Finite Group Databasebeta

← #15682656

Subgroups — #15682656

Order 2944.

754 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 141 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#67292C211yesmaximal, characteristic
2C2×#10625089C212yesmaximal, complemented
3C22⋊Q8×D23C211yesmaximal, characteristic, complemented
4#13092582C212yesmaximal, complemented
5#13092578C212yesmaximal, complemented
6#13092573C211yesmaximal, characteristic, complemented
7#13092570C211yesmaximal, characteristic, complemented
8#13092567C211yesmaximal, characteristic, complemented
9#13092417C212yesmaximal, complemented
10#13092379C211yesmaximal, characteristic, complemented
11C23×#451C211yesmaximal, characteristic, complemented
12C22×Q8×C23C2211yescharacteristic
13C2×#67292—28no—
14C2×#67292—21no—
15C2×#67292—21no—
16C2×#67292C2212yes—
17C2×#67292C2212yes—
18C2×Q8×D23C2211yescharacteristic, complemented
19C22×D92—21no—
20C22×D92C2211yescharacteristic
21C22×C4×D23—21no—
22C22×C4×D23C2211yescharacteristic
23C2×Q16×C23C2212yescomplemented
24C23×C22⋊Q8C2211yescharacteristic
25#10625117C2211yescharacteristic, complemented
26C2×#67275C2212yescomplemented
27C2×#67274C2212yescomplemented
28#10625095C2211yescharacteristic
29#10625089—28no—
30C2×#67263C2212yescomplemented
31C2×#67262C2212yescomplemented
32#10625068C2211yescharacteristic, complemented
33#10625067—21no—
34C4⋊C4×D23—21no—
35C4⋊C4×D23C2211yescharacteristic, complemented
36C22⋊C4×D23—21no—
37#10625049—21no—
38C23×#103C2212yescomplemented
39C23×C22⋊C8C2211yescharacteristic
40#10625017C2212yescomplemented
41C22⋊(C23⋊C8)C2211yescharacteristic
42#10625003C2212yescomplemented
43(C2×D23)⋊C8C2212yescomplemented
44#10624991C2212yescomplemented
45C23×D23—21no—
46C23×D23D411yescharacteristic
47C2×Q8×C23—24no—
48C2×Q8×C23—21no—
49C2×Q8×C23C2311yescharacteristic
50C2×Q8×C23C2312yes—
51C22×C4×C23—21no—
52C22×C4×C23C2311yescharacteristic
53C22×Dic23D411yescharacteristic
54#67292—28no—
55#67292—28no—
56#67292—24no—
57#67292—24no—
58#67292—24no—
59#67292D418yes—
60Q8×D23—24no—
61C2×D92—28no—
62C2×D92—22no—
63C2×D92—22no—
64C2×D92—21no—
65C2×D92—24no—
66C2×D92C2312yes—
67C2×C4×D23—28no—
68C2×C4×D23—22no—
69C2×C4×D23—22no—
70C2×C4×D23—24no—
71C2×C4×D23—21no—
72C2×C4×D23—21no—
73C2×C4×D23D411yescharacteristic
74C2×C4×D23D411yescharacteristic
75C2×C4×D23C2311yescharacteristic
76C2×C4×D23C2311yescharacteristic
77C2×Dic46—21no—
78C2×Dic46C2311yescharacteristic
79Q16×C23—24no—
80C2×C8×C23C2312yes—
81C4⋊C4×C23—21no—
82C4⋊C4×C23C2311yescharacteristic
83C22⋊C4×C23—21no—
84C22⋊Dic23—21no—
85#67275—24no—
86#67274—24no—
87(C2×D23)⋊C4—21no—
88(C2×D23)⋊C4—21no—
89C4⋊Dic23—21no—
90C4⋊Dic23C2311yescharacteristic
91Dic23⋊C4—21no—
92Dic23⋊C4—21no—
93Dic23⋊C4—21no—
94C2×C23⋊C8C2312yes—
95#67263—24no—
96#67262—24no—
97C23×C23C2×D411yescharacteristic
98C22×D23—24no—
99C22×D23—22no—
