Finite Group Databasebeta

← #37584059

Subgroups — #37584059

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×#20259343C211yesmaximal, characteristic, complemented
3#27133333C211yesmaximal, characteristic, complemented
4#27133306C211yesmaximal, characteristic, complemented
5#27132906C211yesmaximal, characteristic, complemented
6#27132896C211yesmaximal, characteristic, complemented
7#27132882C211yesmaximal, characteristic, complemented
8#27132855C211yesmaximal, characteristic, complemented
9#27132822C211yesmaximal, characteristic, complemented
10#27132807C211yesmaximal, characteristic, complemented
11#27132805C211yesmaximal, characteristic, complemented
12#27132795C211yesmaximal, characteristic, complemented
13#27132765C211yesmaximal, characteristic, complemented
14#27132736C211yesmaximal, characteristic, complemented
15#496×C191C211yesmaximal, characteristic, complemented
16C2×#15690288C2211yescharacteristic
17C2×#15690286C2211yescharacteristic
18C2×#15690286C2211yescharacteristic, complemented
19C2×D4×D191C2211yescharacteristic
20#20259346—24no—
21#20259343—24no—
22C2×#15690277C2211yescharacteristic, complemented
23C2×#15690276C2211yescharacteristic
24C2×#15690275C2211yescharacteristic
25C2×#15690272C2211yescharacteristic
26C2×#15690272C2211yescharacteristic
27C2×#15690270C2211yescharacteristic
28C2×C8×D191C2211yescharacteristic, complemented
29#20259318—21no—
30#20259304—21no—
31#20259298C2211yescharacteristic
32#20259295—21no—
33#20259294—21no—
34#20259294—21no—
35#20259293C2211yescharacteristic
36#20259293C2211yescharacteristic
37#20259293C2211yescharacteristic, complemented
38#20259292—21no—
39C4×Dic382C2211yescharacteristic, complemented
40C4×D764C2211yescharacteristic
41C4×D764C2211yescharacteristic
42D4×Dic191—21no—
43C22⋊C4×D191—21no—
44#20259256C2211yescharacteristic, complemented
45C4⋊(C191⋊C8)C2211yescharacteristic, complemented
46#20259247C2211yescharacteristic, complemented
47#20259246C2211yescharacteristic, complemented
48#20259244C2211yescharacteristic
49#20259243C2211yescharacteristic
50#20259243C2211yescharacteristic, complemented
51#20259243C2211yescharacteristic, complemented
52(C2×D191)⋊C8C2211yescharacteristic
53(C2×D191)⋊C8C2211yescharacteristic, complemented
54C2×SD16×C191C2211yescharacteristic, complemented
55C2×SD16×C191C2211yescharacteristic, complemented
56C2×D8×C191C2211yescharacteristic
57C2×D8×C191C2211yescharacteristic
58#127×C191C2211yescharacteristic, complemented
59#124×C191C2211yescharacteristic
60#97×C191C2211yescharacteristic, complemented
61C23×D191—21no—
62#15690288—22no—
63#15690288—41no—
64#15690286—22no—
65#15690286—22no—
66#15690286—41no—
67#15690286—41no—
68C2×Dic382C2311yescharacteristic
69C2×D764—21no—
70C2×D764D411yescharacteristic
71C2×D764C2311yescharacteristic
72C2×C22⋊D191—21no—
73C2×C22⋊D191—21no—
74C2×C22⋊D191—21no—
75C2×C22⋊D191—21no—
76C2×C22⋊D191—21no—
77C2×C22⋊D191—21no—
78D4×D191—22no—
79D4×D191—41no—
80C22×Dic191—21no—
81C22×Dic191—21no—
82C2×C4×D191—21no—
83C2×C4×D191—21no—
84C2×C4×D191C2311yescharacteristic
85C2×C4×D191C2311yescharacteristic
86#15690277—22no—
87#15690276D414yescomplemented
88#15690275—22no—
89#15690272—22no—
90#15690272—22no—
91#15690270—22no—
92C2×C191⋊C8C2311yescharacteristic
93C2×C191⋊C8C2311yescharacteristic
94C8×D191—22no—
95C4⋊Dic191D411yescharacteristic
96C4⋊Dic191C2311yescharacteristic
97C4⋊Dic191C2311yescharacteristic
