Finite Group Databasebeta

← #46826111

Subgroups — #46826111

Order 34432.

456 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 84 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
1D269⋊(C22⋊C8)C212yesmaximal, complemented
2#31771449C212yesmaximal, complemented
3#31771448C212yesmaximal, complemented
4C2×C22⋊C4×D269C211yesmaximal, characteristic
5C2×D269⋊C8—22no—
6#22575510—22no—
7C22⋊#14524137C2214yescomplemented
8C24×D269C411yescharacteristic
9C22×C4×D269C411yescharacteristic
10C22×C4×D269C2211yescharacteristic
11C2×C22⋊Dic269C2211yescharacteristic
12C2×#16864379C411yescharacteristic
13C2×#16864379C2211yescharacteristic
14C22⋊C4×D269—22no—
15C22⋊C4×D269—22no—
16C2×C22⋊C4×C269C411yescharacteristic
17D269⋊C8—44no—
18C2×#14524137—24no—
19C23×D269—21no—
20C23×D269—21no—
21C23×D269—22no—
22C23×D269—42no—
23C23×D269—42no—
24C23×D269C814yescomplemented
25C23×D269C2×C411yescharacteristic
26C23×D269C2×C411yescharacteristic
27C22×Dic269C2×C411yescharacteristic
28C22×Dic269C2311yescharacteristic
29C2×C4×D269—21no—
30C2×C4×D269—21no—
31C2×C4×D269—42no—
32C2×C4×D269—42no—
33C2×C4×D269D412yes—
34C2×C4×D269D412yes—
35C22⋊Dic269—22no—
36#16864379—22no—
37#16864379—22no—
38C24×C269C2×C411yescharacteristic
39C22×C4×C269C2×C411yescharacteristic
40C22×C4×C269C2×C411yescharacteristic
41C22⋊C4×C269—22no—
42#14524137—44no—
43C22×D269—22no—
44C22×D269—22no—
45C22×D269—21no—
46C22×D269—21no—
47C22×D269—24no—
48C22×D269—24no—
49C22×D269—42no—
50C22×D269—42no—
51C22×D269—48no—
52C22×D269—48no—
53C22×D269—42no—
54C22×D269—42no—
55C22×D269—41no—
56C22×D269—41no—
57C22×D269C22⋊C411yescharacteristic
58C22×D269C22⋊C411yescharacteristic
59C22×D269C2×C812yes—
60C22×D269C2×C812yescomplemented
61C22×D269M1612yes—
62C2×Dic269—21no—
63C2×Dic269—41no—
64C2×Dic269—41no—
65C2×Dic269C2×D412yes—
66C4×D269—24no—
67C4×D269—42no—
68C4×D269—42no—
69C4×D269—42no—
70C23×C269—21no—
71C23×C269—21no—
72C23×C269—41no—
73C23×C269—41no—
74C23×C269C2×C812yescomplemented
75C23×C269C22×C411yescharacteristic
76C2×C4×C269—21no—
77C2×C4×C269—41no—
78C2×C4×C269—41no—
79C2×C4×C269C22⋊C412yes—
80C22×C269—21no—
81C22×C269—21no—
82C22×C269—22no—
83C22×C269—42no—
84C22×C269—42no—
85C22×C269—41no—
86C22×C269—41no—
87C22×C269C2×C22⋊C411yescharacteristic
88C22×C269C22×C811yescharacteristic
89C22×C269C2×M1611yescharacteristic
90C2×D269—21no—
91C2×D269—21no—
92C2×D269—24no—
93C2×D269—24no—
94C2×D269—22no—
95C2×D269—48no—
96C2×D269—42no—
97C2×D269—41no—
98C2×D269—41no—
99C2×D269—48no—
100C2×D269—42no—
101C2×D269C22⋊C812yes—
102C2×D269C22⋊C812yes—
103C2×D269#9912yes—
104C2×D269#10012yes—
105C4×C269—22no—
106C4×C269—41no—
107Dic269—22no—
108Dic269—41no—
109C2×C269—21no—
110C2×C269—21no—
111C2×C269—42no—
112C2×C269C2×C22⋊C811yescharacteristic
113C2×C269C2×#9911yescharacteristic
114C2×C269C2×#10011yescharacteristic
115D269—22no—
116D269—42no—
117D269#32314yescomplemented
118C269C2×#32311yescharacteristic, complemented, Sylow
