Finite Group Databasebeta

← #37584127

Subgroups — #37584127

Order 24448.

520 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20259348C211yesmaximal, characteristic, complemented
2C2×#20259347C211yesmaximal, characteristic, complemented
3#27133309C211yesmaximal, characteristic, complemented
4#27133045C211yesmaximal, characteristic, complemented
5#27132911C211yesmaximal, characteristic, complemented
6#27132907C211yesmaximal, characteristic, complemented
7#27132878C211yesmaximal, characteristic, complemented
8#27132875C211yesmaximal, characteristic, complemented
9#27132825C211yesmaximal, characteristic, complemented
10#27132823C211yesmaximal, characteristic, complemented
11#27132797C211yesmaximal, characteristic, complemented
12#27132796C211yesmaximal, characteristic, complemented
13C4×#15690272C211yesmaximal, characteristic, complemented
14#27132741C211yesmaximal, characteristic, complemented
15#546×C191C211yesmaximal, characteristic, complemented
16C2×#15690287C2211yescharacteristic
17C22×D764C2211yescharacteristic, complemented
18#20259348—24no—
19#20259347—24no—
20C2×#15690277C2211yescharacteristic, complemented
21C2×#15690275C2211yescharacteristic, complemented
22C2×#15690275C2211yescharacteristic, complemented
23C2×#15690274C2211yescharacteristic, complemented
24C2×#15690274C2211yescharacteristic, complemented
25C2×#15690272C2211yescharacteristic
26C2×#15690272C2211yescharacteristic
27C2×#15690272C2211yescharacteristic
28C2×#15690272C2211yescharacteristic
29C2×#15690271C2211yescharacteristic, complemented
30C2×#15690271C2211yescharacteristic, complemented
31#20259310C2211yescharacteristic, complemented
32#20259307—21no—
33#20259301C2211yescharacteristic, complemented
34#20259296C2211yescharacteristic
35#20259295—21no—
36#20259290—21no—
37C4×D764C2211yescharacteristic, complemented
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, complemented
46#20259242C2211yescharacteristic, complemented
47C22⋊(C191⋊C8)C2211yescharacteristic, complemented
48C22⋊(C191⋊C8)C2211yescharacteristic, complemented
49C4×C191⋊C8C2211yescharacteristic, complemented
50C2×#41×C191C2211yescharacteristic
51C22×D4×C191C2211yescharacteristic, complemented
52#124×C191C2211yescharacteristic, complemented
53#123×C191—21no—
54C22⋊Q8×C191C2211yescharacteristic, complemented
55#121×C191—21no—
56#121×C191C2211yescharacteristic
57#120×C191—21no—
58C4×D4×C191C2211yescharacteristic
59C4×D4×C191C2211yescharacteristic, complemented
60C2×C22⋊C4×C191—21no—
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—41no—
75#15690275—22no—
76#15690275—22no—
77#15690274—22no—
78#15690274—22no—
79#15690272—22no—
80#15690272—22no—
81#15690272—41no—
82#15690272D414yescomplemented
83#15690271—22no—
84#15690271—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—
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—
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×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—22no—
190C2×D191—41no—
191C2×D191—41no—
192C2×D191—41no—
193C2×D191—41no—
194C2×D191—41no—
195C2×D191—84no—
196C4×C191—21no—
197C4×C191—21no—
198C4×C191—21no—
199C4×C191—21no—
200C4×C191—41no—
201C4×C191—41no—
202C4×C191C22×D411yescharacteristic
203C4×C191H(5,𝔽2)11yescharacteristic
204Dic191—41no—
205Dic191—41no—
206C2×C191—41no—
207C2×C191—41no—
208C2×C191—42no—
209C2×C191—81no—
210C2×C191D4211yescharacteristic
211C2×C191#57511yescharacteristic
212C2×C191#57611yescharacteristic
213D191—42no—
214D191—81no—
215C191#327311yescharacteristic, complemented, Sylow
216#3273—1911nomaximal, Sylow
217C2×AGL(1,ℤ/8)—1911no—
218C2×AGL(1,ℤ/8)—1911no—
219#546—1911no—
220#546D19111yescharacteristic, complemented
221#495—1911no—
222#468—1911no—
223#468—1911no—
224#463—1911no—
225#463—1911no—
226#452—1911no—
227#452—1911no—
228#447—1911no—
229#447—1911no—
230C4×D8—1911no—
231#436—1911no—
232C2×#41—1911no—
233C2×#41C2×D19111yescharacteristic
234C22×D4—1911no—
235C22×D4C2×D19111yescharacteristic, complemented
236AGL(1,ℤ/8)—3824no—
237AGL(1,ℤ/8)—3824no—
238C2×Q16—1911no—
239C2×SD16—1911no—
240C2×SD16—1911no—
241C2×SD16—1911no—
242C2×SD16—1911no—
243C2×D8—1911no—
244C2×D8—1911no—
245C2×D8—1911no—
246C2×D8—1911no—
247C2×M16—1911no—
248C2×M16—1911no—
249#124—1911no—
250#124C2×D19111yescharacteristic, complemented
251#123—21no—
252#123—3821no—
253C22⋊Q8—1911no—
254C22⋊Q8C2×D19111yescharacteristic, complemented
255#121—21no—
256#121—3821no—
257#121—1911no—
258#121C2×D19111yescharacteristic
259#120—21no—
260#120—3821no—
261C4×D4—1911no—
262C4×D4C2×D19111yescharacteristic
