Finite Group Databasebeta

← #115190

Subgroups — #115190

Order 512.

862 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 862 conjugacy classes of subgroups — too many to draw. Only its 134 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×#2634C211yesmaximal, characteristic
2#21830C211yesmaximal, characteristic
3#17449C212yesmaximal
4#17394C211yesmaximal, characteristic
5#7975C212yesmaximal
6C22×#125C2211yescharacteristic
7C22×C4×Q8C2211yescharacteristic
8#2920C2212yes—
9#2920C2212yes—
10#2634—24no—
11#2634—24no—
12C2×#400—21no—
13C2×#400C412yes—
14C2×#398—21no—
15C2×#395C412yes—
16C2×#387—22no—
17C2×#383C2211yescharacteristic
18#1998—24no—
19#1821—21no—
20#1821—21no—
21#1491—24no—
22#1487—24no—
23#1479—24no—
24#1470—24no—
25#1463—22no—
26#1288—24no—
27C23×Q8C2×C411yescharacteristic
28C2×#125—22no—
29C2×#125—22no—
30C2×#125—22no—
31C2×#125—22no—
32C2×#125—22no—
33C2×#125—22no—
34C2×C4×Q8—22no—
35C2×C4×Q8—22no—
36C2×C4×Q8—22no—
37C2×C4×Q8—22no—
38C2×C4×Q8—22no—
39C2×C4×Q8D414yes—
40C22×C4⋊C4—22no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4C2×C412yes—
43C22×C4⋊C4C2×C411yescharacteristic
44C22×C4⋊C4D411yescharacteristic
45C22×C4⋊C4D411yescharacteristic
46C22×C4⋊C4D412yes—
47C22×C42—21no—
48C22×C42C2311yescharacteristic
49#432—22no—
50#432—22no—
51#432—22no—
52#432—22no—
53#431—22no—
54#431—22no—
55#431—22no—
56#431D414yes—
57#423—22no—
58#423—22no—
59#407—22no—
60#407—22no—
61C4×M16—22no—
62C4×M16D414yes—
63#400—28no—
64#400—24no—
65#400—24no—
66#398—28no—
67#395—24no—
68#395—24no—
69#387—216no—
70#383—22no—
71#383D414yes—
72C2×#95—22no—
73C2×#95—22no—
74C2×#95—22no—
75C2×#95—21no—
76C2×#95—21no—
77C2×#95C2×C412yes—
78#343—24no—
79#343—24no—
80Q8⋊C8—216no—
81#324—28no—
82C22×Q8—21no—
83C22×Q8—22no—
84C22×Q8—22no—
85C22×Q8—44no—
86C22×Q8C22⋊C412yes—
87C23×C4—21no—
88C23×C4—21no—
89C23×C4—22no—
90C23×C4C22⋊C412yes—
91C23×C4C22⋊C412yes—
92C23×C4C22×C411yescharacteristic
93C23×C4C2×D411yescharacteristic
94C23×C4C2×D411yescharacteristic
95C2×M16—24no—
96C2×M16—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—
105C4×Q8—24no—
106C4×Q8—24no—
107C4×Q8—24no—
108C4×Q8—24no—
109C4×Q8—42no—
110C4×Q8—42no—
111C4×Q8—42no—
112C4×Q8—42no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—28no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—48no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4C22⋊C412yes—
139C2×C42—24no—
140C2×C42—41no—
141C2×C42—42no—
142C2×C42—41no—
143C2×C42C2×D412yes—
144C2×C42C2×D412yes—
145C2×C42C2×D412yes—
146C2×C42C2×D412yes—
147C2×C42#4112yes—
148C2×C42#4112yes—
149C4⋊C8—24no—
150C4⋊C8—42no—
151C4⋊C8—42no—
152C4⋊C8—42no—
153C22⋊C8—24no—
154C22⋊C8—42no—
155C22⋊C8—42no—
156C22⋊C8—42no—
157#97—24no—
158#97—42no—
159C4×C8—24no—
160C4×C8—24no—
161#95—28no—
162#95—28no—
163#95—28no—
164#95—22no—
165#95—24no—
166#95—24no—
167#95C22⋊C414yes—
168C24C2×C22⋊C411yescharacteristic
169C2×Q8—28no—
170C2×Q8—24no—
171C2×Q8—44no—
172C2×Q8—44no—
173C2×Q8—44no—
174C2×Q8—48no—
175C2×Q8—42no—
176C22×C4—24no—
177C22×C4—22no—
178C22×C4—24no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—44no—
193C22×C4—44no—
194C22×C4—44no—
195C22×C4—44no—
196C22×C4—44no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—44no—
200C22×C4C2×C22⋊C411yescharacteristic
201C22×C4C2×C22⋊C411yescharacteristic
202C22×C4C4×D412yes—
203C22×C4C4×D412yes—
204C22×C4#12011yescharacteristic
205C22×C4#12012yes—
206C22×C4#12011yescharacteristic
207C22×C4#12012yes—
208C22×C4#12112yes—
209C22×C4#12112yes—
210C22×C4#12311yescharacteristic
211C22×C4#12311yescharacteristic
212M16—48no—
213M16—84no—
214C2×C8—44no—
215C2×C8—44no—
216C4⋊C4—24no—
217C4⋊C4—416no—
218C4⋊C4—44no—
219C4⋊C4—42no—
220C4⋊C4—44no—
221C4⋊C4—44no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—44no—
225C4⋊C4—44no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—44no—
229C42—44no—
230C42—44no—
231C42#12012yes—
232C42#12112yes—
233C42#12112yes—
234C42#12112yes—
235C42#12112yes—
236C42#12312yes—
237C42#12412yes—
238C42#12712yes—
239C23—22no—
240C23—22no—
241C23—22no—
242C23#37911yescharacteristic
243C23#38611yescharacteristic
244C23#38611yescharacteristic
245Q8—28no—
246Q8—44no—
247Q8—48no—
248C2×C4—24no—
249C2×C4—24no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—24no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—44no—
263C2×C4—44no—
264C2×C4—42no—
265C2×C4—44no—
266C2×C4—44no—
267C2×C4—44no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—44no—
271C2×C4—44no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—88no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4#38612yes—
282C2×C4#39012yes—
283C2×C4#39212yes—
284C2×C4#39212yes—
285C2×C4#39412yes—
286C2×C4#39412yes—
287C8—84no—
288C8—84no—
289C22—28no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22#143811yescharacteristic
297C22#179612yes—
298C22#189112yes—
299C22#191212yes—
300C4—22no—
301C4—22no—
302C4—22no—
303C4—22no—
304C4—22no—
305C4—22no—
306C4—44no—
307C4—84no—
308C4—82no—
309C4—82no—
310C4—82no—
311C2—22no—
312C2—22no—
313C2—22no—
314C2#1098511yescharacteristic
315C2#1099011yescharacteristic
316C2#1099311yescharacteristic
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×#26342
#218302
#174492
#173942
#79752