Finite Group Databasebeta

← #37584008

Subgroups — #37584008

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×#20259349C211yesmaximal, characteristic, complemented
2C22×#15690274C211yesmaximal, characteristic, complemented
3#27133324C211yesmaximal, characteristic, complemented
4#27133297C211yesmaximal, characteristic, complemented
5#27133049C211yesmaximal, characteristic, complemented
6#27132912C211yesmaximal, characteristic, complemented
7#27132897C211yesmaximal, characteristic, complemented
8#27132881C211yesmaximal, characteristic, complemented
9#27132861C211yesmaximal, characteristic, complemented
10#27132812C211yesmaximal, characteristic, complemented
11#27132811C211yesmaximal, characteristic, complemented
12#27132799C211yesmaximal, characteristic, complemented
13#27132766C211yesmaximal, characteristic, complemented
14#27132731C211yesmaximal, characteristic, complemented
15#448×C191C211yesmaximal, characteristic, complemented
16C2×#15690288C2211yescharacteristic, complemented
17C22×D764C2211yescharacteristic
18C2×D4×D191—21no—
19C2×D4×D191C2211yescharacteristic
20#20259349—24no—
21C2×#15690274—24no—
22C2×#15690274C2211yescharacteristic
23C2×#15690274C2211yescharacteristic
24C2×#15690274C2211yescharacteristic, complemented
25C2×#15690274C2211yescharacteristic, complemented
26C2×D1528C2211yescharacteristic
27C2×D1528C2211yescharacteristic, complemented
28C2×#15690272C2211yescharacteristic, complemented
29C2×#15690270C2211yescharacteristic, complemented
30C22×C191⋊C8C2211yescharacteristic, complemented
31#20259318C2211yescharacteristic
32#20259311—21no—
33#20259311C2211yescharacteristic, complemented
34#20259298—21no—
35#20259298C2211yescharacteristic, complemented
36#20259297C2211yescharacteristic
37#20259296—21no—
38#20259296C2211yescharacteristic, complemented
39#20259295—21no—
40#20259292—21no—
41#20259290—21no—
42Q8×Dic191C2211yescharacteristic, complemented
43#20259284C2211yescharacteristic
44C4×C22⋊D191—21no—
45C4×C22⋊D191C2211yescharacteristic
46#20259255C2211yescharacteristic, complemented
47Dic191⋊C8C2211yescharacteristic, complemented
48#20259244C2211yescharacteristic
49#20259244C2211yescharacteristic, complemented
50#20259243C2211yescharacteristic, complemented
51#20259242C2211yescharacteristic, complemented
52(C2×D191)⋊C8C2211yescharacteristic, complemented
53#20259232C2211yescharacteristic, complemented
54C22×Q8×C191C2211yescharacteristic, complemented
55C2×SD16×C191C2211yescharacteristic, complemented
56C2×SD16×C191C2211yescharacteristic, complemented
57#121×C191C2211yescharacteristic
58#103×C191C2211yescharacteristic, complemented
59#103×C191C2211yescharacteristic, complemented
60C22⋊C8×C191C2211yescharacteristic, complemented
61C23×D191—21no—
62C23×D191—21no—
63#15690288—22no—
64#15690288—41no—
65C2×D764—22no—
66C2×D764—21no—
67C2×D764—21no—
68C2×D764—21no—
69C2×D764—41no—
70C2×D764C2311yescharacteristic
71C2×D764C2311yescharacteristic
72C2×D764C2311yescharacteristic
73C2×C22⋊D191—21no—
74C2×C22⋊D191—21no—
75C2×C22⋊D191—21no—
