Finite Group Databasebeta

← #37584002

Subgroups — #37584002

Order 24448.

568 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20259346C211yesmaximal, characteristic, complemented
2C2×D8×D191C211yesmaximal, characteristic, complemented
3#27133308C211yesmaximal, characteristic, complemented
4D4×D764C211yesmaximal, characteristic, complemented
5#27133052C211yesmaximal, characteristic, complemented
6#27132917C211yesmaximal, characteristic, complemented
7#27132893C211yesmaximal, characteristic, complemented
8#27132878C211yesmaximal, characteristic, complemented
9#27132858C211yesmaximal, characteristic, complemented
10#27132815C211yesmaximal, characteristic, complemented
11#27132809C211yesmaximal, characteristic, complemented
12#27132798C211yesmaximal, characteristic, complemented
13#27132798C211yesmaximal, characteristic, complemented
14#27132736C211yesmaximal, characteristic, complemented
15#442×C191C211yesmaximal, characteristic, complemented
16C2×#15690286C2211yescharacteristic, complemented
17C22×D764—21no—
18C2×D4×D191C2211yescharacteristic
19C2×D4×D191C2211yescharacteristic, complemented
20C2×D4×D191C2211yescharacteristic, complemented
21#20259346—24no—
22C2×#15690276C2211yescharacteristic, complemented
23C2×#15690276C2211yescharacteristic, complemented
24C2×#15690275C2211yescharacteristic, complemented
25C2×D1528C2211yescharacteristic, complemented
26C2×D1528C2211yescharacteristic, complemented
27C2×#15690272C2211yescharacteristic
28C2×#15690272C2211yescharacteristic, complemented
29C2×#15690272C2211yescharacteristic, complemented
30D8×D191—24no—
31C2×#15690270C2211yescharacteristic, complemented
32C2×C8×D191C2211yescharacteristic, complemented
33#20259319C2211yescharacteristic, complemented
34#20259310C2211yescharacteristic, complemented
35#20259306—21no—
36#20259302C2211yescharacteristic, complemented
37#20259297C2211yescharacteristic
38#20259297C2211yescharacteristic, complemented
39#20259297C2211yescharacteristic, complemented
40#20259296—21no—
41#20259296—21no—
42#20259292—21no—
43#20259292—21no—
44#20259292—21no—
45C4×D764C2211yescharacteristic
46C2×(C2×D191)⋊C4—21no—
47C4⋊(C191⋊C8)C2211yescharacteristic, complemented
48#20259245C2211yescharacteristic, complemented
49#20259242C2211yescharacteristic, complemented
50#20259242C2211yescharacteristic, complemented
51#20259242C2211yescharacteristic, complemented
52(C2×D191)⋊C8C2211yescharacteristic, complemented
53(C2×D191)⋊C8C2211yescharacteristic, complemented
54C2×D8×C191C2211yescharacteristic
55C4×D4×C191C2211yescharacteristic, complemented
56C4×D4×C191C2211yescharacteristic, complemented
57#106×C191C2211yescharacteristic, complemented
58#102×C191C2211yescharacteristic, complemented
59#102×C191C2211yescharacteristic, complemented
60#97×C191C2211yescharacteristic, complemented
61C23×D191—21no—
62C23×D191—21no—
63C23×D191—21no—
64#15690286—22no—
65#15690286—41no—
66C2×Dic382C2311yescharacteristic
67C2×D764—21no—
68C2×D764—21no—
69C2×D764—21no—
70C2×D764—21no—
71C2×D764—21no—
72C2×D764—44no—
73C2×D764D411yescharacteristic
74C2×D764C2311yescharacteristic
75C2×D764C2311yescharacteristic
76C2×D764C2311yescharacteristic
