Finite Group Databasebeta

← #134755

Subgroups — #134755

Order 512.

496 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 90 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
1C8×#127C211yesmaximal, characteristic
2#12797C212yesmaximal
3#8609C212yesmaximal, complemented
4#7237C212yesmaximal, complemented
5C2×C8×D4—22no—
6C2×C8×D4C411yescharacteristic, complemented
7C2×C8×D4C2211yescharacteristic
8C4×#127C2211yescharacteristic
9C2×#353—22no—
10#1954—22no—
11#1954C411yescharacteristic, complemented
12#1954C2211yescharacteristic
13#1842C411yescharacteristic, complemented
14#1808—22no—
15#1799C2214yes—
16C42×C8C411yescharacteristic, complemented
17#1393—22no—
18#1289—22no—
19C23×C8—21no—
20C23×C8D412yes—
21C2×#127D411yescharacteristic
22C2×C4×D4—22no—
23C2×C4×D4C2×C411yescharacteristic
24C2×C4×D4C2311yescharacteristic
25C8×D4—41no—
26C8×D4—41no—
27C8×D4—41no—
28C8×D4—41no—
29C8×D4—48no—
30C2×#107—42no—
31C2×C4⋊C8—22no—
32C2×C4⋊C8C2×C411yescharacteristic
33C2×C4⋊C8C2×C411yescharacteristic
34C2×#99—24no—
35C2×C22⋊C8—22no—
36C2×C22⋊C8—22no—
37C2×C22⋊C8—21no—
38C2×C22⋊C8D412yes—
39C2×C4×C8—21no—
40C2×C4×C8—21no—
41C2×C4×C8C2×C411yescharacteristic
42C2×C4×C8C2×C411yescharacteristic
43#390—22no—
44#390D411yescharacteristic
45#390Q811yescharacteristic
46#384Q811yescharacteristic
47#380—24no—
48C43C2×C411yescharacteristic
49#353—44no—
50#342—24no—
51#340—42no—
52C22×D4—22no—
53C22×D4C4⋊C411yescharacteristic
54C22×D4C2×D411yescharacteristic
55C23×C4—21no—
56C23×C4C2×D412yes—
57C22×C8—21no—
58C22×C8—22no—
59C22×C8—22no—
60C22×C8—21no—
61C22×C8—44no—
62C22×C8—44no—
63C22×C8—41no—
64C22×C8—41no—
65C22×C8C22⋊C411yescharacteristic
66C22×C8C22⋊C412yes—
67C22×C8C22⋊C411yescharacteristic
68#127—22no—
69#127D812yes—
70#127Q1612yes—
71C4×D4—41no—
72C4×D4—41no—
73C4×D4—44no—
74C4×D4—44no—
75C4×D4—41no—
76C4×D4—41no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—44no—
79C2×C4⋊C4C4⋊C411yescharacteristic
80C2×C4⋊C4C2×Q811yescharacteristic
81C2×C22⋊C4—22no—
82C2×C22⋊C4—21no—
83C2×C22⋊C4—22no—
84C2×C22⋊C4—44no—
85C2×C22⋊C4#4112yes—
86C2×C42—21no—
87C2×C42—21no—
88C2×C42C4⋊C411yescharacteristic
89C2×C42C4⋊C411yescharacteristic
90C2×C42C22×C411yescharacteristic
91#107—84no—
92C4⋊C8—41no—
93C4⋊C8—41no—
94C4⋊C8—44no—
95#99—44no—
96#99—44no—
97C22⋊C8—42no—
98C22⋊C8—44no—
99C22⋊C8—44no—
100C22⋊C8—42no—
101C4×C8—24no—
102C4×C8—21no—
103C4×C8—21no—
104C4×C8—41no—
105C4×C8—42no—
106#95—44no—
107C24—21no—
108C24#12112yes—
109C2×D4—24no—
110C2×D4—24no—
111C2×D4—21no—
112C2×D4—41no—
113C2×D4—41no—
114C2×D4—41no—
115C2×D4—41no—
116C2×D4—44no—
117C2×D4—44no—
118C2×D4#10712yescomplemented
119C2×D4C2×D811yescharacteristic
