Finite Group Databasebeta

← #142005

Subgroups — #142005

Order 512.

424 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 118 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
1#33209C211yesmaximal, characteristic
2C4×C22⋊C16C211yesmaximal, characteristic
3#10864C211yesmaximal, characteristic
4#7259C214yesmaximal
5C2×#500C411yescharacteristic
6C22×#108—21no—
7C22×C4×C8C2211yescharacteristic
8#2857—21no—
9C2×C22⋊C16C2212yes—
10C2×C4×C16—21no—
11#1990—21no—
12#1942—21no—
13#1942C412yes—
14#1845C411yescharacteristic
15#1808—21no—
16#1808C2212yes—
17C2×#341C2212yes—
18#1728—21no—
19#1374—21no—
20C22×M16—22no—
21C23×C8D411yescharacteristic
22C23×C8C2311yescharacteristic
23C2×C4⋊Q8—21no—
24C2×C22⋊Q8—41no—
25C22×C42D411yescharacteristic
26C22×C16—22no—
27#500—42no—
28C2×#108—24no—
29C2×#108—42no—
30C2×#108—42no—
31C2×#107—21no—
32C2×#107—21no—
33C2×#107C2×C412yes—
34C2×#107C2×C412yes—
35C2×C4×C8—42no—
36C2×C4×C8C2×C411yescharacteristic
37C2×C4×C8C2×C411yescharacteristic
38C2×C4×C8D411yescharacteristic
39C2×C4×C8D411yescharacteristic
40#391—41no—
41#389—22no—
42#389—41no—
43#384—21no—
44#380—22no—
45#380—41no—
46C22⋊C16—22no—
47C22⋊C16Q814yescomplemented
48C4×C16—42no—
49#341—24no—
50#340—22no—
51#340—22no—
52#336—41no—
53#336—41no—
54C22×Q8—41no—
55C23×C4SD1611yescharacteristic
56C23×C4SD1611yescharacteristic
57C23×C4C2×D411yescharacteristic
58C2×M16—24no—
59C2×M16—42no—
60C2×M16—42no—
61C2×M16—42no—
62C2×M16—42no—
63C2×M16—42no—
64C22×C8—22no—
65C22×C8—22no—
66C22×C8—41no—
67C22×C8—41no—
68C22×C8—42no—
69C22×C8C22⋊C411yescharacteristic
70C22×C8C22⋊C411yescharacteristic
71C22×C8SD1612yes—
72C22×C8Q1612yes—
73C22×C8C22×C411yescharacteristic
74C22×C8C2×D411yescharacteristic
75C22×C8C2×Q812yes—
76C22×C8#4112yes—
77C4⋊Q8—42no—
78C4⋊Q8—42no—
79C22⋊Q8—84no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C22⋊C4—42no—
87C2×C42—41no—
88C2×C42—41no—
89C2×C42C22⋊C411yescharacteristic
90C2×C42C22⋊C411yescharacteristic
91C2×C42D811yescharacteristic
92C2×C42D811yescharacteristic
93C2×C16—24no—
94C2×C16—42no—
95C2×C16—42no—
96#108—48no—
97#107—42no—
98#107—42no—
99#107—42no—
100#107—42no—
101C4×C8—22no—
102C4×C8—22no—
103C4×C8—41no—
104C4×C8—41no—
105#95—42no—
106#95—42no—
107C24C2×SD1611yescharacteristic
108C2×Q8—82no—
109C2×Q8—82no—
110C2×Q8—82no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—41no—
120C22×C4—41no—
121C22×C4—42no—
122C22×C4—42no—
123C22×C4#10211yescharacteristic
124C22×C4#10211yescharacteristic
125C22×C4#10211yescharacteristic
126C22×C4#10211yescharacteristic
127C22×C4C2×C22⋊C411yescharacteristic
128C22×C4#12411yescharacteristic
129C22×C4C4⋊Q811yescharacteristic
130C22×C4C2×D811yescharacteristic
131C22×C4C2×SD1611yescharacteristic
132C22×C4C2×Q1611yescharacteristic
133M16—44no—
134M16—44no—
135M16—82no—
136M16—82no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—24no—
144C2×C8—24no—
145C2×C8—42no—
146C2×C8—42no—
147C2×C8—44no—
148C2×C8—82no—
149C2×C8—82no—
150C2×C8#9912yes—
151C2×C8#10312yes—
152C2×C8#10312yes—
153C2×C8#10412yes—
154C2×C8C4×Q812yes—
155C2×C8C22⋊Q812yes—
156C4⋊C4—42no—
157C4⋊C4—42no—
158C4⋊C4—42no—
159C4⋊C4—42no—
160C4⋊C4—82no—
161C4⋊C4—82no—
162C4⋊C4—82no—
163C4⋊C4—84no—
164C22⋊C4—84no—
165C42—21no—
166C42—21no—
167C42—21no—
168C42—21no—
169C42—41no—
170C42—41no—
171C16—44no—
172C23—21no—
173C23—21no—
174C23—21no—
175C23—21no—
176C23—21no—
177C23—21no—
178C23C2×#10211yescharacteristic
179C23#48711yescharacteristic
180C23#49811yescharacteristic
181Q8—164no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—44no—
201C2×C4—44no—
202C2×C4—82no—
203C2×C4—82no—
204C2×C4—82no—
205C2×C4—82no—
206C2×C4—82no—
207C2×C4—82no—
208C2×C4#35912yes—
209C2×C4#35912yes—
210C2×C4#36112yes—
211C2×C4#36212yes—
212C2×C4#39111yescharacteristic
213C2×C4C2×#9911yescharacteristic
214C2×C4#41011yescharacteristic
215C2×C4C2×#10311yescharacteristic
216C2×C4C2×#10411yescharacteristic
217C2×C4#42111yescharacteristic
218C2×C4#48611yescharacteristic
219C2×C4#50011yescharacteristic
220C8—22no—
221C8—22no—
222C8—22no—
223C8—84no—
224C8#32814yes—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—21no—
230C22—21no—
231C22—41no—
232C22—41no—
233C22—41no—
234C22—41no—
235C22#150211yescharacteristic
236C22#151911yescharacteristic
237C22#199011yescharacteristic
238C22C2×#35911yescharacteristic
239C22C2×#35911yescharacteristic
240C22C2×#36111yescharacteristic
241C22C2×#36211yescharacteristic
242C4—22no—
243C4—22no—
244C4—22no—
245C4—22no—
246C4—21no—
247C4—21no—
248C4—162no—
249C4—162no—
250C4C2×#32811yescharacteristic
251C4#151811yescharacteristic
252C4#198611yescharacteristic
253C4#199611yescharacteristic
254C2—41no—
255C2—41no—
256C2#988411yescharacteristic
257C2#1174711yescharacteristic
258C2#1174811yescharacteristic
259C2#1189111yescharacteristic
260C2#1189211yescharacteristic
261C2#1328511yescharacteristic
262C2#1329011yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC24C2×SD16
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
47. C22⋊C16Q8semidirect

Maximal subgroups

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

subgroupindex
#332092
C4×C22⋊C162
#108642
#72592