Finite Group Databasebeta

← #142196

Subgroups — #142196

Order 512.

434 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 114 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
2#12783C211yesmaximal, characteristic
3#10820C211yesmaximal, characteristic
4#7258C214yesmaximal
5C2×#500C411yescharacteristic, complemented
6C22×C4×C8C2211yescharacteristic
7#2857—21no—
8C2×C4⋊C16—21no—
9C2×C22⋊C16C2212yes—
10#1990—21no—
11#1942—21no—
12#1942C412yescomplemented
13#1845C411yescharacteristic, complemented
14#1808—21no—
15#1808C2212yes—
16C2×#340—21no—
17C2×#340C2212yes—
18C4⋊#95—21no—
19#1374—21no—
20C23×C8D411yescharacteristic
21C23×C8C2311yescharacteristic
22C2×C4⋊Q8—21no—
23C2×C22⋊Q8—41no—
24C22×C4⋊C4—22no—
25C22×C42D411yescharacteristic
26C22×C16—22no—
27#500—42no—
28C2×#107—21no—
29C2×#107—21no—
30C2×#107C2×C412yes—
31C2×#107C2×C412yes—
32C2×C4×C8—42no—
33C2×C4×C8C2×C411yescharacteristic
34C2×C4×C8C2×C411yescharacteristic
35C2×C4×C8D411yescharacteristic
36C2×C4×C8D411yescharacteristic
37#391—41no—
38#389—22no—
39#389—41no—
40#384—21no—
41#380—22no—
42#380—41no—
43C2×#95—41no—
44C2×#95—41no—
45C4⋊C16—42no—
46C22⋊C16—24no—
47#340—22no—
48#340—22no—
49#340—22no—
50#340—42no—
51#340Q814yescomplemented
52C22×Q8—41no—
53C23×C4—42no—
54C23×C4D811yescharacteristic
55C23×C4D811yescharacteristic
56C23×C4C2×D411yescharacteristic
57C22×C8—22no—
58C22×C8—22no—
59C22×C8—41no—
60C22×C8—41no—
61C22×C8—41no—
62C22×C8C22⋊C411yescharacteristic
63C22×C8C22⋊C411yescharacteristic
64C22×C8C22×C411yescharacteristic
65C22×C8C2×D411yescharacteristic
66C22×C8C2×Q812yes—
67C22×C8#4112yes—
68C4⋊Q8—42no—
69C4⋊Q8—42no—
70C22⋊Q8—84no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—24no—
74C2×C4⋊C4—42no—
75C2×C4⋊C4—42no—
76C2×C4⋊C4—42no—
77C2×C4⋊C4—42no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—42no—
83C2×C22⋊C4—42no—
84C2×C42—41no—
85C2×C42—41no—
86C2×C42C22⋊C411yescharacteristic
87C2×C42C22⋊C411yescharacteristic
88C2×C42SD1611yescharacteristic
89C2×C42SD1611yescharacteristic
90C2×C16—42no—
91C2×C16—42no—
92C2×C16—42no—
93#107—42no—
94#107—42no—
95#107—42no—
96#107—42no—
97C4×C8—22no—
98C4×C8—22no—
99C4×C8—41no—
100C4×C8—41no—
101#95—42no—
102#95—42no—
103#95—82no—
104#95—82no—
105C24C2×D811yescharacteristic
106C2×Q8—82no—
107C2×Q8—82no—
108C2×Q8—82no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—41no—
116C22×C4—41no—
117C22×C4—41no—
118C22×C4—42no—
119C22×C4—42no—
120C22×C4—44no—
121C22×C4—82no—
122C22×C4—82no—
123C22×C4—82no—
124C22×C4—82no—
125C22×C4—82no—
126C22×C4—82no—
127C22×C4#10211yescharacteristic
128C22×C4#10211yescharacteristic
129C22×C4#10211yescharacteristic
130C22×C4#10211yescharacteristic
131C22×C4SD3212yes—
132C22×C4Q3212yes—
133C22×C4C2×C22⋊C411yescharacteristic
134C22×C4C22⋊Q812yes—
135C22×C4C2×SD1611yescharacteristic
136C22×C4#13712yes—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—42no—
147C2×C8—42no—
148C2×C8—44no—
149C2×C8#9912yes—
150C2×C8C4×Q812yes—
151C2×C8#12411yescharacteristic
152C2×C8C4⋊Q811yescharacteristic
153C4⋊C4—44no—
154C4⋊C4—44no—
155C4⋊C4—42no—
156C4⋊C4—42no—
157C4⋊C4—42no—
158C4⋊C4—42no—
159C4⋊C4—82no—
160C4⋊C4—82no—
161C4⋊C4—82no—
162C4⋊C4—82no—
163C4⋊C4—84no—
164C4⋊C4—82no—
165C22⋊C4—84no—
166C42—21no—
167C42—21no—
168C42—21no—
169C42—21no—
170C42—41no—
171C42—41no—
172C16—82no—
173C23—21no—
174C23—21no—
175C23—21no—
176C23—21no—
177C23C2×#10211yescharacteristic
178C23#47412yes—
179C23C2×SD3211yescharacteristic
180C23C2×Q3211yescharacteristic
181C23#50912yes—
182Q8—164no—
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—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—44no—
201C2×C4—84no—
202C2×C4—84no—
203C2×C4—82no—
204C2×C4—82no—
205C2×C4—82no—
206C2×C4—82no—
207C2×C4—84no—
208C2×C4—82no—
209C2×C4—82no—
210C2×C4—162no—
211C2×C4—162no—
212C2×C4—162no—
213C2×C4—162no—
214C2×C4#35812yes—
215C2×C4#35812yes—
216C2×C4#36012yes—
217C2×C4#39111yescharacteristic
218C2×C4C2×#9911yescharacteristic
219C2×C4#41011yescharacteristic
220C2×C4#41911yescharacteristic
221C2×C4#42111yescharacteristic
222C2×C4#42111yescharacteristic
223C2×C4#47512yes—
224C8—21no—
225C8—22no—
226C8—21no—
227C8—22no—
228C8—41no—
229C22—21no—
230C22—21no—
231C22—22no—
232C22—21no—
233C22—21no—
234C22—22no—
235C22—22no—
236C22—22no—
237C22#151711yescharacteristic
238C22#151911yescharacteristic
239C22#198811yescharacteristic
240C22C2×#35811yescharacteristic
241C22#214011yescharacteristic
242C22C2×#36011yescharacteristic
243C22#214511yescharacteristic
244C22#222612yes—
245C22#222712yes—
246C4—22no—
247C4—22no—
248C4—22no—
249C4—22no—
250C4—41no—
251C4—164no—
252C4—162no—
253C4—162no—
254C4#151111yescharacteristic
255C4#151511yescharacteristic
256C4#199511yescharacteristic
257C4#199611yescharacteristic
258C2—22no—
259C2—22no—
260C2#991211yescharacteristic
261C2#1185711yescharacteristic
262C2#1185811yescharacteristic
263C2#1188711yescharacteristic
264C2#1188811yescharacteristic
265C2#1327611yescharacteristic
266C2#1327811yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC24C2×D8
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
5. C2×#500C4semidirect
12. #1942C4semidirect
13. #1845C4semidirect
51. #340Q8semidirect

Maximal subgroups

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

subgroupindex
#332092
#127832
#108202
#72582