Finite Group Databasebeta

← #37581803

Subgroups — #37581803

Order 24448.

732 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20259369C211yesmaximal, characteristic, complemented
2C2×#20259346C211yesmaximal, characteristic, complemented
3C2×SD16×D191C211yesmaximal, characteristic, complemented
4#27133229C211yesmaximal, characteristic, complemented
5#27132809C211yesmaximal, characteristic, complemented
6#27132808C211yesmaximal, characteristic, complemented
7#27132805C211yesmaximal, characteristic, complemented
8#27132804C211yesmaximal, characteristic, complemented
9#27132803C211yesmaximal, characteristic, complemented
10#27132801C211yesmaximal, characteristic, complemented
11#27132800C211yesmaximal, characteristic, complemented
12#27132670C211yesmaximal, characteristic, complemented
13#27132656C211yesmaximal, characteristic, complemented
14#27132628C211yesmaximal, characteristic, complemented
15#449×C191C211yesmaximal, characteristic, complemented
16#20259369—24no—
17#20259369—24no—
18C2×#15690287—21no—
19C2×#15690287C2211yescharacteristic
20C2×#15690286—21no—
21C2×#15690286—21no—
22C2×#15690286C2211yescharacteristic
23C2×#15690286C2211yescharacteristic, complemented
24C22×Dic382—21no—
25C22×Dic382C2211yescharacteristic, complemented
26C2×Q8×D191C2211yescharacteristic
27C2×Q8×D191C2211yescharacteristic, complemented
28C2×D4×D191C2211yescharacteristic
29#20259346—24no—
30C2×#15690276C2211yescharacteristic
31C2×#15690276C2211yescharacteristic
32C2×#15690275C2211yescharacteristic
33C2×#15690275C2211yescharacteristic
34C2×#15690274C2211yescharacteristic
35SD16×D191—24no—
36C2×#15690272C2211yescharacteristic
37C2×#15690270C2211yescharacteristic
38C2×C8×D191C2211yescharacteristic
39#20259318—21no—
40#20259308—21no—
41#20259297C2211yescharacteristic
42#20259296C2211yescharacteristic
43#20259293C2211yescharacteristic
44#20259292—21no—
45#20259290—21no—
46#20259280C2211yescharacteristic
47#20259247C2211yescharacteristic, complemented
48#20259247C2211yescharacteristic, complemented
49#20259246C2211yescharacteristic, complemented
50#20259245C2211yescharacteristic, complemented
51#20259245C2211yescharacteristic, complemented
52#20259243C2211yescharacteristic, complemented
53C22⋊(C191⋊C8)C2211yescharacteristic, complemented
54(C2×D191)⋊C8C2211yescharacteristic, complemented
55(C2×D191)⋊C8C2211yescharacteristic, complemented
56C2×#41×C191C2211yescharacteristic, complemented
57C2×SD16×C191C2211yescharacteristic
58C2×D8×C191C2211yescharacteristic
59#121×C191C2211yescharacteristic
60#103×C191C2211yescharacteristic, complemented
61#102×C191C2211yescharacteristic, complemented
62C22⋊C8×C191C2211yescharacteristic, complemented
63C23×D191—21no—
64#15690287—28no—
65#15690287—22no—
66#15690287—22no—
67#15690286—28no—
68#15690286—28no—
69#15690286—22no—
70#15690286—22no—
71#15690286—22no—
72#15690286D414yescomplemented
73C2×Dic382—28no—
74C2×Dic382—22no—
75C2×Dic382—22no—
76C2×Dic382—21no—
77C2×Dic382—21no—
78C2×Dic382—21no—
79C2×Dic382D411yescharacteristic
80C2×Dic382C2311yescharacteristic
