Finite Group Databasebeta

← #150276

Subgroups — #150276

Order 512.

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

normal subgroups only

This group has 453 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
1C2×#2028C211yesmaximal, characteristic
2C2×#1394C211yesmaximal, characteristic
3#7595C211yesmaximal, characteristic
4#7431C214yesmaximal, complemented
5C2×#544C411yescharacteristic
6C2×#419—21no—
7C22×#101—21no—
8C2×#391C2211yescharacteristic
9#2028—21no—
10#2028—21no—
11#2028C412yes—
12#2028C2212yes—
13#1817—21no—
14C2×#343C411yescharacteristic
15C2×#339—21no—
16#1394—21no—
17#1394C2212yes—
18#1272—21no—
19#1272C2212yes—
20C23×Q8D411yescharacteristic
21C22×M16—21no—
22C22×M16—41no—
23#544—21no—
24#544—41no—
25#544—42no—
26#544C2×C412yes—
27C2×C4⋊Q8—41no—
28C2×#125—41no—
29C2×C22⋊Q8—41no—
30C2×#117—21no—
31C22×C4⋊C4C2×C411yescharacteristic
32C22×C42D411yescharacteristic
33#419—22no—
34#419—22no—
35#419—42no—
36C2×#103—41no—
37C2×#103—41no—
38C2×#101—24no—
39C2×#101—22no—
40C2×#101—41no—
41C2×#101—42no—
42C2×C22⋊C8—41no—
43#391—41no—
44#391C2×C411yescharacteristic
45#391D411yescharacteristic
46#391D411yescharacteristic
47#391C2311yescharacteristic
48C2×#95—41no—
49#343—21no—
50#343C2×C412yes—
51#339—22no—
52#339—22no—
53#336—22no—
54#336—41no—
55C22×Q8—21no—
56C22×Q8—22no—
57C22×Q8—42no—
58C22×Q8—41no—
59C22×Q8—41no—
60C22×Q8—41no—
61C22×Q8C22⋊C411yescharacteristic
62C22×Q8SD1612yes—
63C22×Q8Q1612yes—
64C22×Q8C2×D411yescharacteristic
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—21no—
68C23×C4C22⋊C411yescharacteristic
69C2×M16—22no—
70C2×M16—41no—
71C2×M16—42no—
72C2×M16—44no—
73C2×M16—42no—
74C2×M16—42no—
75C2×M16—81no—
76C2×M16—82no—
77C22×C8—41no—
78C22×C8—42no—
79C22×C8—81no—
80C4⋊Q8—42no—
81C4⋊Q8—42no—
82C4⋊Q8—44no—
83#125—42no—
84#125—42no—
85#125—44no—
86C22⋊Q8—42no—
87C22⋊Q8—42no—
88C22⋊Q8—42no—
89C22⋊Q8—42no—
90#117—22no—
91#117—22no—
92#117—41no—
93#117—41no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4C22⋊C411yescharacteristic
102C2×C4⋊C4C22×C411yescharacteristic
103C2×C22⋊C4—41no—
104C2×C42—21no—
105C2×C42—41no—
106C2×C42—42no—
107C2×C42C22⋊C411yescharacteristic
108C2×C42C2×D411yescharacteristic
109#103—42no—
110#103—42no—
111#103—42no—
112#103—42no—
113#101—24no—
114#101—44no—
115#101—42no—
116C22⋊C8—82no—
117#95—41no—
118#95—41no—
119#95—41no—
120#95—41no—
121C24#9911yescharacteristic
122C2×Q8—22no—
123C2×Q8—22no—
124C2×Q8—21no—
125C2×Q8—21no—
126C2×Q8—41no—
127C2×Q8—41no—
128C2×Q8—41no—
129C2×Q8—42no—
130C2×Q8—42no—
131C2×Q8—42no—
132C2×Q8—42no—
133C2×Q8—41no—
134C2×Q8—42no—
135C2×Q8—42no—
136C2×Q8—81no—
137C2×Q8—82no—
138C2×Q8—82no—
139C2×Q8—81no—
140C2×Q8—162no—
141C2×Q8#10312yes—
142C2×Q8C2×SD1611yescharacteristic
143C2×Q8C2×Q1611yescharacteristic
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—42no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—42no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—42no—
163C22×C4—41no—
164C22×C4—81no—
165C22×C4—82no—
166C22×C4#9911yescharacteristic
167C22×C4#10312yes—
168C22×C4#10412yes—
169C22×C4C2×C22⋊C411yescharacteristic
170M16—44no—
171M16—42no—
172M16—42no—
173M16—82no—
174M16—84no—
175M16—82no—
176C2×C8—42no—
177C2×C8—82no—
178C2×C8—82no—
179C2×C8—84no—
180C2×C8—82no—
181C2×C8—82no—
182C2×C8—82no—
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—41no—
186C4⋊C4—41no—
187C4⋊C4—42no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—44no—
193C4⋊C4—44no—
194C22⋊C4—42no—
195C22⋊C4—42no—
196C42—41no—
197C42—41no—
198C42—42no—
199C42—42no—
200C42—81no—
201C23—41no—
202C23—41no—
203C23#32812yes—
204C23AΣL(1,𝔽16)11yescharacteristic
205C23AΣL(1,𝔽16)11yescharacteristic
206C23#35412yes—
207C23C2×#9911yescharacteristic
208Q8—42no—
209Q8—41no—
210Q8—41no—
211Q8—81no—
212Q8—81no—
213Q8—82no—
214Q8—82no—
215Q8—162no—
216Q8—162no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—21no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—84no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—161no—
253C2×C4#32812yes—
254C2×C4C2×#10311yescharacteristic
255C2×C4C2×#10411yescharacteristic
256C8—82no—
257C8—84no—
258C8—162no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22—42no—
266C22—42no—
267C22#135112yes—
268C22#135112yes—
269C22C2×#32811yescharacteristic
270C22C2×AΣL(1,𝔽16)11yescharacteristic
271C22C2×#35411yescharacteristic
272C4—21no—
273C4—22no—
274C4—21no—
275C4—42no—
276C4—41no—
277C4—42no—
278C4—41no—
279C4—81no—
280C4—82no—
281C4—81no—
282C4—81no—
283C4—82no—
284C2—21no—
285C2—22no—
286C2—22no—
287C2—21no—
288C2C2×#135111yescharacteristic
289C2#1356511yescharacteristic
290C2#1357311yescharacteristic
subgroupG/H
center, socleC22C2×AΣL(1,𝔽16)
commutatorC2×C4⋊C4C22×C4
Frattini, cosocle#391C23
prime radicalC24#99
LCSG ▷ C2×C4⋊C4 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#391 ◁ G
LECSG ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7431C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20282
C2×#13942
#75952
#74312