Finite Group Databasebeta

← #37584011

Subgroups — #37584011

Order 24448.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20259348C211yesmaximal, characteristic, complemented
2C22×#15690275C211yesmaximal, characteristic, complemented
3#27133322C211yesmaximal, characteristic, complemented
4#27133050C211yesmaximal, characteristic, complemented
5#27132907C211yesmaximal, characteristic, complemented
6#27132897C211yesmaximal, characteristic, complemented
7#27132878C211yesmaximal, characteristic, complemented
8#27132860C211yesmaximal, characteristic, complemented
9#27132814C211yesmaximal, characteristic, complemented
10#27132806C211yesmaximal, characteristic, complemented
11#27132801C211yesmaximal, characteristic, complemented
12#27132797C211yesmaximal, characteristic, complemented
13#27132763C211yesmaximal, characteristic, complemented
14#27132735C211yesmaximal, characteristic, complemented
15D42×C191C211yesmaximal, characteristic, complemented
16C2×#15690287C2211yescharacteristic, complemented
17C22×Dic382C2211yescharacteristic, complemented
18#20259348—24no—
19C2×#15690275—24no—
20C2×#15690275C2211yescharacteristic
21C2×#15690275C2211yescharacteristic
22C2×#15690275C2211yescharacteristic, complemented
23C2×#15690275C2211yescharacteristic, complemented
24C2×#15690275C2211yescharacteristic, complemented
25C2×#15690275C2211yescharacteristic, complemented
26C2×#15690272C2211yescharacteristic
27C2×#15690272C2211yescharacteristic, complemented
28C2×#15690272C2211yescharacteristic, complemented
29C2×#15690271C2211yescharacteristic, complemented
30C22×C191⋊C8C2211yescharacteristic, complemented
31#20259310C2211yescharacteristic, complemented
32#20259308—21no—
33#20259301C2211yescharacteristic, complemented
34#20259300—21no—
35#20259296C2211yescharacteristic, complemented
36#20259294—21no—
37C4×Dic382C2211yescharacteristic, complemented
38(C22⋊D191)⋊C4—21no—
39#20259255C2211yescharacteristic, complemented
40C4⋊(C191⋊C8)C2211yescharacteristic, complemented
41#20259245C2211yescharacteristic, complemented
42#20259245C2211yescharacteristic, complemented
43#20259243C2211yescharacteristic
44#20259243C2211yescharacteristic, complemented
45#20259243C2211yescharacteristic, complemented
46#20259242C2211yescharacteristic, complemented
47C22⋊(C191⋊C8)C2211yescharacteristic, complemented
48C22⋊(C191⋊C8)C2211yescharacteristic, complemented
49#20259233C2211yescharacteristic, complemented
50C22×D4×C191—21no—
51C22×D4×C191C2211yescharacteristic
52C22×D4×C191C2211yescharacteristic
53#127×C191C2211yescharacteristic
54#121×C191—21no—
55#121×C191C2211yescharacteristic
56#121×C191C2211yescharacteristic
57#120×C191—21no—
58#120×C191—21no—
59C4×D4×C191C2211yescharacteristic
60C4×D4×C191C2211yescharacteristic
61#15690287—22no—
62#15690287—41no—
63C2×Dic382—22no—
64C2×Dic382—21no—
65C2×Dic382—41no—
66C2×Dic382C2311yescharacteristic
67C2×Dic382C2311yescharacteristic
68C2×D764C2311yescharacteristic
69C2×C22⋊D191—21no—
