Finite Group Databasebeta

← #111928

Subgroups — #111928

Order 512.

900 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 900 conjugacy classes of subgroups — too many to draw. Only its 150 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×#2660C211yesmaximal, characteristic
2#17742C212yesmaximal
3#17394C211yesmaximal, characteristic
4#17394C211yesmaximal, characteristic
5#7855C212yesmaximal
6C22×C4⋊Q8C2211yescharacteristic
7C22×#125C2211yescharacteristic
8#2920C2212yes—
9#2920C2212yes—
10#2660—24no—
11#2660—24no—
12C2×#398—22no—
13C2×#395—22no—
14C2×#384C412yes—
15C2×#382C412yes—
16C2×#376C2211yescharacteristic
17#1556—24no—
18#1549—24no—
19#1546—24no—
20#1470—24no—
21#1470—24no—
22#1463—22no—
23#1463—22no—
24#1274—24no—
25C23×Q8—21no—
26C2×C4⋊Q8—22no—
27C2×C4⋊Q8—22no—
28C2×C4⋊Q8—22no—
29C2×C4⋊Q8—22no—
30C2×C4⋊Q8—22no—
31C2×C4⋊Q8D414yes—
32C2×#125—22no—
33C2×#125—22no—
34C2×#125—22no—
35C2×#125—22no—
36C2×#125—22no—
37C2×#125—22no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4—24no—
40C22×C4⋊C4C2×C411yescharacteristic
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C4⋊C4C2×C412yes—
43C22×C4⋊C4D412yes—
44C22×C4⋊C4D412yes—
45C22×C42C2×C412yes—
46C22×C42C2311yescharacteristic
47#432—22no—
48#432—22no—
49#432—22no—
50#432—22no—
51#431—22no—
52#431—22no—
53#431D414yes—
54#431D414yes—
55#423—22no—
56#423—22no—
57#407—22no—
58#407—22no—
59C4×M16—22no—
60C4×M16—22no—
61#398—216no—
62#395—216no—
63#384—24no—
64#384—24no—
65#382—24no—
66#382—24no—
67#376—22no—
68#376D414yes—
69C2×#95—24no—
70C2×#95—21no—
71C2×#95—21no—
72#332—216no—
73#324—28no—
74#324—28no—
75C22×Q8—24no—
76C22×Q8—41no—
77C22×Q8—42no—
78C22×Q8—41no—
79C22×Q8—84no—
80C23×C4—22no—
81C23×C4—22no—
82C23×C4C22⋊C412yes—
83C23×C4C22⋊C412yes—
84C23×C4C22×C411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C23×C4C2×D411yescharacteristic
87C2×M16—42no—
88C2×M16—42no—
89C4⋊Q8—24no—
90C4⋊Q8—24no—
91C4⋊Q8—24no—
92C4⋊Q8—24no—
93C4⋊Q8—42no—
94C4⋊Q8—42no—
95C4⋊Q8—42no—
96C4⋊Q8—42no—
97#125—42no—
98#125—42no—
99#125—42no—
100#125—42no—
101#125—42no—
102#125—42no—
103#125—42no—
104#125—42no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—216no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—88no—
130C2×C4⋊C4C22⋊C414yes—
131C2×C42—22no—
132C2×C42—28no—
133C2×C42C22⋊C414yes—
134C2×C42C2×D412yes—
135C2×C42C2×D412yes—
136C2×C42C2×D412yes—
137C2×C42C2×D412yes—
138C2×C42#4112yes—
139C2×C42#4112yes—
140C4⋊C8—24no—
141C4⋊C8—24no—
142C4⋊C8—42no—
143C4⋊C8—42no—
144C22⋊C8—24no—
145C22⋊C8—24no—
146C22⋊C8—42no—
147C22⋊C8—42no—
148#97—42no—
149#97—42no—
150C4×C8—24no—
151C4×C8—24no—
152#95—216no—
153#95—24no—
154#95—24no—
155C24C2×C22⋊C411yescharacteristic
156C2×Q8—44no—
157C2×Q8—44no—
158C2×Q8—88no—
159C2×Q8—88no—
160C2×Q8—84no—
161C2×Q8—84no—
162C22×C4—24no—
163C22×C4—28no—
164C22×C4—24no—
165C22×C4—22no—
166C22×C4—24no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—44no—
175C22×C4—44no—
176C22×C4—44no—
177C22×C4—44no—
178C22×C4—48no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4C2×C22⋊C411yescharacteristic
182C22×C4C2×C22⋊C411yescharacteristic
183C22×C4C4×D412yes—
184C22×C4C4×D412yes—
185C22×C4#12011yescharacteristic
186C22×C4#12011yescharacteristic
187C22×C4#12012yes—
188C22×C4#12012yes—
189C22×C4#12112yes—
190C22×C4#12112yes—
191C22×C4#12311yescharacteristic
192C22×C4#12311yescharacteristic
193M16—84no—
194M16—84no—
195C2×C8—44no—
196C2×C8—44no—
197C4⋊C4—28no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—44no—
203C4⋊C4—44no—
204C4⋊C4—44no—
205C4⋊C4—44no—
206C4⋊C4—44no—
207C4⋊C4—44no—
208C4⋊C4—44no—
209C4⋊C4—816no—
210C42—28no—
211C42—24no—
212C42#10118yes—
213C42#12012yes—
214C42#12112yes—
215C42#12112yes—
216C42#12112yes—
217C42#12112yes—
218C42#12312yes—
219C42#12412yes—
220C42#12712yes—
221C23—22no—
222C23—22no—
223C23—22no—
224C23#37911yescharacteristic
225C23#38611yescharacteristic
226C23#38611yescharacteristic
227Q8—88no—
228Q8—88no—
229C2×C4—24no—
230C2×C4—24no—
231C2×C4—22no—
232C2×C4—24no—
233C2×C4—24no—
234C2×C4—28no—
235C2×C4—41no—
236C2×C4—42no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—44no—
242C2×C4—416no—
243C2×C4—44no—
244C2×C4—44no—
245C2×C4—48no—
246C2×C4—44no—
247C2×C4—44no—
248C2×C4—44no—
249C2×C4—44no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—84no—
253C2×C4—84no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4#38612yes—
257C2×C4#39012yes—
258C2×C4#39212yes—
259C2×C4#39212yes—
260C2×C4#39412yes—
261C2×C4#39412yes—
262C2×C4C2×#10114yes—
263C8—84no—
264C8—84no—
265C22—28no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22#143811yescharacteristic
273C22#178412yes—
274C22#190912yes—
275C22#191012yes—
276C4—22no—
277C4—22no—
278C4—22no—
279C4—22no—
280C4—22no—
281C4—48no—
282C4—82no—
283C4—84no—
284C4—82no—
285C4#196614yes—
286C2—22no—
287C2—22no—
288C2—22no—
289C2#1098311yescharacteristic
290C2#1098711yescharacteristic
291C2#1098811yescharacteristic
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×#26602
#177422
#173942
#78552