Finite Group Databasebeta

← #15676727

Subgroups — #15676727

Order 2944.

864 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 159 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×#10625054C212yesmaximal, complemented
3C2×C4⋊C4×D23C211yesmaximal, characteristic, complemented
4#13092338C211yesmaximal, characteristic, complemented
5#13092337C211yesmaximal, characteristic, complemented
6#13092331C211yesmaximal, characteristic, complemented
7#13092293C212yesmaximal, complemented
8#13092292C212yesmaximal, complemented
9#13092289C212yesmaximal, complemented
10#13092279C211yesmaximal, characteristic, complemented
11C23×#394C211yesmaximal, characteristic, complemented
12C22×D4×C23C2211yescharacteristic
13C22×C22⋊D23C2212yes—
14C23×Dic23C2212yes—
15C2×#67290—28no—
16C2×#67290—24no—
17C22×C4×D23C2211yescharacteristic
18C22×C4×D23C2211yescharacteristic
19C22×C4×D23C2211yescharacteristic
20C22×Dic46C2211yescharacteristic
21C2×C4⋊C4×C23C2211yescharacteristic
22C2×C22⋊C4×C23C2212yes—
23C2×C22⋊C4×C23C2212yes—
24C2×C22⋊Dic23C2212yes—
25C2×C22⋊Dic23C2212yes—
26C2×(C2×D23)⋊C4C2212yes—
27C2×C4⋊Dic23C2211yescharacteristic
28C2×C4⋊Dic23C2212yes—
29C2×Dic23⋊C4C2211yescharacteristic
30C2×Dic23⋊C4C2211yescharacteristic
31C2×Dic23⋊C4C2212yes—
32C4⋊C4×D23—24no—
33#10625054—28no—
34C23×#95C2211yescharacteristic
35C23⋊#95C2212yes—
36C23⋊#95C2211yescharacteristic
37C23⋊#95C2212yes—
38C23⋊#95C2212yes—
39C24×C23C2312yes—
40C23×D23C2311yescharacteristic
41C2×D4×C23—24no—
42C2×D4×C23—22no—
43C22×C4×C23C2311yescharacteristic
44C22×C4×C23C2312yes—
45C22×C4×C23C2311yescharacteristic
46C22×C4×C23C2311yescharacteristic
47C2×C22⋊D23—24no—
48C2×C22⋊D23—24no—
49C2×C22⋊D23D418yes—
50C22×Dic23—24no—
51C22×Dic23—24no—
52C22×Dic23—24no—
53C22×Dic23C2311yescharacteristic
54C22×Dic23C2312yes—
55C22×Dic23C2312yes—
56C22×Dic23C2311yescharacteristic
57C22×Dic23C2311yescharacteristic
58#67290—232no—
59C2×C4×D23—22no—
60C2×C4×D23—24no—
61C2×C4×D23—22no—
62C2×C4×D23—22no—
63C2×C4×D23—24no—
64C2×C4×D23—22no—
65C2×C4×D23D414yes—
66C2×Dic46—22no—
67C2×Dic46—24no—
68C4⋊C4×C23—22no—
69C22⋊C4×C23—24no—
70C22⋊C4×C23—24no—
71C22⋊Dic23—24no—
72C22⋊Dic23—24no—
73(C2×D23)⋊C4—24no—
74C4⋊Dic23—24no—
75C4⋊Dic23—22no—
76Dic23⋊C4—22no—
77Dic23⋊C4—24no—
78Dic23⋊C4—22no—
79C23×C23—22no—
80C23×C23—22no—
81C23×C23—22no—
82C23×C23—22no—
83C23×C23—22no—
84C23×C23—22no—
85C23×C23C2×D414yes—
86C23×C23C2411yescharacteristic
87C22×D23—21no—
88C22×D23—21no—
89C22×D23—22no—
90C22×D23C2×D414yes—
91C22×D23C2×D412yes—
92D4×C23—28no—
93C2×C4×C23—21no—
94C2×C4×C23—22no—
95C2×C4×C23—21no—
96C2×C4×C23—22no—
97C2×C4×C23—21no—
98C2×C4×C23—22no—
99C2×C4×C23—22no—
100C2×C4×C23—22no—
101C2×C4×C23C2×D412yes—
102C2×C4×C23#4112yescomplemented
103C22⋊D23—216no—
104C2×Dic23—216no—
105C2×Dic23—21no—
106C2×Dic23—22no—
107C2×Dic23—22no—
108C2×Dic23—22no—
109C2×Dic23—21no—
110C2×Dic23—21no—
111C2×Dic23—22no—
112C2×Dic23—22no—
113C2×Dic23—22no—
114C2×Dic23—22no—
115C2×Dic23C2×D414yes—
116C2×Dic23C2×D412yes—
117C2×Dic23#4112yescomplemented
118C4×D23—28no—
119C4×D23—28no—
120C4×D23—44no—
121Dic46—28no—
122#2406—231nomaximal, Sylow
123C22×C23—24no—
124C22×C23—24no—
125C22×C23—24no—
126C22×C23—42no—
127C22×C23—42no—
128C22×C23—42no—
129C22×C23—42no—
130C22×C23C22×D411yescharacteristic
131C22×C23C22×D412yes—
132C22×C23C2×#4111yescharacteristic
133C22×C23C2×#4111yescharacteristic
134C22×C232−1+412yes—
135C2×D23—22no—
136C2×D23—22no—
137C2×D23—24no—
138C2×D23—24no—
139C2×D23#12014yes—
140C4×C23—22no—
141C4×C23—22no—
142C4×C23—41no—
143C4×C23—42no—
144Dic23—24no—
145Dic23—22no—
146Dic23—22no—
147Dic23—41no—
148Dic23—42no—
149C22×#41—231no—
150C2×#123—232no—
