Finite Group Databasebeta

← #37584056

Subgroups — #37584056

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×#20259348C211yesmaximal, characteristic, complemented
2C2×#20259342C211yesmaximal, characteristic, complemented
3#27133334C211yesmaximal, characteristic, complemented
4#27133053C211yesmaximal, characteristic, complemented
5#27132911C211yesmaximal, characteristic, complemented
6#27132897C211yesmaximal, characteristic, complemented
7#27132882C211yesmaximal, characteristic, complemented
8#27132858C211yesmaximal, characteristic, complemented
9#27132823C211yesmaximal, characteristic, complemented
10#27132808C211yesmaximal, characteristic, complemented
11#27132806C211yesmaximal, characteristic, complemented
12#27132797C211yesmaximal, characteristic, complemented
13#27132763C211yesmaximal, characteristic, complemented
14#27132735C211yesmaximal, characteristic, complemented
15#546×C191C211yesmaximal, characteristic, complemented
16C2×#15690287C2211yescharacteristic
17C2×#15690287C2211yescharacteristic, complemented
18#20259348—24no—
19#20259342—24no—
20C2×#15690277C2211yescharacteristic, complemented
21C2×#15690275C2211yescharacteristic
22C2×#15690275C2211yescharacteristic
23C2×#15690274C2211yescharacteristic, complemented
24C2×#15690274C2211yescharacteristic, complemented
25C2×#15690272C2211yescharacteristic
26C2×#15690272C2211yescharacteristic
27C2×#15690272C2211yescharacteristic
28C2×#15690272C2211yescharacteristic
29C2×#15690271C2211yescharacteristic
30C22×C191⋊C8C2211yescharacteristic, complemented
31#20259319C2211yescharacteristic, complemented
32#20259318—21no—
33#20259296C2211yescharacteristic
34#20259296C2211yescharacteristic, complemented
35#20259295—21no—
36#20259295—21no—
37C4×Dic382C2211yescharacteristic, complemented
38#20259284—21no—
39#20259255C2211yescharacteristic, complemented
40C4⋊(C191⋊C8)C2211yescharacteristic, complemented
41#20259246C2211yescharacteristic, complemented
42#20259245C2211yescharacteristic, complemented
43#20259243C2211yescharacteristic
44#20259243C2211yescharacteristic
45#20259242C2211yescharacteristic, complemented
46#20259242C2211yescharacteristic, complemented
47C22⋊(C191⋊C8)C2211yescharacteristic
48C22⋊(C191⋊C8)C2211yescharacteristic, complemented
49#20259233C2211yescharacteristic, complemented
50C2×#41×C191C2211yescharacteristic
51C22×D4×C191C2211yescharacteristic
52#124×C191C2211yescharacteristic
53#123×C191—21no—
54C22⋊Q8×C191C2211yescharacteristic
55#121×C191—21no—
56#121×C191C2211yescharacteristic
57#120×C191—21no—
58C4×D4×C191C2211yescharacteristic
59C4×D4×C191C2211yescharacteristic
60C2×C22⋊C4×C191—21no—
61#15690287—22no—
62#15690287—22no—
63#15690287—41no—
64#15690287—41no—
65C2×Dic382C2311yescharacteristic
66C2×D764—21no—
67C2×D764—21no—
68C2×D764C2311yescharacteristic
69C2×D764C2311yescharacteristic
70C2×C22⋊D191—21no—
71C2×C22⋊D191—21no—
72C2×C22⋊D191—21no—
73C2×C22⋊D191—21no—
74C2×C4×D191—21no—
75C2×C4×D191—21no—
76#15690277—22no—
77#15690275—22no—
78#15690275D414yescomplemented
79#15690274—22no—
80#15690274—41no—
81#15690272—22no—
82#15690272—22no—
