Finite Group Databasebeta

← #15685216

Subgroups — #15685216

Order 2944.

462 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13093125C211yesmaximal, characteristic, complemented
2#13093051C211yesmaximal, characteristic
3#13092762C211yesmaximal, characteristic, complemented
4#13092737C211yesmaximal, characteristic, complemented
5#13092733C211yesmaximal, characteristic, complemented
6#13092706C211yesmaximal, characteristic, complemented
7#13092649C211yesmaximal, characteristic, complemented
8#13092615C211yesmaximal, characteristic, complemented
9#13092567C211yesmaximal, characteristic, complemented
10#13092566C211yesmaximal, characteristic, complemented
11#13092502C211yesmaximal, characteristic, complemented
12C8×C22⋊D23C211yesmaximal, characteristic, complemented
13C2×#10625003C211yesmaximal, characteristic, complemented
14#102×D23C211yesmaximal, characteristic, complemented
15C23×#465C211yesmaximal, characteristic, complemented
16C2×D4×D23—21no—
17C2×D4×D23C2211yescharacteristic
18C22×D92C2211yescharacteristic
19C2×SD16×C23C2211yescharacteristic, complemented
20C22×C8×C23C2211yescharacteristic, complemented
21C23×C22⋊Q8C2211yescharacteristic, complemented
22C23×#121C2211yescharacteristic
23#10625117C2211yescharacteristic, complemented
24Q8×Dic23C2211yescharacteristic, complemented
25C2×#67274C2211yescharacteristic
26C2×#67274C2211yescharacteristic, complemented
27#10625111C2211yescharacteristic
28#10625108—21no—
29C4×C22⋊D23C2211yescharacteristic
30C2×C4⋊Dic23C2211yescharacteristic, complemented
31C2×C8×D23C2211yescharacteristic, complemented
32#10625068—21no—
33#10625066C2211yescharacteristic
34#10625064C2211yescharacteristic
35C4⋊C4×D23C2211yescharacteristic, complemented
36#10625060—21no—
37#10625059C2211yescharacteristic, complemented
38#10625055—21no—
39#10625053—21no—
40(C22⋊D23)⋊C4—21no—
41#10625049—21no—
42C23×#106C2211yescharacteristic, complemented
43C23×#103C2211yescharacteristic, complemented
44C23×#102C2211yescharacteristic, complemented
45#10625014C2211yescharacteristic, complemented
46#10625014C2211yescharacteristic, complemented
47C22⋊(C23⋊C8)C2211yescharacteristic, complemented
48#10625003C2211yescharacteristic
49#10625003C2211yescharacteristic
50#10625003C2211yescharacteristic, complemented
51#10625003C2211yescharacteristic, complemented
52(C2×D23)⋊C8C2211yescharacteristic, complemented
53Dic23⋊C8C2211yescharacteristic, complemented
54C8×Dic23C2211yescharacteristic, complemented
55#10624991C2211yescharacteristic
56#10624991C2211yescharacteristic, complemented
57#10624990C2211yescharacteristic, complemented
58#10624990C2211yescharacteristic, complemented
59C23×D23—21no—
60C23×D23—21no—
61C2×Q8×C23C2311yescharacteristic
62C2×D4×C23—21no—
63C2×D4×C23C2311yescharacteristic
64C22×C4×C23C2311yescharacteristic
65C2×C22⋊D23—21no—
66C2×C22⋊D23—21no—
67C2×C22⋊D23—21no—
68C2×C22⋊D23D411yescharacteristic
69C22×Dic23—21no—
70D4×D23—22no—
71D4×D23—41no—
72D4×D23—44no—
73C2×D92—22no—
74C2×D92—21no—
75C2×D92—21no—
76C2×D92—41no—
77C2×D92C2311yescharacteristic
78C2×D92C2311yescharacteristic
79C2×C4×D23—21no—
80C2×C4×D23—21no—
81C2×C4×D23C2311yescharacteristic
82C2×Dic46—21no—
83SD16×C23—41no—
84C2×C8×C23—41no—
85C2×C8×C23C2311yescharacteristic
86C2×C8×C23C2311yescharacteristic
87C4⋊C4×C23—21no—
88C4⋊C4×C23C2311yescharacteristic
89C4⋊C4×C23C2311yescharacteristic
90C22⋊C4×C23—21no—
91C22⋊C4×C23—21no—
92C22⋊Dic23D411yescharacteristic
93#67274—22no—
94#67274—41no—
95(C2×D23)⋊C4—21no—
96(C2×D23)⋊C4—21no—
97(C2×D23)⋊C4—21no—
98(C2×D23)⋊C4D411yescharacteristic
99C4⋊Dic23—21no—
100C4⋊Dic23—21no—
101C4⋊Dic23C2311yescharacteristic
