Finite Group Databasebeta

← #37584007

Subgroups — #37584007

Order 24448.

568 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20259350C211yesmaximal, characteristic, complemented
2C2×D8×D191C211yesmaximal, characteristic, complemented
3#27133310C211yesmaximal, characteristic, complemented
4#27133297C211yesmaximal, characteristic, complemented
5#27133051C211yesmaximal, characteristic, complemented
6#27132893C211yesmaximal, characteristic, complemented
7#27132877C211yesmaximal, characteristic, complemented
8#27132857C211yesmaximal, characteristic, complemented
9#27132815C211yesmaximal, characteristic, complemented
10#27132806C211yesmaximal, characteristic, complemented
11#27132798C211yesmaximal, characteristic, complemented
12#27132796C211yesmaximal, characteristic, complemented
13#27132774C211yesmaximal, characteristic, complemented
14#27132732C211yesmaximal, characteristic, complemented
15#469×C191C211yesmaximal, characteristic, complemented
16C2×#15690287C2211yescharacteristic, complemented
17C22×D764C2211yescharacteristic, complemented
18C2×D4×D191—21no—
19C2×D4×D191C2211yescharacteristic
20C2×D4×D191C2211yescharacteristic, complemented
21#20259350—24no—
22C2×#15690276C2211yescharacteristic, complemented
23C2×#15690276C2211yescharacteristic, complemented
24C2×#15690275C2211yescharacteristic, complemented
25C2×D1528C2211yescharacteristic
26C2×D1528C2211yescharacteristic, complemented
27C2×#15690272C2211yescharacteristic
28C2×#15690272C2211yescharacteristic, complemented
29C2×#15690272C2211yescharacteristic, complemented
30D8×D191—24no—
31C2×C8×D191C2211yescharacteristic, complemented
32#20259318C2211yescharacteristic, complemented
33#20259309C2211yescharacteristic, complemented
34#20259304—21no—
35#20259302C2211yescharacteristic, complemented
36#20259297C2211yescharacteristic
37#20259295—21no—
38#20259293—21no—
39#20259292—21no—
40#20259290—21no—
41#20259290—21no—
42#20259284C2211yescharacteristic, complemented
43#20259284C2211yescharacteristic, complemented
44C4×C22⋊D191C2211yescharacteristic
45C22⋊C4×D191—21no—
46Dic191⋊C8C2211yescharacteristic, complemented
47#20259245C2211yescharacteristic, complemented
48#20259242C2211yescharacteristic, complemented
49#20259242C2211yescharacteristic, complemented
50#20259242C2211yescharacteristic, complemented
51C22⋊(C191⋊C8)C2211yescharacteristic, complemented
52(C2×D191)⋊C8C2211yescharacteristic, complemented
53#20259232C2211yescharacteristic, complemented
54C2×D8×C191C2211yescharacteristic, complemented
55C2×M16×C191C2211yescharacteristic, complemented
56#121×C191C2211yescharacteristic, complemented
57#121×C191C2211yescharacteristic, complemented
58#106×C191C2211yescharacteristic, complemented
59#102×C191C2211yescharacteristic, complemented
60#102×C191C2211yescharacteristic, complemented
61C23×D191—21no—
62C23×D191—21no—
63C23×D191—21no—
64#15690287—22no—
65#15690287—41no—
66C2×Dic382C2311yescharacteristic
67C2×D764—22no—
68C2×D764—21no—
69C2×D764—21no—
70C2×D764—41no—
71C2×D764C2311yescharacteristic
72C2×D764C2311yescharacteristic
73C2×D764C2311yescharacteristic
