Finite Group Databasebeta

← #15676632

Subgroups — #15676632

Order 2944.

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

normal subgroups only

This group has 696 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
1C2×D4×Dic23C212yesmaximal, complemented
2C2×D4×Dic23C211yesmaximal, characteristic, complemented
3C2×#10625048C211yesmaximal, characteristic, complemented
4#13092349C211yesmaximal, characteristic, complemented
5#13092342C211yesmaximal, characteristic, complemented
6#13092340C212yesmaximal, complemented
7#13092331C211yesmaximal, characteristic, complemented
8#13092331C212yesmaximal, complemented
9#13092268C211yesmaximal, characteristic, complemented
10C22⋊C4×Dic23C212yesmaximal, complemented
11C23×#392C211yesmaximal, characteristic
12C22×D4×C23C2211yescharacteristic
13C22×D4×C23C2212yes—
14C23×Dic23C2212yes—
15C23×Dic23C2211yescharacteristic
16C2×C22⋊C4×C23C2211yescharacteristic
17C2×C22⋊C4×C23C2212yes—
18C2×C22⋊Dic23C2211yescharacteristic
19C2×C22⋊Dic23C2212yes—
20C2×C22⋊Dic23C2212yes—
21C2×C22⋊Dic23C2212yes—
22C2×C22⋊Dic23C2212yes—
23D4×Dic23—28no—
24D4×Dic23—24no—
25C2×C4⋊Dic23C2212yes—
26C2×C4⋊Dic23C2211yescharacteristic
27C2×Dic23⋊C4C2211yescharacteristic
28C2×Dic23⋊C4C2211yescharacteristic
29C2×Dic23⋊C4C2212yes—
30C2×C4×Dic23C2211yescharacteristic
31C2×C4×Dic23C2212yes—
32C2×C4×Dic23C2211yescharacteristic
33#10625048—24no—
34C23×#95C2211yescharacteristic
35C23⋊#95C2212yes—
36C23⋊#95C2212yes—
37C23⋊#95C2211yescharacteristic
38C24×C23C2311yescharacteristic
39C24×C23C2312yes—
40C2×D4×C23—24no—
41C2×D4×C23—24no—
42C2×D4×C23—24no—
43C2×D4×C23—22no—
44C2×D4×C23—24no—
45C22×C4×C23C2311yescharacteristic
46C22×C4×C23C2311yescharacteristic
47C22×C4×C23C2312yes—
48C22×Dic23—22no—
49C22×Dic23—22no—
50C22×Dic23—24no—
51C22×Dic23—24no—
52C22×Dic23—24no—
53C22×Dic23D414yescomplemented
54C22×Dic23C2311yescharacteristic
55C22×Dic23C2312yes—
56C22×Dic23C2312yes—
57C22×Dic23C2311yescharacteristic
58C22×Dic23C2311yescharacteristic
59C22×Dic23C2311yescharacteristic
60C22⋊C4×C23—22no—
61C22⋊C4×C23—24no—
62C22⋊Dic23—24no—
63C22⋊Dic23—24no—
64C22⋊Dic23—22no—
65C22⋊Dic23—24no—
66C22⋊Dic23—24no—
67C4⋊Dic23—22no—
68C4⋊Dic23—24no—
69Dic23⋊C4—22no—
70Dic23⋊C4—24no—
71Dic23⋊C4—22no—
72C4×Dic23—22no—
73C4×Dic23—24no—
74C4×Dic23D414yescomplemented
75C23×C23—21no—
76C23×C23—22no—
77C23×C23—22no—
78C23×C23—22no—
79C23×C23—21no—
80C23×C23—22no—
81C23×C23—22no—
82C23×C23—22no—
83C23×C23—22no—
84C23×C23—22no—
85C23×C23C2×D412yes—
86C23×C23#4114yes—
87C23×C23C2411yescharacteristic
88D4×C23—28no—
89D4×C23—216no—
90C2×C4×C23—22no—
91C2×C4×C23—22no—
92C2×C4×C23—22no—
93C2×C4×C23—21no—
94C2×C4×C23—22no—
95C2×C4×C23C2×D412yes—
96C2×C4×C23#4114yes—
97C2×Dic23—21no—
98C2×Dic23—22no—
99C2×Dic23—21no—
100C2×Dic23—28no—
101C2×Dic23—21no—
102C2×Dic23—22no—
103C2×Dic23—22no—
104C2×Dic23—22no—
105C2×Dic23—22no—
106C2×Dic23—22no—
107C2×Dic23—22no—
108C2×Dic23—22no—
109C2×Dic23—48no—
110C2×Dic23C2×D412yes—
111C2×Dic23C2×D412yes—
112C2×Dic23C2×D412yes—
113C2×Dic23C2×D412yes—
114C2×Dic23#4112yes—
115#2392—231nomaximal, Sylow
116C22×C23—22no—
117C22×C23—24no—
118C22×C23—24no—
119C22×C23—24no—
120C22×C23—24no—
121C22×C23—42no—
122C22×C23—41no—
123C22×C23—42no—
124C22×C23—42no—
125C22×C23—42no—
126C22×C23—41no—
127C22×C23—42no—
128C22×C23C22×D411yescharacteristic
129C22×C23C22×D411yescharacteristic
130C22×C23C2×#4112yes—
131C22×C23C2×#4112yes—
132C22×C23C2×#4111yescharacteristic
133C4×C23—22no—
134C4×C23—24no—
135C4×C23—41no—
136Dic23—22no—
137Dic23—22no—
138Dic23—42no—
