yescharacteristic
128C22×D191C2×D411yescharacteristic
129Dic382—21no—
130Dic382—42no—
131Dic382—42no—
132Dic382C2×D412yes—
133D764—22no—
134D764—21no—
135D764—42no—
136D764C2×D412yes—
137C22⋊D191—42no—
138C22⋊D191—81no—
139C2×Dic191—21no—
140C2×Dic191—21no—
141C2×Dic191—21no—
142C2×Dic191C2×D411yescharacteristic
143C2×Dic191C2×D411yescharacteristic
144C4×D191—21no—
145C4×D191—21no—
146C4×D191—21no—
147C4×D191—21no—
148C4×D191—42no—
149C4×D191—42no—
150C4×D191—42no—
151C4×D191—81no—
152C191⋊C8—21no—
153C191⋊C8—21no—
154C23×C191—21no—
155Q8×C191—21no—
156Q8×C191—21no—
157Q8×C191—21no—
158Q8×C191—21no—
159Q8×C191—21no—
160Q8×C191—21no—
161D4×C191—21no—
162D4×C191—21no—
163C2×C4×C191—21no—
164C2×C4×C191—21no—
165C2×C4×C191—21no—
166C2×C4×C191C2×D411yescharacteristic
167C2×C4×C191C2×D411yescharacteristic
168C2×C4×C191C2411yescharacteristic
169C8×C191—21no—
170C8×C191C2×D412yes—
171C22×C191—41no—
172C22×C191—41no—
173C22×C191—41no—
174C22×C191C22×D411yescharacteristic
175C2×D191—21no—
176C2×D191—21no—
177C2×D191—21no—
178C2×D191—22no—
179C2×D191—21no—
180C2×D191—21no—
181C2×D191—41no—
182C2×D191—41no—
183C2×D191—84no—
184C2×D191#13712yescomplemented
185C4×C191—21no—
186C4×C191—21no—
187C4×C191—41no—
188C4×C191—41no—
189C4×C191—41no—
190C4×C191C22×D411yescharacteristic
191C4×C191H(5,𝔽2)11yescharacteristic
192Dic191—21no—
193Dic191—21no—
194Dic191—22no—
195Dic191—41no—
196Dic191—41no—
197C2×C191—81no—
198C2×C191D4211yescharacteristic
199C2×C191C2×#13711yescharacteristic
200C2×C191#57711yescharacteristic
201D191—22no—
202D191—41no—
203D191—42no—
204C191#326611yescharacteristic, complemented, Sylow
205#3266—1911nomaximal, Sylow
206C2×#137—1911no—
207C22×SD16—1911no—
208D4×Q8—1911no—
209#546—1911no—
210#497D19111yescharacteristic, complemented
211#468—1911no—
212#467—1911no—
213#464—1911no—
214#461—1911no—
215#452—1911no—
216#451—1911no—
217#450—1911no—
218#448—1911no—
219#440—1911no—
220#435—1911no—
221C2×#41—1911no—
222C22×Q8—1911no—
223C22×Q8—1911no—
224C22×D4—1911no—
225#137—3824no—
226C2×Q16—1911no—
227C2×Q16C2×D19111yescharacteristic, complemented
228C2×Q16C2×D19111yescharacteristic, complemented
229C2×SD16—3824no—
230C2×SD16—1911no—
231C2×SD16—1911no—
232C2×SD16—1911no—
233C2×SD16—1911no—
234C2×SD16C2×D19111yescharacteristic
235C2×SD16C2×D19111yescharacteristic
236C2×M16—1911no—
237C22×C8—1911no—
238C4⋊Q8—3821no—
239C4⋊Q8C2×D19111yescharacteristic, complemented
240#124C2×D19111yescharacteristic
241#123—3821no—
242C22⋊Q8—3821no—
243C22⋊Q8—3821no—
244C22⋊Q8—1911no—
245C22⋊Q8—1911no—
246C22⋊Q8—1911no—
247#121—3821no—
248#121—1911no—
249#120—3821no—
250C4×Q8—1911no—
251C4×D4—3821no—
252C4×D4—1911no—
253C4×D4—1911no—
254C2×C22⋊C4—3821no—
255#107—1911no—
256C4⋊C8—1911no—
257#103—1911no—
258#103—1911no—
259#103—1911no—
260#103—1911no—
261#102—1911no—
262#102—1911no—
263C22⋊C8—1911no—
264C22⋊C8—1911no—
265#97C2×D19111yescharacteristic, complemented
266C24—3821no—
267#41—3822no—
268#41—7641no—
269C2×Q8—3821no—
270C2×Q8—3821no—
271C2×Q8—3822no—
272C2×Q8—3822no—
273C2×Q8—1911no—
274C2×Q8—7641no—
275C2×Q8—7641no—
276C2×Q8C22×D19111yescharacteristic
277C2×Q8C22×D19111yescharacteristic
278C2×Q8C22×D19111yescharacteristic
279C2×D4—3821no—
280C2×D4—3822no—
281C2×D4—3821no—
282C2×D4—3821no—
283C2×D4—1911no—
284C2×D4—1911no—
285C2×D4—7641no—
286C2×D4C22×D19111yescharacteristic
287C22×C4—3821no—
288C22×C4—3821no—
289C22×C4—3821no—
290C22×C4—3821no—
291C22×C4—1911no—
292C22×C4—1911no—