74C2×C22⋊D191—21no—
75C2×C22⋊D191—21no—
76C2×C22⋊D191—21no—
77C2×C22⋊D191—21no—
78C2×C22⋊D191—21no—
79C2×C22⋊D191D411yescharacteristic
80D4×D191—22no—
81D4×D191—22no—
82D4×D191—41no—
83D4×D191—41no—
84D4×D191—44no—
85C2×C4×D191—21no—
86C2×C4×D191—21no—
87C2×C4×D191—21no—
88C2×C4×D191C2311yescharacteristic
89#15690276—22no—
90#15690276—22no—
91#15690275—41no—
92D1528—22no—
93D1528D414yescomplemented
94#15690272—22no—
95#15690272—22no—
96#15690272—41no—
97C2×C191⋊C8C2311yescharacteristic
98C2×C191⋊C8C2311yescharacteristic
99C8×D191—22no—
100C4⋊Dic191—21no—
101Dic191⋊C4D411yescharacteristic
102C22⋊Dic191—21no—
103C22⋊Dic191—21no—
104C22⋊Dic191D411yescharacteristic
105(C2×D191)⋊C4—21no—
106(C2×D191)⋊C4—21no—
107(C2×D191)⋊C4—21no—
108(C2×D191)⋊C4—21no—
109(C2×D191)⋊C4D411yescharacteristic
110C4×Dic191C2311yescharacteristic
111C2×D4×C191—21no—
112C2×D4×C191—21no—
113C2×D4×C191C2311yescharacteristic
114C2×D4×C191C2311yescharacteristic
115C22×C4×C191C2311yescharacteristic
116D8×C191—22no—
117M16×C191—22no—
118C2×C8×C191C2311yescharacteristic
119C2×C8×C191C2311yescharacteristic
120C4⋊C4×C191C2311yescharacteristic
121C4⋊C4×C191C2311yescharacteristic
122C22⋊C4×C191—21no—
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—42no—
132C22×D191—42no—
133C22×D191—81no—
134C22×D191—81no—
135C22×D191—81no—
136C22×D191C2×D411yescharacteristic
137Dic382—21no—
138Dic382—21no—
139D764—21no—
140D764—21no—
141D764—21no—
142D764—21no—
143D764—42no—
144D764—42no—
145D764C2×D412yes—
146D764C2×D412yescomplemented
147C22⋊D191—22no—
148C22⋊D191—42no—
149C22⋊D191—42no—
150C22⋊D191—42no—
151C22⋊D191—42no—
152C22⋊D191—81no—
153C2×Dic191—21no—
154C2×Dic191C2×D411yescharacteristic
155C2×Dic191C2×D411yescharacteristic
156C2×Dic191C2×D411yescharacteristic
157C4×D191—21no—
158C4×D191—21no—
159C4×D191—42no—
160C4×D191—42no—
161C4×D191—81no—
162C191⋊C8—21no—
163C191⋊C8—21no—
164C23×C191—21no—
165C23×C191—21no—
166C23×C191C2×D411yescharacteristic
167D4×C191—21no—
168D4×C191—21no—
169D4×C191—21no—
170D4×C191—21no—
171D4×C191—42no—
172D4×C191—81no—
173C2×C4×C191—21no—
174C2×C4×C191—21no—
175C2×C4×C191—21no—
176C2×C4×C191—21no—
177C2×C4×C191C2×D411yescharacteristic
178C2×C4×C191C2411yescharacteristic
179C8×C191—21no—
180C8×C191C2×D412yescomplemented
181C22×C191—21no—
182C22×C191—21no—
183C22×C191—21no—
184C22×C191—41no—
185C22×C191—41no—
186C22×C191—41no—
187C22×C191—41no—
188C22×C191—41no—
189C22×C191—41no—
190C22×C191C22×D411yescharacteristic
191C2×D191—21no—
192C2×D191—21no—
193C2×D191—22no—
194C2×D191—22no—
195C2×D191—41no—
196C2×D191—41no—
197C2×D191—41no—
198C2×D191—41no—
199C2×D191—41no—
200C2×D191—41no—
201C2×D191—41no—
202C2×D191—84no—
203C2×D191—84no—
204C2×D191—84no—
205C2×D191AGL(1,ℤ/8)12yescomplemented
206C4×C191—21no—
207C4×C191—41no—
208C4×C191—41no—
209C4×C191C22×D411yescharacteristic
210C4×C191H(5,𝔽2)11yescharacteristic
211Dic191—21no—
212Dic191—21no—
213Dic191—21no—
214Dic191—41no—
215C2×C191—41no—
216C2×C191—81no—
217C2×C191—81no—
218C2×C191D4211yescharacteristic
219C2×C191C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C191#57611yescharacteristic
