Finite Group Databasebeta

← #97017

Subgroups — #97017

Order 512.

1028 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 1028 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×#2316C211yesmaximal, characteristic
2#8300C212yesmaximal
3#7593C214yesmaximal, complemented
4C22×#117C411yescharacteristic
5C22×#117C2211yescharacteristic
6C23×C4⋊C4C411yescharacteristic
7#2316—28no—
8C2×#389—22no—
9C2×#389C412yes—
10C2×#389C2212yes—
11C2×#380—22no—
12#1817C2214yes—
13C2×#324—22no—
14C2×#323—22no—
15#1271—28no—
16C24×C4—21no—
17C24×C4C2×C411yescharacteristic
18C2×#117—24no—
19C2×#117—24no—
20C2×#117—42no—
21C2×#117—42no—
22C2×#117—42no—
23C2×#117—42no—
24C22×C4⋊C4—21no—
25C22×C4⋊C4—22no—
26C22×C4⋊C4—24no—
27C22×C4⋊C4—24no—
28C22×C4⋊C4—22no—
29C22×C4⋊C4—22no—
30C22×C4⋊C4C814yescomplemented
31C22×C4⋊C4C2×C412yes—
32C22×C4⋊C4C2×C412yes—
33C22×C4⋊C4D412yes—
34C22×C22⋊C4—21no—
35C22×C22⋊C4D412yes—
36C22×C42—21no—
37C22×C42C2×C411yescharacteristic
38C22×C42C2311yescharacteristic
39C2×C22⋊C8—24no—
40#389—216no—
41#389—24no—
42#389—24no—
43#389—24no—
44#389D418yes—
45#380—216no—
46C2×#95—22no—
47C2×#95—22no—
48#336—28no—
49#324—44no—
50#323—44no—
51C25C22⋊C411yescharacteristic
52C23×C4—22no—
53C23×C4—22no—
54C23×C4—22no—
55C23×C4—22no—
56C23×C4—22no—
57C23×C4—22no—
58C23×C4—22no—
59C23×C4—24no—
60C23×C4—24no—
61C23×C4—24no—
62C23×C4—22no—
63C23×C4—84no—
64C23×C4C22⋊C411yescharacteristic
65C23×C4C22⋊C412yes—
66C23×C4C2×C812yes—
67C23×C4M1612yes—
68C23×C4C22×C411yescharacteristic
69C23×C4C2×D412yes—
70C22×C8—44no—
71#117—416no—
72#117—416no—
73C2×C4⋊C4—24no—
74C2×C4⋊C4—216no—
75C2×C4⋊C4—24no—
76C2×C4⋊C4—24no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—216no—
81C2×C4⋊C4—24no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—24no—
84C2×C4⋊C4—28no—
85C2×C4⋊C4—28no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—84no—
90C2×C4⋊C4—84no—
91C2×C4⋊C4C22⋊C414yes—
92C2×C4⋊C4C22⋊C414yes—
93C2×C4⋊C4C2×C814yes—
94C2×C4⋊C4M1614yes—
95C2×C22⋊C4—24no—
96C2×C22⋊C4—24no—
97C2×C22⋊C4—24no—
98C2×C22⋊C4—42no—
99C2×C22⋊C4—42no—
100C2×C22⋊C4—42no—
101C2×C42—24no—
102C2×C42—22no—
103C2×C42—41no—
104C2×C42—41no—
105C2×C42—41no—
106C2×C42—41no—
107C2×C42—42no—
108C2×C42—42no—
109C2×C42C2×D414yes—
110C22⋊C8—44no—
111C22⋊C8—44no—
112#95—28no—
113#95—28no—
114C24—21no—
115C24—21no—
116C24—21no—
117C24—21no—
118C24—42no—
119C24—42no—
120C24C22⋊C812yes—
121C24#9912yes—
122C24#10112yes—
123C24C2×C22⋊C411yescharacteristic
124C22×C4—28no—
125C22×C4—28no—
126C22×C4—28no—
127C22×C4—28no—
128C22×C4—24no—
129C22×C4—24no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—24no—
139C22×C4—24no—
140C22×C4—24no—
141C22×C4—24no—
142C22×C4—24no—
143C22×C4—24no—
144C22×C4—24no—
145C22×C4—216no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—24no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—44no—
155C22×C4—44no—
156C22×C4—44no—
157C22×C4—44no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—88no—
163C22×C4—88no—
164C22×C4—88no—
165C22×C4C22⋊C812yes—
166C22×C4#9912yes—
167C22×C4#10112yes—
168C22×C4C2×C22⋊C412yes—
169C22×C4#12014yes—
170C22×C4C22×C811yescharacteristic
171C22×C4C2×M1612yes—
172C22×C4C2×M1611yescharacteristic
173C2×C8—84no—
174C2×C8—84no—
175C2×C8—84no—
176C4⋊C4—216no—
177C4⋊C4—24no—
178C4⋊C4—24no—
179C4⋊C4—24no—
180C4⋊C4—48no—
181C4⋊C4—44no—
182C4⋊C4—44no—
183C4⋊C4—88no—
184C4⋊C4—88no—
185C4⋊C4C22⋊C818yes—
186C22⋊C4—48no—
187C22⋊C4—48no—
188C42—44no—
189C42—48no—
190C42—44no—
191C23—22no—
192C23—22no—
193C23—22no—
194C23—22no—
195C23—22no—
196C23—22no—
197C23—22no—
198C23—22no—
199C23—42no—
200C23—42no—
201C23—44no—
202C23—44no—
203C23—42no—
204C23—44no—
205C23—44no—
206C23—42no—
207C23—42no—
208C23—42no—
209C23#32414yes—
210C23#37911yescharacteristic
211C23C2×C22⋊C811yescharacteristic
212C23#40711yescharacteristic
213C23C2×#9911yescharacteristic
214C23C2×#9911yescharacteristic
215C23C2×#10111yescharacteristic
216C23C2×#10111yescharacteristic
217C2×C4—216no—
218C2×C4—216no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—24no—
234C2×C4—24no—
235C2×C4—28no—
236C2×C4—24no—
237C2×C4—48no—
238C2×C4—48no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—42no—
242C2×C4—44no—
243C2×C4—42no—
244C2×C4—44no—
245C2×C4—42no—
246C2×C4—44no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4—816no—
250C2×C4—84no—
251C2×C4—84no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4#32414yes—
255C2×C4C2×C22⋊C812yes—
256C2×C4#40712yes—
257C8—164no—
258C22—22no—
259C22—22no—
260C22—22no—
261C22—22no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22—21no—
266C22—24no—
267C22—44no—
268C22—44no—
269C22—44no—
270C22—44no—
271C22—44no—
272C22—44no—
273C22—42no—
274C22—42no—
275C22C2×#32411yescharacteristic
276C22C2×#32411yescharacteristic
277C22#146311yescharacteristic
278C22#146311yescharacteristic
279C22#177911yescharacteristic
280C22#178111yescharacteristic
281C22#178411yescharacteristic
282C22#179512yes—
283C22#179612yes—
284C4—24no—
285C4—22no—
286C4—22no—
287C4—44no—
288C4—48no—
289C4—82no—
290C4—84no—
291C2—21no—
292C2—21no—
293C2—21no—
294C2—21no—
295C2—44no—
296C2#829911yescharacteristic
297C2#832211yescharacteristic
298C2#832311yescharacteristic
299C2#851712yes—
300C2#852512yes—
subgroupG/H
center, socleC23#379
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7593C2semidirect
30. C22×C4⋊C4C8semidirect

Maximal subgroups

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

subgroupindex
C2×#23162
#83002
#75932