Finite Group Databasebeta

← #142161

Subgroups — #142161

Order 512.

506 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 126 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
3C4×#340C211yesmaximal, characteristic
4#7258C214yesmaximal
5C2×#500C411yescharacteristic, complemented
6C22×C4×C8C2211yescharacteristic
7#2857—21no—
8C2×C4×C4⋊C4—21no—
9C2×C4⋊C16—21no—
10C2×C22⋊C16C2212yes—
11#1990—21no—
12#1942—21no—
13#1942C412yescomplemented
14#1845C411yescharacteristic, complemented
15#1808—21no—
16#1808C2212yes—
17C2×#340—21no—
18C2×#340C2212yes—
19#1374—21no—
20C23×C8D411yescharacteristic
21C23×C8C2311yescharacteristic
22C2×C4⋊Q8—21no—
23C2×C22⋊Q8—41no—
24C22×C4⋊C4—22no—
25C22×C42—41no—
26C22×C42D411yescharacteristic
27C22×C16—22no—
28#500—42no—
29C2×#107—21no—
30C2×#107—21no—
31C2×#107C2×C412yes—
32C2×#107C2×C412yes—
33C2×C4×C8—42no—
34C2×C4×C8C2×C411yescharacteristic
35C2×C4×C8C2×C411yescharacteristic
36C2×C4×C8D411yescharacteristic
37C2×C4×C8D411yescharacteristic
38#391—41no—
39#389—22no—
40#389—41no—
41#384—21no—
42#380—22no—
43#380—41no—
44C4×C4⋊C4—24no—
45C4×C4⋊C4—42no—
46C2×#95—41no—
47C4⋊C16—42no—
48C22⋊C16—24no—
49#340—24no—
50#340—22no—
51#340—22no—
52#340—22no—
53#340Q814yescomplemented
54C22×Q8—41no—
55C23×C4—42no—
56C23×C4D811yescharacteristic
57C23×C4D811yescharacteristic
58C23×C4C2×D411yescharacteristic
59C22×C8—22no—
60C22×C8—22no—
61C22×C8—41no—
62C22×C8—41no—
63C22×C8—41no—
64C22×C8C22⋊C411yescharacteristic
65C22×C8C22⋊C411yescharacteristic
66C22×C8C22×C411yescharacteristic
67C22×C8C2×D411yescharacteristic
68C22×C8C2×Q812yes—
69C22×C8#4112yes—
70C4⋊Q8—42no—
71C4⋊Q8—42no—
72C22⋊Q8—82no—
73C22⋊Q8—82no—
74C2×C4⋊C4—24no—
75C2×C4⋊C4—24no—
76C2×C4⋊C4—24no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C22⋊C4—42no—
87C2×C42—22no—
88C2×C42—22no—
89C2×C42—44no—
90C2×C42—82no—
91C2×C42—82no—
92C2×C42C22⋊C411yescharacteristic
93C2×C42C22⋊C411yescharacteristic
94C2×C42SD1611yescharacteristic
95C2×C42SD1611yescharacteristic
96C2×C16—42no—
97C2×C16—42no—
98C2×C16—42no—
99#107—42no—
100#107—42no—
101#107—42no—
102#107—42no—
103C4×C8—22no—
104C4×C8—22no—
105C4×C8—41no—
106C4×C8—41no—
107#95—44no—
108#95—42no—
109#95—42no—
110C24C2×D811yescharacteristic
111C2×Q8—82no—
112C2×Q8—82no—
113C2×Q8—82no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—41no—
123C22×C4—44no—
124C22×C4—44no—
125C22×C4—44no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4—82no—
129C22×C4—82no—
130C22×C4—82no—
131C22×C4—82no—
132C22×C4#10211yescharacteristic
133C22×C4#10211yescharacteristic
134C22×C4#10211yescharacteristic
135C22×C4#10211yescharacteristic
136C22×C4D1612yes—
137C22×C4SD3212yes—
138C22×C4C2×C22⋊C411yescharacteristic
139C22×C4C22⋊Q812yes—
140C22×C4C2×SD1611yescharacteristic
141C22×C4#13712yes—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—42no—
152C2×C8—42no—
153C2×C8—44no—
154C2×C8#10412yes—
155C2×C8C4×Q812yes—
156C2×C8#12411yescharacteristic
157C2×C8C4⋊Q811yescharacteristic
158C4⋊C4—28no—
159C4⋊C4—44no—
160C4⋊C4—44no—
161C4⋊C4—44no—
162C4⋊C4—42no—
163C4⋊C4—42no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4—82no—
167C4⋊C4—82no—
168C4⋊C4—82no—
169C4⋊C4—84no—
170C22⋊C4—82no—
171C22⋊C4—82no—
172C42—22no—
173C42—22no—
174C42—21no—
175C42—21no—
176C42—88no—
177C42#9912yes—
178C42#10412yes—
179C16—82no—
180C23—21no—
181C23—21no—
182C23—21no—
183C23—21no—
184C23C2×#10211yescharacteristic
185C23#47412yes—
186C23C2×D1611yescharacteristic
187C23C2×SD3211yescharacteristic
188C23#51012yes—
189Q8—164no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—24no—
197C2×C4—24no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—44no—
206C2×C4—48no—
207C2×C4—84no—
208C2×C4—84no—
209C2×C4—84no—
210C2×C4—84no—
211C2×C4—84no—
212C2×C4—84no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4—82no—
216C2×C4—82no—
217C2×C4—82no—
218C2×C4—82no—
219C2×C4#32712yes—
220C2×C4#32712yes—
221C2×C4#35712yes—
222C2×C4#35712yes—
223C2×C4#36012yes—
224C2×C4#39111yescharacteristic
225C2×C4C2×#9911yescharacteristic
226C2×C4#41011yescharacteristic
227C2×C4#41911yescharacteristic
228C2×C4C2×#10411yescharacteristic
229C2×C4C2×#10411yescharacteristic
230C2×C4#47512yes—
231C8—21no—
232C8—22no—
233C8—21no—
234C8—22no—
235C8—41no—
236C22—22no—
237C22—21no—
238C22—21no—
239C22—22no—
240C22—21no—
241C22—21no—
242C22—22no—
243C22—22no—
244C22C2×#32711yescharacteristic
245C22#150211yescharacteristic
246C22#198811yescharacteristic
247C22C2×#35711yescharacteristic
248C22#214111yescharacteristic
249C22C2×#36011yescharacteristic
250C22#214511yescharacteristic
251C22#222412yes—
252C22#222512yes—
253C4—22no—
254C4—22no—
255C4—22no—
256C4—41no—
257C4—88no—
258C4—162no—
259C4—162no—
260C4#134714yes—
261C4#150611yescharacteristic
262C4#150811yescharacteristic
263C4#199411yescharacteristic
264C4#199611yescharacteristic
265C2—22no—
266C2—22no—
267C2#990911yescharacteristic
268C2C2×#134711yescharacteristic
269C2#1170611yescharacteristic
270C2#1174211yescharacteristic
271C2#1174511yescharacteristic
272C2#1327511yescharacteristic
273C2#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
13. #1942C4semidirect
14. #1845C4semidirect
53. #340Q8semidirect

Maximal subgroups

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

subgroupindex
#332092
#127832
C4×#3402
#72582