70C2×C22⋊D191—21no—
71C22×Dic191—21no—
72C2×C4×D191—21no—
73#15690275—22no—
74#15690275—22no—
75#15690275—22no—
76#15690275—22no—
77#15690275—41no—
78#15690275D414yescomplemented
79#15690272—22no—
80#15690272—22no—
81#15690272—41no—
82#15690271—22no—
83C2×C191⋊C8—22no—
84C2×C191⋊C8C2311yescharacteristic
85C2×C191⋊C8C2311yescharacteristic
86C2×C191⋊C8C2311yescharacteristic
87C2×C191⋊C8C2311yescharacteristic
88C4⋊Dic191C2311yescharacteristic
89Dic191⋊C4—21no—
90Dic191⋊C4—21no—
91C22⋊Dic191—21no—
92(C2×D191)⋊C4—21no—
93(C2×D191)⋊C4—21no—
94(C2×D191)⋊C4—21no—
95C4×Dic191—21no—
96C24×C191—21no—
97C24×C191—21no—
98C2×D4×C191—22no—
99C2×D4×C191—22no—
100C2×D4×C191—21no—
101C2×D4×C191—21no—
102C2×D4×C191—21no—
103C2×D4×C191—21no—
104C2×D4×C191—21no—
105C2×D4×C191—21no—
106C2×D4×C191—41no—
107C2×D4×C191—41no—
108C2×D4×C191—44no—
109C2×D4×C191D411yescharacteristic
110C2×D4×C191C2311yescharacteristic
111C2×D4×C191C2311yescharacteristic
112C2×D4×C191C2311yescharacteristic
113C22×C4×C191—21no—
114C22×C4×C191C2311yescharacteristic
115C22×C4×C191C2311yescharacteristic
116C4⋊C4×C191D411yescharacteristic
117C4⋊C4×C191C2311yescharacteristic
118C22⋊C4×C191—21no—
119C22⋊C4×C191—21no—
120C22⋊C4×C191—21no—
121C22⋊C4×C191D411yescharacteristic
122C22⋊C4×C191D411yescharacteristic
123C42×C191C2311yescharacteristic
124C22×D191—21no—
125Dic382—21no—
126Dic382—21no—
127Dic382—21no—
128Dic382—42no—
129Dic382C2×D412yescomplemented
130D764—21no—
131D764—21no—
132C22⋊D191—42no—
133C22⋊D191—42no—
134C2×Dic191—21no—
135C2×Dic191—21no—
136C2×Dic191—21no—
137C2×Dic191—42no—
138C4×D191—42no—
139C191⋊C8—21no—
140C191⋊C8—21no—
141C191⋊C8—21no—
142C191⋊C8C2×D412yescomplemented
143C23×C191—21no—
144C23×C191—21no—
145C23×C191—21no—
146C23×C191—42no—
147C23×C191—42no—
148C23×C191—42no—
149C23×C191—42no—
150C23×C191—81no—
151C23×C191—81no—
152C23×C191C2×D411yescharacteristic
153C23×C191C2×D411yescharacteristic
154D4×C191—22no—
155D4×C191—21no—
156D4×C191—21no—
157D4×C191—21no—
158D4×C191—21no—
159D4×C191—21no—
160D4×C191—42no—
161D4×C191—42no—
162D4×C191—42no—
163D4×C191—42no—
164D4×C191—42no—
165D4×C191—81no—
166D4×C191C2×D412yes—
167C2×C4×C191—21no—
168C2×C4×C191—21no—
169C2×C4×C191—21no—
170C2×C4×C191—21no—
171C2×C4×C191—21no—
172C2×C4×C191—42no—
173C2×C4×C191C2×D411yescharacteristic
174C2×C4×C191C2×D411yescharacteristic
175C2×C4×C191C2×D411yescharacteristic
176C2×C4×C191C2×D411yescharacteristic
177C2×C4×C191C2411yescharacteristic
178C22×C191—21no—
179C22×C191—21no—
180C22×C191—21no—
181C22×C191—22no—
182C22×C191—21no—
183C22×C191—21no—
184C22×C191—41no—
185C22×C191—41no—
186C22×C191—41no—
187C22×C191—41no—
188C22×C191—41no—
189C22×C191—41no—
190C22×C191—41no—
191C22×C191—41no—
192C22×C191—84no—
193C22×C191—84no—
194C22×C191AGL(1,ℤ/8)12yescomplemented
195C22×C191C22×D411yescharacteristic
196C2×D191—41no—
197C2×D191—41no—
198C2×D191—41no—
199C4×C191—21no—
200C4×C191—21no—
201C4×C191—21no—
202C4×C191—21no—
203C4×C191—41no—
