Finite Group Databasebeta

← #37584061

Subgroups — #37584061

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×#20259347C211yesmaximal, characteristic, complemented
2C2×#20259342C211yesmaximal, characteristic, complemented
3#27133309C211yesmaximal, characteristic, complemented
4#27133050C211yesmaximal, characteristic, complemented
5#27132907C211yesmaximal, characteristic, complemented
6#27132897C211yesmaximal, characteristic, complemented
7#27132885C211yesmaximal, characteristic, complemented
8#27132855C211yesmaximal, characteristic, complemented
9#27132825C211yesmaximal, characteristic, complemented
10#27132808C211yesmaximal, characteristic, complemented
11#27132806C211yesmaximal, characteristic, complemented
12#27132796C211yesmaximal, characteristic, complemented
13#27132761C211yesmaximal, characteristic, complemented
14#27132735C211yesmaximal, characteristic, complemented
15#550×C191C211yesmaximal, characteristic, complemented
16C2×#15690287C2211yescharacteristic
17C22×D764C2211yescharacteristic
18#20259347—24no—
19#20259342—24no—
20C2×#15690277C2211yescharacteristic, complemented
21C2×#15690275C2211yescharacteristic, complemented
22C2×#15690275C2211yescharacteristic, complemented
23C2×#15690274C2211yescharacteristic
24C2×#15690274C2211yescharacteristic
25C2×#15690272C2211yescharacteristic
26C2×#15690272C2211yescharacteristic
27C2×#15690272C2211yescharacteristic
28C2×#15690272C2211yescharacteristic
29C2×#15690271C2211yescharacteristic
30C22×C191⋊C8C2211yescharacteristic, complemented
31#20259310C2211yescharacteristic
32#20259307—21no—
33#20259301C2211yescharacteristic
34#20259296C2211yescharacteristic
35#20259295—21no—
36#20259290—21no—
37C4×D764C2211yescharacteristic
38C22⋊C4×D191—21no—
39#20259255C2211yescharacteristic, complemented
40C4⋊(C191⋊C8)C2211yescharacteristic, complemented
41#20259246C2211yescharacteristic, complemented
42#20259245C2211yescharacteristic, complemented
43#20259243C2211yescharacteristic, complemented
44#20259243C2211yescharacteristic, complemented
45#20259242C2211yescharacteristic
46#20259242C2211yescharacteristic
47C22⋊(C191⋊C8)C2211yescharacteristic
48C22⋊(C191⋊C8)C2211yescharacteristic, complemented
49#20259233C2211yescharacteristic, complemented
50C2×#41×C191C2211yescharacteristic
51C2×#41×C191C2211yescharacteristic, complemented
52#127×C191—21no—
53#127×C191C2211yescharacteristic, complemented
54#121×C191—21no—
55#121×C191—21no—
56#121×C191C2211yescharacteristic
57#121×C191C2211yescharacteristic, complemented
58C4×Q8×C191C2211yescharacteristic, complemented
59C4×D4×C191—21no—
60C4×D4×C191C2211yescharacteristic
61C23×D191—21no—
62#15690287—22no—
63#15690287—41no—
64C2×Dic382C2311yescharacteristic
65C2×D764—22no—
66C2×D764—21no—
67C2×D764—41no—
68C2×D764C2311yescharacteristic
69C2×D764C2311yescharacteristic
70C2×C22⋊D191—21no—
71C2×C22⋊D191—21no—
72C2×C4×D191—21no—
73C2×C4×D191—21no—
74#15690277—22no—
75#15690275—22no—
76#15690275—41no—
77#15690274—22no—
78#15690274D414yescomplemented
79#15690272—22no—
80#15690272—22no—
81#15690272—22no—
82#15690272—41no—
83#15690271—22no—
84C2×C191⋊C8—22no—
85C2×C191⋊C8C2311yescharacteristic
86C2×C191⋊C8C2311yescharacteristic
87C2×C191⋊C8C2311yescharacteristic
88C2×C191⋊C8C2311yescharacteristic
89C4⋊Dic191C2311yescharacteristic
90Dic191⋊C4—21no—
91C22⋊Dic191—21no—
92(C2×D191)⋊C4—21no—
93(C2×D191)⋊C4—21no—
94(C2×D191)⋊C4—21no—
95(C2×D191)⋊C4—21no—
96#41×C191—22no—
97#41×C191—22no—
98#41×C191—41no—
99#41×C191—41no—
100C2×Q8×C191C2311yescharacteristic