102C4⋊Dic23C2311yescharacteristic
103Dic23⋊C4—21no—
104Dic23⋊C4—21no—
105Dic23⋊C4—21no—
106Dic23⋊C4D411yescharacteristic
107C4×Dic23—21no—
108C4×Dic23C2311yescharacteristic
109C2×C23⋊C8C2311yescharacteristic
110C2×C23⋊C8C2311yescharacteristic
111C8×D23—41no—
112C23×C23—21no—
113C23×C23C2×D411yescharacteristic
114C22×D23—21no—
115C22×D23—21no—
116C22×D23—42no—
117C22×D23—42no—
118C22×D23—42no—
119C22×D23—42no—
120C22×D23—81no—
121C22×D23—81no—
122C22×D23C2×D411yescharacteristic
123Q8×C23—21no—
124Q8×C23—21no—
125D4×C23—21no—
126D4×C23—21no—
127D4×C23—42no—
128C2×C4×C23—21no—
129C2×C4×C23—21no—
130C2×C4×C23—21no—
131C2×C4×C23—21no—
132C2×C4×C23—21no—
133C2×C4×C23C2×D411yescharacteristic
134C2×C4×C23C2411yescharacteristic
135C22⋊D23—42no—
136C22⋊D23—42no—
137C22⋊D23—81no—
138C22⋊D23SD1614yescomplemented
139C2×Dic23—21no—
140C2×Dic23—21no—
141C2×Dic23—81no—
142C2×Dic23C2×D411yescharacteristic
143C2×Dic23C2×D411yescharacteristic
144C2×Dic23C2×D411yescharacteristic
145D92—21no—
146D92—21no—
147D92—21no—
148D92—42no—
149D92—42no—
150D92#4112yes—
151C4×D23—21no—
152C4×D23—21no—
153C4×D23—42no—
154C4×D23—81no—
155Dic46—81no—
156C8×C23—21no—
157C8×C23—21no—
158C23⋊C8—21no—
159C23⋊C8—21no—
160#3295—231nomaximal, Sylow
161C22×C23—21no—
162C22×C23—21no—
163C22×C23—41no—
164C22×C23—41no—
165C22×C23—41no—
166C22×C23C2×SD1612yes—
167C22×C23C22×D411yescharacteristic
168C2×D23—21no—
169C2×D23—21no—
170C2×D23—22no—
171C2×D23—41no—
172C2×D23—41no—
173C2×D23—41no—
174C2×D23—41no—
175C2×D23—41no—
176C2×D23—84no—
177C2×D23—84no—
178C2×D23C2×SD1612yes—
179C4×C23—21no—
180C4×C23—41no—
181C4×C23—41no—
182C4×C23—41no—
183C4×C23C2×#4111yescharacteristic
184C4×C23H(5,𝔽2)11yescharacteristic
185Dic23—21no—
186Dic23—21no—
187Dic23—41no—
188Dic23—41no—
189Dic23C2×SD1612yes—
190#554—231no—
191D42—231no—
192#486—231no—
193#481—231no—
194#481—231no—
195#476—231no—
196#465—231no—
197#465D2311yescharacteristic, complemented
198#460—231no—
199#450—231no—
200#450—231no—
201C4×SD16—231no—
202C8×D4—231no—
203C2×#102—231no—
204C2×#102—231no—
205C2×C23—22no—
206C2×C23—81no—
207C2×C23#54611yescharacteristic
208C2×C23C22×SD1611yescharacteristic
209C2×C23#57611yescharacteristic
210D23—22no—
211D23—42no—
212D23—81no—
213C22×D4—231no—
214C22×D4—231no—
215C22×D4—461no—
216C2×SD16—231no—
217C2×SD16—231no—
218C2×SD16C2×D2311yescharacteristic, complemented
219C22×C8—231no—
220C22×C8C2×D2311yescharacteristic, complemented
221C4⋊Q8—231no—
222#127—231no—
223#125—461no—
224C22⋊Q8—231no—
225C22⋊Q8—461no—
226C22⋊Q8—461no—
227C22⋊Q8C2×D2311yescharacteristic, complemented
228#121—231no—
229#121—461no—
230#121C2×D2311yescharacteristic
231#120—461no—
232#120—461no—
233C4×Q8—231no—
234C4×D4—231no—
235C4×D4—231no—
236C4×D4—461no—
237C2×C4⋊C4—231no—
238C2×C4⋊C4—231no—
239#106—231no—
240#106—231no—
241#106C2×D2311yescharacteristic, complemented
242C4⋊C8—231no—
243#103C2×D2311yescharacteristic, complemented
244#102—231no—
245#102—231no—
246#102—231no—
247#102—231no—
248#102—231no—
249#102—231no—
250#102—231no—
251#102—231no—
252#102C2×D2311yescharacteristic, complemented
253C22⋊C8—231no—
254C22⋊C8—231no—
255C4×C8—231no—
256C23#329511yescharacteristic, complemented, Sylow
257C24—461no—
258C24—461no—
259C2×Q8—461no—
260C2×Q8C22×D2311yescharacteristic
261C2×D4—21no—
262C2×D4—231no—
263C2×D4—231no—
264C2×D4—231no—
265C2×D4—462no—
266C2×D4—461no—
267C2×D4—461no—
268C2×D4—461no—
269C2×D4—461no—
270C2×D4—462no—
271C2×D4—461no—
272C2×D4—921no—
273C2×D4—921no—
274C2×D4—924no—
275C2×D4C22×D2311yescharacteristic
276C22×C4—231no—
277C22×C4—461no—
278C22×C4—461no—
279C22×C4—461no—
280C22×C4C22×D2311yescharacteristic
