Finite Group Databasebeta

← #15682652

Subgroups — #15682652

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×#67290C211yesmaximal, characteristic
2C2×#10625086C212yesmaximal, complemented
3C22⋊Q8×D23C211yesmaximal, characteristic, complemented
4#13092580C212yesmaximal, complemented
5#13092576C212yesmaximal, complemented
6#13092575C211yesmaximal, characteristic, complemented
7#13092572C211yesmaximal, characteristic, complemented
8#13092568C211yesmaximal, characteristic, complemented
9#13092415C212yesmaximal, complemented
10#13092379C211yesmaximal, characteristic, complemented
11C23×#450C211yesmaximal, characteristic, complemented
12C22×D4×C23C2211yescharacteristic
13C22×C22⋊D23—21no—
14C23×Dic23—21no—
15C2×Q8×D23C2211yescharacteristic, complemented
16C2×#67290—28no—
17C2×#67290—21no—
18C2×#67290—21no—
19C2×#67290C2212yes—
20C2×#67290C2212yes—
21C22×C4×D23C2211yescharacteristic
22C22×Dic46C2211yescharacteristic
23C2×SD16×C23C2212yescomplemented
24C23×C22⋊Q8C2211yescharacteristic
25#10625117C2211yescharacteristic, complemented
26C2×#67275C2212yescomplemented
27C2×#67273C2212yescomplemented
28#10625095C2211yescharacteristic
29#10625086—28no—
30C2×Dic92C2212yescomplemented
31C2×#67262C2212yescomplemented
32#10625068C2211yescharacteristic, complemented
33#10625067—21no—
34C4⋊C4×D23—21no—
35C4⋊C4×D23C2211yescharacteristic, complemented
36C22⋊C4×D23—21no—
37#10625049—21no—
38C23×#102C2212yescomplemented
39C23×C22⋊C8C2211yescharacteristic
40#10625014C2212yescomplemented
41C22⋊(C23⋊C8)C2211yescharacteristic
42(C2×D23)⋊C8C2212yescomplemented
43#10624998C2212yescomplemented
44#10624992C2212yescomplemented
45C24×C23—21no—
46C23×D23D411yescharacteristic
47C2×Q8×C23C2311yescharacteristic
48C2×D4×C23—24no—
49C2×D4×C23—21no—
50C2×D4×C23C2312yes—
51C22×C4×C23C2311yescharacteristic
52C2×C22⋊D23—28no—
53C2×C22⋊D23—22no—
54C2×C22⋊D23—22no—
55C22×Dic23—28no—
56C22×Dic23—21no—
57C22×Dic23—22no—
58C22×Dic23—22no—
59C22×Dic23D411yescharacteristic
60Q8×D23—24no—
61#67290—28no—
62#67290—28no—
63#67290—24no—
64#67290—24no—
65#67290—24no—
66#67290D418yes—
67C2×C4×D23—24no—
68C2×C4×D23—21no—
69C2×C4×D23—21no—
70C2×C4×D23D411yescharacteristic
71C2×C4×D23D411yescharacteristic
72C2×C4×D23C2311yescharacteristic
73C2×C4×D23C2311yescharacteristic
74C2×Dic46—21no—
75C2×Dic46—24no—
76C2×Dic46—21no—
77C2×Dic46C2311yescharacteristic
78C2×Dic46C2312yes—
79SD16×C23—24no—
80C2×C8×C23C2312yes—
81C4⋊C4×C23—21no—
82C4⋊C4×C23C2311yescharacteristic
83C22⋊C4×C23—21no—
84C22⋊Dic23—21no—
85#67275—24no—
86#67273—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—
95Dic92—24no—
96#67262—24no—
97C23×C23—24no—
98C23×C23—22no—
99C23×C23—42no—
100C23×C23—42no—
101C23×C23C2×D411yescharacteristic
102C22×D23—22no—
103C22×D23—22no—
104C22×D23—41no—
105C22×D23C2×D411yescharacteristic
106C22×D23C2×D411yescharacteristic
107Q8×C23—22no—
108D4×C23—24no—
109D4×C23—22no—
110D4×C23C2×D414yes—
111C2×C4×C23—22no—
112C2×C4×C23—21no—
113C2×C4×C23—21no—
114C2×C4×C23C2×D411yescharacteristic
115C2×C4×C23C2411yescharacteristic
116C22⋊D23—28no—
117C22⋊D23—28no—
118C2×Dic23—24no—
119C2×Dic23—28no—
120C2×Dic23—22no—
121C2×Dic23—22no—
122C2×Dic23—28no—
123C2×Dic23—21no—
124C2×Dic23—21no—
125C2×Dic23C2×D411yescharacteristic
126C2×Dic23C2×D411yescharacteristic
127C4×D23—22no—
128C4×D23—22no—
129C4×D23—22no—
130C4×D23—42no—
131C4×D23—42no—
132C4×D23C2×D414yes—
133Dic46—24no—
134Dic46—22no—
135Dic46—22no—
136Dic46—42no—
137Dic46C2×D414yes—
138C8×C23—22no—
139C23⋊C8—22no—
140#3007—231nomaximal, Sylow
141C22×C23—21no—
142C22×C23—21no—
143C22×C23—24no—
144C22×C23—48no—
145C22×C23—42no—
146C22×C23—42no—
147C22×C23#13712yescomplemented
148C22×C23C22×D411yescharacteristic
149C2×D23—21no—
150C2×D23—21no—
151C2×D23—21no—
152C2×D23—21no—
153C2×D23—21no—
154C2×D23—44no—
155C2×D23#12012yes—
156C4×C23—21no—
157C4×C23—41no—
158C4×C23—41no—