204C4×C191C22×D411yescharacteristic
205C4×C191H(5,𝔽2)11yescharacteristic
206Dic191—22no—
207Dic191—41no—
208Dic191—41no—
209C2×C191—22no—
210C2×C191—41no—
211C2×C191—42no—
212C2×C191—81no—
213C2×C191—81no—
214C2×C191D4211yescharacteristic
215C2×C191C2×AGL(1,ℤ/8)11yescharacteristic
216C2×C191#57711yescharacteristic
217D191—81no—
218C191#326511yescharacteristic, complemented, Sylow
219#3265—1911nomaximal, Sylow
220C2×AGL(1,ℤ/8)—1911no—
221C22×SD16—1911no—
222#548—1911no—
223D42D19111yescharacteristic, complemented
224#496—1911no—
225#468—1911no—
226#466—1911no—
227#463—1911no—
228#460—1911no—
229#450—1911no—
230#449—1911no—
231#448—1911no—
232#447—1911no—
233#440—1911no—
234#435—1911no—
235C2×#41—1911no—
236C22×Q8—1911no—
237C22×D4—21no—
238C22×D4C2×D19111yescharacteristic
239C22×D4C2×D19111yescharacteristic
240AGL(1,ℤ/8)—3824no—
241C2×SD16—3824no—
242C2×SD16—1911no—
243C2×SD16—1911no—
244C2×SD16—1911no—
245C2×SD16—1911no—
246C2×SD16—1911no—
247C2×SD16—1911no—
248C2×D8—1911no—
249C2×D8—1911no—
250C2×D8—1911no—
251C2×M16—1911no—
252C22×C8—1911no—
253#127C2×D19111yescharacteristic
254#124—3821no—
255#124—1911no—
256C22⋊Q8—3821no—
257C22⋊Q8—1911no—
258#121—21no—
259#121—3821no—
260#121—1911no—
261#121C2×D19111yescharacteristic
262#121C2×D19111yescharacteristic
263#120—21no—
264#120—21no—
265C4×Q8—1911no—
266C4×D4—3821no—
267C4×D4C2×D19111yescharacteristic
268C4×D4C2×D19111yescharacteristic
269#107—1911no—
270C4⋊C8—1911no—
271#103—1911no—
272#103—1911no—
273#102—1911no—
274#102—1911no—
275#102—1911no—
276#102—1911no—
277C22⋊C8—1911no—
278C22⋊C8—1911no—
279#97—1911no—
280C24—21no—
281C24—21no—
282#41—3822no—
283#41—7641no—
284C2×Q8—3822no—
285C2×Q8—3821no—
286C2×Q8—1911no—
287C2×Q8—1911no—
288C2×Q8—7641no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—44no—
300C2×D4—3821no—
301C2×D4—3821no—
302C2×D4—1911no—
303C2×D4C22⋊D19111yescharacteristic
304C2×D4C22×D19111yescharacteristic
305C2×D4C22×D19111yescharacteristic
306C2×D4C22×D19111yescharacteristic
307C22×C4—21no—
308C22×C4—3821no—
309C22×C4—3821no—
310C22×C4C22×D19111yescharacteristic
311C22×C4C22×D19111yescharacteristic
312SD16—3822no—
313SD16—3822no—
314SD16—3822no—
315SD16—3822no—
316SD16—1914no—
317SD16—7641no—
318D8—3822no—
319D8—3822no—
320D8—7641no—
321M16—3822no—
322C2×C8—3822no—
323C2×C8—1911no—
324C2×C8—1911no—
325C2×C8—1911no—
326C2×C8—1911no—
327C4⋊C4—3821no—
328C4⋊C4—3821no—
329C4⋊C4—1911no—
330C4⋊C4C22⋊D19111yescharacteristic
331C4⋊C4C22×D19111yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—21no—
335C22⋊C4—3821no—
336C22⋊C4—3821no—
337C22⋊C4—3821no—
338C22⋊C4—3821no—
339C22⋊C4C22⋊D19111yescharacteristic
340C22⋊C4C22⋊D19111yescharacteristic
341C42—3821no—
342C42C22×D19111yescharacteristic
343C23—21no—
344C23—21no—
345C23—21no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23—42no—
350C23—81no—
351C23—81no—
352C23—3821no—
353C23C2×C22⋊D19111yescharacteristic
354C23C2×C22⋊D19111yescharacteristic
355Q8—3821no—
356Q8—3821no—
