Finite Group Databasebeta

← #117141

Subgroups — #117141

Order 512.

700 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 178 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×#1960C211yesmaximal, characteristic
2C2×#1838C211yesmaximal, characteristic
3C2×#1546C212yesmaximal
4C2×#1286C212yesmaximal
5C2×#1281C211yesmaximal, characteristic
6C22×C4⋊Q8C2211yescharacteristic
7C2×#431C2212yes—
8C2×#423C2212yes—
9C22×#103—22no—
10C22×#97C2211yescharacteristic
11C2×#382C2211yescharacteristic
12#1960—22no—
13#1960C414yes—
14#1838—24no—
15C2×#341—22no—
16#1546—24no—
17#1546—24no—
18C2×#332—24no—
19#1286—24no—
20#1281—22no—
21C23×Q8—21no—
22C22×M16—22no—
23C23×C8D412yes—
24C2×C4⋊Q8—22no—
25C2×C4⋊Q8—22no—
26C2×C4⋊Q8—21no—
27C2×C4⋊Q8—22no—
28C2×C4⋊Q8—21no—
29C2×C4⋊Q8—21no—
30C2×C4⋊Q8C2×C412yes—
31C2×C4⋊Q8D412yes—
32C2×C4⋊Q8Q812yes—
33C22×C4⋊C4—22no—
34C22×C4⋊C4—21no—
35C22×C42C2311yescharacteristic
36#431—22no—
37#431—22no—
38#431—22no—
39#431D414yes—
40#423—22no—
41#423—22no—
42#423—22no—
43#423D414yes—
44C2×C4⋊C8—22no—
45C2×C4⋊C8—22no—
46C2×C4⋊C8—22no—
47C2×#103—28no—
48C2×#103—48no—
49C2×C22⋊C8—22no—
50C2×#97—22no—
51C2×#97—22no—
52C2×#97C2×C412yes—
53C2×#97D412yes—
54C2×C4×C8—22no—
55#382—22no—
56#382—21no—
57#382C2×C412yes—
58C2×#95—22no—
59#341—28no—
60#332—216no—
61C22×Q8—22no—
62C22×Q8—22no—
63C22×Q8—22no—
64C22×Q8—41no—
65C22×Q8—41no—
66C22×Q8—41no—
67C22×Q8—44no—
68C22×Q8—44no—
69C23×C4—41no—
70C23×C4—41no—
71C23×C4C2×D411yescharacteristic
72C23×C4#4112yes—
73C2×M16—24no—
74C2×M16—42no—
75C2×M16—44no—
76C2×M16—44no—
77C22×C8—22no—
78C22×C8—22no—
79C22×C8—24no—
80C22×C8—24no—
81C22×C8—42no—
82C22×C8—42no—
83C22×C8C22⋊C414yes—
84C4⋊Q8—22no—
85C4⋊Q8—24no—
86C4⋊Q8—24no—
87C4⋊Q8—22no—
88C4⋊Q8—24no—
89C4⋊Q8—22no—
90C4⋊Q8—42no—
91C4⋊Q8—42no—
92C4⋊Q8—42no—
93C4⋊Q8C4⋊C414yes—
94C2×C4⋊C4—24no—
95C2×C4⋊C4—24no—
96C2×C4⋊C4—24no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—44no—
102C2×C4⋊C4—41no—
103C2×C42C22×C411yescharacteristic
104C2×C42C2×D411yescharacteristic
105C2×C42C2×D412yes—
106C2×C42C2×D412yes—
107C2×C42C2×D411yescharacteristic
108C2×C42C2×Q811yescharacteristic
109C2×C42#4111yescharacteristic
110C2×C42#4111yescharacteristic
111C2×C42#4112yes—
112C4⋊C8—24no—
113C4⋊C8—24no—
114C4⋊C8—24no—
115C4⋊C8—42no—
116C4⋊C8—42no—
117C4⋊C8—42no—
118#103—416no—
119C22⋊C8—24no—
120C22⋊C8—42no—
121#97—24no—
122#97—24no—
123C4×C8—24no—
124C4×C8—42no—
125#95—24no—
126#95—42no—
127C24#12411yescharacteristic
128C2×Q8—24no—
129C2×Q8—42no—
130C2×Q8—42no—
