Finite Group Databasebeta

← #134757

Subgroups — #134757

Order 512.

435 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 435 conjugacy classes of subgroups — too many to draw. Only its 88 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
1#22260C211yesmaximal, characteristic
2#12797C212yesmaximal
3#8609C212yesmaximal, complemented
4#7237C212yesmaximal, complemented
5C2×C8×D4C411yescharacteristic, complemented
6C2×C8×D4C2211yescharacteristic
7C4×#127C2211yescharacteristic
8C2×#353—22no—
9#1954C411yescharacteristic, complemented
10#1954C2211yescharacteristic
11#1869—22no—
12#1869—22no—
13#1831C411yescharacteristic, complemented
14#1808—22no—
15#1799C2214yes—
16#1744C411yescharacteristic, 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—42no—
26C8×D4—42no—
27C2×#107—42no—
28C2×C4⋊C8C2×C411yescharacteristic
29C2×C4⋊C8C2×C411yescharacteristic
30C2×#99—24no—
31C2×C22⋊C8—21no—
32C2×C22⋊C8—22no—
33C2×C22⋊C8—22no—
34C2×C22⋊C8D412yes—
35C2×C4×C8C2×C411yescharacteristic
36C2×C4×C8C2×C411yescharacteristic
37#390—22no—
38#390D411yescharacteristic
39#390Q811yescharacteristic
40#384Q811yescharacteristic
41#380—24no—
42C43C2×C411yescharacteristic
43#353—44no—
44#342—24no—
45#340—42no—
46#336—22no—
47#336—21no—
48#336—21no—
49C22×D4—22no—
50C22×D4C4⋊C411yescharacteristic
51C22×D4C2×D411yescharacteristic
52C23×C4—21no—
53C23×C4C2×D412yes—
54C22×C8—22no—
55C22×C8—22no—
56C22×C8—21no—
57C22×C8—21no—
58C22×C8—42no—
59C22×C8—82no—
60C22×C8—82no—
61C22×C8C22⋊C411yescharacteristic
62C22×C8C22⋊C412yes—
63C22×C8C22⋊C411yescharacteristic
64#127—22no—
65#127D812yes—
66#127Q1612yes—
67C4×D4—41no—
68C4×D4—41no—
69C4×D4—41no—
70C4×D4—41no—
71C4×D4—82no—
72C4×D4—82no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—44no—
75C2×C4⋊C4C4⋊C411yescharacteristic
76C2×C4⋊C4C2×Q811yescharacteristic
77C2×C22⋊C4—21no—
78C2×C22⋊C4—22no—
79C2×C22⋊C4—22no—
80C2×C22⋊C4—44no—
81C2×C22⋊C4#4112yes—
82C2×C42—21no—
83C2×C42—21no—
84C2×C42C4⋊C411yescharacteristic
85C2×C42C4⋊C411yescharacteristic
86C2×C42C22×C411yescharacteristic
87#107—84no—
88C4⋊C8—41no—
89C4⋊C8—41no—
90#99—44no—
91#99—44no—
92C22⋊C8—42no—
93C22⋊C8—42no—
94C22⋊C8—82no—
95C22⋊C8—82no—
96C4×C8—22no—
97C4×C8—41no—
98#95—44no—
99C24—21no—
100C24#12112yes—
101C2×D4—24no—
102C2×D4—24no—
103C2×D4—21no—
104C2×D4—41no—
105C2×D4—41no—
106C2×D4—41no—
107C2×D4—41no—
108C2×D4—82no—
109C2×D4—82no—
110C2×D4#10712yescomplemented
111C2×D4C2×D811yescharacteristic
112C2×D4C2×Q1611yescharacteristic
113C2×D4#13512yes—
114C22×C4—21no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—42no—
120C22×C4—42no—
121C22×C4—42no—
122C22×C4—42no—
123C22×C4—41no—
124C22×C4—41no—
125C22×C4—84no—
126C22×C4C2×C22⋊C411yescharacteristic
127C22×C4C2×C4⋊C411yescharacteristic
128C22×C4#11711yescharacteristic
129C22×C4C22⋊Q812yes—
130C2×C8—21no—
131C2×C8—22no—
132C2×C8—41no—
133C2×C8—44no—
134C2×C8—42no—
135C2×C8—41no—
136C2×C8—41no—
137C2×C8—41no—
138C2×C8—42no—
139C2×C8—41no—
140C2×C8—41no—
141C2×C8—82no—
142C2×C8—84no—
143C2×C8#9912yes—
144C4⋊C4—41no—
145C4⋊C4—41no—
146C4⋊C4—82no—
147C4⋊C4—84no—
148C4⋊C4—84no—
149C22⋊C4—42no—
150C22⋊C4—42no—
151C22⋊C4—82no—
152C22⋊C4—82no—
153C22⋊C4—84no—
154C22⋊C4—84no—
155C42—21no—
156C42—21no—
157C42—21no—
158C42—41no—
159C42—41no—
160C42—41no—
161C42—81no—
162C42#10712yescomplemented
163C23—22no—
164C23—22no—
165C23—22no—
166C23—42no—
167C23—42no—
168C23—42no—
169C23—42no—
170C23—41no—
171C23—41no—
172C23#38011yescharacteristic
173C23#46612yes—
174C23#46712yes—
175D4—41no—
176D4—41no—
177D4—42no—
178D4—42no—
179D4—42no—
180D4—84no—
181D4—84no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4—81no—
203C2×C4—81no—
204C2×C4—164no—
205C2×C4—164no—
206C2×C4—164no—
207C2×C4C2×#9911yescharacteristic
208C2×C4#41011yescharacteristic
209C2×C4C2×#10711yescharacteristic
210C2×C4#42711yescharacteristic
211C8—22no—
212C8—42no—
213C8—41no—
214C8—81no—
215C8—81no—
216C22—21no—
217C22—21no—
218C22—22no—
219C22—22no—
220C22—42no—
221C22—42no—
222C22—44no—
223C22—42no—
224C22—42no—
225C22—42no—
226C22—82no—
227C22—82no—
228C22—81no—
229C22—81no—
230C22#179911yescharacteristic
231C22#180811yescharacteristic
232C22#181211yescharacteristic
233C4—21no—
234C4—21no—
235C4—21no—
236C4—42no—
237C4—41no—
238C4—42no—
239C4—81no—
240C4—81no—
241C4—324no—
242C4#212711yescharacteristic
243C4#212811yescharacteristic
244C2—21no—
245C2—21no—
246C2—42no—
247C2—42no—
248C2—82no—
249C2#860911yescharacteristic
250C2#1279911yescharacteristic
251C2#1280811yescharacteristic
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
5. C2×C8×D4C4semidirect
9. #1954C4semidirect
13. #1831C4semidirect
16. #1744C4semidirect
110. C2×D4#107semidirect
162. C42#107semidirect

Maximal subgroups

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

subgroupindex
#222602
#127972
#86092
#72372