Finite Group Databasebeta

← #150768

Subgroups — #150768

Order 512.

369 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 369 conjugacy classes of subgroups — too many to draw. Only its 56 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#28304C211yesmaximal, characteristic
2#12860C211yesmaximal, characteristic
3C2×#1319C211yesmaximal, characteristic
4#7436C212yesmaximal, complemented
5#7405C212yesmaximal, complemented
6C2×#484—21no—
7C2×#451—21no—
8#2857C411yescharacteristic
9C2×#391C2211yescharacteristic
10#2095C411yescharacteristic
11#2095C2211yescharacteristic
12#2016C411yescharacteristic
13#2016C2211yescharacteristic
14#1969—21no—
15#1969—21no—
16#1784—21no—
17C4×C22⋊C8C411yescharacteristic
18C2×#324—21no—
19#1394—21no—
20#1394C2212yes—
21#1319—21no—
22#1319C2212yes—
23C23×Q8D411yescharacteristic
24C22×Q16—21no—
25C22×Q16—21no—
26C2×C4⋊Q8C2×C411yescharacteristic
27C2×C22⋊Q8—21no—
28C22×C4⋊C4D411yescharacteristic
29C22×C42C2×C411yescharacteristic
30#484—42no—
31#484—42no—
32#451—42no—
33#451—42no—
34C2×#107—21no—
35C2×#107—21no—
36C2×#103—21no—
37C2×#103—21no—
38C2×#103—21no—
39C2×#103—21no—
40C2×#101—22no—
41C2×#101—41no—
42#407—41no—
43C2×C22⋊C8—21no—
44C2×C22⋊C8—41no—
45C2×C4×C8—21no—
46#391—21no—
47#391—41no—
48#391C2×C411yescharacteristic
49#391D411yescharacteristic
50#391D411yescharacteristic
51#391C2311yescharacteristic
52#389—41no—
53#384C2×C411yescharacteristic
54#380—41no—
55C2×#95—41no—
56#336C2×C411yescharacteristic
57#336C2×C411yescharacteristic
58#324—22no—
59#324—41no—
60C22×Q8—21no—
61C22×Q8—22no—
62C22×Q8—21no—
63C22×Q8—41no—
64C22×Q8—41no—
65C22×Q8—41no—
66C22×Q8—81no—
67C22×Q8C22⋊C411yescharacteristic
68C22×Q8C2×D411yescharacteristic
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4C22⋊C411yescharacteristic
73C2×Q16—42no—
74C2×Q16—42no—
75C2×Q16—42no—
76C2×Q16—42no—
77C2×Q16—81no—
78C2×Q16—81no—
79C2×M16—42no—
80C22×C8—21no—
81C22×C8—21no—
82C22×C8—81no—
83C4⋊Q8—81no—
84C22⋊Q8—41no—
85C22⋊Q8—41no—
86C22⋊Q8—41no—
87C22⋊Q8—41no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4C22⋊C411yescharacteristic
96C2×C4⋊C4C2×D411yescharacteristic
97C2×C22⋊C4—41no—
98C2×C42—21no—
99C2×C42—81no—
100C2×C42C22⋊C411yescharacteristic
101C2×C42C22×C411yescharacteristic
102#107—42no—
103#107—42no—
104#103—42no—
105#103—42no—
106#103—42no—
107#103—42no—
108#101—42no—
109#101—42no—
110#101—82no—
111C22⋊C8—22no—
112C22⋊C8—22no—
113C22⋊C8—42no—
114C22⋊C8—81no—
115C22⋊C8—81no—
116C22⋊C8—81no—
117C4×C8—81no—
118#95—41no—
119#95—41no—
120#95—41no—
121#95—41no—
122#95—42no—
123C24#9911yescharacteristic
124C2×Q8—21no—
125C2×Q8—21no—
126C2×Q8—21no—
127C2×Q8—42no—
128C2×Q8—42no—
129C2×Q8—41no—
130C2×Q8—41no—
131C2×Q8—41no—
132C2×Q8—41no—
133C2×Q8—41no—
134C2×Q8—41no—
135C2×Q8—41no—
136C2×Q8—41no—
137C2×Q8—41no—
138C2×Q8—41no—
139C2×Q8—81no—
140C2×Q8—82no—
141C2×Q8—82no—
142C2×Q8—82no—
143C2×Q8—82no—
144C2×Q8#13512yes—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—24no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—41no—
157C22×C4—42no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—81no—
165C22×C4—81no—
166C22×C4—81no—
167C22×C4—82no—
168C22×C4#9911yescharacteristic
169C22×C4C2×C22⋊C411yescharacteristic
170Q16—84no—
171Q16—84no—
172M16—82no—
173M16—82no—
174C2×C8—22no—
175C2×C8—22no—
176C2×C8—41no—
177C2×C8—41no—
178C2×C8—41no—
179C2×C8—41no—
180C2×C8—82no—
181C2×C8—82no—
182C2×C8—162no—
183C4⋊C4—41no—
184C4⋊C4—41no—
185C4⋊C4—41no—
186C4⋊C4—41no—
187C4⋊C4—81no—
188C4⋊C4—81no—
189C4⋊C4—81no—
190C4⋊C4—82no—
191C4⋊C4—82no—
192C22⋊C4—81no—
193C22⋊C4—81no—
194C42—41no—
195C42—41no—
196C42—81no—
197C23—21no—
198C23—22no—
199C23—21no—
200C23—41no—
201C23#35311yescharacteristic
202C23#35311yescharacteristic
203C23C2×#9911yescharacteristic
204Q8—42no—
205Q8—42no—
206Q8—81no—
207Q8—81no—
208Q8—81no—
209Q8—81no—
210Q8—81no—
211Q8—81no—
212Q8—84no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—84no—
237C2×C4—82no—
238C2×C4—82no—
239C2×C4—82no—
240C2×C4—81no—
241C2×C4—81no—
242C2×C4—82no—
243C2×C4—81no—
244C2×C4—81no—
245C2×C4—81no—
246C2×C4—81no—
247C2×C4—161no—
248C2×C4#41611yescharacteristic
249C2×C4#42111yescharacteristic
250C8—42no—
251C8—42no—
252C8—162no—
253C8—162no—
254C22—21no—
255C22—21no—
256C22—21no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—42no—
262C22—42no—
263C22#141112yes—
264C22#151611yescharacteristic
265C22#212111yescharacteristic
266C22C2×#35311yescharacteristic
267C4—21no—
268C4—21no—
269C4—42no—
270C4—42no—
271C4—42no—
272C4—41no—
273C4—81no—
274C4—81no—
275C4—81no—
276C4—161no—
277C4—161no—
278C2—21no—
279C2—22no—
280C2—21no—
281C2—41no—
282C2#1184311yescharacteristic
283C2#1357811yescharacteristic
284C2C2×#141111yescharacteristic
subgroupG/H
center, socleC22C2×#353
commutatorC2×C42C22×C4
Frattini, cosocle#391C23
prime radicalC24#99
LCSG ▷ C2×C42 ▷ 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. #7436C2semidirect
5. #7405C2semidirect

Maximal subgroups

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

subgroupindex
#283042
#128602
C2×#13192
#74362
#74052