Finite Group Databasebeta

← #96937

Subgroups — #96937

Order 512.

1106 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 1106 conjugacy classes of subgroups — too many to draw. Only its 170 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×#2851C211yesmaximal, characteristic
2#10712C212yesmaximal
3C2×#1271C214yesmaximal, complemented
4C22×#125C411yescharacteristic
5C23×C42C2211yescharacteristic
6C22×C22⋊C8—22no—
7#2851—28no—
8C2×#389C2212yes—
9C2×#387—22no—
10C2×#385—22no—
11C2×#382C411yescharacteristic
12C2×#380C412yes—
13C2×#336—22no—
14C2×#336C2214yes—
15#1271—28no—
16C24×C4C2×C411yescharacteristic
17C24×C4D412yes—
18C23×C8—24no—
19C2×#125—24no—
20C2×#125—44no—
21C22×C4⋊C4—22no—
22C22×C4⋊C4—22no—
23C22×C4⋊C4C2×C412yes—
24C22×C4⋊C4C2×C412yes—
25C22×C22⋊C4—22no—
26C22×C42—24no—
27C22×C42—44no—
28C22×C42C2×C411yescharacteristic
29C22×C42D412yes—
30C22×C42C2311yescharacteristic
31C2×C22⋊C8—28no—
32C2×C22⋊C8—44no—
33C2×C22⋊C8—44no—
34#389—24no—
35#389D418yes—
36#387—216no—
37#385—216no—
38#382—24no—
39#380—24no—
40#380—24no—
41C2×#95—22no—
42C2×#95—22no—
43C2×#95—22no—
44#336—28no—
45#336—44no—
46C25C22⋊C411yescharacteristic
47C23×C4—22no—
48C23×C4—24no—
49C23×C4—24no—
50C23×C4—22no—
51C23×C4—22no—
52C23×C4—22no—
53C23×C4—22no—
54C23×C4—22no—
55C23×C4—22no—
56C23×C4—24no—
57C23×C4—24no—
58C23×C4—24no—
59C23×C4—42no—
60C23×C4—42no—
61C23×C4—42no—
62C23×C4C22⋊C412yes—
63C23×C4C22⋊C411yescharacteristic
64C23×C4C22×C411yescharacteristic
65C23×C4C2×D412yes—
66C22×C8—28no—
67C22×C8—44no—
68C22×C8—44no—
69C22×C8—44no—
70C22×C8—44no—
71C22×C8—44no—
72C22×C8—44no—
73#125—416no—
74C2×C4⋊C4—24no—
75C2×C4⋊C4—24no—
76C2×C4⋊C4—28no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—28no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—44no—
83C2×C4⋊C4—44no—
84C2×C4⋊C4—44no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4C22⋊C414yes—
88C2×C4⋊C4C22⋊C414yes—
89C2×C22⋊C4—28no—
90C2×C22⋊C4—44no—
91C2×C22⋊C4—44no—
92C2×C42—24no—
93C2×C42—22no—
94C2×C42—24no—
95C2×C42—24no—
96C2×C42—416no—
97C2×C42—42no—
98C2×C42—42no—
99C2×C42C2×D414yes—
100C22⋊C8—416no—
101#95—28no—
102#95—28no—
103#95—28no—
104C24—21no—
105C24—22no—
106C24—22no—
107C24—21no—
108C24—22no—
109C24—22no—
110C24—22no—
111C24#10112yes—
112C24#10414yes—
113C24C2×C22⋊C411yescharacteristic
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—28no—
122C22×C4—28no—
123C22×C4—28no—
124C22×C4—21no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—21no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—24no—
135C22×C4—24no—
136C22×C4—48no—
137C22×C4—48no—
138C22×C4—48no—
139C22×C4—48no—
140C22×C4—48no—
141C22×C4—48no—
142C22×C4—44no—
143C22×C4—44no—
144C22×C4—44no—
145C22×C4—44no—
146C22×C4—44no—
147C22×C4—44no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4#9914yes—
161C22×C4#10112yes—
162C22×C4C2×C22⋊C412yes—
163C22×C4#12014yes—
164C22×C4#13114yes—
165C22×C4#13514yes—
166C22×C4AGL(1,ℤ/8)12yes—
167C22×C4#13712yes—
168C2×C8—48no—
169C2×C8—48no—
170C2×C8—48no—
171C2×C8—84no—
172C2×C8—84no—
173C2×C8—84no—
174C2×C8—84no—
175C4⋊C4—28no—
176C4⋊C4—44no—
177C4⋊C4—44no—
178C4⋊C4—416no—
179C4⋊C4—44no—
180C4⋊C4—48no—
181C4⋊C4—48no—
182C22⋊C4—416no—
183C42—28no—
184C42—44no—
185C42—48no—
186C23—24no—
187C23—24no—
188C23—24no—
189C23—24no—
190C23—24no—
191C23—24no—
192C23—24no—
193C23—22no—
194C23—24no—
195C23—22no—
196C23—41no—
197C23—41no—
198C23—42no—
199C23—42no—
200C23—42no—
201C23—42no—
202C23—42no—
203C23—41no—
204C23—41no—
205C23—42no—
206C23#33014yes—
207C23#37911yescharacteristic
208C23#40812yes—
209C23C2×#9912yes—
210C23C2×#10111yescharacteristic
211C23C2×#10111yescharacteristic
212C23#41612yes—
213C23#41712yes—
214C23C2×#10412yes—
215C23#42112yes—
216C2×C4—24no—
217C2×C4—24no—
218C2×C4—24no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—24no—
224C2×C4—22no—
225C2×C4—24no—
226C2×C4—22no—
227C2×C4—24no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—24no—
231C2×C4—24no—
232C2×C4—22no—
233C2×C4—416no—
234C2×C4—416no—
235C2×C4—416no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—44no—
244C2×C4—44no—
245C2×C4—44no—
246C2×C4—44no—
247C2×C4—44no—
248C2×C4—44no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4#40814yes—
258C2×C4#44914yes—
259C2×C4#45214yes—
260C8—88no—
261C22—28no—
262C22—24no—
263C22—44no—
264C22—44no—
265C22—42no—
266C22—44no—
267C22—44no—
268C22—42no—
269C22—44no—
270C22#147212yes—
271C22#147212yes—
272C22#148212yes—
273C22#148312yes—
274C22#149412yes—
275C22#149512yes—
276C22#150112yes—
277C22#150412yes—
278C22#151312yes—
279C22#151612yes—
280C22C2×#33011yescharacteristic
281C22#152812yes—
282C22#153411yescharacteristic
283C22#154411yescharacteristic
284C22#155212yes—
285C22#156111yescharacteristic
286C22#178011yescharacteristic
287C22#178111yescharacteristic
288C22#178411yescharacteristic
289C22#178812yes—
290C22#178912yes—
291C4—24no—
292C4—24no—
293C4—24no—
294C4—42no—
295C4—44no—
296C4—84no—
297C4—84no—
298C2—44no—
299C2#767112yes—
300C2#769112yes—
301C2#771112yes—
302C2#773112yes—
303C2#830811yescharacteristic
304C2#831512yes—
305C2#833112yes—
306C2#833511yescharacteristic
307C2#834511yescharacteristic
subgroupG/H
center, socleC24C2×C22⋊C4
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1271C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28512
#107122
C2×#12712