77C2×C22⋊D191—21no—
78C2×C22⋊D191—21no—
79C2×C22⋊D191—21no—
80C2×C22⋊D191—21no—
81D4×D191—22no—
82D4×D191—22no—
83D4×D191—22no—
84D4×D191—41no—
85D4×D191—41no—
86D4×D191—41no—
87C22×Dic191—21no—
88C2×C4×D191C2311yescharacteristic
89C2×C4×D191C2311yescharacteristic
90#15690276—22no—
91#15690276—41no—
92#15690275—22no—
93D1528—22no—
94D1528—41no—
95#15690272—22no—
96#15690272—22no—
97#15690272—22no—
98#15690270—22no—
99C2×C191⋊C8C2311yescharacteristic
100C2×C191⋊C8C2311yescharacteristic
101C8×D191—22no—
102C4⋊Dic191—21no—
103C4⋊Dic191D411yescharacteristic
104(C2×D191)⋊C4—21no—
105(C2×D191)⋊C4—21no—
106(C2×D191)⋊C4—21no—
107(C2×D191)⋊C4—21no—
108(C2×D191)⋊C4—21no—
109(C2×D191)⋊C4—21no—
110(C2×D191)⋊C4D411yescharacteristic
111(C2×D191)⋊C4D411yescharacteristic
112C2×D4×C191C2311yescharacteristic
113C2×D4×C191C2311yescharacteristic
114C22×C4×C191—21no—
115C22×C4×C191—21no—
116D8×C191D414yescomplemented
117C2×C8×C191C2311yescharacteristic
118C2×C8×C191C2311yescharacteristic
119C4⋊C4×C191C2311yescharacteristic
120C4⋊C4×C191C2311yescharacteristic
121C22⋊C4×C191—21no—
122C22⋊C4×C191—21no—
123C42×C191C2311yescharacteristic
124C22×D191—21no—
125C22×D191—21no—
126C22×D191—21no—
127C22×D191—42no—
128C22×D191—42no—
129C22×D191—42no—
130C22×D191—42no—
131C22×D191—42no—
132C22×D191—42no—
133C22×D191—81no—
134C22×D191—81no—
135C22×D191—81no—
136C22×D191C2×D411yescharacteristic
137C22×D191C2×D411yescharacteristic
138Dic382—21no—
139Dic382—21no—
140D764—22no—
141D764—21no—
142D764—21no—
143D764—21no—
144D764—21no—
145D764—21no—
146D764—21no—
147D764—42no—
148D764—42no—
149D764—42no—
150D764—81no—
151D764—81no—
152C22⋊D191—42no—
153C22⋊D191—42no—
154C22⋊D191—42no—
155C22⋊D191—42no—
156C2×Dic191—21no—
157C2×Dic191—42no—
158C2×Dic191C2×D411yescharacteristic
159C2×Dic191C2×D411yescharacteristic
160C4×D191—21no—
161C4×D191—21no—
162C4×D191—21no—
163C4×D191—21no—
164C191⋊C8—21no—
165C191⋊C8—21no—
166C23×C191—21no—
167C23×C191—21no—
168D4×C191—21no—
169D4×C191—21no—
170D4×C191C2×D412yes—
171D4×C191C2×D412yescomplemented
172C2×C4×C191—21no—
173C2×C4×C191—21no—
174C2×C4×C191—42no—
175C2×C4×C191—81no—
176C2×C4×C191C2×D411yescharacteristic
177C2×C4×C191C2×D411yescharacteristic
178C2×C4×C191C2411yescharacteristic
179C8×C191—21no—
180C8×C191C2×D412yescomplemented
181C22×C191—22no—
182C22×C191—22no—
183C22×C191—41no—
184C22×C191—41no—
185C22×C191—41no—
186C22×C191—41no—
187C22×C191C22×D411yescharacteristic
188C2×D191—21no—
189C2×D191—21no—
190C2×D191—21no—
191C2×D191—21no—
192C2×D191—21no—
193C2×D191—41no—
194C2×D191—41no—
195C2×D191—41no—
196C2×D191—41no—
197C2×D191—41no—
198C2×D191—41no—
199C2×D191—41no—
200C2×D191—41no—
201C2×D191—41no—
202C2×D191—84no—
203C2×D191—84no—
204C2×D191—84no—
205C2×D191AGL(1,ℤ/8)12yescomplemented
206C4×C191—21no—
207C4×C191—21no—
208C4×C191—41no—
209C4×C191—41no—
210C4×C191C22×D411yescharacteristic
211C4×C191H(5,𝔽2)11yescharacteristic
212Dic191—21no—
213Dic191—21no—
214Dic191—41no—
215C2×C191—42no—
216C2×C191—42no—
217C2×C191D4211yescharacteristic
218C2×C191C2×AGL(1,ℤ/8)11yescharacteristic
219C2×C191#57611yescharacteristic
220D191—22no—
221D191—41no—
222D191—81no—
223D191—81no—
224D191—81no—
