Finite Group Databasebeta

← #15676664

Subgroups — #15676664

Order 2944.

528 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 155 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×Q8×Dic23C211yesmaximal, characteristic, complemented
2C2×#10625048C211yesmaximal, characteristic
3#13092357C211yesmaximal, characteristic, complemented
4#13092353C211yesmaximal, characteristic
5#13092351C212yesmaximal, complemented
6#13092340C212yesmaximal
7#13092336C211yesmaximal, characteristic, complemented
8#13092268C212yesmaximal, complemented
9#13092268C211yesmaximal, characteristic, complemented
10C22⋊C4×Dic23C212yesmaximal
11C23×#393C211yesmaximal, characteristic
12C22×Q8×C23C2211yescharacteristic
13C23×Dic23C2211yescharacteristic
14C2×C22⋊C4×C23C2211yescharacteristic
15C2×C22⋊C4×C23C2212yes—
16C2×C22⋊Dic23C2212yes—
17C2×C22⋊Dic23C2211yescharacteristic
18Q8×Dic23—24no—
19C2×C4⋊Dic23C2211yescharacteristic
20C2×C4⋊Dic23C2212yes—
21C2×Dic23⋊C4C2212yes—
22C2×Dic23⋊C4C2211yescharacteristic
23C2×Dic23⋊C4C2211yescharacteristic
24C2×Dic23⋊C4C2212yes—
25C2×Dic23⋊C4C2212yes—
26C2×C4×Dic23C2211yescharacteristic
27C2×C4×Dic23C2212yes—
28C2×C4×Dic23C2212yes—
29C2×C4×Dic23C2211yescharacteristic
30#10625048—24no—
31C23×#95C2211yescharacteristic
32C23×#95C2212yes—
33C23⋊#95C2211yescharacteristic
34C23⋊#95C2212yes—
35C23⋊#95C2212yes—
36C23⋊#95C2212yes—
37C24×C23C2311yescharacteristic
38C2×Q8×C23—22no—
39C2×Q8×C23—24no—
40C22×C4×C23C2311yescharacteristic
41C22×C4×C23C2312yes—
42C22×C4×C23C2311yescharacteristic
43C22×C4×C23C2312yes—
44C22×Dic23—22no—
45C22×Dic23—22no—
46C22×Dic23Q814yescomplemented
47C22×Dic23C2311yescharacteristic
48C22×Dic23C2312yes—
49C22×Dic23C2311yescharacteristic
50C22×Dic23C2311yescharacteristic
51C22×Dic23C2312yes—
52C22×Dic23C2311yescharacteristic
53C22⋊C4×C23—22no—
54C22⋊C4×C23—24no—
55C22⋊Dic23—22no—
56C22⋊Dic23—24no—
57C4⋊Dic23—22no—
58C4⋊Dic23—24no—
59Dic23⋊C4—22no—
60Dic23⋊C4—24no—
61Dic23⋊C4—22no—
62Dic23⋊C4—24no—
63Dic23⋊C4—24no—
64C4×Dic23—22no—
65C4×Dic23—24no—
66C4×Dic23—24no—
67C4×Dic23D414yes—
68C23×C23—21no—
69C23×C23—22no—
70C23×C23—22no—
71C23×C23—21no—
72C23×C23C2×Q812yes—
73C23×C23C2411yescharacteristic
74Q8×C23—28no—
75C2×C4×C23—22no—
76C2×C4×C23—22no—
77C2×C4×C23—22no—
78C2×C4×C23—21no—
79C2×C4×C23—22no—
80C2×C4×C23—22no—
81C2×C4×C23—22no—
82C2×C4×C23—22no—
83C2×C4×C23C2×D412yes—
84C2×C4×C23#4114yes—
85C2×Dic23—21no—
86C2×Dic23—22no—
87C2×Dic23—22no—
88C2×Dic23—22no—
89C2×Dic23—21no—
90C2×Dic23—28no—
91C2×Dic23—21no—
92C2×Dic23—22no—
93C2×Dic23—22no—
94C2×Dic23—22no—
95C2×Dic23—22no—
96C2×Dic23C2×D412yes—
97C2×Dic23C2×D412yes—
98C2×Dic23C2×Q812yes—
99C2×Dic23C2×Q812yes—
100C2×Dic23#4114yes—
101C2×Dic23#4112yes—
102#2396—231nomaximal, Sylow
103C22×C23—22no—
104C22×C23—24no—
105C22×C23—41no—
106C22×C23—42no—
107C22×C23—41no—
108C22×C23C22×D411yescharacteristic
109C22×C23C22×Q811yescharacteristic
110C22×C23C2×#4112yes—
111C22×C23C2×#4111yescharacteristic
112C22×C23C2×#4112yes—
113C4×C23—22no—
114C4×C23—24no—
115C4×C23—42no—
116C4×C23—41no—
117Dic23—22no—
118Dic23—24no—
119Dic23—22no—
120Dic23—41no—