76C2×C22⋊D191—21no—
77C2×C22⋊D191D411yescharacteristic
78D4×D191—22no—
79D4×D191—41no—
80D4×D191—44no—
81C2×C4×D191—21no—
82C2×C4×D191—21no—
83C2×C4×D191C2311yescharacteristic
84#15690274—22no—
85#15690274—22no—
86#15690274—22no—
87#15690274D414yescomplemented
88D1528—22no—
89D1528—41no—
90#15690272—22no—
91#15690270—22no—
92C2×C191⋊C8—22no—
93C2×C191⋊C8C2311yescharacteristic
94C2×C191⋊C8C2311yescharacteristic
95C4⋊Dic191—21no—
96C4⋊Dic191C2311yescharacteristic
97Dic191⋊C4—21no—
98Dic191⋊C4D411yescharacteristic
99C22⋊Dic191D411yescharacteristic
100(C2×D191)⋊C4—21no—
101(C2×D191)⋊C4—21no—
102(C2×D191)⋊C4—21no—
103(C2×D191)⋊C4—21no—
104(C2×D191)⋊C4—21no—
105(C2×D191)⋊C4—21no—
106(C2×D191)⋊C4D411yescharacteristic
107C4×Dic191—21no—
108C4×Dic191C2311yescharacteristic
109C2×Q8×C191—21no—
110C2×Q8×C191—22no—
111C2×Q8×C191—41no—
112C2×Q8×C191C2311yescharacteristic
113C2×Q8×C191C2311yescharacteristic
114C2×D4×C191—21no—
115C2×D4×C191C2311yescharacteristic
116C22×C4×C191—21no—
117C22×C4×C191C2311yescharacteristic
118SD16×C191—22no—
119SD16×C191—41no—
120C2×C8×C191C2311yescharacteristic
121C2×C8×C191C2311yescharacteristic
122C4⋊C4×C191C2311yescharacteristic
123C22⋊C4×C191—21no—
124C22×D191—21no—
125C22×D191—21no—
126C22×D191—21no—
127C22×D191—42no—
128C22×D191—42no—
129C22×D191—42no—
130C22×D191—42no—
131C22×D191—81no—
132C22×D191—81no—
133C22×D191C2×D411yescharacteristic
134D764—21no—
135D764—21no—
136D764—21no—
137D764—21no—
138D764—21no—
139D764—42no—
140D764—42no—
141D764—42no—
142D764C2×D412yes—
143C22⋊D191—22no—
144C22⋊D191—42no—
145C22⋊D191—42no—
146C22⋊D191—42no—
147C22⋊D191—81no—
148C2×Dic191—21no—
149C2×Dic191C2×D411yescharacteristic
150C2×Dic191C2×D411yescharacteristic
151C2×Dic191C2×D411yescharacteristic
152C4×D191—21no—
153C4×D191—21no—
154C4×D191—42no—
155C4×D191—42no—
156C191⋊C8—21no—
157C191⋊C8C2×D412yescomplemented
158C23×C191—21no—
159C23×C191C2×D411yescharacteristic
160Q8×C191—21no—
161Q8×C191—21no—
162Q8×C191—21no—
163Q8×C191—42no—
164Q8×C191C2×D412yescomplemented
165D4×C191—21no—
166D4×C191—21no—
167D4×C191—42no—
168C2×C4×C191—21no—
169C2×C4×C191—21no—
170C2×C4×C191—21no—
171C2×C4×C191—21no—
172C2×C4×C191—21no—
173C2×C4×C191—42no—
174C2×C4×C191C2×D411yescharacteristic
175C2×C4×C191C2411yescharacteristic
176C8×C191—21no—
177C8×C191—21no—
178C22×C191—21no—
179C22×C191—21no—
180C22×C191—41no—
181C22×C191—41no—
182C22×C191—41no—
183C22×C191AGL(1,ℤ/8)12yescomplemented
184C22×C191C22×D411yescharacteristic
185C2×D191—21no—
186C2×D191—22no—
187C2×D191—21no—
188C2×D191—21no—
189C2×D191—41no—
190C2×D191—41no—
191C2×D191—41no—
192C2×D191—41no—
193C2×D191—41no—
194C2×D191—41no—
195C2×D191—41no—
196C2×D191—41no—
197C2×D191—84no—
198C2×D191—84no—
199C4×C191—21no—
200C4×C191—22no—
201C4×C191—41no—
202C4×C191—41no—
203C4×C191C22×D411yescharacteristic
204C4×C191H(5,𝔽2)11yescharacteristic
205Dic191—21no—
206Dic191—21no—
207Dic191—21no—
208Dic191—41no—
209C2×C191—22no—
210C2×C191—81no—
211C2×C191D4211yescharacteristic
212C2×C191C2×AGL(1,ℤ/8)11yescharacteristic
