Finite Group Databasebeta

← #15673692

Subgroups — #15673692

Order 2944.

408 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 109 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×#67267C211yesmaximal, characteristic, complemented
2C2×C4×C22⋊D23C211yesmaximal, characteristic
3C2×(C2×D23)⋊C8C212yesmaximal, complemented
4#13092171C212yesmaximal, complemented
5C2×C23×C22⋊C8C211yesmaximal, characteristic, complemented
6C23×C4×C23C2211yescharacteristic
7C22×C22⋊D23C411yescharacteristic
8C22×C4×D23C2211yescharacteristic
9C22×C8×C23C2212yes—
10C2×C22⋊Dic23C411yescharacteristic
11C4×C22⋊D23—22no—
12C4×C22⋊D23—22no—
13C2×(C2×D23)⋊C4C411yescharacteristic
14C2×Dic23⋊C4C411yescharacteristic
15C2×C4×Dic23C2211yescharacteristic
16C2×#67267—22no—
17C2×#67267—24no—
18C22×C23⋊C8C2212yes—
19C23×C22⋊C8—22no—
20(C2×D23)⋊C8—24no—
21C24×C23C2×C411yescharacteristic
22C23×D23C2×C411yescharacteristic
23C22×C4×C23—22no—
24C22×C4×C23—22no—
25C22×C4×C23C2×C411yescharacteristic
26C22×C4×C23D412yes—
27C22×C4×C23D412yes—
28C22×C4×C23C2311yescharacteristic
29C2×C22⋊D23—22no—
30C2×C22⋊D23—22no—
31C2×C22⋊D23—22no—
32C22×Dic23C2×C411yescharacteristic
33C22×Dic23C2×C411yescharacteristic
34C22×Dic23C2×C411yescharacteristic
35C2×C4×D23—21no—
36C2×C4×D23—21no—
37C2×C4×D23—22no—
38C2×C4×D23—22no—
39C2×C8×C23—22no—
40C2×C8×C23—22no—
41C2×C8×C23—22no—
42C22⋊Dic23—22no—
43(C2×D23)⋊C4—22no—
44Dic23⋊C4—22no—
45C4×Dic23—21no—
46C4×Dic23—21no—
47#67267—28no—
48C2×C23⋊C8—22no—
49C2×C23⋊C8—22no—
50C2×C23⋊C8D414yescomplemented
51C23×C23—22no—
52C23×C23—21no—
53C23×C23—21no—
54C23×C23—22no—
55C23×C23C22⋊C412yes—
56C23×C23C22×C411yescharacteristic
57C22×D23—21no—
58C22×D23—22no—
59C22×D23—21no—
60C22×D23—22no—
61C22×D23M1612yescomplemented
62C2×C4×C23—22no—
63C2×C4×C23—22no—
64C2×C4×C23—24no—
65C2×C4×C23—22no—
66C2×C4×C23C22⋊C412yes—
67C2×C4×C23C2×D411yescharacteristic
68C2×C4×C23C2×D412yes—
69C2×C4×C23C2×D411yescharacteristic
70C2×C4×C23#4112yes—
71C22⋊D23—28no—
72C2×Dic23—21no—
73C2×Dic23—21no—
74C2×Dic23—21no—
75C2×Dic23—21no—
76C2×Dic23—22no—
77C2×Dic23—22no—
78C2×Dic23M1612yescomplemented
79C4×D23—41no—
80C4×D23—42no—
81C4×D23—41no—
82C8×C23—42no—
83C23⋊C8—24no—
84#1870—231nomaximal, Sylow
85C22×C23—22no—
86C22×C23—22no—
87C22×C23—22no—
88C22×C23—42no—
89C22×C23—42no—
90C22×C23C2×C22⋊C411yescharacteristic
91C22×C23C4×D412yes—
92C22×C23C2×M1611yescharacteristic
93C22×C23C2×M1611yescharacteristic
94C22×C23#13112yes—
95C2×D23—22no—
96C2×D23—24no—
97C2×D23—41no—
98C2×D23—41no—
99C2×D23—42no—
100C4×C23—22no—
101C4×C23#12011yescharacteristic
102C4×C23#12112yes—
103C4×C23#12311yescharacteristic
104Dic23—22no—
105Dic23—41no—
106Dic23—41no—
107C22×M16—231no—
108C2×C4×D4—231no—
109C2×C22⋊C8—232no—
110C2×C22⋊C8D2311yescharacteristic, complemented
111#336—232no—
112C2×C23—42no—
113C2×C23#38611yescharacteristic
114C2×C23#40711yescharacteristic