357Q8—3821no—
358Q8—1912no—
359Q8—7642no—
360D4—22no—
361D4—21no—
362D4—21no—
363D4—21no—
364D4—21no—
365D4—21no—
366D4—42no—
367D4—42no—
368D4—42no—
369D4—42no—
370D4—42no—
371D4—81no—
372D4—3821no—
373D4—3821no—
374D4—7642no—
375D4—7642no—
376D4D4×D19112yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—42no—
383C2×C4—3821no—
384C2×C4—3821no—
385C2×C4—3821no—
386C2×C4—7642no—
387C2×C4—7642no—
388C2×C4C2×C22⋊D19111yescharacteristic
389C2×C4C2×C22⋊D19111yescharacteristic
390C2×C4C2×C22⋊D19111yescharacteristic
391C2×C4C2×C22⋊D19111yescharacteristic
392C2×C4C23×D19111yescharacteristic
393C8—3821no—
394C8—3821no—
395C8—3821no—
396C8—1912no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—22no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—84no—
412C22—84no—
413C22—7641no—
414C22—7641no—
415C22—7641no—
416C22#2025934812yescomplemented
417C22C22×C22⋊D19111yescharacteristic
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—41no—
423C4—3822no—
424C4—7641no—
425C4—7641no—
426C4C2×D4×D19111yescharacteristic
427C4#2025936711yescharacteristic
428C2—22no—
429C2—41no—
430C2—42no—
431C2—81no—
432C2—81no—
433C2—15281no—
434C2D4×C22⋊D19111yescharacteristic
435C2C2×#2025934811yescharacteristic
436C2#2713347411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
FittingD42×C191C2
socleC22×C191C22×D4
hypercenterD42D191
hypocenterC191#3265
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C2×#20259348C2semidirect
2. C22×#15690275C2semidirect
3. #27133322C2semidirect
4. #27133050C2semidirect
5. #27132907C2semidirect
6. #27132897C2semidirect
7. #27132878C2semidirect
8. #27132860C2semidirect
9. #27132814C2semidirect
10. #27132806C2semidirect
11. #27132801C2semidirect
12. #27132797C2semidirect
13. #27132763C2semidirect
14. #27132735C2semidirect
15. D42×C191C2semidirect
16. C2×#15690287C22semidirect
17. C22×Dic382C22semidirect
22. C2×#15690275C22semidirect
23. C2×#15690275C22semidirect
24. C2×#15690275C22semidirect
25. C2×#15690275C22semidirect
27. C2×#15690272C22semidirect
28. C2×#15690272C22semidirect
29. C2×#15690271C22semidirect
30. C22×C191⋊C8C22semidirect
31. #20259310C22semidirect
33. #20259301C22semidirect
35. #20259296C22semidirect
37. C4×Dic382C22semidirect
39. #20259255C22semidirect
40. C4⋊(C191⋊C8)C22semidirect
41. #20259245C22semidirect
42. #20259245C22semidirect
44. #20259243C22semidirect
45. #20259243C22semidirect
46. #20259242C22semidirect
47. C22⋊(C191⋊C8)C22semidirect
48. C22⋊(C191⋊C8)C22semidirect
49. #20259233C22semidirect
78. #15690275D4semidirect
129. Dic382C2×D4semidirect
142. C191⋊C8C2×D4semidirect
194. C22×C191AGL(1,ℤ/8)semidirect
218. C191#3265semidirect
223. D42D191semidirect
416. C22#20259348semidirect

Maximal subgroups

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

subgroupindex
C2×#202593482
C22×#156902752
#271333222
#271330502
#271329072
#271328972
#271328782
#271328602
#271328142
#271328062
#271328012
#271327972
#271327632
#271327352
D42×C1912
#3265191