Finite Group Databasebeta

← #192087

Subgroups — #192087

Order 512.

1056 conjugacy classes of subgroups; only the 266 normal subgroups are shown.

normal subgroups only

This group has 1056 conjugacy classes of subgroups — too many to draw. Only its 266 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#34187C211yesmaximal, characteristic
2#21365C212yesmaximal
3C2×#1467C211yesmaximal, characteristic
4#13850C211yesmaximal, characteristic
5#8855C218yesmaximal
6#8022C212yesmaximal
7C22×C4×Q8C2211yescharacteristic
8C2×#432C2212yes—
9#2356—24no—
10#2355—22no—
11#2355—22no—
12#2313C2214yes—
13#2310—21no—
14#2285C2212yes—
15#2283—22no—
16#2274—21no—
17C2×C4×C4⋊C4C2211yescharacteristic
18C2×#376C2211yescharacteristic
19#1948—22no—
20#1947—22no—
21#1853C2214yes—
22#1852C2214yes—
23#1776—24no—
24#1467C418yescomplemented
25C2×#324C2214yes—
26#1446—21no—
27#1445—21no—
28#1441C2214yes—
29C4×#95—22no—
30#1294C2218yes—
31C23×Q8D411yescharacteristic
32C2×C4×Q8—22no—
33C2×C4×Q8—22no—
34C2×C4×Q8—24no—
35C2×C4×Q8C2×C414yes—
36C2×C4×Q8Q814yes—
37C2×#117—22no—
38C22×C4⋊C4D411yescharacteristic
39C22×C4⋊C4C2312yes—
40C22×C4⋊C4C2312yes—
41C22×C42—21no—
42C22×C42—21no—
43C22×C42D412yes—
44C22×C42C2311yescharacteristic
45#432—24no—
46#432C2×C418yes—
47C2×#107—24no—
48C2×#106—24no—
49C2×C4⋊C8—22no—
50C2×C22⋊C8C2314yes—
51C2×#97—22no—
52#391—24no—
53#391—24no—
54#387—28no—
55#387—28no—
56#384D412yes—
57#384D412yes—
58#383—24no—
59#382—22no—
60#382—22no—
61#382—22no—
62#381—24no—
63#380—22no—
64#380C2314yes—
65C4×C4⋊C4—24no—
66C4×C4⋊C4—24no—
67C4×C4⋊C4—24no—
68C4×C4⋊C4—22no—
69C4×C4⋊C4—22no—
70C4×C4⋊C4—22no—
71C4×C4⋊C4D414yes—
72C4×C22⋊C4—24no—
73#376—24no—
74#376—24no—
75C2×#95—24no—
76C2×#95—24no—
77C2×#95—21no—
78C2×#95—21no—
79C2×#95C2312yes—
80#340—24no—
81#340—24no—
82#324C2×C4116yes—
83C22×Q8—22no—
84C22×Q8—22no—
85C22×Q8—44no—
86C22×Q8C4⋊C414yes—
87C23×C4—22no—
88C23×C4—22no—
89C23×C4C2×D412yes—
90C23×C4C2×D411yescharacteristic
91C23×C4C2×D411yescharacteristic
92C23×C4C2×D412yes—
93C23×C4C2411yescharacteristic
94C22×C8—24no—
95C4×Q8—28no—
96C4×Q8—28no—
97C4×Q8—28no—
98C4×Q8—28no—
99#117—48no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—44no—
111C2×C4⋊C4C4⋊C414yes—
112C2×C4⋊C4C22×C414yes—
113C2×C4⋊C4C2×D414yes—
114C2×C4⋊C4C2×D414yes—
115C2×C4⋊C4C2×Q814yes—
116C2×C22⋊C4—24no—
117C2×C42—22no—
118C2×C42—22no—
119C2×C42—24no—
120C2×C42—24no—
121C2×C42—24no—
122C2×C42—28no—
123C2×C42—42no—
124C2×C42—42no—
125C2×C42—44no—
126C2×C42—41no—
127C2×C42—41no—
128C2×C42C4⋊C418yes—
129C2×C42C22×C412yes—
130C2×C42C2×D412yes—
131C2×C42C2×D412yes—
132C2×C42C2×Q812yes—
133C2×C42#4114yes—
134#107—48no—
135#106—48no—
136C4⋊C8—28no—
137C22⋊C8—24no—
138C22⋊C8C22×C418yes—
139#97—28no—
140#95—24no—
141#95—24no—
142#95—24no—
143#95—24no—
144#95—28no—
145#95—24no—
146#95—24no—
147#95—24no—
148#95—48no—
