Finite Group Databasebeta

← #106152

Subgroups — #106152

Order 512.

1210 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1210 conjugacy classes of subgroups — too many to draw. Only its 210 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×#2296C211yesmaximal, characteristic
2C2×#1464C212yesmaximal
3C2×#1285C214yesmaximal, complemented
4C22×#117C411yescharacteristic
5C22×#117C2211yescharacteristic
6C23×C42C411yescharacteristic
7C2×#432C2214yes—
8#2296—28no—
9C2×#389—22no—
10C2×#385—22no—
11C2×C4×C4⋊C4C412yes—
12C2×C4×C4⋊C4C2212yes—
13#1464—24no—
14#1464C418yescomplemented
15C2×#324—22no—
16C2×#323—22no—
17#1285—28no—
18C24×C4—21no—
19C24×C4C2×C411yescharacteristic
20C2×#117—24no—
21C2×#117—24no—
22C2×#117—24no—
23C2×#117—42no—
24C2×#117—42no—
25C2×#117C2×C414yes—
26C2×#117D412yes—
27C2×#117Q812yes—
28C22×C4⋊C4—21no—
29C22×C4⋊C4—22no—
30C22×C4⋊C4—22no—
31C22×C4⋊C4D412yes—
32C22×C22⋊C4—21no—
33C22×C22⋊C4D412yes—
34C22×C42—21no—
35C22×C42—24no—
36C22×C42—42no—
37C22×C42—42no—
38C22×C42C2×C411yescharacteristic
39C22×C42C2×C412yes—
40C22×C42C2×C412yes—
41C22×C42C2311yescharacteristic
42#432—24no—
43#432—24no—
44#432—24no—
45#432C2×C418yescomplemented
46C2×C4⋊C8—24no—
47C2×C22⋊C8—24no—
48C2×#97—24no—
49#389—216no—
50#385—216no—
51C4×C4⋊C4—24no—
52C4×C4⋊C4—24no—
53C4×C4⋊C4—24no—
54C4×C4⋊C4D418yes—
55C2×#95—22no—
56C2×#95—22no—
57#324—28no—
58#323—28no—
59C25C22⋊C411yescharacteristic
60C23×C4—22no—
61C23×C4—22no—
62C23×C4—22no—
63C23×C4—22no—
64C23×C4—22no—
65C23×C4—22no—
66C23×C4—22no—
67C23×C4—24no—
68C23×C4—22no—
69C23×C4—48no—
70C23×C4—42no—
71C23×C4—42no—
72C23×C4C4212yes—
73C23×C4C22⋊C411yescharacteristic
74C23×C4C22⋊C412yes—
75C23×C4C4⋊C412yes—
76C23×C4C22×C411yescharacteristic
77C23×C4C2×D412yes—
78C22×C8—44no—
79#117—28no—
80#117—28no—
81#117—28no—
82#117—28no—
83#117—416no—
84C2×C4⋊C4—24no—
85C2×C4⋊C4—24no—
86C2×C4⋊C4—24no—
87C2×C4⋊C4—28no—
88C2×C4⋊C4—28no—
89C2×C4⋊C4—48no—
90C2×C4⋊C4—48no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4C22⋊C414yes—
94C2×C4⋊C4C4⋊C414yes—
95C2×C22⋊C4—24no—
96C2×C22⋊C4—24no—
97C2×C22⋊C4—24no—
98C2×C22⋊C4—42no—
99C2×C22⋊C4—42no—
100C2×C22⋊C4C22⋊C414yes—
101C2×C22⋊C4C4⋊C414yes—
102C2×C42—22no—
103C2×C42—22no—
104C2×C42—24no—
105C2×C42—22no—
106C2×C42—22no—
107C2×C42—42no—
108C2×C42—42no—
109C2×C42—42no—
110C2×C42—42no—
111C2×C42—416no—
112C2×C42—42no—
113C2×C42—42no—
114C2×C42C4212yes—
115C2×C42C22⋊C414yes—
116C2×C42C22⋊C414yes—
117C2×C42C4⋊C412yes—
118C2×C42C22×C412yes—
119C2×C42C2×D414yes—
120C2×C42C2×D411yescharacteristic
121C2×C42C2×Q811yescharacteristic
122C2×C42#4114yes—
123C4⋊C8—28no—
124C4⋊C8—44no—
125C22⋊C8—28no—
126C22⋊C8—44no—
127#97—28no—
128#97—44no—
129#95—28no—
130#95—28no—
131C24—21no—
132C24—21no—
133C24—21no—
134C24—21no—
135C24—44no—
136C24#9512yes—
137C24#10012yes—
138C24#10112yes—
139C24C2×C22⋊C411yescharacteristic
140C22×C4—28no—
141C22×C4—28no—
142C22×C4—28no—
143C22×C4—28no—