101C2×D4×C191—21no—
102C2×D4×C191—21no—
103C2×D4×C191—21no—
104C2×D4×C191—21no—
105C2×D4×C191—21no—
106C2×D4×C191—21no—
107C2×D4×C191D411yescharacteristic
108C2×D4×C191C2311yescharacteristic
109C2×D4×C191C2311yescharacteristic
110C22×C4×C191—21no—
111C22×C4×C191—21no—
112C22×C4×C191C2311yescharacteristic
113C22×C4×C191C2311yescharacteristic
114C4⋊C4×C191—21no—
115C4⋊C4×C191D411yescharacteristic
116C4⋊C4×C191C2311yescharacteristic
117C22⋊C4×C191—21no—
118C22⋊C4×C191—21no—
119C22⋊C4×C191D411yescharacteristic
120C22⋊C4×C191D411yescharacteristic
121C42×C191—21no—
122C42×C191C2311yescharacteristic
123C22×D191—21no—
124C22×D191—21no—
125C22×D191—42no—
126C22×D191—42no—
127C22×D191—81no—
128Dic382—21no—
129Dic382—21no—
130D764—21no—
131D764—21no—
132D764—21no—
133D764—42no—
134D764C2×D412yes—
135C22⋊D191—42no—
136C22⋊D191—81no—
137C2×Dic191—21no—
138C2×Dic191—21no—
139C4×D191—42no—
140C4×D191—81no—
141C191⋊C8—21no—
142C191⋊C8—21no—
143C191⋊C8—21no—
144C191⋊C8C2×D412yes—
145C23×C191—21no—
146C23×C191—21no—
147C23×C191C2×D411yescharacteristic
148C23×C191C2×D411yescharacteristic
149Q8×C191—21no—
150Q8×C191C2×D412yes—
151D4×C191—21no—
152D4×C191—21no—
153D4×C191—21no—
154D4×C191—21no—
155D4×C191—41no—
156D4×C191—42no—
157D4×C191—42no—
158D4×C191—42no—
159D4×C191—81no—
160D4×C191—81no—
161D4×C191—81no—
162C2×C4×C191—21no—
163C2×C4×C191—21no—
164C2×C4×C191—21no—
165C2×C4×C191—21no—
166C2×C4×C191—21no—
167C2×C4×C191—21no—
168C2×C4×C191—42no—
169C2×C4×C191—42no—
170C2×C4×C191C2×D411yescharacteristic
171C2×C4×C191C2×D411yescharacteristic
172C2×C4×C191C2×D411yescharacteristic
173C2×C4×C191C2×D411yescharacteristic
174C2×C4×C191C2411yescharacteristic
175C22×C191—21no—
176C22×C191—21no—
177C22×C191—21no—
178C22×C191—21no—
179C22×C191—21no—
180C22×C191—21no—
181C22×C191—41no—
182C22×C191—41no—
183C22×C191—41no—
184C22×C191—41no—
185C22×C191—41no—
186C22×C191—41no—
187C22×C191C22×D411yescharacteristic
188C2×D191—22no—
189C2×D191—41no—
190C2×D191—41no—
191C2×D191—41no—
192C2×D191—41no—
193C2×D191—41no—
194C2×D191—84no—
195C4×C191—21no—
196C4×C191—21no—
197C4×C191—22no—
198C4×C191—21no—
199C4×C191—21no—
200C4×C191—41no—
201C4×C191C22×D411yescharacteristic
202C4×C191H(5,𝔽2)11yescharacteristic
203Dic191—41no—
204Dic191—41no—
205C2×C191—41no—
206C2×C191—41no—
207C2×C191—81no—
208C2×C191—81no—
209C2×C191D4211yescharacteristic
210C2×C191#57511yescharacteristic
211C2×C191#57611yescharacteristic
212D191—42no—
213D191—81no—
214C191#326811yescharacteristic, complemented, Sylow
215#3268—1911nomaximal, Sylow
216C2×AGL(1,ℤ/8)—1911no—
217C2×#135—1911no—
218#550D19111yescharacteristic, complemented
219#546—1911no—
220#496—1911no—
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×#41C2×D19111yescharacteristic
233C2×#41C2×D19111yescharacteristic, complemented
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×D8—1911no—
246C2×M16—1911no—
247C22×C8—1911no—
248#127—21no—
249#127C2×D19111yescharacteristic, complemented
250#124—1911no—
251#123—3821no—
252C22⋊Q8—1911no—
253#121—21no—
254#121—21no—
255#121—3821no—
256#121—1911no—
257#121C2×D19111yescharacteristic
258#121C2×D19111yescharacteristic, complemented
259#120—3821no—
260C4×Q8C2×D19111yescharacteristic, complemented
261C4×D4—21no—
262C4×D4—1911no—
263C4×D4C2×D19111yescharacteristic
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#97—1911no—
276C24—3821no—
277#41—22no—
278#41—22no—
279#41—41no—
280#41—41no—
281#41—3822no—
282#41—7641no—
283C2×Q8—1911no—
284C2×Q8C22×D19111yescharacteristic
