Finite Group Databasebeta

← #147608

Subgroups — #147608

Order 512.

900 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 900 conjugacy classes of subgroups — too many to draw. Only its 138 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#35674C211yesmaximal, characteristic
2C22×#356C212yesmaximal
3C2×#1830C212yesmaximal
4#12586C212yesmaximal
5C22×C4⋊Q8C2211yescharacteristic
6C2×#519C2211yescharacteristic
7C22×#101C2214yes—
8#2484—24no—
9#2446—24no—
10#2446—24no—
11#2310C411yescharacteristic
12#2310C411yescharacteristic
13#2310C2211yescharacteristic
14#2296C411yescharacteristic
15#2296C411yescharacteristic
16C2×#356—28no—
17C2×#356—24no—
18#1830—28no—
19C2×#344—24no—
20#1394—24no—
21C23×Q8C2×C411yescharacteristic
22C23×Q8C2311yescharacteristic
23C22×M16—24no—
24C2×C4⋊Q8—22no—
25C2×C4⋊Q8—24no—
26C2×C4⋊Q8—24no—
27C2×C4⋊Q8D414yes—
28#519—24no—
29#519—24no—
30C2×C4×Q8—22no—
31C2×C4×Q8—22no—
32C2×#117C2×C411yescharacteristic
33C2×#117C2×C411yescharacteristic
34C2×#117C2×C411yescharacteristic
35C2×#117C2×C411yescharacteristic
36C2×#117D414yes—
37C22×C4⋊C4—22no—
38C22×C42C2×C411yescharacteristic
39C2×#101—24no—
40C2×#101—24no—
41C2×#101—24no—
42C2×#101—24no—
43C2×#101—24no—
44C2×#101D418yes—
45#400—28no—
46#398—28no—
47#391—22no—
48#391—22no—
49#391—22no—
50#391—22no—
51#391—22no—
52#391D414yes—
53#389—22no—
54#389—22no—
55#387—28no—
56#387—28no—
57#385—22no—
58#385—22no—
59#381—22no—
60#381—22no—
61#381—22no—
62C4×C4⋊C4—22no—
63C4×C4⋊C4—22no—
64#356—216no—
65#344—216no—
66C22×Q8—21no—
67C22×Q8—24no—
68C22×Q8—42no—
69C22×Q8—42no—
70C22×Q8—42no—
71C22×Q8—42no—
72C22×Q8C22⋊C412yes—
73C22×Q8C2×D412yes—
74C22×Q8C2×D412yes—
75C22×Q8C2×D414yes—
76C22×Q8#4114yes—
77C23×C4—21no—
78C23×C4—21no—
79C23×C4C22×C411yescharacteristic
80C23×C4C2×D412yes—
81C2×M16—28no—
82C2×M16—44no—
83C2×M16—44no—
84C2×M16—44no—
85C2×M16—44no—
86C2×M16—44no—
87C22×C8—44no—
88C4⋊Q8—24no—
89C4⋊Q8—28no—
90C4⋊Q8—24no—
91C4⋊Q8—44no—
92C4⋊Q8—42no—
93C4⋊Q8—42no—
94C4×Q8—42no—
95C4×Q8—44no—
96C4×Q8—44no—
97C4×Q8—42no—
98C4×Q8—42no—
99C4×Q8—42no—
100#117—24no—
101#117—24no—
102#117—24no—
103#117—24no—
104#117—24no—
105#117—24no—
106#117—42no—
107#117—42no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—28no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—44no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4C22⋊C412yes—
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4C22⋊C412yes—
122C2×C22⋊C4C22⋊C412yes—
123C2×C42—24no—
124C2×C42—21no—
125C2×C42—21no—
126C2×C42—21no—
127C2×C42—21no—
128C2×C42—24no—
129C2×C42C22⋊C412yes—
130C2×C42C22⋊C412yes—
131#101—28no—
132#101—28no—
133#101—44no—
134#101—44no—
135#95—24no—
136#95—24no—
137#95—28no—
138#95—28no—
139#95—28no—
140C24C2×C22⋊C411yescharacteristic
141C2×Q8—24no—
142C2×Q8—44no—
143C2×Q8—44no—
144C2×Q8—42no—
145C2×Q8—42no—
146C2×Q8—42no—
147C2×Q8—42no—
148C2×Q8—44no—
149C2×Q8—44no—
150C2×Q8—42no—
151C2×Q8—42no—
152C2×Q8—42no—
153C2×Q8—42no—
154C2×Q8—42no—
155C2×Q8—84no—
156C2×Q8—84no—
157C2×Q8#12012yes—
158C2×Q8#12114yes—
159C2×Q8#12114yes—
160C2×Q8#12312yes—
161C2×Q8#12412yes—
162C2×Q8#12712yes—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—24no—
166C22×C4—24no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—44no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4C2×C22⋊C411yescharacteristic
179C22×C4C2×C22⋊C411yescharacteristic
180C22×C4C4×D414yes—
181C22×C4#12012yes—
182C22×C4#12014yes—
183C22×C4#12114yes—
184C22×C4#12312yes—
185M16—48no—
186M16—48no—
187M16—84no—
188M16—84no—
189C2×C8—48no—
190C2×C8—84no—
191C2×C8—84no—
192C4⋊C4—42no—
193C4⋊C4—48no—
194C4⋊C4—48no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4—48no—
199C4⋊C4—44no—
200C4⋊C4—48no—
201C4⋊C4—44no—
202C22⋊C4—24no—
203C22⋊C4—24no—
204C22⋊C4—42no—
205C22⋊C4—42no—
206C42—24no—
207C42—42no—
208C42—42no—
209C42—42no—
210C42—42no—
211C42—42no—
212C42—44no—
213C42—44no—
214C23—42no—
215C23#37911yescharacteristic
216C23#38612yes—
217C23#41012yes—
218C23#41012yes—
219Q8—42no—
220Q8—44no—
221Q8—44no—
222Q8—42no—
223Q8—42no—
224Q8—42no—
225Q8—84no—
226Q8—84no—
227C2×C4—24no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—24no—
231C2×C4—24no—
232C2×C4—24no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—24no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—24no—
239C2×C4—24no—
240C2×C4—42no—
241C2×C4—48no—
242C2×C4—42no—
243C2×C4—44no—
244C2×C4—44no—
245C2×C4—44no—
246C2×C4—44no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—84no—
255C2×C4#38612yes—
256C2×C4#39012yes—
257C2×C4#39214yes—
258C2×C4#39414yes—
259C8—88no—
260C22—22no—
261C22—24no—
262C22—42no—
263C22—42no—
264C22#143811yescharacteristic
265C22#178211yescharacteristic
266C22#178211yescharacteristic
267C22#189914yes—
268C4—24no—
269C4—22no—
270C4—22no—
271C4—24no—
272C4—44no—
273C4—82no—
274C4—82no—
275C4—82no—
276C4—82no—
277C4—82no—
278C2—21no—
279C2—21no—
280C2—24no—
281C2#1096611yescharacteristic
282C2#1273012yes—
subgroupG/H
center, socleC22#1782
commutatorC23×C4C22×C4
Frattini, cosocleC23×Q8C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C2×#519 ◁ G
LECSG ▷ C23×Q8 ▷ C22 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#356742
C22×#3562
C2×#18302
#125862