98Dic191⋊C4—21no—
99Dic191⋊C4—21no—
100C22⋊Dic191—21no—
101C22⋊Dic191—21no—
102C22⋊Dic191—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×D4×C191—21no—
110C2×D4×C191—21no—
111C2×D4×C191C2311yescharacteristic
112C2×D4×C191C2311yescharacteristic
113C2×D4×C191C2311yescharacteristic
114SD16×C191—22no—
115SD16×C191—41no—
116D8×C191—22no—
117D8×C191—41no—
118C2×C8×C191C2311yescharacteristic
119C2×C8×C191C2311yescharacteristic
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—
130Dic382C2×D412yes—
131D764—21no—
132D764—41no—
133D764—42no—
134D764C2×D412yes—
135C22⋊D191—42no—
136C22⋊D191—42no—
137C22⋊D191—42no—
138C22⋊D191—81no—
139C22⋊D191—81no—
140C22⋊D191—81no—
141C2×Dic191—21no—
142C2×Dic191—21no—
143C2×Dic191—21no—
144C2×Dic191—42no—
145C2×Dic191—42no—
146C2×Dic191C2×D411yescharacteristic
147C2×Dic191C2×D411yescharacteristic
148C4×D191—21no—
149C4×D191—21no—
150C4×D191—21no—
151C4×D191—21no—
152C4×D191—42no—
153C4×D191—81no—
154C191⋊C8—21no—
155C191⋊C8—21no—
156C23×C191—21no—
157C23×C191—21no—
158C23×C191—21no—
159Q8×C191—21no—
160Q8×C191—21no—
161D4×C191—21no—
162D4×C191—21no—
163D4×C191—21no—
164D4×C191—21no—
165D4×C191—21no—
166D4×C191—21no—
167D4×C191—42no—
168D4×C191—81no—
169C2×C4×C191—21no—
170C2×C4×C191C2×D411yescharacteristic
171C2×C4×C191C2×D411yescharacteristic
172C2×C4×C191C2411yescharacteristic
173C8×C191—21no—
174C8×C191C2×D412yes—
175C22×C191—41no—
176C22×C191—41no—
177C22×C191—41no—
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—21no—
187C2×D191—22no—
188C2×D191—21no—
189C2×D191—21no—
190C2×D191—21no—
191C2×D191—21no—
192C2×D191—41no—
193C2×D191—41no—
194C2×D191—84no—
195C4×C191—21no—
196C4×C191—21no—
197C4×C191—41no—
198C4×C191C22×D411yescharacteristic
199C4×C191H(5,𝔽2)11yescharacteristic
200Dic191—21no—
201Dic191—21no—
202Dic191—22no—
203Dic191—41no—
204Dic191—41no—
205C2×C191—81no—
206C2×C191—81no—
207C2×C191—81no—
208C2×C191D4211yescharacteristic
209C2×C191#57511yescharacteristic
210C2×C191#57611yescharacteristic
211D191—41no—
212D191—41no—
213D191—42no—
214C191#326811yescharacteristic, complemented, Sylow
215#3268—1911nomaximal, Sylow
216C2×AGL(1,ℤ/8)—1911no—
217C2×#135—1911no—
218#550—1911no—
219#546—1911no—
220#496D19111yescharacteristic, complemented
221#468—1911no—
222#466—1911no—
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×SD16C2×D19111yescharacteristic, complemented
241C2×SD16C2×D19111yescharacteristic, complemented
242C2×D8—1911no—
243C2×D8—1911no—
244C2×D8C2×D19111yescharacteristic
245C2×D8C2×D19111yescharacteristic
246C2×M16—1911no—
247C22×C8—1911no—
248#127—3821no—
249#127C2×D19111yescharacteristic, complemented
250#124C2×D19111yescharacteristic
251#123—3821no—
252C22⋊Q8—1911no—
253#121—3821no—
254#121—3821no—
255#121—3821no—
256#121—1911no—
257#121—1911no—
258#121—1911no—
259#120—3821no—
260C4×Q8—1911no—
261C4×D4—3821no—
262C4×D4—1911no—
263C4×D4—1911no—
264C2×C22⋊C4—3821no—
265#107—1911no—
266C4⋊C8—1911no—
267#103—1911no—
268#103—1911no—
269#102—1911no—
270#102—1911no—
271#102—1911no—
272#102—1911no—
273C22⋊C8—1911no—
274C22⋊C8—1911no—
275#97C2×D19111yescharacteristic, complemented
276C24—3821no—
277#41—3822no—
278#41—3822no—
279#41—3822no—
280#41—7641no—
281#41—7641no—
282#41—7641no—
283C2×Q8—1911no—
284C2×Q8C22×D19111yescharacteristic
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—7641no—
298C2×D4C22×D19111yescharacteristic
299C2×D4C22×D19111yescharacteristic