119C2×#323—2691nomaximal, Sylow
120C22×C22⋊C4—2691no—
121C2×C22⋊C8—2692no—
122#323—2692no—
123#323—2692no—
124C25—2691no—
125C23×C4—2691no—
126C23×C4—2691no—
127C22×C8—5382no—
128C2×C22⋊C4—2691no—
129C2×C22⋊C4—2691no—
130C2×C22⋊C4—2691no—
131C2×C22⋊C4—5382no—
132C2×C22⋊C4—5382no—
133C2×C22⋊C4#1176164011yescharacteristic
134C22⋊C8—2694no—
135C22⋊C8—5382no—
136C24—2691no—
137C24—2694no—
138C24—2691no—
139C24—5381no—
140C24—5381no—
141C24—5382no—
142C24—10762no—
143C24—10762no—
144C24C2×#1176164011yescharacteristic
145C22×C4—2691no—
146C22×C4—2692no—
147C22×C4—2692no—
148C22×C4—2691no—
149C22×C4—5381no—
150C22×C4—5381no—
151C22×C4—10762no—
152C22×C4—10762no—
153C22×C4C2×#1176164011yescharacteristic
154C22×C4C2×#1176164011yescharacteristic
155C2×C8—5384no—
156C2×C8—10764no—
157C22⋊C4—22no—
158C22⋊C4—5382no—
159C22⋊C4—5382no—
160C22⋊C4—5382no—
161C23—21no—
162C23—21no—
163C23—41no—
164C23—41no—
165C23—2691no—
166C23—2691no—
167C23—2692no—
168C23—2692no—
169C23—2692no—
170C23—5382no—
171C23—5382no—
172C23—5384no—
173C23—5384no—
174C23—5381no—
175C23—5381no—
176C23—10761no—
177C23—10761no—
178C23—10762no—
179C23—10762no—
180C23—10768no—
181C23—10768no—
182C23—10762no—
183C23—10762no—
184C23D269⋊C812yescomplemented
185C23C22×#1176164011yescharacteristic
186C2×C4—21no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—2692no—
190C2×C4—5384no—
191C2×C4—5381no—
192C2×C4—10762no—
193C2×C4—10762no—
194C2×C4—10762no—
195C2×C4—10761no—
196C2×C4—10761no—
197C2×C4C22⋊#1176164012yes—
198C8—10764no—
199C22—21no—
200C22—21no—
201C22—22no—
202C22—42no—
203C22—42no—
204C22—41no—
205C22—41no—
206C22—2692no—
207C22—2692no—
208C22—2692no—
209C22—2692no—
210C22—5384no—
211C22—5384no—
212C22—5381no—
213C22—5381no—
214C22—5382no—
215C22—10762no—
216C22—10762no—
217C22—10768no—
218C22—10768no—
219C22—10761no—
220C22—10761no—
221C22C2×C22⋊#1176164011yescharacteristic
222C22C2×D269⋊C811yescharacteristic
223C22D269⋊M1611yescharacteristic
224C4—22no—
225C4—41no—
226C4—5382no—
227C4—10761no—
228C2—21no—
229C2—21no—
230C2—42no—
231C2—2694no—
232C2—5382no—
233C2—10762no—
234C2D269⋊(C22⋊C8)11yescharacteristic
235C2D269⋊#9911yescharacteristic
236C2D269⋊#10011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#11761640
commutatorC22×C269C22×C8
FrattiniC23C22×#11761640
FittingC2×C22⋊C4×C269C4
socleC22×C269C2×C22⋊C4
prime radicalC24×D269C4
cosocleC22×Dic269C23
hypercenterC2×C22⋊C4#11761640
hypocenterC269C2×#323
LCSG ▷ C22×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. D269⋊(C22⋊C8)C2semidirect
2. #31771449C2semidirect
3. #31771448C2semidirect
7. C22⋊#14524137C22semidirect
24. C23×D269C8semidirect
60. C22×D269C2×C8semidirect
74. C23×C269C2×C8semidirect
117. D269#323semidirect
118. C269C2×#323semidirect
184. C23D269⋊C8semidirect

Maximal subgroups

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

subgroupindex
D269⋊(C22⋊C8)2
#317714492
#317714482
C2×C22⋊C4×D2692
C2×#323269