Finite Group Databasebeta

← #510369

Subgroups — #510369

Order 512.

512 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 138 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#25035C212yesmaximal
2#13538C212yesmaximal, complemented
3#13493C211yesmaximal, characteristic
4#13475C212yesmaximal, complemented
5#13306C211yesmaximal, characteristic
6#13060C212yesmaximal
7#13015C212yesmaximal, complemented
8#12912C211yesmaximal, characteristic
9#12812C212yesmaximal, complemented
10C22×Q32C2211yescharacteristic
11C23×C16C2211yescharacteristic
12C2×#467C2212yes—
13C2×C4×Q16C2212yes—
14#2423—22no—
15C2×#366C2211yescharacteristic
16C2×C4⋊C16C2212yes—
17C2×#358C2212yes—
18C2×#358C2212yes—
19C2×#358C2212yes—
20C2×C22⋊C16C2211yescharacteristic
21#2072—22no—
22#2036—22no—
23#1969C2212yes—
24#1942C2212yes—
25#1863—22no—
26#1845C2212yes—
27#1808C2212yes—
28C8×C22⋊C4C2211yescharacteristic
29#1380C2212yes—
30#1380C2212yes—
31#1380C2211yescharacteristic
32#1380C2212yes—
33#1380C2211yescharacteristic
34#1374C2212yes—
35C22×Q16C2312yes—
36C23×C8C2311yescharacteristic
37C2×C22⋊Q8—22no—
38C2×C4×Q8—22no—
39C2×Q32—24no—
40C2×Q32—41no—
41C22×C16—41no—
42C22×C16—42no—
43C22×C16C2311yescharacteristic
44C22×C16C2311yescharacteristic
45C22×C16C2312yes—
46#467—24no—
47#467—42no—
48C4×Q16—24no—
49C4×Q16—42no—
50C2×#107—22no—
51C2×#107C2312yes—
52C2×#107C2312yes—
53C2×#107C2312yes—
54C2×#103—22no—
55C2×#103—22no—
56C2×#103—22no—
57C2×C22⋊C8D411yescharacteristic
58C2×C4×C8C2312yes—
59#399—42no—
60#398—42no—
61#391—22no—
62#389—22no—
63#388—42no—
64#387—42no—
65#384—22no—
66#380—22no—
67C4×C22⋊C4D411yescharacteristic
68#366—41no—
69C4⋊C16—42no—
70#358—42no—
71#358—42no—
72#358—42no—
73C22⋊C16—41no—
74#340—22no—
75#340—22no—
76#336D412yes—
77C22×Q8—22no—
78C23×C4C2×D411yescharacteristic
79C2×Q16—22no—
80C2×Q16—22no—
81C2×Q16—22no—
82C2×Q16—22no—
83C2×Q16—22no—
84C2×Q16#4114yes—
85C22×C8—22no—
86C22×C8—21no—
87C22×C8—22no—
88C22×C8—21no—
89C22×C8C2×D411yescharacteristic
90C22×C8C2×D412yes—
91C22×C8C2411yescharacteristic
92C22⋊Q8—44no—
93C22⋊Q8—82no—
94C4×Q8—44no—
95C4×Q8—82no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—42no—
104C2×C22⋊C4—42no—
105C2×C22⋊C4—42no—
106C2×C22⋊C4D811yescharacteristic
107C2×C22⋊C4D811yescharacteristic
108C2×C42—42no—
109C2×C42C2×D412yes—
110Q32—44no—
111C2×C16—22no—
112C2×C16—21no—
113C2×C16—21no—
114C2×C16—22no—
115C2×C16—22no—
116C2×C16—22no—
117C2×C16—22no—
118C2×C16—44no—
119#107—22no—
120#107—22no—
121#107—22no—
122#107—22no—
123#107—22no—
124#107—22no—
125#107—82no—
126#103—44no—
127#103—44no—
128#103—44no—
129C22⋊C8—41no—
130C4×C8—22no—
131C4×C8—22no—
132#95—42no—
133#95—42no—
134#95—42no—
135#95—42no—
136#95D812yes—
137C24C2×D811yescharacteristic
138C2×Q8—24no—
139C2×Q8—42no—
140C2×Q8—42no—
141C2×Q8—42no—
142C2×Q8—42no—
143C2×Q8—42no—
144C22×C4—21no—
145C22×C4—22no—
146C22×C4—21no—
147C22×C4—22no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4C2×D811yescharacteristic
153C22×C4C2×D812yes—
154C22×C4C2×D812yes—
155C22×C4C22×D411yescharacteristic
156Q16—24no—
157Q16—24no—
158Q16—42no—
159Q16—42no—
160C2×C8—21no—
161C2×C8—22no—
162C2×C8—22no—
163C2×C8—22no—
164C2×C8—22no—
165C2×C8—42no—
166C2×C8—41no—
167C2×C8—41no—
168C2×C8C2×#4112yes—
169C2×C8H(5,𝔽2)12yes—
170C2×C8H(5,𝔽2)11yescharacteristic
171C2×C82−1+411yescharacteristic
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—84no—
179C4⋊C4—84no—
180C4⋊C4—84no—
181C4⋊C4—162no—
182C4⋊C4—162no—
183C22⋊C4—41no—
184C22⋊C4—84no—
185C22⋊C4—162no—
186C22⋊C4Q3214yescomplemented
187C42—22no—
188C42—22no—
189C42—84no—
190C16—41no—
191C16—41no—
192C16—42no—
193C23—21no—
194C23—22no—
195C23—21no—
196C23—22no—
197C23C2×Q3212yes—
198C23C22×D811yescharacteristic
199Q8—44no—
200Q8—44no—
201Q8—82no—
202Q8—82no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—42no—
213C2×C4—44no—
214C2×C4—82no—
215C2×C4—82no—
216C2×C4—82no—
217C2×C4—82no—
218C2×C4—82no—
219C2×C4—82no—
220C2×C4—82no—
221C2×C4—82no—
222C2×C4—82no—
223C2×C4—82no—
224C2×C4—82no—
225C2×C4C2×Q3214yes—
226C2×C4#53711yescharacteristic
227C2×C4#54612yes—
228C2×C4#57511yescharacteristic
229C2×C4#57712yes—
230C8—41no—
231C8—42no—
232C8#53912yes—
233C8#55111yescharacteristic
234C8#56011yescharacteristic
235C22—22no—
236C22—24no—
237C22—41no—
238C22—41no—
239C22—42no—
240C22#324511yescharacteristic
241C22#329412yes—
242C22C22×Q3211yescharacteristic
243C22#341411yescharacteristic
244C22#341612yes—
245C4—24no—
246C4—41no—
247C4—42no—
248C4—84no—
249C4—162no—
250C4—162no—
251C4—162no—
252C4#269311yescharacteristic
253C4#326311yescharacteristic
254C4#330712yes—
255C2—42no—
256C2#3155011yescharacteristic
257C2#3381511yescharacteristic
258C2#3382911yescharacteristic
259C2#3386612yes—
260C2#3388412yes—
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #13538C2semidirect
4. #13475C2semidirect
7. #13015C2semidirect
9. #12812C2semidirect
186. C22⋊C4Q32semidirect

Maximal subgroups

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

subgroupindex
#250352
#135382
#134932
#134752
#133062
#130602
#130152
#129122
#128122