Finite Group Databasebeta

← #37584054

Subgroups — #37584054

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×#20259343C211yesmaximal, characteristic, complemented
2#27133332C211yesmaximal, characteristic, complemented
3#27133307C211yesmaximal, characteristic, complemented
4#27133051C211yesmaximal, characteristic, complemented
5#27132909C211yesmaximal, characteristic, complemented
6#27132896C211yesmaximal, characteristic, complemented
7#27132883C211yesmaximal, characteristic, complemented
8#27132856C211yesmaximal, characteristic, complemented
9#27132825C211yesmaximal, characteristic, complemented
10#27132807C211yesmaximal, characteristic, complemented
11#27132805C211yesmaximal, characteristic, complemented
12#27132797C211yesmaximal, characteristic, complemented
13#27132764C211yesmaximal, characteristic, complemented
14#27132732C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C191C211yesmaximal, characteristic, complemented
16C2×#15690288C2211yescharacteristic, complemented
17C2×#15690286C2211yescharacteristic
18#20259343—24no—
19C2×#15690277C2211yescharacteristic, complemented
20C2×#15690276C2211yescharacteristic, complemented
21C2×#15690275C2211yescharacteristic, complemented
22C2×#15690272C2211yescharacteristic
23C2×#15690272C2211yescharacteristic
24C2×C8×D191C2211yescharacteristic, complemented
25#20259318—21no—
26#20259318C2211yescharacteristic, complemented
27#20259309C2211yescharacteristic
28#20259303—21no—
29#20259301C2211yescharacteristic
30#20259296—21no—
31#20259296C2211yescharacteristic
32#20259294—21no—
33#20259293—21no—
34#20259293C2211yescharacteristic
35#20259293C2211yescharacteristic, complemented
36#20259291—21no—
37Q8×Dic191C2211yescharacteristic, complemented
38D4×Dic191C2211yescharacteristic
39C4×C22⋊D191—21no—
40C4×C22⋊D191C2211yescharacteristic
41C2×C22⋊Dic191—21no—
42#20259256C2211yescharacteristic, complemented
43Dic191⋊C8C2211yescharacteristic, complemented
44#20259247C2211yescharacteristic, complemented
45#20259246C2211yescharacteristic, complemented
46#20259244C2211yescharacteristic, complemented
47#20259243C2211yescharacteristic
48#20259243C2211yescharacteristic
49#20259243C2211yescharacteristic, complemented
50C22⋊(C191⋊C8)C2211yescharacteristic
51(C2×D191)⋊C8C2211yescharacteristic, complemented
52#20259232C2211yescharacteristic, complemented
53C2×#41×C191C2211yescharacteristic
54C22×D4×C191C2211yescharacteristic
55AGL(1,ℤ/8)×C191—24no—
56C2×SD16×C191C2211yescharacteristic
57C2×SD16×C191C2211yescharacteristic
58C2×D8×C191C2211yescharacteristic
59C2×D8×C191C2211yescharacteristic
60C2×M16×C191C2211yescharacteristic
61#15690288—22no—
62#15690288—41no—
63#15690286—22no—
64#15690286—41no—
65C2×Dic382C2311yescharacteristic
66C2×D764—21no—
67C2×D764C2311yescharacteristic
68C2×C22⋊D191—21no—
69C2×C22⋊D191—21no—
70C2×C22⋊D191—21no—
71C2×C22⋊D191—21no—
72C2×C22⋊D191—21no—
73C2×C22⋊D191—21no—
74C2×C22⋊D191D411yescharacteristic
75C22×Dic191—21no—
76C22×Dic191—21no—
77C2×C4×D191—21no—
78C2×C4×D191C2311yescharacteristic
79#15690277—22no—
80#15690276—22no—
81#15690275—41no—
82#15690272—22no—
83#15690272—41no—
84C2×C191⋊C8C2311yescharacteristic
85C2×C191⋊C8C2311yescharacteristic
86C8×D191—22no—
87C4⋊Dic191—21no—
88C4⋊Dic191C2311yescharacteristic
89C4⋊Dic191C2311yescharacteristic
90Dic191⋊C4—21no—
91Dic191⋊C4D411yescharacteristic
92C22⋊Dic191—21no—
93C22⋊Dic191—21no—
94C22⋊Dic191—21no—
95C22⋊Dic191—21no—
96C22⋊Dic191—21no—
97C22⋊Dic191—21no—
98C22⋊Dic191D411yescharacteristic
99(C2×D191)⋊C4—21no—
