Finite Group Databasebeta

← #15676668

Subgroups — #15676668

Order 2944.

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

normal subgroups only

This group has 528 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×Q8×Dic23C211yesmaximal, characteristic, complemented
2C2×#10625063C211yesmaximal, characteristic
3#13092356C211yesmaximal, characteristic, complemented
4#13092354C211yesmaximal, characteristic
5#13092349C212yesmaximal, complemented
6#13092339C212yesmaximal
7#13092328C211yesmaximal, characteristic
8C4⋊C4×Dic23C212yesmaximal, complemented
9#13092268C211yesmaximal, characteristic, complemented
10#13092264C212yesmaximal
11C23×#395C211yesmaximal, characteristic, complemented
12C22×Q8×C23C2211yescharacteristic
13C22×C4×D23C2211yescharacteristic
14C2×C4⋊C4×C23C2211yescharacteristic
15C2×C4⋊C4×C23C2212yes—
16Q8×Dic23—24no—
17C2×(C2×D23)⋊C4C2211yescharacteristic
18C2×(C2×D23)⋊C4C2212yes—
19C2×(C2×D23)⋊C4C2211yescharacteristic
20C2×(C2×D23)⋊C4C2212yes—
21C2×C4⋊Dic23C2211yescharacteristic
22C2×C4⋊Dic23C2212yes—
23C2×C4⋊Dic23C2211yescharacteristic
24C2×C4⋊Dic23C2212yes—
25C2×Dic23⋊C4C2212yes—
26C2×C4×Dic23C2211yescharacteristic
27C2×C4×Dic23C2212yes—
28C2×C4×Dic23C2212yes—
29C2×C4×Dic23C2211yescharacteristic
30#10625063—24no—
31C23×#95C2211yescharacteristic
32C23×#95C2212yes—
33C23⋊#95C2212yes—
34C23⋊#95C2211yescharacteristic
35C23⋊#95C2212yes—
36C23⋊#95C2212yes—
37C23×D23C2311yescharacteristic
38C2×Q8×C23—22no—
39C2×Q8×C23—24no—
40C22×C4×C23C2311yescharacteristic
41C22×C4×C23C2311yescharacteristic
42C22×C4×C23C2311yescharacteristic
43C22×C4×C23C2312yes—
44C22×C4×C23C2312yes—
45C22×Dic23C2312yes—
46C22×Dic23C2312yes—
47C22×Dic23C2311yescharacteristic
48C22×Dic23C2311yescharacteristic
49C22×Dic23C2311yescharacteristic
50C2×C4×D23—22no—
51C2×C4×D23—22no—
52C2×C4×D23Q814yescomplemented
53C4⋊C4×C23—22no—
54C4⋊C4×C23—24no—
55(C2×D23)⋊C4—22no—
56(C2×D23)⋊C4—24no—
57(C2×D23)⋊C4—24no—
58(C2×D23)⋊C4—22no—
59C4⋊Dic23—22no—
60C4⋊Dic23—24no—
61C4⋊Dic23—22no—
62C4⋊Dic23—24no—
63Dic23⋊C4—24no—
64C4×Dic23—22no—
65C4×Dic23—24no—
66C4×Dic23—24no—
67C4×Dic23D414yes—
68C23×C23C2411yescharacteristic
69C22×D23—21no—
70C22×D23—22no—
71C22×D23—22no—
72C22×D23—21no—
73C22×D23C2×Q812yes—
74Q8×C23—28no—
75C2×C4×C23—22no—
76C2×C4×C23—21no—
77C2×C4×C23—22no—
78C2×C4×C23—22no—
79C2×C4×C23—21no—
80C2×C4×C23—22no—
81C2×C4×C23—22no—
82C2×C4×C23—21no—
83C2×C4×C23—22no—
84C2×C4×C23C2×D412yes—
85C2×C4×C23C2×Q812yes—
86C2×C4×C23#4114yes—
87C2×C4×C23#4114yes—
88C2×Dic23—22no—
89C2×Dic23—22no—
90C2×Dic23—21no—
91C2×Dic23—22no—
92C2×Dic23—22no—
93C2×Dic23—22no—
94C2×Dic23—22no—
95C2×Dic23—22no—
96C2×Dic23—22no—
97C2×Dic23C2×D412yes—
98C2×Dic23C2×D412yes—
99C2×Dic23C2×Q812yes—
100C2×Dic23#4112yes—
101C4×D23—28no—
102#2396—231nomaximal, Sylow
103C22×C23C22×D411yescharacteristic
104C22×C23C22×Q811yescharacteristic
105C22×C23C2×#4112yes—
106C22×C23C2×#4112yes—
107C22×C23C2×#4111yescharacteristic
108C2×D23—22no—
109C2×D23—24no—
110C2×D23—41no—
111C2×D23—42no—
112C2×D23—41no—
113C4×C23—22no—
