Finite Group Databasebeta

← C2×#8306 #184756

Subgroups — C2×#8306 #184756

Order 512.

2618 conjugacy classes of subgroups; only the 418 normal subgroups are shown.

normal subgroups only

This group has 2618 conjugacy classes of subgroups — too many to draw. Only its 418 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
1C23×#125C211yesmaximal, characteristic
2C2×#1817C212yesmaximal
3C22×#324C214yesmaximal
4#8306C218yesmaximal, direct product factor
5C22×#125—24no—
6C22×#125—24no—
7C22×#125—216no—
8C22×#125C414yes—
9C22×#125C2214yes—
10C23×C4⋊C4C411yescharacteristic
11C23×C4⋊C4C412yes—
12C23×C4⋊C4C411yescharacteristic
13C23×C4⋊C4C2212yes—
14C23×C42C2211yescharacteristic
15C22×C22⋊C8C2214yes—
16#1817—24no—
17#1817C2218yes—
18C2×#336—24no—
19C2×#324—216no—
20C2×#324C22116yes—
21C24×C4C2×C411yescharacteristic
22C24×C4C2×C411yescharacteristic
23C24×C4C2×C412yes—
24C24×C4C2×C412yes—
25C24×C4C2311yescharacteristic
26C23×C8—24no—
27C2×#125—264no—
28C2×#125—264no—
29C2×#125—232no—
30C2×#125—232no—
31C2×#125—216no—
32C2×#125—216no—
33C22×C4⋊C4—28no—
34C22×C4⋊C4—28no—
35C22×C4⋊C4—22no—
36C22×C4⋊C4—22no—
37C22×C4⋊C4—24no—
38C22×C4⋊C4—28no—
39C22×C4⋊C4—28no—
40C22×C4⋊C4—24no—
41C22×C4⋊C4—22no—
42C22×C4⋊C4—22no—
43C22×C4⋊C4—24no—
44C22×C4⋊C4—28no—
45C22×C4⋊C4—24no—
46C22×C4⋊C4C2×C414yes—
47C22×C4⋊C4C2×C418yes—
48C22×C4⋊C4C2×C414yes—
49C22×C4⋊C4C2×C414yes—
50C22×C4⋊C4D4116yes—
51C22×C4⋊C4C2314yes—
52C22×C42—28no—
53C22×C42C2×C412yes—
54C22×C42D414yes—
55C22×C42D414yes—
56C22×C42C2312yes—
57C2×C22⋊C8—216no—
58C2×C22⋊C8—24no—
59C2×C22⋊C8C2318yes—
60#336—216no—
61#324—232no—
62C25C22×C411yescharacteristic
63C23×C4—24no—
64C23×C4—28no—
65C23×C4—24no—
66C23×C4—22no—
67C23×C4—24no—
68C23×C4—22no—
69C23×C4—24no—
70C23×C4—22no—
71C23×C4—22no—
72C23×C4—28no—
73C23×C4—216no—
74C23×C4—44no—
75C23×C4—48no—
76C23×C4C22⋊C418yes—
77C23×C4C22⋊C412yes—
78C23×C4C22⋊C412yes—
79C23×C4C22⋊C412yes—
80C23×C4C22⋊C412yes—
81C23×C4C22⋊C418yes—
82C23×C4C22×C414yes—
83C23×C4C22×C412yes—
84C23×C4C22×C412yes—
85C23×C4C22×C414yes—
86C23×C4C22×C411yescharacteristic
87C23×C4C2×D412yes—
88C23×C4C2×D412yes—
89C23×C4C2×D418yes—
90C23×C4C2411yescharacteristic
91C22×C8—28no—
92C22×C8—416no—
93C22×C8—48no—
94#125—2128no—
95#125—2128no—
96C2×C4⋊C4—216no—
97C2×C4⋊C4—216no—
98C2×C4⋊C4—216no—
99C2×C4⋊C4—216no—
100C2×C4⋊C4—232no—
101C2×C4⋊C4—232no—
102C2×C4⋊C4—216no—
103C2×C4⋊C4—216no—
104C2×C4⋊C4—216no—
105C2×C4⋊C4—216no—
106C2×C4⋊C4—216no—
107C2×C4⋊C4—28no—
108C2×C4⋊C4—28no—
109C2×C4⋊C4—28no—
110C2×C4⋊C4—28no—
111C2×C4⋊C4—232no—
112C2×C4⋊C4—232no—
113C2×C4⋊C4—28no—
114C2×C4⋊C4—28no—
115C2×C4⋊C4C22⋊C4116yes—
116C2×C4⋊C4C2×D4116yes—
117C2×C42—28no—
118C2×C42—216no—
119C2×C42—216no—
120C2×C42—28no—
121C2×C42C22⋊C418yes—
122C2×C42C2×D418yes—
123C22⋊C8—216no—
