Finite Group Databasebeta

← #96920

Subgroups — #96920

Order 512.

912 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 912 conjugacy classes of subgroups — too many to draw. Only its 194 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×C4×#125C211yesmaximal, characteristic
2#10795C212yesmaximal
3C2×#1271C214yesmaximal
4C22×#125C411yescharacteristic
5C4×#125—28no—
6C2×#389C2212yes—
7C2×#387—22no—
8C2×#382C411yescharacteristic
9C2×C4×C4⋊C4—22no—
10C2×C4×C4⋊C4C412yes—
11C2×C43C2211yescharacteristic
12C2×#336—22no—
13C2×#336—22no—
14C2×#336C2214yes—
15#1271—28no—
16C23×C8—24no—
17C2×#125—24no—
18C2×#125—24no—
19C2×#125—22no—
20C2×#125C814yescomplemented
21C22×C4⋊C4—22no—
22C22×C4⋊C4—22no—
23C22×C4⋊C4C2×C412yes—
24C22×C4⋊C4C2×C412yes—
25C22×C42—22no—
26C22×C42C2×C411yescharacteristic
27C22×C42C2×C411yescharacteristic
28C22×C42D412yes—
29C22×C42D412yes—
30C22×C42C2311yescharacteristic
31#389—24no—
32#389D418yes—
33#387—216no—
34#382—24no—
35C4×C4⋊C4—216no—
36C4×C4⋊C4—24no—
37C4×C4⋊C4—24no—
38C2×#95—22no—
39C2×#95—22no—
40C2×#95—22no—
41C43—24no—
42#336—28no—
43#336—28no—
44#336—44no—
45C23×C4—22no—
46C23×C4—22no—
47C23×C4C22⋊C412yes—
48C23×C4C22⋊C411yescharacteristic
49C23×C4C22⋊C411yescharacteristic
50C23×C4C22×C411yescharacteristic
51C23×C4C2×D412yes—
52C22×C8—28no—
53C22×C8—44no—
54C22×C8—44no—
55C22×C8—44no—
56C22×C8—44no—
57C22×C8—44no—
58C22×C8—44no—
59#125—216no—
60#125—216no—
61C2×C4⋊C4—28no—
62C2×C4⋊C4—28no—
63C2×C4⋊C4—28no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—28no—
69C2×C4⋊C4—42no—
70C2×C4⋊C4—42no—
71C2×C4⋊C4—44no—
72C2×C4⋊C4—42no—
73C2×C4⋊C4—42no—
74C2×C4⋊C4C22⋊C414yes—
75C2×C4⋊C4C22⋊C414yes—
76C2×C4⋊C4C2×C814yes—
77C2×C4⋊C4M1614yes—
78C2×C42—22no—
79C2×C42—22no—
80C2×C42—22no—
81C2×C42—22no—
82C2×C42—22no—
83C2×C42—28no—
84C2×C42—24no—
85C2×C42—24no—
86C2×C42—24no—
87C2×C42—24no—
88C2×C42—22no—
89C2×C42—22no—
90C2×C42—42no—
91C2×C42—41no—
92C2×C42—41no—
93C2×C42—44no—
94C2×C42—44no—
95C2×C42C2×C812yes—
96C2×C42M1612yes—
97C2×C42C2×D414yes—
98#95—28no—
99#95—28no—
100#95—28no—
101C24C2×C22⋊C411yescharacteristic
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—21no—
114C22×C4—22no—
115C22×C4—24no—
116C22×C4—24no—
117C22×C4—22no—
118C22×C4—24no—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—24no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4—42no—
131C22×C4—42no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4C22⋊C814yes—
138C22×C4#10112yes—
139C22×C4#10112yes—
140C22×C4#10414yes—
141C22×C4C2×C22⋊C412yes—
142C22×C4#12014yes—
143C22×C4C22×C811yescharacteristic
144C22×C4C2×M1612yes—
145C22×C4C2×M1611yescharacteristic
146C22×C4#13514yes—
147C22×C4AGL(1,ℤ/8)12yes—
148C22×C4#13712yes—
149C2×C8—48no—
150C2×C8—48no—
151C2×C8—48no—
152C2×C8—84no—
153C2×C8—84no—
154C2×C8—84no—
155C2×C8—84no—
156C4⋊C4—232no—
157C4⋊C4—24no—
158C4⋊C4—24no—
159C4⋊C4—24no—
160C4⋊C4—44no—
161C4⋊C4—44no—
162C4⋊C4—48no—
163C4⋊C4—48no—
164C4⋊C4C22⋊C818yes—
165C42—28no—
166C42—24no—
167C42—24no—
168C42—44no—
169C42—48no—
170C42—416no—
171C42—44no—
172C42—44no—
173C23#37911yescharacteristic
174C23C2×C22⋊C811yescharacteristic
175C23#40711yescharacteristic
176C23#41011yescharacteristic
177C23#41011yescharacteristic
178C23C2×#10111yescharacteristic
179C23C2×#10111yescharacteristic
180C23#41612yes—
181C23#41712yes—
182C23C2×#10412yes—
183C23#42112yes—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—24no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—24no—
196C2×C4—22no—
197C2×C4—24no—
198C2×C4—24no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—22no—
204C2×C4—24no—
205C2×C4—24no—
206C2×C4—24no—
207C2×C4—24no—
208C2×C4—28no—
209C2×C4—22no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—44no—
213C2×C4—44no—
214C2×C4—42no—
215C2×C4—44no—
216C2×C4—42no—
217C2×C4—44no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—44no—
221C2×C4—42no—
222C2×C4—44no—
223C2×C4—44no—
224C2×C4—44no—
225C2×C4—44no—
226C2×C4—82no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4—82no—
230C2×C4#33014yes—
231C2×C4C2×C22⋊C812yes—
232C2×C4#40712yes—
233C2×C4#44914yes—
234C2×C4#45214yes—
235C8—88no—
236C22#146111yescharacteristic
237C22#146111yescharacteristic
238C22#146711yescharacteristic
239C22#146711yescharacteristic
240C22#147612yes—
241C22#147712yes—
242C22#148812yes—
243C22#148912yes—
244C22#150012yes—
245C22#150912yes—
246C22#151212yes—
247C22#152112yes—
248C22C2×#33011yescharacteristic
249C22#152812yes—
250C22#153411yescharacteristic
251C22#154411yescharacteristic
252C22#155212yes—
253C22#156111yescharacteristic
254C22#177911yescharacteristic
255C22#178211yescharacteristic
256C22#178411yescharacteristic
257C22#178812yes—
258C22#178912yes—
259C4—22no—
260C4—22no—
261C4—24no—
262C4—24no—
263C4—24no—
264C4—44no—
265C4—48no—
266C4—42no—
267C4—84no—
268C2#756811yescharacteristic
269C2#758512yes—
270C2#760211yescharacteristic
271C2#761912yes—
272C2#763611yescharacteristic
273C2#765011yescharacteristic
274C2#830311yescharacteristic
275C2#831212yes—
276C2#833012yes—
277C2#833511yescharacteristic
278C2#834511yescharacteristic
subgroupG/H
center, socle, prime radicalC24C2×C22⋊C4
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
20. C2×#125C8semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1252
#107952
C2×#12712