100C22×D23—22no—
101C22×D23—22no—
102C22×D23—41no—
103C22×D23—42no—
104C22×D23—42no—
105C22×D23C2×D411yescharacteristic
106C22×D23C2×D411yescharacteristic
107Q8×C23—24no—
108Q8×C23—22no—
109Q8×C23—22no—
110Q8×C23C2×D414yes—
111C2×C4×C23—24no—
112C2×C4×C23—22no—
113C2×C4×C23—22no—
114C2×C4×C23—21no—
115C2×C4×C23—21no—
116C2×C4×C23C2×D411yescharacteristic
117C2×C4×C23C2411yescharacteristic
118C2×Dic23—22no—
119C2×Dic23—21no—
120C2×Dic23—21no—
121C2×Dic23C2×D411yescharacteristic
122C2×Dic23C2×D411yescharacteristic
123D92—28no—
124D92—24no—
125D92—28no—
126D92—22no—
127D92C2×D414yes—
128C4×D23—28no—
129C4×D23—22no—
130C4×D23—22no—
131C4×D23—28no—
132C4×D23—22no—
133C4×D23—42no—
134C4×D23—42no—
135C4×D23C2×D414yes—
136Dic46—22no—
137Dic46—42no—
138C8×C23—22no—
139C23⋊C8—22no—
140#3007—231nomaximal, Sylow
141C22×C23—21no—
142C22×C23—21no—
143C22×C23#13712yescomplemented
144C22×C23C22×D411yescharacteristic
145C2×D23—21no—
146C2×D23—21no—
147C2×D23—21no—
148C2×D23—24no—
149C2×D23—21no—
150C2×D23—21no—
151C2×D23—44no—
152C2×D23—48no—
153C2×D23—42no—
154C2×D23—42no—
155C2×D23#12012yes—
156C4×C23—24no—
157C4×C23—21no—
158C4×C23—41no—
159C4×C23—41no—
160C4×C23C22×D412yes—
161Dic23—21no—
162Dic23—41no—
163Dic23—41no—
164Dic23#12012yes—
165C22×#41—231no—
166C2×#137—232no—
167C2×C22⋊Q8—231no—
168#452—232no—
169#452—232no—
170#451—231no—
171#451D2311yescharacteristic, complemented
172#450—231no—
173#450—231no—
174#417—232no—
175#407—231no—
176C2×C23—22no—
177C2×C23C2×#12011yescharacteristic
178C2×C23C2×#13711yescharacteristic
179C2×C23#57511yescharacteristic
180D23—22no—
181D23—41no—
182D23—44no—
183C2×#41—232no—
184C2×#41—232no—
185C2×#41—461no—
186C2×#41—461no—
187C2×#41—468no—
188C22×Q8—231no—
189C22×Q8C2×D2311yescharacteristic
190C22×D4—231no—
191C22×D4—461no—
192C23×C4—231no—
193C23×C4—461no—
194#137—468no—
195C2×Q16—232no—
196C2×Q16C2×D2312yescomplemented
197C2×SD16—232no—
198C2×SD16—232no—
199C2×M16—232no—
200C22⋊Q8—231no—
201C22⋊Q8—231no—
202C22⋊Q8—231no—
203C22⋊Q8—461no—
204C22⋊Q8—461no—
205C22⋊Q8C2×D2311yescharacteristic
206C2×C4⋊C4—231no—
207C2×C4⋊C4—461no—
208C2×C22⋊C4—461no—
209#103—232no—
210#103C2×D2312yescomplemented
211#102—232no—
212#102—232no—
213C22⋊C8—231no—
214C22⋊C8—232no—
215C22⋊C8C2×D2311yescharacteristic
216C23#300711yescharacteristic, complemented, Sylow
217C24—231no—
218C24—461no—
219#41—238no—
220#41—464no—
221#41—464no—
222#41—468no—
223#41—468no—
224#41—464no—
225C2×Q8—24no—
226C2×Q8—21no—
227C2×Q8—231no—
228C2×Q8—461no—
229C2×Q8—464no—
230C2×Q8C22×D2311yescharacteristic
231C2×Q8C22×D2312yes—
232C2×D4—232no—
233C2×D4—464no—
234C2×D4—462no—
235C2×D4—462no—
236C2×D4—468no—
237C2×D4—461no—
238C22×C4—21no—
239C22×C4—231no—
240C22×C4—231no—
241C22×C4—231no—
242C22×C4—231no—
243C22×C4—231no—
244C22×C4—464no—
245C22×C4—462no—
246C22×C4—462no—
247C22×C4—468no—