139Dic23—41no—
140Dic23—42no—
141Dic23#12114yes—
142C2×C4×D4—231no—
143C2×C4×D4—232no—
144C2×#117—231no—
145#398—231no—
146#397—231no—
147#396—232no—
148#394—232no—
149#394—231no—
150#392D2311yescharacteristic
151C4×C4⋊C4—231no—
152C4×C22⋊C4—232no—
153C2×C23—42no—
154C2×C23—44no—
155C2×C23C2×#12111yescharacteristic
156C2×C23#52511yescharacteristic
157C2×C23#52512yes—
158C2×C23#52912yes—
159C2×C23#53211yescharacteristic
160C22×D4C2×D2311yescharacteristic
161C22×D4C2×D2312yes—
162C23×C4—231no—
163C23×C4—232no—
164C4×D4—464no—
165C4×D4—468no—
166#117—464no—
167C2×C4⋊C4—231no—
168C2×C4⋊C4—231no—
169C2×C4⋊C4—232no—
170C2×C4⋊C4—231no—
171C2×C4⋊C4—232no—
172C2×C22⋊C4—231no—
173C2×C22⋊C4—232no—
174C2×C22⋊C4—232no—
175C2×C22⋊C4—232no—
176C2×C22⋊C4—232no—
177C2×C22⋊C4C2×D2311yescharacteristic
178C2×C22⋊C4C2×D2312yes—
179C2×C42—231no—
180C2×C42—231no—
181C2×C42—232no—
182#95—232no—
183#95—232no—
184#95—231no—
185#95C2×D2311yescharacteristic
186C23#239211yescharacteristic, complemented, Sylow
187C24C22×D2311yescharacteristic
188C24C22×D2312yes—
189C2×D4—24no—
190C2×D4—24no—
191C2×D4—24no—
192C2×D4—22no—
193C2×D4—24no—
194C22×C4—231no—
195C22×C4—234no—
196C22×C4—231no—
197C22×C4—231no—
198C22×C4—232no—
199C22×C4—231no—
200C22×C4—232no—
201C22×C4—462no—
202C22×C4—462no—
203C22×C4—464no—
204C22×C4—464no—
205C22×C4—464no—
206C22×C4C22×D2311yescharacteristic
207C22×C4C22×D2311yescharacteristic
208C22×C4C22×D2312yes—
209C4⋊C4—462no—
210C4⋊C4—464no—
211C4⋊C4—462no—
212C4⋊C4—462no—
213C4⋊C4—464no—
214C22⋊C4—22no—
215C22⋊C4—24no—
216C22⋊C4—462no—
217C22⋊C4—464no—
218C22⋊C4—464no—
219C22⋊C4—464no—
220C22⋊C4—464no—
221C42—234no—
222C42—462no—
223C42—464no—
224C23—21no—
225C23—22no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—21no—
233C23—22no—
234C23D4×D2312yes—
235C23#6729014yes—
236C23C23×D2311yescharacteristic
237D4—28no—
238D4—216no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—232no—
245C2×C4—232no—
246C2×C4—232no—
247C2×C4—232no—
248C2×C4—232no—
249C2×C4—468no—
250C2×C4—461no—
251C2×C4—462no—
252C2×C4—461no—
253C2×C4—462no—
254C2×C4—461no—
255C2×C4—462no—
256C2×C4—462no—
257C2×C4—462no—
258C2×C4—462no—
259C2×C4—462no—
260C2×C4—462no—
261C2×C4—928no—
262C2×C4D4×D2312yes—
263C2×C4#6729014yes—
264C22—22no—
265C22—24no—
266C22—24no—
267C22—24no—
268C22—24no—
269C22—42no—
270C22—41no—
271C22—42no—
272C22—42no—
273C22—42no—
274C22—41no—
275C22—42no—
276C22C2×D4×D2311yescharacteristic
277C22C2×D4×D2311yescharacteristic
278C22C2×#6729012yes—
279C22C2×#6729012yes—
280C22#41×D2311yescharacteristic
281C4—22no—
282C4—24no—
283C4—41no—
284C4—234no—
285C4—462no—
286C4—462no—
287C4—922no—
288C4—922no—
289C4—921no—
290C2—42no—
291C2—44no—
292C2#121×D2311yescharacteristic
293C2#1309292311yescharacteristic
294C2#1309292312yes—
295C2#1309294812yes—
296C2#1309296511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D23
commutator, socle, cosocleC23×C23C24
Fitting, prime radicalC23×#392C2
hypercenter#392D23
hypocenterC23#2392
LCSG ▷ C23×C23 ▷ C23
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×D4×Dic23C2semidirect
2. C2×D4×Dic23C2semidirect
3. C2×#10625048C2semidirect
4. #13092349C2semidirect
5. #13092342C2semidirect
6. #13092340C2semidirect
7. #13092331C2semidirect
8. #13092331C2semidirect
9. #13092268C2semidirect
10. C22⋊C4×Dic23C2semidirect
53. C22×Dic23D4semidirect
74. C4×Dic23D4semidirect
186. C23#2392semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic232
C2×#106250482
#130923492
#130923422
#130923402
#130923312
#130922682
C22⋊C4×Dic232
C23×#3922
#239223