221D191—22no—
222D191—42no—
223D191—42no—
224D191—81no—
225C191#326411yescharacteristic, complemented, Sylow
226#3264—1911nomaximal, Sylow
227C2×AGL(1,ℤ/8)—1911no—
228C22×D8—1911no—
229#546—1911no—
230D42—1911no—
231#496—1911no—
232#469D19111yescharacteristic, complemented
233#465—1911no—
234#463—1911no—
235#459—1911no—
236#450—1911no—
237#449—1911no—
238#447—1911no—
239#447—1911no—
240#442—1911no—
241#435—1911no—
242C2×#41—1911no—
243C22×D4—3821no—
244C22×D4—1911no—
245C22×D4—1911no—
246C22×D4—1911no—
247AGL(1,ℤ/8)—3824no—
248C2×SD16—1911no—
249C2×SD16—1911no—
250C2×SD16—1911no—
251C2×D8—3824no—
252C2×D8—1911no—
253C2×D8—1911no—
254C2×D8—1911no—
255C2×D8—1911no—
256C2×D8—1911no—
257C2×D8C2×D19111yescharacteristic, complemented
258C2×M16C2×D19111yescharacteristic, complemented
259C22×C8—1911no—
260#127—1911no—
261#124—1911no—
262#123—3821no—
263C22⋊Q8—1911no—
264#121—3821no—
265#121—3821no—
266#121—1911no—
267#121C2×D19111yescharacteristic, complemented
268#121C2×D19111yescharacteristic, complemented
269#120—3821no—
270#120—3821no—
271#120—3821no—
272C4×D4—1911no—
273C4×D4—1911no—
274C4×D4—1911no—
275C2×C22⋊C4—3821no—
276#106C2×D19111yescharacteristic, complemented
277C4⋊C8—1911no—
278#103—1911no—
279#102—1911no—
280#102—1911no—
281#102—1911no—
282#102C2×D19111yescharacteristic, complemented
283#102C2×D19111yescharacteristic, complemented
284C22⋊C8—1911no—
285C22⋊C8—1911no—
286#97—1911no—
287C24—3821no—
288C24—3821no—
289C24—3821no—
290#41—3822no—
291#41—7641no—
292C2×Q8—1911no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—3822no—
296C2×D4—3821no—
297C2×D4—3822no—
298C2×D4—3822no—
299C2×D4—3821no—
300C2×D4—3821no—
301C2×D4—3821no—
302C2×D4—3821no—
303C2×D4—3821no—
304C2×D4—3821no—
305C2×D4—1911no—
306C2×D4—1911no—
307C2×D4—1911no—
308C2×D4—1911no—
309C2×D4—7641no—
310C2×D4—7641no—
311C2×D4—7641no—
312C2×D4—7644no—
313C2×D4C22×D19111yescharacteristic
314C2×D4C22×D19111yescharacteristic
315C22×C4—3821no—
316C22×C4—3821no—
317C22×C4—3821no—
318C22×C4—1911no—
319C22×C4C22×D19111yescharacteristic
320SD16—3822no—
321SD16—3822no—
322SD16—7641no—
323D8—22no—
324D8—3822no—
325D8—3822no—
326D8—3822no—
327D8—1914no—
328D8—7641no—
329M16—22no—
330C2×C8—3822no—
331C2×C8—1911no—
332C2×C8—1911no—
333C2×C8C22×D19111yescharacteristic
334C2×C8C22×D19111yescharacteristic
335C4⋊C4—3821no—
336C4⋊C4—1911no—
337C4⋊C4C22×D19111yescharacteristic
338C4⋊C4C22×D19111yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—3821no—
342C22⋊C4—3821no—
343C22⋊C4—3821no—
344C22⋊C4—3821no—
345C22⋊C4—3821no—
346C22⋊C4—3821no—
347C22⋊C4—1911no—
348C22⋊C4—1911no—
349C42—1911no—
350C23—21no—
351C23—21no—
352C23—3821no—
353C23—3821no—
354C23—3821no—
355C23—1911no—
356C23—15281no—
357C23—15281no—
358C23—15281no—
359C23—7642no—
360C23—7642no—
361C23—7642no—
362C23—7642no—
363C23—7642no—
364C23—7642no—
365C23D4×D19111yescharacteristic
366Q8—3821no—
367Q8—3821no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—42no—
373D4—81no—
374D4—3822no—
375D4—3821no—
376D4—3821no—
377D4—3821no—
378D4—3821no—
379D4—1912no—
380D4—1912no—
381D4—15281no—
382D4—7642no—
