Finite Group Databasebeta

← #513299

Subgroups — #513299

Order 512.

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

normal subgroups only

This group has 323 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#33962C211yesmaximal, characteristic, complemented
2#33960C211yesmaximal, characteristic
3#13730C212yesmaximal
4#13726C211yesmaximal, characteristic, complemented
5#13690C212yesmaximal
6#13688C211yesmaximal, characteristic, complemented
7#13688C212yesmaximal, complemented
8#13687C211yesmaximal, characteristic
9#13660C211yesmaximal, characteristic, complemented
10#13653C212yesmaximal, complemented
11C4×SD64C211yesmaximal, characteristic
12#3395—21no—
13#3393—21no—
14C2×Q64C2212yes—
15C2×Q64C2212yes—
16C2×Q64C2212yes—
17C2×SD64C2211yescharacteristic
18C2×SD64C2212yes—
19#2249C2212yes—
20#2247C2211yescharacteristic
21#2247C2211yescharacteristic
22#2247C2212yes—
23#2211—22no—
24#2209—21no—
25#2209—22no—
26#2208—21no—
27Q8×C16C2211yescharacteristic
28C4×Q32C2211yescharacteristic
29C4×Q32C2212yes—
30C4×Q32C2212yes—
31C4×D16C2211yescharacteristic
32#1424C2211yescharacteristic
33C4⋊C32C2211yescharacteristic
34C4⋊C32C2212yes—
35#1416C2211yescharacteristic
36#1416C2212yes—
37#1416C2212yes—
38#1415C2211yescharacteristic
39#1397C2212yes—
40C4×C32C2211yescharacteristic
41#557—41no—
42#556—41no—
43C2×Q32—21no—
44C2×Q32—22no—
45C2×Q32—21no—
46C2×Q32C2311yescharacteristic
47C2×Q32C2312yes—
48C2×Q32C2312yes—
49C2×SD32—22no—
50C2×D16C2311yescharacteristic
51#495—22no—
52#494—21no—
53#494—21no—
54#494—22no—
55#464—42no—
56#462—41no—
57#462—42no—
58#461—41no—
59C8×Q8D411yescharacteristic
60C4×Q16—22no—
61C4×Q16—21no—
62C4×Q16—22no—
63C4×D8—21no—
64Q64—24no—
65Q64—42no—
66Q64—42no—
67SD64—22no—
68SD64—42no—
69C2×C32C2311yescharacteristic
70C2×C32C2312yes—
71C2×C32C2311yescharacteristic
72#366C2311yescharacteristic
73#366C2311yescharacteristic
74C4⋊C16D411yescharacteristic
75C4⋊C16D412yes—
76#358—22no—
77#358—21no—
78#358—22no—
79#357—21no—
80C4×C16C2311yescharacteristic
81C4×C16C2312yes—
82C2×Q16—21no—
83C2×Q16—22no—
84C2×Q16—22no—
85C2×Q16—41no—
86C2×Q16—42no—
87C2×Q16—41no—
88C2×SD16—42no—
89C2×D8—21no—
90C4⋊Q8—41no—
91C4⋊Q8—41no—
92C4⋊Q8—42no—
93#126—82no—
94#125—81no—
95#125—82no—
96#124—42no—
97C22⋊Q8—81no—
98C4×Q8—42no—
99C4×Q8—41no—
100C4×Q8—42no—
101C4×Q8—81no—
102C4×Q8D811yescharacteristic
103C4×D4—41no—
104#117—81no—
105Q32—21no—
106Q32—22no—
107Q32—22no—
108Q32—22no—
109Q32—21no—
110Q32—22no—
111Q32—82no—
112Q32—81no—
113Q32—81no—
114SD32—82no—
115D16—21no—
116D16—21no—
117C2×C16C2×D411yescharacteristic
118C2×C16C2×D412yes—
119C2×C16C2411yescharacteristic
120#107—21no—
121#107—21no—
122C4⋊C8D811yescharacteristic
123C4⋊C8D812yes—
124#103—42no—
125#103—41no—
126#103—42no—
127#102—41no—
128C4×C8C2×D411yescharacteristic
129C4×C8C2×D412yes—
130C32—22no—
131C32—21no—
132C32#4112yes—
133C2×Q8—42no—
134C2×Q8—41no—
135C2×Q8—42no—
136C2×Q8D1611yescharacteristic
137C2×D4—41no—
138C22×C4—81no—
139Q16—41no—
140Q16—42no—
141Q16—42no—
142Q16—42no—
143Q16—41no—
144Q16—42no—
145Q16—162no—
146Q16—161no—
147Q16—161no—
148SD16—162no—
149D8—41no—
150D8—41no—
151C2×C8C2×D811yescharacteristic
152C2×C8C2×D812yes—
153C2×C8C22×D411yescharacteristic
154C4⋊C4—41no—
155C4⋊C4—41no—
156C4⋊C4—81no—
157C4⋊C4—82no—
158C4⋊C4—82no—
159C4⋊C4—81no—
160C4⋊C4—82no—
161C4⋊C4—81no—
162C4⋊C4—82no—
163C4⋊C4—81no—
164C4⋊C4—81no—
165C4⋊C4D1611yescharacteristic
166C4⋊C4D1612yes—
167C22⋊C4—82no—
168C22⋊C4—81no—
169C22⋊C4—82no—
170C42—82no—
171C42—82no—
172C42—81no—
173C42C2×D811yescharacteristic
174C42C2×D812yes—
175C16—22no—
176C16—21no—
177C16C2×#4111yescharacteristic
178C16H(5,𝔽2)11yescharacteristic
179C23—81no—
180Q8—22no—
181Q8—82no—
182Q8—81no—
183Q8—82no—
184Q8—82no—
185Q8—81no—
186Q8—82no—
187D4—81no—
188D4—81no—
189C2×C4—81no—
190C2×C4—81no—
191C2×C4—82no—
192C2×C4—82no—
193C2×C4—81no—
194C2×C4—162no—
195C2×C4C2×D1612yes—
196C2×C4C2×D1612yes—
197C2×C4C2×D1611yescharacteristic
198C2×C4C2×D1611yescharacteristic
199C2×C4C22×D811yescharacteristic
200C8—22no—
201C8—21no—
202C8#55011yescharacteristic
203C8#57611yescharacteristic
204C22—161no—
205C22—161no—
206C22—161no—
207C22C22×D1611yescharacteristic
208C4—22no—
209C4—22no—
210C4—21no—
211C4—162no—
212C4—162no—
213C4—161no—
214C4—161no—
215C4—161no—
216C4#226412yes—
217C4#339111yescharacteristic
218C4#341511yescharacteristic
219C2—321no—
220C2#3395811yescharacteristic
221C2C2×#226411yescharacteristic
222C2#3398411yescharacteristic
subgroupG/H
center, socleC22C22×D16
commutator, Frattini, cosocleC2×C16C24
prime radicalC2×D16C23
LCSG ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C4×Q8 ◁ C8×Q8 ◁ Q8×C16 ◁ G
LECSG ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33962C2semidirect
4. #13726C2semidirect
6. #13688C2semidirect
7. #13688C2semidirect
9. #13660C2semidirect
10. #13653C2semidirect

Maximal subgroups

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

subgroupindex
#339622
#339602
#137302
#137262
#136902
#136882
#136872
#136602
#136532
C4×SD642