Finite Group Databasebeta

← #150265

Subgroups — #150265

Order 512.

464 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 74 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#23453C211yesmaximal, characteristic
2#12581C211yesmaximal, characteristic
3#11431C211yesmaximal, characteristic
4#7428C214yesmaximal, complemented
5C22×#125C411yescharacteristic, complemented
6C2×#417—21no—
7C2×#410—21no—
8#2297C2211yescharacteristic
9#2026—21no—
10#2026C412yescomplemented
11#2025—21no—
12#2025C2212yes—
13#1736—21no—
14#1728C411yescharacteristic, complemented
15#1458—21no—
16#1393—21no—
17#1393C2212yes—
18#1319—21no—
19#1318C2212yes—
20C22×#41D411yescharacteristic
21C22×M16—21no—
22C2×#125—21no—
23C2×#125—41no—
24C2×#125—44no—
25C2×#125C2×C412yes—
26C2×#117—41no—
27C2×#117D411yescharacteristic
28C22×C4⋊C4—21no—
29C22×C4⋊C4—41no—
30C22×C42C2×C411yescharacteristic
31#431—41no—
32#417—44no—
33C2×#103—22no—
34C2×#102—22no—
35#410—24no—
36#410—42no—
37C2×#99—22no—
38C2×#99—41no—
39#391C2×C411yescharacteristic
40#391D411yescharacteristic
41#390D411yescharacteristic
42#390C2311yescharacteristic
43#386—41no—
44C4×C22⋊C4—41no—
45#340—22no—
46#340—22no—
47#336—21no—
48#336C2×C412yes—
49#324—22no—
50#323—41no—
51C2×#41—22no—
52C2×#41—41no—
53C2×#41—41no—
54C2×#41—42no—
55C2×#41SD1612yes—
56C2×#41Q1612yes—
57C22×Q8C22⋊C411yescharacteristic
58C22×D4—21no—
59C22×D4C2×D411yescharacteristic
60C23×C4—21no—
61C23×C4—21no—
62C23×C4—41no—
63C23×C4C22⋊C411yescharacteristic
64C2×M16—41no—
65C2×M16—41no—
66C2×M16—42no—
67C2×M16—42no—
68C22×C8—22no—
69#125—44no—
70#125—44no—
71#125—44no—
72#125—44no—
73#117—41no—
74#117—41no—
75#117—44no—
76#117—82no—
77C2×C4⋊C4—24no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—48no—
83C2×C4⋊C4—41no—
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×C42—21no—
92C2×C42—21no—
93C2×C42—42no—
94C2×C42—81no—
95C2×C42C22⋊C411yescharacteristic
96C2×C42C22×C411yescharacteristic
97C4⋊C8—81no—
98#103—44no—
99#102—44no—
100#99—24no—
101#99—44no—
102#99—42no—
103C22⋊C8—42no—
104C4×C8—81no—
105#95—41no—
106#95—41no—
107C24—21no—
108C24#9911yescharacteristic
109#41—42no—
110#41—42no—
111#41—42no—
112#41—42no—
113#41—82no—
114#41—82no—
115C2×Q8—21no—
116C2×Q8—41no—
117C2×Q8—41no—
118C2×Q8#10312yes—
119C2×D4—22no—
120C2×D4—22no—
121C2×D4—21no—
122C2×D4—41no—
123C2×D4—41no—
124C2×D4—81no—
125C2×D4—81no—
126C2×D4C2×SD1611yescharacteristic
127C2×D4C2×Q1611yescharacteristic
128C22×C4—22no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—42no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4—42no—
142C22×C4—44no—
143C22×C4—42no—
144C22×C4—81no—
145C22×C4—81no—
146C22×C4—82no—
147C22×C4—82no—
148C22×C4—82no—
149C22×C4#9911yescharacteristic
150C22×C4#10312yes—
151C22×C4#10412yes—
152C22×C4C2×C22⋊C411yescharacteristic
153M16—82no—
154M16—82no—
155C2×C8—42no—
156C2×C8—42no—
157C2×C8—42no—
158C2×C8—82no—
159C4⋊C4—24no—
160C4⋊C4—41no—
161C4⋊C4—41no—
162C4⋊C4—42no—
163C4⋊C4—44no—
164C4⋊C4—44no—
165C4⋊C4—48no—
166C4⋊C4—44no—
167C4⋊C4—84no—
168C22⋊C4—44no—
169C22⋊C4—81no—
170C22⋊C4—82no—
171C22⋊C4—82no—
172C22⋊C4—82no—
173C42—41no—
174C42—41no—
175C42—42no—
176C42—41no—
177C42—41no—
178C42—84no—
179C23—21no—
180C23—21no—
181C23—21no—
182C23—21no—
183C23—41no—
184C23—41no—
185C23—41no—
186C23—41no—
187C23—81no—
188C23—81no—
189C23#32812yes—
190C23C2×#9911yescharacteristic
191Q8—42no—
192Q8—81no—
193D4—41no—
194D4—41no—
195D4—81no—
196D4—82no—
197D4—82no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—44no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—82no—
221C2×C4—82no—
222C2×C4—84no—
223C2×C4—88no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4—84no—
227C2×C4—162no—
228C2×C4—162no—
229C2×C4#32812yes—
230C2×C4AΣL(1,𝔽16)12yes—
231C2×C4#35411yescharacteristic
232C2×C4#35411yescharacteristic
233C2×C4C2×#10311yescharacteristic
234C2×C4C2×#10411yescharacteristic
235C8—82no—
236C8—162no—
237C22—21no—
238C22—21no—
239C22—22no—
240C22—42no—
241C22—42no—
242C22—41no—
243C22—41no—
244C22—41no—
245C22—41no—
246C22—81no—
247C22—81no—
248C22—81no—
249C22—81no—
250C22—161no—
251C22C2×#32811yescharacteristic
252C22C2×AΣL(1,𝔽16)11yescharacteristic
253C22C2×#35411yescharacteristic
254C4—21no—
255C4—21no—
256C4—42no—
257C4—81no—
258C4—81no—
259C4—84no—
260C4—81no—
261C4—81no—
262C4—164no—
263C4#135112yes—
264C4#135112yes—
265C2—21no—
266C2—21no—
267C2—42no—
268C2—81no—
269C2—81no—
270C2C2×#135111yescharacteristic
271C2#1359611yescharacteristic
272C2#1360411yescharacteristic
subgroupG/H
center, socleC22C2×#354
commutatorC2×C42C22×C4
Frattini, cosocle#390C23
prime radicalC22×#41D4
LCSG ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ #2297 ◁ G
LECSG ▷ #390 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7428C2semidirect
5. C22×#125C4semidirect
10. #2026C4semidirect
14. #1728C4semidirect

Maximal subgroups

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

subgroupindex
#234532
#125812
#114312
#74282