81C2×Dic382C2311yescharacteristic
82Q8×D191—22no—
83Q8×D191—22no—
84Q8×D191—22no—
85Q8×D191D414yescomplemented
86C2×D764—21no—
87C2×D764D411yescharacteristic
88C2×D764C2311yescharacteristic
89C2×C22⋊D191—21no—
90C2×C22⋊D191—21no—
91C2×C22⋊D191—21no—
92C2×C22⋊D191—21no—
93C2×C22⋊D191—21no—
94C2×C22⋊D191D411yescharacteristic
95C2×C22⋊D191D411yescharacteristic
96D4×D191—22no—
97D4×D191—22no—
98C22×Dic191—21no—
99C22×Dic191—21no—
100C22×Dic191—21no—
101C2×C4×D191—21no—
102C2×C4×D191—21no—
103C2×C4×D191C2311yescharacteristic
104C2×C4×D191C2311yescharacteristic
105#15690276—22no—
106#15690276—22no—
107#15690275—22no—
108#15690275—22no—
109#15690274—22no—
110#15690272—22no—
111#15690270—22no—
112C2×C191⋊C8C2311yescharacteristic
113C2×C191⋊C8C2311yescharacteristic
114C8×D191—22no—
115C4⋊Dic191C2311yescharacteristic
116C22⋊Dic191—21no—
117(C2×D191)⋊C4—21no—
118(C2×D191)⋊C4—21no—
119(C2×D191)⋊C4—21no—
120C4×Dic191—21no—
121#41×C191—22no—
122#41×C191—22no—
123C2×Q8×C191C2311yescharacteristic
124C2×D4×C191—21no—
125C2×D4×C191—21no—
126C2×D4×C191C2311yescharacteristic
127C2×D4×C191C2311yescharacteristic
128C22×C4×C191—21no—
129C22×C4×C191C2311yescharacteristic
130SD16×C191—22no—
131D8×C191—22no—
132C2×C8×C191C2311yescharacteristic
133C2×C8×C191C2311yescharacteristic
134C4⋊C4×C191C2311yescharacteristic
135C22⋊C4×C191—21no—
136C22×D191—21no—
137C22×D191—42no—
138C22×D191—81no—
139C22×D191—81no—
140C22×D191C2×D411yescharacteristic
141C22×D191C2×D411yescharacteristic
142Dic382—24no—
143Dic382—24no—
144Dic382—21no—
145Dic382—21no—
146Dic382—24no—
147Dic382—22no—
148Dic382C2×D412yes—
149Dic382C2×D412yes—
150D764—22no—
151D764—21no—
152D764—21no—
153D764—81no—
154C22⋊D191—24no—
155C22⋊D191—24no—
156C22⋊D191—22no—
157C22⋊D191—22no—
158C22⋊D191—42no—
159C22⋊D191—81no—
160C22⋊D191—81no—
161C2×Dic191—24no—
162C2×Dic191—24no—
163C2×Dic191—21no—
164C2×Dic191—21no—
165C2×Dic191—24no—
166C2×Dic191—21no—
167C2×Dic191C2×D411yescharacteristic
168C2×Dic191C2×D411yescharacteristic
169C4×D191—24no—
170C4×D191—24no—
171C4×D191—21no—
172C4×D191—21no—
173C4×D191C2×D412yes—
174C4×D191C2×D412yes—
175C191⋊C8—21no—
176C191⋊C8—21no—
177C23×C191—21no—
178C23×C191—21no—
179C23×C191C2×D411yescharacteristic
180Q8×C191—21no—
181Q8×C191C2×D412yes—
182D4×C191—24no—
183D4×C191—21no—
184D4×C191—21no—
185D4×C191—21no—
186D4×C191—81no—
187D4×C191C2×D412yes—
188C2×C4×C191—24no—
189C2×C4×C191—21no—
190C2×C4×C191—21no—
191C2×C4×C191—21no—
192C2×C4×C191—21no—
193C2×C4×C191C2×D411yescharacteristic
194C2×C4×C191C2411yescharacteristic
195C8×C191—21no—
196C8×C191—21no—
197C22×C191—21no—
198C22×C191—22no—
199C22×C191—21no—
200C22×C191—21no—
201C22×C191—41no—
202C22×C191—41no—
203C22×C191—41no—
204C22×C191—41no—
205C22×C191—41no—
206C22×C191C22×D411yescharacteristic
207C2×D191—21no—
208C2×D191—21no—
209C2×D191—21no—
210C2×D191—21no—
211C2×D191—21no—
212C2×D191—41no—
213C2×D191—41no—
214C2×D191—41no—
215C2×D191—84no—
216C2×D191#12012yes—
217C4×C191—21no—
218C4×C191—22no—
219C4×C191—41no—
220C4×C191C22×D411yescharacteristic
