Finite Group Databasebeta

← #15676611

Subgroups — #15676611

Order 2944.

944 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 944 conjugacy classes of subgroups — too many to draw. Only its 163 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, complemented
2C2×#10625064C211yesmaximal, characteristic, complemented
3C2×(C22⋊D23)⋊C4C212yesmaximal, complemented
4#13092337C211yesmaximal, characteristic, complemented
5#13092332C212yesmaximal, complemented
6#13092330C211yesmaximal, characteristic, complemented
7#13092326C212yesmaximal, complemented
8#13092325C211yesmaximal, characteristic, complemented
9C22⋊C4×Dic23C212yesmaximal, complemented
10#13092264C211yesmaximal, characteristic, complemented
11C23×#394C211yesmaximal, characteristic, complemented
12C22×D4×C23C2211yescharacteristic
13C22×C22⋊D23C2212yes—
14C22×C22⋊D23C2212yes—
15C23×Dic23C2212yes—
16C2×#67290—28no—
17C2×#67290—24no—
18C22×D92C2211yescharacteristic
19C22×C4×D23C2211yescharacteristic
20C22×C4×D23C2211yescharacteristic
21C22×Dic46C2211yescharacteristic
22C2×C4⋊C4×C23C2211yescharacteristic
23C2×C22⋊C4×C23C2212yes—
24C2×C22⋊C4×C23C2212yes—
25C2×C22⋊Dic23C2212yes—
26C2×(C2×D23)⋊C4C2212yes—
27C2×(C2×D23)⋊C4C2211yescharacteristic
28C2×(C2×D23)⋊C4C2211yescharacteristic
29C2×(C2×D23)⋊C4C2212yes—
30C2×Dic23⋊C4C2212yes—
31C2×C4×Dic23C2212yes—
32C2×C4×Dic23C2211yescharacteristic
33#10625064—24no—
34(C22⋊D23)⋊C4—28no—
35C23×#95C2211yescharacteristic
36C23⋊#95C2211yescharacteristic
37C23⋊#95C2212yes—
38C23⋊#95C2212yes—
39C24×C23C2312yes—
40C23×D23C2311yescharacteristic
41C23×D23C2311yescharacteristic
42C2×D4×C23—24no—
43C2×D4×C23—22no—
44C22×C4×C23C2312yes—
45C22×C4×C23C2311yescharacteristic
46C22×C4×C23C2311yescharacteristic
47C22×C4×C23C2311yescharacteristic
48C2×C22⋊D23—24no—
49C2×C22⋊D23—24no—
50C2×C22⋊D23—24no—
51C2×C22⋊D23—24no—
52C2×C22⋊D23—24no—
53C2×C22⋊D23—24no—
54C22×Dic23—24no—
55C22×Dic23—24no—
56C22×Dic23D418yescomplemented
57C22×Dic23C2311yescharacteristic
58C22×Dic23C2312yes—
59C22×Dic23C2312yes—
60C22×Dic23C2311yescharacteristic
61#67290—232no—
62C2×D92—22no—
63C2×D92—22no—
64C2×D92—22no—
65C2×C4×D23—24no—
66C2×C4×D23—24no—
67C2×C4×D23—22no—
68C2×C4×D23D414yescomplemented
69C2×Dic46—22no—
70C2×Dic46—24no—
71C4⋊C4×C23—22no—
72C22⋊C4×C23—24no—
73C22⋊C4×C23—24no—
74C22⋊Dic23—24no—
75(C2×D23)⋊C4—22no—
76(C2×D23)⋊C4—22no—
77(C2×D23)⋊C4—24no—
78(C2×D23)⋊C4—24no—
79Dic23⋊C4—24no—
80C4×Dic23—24no—
81C4×Dic23—22no—
82C23×C23—22no—
83C23×C23—22no—
84C23×C23—22no—
85C23×C23—22no—
86C23×C23—22no—
87C23×C23—22no—
88C23×C23C2×D414yes—
89C23×C23C2411yescharacteristic
90C22×D23—21no—