151C22×C4⋊C4—231no—
152#395—231no—
153#395—231no—
154#394—231no—
155#394D2311yescharacteristic, complemented
156#386—232no—
157#385—232no—
158#385—232no—
159#381—231no—
160C2×C23—44no—
161C2×C23C2×#12011yescharacteristic
162C2×C23#52511yescharacteristic
163C2×C23#53611yescharacteristic
164C2×C23#54712yes—
165C2×C23#54712yes—
166D23—24no—
167C2×#41—464no—
168C2×#41—468no—
169C22×Q8—231no—
170C22×D4—232no—
171C22×D4C2×D2311yescharacteristic
172C23×C4—231no—
173C23×C4—232no—
174C23×C4—231no—
175C23×C4—231no—
176#123—468no—
177C2×C4⋊C4—231no—
178C2×C4⋊C4—231no—
179C2×C4⋊C4—231no—
180C2×C4⋊C4—232no—
181C2×C4⋊C4—232no—
182C2×C4⋊C4—464no—
183C2×C4⋊C4C2×D2311yescharacteristic
184C2×C22⋊C4—232no—
185C2×C22⋊C4—232no—
186C2×C22⋊C4—232no—
187C2×C22⋊C4C2×D2312yes—
188C2×C22⋊C4C2×D2312yes—
189#95—232no—
190#95—231no—
191#95—232no—
192#95—232no—
193#95C2×D2311yescharacteristic
194C23#240611yescharacteristic, complemented, Sylow
195C24—231no—
196C24C22×D2312yes—
197#41—4632no—
198C2×Q8—464no—
199C2×Q8—462no—
200C2×D4—24no—
201C2×D4—22no—
202C2×D4—238no—
203C2×D4—464no—
204C2×D4—464no—
205C22×C4—231no—
206C22×C4—231no—
207C22×C4—232no—
208C22×C4—234no—
209C22×C4—232no—
210C22×C4—231no—
211C22×C4—462no—
212C22×C4—462no—
213C22×C4—464no—
214C22×C4—462no—
215C22×C4—464no—
216C22×C4—464no—
217C22×C4—464no—
218C22×C4—464no—
219C22×C4—462no—
220C22×C4C22×D2312yes—
221C22×C4C22×D2311yescharacteristic
222C22×C4C22×D2311yescharacteristic
223C22×C4C22×D2311yescharacteristic
224C4⋊C4—22no—
225C4⋊C4—462no—
226C4⋊C4—462no—
227C4⋊C4—464no—
228C4⋊C4—464no—
229C4⋊C4—462no—
230C22⋊C4—24no—
231C22⋊C4—24no—
232C22⋊C4—464no—
233C22⋊C4—464no—
234C22⋊C4—464no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—234no—
242C23—232no—
243C23—461no—
244C23—461no—
245C23—462no—
246C23D4×D2314yes—
247C23C23×D2311yescharacteristic
248Q8—468no—
249D4—28no—
250D4—4616no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—232no—
260C2×C4—234no—
261C2×C4—232no—
262C2×C4—468no—
263C2×C4—468no—
264C2×C4—4616no—
265C2×C4—461no—
266C2×C4—462no—
267C2×C4—462no—
268C2×C4—462no—
269C2×C4—461no—
270C2×C4—462no—
271C2×C4—462no—
272C2×C4—461no—
273C2×C4—462no—
274C2×C4—462no—
275C2×C4—924no—
276C2×C4D4×D2312yes—
277C2×C4#6729012yescomplemented
278C22—24no—
279C22—24no—
280C22—24no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—234no—
286C22—462no—
287C22—462no—
288C22—464no—
289C22—464no—
290C22C2×D4×D2311yescharacteristic
291C22C2×D4×D2312yes—
292C22C2×#6729011yescharacteristic
293C22#41×D2311yescharacteristic
294C22#1062516112yes—
295C4—22no—
296C4—22no—
297C4—41no—
298C4—42no—
299C4—462no—
300C4—464no—
301C4—462no—
302C4—921no—
303C4—922no—
304C2—44no—
305C2—464no—
306C2#120×D2311yescharacteristic
307C2#1309292711yescharacteristic
308C2#1309298711yescharacteristic
309C2#1309306512yes—
310C2#1309306712yes—
subgroupG/H
center, FrattiniC23C23×D23
commutator, socle, cosocleC23×C23C24
FittingC23×#394C2
prime radicalC22×#67290C2
hypercenter#394D23
hypocenterC23#2406
LCSG ▷ C23×C23 ▷ C23
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#10625054C2semidirect
3. C2×C4⋊C4×D23C2semidirect
4. #13092338C2semidirect
5. #13092337C2semidirect
6. #13092331C2semidirect
7. #13092293C2semidirect
8. #13092292C2semidirect
9. #13092289C2semidirect
10. #13092279C2semidirect
11. C23×#394C2semidirect
102. C2×C4×C23#41semidirect
117. C2×Dic23#41semidirect
155. #394D23semidirect
194. C23#2406semidirect
277. C2×C4#67290semidirect

Maximal subgroups

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

subgroupindex
C22×#672902
C2×#106250542
C2×C4⋊C4×D232
#130923382
#130923372
#130923312
#130922932
#130922922
#130922892
#130922792
C23×#3942
#240623