144C22×C4—24no—
145C22×C4—24no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—24no—
155C22×C4—24no—
156C22×C4—24no—
157C22×C4—24no—
158C22×C4—28no—
159C22×C4—24no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—24no—
163C22×C4—44no—
164C22×C4—416no—
165C22×C4—48no—
166C22×C4—44no—
167C22×C4—416no—
168C22×C4—416no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—44no—
172C22×C4—44no—
173C22×C4—44no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4#9512yes—
177C22×C4#9514yes—
178C22×C4#10012yes—
179C22×C4#10112yes—
180C22×C4C2×C4211yescharacteristic
181C22×C4C2×C22⋊C412yes—
182C22×C4C2×C22⋊C412yes—
183C22×C4C2×C4⋊C411yescharacteristic
184C22×C4C2×C4⋊C412yes—
185C22×C4#11712yes—
186C22×C4#12014yes—
187C22×C4#12314yes—
188C2×C8—48no—
189C2×C8—88no—
190C4⋊C4—28no—
191C4⋊C4—28no—
192C4⋊C4—416no—
193C4⋊C4—416no—
194C4⋊C4—48no—
195C22⋊C4—28no—
196C22⋊C4—28no—
197C22⋊C4—48no—
198C42—24no—
199C42—24no—
200C42—44no—
201C42—44no—
202C42—44no—
203C42—48no—
204C42—44no—
205C42C4×D414yes—
206C42C4×D414yes—
207C42#12114yes—
208C42C22⋊Q814yes—
209C23—24no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—22no—
214C23—22no—
215C23—22no—
216C23—44no—
217C23—44no—
218C23—44no—
219C23—44no—
220C23—44no—
221C23—44no—
222C23—44no—
223C23#34314yes—
224C23C2×#9511yescharacteristic
225C23#37911yescharacteristic
226C23#38111yescharacteristic
227C23C2×#10011yescharacteristic
228C23C2×#10011yescharacteristic
229C23C2×#10111yescharacteristic
230C23C2×#10111yescharacteristic
231C2×C4—216no—
232C2×C4—216no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—28no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—28no—
249C2×C4—24no—
250C2×C4—44no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—44no—
254C2×C4—44no—
255C2×C4—48no—
256C2×C4—48no—
257C2×C4—432no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—44no—
261C2×C4—44no—
262C2×C4—44no—
263C2×C4—44no—
264C2×C4—84no—
265C2×C4—84no—
266C2×C4#34314yes—
267C2×C4C4×C22⋊C412yes—
268C2×C4#38012yes—
269C2×C4#38514yes—
270C2×C4#38614yes—
271C8—88no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—22no—
277C22—22no—
278C22—24no—
279C22—44no—
280C22—44no—
281C22—44no—
282C22—44no—
283C22—44no—
284C22—44no—
285C22—44no—
286C22C22⋊#9511yescharacteristic
287C22C2×#34311yescharacteristic
288C22C2×#34311yescharacteristic
289C22#178311yescharacteristic
290C22#178411yescharacteristic
291C22#182111yescharacteristic
292C22#182111yescharacteristic
293C22#182714yes—
294C4—24no—
295C4—24no—
296C4—24no—
297C4—22no—
298C4—22no—
299C4—48no—
300C4—44no—
301C4—84no—
302C2—21no—
303C2—21no—
304C2—21no—
305C2—21no—
306C2—44no—
307C2#874212yes—
308C2#874312yes—
309C2#1072711yescharacteristic
310C2#1075812yes—
subgroupG/H
center, socleC23#381
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1285C2semidirect
14. #1464C4semidirect
45. #432C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22962
C2×#14642
C2×#12852