114C4×C23—22no—
115C4×C23—24no—
116C4×C23—24no—
117C4×C23—41no—
118Dic23—22no—
119Dic23—42no—
120Dic23—42no—
121Dic23—41no—
122Dic23C22⋊Q814yes—
123C2×C4×Q8—231no—
124C2×#117—231no—
125#400—231no—
126#399—231no—
127#398—232no—
128#396—232no—
129#395D2311yescharacteristic, complemented
130#393—231no—
131C4×C4⋊C4—232no—
132C4×C4⋊C4—231no—
133C4×C22⋊C4—232no—
134C2×C23C2×C22⋊Q811yescharacteristic
135C2×C23#52511yescharacteristic
136C2×C23#52912yes—
137C2×C23#53211yescharacteristic
138C2×C23#53312yes—
139D23—42no—
140C22×Q8C2×D2311yescharacteristic
141C23×C4—231no—
142C4×Q8—464no—
143#117—464no—
144C2×C4⋊C4—231no—
145C2×C4⋊C4—232no—
146C2×C4⋊C4—231no—
147C2×C4⋊C4—232no—
148C2×C4⋊C4—232no—
149C2×C4⋊C4C2×D2311yescharacteristic
150C2×C4⋊C4C2×D2312yes—
151C2×C22⋊C4—231no—
152C2×C22⋊C4—231no—
153C2×C22⋊C4—232no—
154C2×C22⋊C4—232no—
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
166C24—231no—
167C2×Q8—22no—
168C2×Q8—24no—
169C22×C4—231no—
170C22×C4—231no—
171C22×C4—234no—
172C22×C4—232no—
173C22×C4—231no—
174C22×C4—232no—
175C22×C4—462no—
176C22×C4—462no—
177C22×C4C22×D2311yescharacteristic
178C22×C4C22×D2311yescharacteristic
179C22×C4C22×D2312yes—
180C22×C4C22×D2311yescharacteristic
181C22×C4C22×D2312yes—
182C4⋊C4—22no—
183C4⋊C4—24no—
184C4⋊C4—462no—
185C4⋊C4—464no—
186C4⋊C4—462no—
187C4⋊C4—464no—
188C4⋊C4—464no—
189C22⋊C4—462no—
190C22⋊C4—462no—
191C22⋊C4—464no—
192C22⋊C4—464no—
193C42—234no—
194C42—462no—
195C42—464no—
196C42—464no—
197C23—232no—
198C23—461no—
199C23—462no—
200C23—462no—
201C23—461no—
202C23C23×D2311yescharacteristic
203Q8—28no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—232no—
214C2×C4—232no—
215C2×C4—232no—
216C2×C4—232no—
217C2×C4—468no—
218C2×C4—462no—
219C2×C4—461no—
220C2×C4—462no—
221C2×C4—462no—
222C2×C4—462no—
223C2×C4—462no—
224C2×C4—462no—
225C2×C4—462no—
226C2×C4—462no—
227C2×C4D4×D2312yes—
228C2×C4#6729014yes—
229C2×C4Q8×D2312yes—
230C2×C4#6729214yes—
231C22—462no—
232C22—464no—
233C22—921no—
234C22—922no—
235C22—921no—
236C22C2×D4×D2311yescharacteristic
237C22C2×#6729012yes—
238C22C2×Q8×D2311yescharacteristic
239C22C2×#6729212yes—
240C22#41×D2311yescharacteristic
241C4—22no—
242C4—22no—
243C4—24no—
244C4—24no—
245C4—41no—
246C4—234no—
247C4—462no—
248C4—922no—
249C4—921no—
250C4—922no—
251C2—922no—
252C2C22⋊Q8×D2311yescharacteristic
253C2#1309292311yescharacteristic
254C2#1309295312yes—
255C2#1309296811yescharacteristic
256C2#1309297212yes—
subgroupG/H
center, FrattiniC23C23×D23
commutator, socle, cosocleC23×C23C24
FittingC23×#395C2
prime radicalC23×D23C23
hypercenter#395D23
hypocenterC23#2396
LCSG ▷ C23×C23 ▷ C23
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
1. C2×Q8×Dic23C2semidirect
3. #13092356C2semidirect
5. #13092349C2semidirect
8. C4⋊C4×Dic23C2semidirect
9. #13092268C2semidirect
11. C23×#395C2semidirect
52. C2×C4×D23Q8semidirect
129. #395D23semidirect
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×#106250632
#130923562
#130923542
#130923492
#130923392
#130923282
C4⋊C4×Dic232
#130922682
#130922642
C23×#3952
#239623