Finite Group Databasebeta

← #112283

Subgroups — #112283

Order 512.

854 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 854 conjugacy classes of subgroups — too many to draw. Only its 142 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×#2642C211yesmaximal, characteristic
2#21830C211yesmaximal, characteristic
3#17742C212yesmaximal
4#17394C211yesmaximal, characteristic
5#8072C212yesmaximal
6C22×C4⋊Q8C2211yescharacteristic
7C22×#125C2211yescharacteristic
8#2920C2212yes—
9#2920C2212yes—
10#2642—24no—
11#2642—24no—
12C2×#400—22no—
13C2×#400C414yes—
14C2×#391—21no—
15C2×#391—21no—
16C2×#376C2211yescharacteristic
17#1998—24no—
18#1821—21no—
19#1821—21no—
20#1556—24no—
21#1549—24no—
22#1546—24no—
23#1470—24no—
24#1463—22no—
25#1300—24no—
26C23×Q8—21no—
27C2×C4⋊Q8—22no—
28C2×C4⋊Q8—22no—
29C2×C4⋊Q8—22no—
30C2×C4⋊Q8—22no—
31C2×C4⋊Q8—22no—
32C2×C4⋊Q8D414yes—
33C2×#125—22no—
34C2×#125—22no—
35C2×#125—22no—
36C2×#125—22no—
37C2×#125—22no—
38C2×#125—22no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4C2×C412yes—
41C22×C4⋊C4C2×C412yes—
42C22×C4⋊C4D411yescharacteristic
43C22×C4⋊C4D411yescharacteristic
44C22×C4⋊C4D412yes—
45C22×C42—21no—
46C22×C42C2311yescharacteristic
47#432—22no—
48#432—22no—
49#432—22no—
50#432D414yes—
51#431—22no—
52#431—22no—
53#431—22no—
54#431—22no—
55#423—22no—
56#423D414yes—
57#407—22no—
58#407—22no—
59C4×M16—22no—
60C4×M16—22no—
61#400—216no—
62#400—28no—
63#400—28no—
64#391—28no—
65#391—28no—
66#376—22no—
67#376D414yes—
68C2×#95—24no—
69C2×#95—24no—
70C2×#95—21no—
71C2×#95C2×C412yes—
72#343—24no—
73#343—24no—
74#332—216no—
75#324—28no—
76C22×Q8—24no—
77C22×Q8—48no—
78C22×Q8—41no—
79C22×Q8—42no—
80C22×Q8—41no—
81C23×C4—22no—
82C23×C4—22no—
83C23×C4C22⋊C412yes—
84C23×C4C22⋊C412yes—
85C23×C4C22×C411yescharacteristic
86C23×C4C2×D411yescharacteristic
87C23×C4C2×D411yescharacteristic
88C2×M16—24no—
89C2×M16—42no—
90C4⋊Q8—24no—
91C4⋊Q8—24no—
92C4⋊Q8—24no—
93C4⋊Q8—24no—
94C4⋊Q8—42no—
95C4⋊Q8—42no—
96C4⋊Q8—42no—
97C4⋊Q8—42no—
98#125—42no—
99#125—42no—
100#125—42no—
101#125—42no—
102#125—42no—
103#125—42no—
104#125—42no—
105#125—42no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4C22⋊C414yes—
130C2×C42—24no—
131C2×C42—44no—
132C2×C42C2×D412yes—
133C2×C42C2×D412yes—
134C2×C42C2×D412yes—
135C2×C42C2×D412yes—
136C2×C42#4112yes—
137C2×C42#4112yes—
138C4⋊C8—24no—
139C4⋊C8—24no—
140C4⋊C8—24no—
141C4⋊C8—42no—
142C22⋊C8—24no—
143C22⋊C8—42no—
144C22⋊C8—42no—
145C22⋊C8—42no—
146#97—24no—
147#97—42no—
148C4×C8—42no—
149C4×C8—42no—
150#95—216no—
151#95—216no—
152#95—22no—
153#95—24no—
154#95C22⋊C414yes—
155C24C2×C22⋊C411yescharacteristic
156C2×Q8—416no—
157C2×Q8—44no—
158C2×Q8—44no—
159C2×Q8—84no—
160C2×Q8—88no—
161C2×Q8—84no—
162C22×C4—24no—
163C22×C4—24no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—24no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—44no—
174C22×C4—44no—
175C22×C4—44no—
176C22×C4—44no—
177C22×C4—48no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—44no—
181C22×C4C2×C22⋊C411yescharacteristic
182C22×C4C2×C22⋊C411yescharacteristic
183C22×C4C4×D412yes—
184C22×C4C4×D412yes—
185C22×C4#12012yes—
186C22×C4#12012yes—
187C22×C4#12011yescharacteristic
188C22×C4#12011yescharacteristic
189C22×C4#12112yes—
190C22×C4#12112yes—
191C22×C4#12311yescharacteristic
192C22×C4#12311yescharacteristic
193M16—48no—
194M16—84no—
195C2×C8—44no—
196C2×C8—44no—
197C4⋊C4—28no—
198C4⋊C4—44no—
199C4⋊C4—44no—
200C4⋊C4—44no—
201C4⋊C4—44no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—44no—
205C4⋊C4—44no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—44no—
209C42—48no—
210C42#12012yes—
211C42#12112yes—
212C42#12112yes—
213C42#12112yes—
214C42#12112yes—
215C42#12312yes—
216C42#12412yes—
217C42#12712yes—
218C23—22no—
219C23—22no—
220C23—22no—
221C23#37911yescharacteristic
222C23#38611yescharacteristic
223C23#38611yescharacteristic
224Q8—88no—
225Q8—88no—
226C2×C4—24no—
227C2×C4—22no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—24no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—42no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—44no—
238C2×C4—44no—
239C2×C4—44no—
240C2×C4—48no—
241C2×C4—44no—
242C2×C4—44no—
243C2×C4—44no—
244C2×C4—44no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—88no—
248C2×C4—84no—
249C2×C4—84no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4#38612yes—
254C2×C4#39012yes—
255C2×C4#39212yes—
256C2×C4#39212yes—
257C2×C4#39412yes—
258C2×C4#39412yes—
259C2×C4#45414yes—
260C8—84no—
261C8—84no—
262C22—28no—
263C22—21no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22#143811yescharacteristic
270C22#179612yes—
271C22#188812yes—
272C22#191512yes—
273C4—22no—
274C4—22no—
275C4—22no—
276C4—22no—
277C4—22no—
278C4—84no—
279C4—82no—
280C4—84no—
281C4—82no—
282C4#200314yes—
283C2—22no—
284C2—22no—
285C2—22no—
286C2#1097411yescharacteristic
287C2#1097611yescharacteristic
288C2#1098511yescharacteristic
subgroupG/H
center, socleC22#1438
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#26422
#218302
#177422
#173942
#80722