Finite Group Databasebeta

← #150301

Subgroups — #150301

Order 512.

358 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 66 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#29964C211yesmaximal, characteristic
2#12583C211yesmaximal, characteristic
3#11432C211yesmaximal, characteristic
4#7434C212yesmaximal, complemented
5#7428C212yesmaximal, complemented
6C2×#410—21no—
7#2629C411yescharacteristic, complemented
8#2311C2211yescharacteristic
9#2095C411yescharacteristic, complemented
10#2093C2211yescharacteristic
11#2068—21no—
12#2067—21no—
13#2026C411yescharacteristic, complemented
14#2025C2211yescharacteristic
15#1852—21no—
16#1798C411yescharacteristic, complemented
17#1788—21no—
18C2×#324—21no—
19#1393—21no—
20#1393C2212yes—
21#1319—21no—
22#1318C2212yes—
23C22×#41D411yescharacteristic
24C2×#125C2×C411yescharacteristic
25C2×#117—41no—
26C2×#117C2×C411yescharacteristic
27C22×C4⋊C4D411yescharacteristic
28C2×#107—21no—
29C2×#107—21no—
30C2×C4⋊C8—21no—
31C2×#103—21no—
32C2×#103—21no—
33C2×#102—21no—
34C2×#102—21no—
35#410—24no—
36#410—42no—
37C2×#99—22no—
38C2×#99—41no—
39C2×C22⋊C8—21no—
40C2×C22⋊C8—41no—
41#398—41no—
42#396—41no—
43#391C2×C411yescharacteristic
44#391D411yescharacteristic
45#390D411yescharacteristic
46#390C2311yescharacteristic
47#389—21no—
48#386—41no—
49#384C2×C411yescharacteristic
50#380—21no—
51#380—41no—
52#340—21no—
53#340—21no—
54#336C2×C411yescharacteristic
55#336C2×C411yescharacteristic
56#324—22no—
57#324—41no—
58C2×#41—22no—
59C2×#41—81no—
60C2×#41—81no—
61C2×#41SD1612yes—
62C2×#41Q1612yes—
63C22×Q8C22⋊C411yescharacteristic
64C22×D4—21no—
65C22×D4C2×D411yescharacteristic
66C23×C4—21no—
67C23×C4—21no—
68C23×C4C22⋊C411yescharacteristic
69C22×C8—21no—
70C22×C8—21no—
71C22×C8—81no—
72#125—81no—
73#117—44no—
74#117—81no—
75#117—82no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—81no—
84C2×C4⋊C4—81no—
85C2×C4⋊C4C22⋊C411yescharacteristic
86C2×C4⋊C4C2×D411yescharacteristic
87C2×C22⋊C4—21no—
88C2×C22⋊C4—42no—
89C2×C22⋊C4—41no—
90C2×C22⋊C4—41no—
91C2×C22⋊C4—41no—
92C2×C42—81no—
93C2×C42C22⋊C411yescharacteristic
94C2×C42C22×C411yescharacteristic
95#107—81no—
96#107—81no—
97C4⋊C8—81no—
98#103—81no—
99#103—81no—
100#102—81no—
101#102—81no—
102#99—24no—
103#99—44no—
104#99—42no—
105C22⋊C8—42no—
106C22⋊C8—42no—
107C22⋊C8—81no—
108#95—41no—
109#95—41no—
110#95—41no—
111#95—41no—
112C24—21no—
113C24#9911yescharacteristic
114#41—82no—
115#41—82no—
116#41—82no—
117#41—82no—
118C2×Q8—21no—
119C2×Q8—41no—
120C2×Q8—41no—
121C2×Q8#10312yes—
122C2×D4—22no—
123C2×D4—22no—
124C2×D4—21no—
125C2×D4—41no—
126C2×D4—41no—
127C2×D4—81no—
128C2×D4—81no—
129C2×D4C2×SD1611yescharacteristic
130C2×D4C2×Q1611yescharacteristic
131C22×C4—21no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4—42no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—82no—
148C22×C4—161no—
149C22×C4#9911yescharacteristic
150C22×C4#10312yes—
151C22×C4#10412yes—
152C22×C4C2×C22⋊C411yescharacteristic
153C2×C8—41no—
154C2×C8—41no—
155C2×C8—41no—
156C2×C8—41no—
157C2×C8—41no—
158C2×C8—41no—
159C2×C8—82no—
160C2×C8—162no—
161C4⋊C4—41no—
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4—41no—
165C4⋊C4—41no—
166C4⋊C4—41no—
167C4⋊C4—81no—
168C4⋊C4—81no—
169C4⋊C4—82no—
170C4⋊C4—84no—
171C4⋊C4—161no—
172C4⋊C4—161no—
173C22⋊C4—44no—
174C22⋊C4—81no—
175C22⋊C4—82no—
176C22⋊C4—161no—
177C22⋊C4—161no—
178C22⋊C4—161no—
179C42—41no—
180C42—41no—
181C42—84no—
182C23—21no—
183C23—21no—
184C23—21no—
185C23—21no—
186C23—41no—
187C23—41no—
188C23—41no—
189C23—41no—
190C23—81no—
191C23—81no—
192C23#32812yes—
193C23C2×#9911yescharacteristic
194Q8—42no—
195Q8—81no—
196D4—41no—
197D4—41no—
198D4—81no—
199D4—82no—
200D4—82no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—82no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4—81no—
227C2×C4—81no—
228C2×C4—81no—
229C2×C4—84no—
230C2×C4—81no—
231C2×C4—81no—
232C2×C4—161no—
233C2×C4—162no—
234C2×C4—162no—
235C2×C4#32812yes—
236C2×C4C2×#10311yescharacteristic
237C2×C4C2×#10411yescharacteristic
238C8—81no—
239C8—81no—
240C8—162no—
241C22—21no—
242C22—21no—
243C22—22no—
244C22—42no—
245C22—41no—
246C22—41no—
247C22—41no—
248C22—41no—
249C22—81no—
250C22—81no—
251C22—81no—
252C22—81no—
253C22—81no—
254C22—161no—
255C22C2×#32811yescharacteristic
256C22#212111yescharacteristic
257C22#212711yescharacteristic
258C4—21no—
259C4—22no—
260C4—21no—
261C4—42no—
262C4—81no—
263C4—81no—
264C4—81no—
265C4—81no—
266C4—161no—
267C4—164no—
268C4—161no—
269C2—21no—
270C2—21no—
271C2—42no—
272C2—81no—
273C2—81no—
274C2#1171611yescharacteristic
275C2#1360811yescharacteristic
276C2#1360911yescharacteristic
subgroupG/H
center, socleC22#2127
commutatorC2×C42C22×C4
Frattini, cosocle#390C23
prime radicalC22×#41D4
LCSG ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C23×C4 ◁ #2311 ◁ G
LECSG ▷ #390 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7434C2semidirect
5. #7428C2semidirect
7. #2629C4semidirect
9. #2095C4semidirect
13. #2026C4semidirect
16. #1798C4semidirect

Maximal subgroups

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

subgroupindex
#299642
#125832
#114322
#74342
#74282