263C4×D4C2×D19111yescharacteristic, complemented
264C2×C22⋊C4—21no—
265C2×C22⋊C4—3821no—
266#107—1911no—
267C4⋊C8—1911no—
268#103—1911no—
269#103—1911no—
270#102—1911no—
271#102—1911no—
272#102—1911no—
273#102—1911no—
274C22⋊C8—1911no—
275C22⋊C8—1911no—
276C4×C8—1911no—
277C24—21no—
278C24—3821no—
279#41—22no—
280#41—41no—
281#41—3822no—
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—3822no—
291C2×D4—3821no—
292C2×D4—3821no—
293C2×D4—3821no—
294C2×D4—1911no—
295C2×D4—1911no—
296C2×D4—7641no—
297C2×D4C22⋊D19111yescharacteristic
298C2×D4C22×D19111yescharacteristic
299C2×D4C22×D19111yescharacteristic
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—3821no—
303C22×C4—3821no—
304C22×C4C22×D19111yescharacteristic
305C22×C4C22×D19111yescharacteristic
306Q16—7641no—
307SD16—3822no—
308SD16—3822no—
309SD16—3822no—
310SD16—3822no—
311D8—3822no—
312D8—3822no—
313D8—1914no—
314D8—7641no—
315M16—3822no—
316M16—3822no—
317C2×C8—1911no—
318C2×C8—1911no—
319C2×C8—1911no—
320C2×C8—1911no—
321C4⋊C4—21no—
322C4⋊C4—3821no—
323C4⋊C4—1911no—
324C4⋊C4C22⋊D19111yescharacteristic
325C4⋊C4C22×D19111yescharacteristic
326C22⋊C4—21no—
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—21no—
330C22⋊C4—21no—
331C22⋊C4—3821no—
332C22⋊C4—3821no—
333C22⋊C4—3821no—
334C22⋊C4—3821no—
335C22⋊C4—3821no—
336C22⋊C4C22⋊D19111yescharacteristic
337C22⋊C4C22⋊D19111yescharacteristic
338C42C22×D19111yescharacteristic
339C23—21no—
340C23—21no—
341C23—42no—
342C23—42no—
343C23—81no—
344C23—3821no—
345C23—3821no—
346C23—15281no—
347C23—7642no—
348C23—7642no—
349C23C2×C22⋊D19111yescharacteristic
350C23C2×C22⋊D19111yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—3821no—
354Q8—3821no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—41no—
359D4—42no—
360D4—42no—
361D4—81no—
362D4—3821no—
363D4—3821no—
364D4—3821no—
365D4—1912no—
366D4—15281no—
367D4—7642no—
368D4—7642no—
369D4D4×D19112yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—42no—
377C2×C4—81no—
378C2×C4—3821no—
379C2×C4—3821no—
380C2×C4—15281no—
381C2×C4—7642no—
382C2×C4C2×C22⋊D19111yescharacteristic
383C2×C4C2×C22⋊D19111yescharacteristic
384C2×C4C2×C22⋊D19111yescharacteristic
385C2×C4C2×C22⋊D19111yescharacteristic
386C2×C4C23×D19111yescharacteristic
387C8—3821no—
388C8—3821no—
389C8—3821no—
390C8—1912no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—22no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—84no—
404C22—3822no—
405C22—15284no—
406C22—7641no—
407C22—7641no—
408C22—7641no—
409C22—7641no—
410C22—7641no—
411C22C22×C22⋊D19111yescharacteristic
412C4—21no—
413C4—21no—
414C4—21no—
415C4—21no—
416C4—41no—
417C4—41no—
418C4—7641no—
419C4—7641no—
420C4C2×D4×D19111yescharacteristic
421C4#2025936711yescharacteristic
422C2—41no—
423C2—41no—
424C2—42no—
425C2—81no—
426C2—15281no—
427C2—7642no—
428C2D4×C22⋊D19111yescharacteristic
429C2#2713346111yescharacteristic
430C2#2713346611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#546×C191C2
socleC22×C191C22×D4
hypercenter#546D191
hypocenterC191#3273
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×#20259347C2semidirect
3. #27133309C2semidirect
4. #27133045C2semidirect
5. #27132911C2semidirect
6. #27132907C2semidirect
7. #27132878C2semidirect
8. #27132875C2semidirect
9. #27132825C2semidirect
10. #27132823C2semidirect
11. #27132797C2semidirect
12. #27132796C2semidirect
13. C4×#15690272C2semidirect
14. #27132741C2semidirect
15. #546×C191C2semidirect
17. C22×D764C22semidirect
20. C2×#15690277C22semidirect
21. C2×#15690275C22semidirect
22. C2×#15690275C22semidirect
23. C2×#15690274C22semidirect
24. C2×#15690274C22semidirect
29. C2×#15690271C22semidirect
30. C2×#15690271C22semidirect
31. #20259310C22semidirect
33. #20259301C22semidirect
37. C4×D764C22semidirect
39. #20259255C22semidirect
40. C4⋊(C191⋊C8)C22semidirect
41. #20259246C22semidirect
42. #20259245C22semidirect
43. #20259243C22semidirect
44. #20259243C22semidirect
45. #20259242C22semidirect
46. #20259242C22semidirect
47. C22⋊(C191⋊C8)C22semidirect
48. C22⋊(C191⋊C8)C22semidirect
49. C4×C191⋊C8C22semidirect
51. C22×D4×C191C22semidirect
52. #124×C191C22semidirect
54. C22⋊Q8×C191C22semidirect
59. C4×D4×C191C22semidirect
82. #15690272D4semidirect
215. C191#3273semidirect
220. #546D191semidirect
235. C22×D4C2×D191semidirect
250. #124C2×D191semidirect
254. C22⋊Q8C2×D191semidirect
263. C4×D4C2×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593482
C2×#202593472
#271333092
#271330452
#271329112
#271329072
#271328782
#271328752
#271328252
#271328232
#271327972
#271327962
C4×#156902722
#271327412
#546×C1912
#3273191