115C2×C23#40811yescharacteristic
116C2×C23#43512yes—
117C2×C23#43612yes—
118D23—42no—
119C22×D4—231no—
120C23×C4—231no—
121C23×C4C2×D2311yescharacteristic
122C2×M16—464no—
123C2×M16—462no—
124C22×C8—232no—
125C22×C8C2×D2312yes—
126C4×D4—462no—
127C4×D4—462no—
128C2×C4⋊C4—231no—
129C2×C22⋊C4—231no—
130C2×C22⋊C4—231no—
131C2×C42—231no—
132C22⋊C8—22no—
133C22⋊C8—464no—
134C23#187011yescharacteristic, complemented, Sylow
135C24—231no—
136C24C4×D2311yescharacteristic
137C2×D4—462no—
138C2×D4—462no—
139C2×D4—462no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—231no—
143C22×C4—231no—
144C22×C4—231no—
145C22×C4—461no—
146C22×C4—462no—
147C22×C4—462no—
148C22×C4—461no—
149C22×C4C4×D2311yescharacteristic
150C22×C4D9212yes—
151C22×C4C22⋊D2312yes—
152C22×C4C22×D2311yescharacteristic
153M16—468no—
154C2×C8—22no—
155C2×C8—22no—
156C2×C8—22no—
157C2×C8—234no—
158C2×C8—462no—
159C2×C8—462no—
160C4⋊C4—462no—
161C22⋊C4—462no—
162C22⋊C4—462no—
163C42—461no—
164C42—461no—
165C23—22no—
166C23—22no—
167C23—21no—
168C23—21no—
169C23—232no—
170C23—462no—
171C23—462no—
172C23—461no—
173C23—461no—
174C23(C2×D23)⋊C412yes—
175C23C2×C4×D2311yescharacteristic
176D4—468no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—24no—
181C2×C4—232no—
182C2×C4—461no—
183C2×C4—461no—
184C2×C4—462no—
185C2×C4—462no—
186C2×C4—461no—
187C2×C4—461no—
188C2×C4—921no—
189C2×C4—921no—
190C2×C4—922no—
191C2×C4(C2×D23)⋊C412yes—
192C2×C4C2×D9211yescharacteristic
193C2×C4D4×D2312yes—
194C2×C4#6729012yes—
195C2×C4C2×C22⋊D2311yescharacteristic
196C8—42no—
197C8—464no—
198C22—22no—
199C22—22no—
200C22—22no—
201C22—42no—
202C22—42no—
203C22—462no—
204C22—464no—
205C22—921no—
206C22—922no—
207C22—921no—
208C22(C22⋊D23)⋊C412yes—
209C22#1062507212yes—
210C22M16×D2311yescharacteristic
211C22M16×D2311yescharacteristic
212C22C2×(C2×D23)⋊C411yescharacteristic
213C4—22no—
214C4—462no—
215C4—921no—
216C4—921no—
217C4#1062505311yescharacteristic
218C4#1062505711yescharacteristic
219C4#1062511012yes—
220C2—42no—
221C2—922no—
222C2#1309229411yescharacteristic
223C2#1309238011yescharacteristic
224C2#1309238311yescharacteristic
225C2#1309249112yes—
226C2#1309249212yes—
subgroupG/H
center, FrattiniC22×C4C22×D23
commutator, socleC23×C23C22×C4
FittingC2×C23×C22⋊C8C2
prime radicalC22×C22⋊D23C4
cosocleC22×C4×C23C23
hypercenterC2×C22⋊C8D23
hypocenterC23#1870
LCSG ▷ C23×C23 ▷ C23
UCSC1 ◁ C22×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
1. C22×#67267C2semidirect
3. C2×(C2×D23)⋊C8C2semidirect
4. #13092171C2semidirect
5. C2×C23×C22⋊C8C2semidirect
50. C2×C23⋊C8D4semidirect
61. C22×D23M16semidirect
78. C2×Dic23M16semidirect
110. C2×C22⋊C8D23semidirect
134. C23#1870semidirect

Maximal subgroups

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

subgroupindex
C22×#672672
C2×C4×C22⋊D232
C2×(C2×D23)⋊C82
#130921712
C2×C23×C22⋊C82
#187023