91C22×D23—22no—
92C22×D23—22no—
93C22×D23—22no—
94C22×D23—21no—
95C22×D23—21no—
96C22×D23—21no—
97C22×D23—22no—
98C22×D23C2×D412yes—
99C22×D23#4112yescomplemented
100D4×C23—28no—
101C2×C4×C23—21no—
102C2×C4×C23—22no—
103C2×C4×C23—21no—
104C2×C4×C23—22no—
105C2×C4×C23—22no—
106C2×C4×C23—21no—
107C2×C4×C23—22no—
108C2×C4×C23—22no—
109C2×C4×C23C2×D412yes—
110C2×C4×C23#4112yescomplemented
111C22⋊D23—216no—
112C22⋊D23—216no—
113C2×Dic23—22no—
114C2×Dic23—22no—
115C2×Dic23—22no—
116C2×Dic23—22no—
117C2×Dic23—216no—
118C2×Dic23—21no—
119C2×Dic23—22no—
120C2×Dic23C2×D414yes—
121C2×Dic23C2×D414yes—
122C2×Dic23C2×D412yes—
123C2×Dic23#4114yescomplemented
124D92—28no—
125C4×D23—28no—
126C4×D23—44no—
127Dic46—28no—
128#2388—231nomaximal, Sylow
129C22×C23—24no—
130C22×C23—24no—
131C22×C23—24no—
132C22×C23—42no—
133C22×C23—42no—
134C22×C23—42no—
135C22×C23—42no—
136C22×C23C22×D411yescharacteristic
137C22×C23C22×D412yes—
138C22×C23C2×#4111yescharacteristic
139C22×C23C2×#4111yescharacteristic
140C22×C23C2×#4112yes—
141C2×D23—22no—
142C2×D23—22no—
143C2×D23—22no—
144C2×D23—24no—
145C2×D23—22no—
146C2×D23—41no—
147C2×D23—42no—
148C2×D23—42no—
149C2×D23—42no—
150C2×D23—41no—
151C4×C23—22no—
152C4×C23—22no—
153C4×C23—41no—
154C4×C23—42no—
155Dic23—24no—
156Dic23—24no—
157Dic23—41no—
158Dic23#12014yes—
159C22×#41—231no—
160C2×C4×D4—231no—
161C2×C4×D4—232no—
162#395—231no—
163#394—232no—
164#394D2311yescharacteristic, complemented
165#393—231no—
166#392—231no—
167#392—232no—
168C4×C22⋊C4—232no—
169C4×C22⋊C4—231no—
170C2×C23—44no—
171C2×C23C2×#12011yescharacteristic
172C2×C23#52512yes—
173C2×C23#52511yescharacteristic
174C2×C23#53212yes—
175C2×C23#53211yescharacteristic
176D23—42no—
177D23—42no—
178C2×#41—468no—
179C2×#41—464no—
180C22×Q8—231no—
181C22×D4—232no—
182C22×D4—232no—
183C22×D4—231no—
184C22×D4C2×D2311yescharacteristic
185C23×C4—232no—
186C23×C4—231no—
187C23×C4—231no—
188C4×D4—468no—
189C4×D4—464no—
190C2×C4⋊C4—232no—
191C2×C4⋊C4C2×D2311yescharacteristic
192C2×C22⋊C4—232no—
193C2×C22⋊C4—232no—
194C2×C22⋊C4—231no—
195C2×C22⋊C4—231no—
196C2×C22⋊C4—232no—
197C2×C22⋊C4C2×D2312yes—
198C2×C22⋊C4C2×D2312yes—
199C2×C42—232no—
200C2×C42—231no—
201#95—231no—
202#95—232no—
203#95—232no—
204#95C2×D2311yescharacteristic
205C23#238811yescharacteristic, complemented, Sylow
206C24—231no—
207C24—231no—
208C24C22×D2312yes—
209#41—4632no—
210C2×Q8—464no—
211C2×Q8—462no—
212C2×D4—24no—
213C2×D4—22no—
214C2×D4—462no—
215C2×D4—462no—
216C2×D4—464no—
217C2×D4—464no—
218C2×D4—464no—
219C2×D4—464no—
220C2×D4—464no—
221C2×D4—464no—
222C2×D4—462no—
223C22×C4—231no—
224C22×C4—238no—