221C4×C191C22×D411yescharacteristic
222Dic191—22no—
223Dic191—22no—
224Dic191—21no—
225Dic191—41no—
226Dic191#12012yes—
227C2×C191—41no—
228C2×C191—42no—
229C2×C191—81no—
230C2×C191C2×#12011yescharacteristic
231C2×C191#57711yescharacteristic
232C2×C191#57711yescharacteristic
233D191—22no—
234D191—41no—
235D191—81no—
236C191#301011yescharacteristic, complemented, Sylow
237#3010—1911nomaximal, Sylow
238C2×2−1+4—1911no—
239C2×AGL(1,ℤ/8)—1911no—
240C22×SD16—1911no—
241#535—1911no—
242#449—1911no—
243#449—1911no—
244#449—1911no—
245#449D19111yescharacteristic, complemented
246#448—1911no—
247#448—1911no—
248#448—1911no—
249#448—1911no—
250#419—1911no—
251#416—1911no—
252#408—1911no—
2532−1+4—3824no—
2542−1+4—3824no—
255C2×#41—3821no—
256C2×#41—3821no—
257C2×#41—3821no—
258C2×#41—1911no—
259C2×#41—1911no—
260C2×#41—1911no—
261C2×#41C2×D19111yescharacteristic, complemented
262C22×Q8—3821no—
263C22×Q8—1911no—
264C22×Q8—1911no—
265C22×Q8—1911no—
266C22×D4—1911no—
267AGL(1,ℤ/8)—3824no—
268C2×SD16—3824no—
269C2×SD16—1911no—
270C2×SD16—1911no—
271C2×SD16—1911no—
272C2×SD16—1911no—
273C2×SD16—1911no—
274C2×SD16C2×D19111yescharacteristic
275C2×D8—1911no—
276C2×D8C2×D19111yescharacteristic
277C2×M16—1911no—
278C22×C8—1911no—
279#127—3821no—
280#124—3821no—
281#121—1911no—
282#121—1911no—
283#121—1911no—
284#121C2×D19111yescharacteristic
285#120—3821no—
286#120—3821no—
287#117—1911no—
288#103—1911no—
289#103—1911no—
290#103—1911no—
291#103—1911no—
292#103—1911no—
293#103C2×D19111yescharacteristic, complemented
294#102—1911no—
295#102C2×D19111yescharacteristic, complemented
296C22⋊C8—1911no—
297C22⋊C8—1911no—
298C22⋊C8—1911no—
299C22⋊C8C2×D19111yescharacteristic, complemented
300C24—3821no—
301#41—22no—
302#41—22no—
303#41—3828no—
304#41—3822no—
305#41—3822no—
306#41—3828no—
307#41—3828no—
308#41—3822no—
309#41—3822no—
310#41—3822no—
311#41—1914no—
312C2×Q8—3828no—
313C2×Q8—3822no—
314C2×Q8—3822no—
315C2×Q8—3821no—
316C2×Q8—3821no—
317C2×Q8—3822no—
318C2×Q8—3822no—
319C2×Q8—3822no—
320C2×Q8—3821no—
321C2×Q8—1914no—
322C2×Q8—1911no—
323C2×Q8—1911no—
324C2×Q8—1911no—
325C2×Q8C22×D19111yescharacteristic
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—3821no—
329C2×D4—3821no—
330C2×D4—3821no—
331C2×D4—3821no—
332C2×D4—3821no—
333C2×D4—3822no—
334C2×D4—3822no—
335C2×D4—3821no—
336C2×D4—1911no—
337C2×D4—1911no—
338C2×D4—1911no—
339C2×D4—1911no—
340C2×D4C22×D19111yescharacteristic
341C2×D4C22×D19111yescharacteristic
342C22×C4—21no—
343C22×C4—3821no—
344C22×C4—3821no—
345C22×C4—3821no—
346C22×C4—3821no—
347C22×C4—3821no—
348C22×C4—1911no—
349C22×C4—1911no—
350C22×C4C22×D19111yescharacteristic
351SD16—22no—
352SD16—3822no—
353SD16—3822no—
354SD16—3822no—
355SD16—3822no—
356SD16—3822no—
357D8—22no—
358D8—3822no—
359M16—3822no—
360C2×C8—3822no—
361C2×C8—1911no—
362C2×C8—1911no—
363C2×C8C22×D19111yescharacteristic
364C2×C8C22×D19111yescharacteristic
365C4⋊C4—1911no—
366C4⋊C4C22×D19111yescharacteristic
367C22⋊C4—21no—
368C22⋊C4—3821no—
369C22⋊C4—3821no—
370C22⋊C4—3821no—
371C22⋊C4—3821no—