225C191#326411yescharacteristic, complemented, Sylow
226#3264—1911nomaximal, Sylow
227C2×AGL(1,ℤ/8)—1911no—
228C22×D8—1911no—
229#546—1911no—
230D42—1911no—
231#496—1911no—
232#469—1911no—
233#465—1911no—
234#463—1911no—
235#459—1911no—
236#450—1911no—
237#449—1911no—
238#447—1911no—
239#447—1911no—
240#442D19111yescharacteristic, complemented
241#435—1911no—
242C2×#41—1911no—
243C22×D4—3821no—
244C22×D4—1911no—
245C22×D4—1911no—
246C22×D4—1911no—
247AGL(1,ℤ/8)—3824no—
248C2×SD16—1911no—
249C2×SD16—1911no—
250C2×SD16—1911no—
251C2×D8—3824no—
252C2×D8—1911no—
253C2×D8—1911no—
254C2×D8—1911no—
255C2×D8—1911no—
256C2×D8—1911no—
257C2×D8C2×D19111yescharacteristic
258C2×M16—1911no—
259C22×C8—1911no—
260#127—1911no—
261#124—1911no—
262#123—3821no—
263C22⋊Q8—1911no—
264#121—3821no—
265#121—3821no—
266#121—1911no—
267#121—1911no—
268#121—1911no—
269#120—3821no—
270#120—3821no—
271#120—3821no—
272C4×D4—1911no—
273C4×D4C2×D19111yescharacteristic, complemented
274C4×D4C2×D19111yescharacteristic, complemented
275C2×C22⋊C4—3821no—
276#106C2×D19111yescharacteristic, complemented
277C4⋊C8—1911no—
278#103—1911no—
279#102—1911no—
280#102—1911no—
281#102—1911no—
282#102C2×D19111yescharacteristic, complemented
283#102C2×D19111yescharacteristic, complemented
284C22⋊C8—1911no—
285C22⋊C8—1911no—
286#97C2×D19111yescharacteristic, complemented
287C24—3821no—
288C24—3821no—
289C24—3821no—
290#41—3822no—
291#41—7641no—
292C2×Q8—1911no—
293C2×D4—3821no—
294C2×D4—3821no—
295C2×D4—3821no—
296C2×D4—3822no—
297C2×D4—3822no—
298C2×D4—3822no—
299C2×D4—3821no—
300C2×D4—3821no—
301C2×D4—3821no—
302C2×D4—3821no—
303C2×D4—3821no—
304C2×D4—3821no—
305C2×D4—1911no—
306C2×D4—1911no—
307C2×D4—1911no—
308C2×D4—1911no—
309C2×D4—7644no—
310C2×D4—7641no—
311C2×D4—7641no—
312C2×D4—7641no—
313C2×D4C22×D19111yescharacteristic
314C2×D4C22×D19111yescharacteristic
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—3821no—
318C22×C4—1911no—
319C22×C4—1911no—
320SD16—3822no—
321SD16—3822no—
322SD16—7641no—
323D8—3822no—
324D8—3822no—
325D8—3822no—
326D8—3822no—
327D8—7641no—
328D8D76414yescomplemented
329M16—3822no—
330C2×C8—3822no—
331C2×C8—1911no—
332C2×C8—1911no—
333C2×C8C22×D19111yescharacteristic
334C2×C8C22×D19111yescharacteristic
335C4⋊C4—3821no—
336C4⋊C4—1911no—
337C4⋊C4C22×D19111yescharacteristic
338C4⋊C4C22×D19111yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—3821no—
342C22⋊C4—3821no—
343C22⋊C4—3821no—
344C22⋊C4—3821no—
345C22⋊C4—3821no—
346C22⋊C4—3821no—
347C22⋊C4—1911no—
348C22⋊C4—1911no—
349C42C22×D19111yescharacteristic
350C23—21no—
351C23—21no—
352C23—3821no—
353C23—3821no—
354C23—3821no—
355C23—1911no—
356C23—1911no—
357C23—15281no—
358C23—15281no—
359C23—15281no—
360C23—7642no—
361C23—7642no—
362C23—7642no—
363C23—7642no—
364C23—7642no—
365C23—7642no—
366Q8—3821no—
367Q8—3821no—
368D4—21no—
369D4—21no—
370D4—3822no—
371D4—3821no—
372D4—3821no—
373D4—3821no—
374D4—3821no—
375D4—3821no—
376D4—3821no—
377D4—15281no—
378D4—15281no—
379D4—7642no—
380D4—7642no—
381D4—7642no—
382D4—7642no—
383D4—7642no—
384D4—7642no—
385D4—7642no—
386D4C2×D76412yes—
387D4C2×D76412yescomplemented