149#95—48no—
150#95—42no—
151#95—42no—
152C24C22×D411yescharacteristic
153C2×Q8—28no—
154C2×Q8—48no—
155C2×Q8—48no—
156C2×Q8—44no—
157C2×Q8—44no—
158C22×C4—24no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—24no—
165C22×C4—24no—
166C22×C4—22no—
167C22×C4—24no—
168C22×C4—24no—
169C22×C4—24no—
170C22×C4—24no—
171C22×C4—42no—
172C22×C4—44no—
173C22×C4—44no—
174C22×C4—42no—
175C22×C4—44no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—44no—
180C22×C4—44no—
181C22×C4C2×C4⋊C414yes—
182C22×C4C2×C4⋊C412yes—
183C22×C4C2×C4⋊C412yes—
184C22×C4C2×C4⋊C414yes—
185C22×C4#12011yescharacteristic
186C22×C4#12012yes—
187C22×C4#12011yescharacteristic
188C22×C4#12312yes—
189C22×C4#12312yes—
190C22×C4C23×C411yescharacteristic
191C22×C4C22×D412yes—
192C22×C4C22×Q811yescharacteristic
193C22×C4C2×#4112yes—
194C2×C8—28no—
195C2×C8—44no—
196C2×C8—44no—
197C4⋊C4—28no—
198C4⋊C4—28no—
199C4⋊C4—28no—
200C4⋊C4—24no—
201C4⋊C4—24no—
202C4⋊C4—44no—
203C4⋊C4—44no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—44no—
207C4⋊C4—44no—
208C4⋊C4—44no—
209C4⋊C4—44no—
210C22⋊C4—48no—
211C42—28no—
212C42—28no—
213C42—48no—
214C42—48no—
215C42—82no—
216C42—88no—
217C42—82no—
218C42C4×D414yes—
219C42C4×D414yes—
220C42C22⋊Q814yes—
221C42C22⋊Q814yes—
222C23—21no—
223C23—21no—
224C23—22no—
225C23C22×C4⋊C411yescharacteristic
226C23C2×#12011yescharacteristic
227C23C2×#12311yescharacteristic
228C23#57514yes—
229Q8—416no—
230C2×C4—24no—
231C2×C4—24no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—24no—
239C2×C4—24no—
240C2×C4—24no—
241C2×C4—24no—
242C2×C4—24no—
243C2×C4—24no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—44no—
248C2×C4—44no—
249C2×C4—44no—
250C2×C4—44no—
251C2×C4—44no—
252C2×C4—48no—
253C2×C4—48no—
254C2×C4—44no—
255C2×C4—44no—
256C2×C4—44no—
257C2×C4—44no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—44no—
261C2×C4—82no—
262C2×C4—84no—
263C2×C4—82no—
264C2×C4—84no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4#38512yes—
269C2×C4#38514yes—
270C2×C4#38512yes—
271C2×C4#45412yes—
272C2×C4#45612yes—
273C2×C4C2×C4×D412yes—
274C2×C4C2×C22⋊Q812yes—
275C8—48no—
276C22—24no—
277C22—24no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22C2×#38511yescharacteristic
283C22#291112yes—
284C22#300512yes—
285C22C2×#45411yescharacteristic
286C22C2×#45611yescharacteristic
287C22#302512yes—
288C22#309312yes—
289C4—24no—
290C4—22no—
291C4—22no—
292C4—22no—
293C4—44no—
294C4—44no—
295C4—84no—
296C4—84no—
297C4—84no—
298C4—84no—
299C4#185714yes—
300C2—22no—
301C2—22no—
302C2#1818212yes—
303C2#1818312yes—
304C2#2084511yescharacteristic
305C2C2×#185711yescharacteristic
306C2#2088811yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
24. #1467C4semidirect

Maximal subgroups

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

subgroupindex
#341872
#213652
C2×#14672
#138502
#88552
#80222