Finite Group Databasebeta

← #134304

Subgroups — #134304

Order 512.

502 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 118 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×C4×C4⋊C8C211yesmaximal, characteristic
2#12788C212yesmaximal
3#11440C212yesmaximal
4#7184C212yesmaximal
5C22×C4⋊C8C411yescharacteristic
6C22×C4⋊C8C2211yescharacteristic
7C22×C4×C8C411yescharacteristic
8C22×C4×C8C2211yescharacteristic
9C2×C43C411yescharacteristic
10C2×#364—22no—
11#2112C2214yes—
12C4×C4⋊C8—22no—
13C4×C4⋊C8—22no—
14C2×#336C411yescharacteristic
15C2×#336C2211yescharacteristic
16#1376—22no—
17#1322—24no—
18C23×C8C2×C411yescharacteristic
19C23×C8C2×C411yescharacteristic
20C23×C8C2×C411yescharacteristic
21C23×C8C2311yescharacteristic
22C22×C42—21no—
23C22×C42—21no—
24C22×C42C2×C411yescharacteristic
25C22×C42C2×C411yescharacteristic
26C22×C42C2×C411yescharacteristic
27C2×M32—24no—
28C2×C4⋊C8—22no—
29C2×C4⋊C8—22no—
30C2×C4⋊C8—22no—
31C2×C4⋊C8—22no—
32C2×C4⋊C8C814yes—
33C2×C4⋊C8D412yes—
34C2×C4⋊C8Q812yes—
35C2×C4×C8—21no—
36C2×C4×C8—21no—
37C2×C4×C8—42no—
38C2×C4×C8—42no—
39C2×C4×C8D412yes—
40C2×C4×C8D412yes—
41C43—21no—
42C43—21no—
43C43—22no—
44#364—44no—
45C22⋊C16—24no—
46C22⋊C16—24no—
47#336—22no—
48#336—22no—
49C23×C4—21no—
50C23×C4—21no—
51C23×C4M1611yescharacteristic
52C23×C4M1611yescharacteristic
53C23×C4C22×C411yescharacteristic
54C22×C8—21no—
55C22×C8—21no—
56C22×C8—21no—
57C22×C8—21no—
58C22×C8—41no—
59C22×C8—41no—
60C22×C8—41no—
61C22×C8—41no—
62C22×C8—41no—
63C22×C8—41no—
64C22×C8—41no—
65C22×C8—41no—
66C22×C8C22⋊C412yes—
67C22×C8C4⋊C412yes—
68C22×C8C2×C812yes—
69C22×C8C2×C812yes—
70C22×C8M1612yes—
71C22×C8C2×D412yes—
72C22×C8C2×D411yescharacteristic
73C22×C8C2×Q811yescharacteristic
74C22×C8#4112yes—
75C2×C42—22no—
76C2×C42—22no—
77C2×C42—28no—
78C2×C42—21no—
79C2×C42—21no—
80C2×C42—21no—
81C2×C42—21no—
82C2×C42—21no—
83C2×C42—22no—
84C2×C42—22no—
85C2×C42—22no—
86C2×C42—22no—
87C2×C42—21no—
88C2×C42—44no—
89C2×C42—41no—
90C2×C42—41no—
91C2×C42C22⋊C411yescharacteristic
92C2×C42C22⋊C411yescharacteristic
93C2×C42C4⋊C412yes—
94C2×C42C2×C812yes—
95C2×C42M1612yes—
96M32—44no—
97M32—44no—
98C2×C16—44no—
99C4⋊C8—28no—
100C4⋊C8—24no—
101C4⋊C8—24no—
102C4×C8—24no—
103C4×C8—42no—
104C4×C8—42no—
105C4×C8—42no—
106C24C2×M1611yescharacteristic
107C22×C4—24no—
108C22×C4—24no—
109C22×C4—24no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—24no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—24no—
120C22×C4—21no—
121C22×C4—22no—
122C22×C4—21no—
123C22×C4—42no—
124C22×C4—42no—
125C22×C4—42no—
126C22×C4C4⋊C812yes—
127C22×C4C4⋊C812yes—
128C22×C4C2×C22⋊C411yescharacteristic
129C22×C4C2×C4⋊C411yescharacteristic
130C22×C4#11711yescharacteristic
131C22×C4C22×C811yescharacteristic
132C22×C4C2×M1611yescharacteristic
133C22×C4C2×M1611yescharacteristic
134C2×C8—22no—
135C2×C8—22no—
136C2×C8—22no—
137C2×C8—22no—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8—42no—
141C2×C8—42no—
142C2×C8—41no—
143C2×C8—41no—
144C2×C8—42no—
145C2×C8—42no—
146C2×C8—42no—
147C2×C8—41no—
148C2×C8—41no—
149C2×C8—41no—
150C2×C8—41no—
151C2×C8—42no—
152C2×C8—42no—
153C2×C8C22⋊C814yes—
154C2×C8#12112yes—
155C2×C8C22⋊Q812yes—
156C42—28no—
157C42—28no—
158C42—24no—
159C42—22no—
160C42—22no—
161C42—21no—
162C42—21no—
163C42—22no—
164C42—22no—
165C42—42no—
166C42—44no—
167C42—44no—
168C42—41no—
169C42—41no—
170C42#9912yes—
171C42#10112yes—
172C16—84no—
173C23—22no—
174C23—22no—
175C23—21no—
176C23—21no—
177C23#40711yescharacteristic
178C23C2×C4⋊C811yescharacteristic
179C23#43211yescharacteristic
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—24no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—216no—
186C2×C4—24no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—24no—
202C2×C4—22no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—48no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4#36414yescomplemented
210C2×C4#38011yescharacteristic
211C2×C4C2×C22⋊C811yescharacteristic
212C2×C4C2×#9911yescharacteristic
213C2×C4#41011yescharacteristic
214C2×C4C2×#10111yescharacteristic
215C2×C4#41311yescharacteristic
216C2×C4#42311yescharacteristic
217C2×C4#43111yescharacteristic
218C8—22no—
219C8—22no—
220C8—24no—
221C8—42no—
222C8—42no—
223C22—24no—
224C22—24no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—21no—
230C22—21no—
231C22#146311yescharacteristic
232C22#146411yescharacteristic
233C22#179711yescharacteristic
234C22C2×#36412yes—
235C22#216512yes—
236C4—24no—
237C4—24no—
238C4—21no—
239C4—21no—
240C4—24no—
241C4—44no—
242C4—42no—
243C4#145911yescharacteristic
244C4#146111yescharacteristic
245C4#179911yescharacteristic
246C4#180111yescharacteristic
247C2—21no—
248C2—21no—
249C2—22no—
250C2—22no—
251C2#856011yescharacteristic
252C2#1278811yescharacteristic
253C2#1279111yescharacteristic
subgroupG/H
centerC2×C4#380
commutatorC22×C4C22×C8
Frattini, cosocleC23×C8C23
socleC22#1797
prime radicalC24C2×M16
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
209. C2×C4#364semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C4⋊C82
#127882
#114402
#71842