131C2×Q8—42no—
132C2×Q8—44no—
133C2×Q8—44no—
134C2×Q8—48no—
135C2×Q8—48no—
136C2×Q8—84no—
137C2×Q8—84no—
138C2×Q8—82no—
139C2×Q8—82no—
140C22×C4—44no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—82no—
146C22×C4—82no—
147C22×C4—84no—
148C22×C4—82no—
149C22×C4C2×C22⋊C411yescharacteristic
150C22×C4C4×D412yes—
151C22×C4#12011yescharacteristic
152C22×C4#12112yes—
153C22×C4#12111yescharacteristic
154C22×C4C22⋊Q812yes—
155C22×C4C22⋊Q811yescharacteristic
156C22×C4#12312yes—
157C22×C4#12311yescharacteristic
158C22×C4#12412yes—
159C22×C4#12612yes—
160C22×C4#12711yescharacteristic
161C22×C4#13514yes—
162C22×C4AGL(1,ℤ/8)14yes—
163M16—48no—
164M16—82no—
165M16—82no—
166C2×C8—24no—
167C2×C8—24no—
168C2×C8—24no—
169C2×C8—44no—
170C2×C8—44no—
171C2×C8—44no—
172C2×C8—44no—
173C2×C8—84no—
174C2×C8—84no—
175C4⋊C4—28no—
176C4⋊C4—44no—
177C4⋊C4—44no—
178C4⋊C4—44no—
179C4⋊C4—44no—
180C4⋊C4—82no—
181C42C2×C4⋊C411yescharacteristic
182C42#11711yescharacteristic
183C42C4×D412yes—
184C42#12012yes—
185C42#12111yescharacteristic
186C42#12112yes—
187C42#12112yes—
188C42C22⋊Q811yescharacteristic
189C42C22⋊Q812yes—
190C42#12312yes—
191C23—24no—
192C23#39011yescharacteristic
193C23#39311yescharacteristic
194C23#39711yescharacteristic
195C23#48812yes—
196C23#49012yes—
197Q8—48no—
198Q8—84no—
199Q8—84no—
200Q8—84no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—44no—
208C2×C4—84no—
209C2×C4—82no—
210C2×C4—82no—
211C2×C4—84no—
212C2×C4—84no—
213C2×C4—82no—
214C2×C4—164no—
215C2×C4#38011yescharacteristic
216C2×C4#38512yes—
217C2×C4#38611yescharacteristic
218C2×C4#38812yes—
219C2×C4#39211yescharacteristic
220C2×C4#39412yes—
221C2×C4#39612yes—
222C2×C4#39911yescharacteristic
223C2×C4#41612yes—
224C2×C4#41812yes—
225C2×C4#44914yes—
226C2×C4#46314yes—
227C2×C4#47814yes—
228C2×C4#48214yes—
229C2×C4#49512yes—
230C2×C4#49612yes—
231C8—44no—
232C8—84no—
233C8—84no—
234C22—22no—
235C22—22no—
236C22—24no—
237C22—24no—
238C22#143911yescharacteristic
239C22#195811yescharacteristic
240C22#196111yescharacteristic
241C22#202512yes—
242C22#207312yes—
243C22#208214yes—
244C4—22no—
245C4—22no—
246C4—22no—
247C4—22no—
248C4—84no—
249C4—162no—
250C4#180412yes—
251C4#187512yes—
252C4#201212yes—
253C4#209212yes—
254C2—21no—
255C2—21no—
256C2—22no—
257C2#1063514yes—
258C2#1105111yescharacteristic
259C2#1113412yes—
subgroupG/H
center, socleC23#397
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#124
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#19602
C2×#18382
C2×#15462
C2×#12862
C2×#12812