300C2×D4C22×D19111yescharacteristic
301C22×C4—3821no—
302C22×C4—3821no—
303C22×C4—3821no—
304C22×C4—3821no—
305C22×C4—1911no—
306C22×C4—1911no—
307Q16—3822no—
308SD16—22no—
309SD16—41no—
310SD16—3822no—
311SD16—1914no—
312D8—22no—
313D8—41no—
314D8—3822no—
315D8—3822no—
316M16—3822no—
317C2×C8—3822no—
318C2×C8—1911no—
319C2×C8—1911no—
320C2×C8C22×D19111yescharacteristic
321C2×C8C22×D19111yescharacteristic
322C4⋊C4—3821no—
323C4⋊C4—3821no—
324C4⋊C4—1911no—
325C4⋊C4—1911no—
326C4⋊C4—1911no—
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—
337C42C22×D19111yescharacteristic
338C23—21no—
339C23—21no—
340C23—21no—
341C23—3821no—
342C23—1911no—
343C23—1911no—
344C23—15281no—
345C23—7642no—
346C23—7642no—
347Q8—21no—
348Q8—21no—
349Q8—3821no—
350Q8—1912no—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—21no—
356D4—21no—
357D4—42no—
358D4—81no—
359D4—3821no—
360D4—1912no—
361D4—15281no—
362D4—15281no—
363D4—15281no—
364D4—7641no—
365D4—7642no—
366D4—7642no—
367D4—7642no—
368D4—7642no—
369C2×C4—21no—
370C2×C4—3821no—
371C2×C4—3821no—
372C2×C4—3821no—
373C2×C4—3821no—
374C2×C4—3821no—
375C2×C4—3821no—
376C2×C4—3821no—
377C2×C4—1911no—
378C2×C4—1911no—
379C2×C4—15281no—
380C2×C4—7642no—
381C2×C4—7642no—
382C2×C4—7642no—
383C2×C4D4×D19111yescharacteristic
384C2×C4D4×D19111yescharacteristic
385C2×C4C23×D19111yescharacteristic
386C8—21no—
387C8—3821no—
388C8—3821no—
389C8D4×D19112yes—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—3821no—
400C22—3821no—
401C22—3821no—
402C22—3821no—
403C22—3821no—
404C22—3821no—
405C22—3822no—
406C22—15284no—
407C22—7641no—
408C22—7641no—
409C22C2×D4×D19111yescharacteristic
410C4—21no—
411C4—21no—
412C4—41no—
413C4—3821no—
414C4—3821no—
415C4—3822no—
416C4—7641no—
417C4—7641no—
418C4C2×D4×D19111yescharacteristic
419C4#2025936711yescharacteristic
420C2—81no—
421C2—81no—
422C2—81no—
423C2—7641no—
424C2—7641no—
425C2—7642no—
426C2#2713329811yescharacteristic
427C2#2713345911yescharacteristic
428C2#2713346711yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#496×C191C2
socleC22×C191C22×D4
hypercenter#496D191
hypocenterC191#3268
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#20259346C2semidirect
2. C2×#20259343C2semidirect
3. #27133333C2semidirect
4. #27133306C2semidirect
5. #27132906C2semidirect
6. #27132896C2semidirect
7. #27132882C2semidirect
8. #27132855C2semidirect
9. #27132822C2semidirect
10. #27132807C2semidirect
11. #27132805C2semidirect
12. #27132795C2semidirect
13. #27132765C2semidirect
14. #27132736C2semidirect
15. #496×C191C2semidirect
18. C2×#15690286C22semidirect
22. C2×#15690277C22semidirect
28. C2×C8×D191C22semidirect
37. #20259293C22semidirect
39. C4×Dic382C22semidirect
44. #20259256C22semidirect
45. C4⋊(C191⋊C8)C22semidirect
46. #20259247C22semidirect
47. #20259246C22semidirect
50. #20259243C22semidirect
51. #20259243C22semidirect
53. (C2×D191)⋊C8C22semidirect
54. C2×SD16×C191C22semidirect
55. C2×SD16×C191C22semidirect
58. #127×C191C22semidirect
60. #97×C191C22semidirect
87. #15690276D4semidirect
214. C191#3268semidirect
220. #496D191semidirect
240. C2×SD16C2×D191semidirect
241. C2×SD16C2×D191semidirect
249. #127C2×D191semidirect
275. #97C2×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593462
C2×#202593432
#271333332
#271333062
#271329062
#271328962
#271328822
#271328552
#271328222
#271328072
#271328052
#271327952
#271327652
#271327362
#496×C1912
#3268191