213C2×C191#57711yescharacteristic
214D191—41no—
215D191—42no—
216D191—81no—
217D191—81no—
218C191#326511yescharacteristic, complemented, Sylow
219#3265—1911nomaximal, Sylow
220C2×AGL(1,ℤ/8)—1911no—
221C22×SD16—1911no—
222#548—1911no—
223D42—1911no—
224#496—1911no—
225#468—1911no—
226#466—1911no—
227#463—1911no—
228#460—1911no—
229#450—1911no—
230#449—1911no—
231#448D19111yescharacteristic, complemented
232#447—1911no—
233#440—1911no—
234#435—1911no—
235C2×#41—1911no—
236C22×Q8C2×D19111yescharacteristic, complemented
237C22×D4—3821no—
238C22×D4—1911no—
239C22×D4—1911no—
240AGL(1,ℤ/8)—3824no—
241C2×SD16—3824no—
242C2×SD16—1911no—
243C2×SD16—1911no—
244C2×SD16—1911no—
245C2×SD16—1911no—
246C2×SD16C2×D19111yescharacteristic, complemented
247C2×SD16C2×D19111yescharacteristic, complemented
248C2×D8—1911no—
249C2×D8—1911no—
250C2×D8—1911no—
251C2×M16—1911no—
252C22×C8—1911no—
253#127—1911no—
254#124—3821no—
255#124—1911no—
256C22⋊Q8—3821no—
257C22⋊Q8—1911no—
258#121—3821no—
259#121—3821no—
260#121—1911no—
261#121—1911no—
262#121C2×D19111yescharacteristic
263#120—3821no—
264#120—3821no—
265C4×Q8—1911no—
266C4×D4—3821no—
267C4×D4—1911no—
268C4×D4—1911no—
269#107—1911no—
270C4⋊C8—1911no—
271#103C2×D19111yescharacteristic, complemented
272#103C2×D19111yescharacteristic, complemented
273#102—1911no—
274#102—1911no—
275#102—1911no—
276#102—1911no—
277C22⋊C8—1911no—
278C22⋊C8C2×D19111yescharacteristic, complemented
279#97—1911no—
280C24—3821no—
281C24—3821no—
282#41—3822no—
283#41—7641no—
284C2×Q8—21no—
285C2×Q8—22no—
286C2×Q8—41no—
287C2×Q8C22×D19111yescharacteristic
288C2×Q8C22×D19111yescharacteristic
289C2×D4—21no—
290C2×D4—3822no—
291C2×D4—3821no—
292C2×D4—3821no—
293C2×D4—3822no—
294C2×D4—3821no—
295C2×D4—3821no—
296C2×D4—3821no—
297C2×D4—3821no—
298C2×D4—3821no—
299C2×D4—1911no—
300C2×D4—1911no—
301C2×D4—1911no—
302C2×D4—1911no—
303C2×D4—7641no—
304C2×D4—7641no—
305C2×D4—7644no—
306C2×D4C22×D19111yescharacteristic
307C22×C4—21no—
308C22×C4—3821no—
309C22×C4—3821no—
310C22×C4—1911no—
311C22×C4C22×D19111yescharacteristic
312SD16—22no—
313SD16—41no—
314SD16—3822no—
315SD16—3822no—
316SD16—3822no—
317SD16—1914no—
318D8—3822no—
319D8—3822no—
320D8—7641no—
321M16—3822no—
322C2×C8—3822no—
323C2×C8—1911no—
324C2×C8—1911no—
325C2×C8C22×D19111yescharacteristic
326C2×C8C22×D19111yescharacteristic
327C4⋊C4—3821no—
328C4⋊C4—3821no—
329C4⋊C4—1911no—
330C4⋊C4—1911no—
331C4⋊C4C22×D19111yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—3821no—
334C22⋊C4—3821no—
335C22⋊C4—3821no—
336C22⋊C4—3821no—
337C22⋊C4—3821no—
338C22⋊C4—3821no—
339C22⋊C4—1911no—
340C22⋊C4—1911no—
341C42—3821no—
342C42—1911no—
343C23—21no—
344C23—3821no—
345C23—3821no—
346C23—3821no—
347C23—1911no—
348C23—15281no—
349C23—15281no—
350C23—7642no—
351C23—7642no—
352C23—7642no—
353C23—7642no—
354C23D4×D19111yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—21no—
358Q8—42no—
359Q8D4×D19112yescomplemented
360D4—21no—
361D4—21no—
362D4—42no—
363D4—3822no—
364D4—3821no—
365D4—3821no—