120C2×D4C2×Q1611yescharacteristic
121C2×D4#13512yes—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—21no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4—42no—
131C22×C4—41no—
132C22×C4—41no—
133C22×C4—84no—
134C22×C4C2×C22⋊C411yescharacteristic
135C22×C4C2×C4⋊C411yescharacteristic
136C22×C4#11711yescharacteristic
137C22×C4C22⋊Q812yes—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8—21no—
141C2×C8—22no—
142C2×C8—41no—
143C2×C8—41no—
144C2×C8—44no—
145C2×C8—41no—
146C2×C8—42no—
147C2×C8—42no—
148C2×C8—41no—
149C2×C8—82no—
150C2×C8—82no—
151C2×C8—82no—
152C2×C8#9911yescharacteristic
153C2×C8#9911yescharacteristic
154C4⋊C4—41no—
155C4⋊C4—41no—
156C4⋊C4—44no—
157C4⋊C4—84no—
158C4⋊C4—84no—
159C22⋊C4—42no—
160C22⋊C4—44no—
161C22⋊C4—44no—
162C22⋊C4—42no—
163C22⋊C4—84no—
164C22⋊C4—84no—
165C42—22no—
166C42—22no—
167C42—21no—
168C42—21no—
169C42—21no—
170C42—41no—
171C42—42no—
172C42#10712yescomplemented
173C23—22no—
174C23—22no—
175C23—22no—
176C23—42no—
177C23—42no—
178C23—42no—
179C23—42no—
180C23—41no—
181C23—41no—
182C23#38011yescharacteristic
183C23#46612yes—
184C23#46712yes—
185D4—41no—
186D4—41no—
187D4—42no—
188D4—42no—
189D4—48no—
190D4—48no—
191D4—42no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—41no—
210C2×C4—82no—
211C2×C4—82no—
212C2×C4—81no—
213C2×C4—81no—
214C2×C4—164no—
215C2×C4—164no—
216C2×C4—164no—
217C2×C4C2×#9911yescharacteristic
218C2×C4#41011yescharacteristic
219C2×C4C2×#10711yescharacteristic
220C2×C4#42711yescharacteristic
221C8—21no—
222C8—42no—
223C8—42no—
224C8—41no—
225C8—81no—
226C8#35312yescomplemented
227C22—21no—
228C22—21no—
229C22—22no—
230C22—22no—
231C22—42no—
232C22—42no—
233C22—44no—
234C22—42no—
235C22—42no—
236C22—42no—
237C22—82no—
238C22—82no—
239C22—81no—
240C22—81no—
241C22#179911yescharacteristic
242C22#180811yescharacteristic
243C22#181211yescharacteristic
244C4—24no—
245C4—21no—
246C4—21no—
247C4—21no—
248C4—42no—
249C4—41no—
250C4—42no—
251C4—81no—
252C4—324no—
253C4C2×#35311yescharacteristic
254C4#212611yescharacteristic
255C2—21no—
256C2—21no—
257C2—42no—
258C2—42no—
259C2—82no—
260C2#860911yescharacteristic
261C2#1279711yescharacteristic
262C2#1280711yescharacteristic
subgroupG/H
center, socleC22#1799
commutatorC2×C42C22×C4
Frattini, cosocleC2×C4×D4C23
prime radicalC2×#127D4
LCSG ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8609C2semidirect
4. #7237C2semidirect
6. C2×C8×D4C4semidirect
11. #1954C4semidirect
13. #1842C4semidirect
16. C42×C8C4semidirect
118. C2×D4#107semidirect
172. C42#107semidirect
226. C8#353semidirect

Maximal subgroups

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

subgroupindex
C8×#1272
#127972
#86092
#72372