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—3822no—
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—21no—
302C22×C4—21no—
303C22×C4—3821no—
304C22×C4—3821no—
305C22×C4C22×D19111yescharacteristic
306C22×C4C22×D19111yescharacteristic
307Q16—3822no—
308SD16—3822no—
309SD16—3822no—
310SD16—1914no—
311SD16—7641no—
312D8—3822no—
313D8—3822no—
314D8—3822no—
315D8—7641no—
316M16—3822no—
317C2×C8—3822no—
318C2×C8—1911no—
319C2×C8—1911no—
320C2×C8—1911no—
321C2×C8—1911no—
322C4⋊C4—21no—
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⋊C4C22⋊D19111yescharacteristic
335C22⋊C4C22⋊D19111yescharacteristic
336C42—21no—
337C42C22×D19111yescharacteristic
338C23—21no—
339C23—21no—
340C23—3821no—
341C23—3821no—
342C23—15281no—
343C23—7642no—
344C23—7642no—
345C23C2×C22⋊D19111yescharacteristic
346C23C2×C22⋊D19111yescharacteristic
347Q8—21no—
348Q8—3821no—
349Q8—3821no—
350Q8D4×D19112yes—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—41no—
356D4—42no—
357D4—42no—
358D4—42no—
359D4—81no—
360D4—81no—
361D4—81no—
362D4—3821no—
363D4—3821no—
364D4—3821no—
365D4—1912no—
366D4—15281no—
367D4—7642no—
368D4—7642no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—3821no—
378C2×C4—3821no—
379C2×C4—15281no—
380C2×C4—7642no—
381C2×C4C2×C22⋊D19111yescharacteristic
382C2×C4C2×C22⋊D19111yescharacteristic
383C2×C4C2×C22⋊D19111yescharacteristic
384C2×C4C2×C22⋊D19111yescharacteristic
385C2×C4C23×D19111yescharacteristic
386C8—3821no—
387C8—3821no—
388C8—3821no—
389C8—1912no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—3822no—
403C22—15284no—
404C22—7641no—
405C22—7641no—
406C22—7641no—
407C22—7641no—
408C22—7641no—
409C22C22×C22⋊D19111yescharacteristic
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—22no—
415C4—41no—
416C4—7641no—
417C4—7641no—
418C4C2×D4×D19111yescharacteristic
419C4#2025936711yescharacteristic
420C2—41no—
421C2—41no—
422C2—81no—
423C2—81no—
424C2—15281no—
425C2—7642no—
426C2D4×C22⋊D19111yescharacteristic
427C2#2713346111yescharacteristic
428C2#2713346611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#550×C191C2
socleC22×C191C22×D4
hypercenter#550D191
hypocenterC191#3268
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C4×D4 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C2×#20259347C2semidirect
2. C2×#20259342C2semidirect
3. #27133309C2semidirect
4. #27133050C2semidirect
5. #27132907C2semidirect
6. #27132897C2semidirect
7. #27132885C2semidirect
8. #27132855C2semidirect
9. #27132825C2semidirect
10. #27132808C2semidirect
11. #27132806C2semidirect
12. #27132796C2semidirect
13. #27132761C2semidirect
14. #27132735C2semidirect
15. #550×C191C2semidirect
20. C2×#15690277C22semidirect
21. C2×#15690275C22semidirect
22. C2×#15690275C22semidirect
30. C22×C191⋊C8C22semidirect
39. #20259255C22semidirect
40. C4⋊(C191⋊C8)C22semidirect
41. #20259246C22semidirect
42. #20259245C22semidirect
43. #20259243C22semidirect
44. #20259243C22semidirect
48. C22⋊(C191⋊C8)C22semidirect
49. #20259233C22semidirect
51. C2×#41×C191C22semidirect
53. #127×C191C22semidirect
57. #121×C191C22semidirect
58. C4×Q8×C191C22semidirect
78. #15690274D4semidirect
214. C191#3268semidirect
218. #550D191semidirect
233. C2×#41C2×D191semidirect
249. #127C2×D191semidirect
258. #121C2×D191semidirect
260. C4×Q8C2×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593472
C2×#202593422
#271333092
#271330502
#271329072
#271328972
#271328852
#271328552
#271328252
#271328082
#271328062
#271327962
#271327612
#271327352
#550×C1912
#3268191