366D4—3821no—
367D4—3821no—
368D4—3821no—
369D4—1912no—
370D4—15281no—
371D4—7642no—
372D4—7642no—
373D4—7642no—
374D4—7642no—
375D4—7642no—
376D4—7642no—
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—1911no—
387C2×C4—1911no—
388C2×C4—1911no—
389C2×C4—7642no—
390C2×C4—7642no—
391C2×C4D4×D19111yescharacteristic
392C2×C4C23×D19111yescharacteristic
393C8—21no—
394C8—21no—
395C8—3821no—
396C8—1912no—
397C22—21no—
398C22—21no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—3821no—
403C22—3821no—
404C22—3821no—
405C22—3822no—
406C22—15284no—
407C22—15284no—
408C22—7641no—
409C22—7641no—
410C22—7641no—
411C22—7641no—
412C22—7641no—
413C22—7641no—
414C22—7641no—
415C22—7641no—
416C22#2025934912yescomplemented
417C22C2×D4×D19111yescharacteristic
418C4—21no—
419C4—22no—
420C4—41no—
421C4—41no—
422C4—3821no—
423C4—3821no—
424C4—3821no—
425C4—7641no—
426C4C2×D4×D19111yescharacteristic
427C4#2025936711yescharacteristic
428C2—22no—
429C2—81no—
430C2—15281no—
431C2—15281no—
432C2—7641no—
433C2—7642no—
434C2#2713329511yescharacteristic
435C2C2×#2025934911yescharacteristic
436C2#2713347211yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#448×C191C2
socleC22×C191C22×D4
hypercenter#448D191
hypocenterC191#3265
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C2×#20259349C2semidirect
2. C22×#15690274C2semidirect
3. #27133324C2semidirect
4. #27133297C2semidirect
5. #27133049C2semidirect
6. #27132912C2semidirect
7. #27132897C2semidirect
8. #27132881C2semidirect
9. #27132861C2semidirect
10. #27132812C2semidirect
11. #27132811C2semidirect
12. #27132799C2semidirect
13. #27132766C2semidirect
14. #27132731C2semidirect
15. #448×C191C2semidirect
16. C2×#15690288C22semidirect
24. C2×#15690274C22semidirect
25. C2×#15690274C22semidirect
27. C2×D1528C22semidirect
28. C2×#15690272C22semidirect
29. C2×#15690270C22semidirect
30. C22×C191⋊C8C22semidirect
33. #20259311C22semidirect
35. #20259298C22semidirect
38. #20259296C22semidirect
42. Q8×Dic191C22semidirect
46. #20259255C22semidirect
47. Dic191⋊C8C22semidirect
49. #20259244C22semidirect
50. #20259243C22semidirect
51. #20259242C22semidirect
52. (C2×D191)⋊C8C22semidirect
53. #20259232C22semidirect
54. C22×Q8×C191C22semidirect
55. C2×SD16×C191C22semidirect
56. C2×SD16×C191C22semidirect
58. #103×C191C22semidirect
59. #103×C191C22semidirect
60. C22⋊C8×C191C22semidirect
87. #15690274D4semidirect
157. C191⋊C8C2×D4semidirect
164. Q8×C191C2×D4semidirect
183. C22×C191AGL(1,ℤ/8)semidirect
218. C191#3265semidirect
231. #448D191semidirect
236. C22×Q8C2×D191semidirect
246. C2×SD16C2×D191semidirect
247. C2×SD16C2×D191semidirect
271. #103C2×D191semidirect
272. #103C2×D191semidirect
278. C22⋊C8C2×D191semidirect
359. Q8D4×D191semidirect
416. C22#20259349semidirect

Maximal subgroups

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

subgroupindex
C2×#202593492
C22×#156902742
#271333242
#271332972
#271330492
#271329122
#271328972
#271328812
#271328612
#271328122
#271328112
#271327992
#271327662
#271327312
#448×C1912
#3265191