121Dic23—42no—
122Dic23C22⋊Q814yes—
123C2×C4×Q8—231no—
124C2×#117—231no—
125#400—231no—
126#399—231no—
127#398—232no—
128#396—232no—
129#395—231no—
130#393D2311yescharacteristic
131C4×C4⋊C4—231no—
132C4×C4⋊C4—232no—
133C4×C22⋊C4—232no—
134C2×C23—42no—
135C2×C23C2×C22⋊Q811yescharacteristic
136C2×C23#52511yescharacteristic
137C2×C23#52912yes—
138C2×C23#53211yescharacteristic
139C2×C23#53312yes—
140C22×Q8C2×D2311yescharacteristic
141C23×C4—231no—
142C4×Q8—464no—
143#117—464no—
144C2×C4⋊C4—231no—
145C2×C4⋊C4—231no—
146C2×C4⋊C4—232no—
147C2×C4⋊C4—231no—
148C2×C4⋊C4—232no—
149C2×C4⋊C4—232no—
150C2×C4⋊C4—232no—
151C2×C22⋊C4—231no—
152C2×C22⋊C4—232no—
153C2×C22⋊C4C2×D2311yescharacteristic
154C2×C22⋊C4C2×D2312yes—
155C2×C42—231no—
156C2×C42—231no—
157C2×C42—232no—
158C2×C42—232no—
159#95—232no—
160#95—232no—
161#95—231no—
162#95—232no—
163#95C2×D2312yes—
164#95C2×D2311yescharacteristic
165C23#239611yescharacteristic, complemented, Sylow
166C24C22×D2311yescharacteristic
167C2×Q8—22no—
168C2×Q8—24no—
169C22×C4—231no—
170C22×C4—234no—
171C22×C4—231no—
172C22×C4—232no—
173C22×C4—231no—
174C22×C4—231no—
175C22×C4—232no—
176C22×C4—462no—
177C22×C4—462no—
178C22×C4C22×D2312yes—
179C22×C4C22×D2311yescharacteristic
180C22×C4C22×D2311yescharacteristic
181C22×C4C22×D2312yes—
182C4⋊C4—462no—
183C4⋊C4—464no—
184C4⋊C4—464no—
185C4⋊C4—464no—
186C4⋊C4—462no—
187C4⋊C4—462no—
188C4⋊C4—464no—
189C22⋊C4—22no—
190C22⋊C4—24no—
191C22⋊C4—462no—
192C22⋊C4—464no—
193C42—234no—
194C42—462no—
195C42—464no—
196C42—464no—
197C23—21no—
198C23—22no—
199C23—22no—
200C23—21no—
201C23Q8×D2312yes—
202C23C23×D2311yescharacteristic
203Q8—28no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—232no—
213C2×C4—232no—
214C2×C4—232no—
215C2×C4—232no—
216C2×C4—232no—
217C2×C4—234no—
218C2×C4—468no—
219C2×C4—461no—
220C2×C4—462no—
221C2×C4—462no—
222C2×C4—462no—
223C2×C4—461no—
224C2×C4—461no—
225C2×C4—462no—
226C2×C4—462no—
227C2×C4—462no—
228C2×C4—462no—
229C2×C4D4×D2312yes—
230C2×C4#6729014yes—
231C22—22no—
232C22—24no—
233C22—41no—
234C22—42no—
235C22—41no—
236C22C2×D4×D2311yescharacteristic
237C22C2×#6729012yes—
238C22C2×Q8×D2311yescharacteristic
239C22#41×D2311yescharacteristic
240C22#41×D2312yes—
241C4—22no—
242C4—24no—
243C4—42no—
244C4—41no—
245C4—234no—
246C4—462no—
247C4—464no—
248C4—462no—
249C4—921no—
250C4—922no—
251C2—42no—
252C2C22⋊Q8×D2311yescharacteristic
253C2#1309292311yescharacteristic
254C2#1309294612yes—
255C2#1309296611yescharacteristic
256C2#1309296912yes—
subgroupG/H
center, FrattiniC23C23×D23
commutator, socle, cosocleC23×C23C24
FittingC23×#393C2
prime radicalC24×C23C23
hypercenter#393D23
hypocenterC23#2396
LCSG ▷ C23×C23 ▷ C23
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×Q8×Dic23C2semidirect
3. #13092357C2semidirect
5. #13092351C2semidirect
7. #13092336C2semidirect
8. #13092268C2semidirect
9. #13092268C2semidirect
46. C22×Dic23Q8semidirect
165. C23#2396semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×Dic232
C2×#106250482
#130923572
#130923532
#130923512
#130923402
#130923362
#130922682
C22⋊C4×Dic232
C23×#3932
#239623