100(C2×D191)⋊C4D411yescharacteristic
101C4×Dic191—21no—
102C4×Dic191C2311yescharacteristic
103C24×C191—21no—
104#41×C191—22no—
105#41×C191—41no—
106C2×Q8×C191C2311yescharacteristic
107C2×D4×C191—22no—
108C2×D4×C191—21no—
109C2×D4×C191—21no—
110C2×D4×C191—41no—
111C2×D4×C191C2311yescharacteristic
112C2×D4×C191C2311yescharacteristic
113C2×D4×C191C2311yescharacteristic
114C22×C4×C191—21no—
115C22×C4×C191C2311yescharacteristic
116SD16×C191—22no—
117SD16×C191D414yescomplemented
118D8×C191—22no—
119D8×C191—22no—
120M16×C191—22no—
121C2×C8×C191C2311yescharacteristic
122C2×C8×C191C2311yescharacteristic
123C22×D191—21no—
124C22×D191C2×D411yescharacteristic
125Dic382—21no—
126Dic382—21no—
127D764—21no—
128D764—21no—
129D764—42no—
130C22⋊D191—41no—
131C22⋊D191—42no—
132C22⋊D191—42no—
133C22⋊D191—81no—
134C22⋊D191—81no—
135C22⋊D191—81no—
136C22⋊D191—81no—
137C2×Dic191—21no—
138C2×Dic191—21no—
139C2×Dic191—21no—
140C2×Dic191—42no—
141C2×Dic191—81no—
142C2×Dic191C2×D411yescharacteristic
143C2×Dic191C2×D411yescharacteristic
144C2×Dic191C2×D411yescharacteristic
145C4×D191—21no—
146C4×D191—21no—
147C4×D191—42no—
148C191⋊C8—21no—
149C191⋊C8—21no—
150C23×C191—21no—
151C23×C191—21no—
152C23×C191—21no—
153C23×C191—42no—
154C23×C191—42no—
155C23×C191—81no—
156C23×C191C2×D411yescharacteristic
157Q8×C191—21no—
158Q8×C191C2×D412yes—
159D4×C191—21no—
160D4×C191—21no—
161D4×C191—21no—
162D4×C191—21no—
163D4×C191—21no—
164D4×C191—42no—
165D4×C191—42no—
166D4×C191C2×D412yes—
167C2×C4×C191—21no—
168C2×C4×C191—21no—
169C2×C4×C191—21no—
170C2×C4×C191—42no—
171C2×C4×C191C2×D411yescharacteristic
172C2×C4×C191C2411yescharacteristic
173C8×C191—21no—
174C8×C191C2×D412yes—
175C22×C191—21no—
176C22×C191—22no—
177C22×C191—21no—
178C22×C191—21no—
179C22×C191—41no—
180C22×C191—41no—
181C22×C191—41no—
182C22×C191—41no—
183C22×C191—41no—
184C22×C191—41no—
185C22×C191—41no—
186C22×C191—41no—
187C22×C191—84no—
188C22×C191C22×D411yescharacteristic
189C2×D191—21no—
190C2×D191—21no—
191C2×D191—21no—
192C2×D191—41no—
193C2×D191—41no—
194C2×D191—41no—
195C4×C191—21no—
196C4×C191—22no—
197C4×C191C22×D411yescharacteristic
198C4×C191H(5,𝔽2)11yescharacteristic
199Dic191—21no—
200Dic191—21no—
201Dic191—21no—
202Dic191—41no—
203Dic191—41no—
204Dic191—41no—
205C2×C191—41no—
206C2×C191—42no—
207C2×C191—81no—
208C2×C191—81no—
209C2×C191D4211yescharacteristic
210C2×C191#57511yescharacteristic
211C2×C191#57611yescharacteristic
212D191—41no—
213D191—81no—
214C191#326811yescharacteristic, complemented, Sylow
215#3268—1911nomaximal, Sylow
216C2×AGL(1,ℤ/8)D19111yescharacteristic, complemented
217C2×#135—1911no—
218#550—1911no—
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×#41—1911no—
233C2×#41C2×D19111yescharacteristic
234C22×D4C2×D19111yescharacteristic
235AGL(1,ℤ/8)—24no—
236#135—3824no—
237C2×Q16—1911no—
238C2×SD16—1911no—
239C2×SD16—1911no—
240C2×SD16C2×D19111yescharacteristic
241C2×SD16C2×D19111yescharacteristic
242C2×D8—1911no—
243C2×D8—1911no—
244C2×D8C2×D19111yescharacteristic
245C2×D8C2×D19111yescharacteristic
246C2×M16C2×D19111yescharacteristic
247C22×C8—1911no—
248#127—3821no—
249#127—1911no—
250#124—1911no—
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#97—1911no—
276C24—21no—
277#41—22no—
278#41—41no—
279#41—3822no—
280#41—3822no—
281#41—7641no—
282#41—7641no—
283C2×Q8—1911no—
284C2×Q8C22×D19111yescharacteristic
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—41no—
289C2×D4—3821no—
290C2×D4—3821no—
291C2×D4—3821no—
292C2×D4—3821no—
293C2×D4—3821no—