124C22⋊C8—48no—
125C24—24no—
126C24—24no—
127C24C2×C22⋊C411yescharacteristic
128C24C2×C22⋊C414yes—
129C24C2×C22⋊C411yescharacteristic
130C24#13118yes—
131C24C23×C411yescharacteristic
132C22×C4—232no—
133C22×C4—24no—
134C22×C4—24no—
135C22×C4—24no—
136C22×C4—28no—
137C22×C4—24no—
138C22×C4—28no—
139C22×C4—28no—
140C22×C4—24no—
141C22×C4—24no—
142C22×C4—24no—
143C22×C4—24no—
144C22×C4—24no—
145C22×C4—216no—
146C22×C4—28no—
147C22×C4—24no—
148C22×C4—24no—
149C22×C4—28no—
150C22×C4—28no—
151C22×C4—28no—
152C22×C4—28no—
153C22×C4—232no—
154C22×C4—28no—
155C22×C4—28no—
156C22×C4—42no—
157C22×C4—44no—
158C22×C4—416no—
159C22×C4—42no—
160C22×C4—48no—
161C22×C4—416no—
162C22×C4—416no—
163C22×C4—48no—
164C22×C4—416no—
165C22×C4—48no—
166C22×C4—48no—
167C22×C4—416no—
168C22×C4—48no—
169C22×C4#10118yes—
170C22×C4C2×C22⋊C414yes—
171C22×C4C2×C22⋊C418yes—
172C22×C4C2×C22⋊C412yes—
173C22×C4C2×C22⋊C414yes—
174C22×C4C2×C22⋊C418yes—
175C22×C4C2×C22⋊C414yes—
176C22×C4#12018yes—
177C22×C4#12018yes—
178C22×C4C22×D412yes—
179C22×C4C22×D414yes—
180C2×C8—416no—
181C2×C8—48no—
182C2×C8—816no—
183C4⋊C4—232no—
184C4⋊C4—232no—
185C4⋊C4—264no—
186C4⋊C4—232no—
187C4⋊C4—232no—
188C42—216no—
189C42—216no—
190C23—22no—
191C23—22no—
192C23—22no—
193C23—22no—
194C23—232no—
195C23—28no—
196C23—28no—
197C23#37912yes—
198C23#37912yes—
199C23#408116yes—
200C23#40814yes—
201C23#40814yes—
202C23#41014yes—
203C23C2×#10114yes—
204C23C22×C22⋊C412yes—
205C23C22×C22⋊C411yescharacteristic
206C23C2×#13114yes—
207C2×C4—216no—
208C2×C4—216no—
209C2×C4—24no—
210C2×C4—24no—
211C2×C4—24no—
212C2×C4—24no—
213C2×C4—216no—
214C2×C4—216no—
215C2×C4—216no—
216C2×C4—28no—
217C2×C4—24no—
218C2×C4—28no—
219C2×C4—28no—
220C2×C4—44no—
221C2×C4—48no—
222C2×C4—44no—
223C2×C4—48no—
224C2×C4—416no—
225C2×C4—48no—
226C2×C4—48no—
227C2×C4—48no—
228C2×C4—48no—
229C2×C4—48no—
230C2×C4—416no—
231C2×C4—48no—
232C2×C4—416no—
233C2×C4—416no—
234C2×C4—48no—
235C2×C4#37914yes—
236C2×C4C2×#10118yes—
237C2×C4C2×#12014yes—
238C8—88no—
239C22—22no—
240C22—22no—
241C22—22no—
242C22—22no—
243C22—232no—
244C22—28no—
245C22—28no—
246C22#1470116yes—
247C22#178014yes—
248C22#178014yes—
249C22#178214yes—
250C22#178414yes—
251C22C2×#37911yescharacteristic
252C22C2×#40814yes—
253C22C2×#40812yes—
254C22C2×#41012yes—
255C22C22×#10112yes—
256C4—24no—
257C4—24no—
258C4—28no—
259C4—44no—
260C4—48no—
261C4—48no—
262C2—24no—
263C2—24no—
264C2#830618yesdirect product factor
265C2C2×#147014yes—
266C2C2×#178011yescharacteristic
267C2C2×#178211yescharacteristic
268C2C2×#178411yescharacteristic
subgroupG/H
center, socleC24C2×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8306C2direct (N × H)
264. C2#8306direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#1252
C2×#18172
C22×#3242
#83062