83#15690272—22no—
84#15690272—41no—
85#15690271—22no—
86C2×C191⋊C8—22no—
87C2×C191⋊C8C2311yescharacteristic
88C2×C191⋊C8C2311yescharacteristic
89C2×C191⋊C8C2311yescharacteristic
90C2×C191⋊C8C2311yescharacteristic
91C4⋊Dic191C2311yescharacteristic
92Dic191⋊C4—21no—
93(C2×D191)⋊C4—21no—
94(C2×D191)⋊C4—21no—
95C4×Dic191—21no—
96C24×C191—21no—
97#41×C191—22no—
98#41×C191—41no—
99C2×Q8×C191C2311yescharacteristic
100C2×D4×C191—22no—
101C2×D4×C191—21no—
102C2×D4×C191—21no—
103C2×D4×C191—21no—
104C2×D4×C191—41no—
105C2×D4×C191D411yescharacteristic
106C2×D4×C191C2311yescharacteristic
107C2×D4×C191C2311yescharacteristic
108C22×C4×C191—21no—
109C22×C4×C191—21no—
110C22×C4×C191C2311yescharacteristic
111C22×C4×C191C2311yescharacteristic
112C4⋊C4×C191—21no—
113C4⋊C4×C191D411yescharacteristic
114C4⋊C4×C191C2311yescharacteristic
115C22⋊C4×C191—21no—
116C22⋊C4×C191—21no—
117C22⋊C4×C191—21no—
118C22⋊C4×C191—21no—
119C22⋊C4×C191—21no—
120C22⋊C4×C191D411yescharacteristic
121C22⋊C4×C191D411yescharacteristic
122C42×C191C2311yescharacteristic
123C22×D191—21no—
124C22×D191—21no—
125Dic382—21no—
126Dic382C2×D412yes—
127D764—21no—
128D764—21no—
129D764—21no—
130D764—21no—
131D764—42no—
132D764—81no—
133C22⋊D191—42no—
134C22⋊D191—42no—
135C22⋊D191—81no—
136C22⋊D191—81no—
137C2×Dic191—21no—
138C2×Dic191—21no—
139C4×D191—42no—
140C4×D191—42no—
141C191⋊C8—21no—
142C191⋊C8—21no—
143C191⋊C8—21no—
144C191⋊C8C2×D412yes—
145C23×C191—21no—
146C23×C191—21no—
147C23×C191—42no—
148C23×C191—42no—
149C23×C191—81no—
150C23×C191C2×D411yescharacteristic
151C23×C191C2×D411yescharacteristic
152Q8×C191—21no—
153Q8×C191—21no—
154D4×C191—21no—
155D4×C191—21no—
156D4×C191—21no—
157D4×C191—41no—
158D4×C191—42no—
159D4×C191—42no—
160D4×C191—81no—
161D4×C191C2×D412yes—
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—81no—
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—22no—
178C22×C191—21no—
179C22×C191—21no—
180C22×C191—21no—
181C22×C191—21no—
182C22×C191—41no—
183C22×C191—41no—
184C22×C191—41no—
185C22×C191—41no—
186C22×C191—41no—
187C22×C191—84no—
188C22×C191C22×D411yescharacteristic
189C2×D191—41no—
190C2×D191—41no—
191C2×D191—41no—
192C2×D191—41no—
193C2×D191—41no—
194C2×D191—41no—
195C4×C191—21no—
196C4×C191—21no—
197C4×C191—21no—
198C4×C191—21no—
199C4×C191—41no—
200C4×C191—41no—
201C4×C191C22×D411yescharacteristic
202C4×C191H(5,𝔽2)11yescharacteristic
203Dic191—22no—
204Dic191—41no—
205C2×C191—41no—
206C2×C191—41no—
207C2×C191—42no—
208C2×C191—81no—
209C2×C191D4211yescharacteristic
210C2×C191#57511yescharacteristic
211C2×C191#57611yescharacteristic
212D191—81no—
213D191—81no—
214C191#326811yescharacteristic, complemented, Sylow
215#3268—1911nomaximal, Sylow
216C2×AGL(1,ℤ/8)—1911no—
217C2×#135—1911no—
218#550—1911no—
219#546D19111yescharacteristic, complemented
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)—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—3821no—
249#127—1911no—
250#124C2×D19111yescharacteristic
251#123—21no—
252C22⋊Q8C2×D19111yescharacteristic
253#121—21no—
254#121—3821no—
255#121—3821no—
256#121—1911no—
