Finite Group Databasebeta

← #15676659

Subgroups — #15676659

Order 2944.

560 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 143 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×(C22⋊D23)⋊C4C211yesmaximal, characteristic
2C2×#10625063C211yesmaximal, characteristic, complemented
3#13092357C211yesmaximal, characteristic, complemented
4#13092353C211yesmaximal, characteristic, complemented
5#13092348C211yesmaximal, characteristic, complemented
6#13092346C212yesmaximal, complemented
7#13092332C212yesmaximal
8#13092268C212yesmaximal, complemented
9C22⋊C4×Dic23C212yesmaximal, complemented
10#13092264C211yesmaximal, characteristic, complemented
11C23×#397C211yesmaximal, characteristic, complemented
12C22×C22⋊D23C2211yescharacteristic
13C23×Dic23C2211yescharacteristic
14C22×C4×D23C2211yescharacteristic
15C2×C4⋊C4×C23C2211yescharacteristic
16C2×C22⋊C4×C23C2211yescharacteristic
17C2×C22⋊C4×C23C2212yes—
18C2×C22⋊Dic23C2212yes—
19C2×(C2×D23)⋊C4C2212yes—
20C2×(C2×D23)⋊C4C2211yescharacteristic
21C2×(C2×D23)⋊C4C2212yes—
22C2×C4⋊Dic23C2211yescharacteristic
23C2×Dic23⋊C4C2212yes—
24C2×Dic23⋊C4C2211yescharacteristic
25C2×Dic23⋊C4C2212yes—
26C2×C4×Dic23C2212yes—
27C2×C4×Dic23C2211yescharacteristic
28C2×C4×Dic23C2212yes—
29#10625063—24no—
30(C22⋊D23)⋊C4—24no—
31C23×#95C2211yescharacteristic
32C23×#95C2212yes—
33C23⋊#95C2211yescharacteristic
34C23⋊#95C2212yes—
35C23⋊#95C2212yes—
36C23⋊#95C2212yes—
37C24×C23C2311yescharacteristic
38C23×D23C2311yescharacteristic
39C22×C4×C23C2311yescharacteristic
40C22×C4×C23C2312yes—
41C22×C4×C23C2312yes—
42C22×C4×C23C2311yescharacteristic
43C2×C22⋊D23—22no—
44C2×C22⋊D23—22no—
45C2×C22⋊D23—22no—
46C22×Dic23—22no—
47C22×Dic23—24no—
48C22×Dic23C2312yes—
49C22×Dic23C2311yescharacteristic
50C22×Dic23C2311yescharacteristic
51C22×Dic23C2312yes—
52C22×Dic23C2311yescharacteristic
53C2×C4×D23—24no—
54C2×C4×D23D414yes—
55C4⋊C4×C23—22no—
56C22⋊C4×C23—24no—
57C22⋊C4×C23—22no—
58C22⋊Dic23—24no—
59(C2×D23)⋊C4—22no—
60(C2×D23)⋊C4—24no—
61(C2×D23)⋊C4—24no—
62C4⋊Dic23—22no—
63Dic23⋊C4—22no—
64Dic23⋊C4—24no—
65Dic23⋊C4—24no—
66C4×Dic23—22no—
67C4×Dic23—24no—
68C4×Dic23—24no—
69C23×C23—21no—
70C23×C23—22no—
71C23×C23—22no—
72C23×C23—21no—
73C23×C23#4112yes—
74C23×C23C2411yescharacteristic
75C22×D23—22no—
76C22×D23—22no—
77C22×D23—21no—
78C22×D23—21no—
79C22×D23C2×D412yes—
80C2×C4×C23—22no—
81C2×C4×C23—21no—
82C2×C4×C23—22no—
83C2×C4×C23—22no—
84C2×C4×C23—22no—
85C2×C4×C23—22no—
86C2×C4×C23—22no—
87C2×C4×C23—22no—
88C2×C4×C23—22no—
89C2×C4×C23C2×D412yes—
90C22⋊D23—28no—
91C2×Dic23—21no—
92C2×Dic23—22no—
93C2×Dic23—22no—
94C2×Dic23—21no—
95C2×Dic23—22no—
96C2×Dic23—21no—
97C2×Dic23—22no—
98C2×Dic23—22no—
99C2×Dic23—22no—
100C2×Dic23—44no—
101C2×Dic23C2×D412yes—
102C2×Dic23#4112yes—
103C2×Dic23#4114yes—
104C2×Dic23#4114yes—
105C4×D23—28no—
106#2395—231nomaximal, Sylow
107C22×C23—22no—
108C22×C23—24no—
109C22×C23—41no—
110C22×C23—42no—
111C22×C23—41no—
112C22×C23C22×D411yescharacteristic
113C22×C23C2×#4111yescharacteristic
114C22×C23C2×#4112yes—
115C22×C23C2×#4111yescharacteristic
116C22×C23C2×#4112yes—
117C2×D23—22no—
118C2×D23—22no—
119C2×D23—22no—
120C2×D23—42no—
121C2×D23—42no—
122C4×C23—22no—
123C4×C23—42no—
124C4×C23—42no—
125C4×C23—41no—
126Dic23—22no—
127Dic23—24no—
128Dic23—41no—
129Dic23—42no—
130Dic23#12314yes—
131C2×C4×D4—231no—