248C22×C4—461no—
249C22×C4—461no—
250C22×C4C22×D2311yescharacteristic
251Q16—24no—
252Q16—464no—
253SD16—464no—
254SD16—464no—
255M16—464no—
256C2×C8—232no—
257C2×C8C22×D2312yes—
258C4⋊C4—21no—
259C4⋊C4—231no—
260C4⋊C4—461no—
261C4⋊C4—461no—
262C4⋊C4—461no—
263C4⋊C4—461no—
264C4⋊C4C22×D2311yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—461no—
267C22⋊C4—461no—
268C22⋊C4—461no—
269C23—231no—
270C23—231no—
271C23—462no—
272C23—464no—
273C23—462no—
274C23—462no—
275C23—921no—
276C23—922no—
277C23—922no—
278C23D4×D2311yescharacteristic
279Q8—24no—
280Q8—22no—
281Q8—22no—
282Q8—462no—
283Q8—922no—
284Q8D4×D2314yes—
285D4—234no—
286D4—468no—
287D4—468no—
288D4—464no—
289D4—462no—
290C2×C4—24no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—231no—
296C2×C4—234no—
297C2×C4—231no—
298C2×C4—462no—
299C2×C4—468no—
300C2×C4—462no—
301C2×C4—468no—
302C2×C4—462no—
303C2×C4—462no—
304C2×C4—461no—
305C2×C4—461no—
306C2×C4—922no—
307C2×C4—922no—
308C2×C4D4×D2311yescharacteristic
309C2×C4C23×D2311yescharacteristic
310C8—22no—
311C8—462no—
312C22—21no—
313C22—21no—
314C22—232no—
315C22—461no—
316C22—461no—
317C22—464no—
318C22—461no—
319C22—461no—
320C22—461no—
321C22—924no—
322C22—928no—
323C22—922no—
324C22—922no—
325C22#1062508912yescomplemented
326C22C2×D4×D2311yescharacteristic
327C4—24no—
328C4—21no—
329C4—41no—
330C4—41no—
331C4—232no—
332C4—461no—
333C4—921no—
334C4—921no—
335C4C2×D4×D2312yes—
336C2—22no—
337C2—462no—
338C2—921no—
339C2—924no—
340C2#120×D2311yescharacteristic
341C2C2×#1062508911yescharacteristic
342C2#1309321611yescharacteristic
subgroupG/H
centerC22C2×D4×D23
commutator, cosocleC2×C4×C23C24
FrattiniC2×C4C23×D23
FittingC23×#451C2
socleC22×C23C22×D4
prime radicalC22×#67292C2
hypercenter#451D23
hypocenterC23#3007
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. C2×#10625089C2semidirect
3. C22⋊Q8×D23C2semidirect
4. #13092582C2semidirect
5. #13092578C2semidirect
6. #13092573C2semidirect
7. #13092570C2semidirect
8. #13092567C2semidirect
9. #13092417C2semidirect
10. #13092379C2semidirect
11. C23×#451C2semidirect
18. C2×Q8×D23C22semidirect
23. C2×Q16×C23C22semidirect
25. #10625117C22semidirect
26. C2×#67275C22semidirect
27. C2×#67274C22semidirect
30. C2×#67263C22semidirect
31. C2×#67262C22semidirect
32. #10625068C22semidirect
35. C4⋊C4×D23C22semidirect
38. C23×#103C22semidirect
40. #10625017C22semidirect
42. #10625003C22semidirect
43. (C2×D23)⋊C8C22semidirect
44. #10624991C22semidirect
143. C22×C23#137semidirect
171. #451D23semidirect
196. C2×Q16C2×D23semidirect
210. #103C2×D23semidirect
216. C23#3007semidirect
325. C22#10625089semidirect

Maximal subgroups

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

subgroupindex
C22×#672922
C2×#106250892
C22⋊Q8×D232
#130925822
#130925782
#130925732
#130925702
#130925672
#130924172
#130923792
C23×#4512
#300723