257#121—1911no—
258#121C2×D19111yescharacteristic
259#120—21no—
260C4×Q8—1911no—
261C4×D4—3821no—
262C4×D4C2×D19111yescharacteristic
263C4×D4C2×D19111yescharacteristic
264C2×C22⋊C4—21no—
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—21no—
289C2×D4—41no—
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—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—21no—
330C22⋊C4—21no—
331C22⋊C4—21no—
332C22⋊C4—3821no—
333C22⋊C4—3821no—
334C22⋊C4C22⋊D19111yescharacteristic
335C22⋊C4C22⋊D19111yescharacteristic
336C42—3821no—
337C42C22×D19111yescharacteristic
338C23—21no—
339C23—21no—
340C23—42no—
341C23—42no—
342C23—81no—
343C23—3821no—
344C23—3821no—
345C23C2×C22⋊D19111yescharacteristic
346C23C2×C22⋊D19111yescharacteristic
347Q8—21no—
348Q8—21no—
349Q8—3821no—
350Q8—1912no—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—41no—
355D4—42no—
356D4—42no—
357D4—81no—
358D4—3821no—
359D4—3821no—
360D4—3821no—
361D4—3821no—
362D4—15281no—
363D4—15281no—
364D4—15281no—
365D4—7642no—
366D4—7642no—
367D4—7642no—
368D4D4×D19112yes—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—42no—
376C2×C4—81no—
377C2×C4—3821no—
378C2×C4—3821no—
379C2×C4—7642no—
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—22no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—84no—
403C22—7641no—
404C22—7641no—
405C22—7641no—
406C22—7641no—
407C22—7641no—
408C22—7641no—
409C22C22×C22⋊D19111yescharacteristic
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—41no—
415C4—41no—
416C4—3822no—
417C4—7641no—
418C4C2×D4×D19111yescharacteristic
419C4#2025936711yescharacteristic
420C2—41no—
421C2—41no—
422C2—42no—
423C2—81no—
424C2—15281no—
425C2—15281no—
426C2D4×C22⋊D19111yescharacteristic
427C2#2713346111yescharacteristic
428C2#2713346611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#546×C191C2
socleC22×C191C22×D4
hypercenter#546D191
hypocenterC191#3268
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. C2×#20259348C2semidirect
2. C2×#20259342C2semidirect
3. #27133334C2semidirect
4. #27133053C2semidirect
5. #27132911C2semidirect
6. #27132897C2semidirect
7. #27132882C2semidirect
8. #27132858C2semidirect
9. #27132823C2semidirect
10. #27132808C2semidirect
11. #27132806C2semidirect
12. #27132797C2semidirect
13. #27132763C2semidirect
14. #27132735C2semidirect
15. #546×C191C2semidirect
17. C2×#15690287C22semidirect
20. C2×#15690277C22semidirect
23. C2×#15690274C22semidirect
24. C2×#15690274C22semidirect
30. C22×C191⋊C8C22semidirect
31. #20259319C22semidirect
34. #20259296C22semidirect
37. C4×Dic382C22semidirect
39. #20259255C22semidirect
40. C4⋊(C191⋊C8)C22semidirect
41. #20259246C22semidirect
42. #20259245C22semidirect
45. #20259242C22semidirect
46. #20259242C22semidirect
48. C22⋊(C191⋊C8)C22semidirect
49. #20259233C22semidirect
78. #15690275D4semidirect
214. C191#3268semidirect
219. #546D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593482
C2×#202593422
#271333342
#271330532
#271329112
#271328972
#271328822
#271328582
#271328232
#271328082
#271328062
#271327972
#271327632
#271327352
#546×C1912
#3268191