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—42no—
391C2×C4—81no—
392C2×C4—3821no—
393C2×C4—3821no—
394C2×C4—3821no—
395C2×C4—3821no—
396C2×C4—3821no—
397C2×C4—1911no—
398C2×C4—1911no—
399C2×C4—7642no—
400C2×C4D4×D19111yescharacteristic
401C2×C4D4×D19111yescharacteristic
402C2×C4C23×D19111yescharacteristic
403C8—21no—
404C8—3821no—
405C8—3821no—
406C8C2×D76412yescomplemented
407C22—22no—
408C22—22no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—3821no—
414C22—3821no—
415C22—3821no—
416C22—3821no—
417C22—3821no—
418C22—1912no—
419C22—15284no—
420C22—15284no—
421C22—15284no—
422C22—7641no—
423C22—7641no—
424C22—7641no—
425C22—7641no—
426C22—7641no—
427C22—7641no—
428C22—7641no—
429C22—7641no—
430C22—7641no—
431C22C2×D4×D19111yescharacteristic
432C4—21no—
433C4—21no—
434C4—41no—
435C4—41no—
436C4—3821no—
437C4—3821no—
438C4—7641no—
439C4C22×D76411yescharacteristic
440C4#2025936811yescharacteristic
441C2—42no—
442C2—42no—
443C2—3822no—
444C2—15281no—
445C2—15281no—
446C2—15281no—
447C2—7641no—
448C2D4×D76411yescharacteristic
449C2AGL(1,ℤ/8)×D19111yescharacteristic
450C2#2713346711yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#442×C191C2
socleC22×C191C22×D4
hypercenter#442D191
hypocenterC191#3264
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. C2×#20259346C2semidirect
2. C2×D8×D191C2semidirect
3. #27133308C2semidirect
4. D4×D764C2semidirect
5. #27133052C2semidirect
6. #27132917C2semidirect
7. #27132893C2semidirect
8. #27132878C2semidirect
9. #27132858C2semidirect
10. #27132815C2semidirect
11. #27132809C2semidirect
12. #27132798C2semidirect
13. #27132798C2semidirect
14. #27132736C2semidirect
15. #442×C191C2semidirect
16. C2×#15690286C22semidirect
19. C2×D4×D191C22semidirect
20. C2×D4×D191C22semidirect
22. C2×#15690276C22semidirect
23. C2×#15690276C22semidirect
24. C2×#15690275C22semidirect
25. C2×D1528C22semidirect
26. C2×D1528C22semidirect
28. C2×#15690272C22semidirect
29. C2×#15690272C22semidirect
31. C2×#15690270C22semidirect
32. C2×C8×D191C22semidirect
33. #20259319C22semidirect
34. #20259310C22semidirect
36. #20259302C22semidirect
38. #20259297C22semidirect
39. #20259297C22semidirect
47. C4⋊(C191⋊C8)C22semidirect
48. #20259245C22semidirect
49. #20259242C22semidirect
50. #20259242C22semidirect
51. #20259242C22semidirect
52. (C2×D191)⋊C8C22semidirect
53. (C2×D191)⋊C8C22semidirect
55. C4×D4×C191C22semidirect
56. C4×D4×C191C22semidirect
57. #106×C191C22semidirect
58. #102×C191C22semidirect
59. #102×C191C22semidirect
60. #97×C191C22semidirect
116. D8×C191D4semidirect
171. D4×C191C2×D4semidirect
180. C8×C191C2×D4semidirect
205. C2×D191AGL(1,ℤ/8)semidirect
225. C191#3264semidirect
240. #442D191semidirect
273. C4×D4C2×D191semidirect
274. C4×D4C2×D191semidirect
276. #106C2×D191semidirect
282. #102C2×D191semidirect
283. #102C2×D191semidirect
286. #97C2×D191semidirect
328. D8D764semidirect
387. D4C2×D764semidirect
406. C8C2×D764semidirect

Maximal subgroups

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

subgroupindex
C2×#202593462
C2×D8×D1912
#271333082
D4×D7642
#271330522
#271329172
#271328932
#271328782
#271328582
#271328152
#271328092
#271327982
#271327362
#442×C1912
#3264191