159C4×C23C22×D412yes—
160Dic23—24no—
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#451—231no—
172#450—231no—
173#450D2311yescharacteristic, complemented
174#417—232no—
175#407—231no—
176C2×C23—22no—
177C2×C23—44no—
178C2×C23C2×#12011yescharacteristic
179C2×C23C2×#13711yescharacteristic
180C2×C23#57511yescharacteristic
181D23—22no—
182D23—41no—
183C2×#41—232no—
184C2×#41—232no—
185C2×#41—461no—
186C2×#41—461no—
187C2×#41—468no—
188C22×Q8—231no—
189C22×Q8—231no—
190C22×D4—461no—
191C22×D4C2×D2311yescharacteristic
192C23×C4—231no—
193C23×C4—461no—
194#137—468no—
195C2×Q16—232no—
196C2×Q16—232no—
197C2×SD16—232no—
198C2×SD16C2×D2312yescomplemented
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#103—232no—
211#102—232no—
212#102C2×D2312yescomplemented
213C22⋊C8—231no—
214C22⋊C8—232no—
215C22⋊C8C2×D2311yescharacteristic
216C23#300711yescharacteristic, complemented, Sylow
217C24—21no—
218C24—231no—
219#41—238no—
220#41—464no—
221#41—464no—
222#41—468no—
223#41—468no—
224#41—464no—
225C2×Q8—232no—
226C2×Q8—231no—
227C2×Q8—461no—
228C2×Q8—464no—
229C2×Q8—464no—
230C2×Q8—461no—
231C2×Q8C22×D2311yescharacteristic
232C2×D4—24no—
233C2×D4—21no—
234C2×D4—462no—
235C2×D4—462no—
236C2×D4—468no—
237C2×D4C22×D2312yes—
238C22×C4—231no—
239C22×C4—231no—
240C22×C4—231no—
241C22×C4—231no—
242C22×C4—231no—
243C22×C4—464no—
244C22×C4—461no—
245C22×C4—461no—
246C22×C4—462no—
247C22×C4—462no—
248C22×C4—468no—
249C22×C4—461no—
250C22×C4C22×D2311yescharacteristic
251Q16—464no—
252Q16—464no—
253SD16—24no—
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—24no—
270C23—22no—
271C23—42no—
272C23—42no—
273C23—231no—
274C23—231no—
275C23—462no—
276C23—462no—
277C23—921no—
278C23D4×D2311yescharacteristic
279Q8—22no—
280Q8—234no—
281Q8—464no—
282Q8—462no—
283Q8—462no—
284Q8—922no—
285D4—24no—
286D4—22no—
287D4—468no—
288D4—468no—
289D4D4×D2314yes—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—231no—
294C2×C4—234no—
295C2×C4—231no—
296C2×C4—462no—
297C2×C4—462no—
298C2×C4—462no—
299C2×C4—468no—
300C2×C4—464no—
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—24no—
315C22—48no—
316C22—42no—
317C22—42no—
318C22—232no—
319C22—461no—
320C22—461no—
321C22—461no—
322C22—461no—
323C22—461no—
324C22—924no—
325C22#1062508612yescomplemented
326C22C2×D4×D2311yescharacteristic
327C4—21no—
328C4—41no—
329C4—41no—
330C4—232no—
331C4—461no—
332C4—464no—
333C4—921no—
334C4—921no—
335C4C2×D4×D2312yes—
336C2—22no—
337C2—44no—
338C2—462no—
339C2—921no—
340C2#120×D2311yescharacteristic
341C2C2×#1062508611yescharacteristic
342C2#1309321311yescharacteristic
subgroupG/H
centerC22C2×D4×D23
commutator, cosocleC2×C4×C23C24
FrattiniC2×C4C23×D23
FittingC23×#450C2
socleC22×C23C22×D4
prime radicalC22×#67290C2
hypercenter#450D23
hypocenterC23#3007
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
2. C2×#10625086C2semidirect
3. C22⋊Q8×D23C2semidirect
4. #13092580C2semidirect
5. #13092576C2semidirect
6. #13092575C2semidirect
7. #13092572C2semidirect
8. #13092568C2semidirect
9. #13092415C2semidirect
10. #13092379C2semidirect
11. C23×#450C2semidirect
15. C2×Q8×D23C22semidirect
23. C2×SD16×C23C22semidirect
25. #10625117C22semidirect
26. C2×#67275C22semidirect
27. C2×#67273C22semidirect
30. C2×Dic92C22semidirect
31. C2×#67262C22semidirect
32. #10625068C22semidirect
35. C4⋊C4×D23C22semidirect
38. C23×#102C22semidirect
40. #10625014C22semidirect
42. (C2×D23)⋊C8C22semidirect
43. #10624998C22semidirect
44. #10624992C22semidirect
147. C22×C23#137semidirect
173. #450D23semidirect
198. C2×SD16C2×D23semidirect
212. #102C2×D23semidirect
216. C23#3007semidirect
325. C22#10625086semidirect

Maximal subgroups

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

subgroupindex
C22×#672902
C2×#106250862
C22⋊Q8×D232
#130925802
#130925762
#130925752
#130925722
#130925682
#130924152
#130923792
C23×#4502
#300723