293Q16—22no—
294Q16—41no—
295Q16—3822no—
296SD16—22no—
297SD16—41no—
298SD16—3822no—
299SD16—3822no—
300SD16—3822no—
301SD16—1914no—
302M16—3822no—
303C2×C8—3822no—
304C2×C8—1911no—
305C2×C8—1911no—
306C2×C8C22×D19111yescharacteristic
307C2×C8C22×D19111yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—3821no—
311C4⋊C4—3821no—
312C4⋊C4—3821no—
313C4⋊C4—3821no—
314C4⋊C4—1911no—
315C4⋊C4—1911no—
316C4⋊C4—1911no—
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—3821no—
320C22⋊C4—3821no—
321C22⋊C4—3821no—
322C22⋊C4—3821no—
323C22⋊C4—3821no—
324C22⋊C4—1911no—
325C22⋊C4—1911no—
326C42—3821no—
327C42C22×D19111yescharacteristic
328C23—21no—
329C23—3821no—
330C23—1911no—
331C23—1911no—
332C23—15281no—
333C23—7642no—
334C23—7642no—
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—21no—
340Q8—21no—
341Q8—3821no—
342Q8—1912no—
343Q8—7642no—
344Q8—7642no—
345D4—21no—
346D4—21no—
347D4—3822no—
348D4—3821no—
349D4—1912no—
350D4—15281no—
351D4—7642no—
352D4—7642no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—3821no—
357C2×C4—3821no—
358C2×C4—3821no—
359C2×C4—3821no—
360C2×C4—3821no—
361C2×C4—3821no—
362C2×C4—3821no—
363C2×C4—1911no—
364C2×C4—1911no—
365C2×C4—15281no—
366C2×C4—7642no—
367C2×C4—7642no—
368C2×C4—7642no—
369C2×C4D4×D19111yescharacteristic
370C2×C4D4×D19111yescharacteristic
371C2×C4C23×D19111yescharacteristic
372C8—21no—
373C8—3821no—
374C8—3821no—
375C8D4×D19112yes—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—3821no—
380C22—3821no—
381C22—3821no—
382C22—3821no—
383C22—3821no—
384C22—3822no—
385C22—1912no—
386C22—15284no—
387C22—7641no—
388C22—7641no—
389C22C2×D4×D19111yescharacteristic
390C4—21no—
391C4—21no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—3821no—
396C4—3821no—
397C4—3822no—
398C4—7641no—
399C4—7641no—
400C4C2×D4×D19111yescharacteristic
401C4#2025936711yescharacteristic
402C2—81no—
403C2—3822no—
404C2—7641no—
405C2—7642no—
406C2#2713329811yescharacteristic
407C2#137×D19111yescharacteristic
408C2#2713347211yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#497×C191C2
socleC22×C191C22×D4
prime radical#27133306C2
hypercenter#497D191
hypocenterC191#3266
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×#20259354C2semidirect
2. C2×SD16×D191C2semidirect
3. #27133328C2semidirect
5. #27132906C2semidirect
6. #27132901C2semidirect
7. #27132882C2semidirect
8. #27132865C2semidirect
9. #27132824C2semidirect
10. #27132817C2semidirect
11. #27132812C2semidirect
12. #27132800C2semidirect
13. #27132765C2semidirect
14. #27132736C2semidirect
15. #497×C191C2semidirect
18. C2×Q8×D191C22semidirect
21. C2×#15690277C22semidirect
27. C2×#15690270C22semidirect
35. #20259298C22semidirect
39. C4×Dic382C22semidirect
44. #20259256C22semidirect
45. C4⋊(C191⋊C8)C22semidirect
46. #20259247C22semidirect
47. #20259246C22semidirect
48. #20259246C22semidirect
49. #20259246C22semidirect
53. (C2×D191)⋊C8C22semidirect
54. C2×Q16×C191C22semidirect
55. C2×Q16×C191C22semidirect
58. C4⋊Q8×C191C22semidirect
60. #97×C191C22semidirect
85. #15690276D4semidirect
184. C2×D191#137semidirect
204. C191#3266semidirect
210. #497D191semidirect
227. C2×Q16C2×D191semidirect
228. C2×Q16C2×D191semidirect
239. C4⋊Q8C2×D191semidirect
265. #97C2×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593542
C2×SD16×D1912
#271333282
#271333062
#271329062
#271329012
#271328822
#271328652
#271328242
#271328172
#271328122
#271328002
#271327652
#271327362
#497×C1912
#3266191