372C42—3821no—
373C23—21no—
374C23—21no—
375C23—3821no—
376C23—1911no—
377C23—1911no—
378C23—15281no—
379C23—15281no—
380C23—7642no—
381C23D4×D19111yescharacteristic
382Q8—21no—
383Q8—3824no—
384Q8—3824no—
385Q8—3822no—
386Q8—3824no—
387Q8—3821no—
388Q8—3821no—
389Q8—1912no—
390Q8—1912no—
391Q8D4×D19112yes—
392D4—24no—
393D4—21no—
394D4—21no—
395D4—21no—
396D4—81no—
397D4—3822no—
398D4—3824no—
399D4—3824no—
400D4—3822no—
401D4—3822no—
402D4—3821no—
403D4—3821no—
404D4—15281no—
405D4—15281no—
406D4—15281no—
407D4—7642no—
408D4D4×D19112yes—
409C2×C4—24no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—3824no—
415C2×C4—3824no—
416C2×C4—3824no—
417C2×C4—3824no—
418C2×C4—3821no—
419C2×C4—3821no—
420C2×C4—3824no—
421C2×C4—3821no—
422C2×C4—3821no—
423C2×C4—3821no—
424C2×C4—1911no—
425C2×C4—1911no—
426C2×C4—1912no—
427C2×C4—1912no—
428C2×C4D4×D19111yescharacteristic
429C2×C4C23×D19111yescharacteristic
430C8—21no—
431C8—21no—
432C8—3821no—
433C8—3821no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—22no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—3821no—
444C22—3821no—
445C22—3821no—
446C22—3821no—
447C22—3821no—
448C22—1912no—
449C22—15284no—
450C22—7641no—
451C22—7641no—
452C22—7641no—
453C22C2×D4×D19111yescharacteristic
454C4—21no—
455C4—22no—
456C4—41no—
457C4—3822no—
458C4—3822no—
459C4—3821no—
460C4—1912no—
461C4—7641no—
462C4C2×D4×D19111yescharacteristic
463C4C2×D4×D19111yescharacteristic
464C2—41no—
465C2—42no—
466C2—81no—
467C2—3822no—
468C2—15281no—
469C2—7641no—
470C2#120×D19111yescharacteristic
471C2#2713347011yescharacteristic
472C2#2713347311yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#449×C191C2
socleC22×C191C22×D4
hypercenter#449D191
hypocenterC191#3010
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#20259369C2semidirect
2. C2×#20259346C2semidirect
3. C2×SD16×D191C2semidirect
4. #27133229C2semidirect
5. #27132809C2semidirect
6. #27132808C2semidirect
7. #27132805C2semidirect
8. #27132804C2semidirect
9. #27132803C2semidirect
10. #27132801C2semidirect
11. #27132800C2semidirect
12. #27132670C2semidirect
13. #27132656C2semidirect
14. #27132628C2semidirect
15. #449×C191C2semidirect
23. C2×#15690286C22semidirect
25. C22×Dic382C22semidirect
27. C2×Q8×D191C22semidirect
47. #20259247C22semidirect
48. #20259247C22semidirect
49. #20259246C22semidirect
50. #20259245C22semidirect
51. #20259245C22semidirect
52. #20259243C22semidirect
53. C22⋊(C191⋊C8)C22semidirect
54. (C2×D191)⋊C8C22semidirect
55. (C2×D191)⋊C8C22semidirect
56. C2×#41×C191C22semidirect
60. #103×C191C22semidirect
61. #102×C191C22semidirect
62. C22⋊C8×C191C22semidirect
72. #15690286D4semidirect
85. Q8×D191D4semidirect
236. C191#3010semidirect
245. #449D191semidirect
261. C2×#41C2×D191semidirect
293. #103C2×D191semidirect
295. #102C2×D191semidirect
299. C22⋊C8C2×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593692
C2×#202593462
C2×SD16×D1912
#271332292
#271328092
#271328082
#271328052
#271328042
#271328032
#271328012
#271328002
#271326702
#271326562
#271326282
#449×C1912
#3010191