294C2×D4—3821no—
295C2×D4—3821no—
296C2×D4—1911no—
297C2×D4—1911no—
298C2×D4C22×D19111yescharacteristic
299C2×D4C22×D19111yescharacteristic
300C2×D4C22×D19111yescharacteristic
301C22×C4—21no—
302C22×C4—3821no—
303C22×C4—3821no—
304C22×C4—3821no—
305C22×C4—1911no—
306C22×C4C22×D19111yescharacteristic
307Q16—3822no—
308SD16—22no—
309SD16—3822no—
310SD16—7641no—
311SD16C22⋊D19114yescomplemented
312D8—22no—
313D8—22no—
314D8—3822no—
315D8—7641no—
316M16—22no—
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—3821no—
328C22⋊C4—3821no—
329C22⋊C4—3821no—
330C22⋊C4—3821no—
331C22⋊C4—3821no—
332C22⋊C4—3821no—
333C22⋊C4—3821no—
334C22⋊C4—1911no—
335C22⋊C4—1911no—
336C42—3821no—
337C42—1911no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—42no—
342C23—42no—
343C23—81no—
344C23—3821no—
345C23—1911no—
346C23D4×D19111yescharacteristic
347Q8—21no—
348Q8—3821no—
349Q8—3821no—
350Q8C2×C22⋊D19112yes—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—21no—
356D4—42no—
357D4—42no—
358D4—3821no—
359D4—3821no—
360D4—15281no—
361D4—15281no—
362D4—15281no—
363D4—15281no—
364D4—7642no—
365D4—7641no—
366D4—7642no—
367D4—7642no—
368D4C2×C22⋊D19112yes—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—42no—
373C2×C4—3821no—
374C2×C4—3821no—
375C2×C4—3821no—
376C2×C4—3821no—
377C2×C4—3821no—
378C2×C4—1911no—
379C2×C4—1911no—
380C2×C4—1911no—
381C2×C4—15281no—
382C2×C4—7642no—
383C2×C4—7642no—
384C2×C4D4×D19111yescharacteristic
385C2×C4C23×D19111yescharacteristic
386C8—21no—
387C8—3821no—
388C8—3821no—
389C8C2×C22⋊D19112yes—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—22no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—84no—
403C22—3821no—
404C22—3821no—
405C22—3821no—
406C22—7641no—
407C22—7641no—
408C22—7641no—
409C22C2×D4×D19111yescharacteristic
410C4—21no—
411C4—22no—
412C4—3821no—
413C4—3821no—
414C4—3821no—
415C4—7641no—
416C4—7641no—
417C4—7641no—
418C4C22×C22⋊D19111yescharacteristic
419C4#2025936711yescharacteristic
420C2—41no—
421C2—42no—
422C2—81no—
423C2—81no—
424C2—15281no—
425C2—7641no—
426C2D4×C22⋊D19111yescharacteristic
427C2#2713345911yescharacteristic
428C2#2713346711yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
FittingC2×AGL(1,ℤ/8)×C191C2
socleC22×C191C22×D4
hypercenterC2×AGL(1,ℤ/8)D191
hypocenterC191#3268
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#20259343C2semidirect
2. #27133332C2semidirect
3. #27133307C2semidirect
4. #27133051C2semidirect
5. #27132909C2semidirect
6. #27132896C2semidirect
7. #27132883C2semidirect
8. #27132856C2semidirect
9. #27132825C2semidirect
10. #27132807C2semidirect
11. #27132805C2semidirect
12. #27132797C2semidirect
13. #27132764C2semidirect
14. #27132732C2semidirect
15. C2×AGL(1,ℤ/8)×C191C2semidirect
16. C2×#15690288C22semidirect
19. C2×#15690277C22semidirect
20. C2×#15690276C22semidirect
21. C2×#15690275C22semidirect
24. C2×C8×D191C22semidirect
26. #20259318C22semidirect
35. #20259293C22semidirect
37. Q8×Dic191C22semidirect
42. #20259256C22semidirect
43. Dic191⋊C8C22semidirect
44. #20259247C22semidirect
45. #20259246C22semidirect
46. #20259244C22semidirect
49. #20259243C22semidirect
51. (C2×D191)⋊C8C22semidirect
52. #20259232C22semidirect
117. SD16×C191D4semidirect
214. C191#3268semidirect
216. C2×AGL(1,ℤ/8)D191semidirect
311. SD16C22⋊D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593432
#271333322
#271333072
#271330512
#271329092
#271328962
#271328832
#271328562
#271328252
#271328072
#271328052
#271327972
#271327642
#271327322
C2×AGL(1,ℤ/8)×C1912
#3268191