225C22×C4—232no—
226C22×C4—232no—
227C22×C4—234no—
228C22×C4—231no—
229C22×C4—464no—
230C22×C4—464no—
231C22×C4—462no—
232C22×C4—464no—
233C22×C4—464no—
234C22×C4C22×D2312yes—
235C22×C4C22×D2311yescharacteristic
236C22×C4C22×D2311yescharacteristic
237C22×C4C22×D2311yescharacteristic
238C4⋊C4—22no—
239C4⋊C4—464no—
240C22⋊C4—24no—
241C22⋊C4—24no—
242C22⋊C4—464no—
243C22⋊C4—464no—
244C22⋊C4—462no—
245C22⋊C4—464no—
246C22⋊C4—462no—
247C42—462no—
248C42—464no—
249C23—22no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—232no—
256C23—232no—
257C23—462no—
258C23—461no—
259C23—462no—
260C23—461no—
261C23—461no—
262C23—462no—
263C23—462no—
264C23—461no—
265C23D4×D2314yes—
266C23C23×D2311yescharacteristic
267Q8—468no—
268D4—28no—
269D4—468no—
270D4—4616no—
271D4—4616no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—234no—
281C2×C4—232no—
282C2×C4—234no—
283C2×C4—234no—
284C2×C4—468no—
285C2×C4—4616no—
286C2×C4—462no—
287C2×C4—462no—
288C2×C4—462no—
289C2×C4—462no—
290C2×C4—461no—
291C2×C4—462no—
292C2×C4—924no—
293C2×C4D4×D2312yes—
294C2×C4#6729012yescomplemented
295C22—24no—
296C22—24no—
297C22—24no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—462no—
303C22—464no—
304C22—462no—
305C22—462no—
306C22—462no—
307C22—921no—
308C22—922no—
309C22—922no—
310C22—922no—
311C22—921no—
312C22C2×D4×D2312yes—
313C22C2×D4×D2311yescharacteristic
314C22C2×#6729011yescharacteristic
315C22#41×D2311yescharacteristic
316C22#41×D2312yes—
317C4—22no—
318C4—22no—
319C4—41no—
320C4—42no—
321C4—234no—
322C4—464no—
323C4—464no—
324C4—921no—
325C2—44no—
326C2—922no—
327C2—922no—
328C2#120×D2311yescharacteristic
329C2#1309292512yes—
330C2#1309292711yescharacteristic
331C2#1309296212yes—
332C2#1309296611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D23
commutator, socle, cosocleC23×C23C24
FittingC23×#394C2
hypercenter#394D23
hypocenterC23#2388
LCSG ▷ C23×C23 ▷ C23
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C22×#67290C2semidirect
2. C2×#10625064C2semidirect
3. C2×(C22⋊D23)⋊C4C2semidirect
4. #13092337C2semidirect
5. #13092332C2semidirect
6. #13092330C2semidirect
7. #13092326C2semidirect
8. #13092325C2semidirect
9. C22⋊C4×Dic23C2semidirect
10. #13092264C2semidirect
11. C23×#394C2semidirect
56. C22×Dic23D4semidirect
68. C2×C4×D23D4semidirect
99. C22×D23#41semidirect
110. C2×C4×C23#41semidirect
123. C2×Dic23#41semidirect
164. #394D23semidirect
205. C23#2388semidirect
294. 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×#106250642
C2×(C22⋊D23)⋊C42
#130923372
#130923322
#130923302
#130923262
#130923252
C22⋊C4×Dic232
#130922642
C23×#3942
#238823