132C2×#117—231no—
133#400—231no—
134#399—231no—
135#398—231no—
136#397—232no—
137#397D2311yescharacteristic, complemented
138#394—232no—
139C4×C4⋊C4—232no—
140C4×C22⋊C4—232no—
141C4×C22⋊C4—231no—
142C2×C23—42no—
143C2×C23C2×#12311yescharacteristic
144C2×C23#52511yescharacteristic
145C2×C23#52912yes—
146C2×C23#52912yes—
147C2×C23#53211yescharacteristic
148D23—42no—
149C22×D4—231no—
150C23×C4—231no—
151C23×C4—231no—
152C4×D4—464no—
153#117—464no—
154C2×C4⋊C4—231no—
155C2×C4⋊C4—231no—
156C2×C4⋊C4—232no—
157C2×C4⋊C4—232no—
158C2×C4⋊C4C2×D2311yescharacteristic
159C2×C22⋊C4—232no—
160C2×C22⋊C4—232no—
161C2×C22⋊C4—231no—
162C2×C22⋊C4—232no—
163C2×C22⋊C4C2×D2311yescharacteristic
164C2×C22⋊C4C2×D2312yes—
165C2×C42—232no—
166C2×C42—231no—
167C2×C42—232no—
168#95—232no—
169#95—232no—
170#95—231no—
171#95—232no—
172#95C2×D2311yescharacteristic
173#95C2×D2312yes—
174C23#239511yescharacteristic, complemented, Sylow
175C24—231no—
176C24C22×D2311yescharacteristic
177C2×D4—462no—
178C2×D4—462no—
179C2×D4—462no—
180C22×C4—231no—
181C22×C4—232no—
182C22×C4—231no—
183C22×C4—234no—
184C22×C4—231no—
185C22×C4—232no—
186C22×C4—464no—
187C22×C4—462no—
188C22×C4—464no—
189C22×C4C22×D2312yes—
190C22×C4C22×D2312yes—
191C22×C4C22×D2311yescharacteristic
192C22×C4C22×D2311yescharacteristic
193C4⋊C4—22no—
194C4⋊C4—464no—
195C4⋊C4—464no—
196C4⋊C4—462no—
197C4⋊C4—462no—
198C22⋊C4—24no—
199C22⋊C4—22no—
200C22⋊C4—464no—
201C22⋊C4—464no—
202C22⋊C4—462no—
203C22⋊C4—464no—
204C42—464no—
205C42—462no—
206C42—464no—
207C23—21no—
208C23—22no—
209C23—22no—
210C23—21no—
211C23—232no—
212C23—461no—
213C23—462no—
214C23—462no—
215C23—461no—
216C23#6729012yes—
217C23C23×D2311yescharacteristic
218D4—468no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—232no—
229C2×C4—234no—
230C2×C4—234no—
231C2×C4—232no—
232C2×C4—468no—
233C2×C4—461no—
234C2×C4—461no—
235C2×C4—462no—
236C2×C4—462no—
237C2×C4—462no—
238C2×C4—461no—
239C2×C4—462no—
240C2×C4—462no—
241C2×C4—462no—
242C2×C4—924no—
243C2×C4D4×D2312yes—
244C22—22no—
245C22—24no—
246C22—41no—
247C22—42no—
248C22—41no—
249C22—462no—
250C22—462no—
251C22—462no—
252C22—922no—
253C22—922no—
254C22C2×D4×D2311yescharacteristic
255C22C2×#6729011yescharacteristic
256C22#41×D2312yes—
257C22#41×D2311yescharacteristic
258C22#41×D2312yes—
259C4—22no—
260C4—42no—
261C4—42no—
262C4—41no—
263C4—234no—
264C4—462no—
265C4—464no—
266C4—921no—
267C4—922no—
268C2—42no—
269C2—922no—
270C2#123×D2311yescharacteristic
271C2#1309292711yescharacteristic
272C2#1309294412yes—
273C2#1309295012yes—
274C2#1309296611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D23
commutator, socle, cosocleC23×C23C24
FittingC23×#397C2
prime radicalC22×C22⋊D23C22
hypercenter#397D23
hypocenterC23#2395
LCSG ▷ C23×C23 ▷ C23
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
2. C2×#10625063C2semidirect
3. #13092357C2semidirect
4. #13092353C2semidirect
5. #13092348C2semidirect
6. #13092346C2semidirect
8. #13092268C2semidirect
9. C22⋊C4×Dic23C2semidirect
10. #13092264C2semidirect
11. C23×#397C2semidirect
137. #397D23semidirect
174. C23#2395semidirect

Maximal subgroups

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

subgroupindex
C2×(C22⋊D23)⋊C42
C2×#106250632
#130923572
#130923532
#130923482
#130923462
#130923322
#130922682
C22⋊C4×Dic232
#130922642
C23×#3972
#239523