Finite Group Databasebeta

← #492436

Subgroups — #492436

Order 512.

391 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 391 conjugacy classes of subgroups — too many to draw. Only its 124 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#33930C212yesmaximal
2#19932C212yesmaximal
3#12504C211yesmaximal, characteristic
4#12488C212yesmaximal
5#12477C212yesmaximal
6#12410C212yesmaximal
7#12334C212yesmaximal
8#12083C211yesmaximal, characteristic
9C16×Q16C211yesmaximal, characteristic
10#3395C2212yes—
11#3387—22no—
12Q8×Q16C2212yes—
13#2252C2211yescharacteristic
14#2247C2211yescharacteristic
15#2241—22no—
16#2241C2212yes—
17#2227—22no—
18#2227C2212yes—
19#2209—22no—
20#2209C2212yes—
21Q8×C16C2212yes—
22C4×Q32—22no—
23C4×Q32C2211yescharacteristic
24#1707C2212yes—
25#1692—22no—
26#1676C2212yes—
27#1670C2212yes—
28#1645—22no—
29#1636—22no—
30#1591C2212yes—
31C8×Q16C2211yescharacteristic
32#1371C2211yescharacteristic
33#1370C2211yescharacteristic
34#1365C2212yes—
35#1362C2212yes—
36Q8⋊C16C2212yes—
37#1332C2212yes—
38C8×C16C2211yescharacteristic
39Q82—22no—
40#557—22no—
41C2×Q32—22no—
42C2×Q32—41no—
43C2×Q32D411yescharacteristic
44#501—42no—
45#500—22no—
46#500C2312yes—
47#498—22no—
48#494—22no—
49#494C2312yes—
50#487—22no—
51#479—42no—
52#477—22no—
53#477—22no—
54#477—42no—
55#475—22no—
56#475—42no—
57#462—22no—
58#462—42no—
59#446—22no—
60C8×Q8C2312yes—
61#441—42no—
62C4×Q16—22no—
63C4×Q16—22no—
64C4×Q16D411yescharacteristic
65C4×Q16C2312yes—
66#366—22no—
67#366—41no—
68#366D411yescharacteristic
69C4⋊C16—22no—
70C4⋊C16C2312yes—
71#358—22no—
72#358—22no—
73#358—22no—
74#358—22no—
75#358D412yes—
76C4×C16—22no—
77C4×C16C2311yescharacteristic
78C4×C16C2311yescharacteristic
79#335D411yescharacteristic
80#335C2312yes—
81#333—22no—
82#332—22no—
83Q8⋊C8—22no—
84Q8⋊C8D412yes—
85C82C2311yescharacteristic
86C2×Q16—22no—
87C2×Q16—22no—
88C2×Q16—42no—
89C2×Q16D811yescharacteristic
90C2×Q16C2×D412yes—
91C4⋊Q8—22no—
92C4⋊Q8—22no—
93C4⋊Q8—42no—
94C4⋊Q8—42no—
95C4⋊Q8—42no—
96C4⋊Q8—82no—
97#125—42no—
98#125—82no—
99C4×Q8—22no—
100C4×Q8—42no—
101C4×Q8—42no—
102C4×Q8—42no—
103C4×Q8—42no—
104C4×Q8C2×D412yes—
105Q32—41no—
106Q32—44no—
107Q32—82no—
108C2×C16—21no—
109C2×C16—22no—
110C2×C16C2×D411yescharacteristic
111C2×C16C2×D411yescharacteristic
112#107—22no—
113#107—22no—
114#107—42no—
115#107—42no—
116#107D811yescharacteristic
117#107C2×D412yes—
118#106—42no—
119#106—42no—
120#106—42no—
121C4⋊C8—22no—
122C4⋊C8—22no—
123C4⋊C8—42no—
124C4⋊C8C2×D412yes—
125#103—22no—
126#103—42no—
127#103—42no—
128#103—42no—
129#103—42no—
130#103—82no—
131#103D812yes—
132#97—42no—
133C4×C8—22no—
134C4×C8C2×D411yescharacteristic
135C4×C8C2×D411yescharacteristic
136C4×C8C2411yescharacteristic
137C2×Q8—22no—
138C2×Q8—42no—
139C2×Q8—42no—
140C2×Q8—82no—
141C2×Q8C2×D812yes—
142Q16—22no—
143Q16—22no—
144Q16—42no—
145Q16—42no—
146Q16—82no—
147Q16—162no—
148Q16Q3214yescomplemented
149C2×C8—22no—
150C2×C8—21no—
151C2×C8—42no—
152C2×C8C2×D811yescharacteristic
153C2×C8C2×D811yescharacteristic
154C2×C8C22×D411yescharacteristic
155C2×C8H(5,𝔽2)11yescharacteristic
156C4⋊C4—22no—
157C4⋊C4—22no—
158C4⋊C4—42no—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C4⋊C4—42no—
162C4⋊C4—42no—
163C4⋊C4—82no—
164C4⋊C4—82no—
165C4⋊C4—82no—
166C4⋊C4—82no—
167C4⋊C4—82no—
168C4⋊C4C2×D812yes—
169C42—22no—
170C42—42no—
171C42—82no—
172C42—82no—
173C42C22×D411yescharacteristic
174C16—22no—
175C16—21no—
176C16—21no—
177C16—42no—
178Q8—22no—
179Q8—42no—
180Q8—42no—
181Q8—82no—
182Q8—82no—
183Q8—162no—
184Q8—322no—
185Q8C2×Q3214yes—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—82no—
191C2×C4—82no—
192C2×C4D4211yescharacteristic
193C2×C4C22×D811yescharacteristic
194C2×C4#57611yescharacteristic
195C8—21no—
196C8—21no—
197C8—41no—
198C8—42no—
199C8—82no—
200C8C2×Q3212yes—
201C8#57511yescharacteristic
202C8#57811yescharacteristic
203C22D4×D811yescharacteristic
204C4—24no—
205C4—21no—
206C4—42no—
207C4—82no—
208C4—82no—
209C4—162no—
210C4—162no—
211C4#327711yescharacteristic
212C4#329111yescharacteristic
213C4C22×Q3211yescharacteristic
214C4#341711yescharacteristic
215C2#1988811yescharacteristic
216C2D4×Q3211yescharacteristic
217C2#3385711yescharacteristic
subgroupG/H
center, socle, prime radicalC22D4×D8
commutator, Frattini, cosocleC4×C8C24
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q16 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
148. Q16Q32semidirect

Maximal subgroups

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

subgroupindex
#339302
#199322
#125042
#124882
#124772
#124102
#123342
#120832
C16×Q162