383D4—7642no—
384D4—7642no—
385D4—7642no—
386D4—7642no—
387D4—7642no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—3821no—
393C2×C4—3821no—
394C2×C4—3821no—
395C2×C4—1911no—
396C2×C4—1911no—
397C2×C4—1911no—
398C2×C4—15281no—
399C2×C4—7642no—
400C2×C4—7642no—
401C2×C4D4×D19111yescharacteristic
402C2×C4C23×D19111yescharacteristic
403C8—21no—
404C8—3821no—
405C8—3821no—
406C8D4×D19112yescomplemented
407C22—21no—
408C22—21no—
409C22—21no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—3821no—
417C22—3821no—
418C22—3822no—
419C22—3822no—
420C22—1912no—
421C22—15284no—
422C22—15284no—
423C22—15284no—
424C22—7641no—
425C22—7641no—
426C22—7641no—
427C22—7641no—
428C22—7641no—
429C22—7641no—
430C22—7641no—
431C22C2×D4×D19111yescharacteristic
432C4—21no—
433C4—41no—
434C4—41no—
435C4—3821no—
436C4—3821no—
437C4—3821no—
438C4—7641no—
439C4C2×D4×D19111yescharacteristic
440C4#2025936811yescharacteristic
441C2—41no—
442C2—81no—
443C2—81no—
444C2—3822no—
445C2—15281no—
446C2—7642no—
447C2—7642no—
448C2#2713329711yescharacteristic
449C2AGL(1,ℤ/8)×D19111yescharacteristic
450C2#2713346711yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#469×C191C2
socleC22×C191C22×D4
hypercenter#469D191
hypocenterC191#3264
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×#20259350C2semidirect
2. C2×D8×D191C2semidirect
3. #27133310C2semidirect
4. #27133297C2semidirect
5. #27133051C2semidirect
6. #27132893C2semidirect
7. #27132877C2semidirect
8. #27132857C2semidirect
9. #27132815C2semidirect
10. #27132806C2semidirect
11. #27132798C2semidirect
12. #27132796C2semidirect
13. #27132774C2semidirect
14. #27132732C2semidirect
15. #469×C191C2semidirect
16. C2×#15690287C22semidirect
17. C22×D764C22semidirect
20. C2×D4×D191C22semidirect
22. C2×#15690276C22semidirect
23. C2×#15690276C22semidirect
24. C2×#15690275C22semidirect
26. C2×D1528C22semidirect
28. C2×#15690272C22semidirect
29. C2×#15690272C22semidirect
31. C2×C8×D191C22semidirect
32. #20259318C22semidirect
33. #20259309C22semidirect
35. #20259302C22semidirect
42. #20259284C22semidirect
43. #20259284C22semidirect
46. Dic191⋊C8C22semidirect
47. #20259245C22semidirect
48. #20259242C22semidirect
49. #20259242C22semidirect
50. #20259242C22semidirect
51. C22⋊(C191⋊C8)C22semidirect
52. (C2×D191)⋊C8C22semidirect
53. #20259232C22semidirect
54. C2×D8×C191C22semidirect
55. C2×M16×C191C22semidirect
56. #121×C191C22semidirect
57. #121×C191C22semidirect
58. #106×C191C22semidirect
59. #102×C191C22semidirect
60. #102×C191C22semidirect
93. D1528D4semidirect
146. D764C2×D4semidirect
180. C8×C191C2×D4semidirect
205. C2×D191AGL(1,ℤ/8)semidirect
225. C191#3264semidirect
232. #469D191semidirect
257. C2×D8C2×D191semidirect
258. C2×M16C2×D191semidirect
267. #121C2×D191semidirect
268. #121C2×D191semidirect
276. #106C2×D191semidirect
282. #102C2×D191semidirect
283. #102C2×D191semidirect
406. C8D4×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593502
C2×D8×D1912
#271333102
#271332972
#271330512
#271328932
#271328772
#271328572
#271328152
#271328062
#271327982
#271327962
#271327742
#271327322
#469×C1912
#3264191