Finite Group Databasebeta

← #112279

Subgroups — #112279

Order 512.

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

normal subgroups only

This group has 898 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×#2660C211yesmaximal, characteristic
2#21830C211yesmaximal, characteristic
3#17742C212yesmaximal
4#17394C211yesmaximal, characteristic
5#8072C212yesmaximal
6C22×C4⋊Q8C2211yescharacteristic
7C22×#125C2211yescharacteristic
8#2920C2212yes—
9#2920C2212yes—
10#2660—24no—
11#2660—24no—
12C2×#398C414yes—
13C2×#395—22no—
14C2×#384—21no—
15C2×#382—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⋊C4—24no—
41C22×C4⋊C4C2×C412yes—
42C22×C4⋊C4C2×C412yes—
43C22×C4⋊C4D411yescharacteristic
44C22×C4⋊C4D411yescharacteristic
45C22×C4⋊C4D412yes—
46C22×C42—21no—
47C22×C42C2311yescharacteristic
48#432—22no—
49#432—22no—
50#432—22no—
51#432D414yes—
52#431—22no—
53#431—22no—
54#431—22no—
55#431—22no—
56#423—22no—
57#423D414yes—
58#407—22no—
59#407—22no—
60C4×M16—22no—
61C4×M16—22no—
62#398—28no—
63#398—28no—
64#395—216no—
65#384—28no—
66#382—28no—
67#376—22no—
68#376D414yes—
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—41no—
78C22×Q8—42no—
79C22×Q8—41no—
80C22×Q8—84no—
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—216no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—88no—
131C2×C4⋊C4C22⋊C414yes—
132C2×C42—28no—
133C2×C42—24no—
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—24no—
143C4⋊C8—42no—
144C22⋊C8—24no—
145C22⋊C8—42no—
146C22⋊C8—42no—
147C22⋊C8—42no—
148#97—24no—
149#97—42no—
150C4×C8—42no—
151C4×C8—42no—
152#95—216no—
153#95—22no—
154#95—24no—
155#95C22⋊C414yes—
156C24C2×C22⋊C411yescharacteristic
157C2×Q8—44no—
158C2×Q8—44no—
159C2×Q8—88no—
160C2×Q8—84no—
161C2×Q8—88no—
162C2×Q8—84no—
163C22×C4—24no—
164C22×C4—28no—
165C22×C4—24no—
166C22×C4—22no—
167C22×C4—24no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—44no—
176C22×C4—44no—
177C22×C4—44no—
178C22×C4—48no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—44no—
182C22×C4C2×C22⋊C411yescharacteristic
183C22×C4C2×C22⋊C411yescharacteristic
184C22×C4C4×D412yes—
185C22×C4C4×D412yes—
186C22×C4#12012yes—
187C22×C4#12012yes—
188C22×C4#12011yescharacteristic
189C22×C4#12011yescharacteristic
190C22×C4#12112yes—
191C22×C4#12112yes—
192C22×C4#12311yescharacteristic
193C22×C4#12311yescharacteristic
194M16—48no—
195M16—84no—
196C2×C8—44no—
197C2×C8—44no—
198C4⋊C4—28no—
199C4⋊C4—44no—
200C4⋊C4—44no—
201C4⋊C4—44no—
202C4⋊C4—44no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—44no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—44no—
209C4⋊C4—44no—
210C4⋊C4—816no—
211C42—216no—
212C42#12012yes—
213C42#12112yes—
214C42#12112yes—
215C42#12112yes—
216C42#12112yes—
217C42#12312yes—
218C42#12412yes—
219C42#12712yes—
220C23—22no—
221C23—22no—
222C23—22no—
223C23#37911yescharacteristic
224C23#38611yescharacteristic
225C23#38611yescharacteristic
226Q8—88no—
227Q8—88no—
228C2×C4—24no—
229C2×C4—22no—
230C2×C4—24no—
231C2×C4—24no—
232C2×C4—24no—
233C2×C4—28no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—42no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—41no—
240C2×C4—416no—
241C2×C4—44no—
242C2×C4—44no—
243C2×C4—44no—
244C2×C4—44no—
245C2×C4—48no—
246C2×C4—44no—
247C2×C4—44no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—84no—
251C2×C4—84no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4#38612yes—
256C2×C4#39012yes—
257C2×C4#39212yes—
258C2×C4#39212yes—
259C2×C4#39412yes—
260C2×C4#39412yes—
261C2×C4#45614yes—
262C8—84no—
263C8—84no—
264C22—28no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22#143811yescharacteristic
272C22#179512yes—
273C22#188612yes—
274C22#191512yes—
275C4—22no—
276C4—22no—
277C4—22no—
278C4—22no—
279C4—22no—
280C4—48no—
281C4—82no—
282C4—82no—
283C4—84no—
284C4#200514yes—
285C2—22no—
286C2—22no—
287C2—22no—
288C2#1097411yescharacteristic
289C2#1097811yescharacteristic
290C2#1098411yescharacteristic
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
#218302
#177422
#173942
#80722