281SD16—41no—
282SD16—462no—
283SD16—921no—
284C2×C8—41no—
285C2×C8—231no—
286C2×C8—231no—
287C2×C8—921no—
288C2×C8C22×D2311yescharacteristic
289C2×C8C22×D2311yescharacteristic
290C4⋊C4—21no—
291C4⋊C4—231no—
292C4⋊C4—231no—
293C4⋊C4—231no—
294C4⋊C4—461no—
295C4⋊C4—461no—
296C4⋊C4—461no—
297C4⋊C4—461no—
298C4⋊C4—461no—
299C4⋊C4C22×D2311yescharacteristic
300C4⋊C4C22×D2311yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—231no—
304C22⋊C4—231no—
305C22⋊C4—461no—
306C22⋊C4—461no—
307C22⋊C4—461no—
308C42—231no—
309C42—461no—
310C23—21no—
311C23—231no—
312C23—461no—
313C23—461no—
314C23—922no—
315C23—922no—
316C23—922no—
317C23—922no—
318C23—1841no—
319C23—1841no—
320C23D4×D2311yescharacteristic
321Q8—21no—
322Q8—21no—
323Q8—1841no—
324D4—21no—
325D4—21no—
326D4—42no—
327D4—234no—
328D4—232no—
329D4—461no—
330D4—461no—
331D4—461no—
332D4—922no—
333D4—922no—
334D4—922no—
335D4—922no—
336D4—1841no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—231no—
343C2×C4—231no—
344C2×C4—231no—
345C2×C4—461no—
346C2×C4—461no—
347C2×C4—461no—
348C2×C4—461no—
349C2×C4—922no—
350C2×C4—1841no—
351C2×C4—1841no—
352C2×C4D4×D2311yescharacteristic
353C2×C4C23×D2311yescharacteristic
354C8—21no—
355C8—21no—
356C8—461no—
357C8—461no—
358C22—21no—
359C22—21no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—232no—
364C22—461no—
365C22—461no—
366C22—462no—
367C22—921no—
368C22—921no—
369C22—921no—
370C22—921no—
371C22—921no—
372C22—1844no—
373C22—1844no—
374C22SD16×D2312yes—
375C22C2×D4×D2311yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—232no—
381C4—461no—
382C4—461no—
383C4—921no—
384C4—921no—
385C4#1062515511yescharacteristic
386C4#41×D2311yescharacteristic
387C2—22no—
388C2—81no—
389C2—462no—
390C2—922no—
391C2—1841no—
392C2#1309305611yescharacteristic
393C2C2×SD16×D2311yescharacteristic
394C2#1309321911yescharacteristic
subgroupG/H
centerC22C2×D4×D23
commutator, cosocleC2×C4×C23C24
FrattiniC2×C4C23×D23
FittingC23×#465C2
socleC22×C23C22×D4
prime radical#13093051C2
hypercenter#465D23
hypocenterC23#3295
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #13093125C2semidirect
3. #13092762C2semidirect
4. #13092737C2semidirect
5. #13092733C2semidirect
6. #13092706C2semidirect
7. #13092649C2semidirect
8. #13092615C2semidirect
9. #13092567C2semidirect
10. #13092566C2semidirect
11. #13092502C2semidirect
12. C8×C22⋊D23C2semidirect
13. C2×#10625003C2semidirect
14. #102×D23C2semidirect
15. C23×#465C2semidirect
19. C2×SD16×C23C22semidirect
20. C22×C8×C23C22semidirect
21. C23×C22⋊Q8C22semidirect
23. #10625117C22semidirect
24. Q8×Dic23C22semidirect
26. C2×#67274C22semidirect
30. C2×C4⋊Dic23C22semidirect
31. C2×C8×D23C22semidirect
35. C4⋊C4×D23C22semidirect
37. #10625059C22semidirect
42. C23×#106C22semidirect
43. C23×#103C22semidirect
44. C23×#102C22semidirect
45. #10625014C22semidirect
46. #10625014C22semidirect
47. C22⋊(C23⋊C8)C22semidirect
50. #10625003C22semidirect
51. #10625003C22semidirect
52. (C2×D23)⋊C8C22semidirect
53. Dic23⋊C8C22semidirect
54. C8×Dic23C22semidirect
56. #10624991C22semidirect
57. #10624990C22semidirect
58. #10624990C22semidirect
138. C22⋊D23SD16semidirect
197. #465D23semidirect
218. C2×SD16C2×D23semidirect
220. C22×C8C2×D23semidirect
227. C22⋊Q8C2×D23semidirect
241. #106C2×D23semidirect
243. #103C2×D23semidirect
252. #102C2×D23semidirect
256. C23#3295semidirect

Maximal subgroups

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

subgroupindex
#130931252
#130930512
#130927622
#130927372
#130927332
#130927062
#130926492
#130926152
#130925672
#130925662
#130925022
C8×C22⋊D232
C2×#106250032
#102×D232
C23×#4652
#329523