Finite Group Databasebeta

← #492792

Subgroups — #492792

Order 512.

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

normal subgroups only

This group has 409 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#33951C212yesmaximal
2#19993C212yesmaximal, complemented
3#12522C211yesmaximal, characteristic, complemented
4#12462C212yesmaximal, complemented
5#12449C212yesmaximal, complemented
6#12411C212yesmaximal, complemented
7#12404C212yesmaximal, complemented
8#12054C211yesmaximal, characteristic, complemented
9C8×D16C211yesmaximal, characteristic
10#3380C2212yes—
11Q8×D8C2212yes—
12#3299—22no—
13#2255—22no—
14#2254C2212yes—
15#2252C2211yescharacteristic
16#2252C2211yescharacteristic
17#2228—22no—
18#2226—22no—
19#2226C2212yes—
20#2224C2212yes—
21#2183C2212yes—
22#2177—22no—
23C4×D16C2211yescharacteristic
24#1710C2212yes—
25#1683—22no—
26#1676C2212yes—
27#1674C2212yes—
28#1644—22no—
29#1637—22no—
30C8×Q16C2211yescharacteristic
31C8×D8C2212yes—
32#1371C2211yescharacteristic
33#1370C2211yescharacteristic
34#1365C2212yes—
35#1364C2212yes—
36Q8⋊C16C2212yes—
37#1331C2212yes—
38C8×C16C2211yescharacteristic
39#554—22no—
40D4×Q8—22no—
41C2×D16D411yescharacteristic
42#500—22no—
43#500C2312yes—
44#500C2312yes—
45#499—22no—
46#487—22no—
47#487—22no—
48#484—42no—
49#478—42no—
50#477—22no—
51#476—22no—
52#474—22no—
53#474—42no—
54#467—42no—
55#462—22no—
56#451—42no—
57C8×Q8C2312yes—
58C4×Q16D411yescharacteristic
59C4×D8—22no—
60C4×D8—22no—
61C4×D8C2312yes—
62C8×D4—22no—
63C2×#103—42no—
64#367—22no—
65#367—22no—
66#367—22no—
67#366—22no—
68#366—41no—
69#366—41no—
70#366D411yescharacteristic
71C4⋊C16—22no—
72C4⋊C16C2312yes—
73#357—22no—
74#357—22no—
75#357D412yes—
76#346—22no—
77C4×C16C2311yescharacteristic
78C4×C16C2311yescharacteristic
79#335D411yescharacteristic
80#335C2312yes—
81#333—22no—
82#332—22no—
83Q8⋊C8D412yes—
84D4⋊C8—22no—
85C82C2311yescharacteristic
86C22×Q8—42no—
87C2×Q16—42no—
88C2×Q16D811yescharacteristic
89C2×D8C2×D412yes—
90C22×C8—42no—
91C4⋊Q8—22no—
92C4⋊Q8—22no—
93C4⋊Q8—42no—
94#125—42no—
95C22⋊Q8—42no—
96C22⋊Q8—42no—
97C22⋊Q8—82no—
98C22⋊Q8—82no—
99C4×Q8C2×D412yes—
100C4×D4—22no—
101C4×D4—42no—
102C4×D4—42no—
103C2×C4⋊C4—42no—
104D16Q1614yescomplemented
105C2×C16—21no—
106C2×C16—22no—
107C2×C16C2×D411yescharacteristic
108C2×C16C2×D411yescharacteristic
109#107—22no—
110#107—22no—
111#107—22no—
112#107—22no—
113#107—42no—
114#107—42no—
115#107D811yescharacteristic
116#107C2×D412yes—
117#106—42no—
118C4⋊C8—22no—
119C4⋊C8—22no—
120C4⋊C8C2×D412yes—
121#103—42no—
122#103—42no—
123#103—42no—
124#103—42no—
125#103—82no—
126#103D812yes—
127#102—22no—
128#102—42no—
129#102—42no—
130C22⋊C8—42no—
131C4×C8—22no—
132C4×C8C2×D411yescharacteristic
133C4×C8C2×D411yescharacteristic
134C4×C8C2411yescharacteristic
135C2×Q8—42no—
136C2×Q8—42no—
137C2×Q8—82no—
138C2×Q8—84no—
139C2×Q8—162no—
140C2×Q8C2×D812yes—
141C2×D4—22no—
142C22×C4—42no—
143C22×C4—82no—
144C22×C4—82no—
145Q16—41no—
146Q16—162no—
147D8—22no—
148D8C2×Q1614yes—
149C2×C8—22no—
150C2×C8—21no—
151C2×C8—42no—
152C2×C8—162no—
153C2×C8C2×D811yescharacteristic
154C2×C8C2×D811yescharacteristic
155C2×C8C22×D411yescharacteristic
156C2×C8H(5,𝔽2)11yescharacteristic
157C4⋊C4—22no—
158C4⋊C4—22no—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C4⋊C4—42no—
162C4⋊C4—42no—
163C4⋊C4—82no—
164C4⋊C4—82no—
165C4⋊C4—82no—
166C4⋊C4C2×D812yes—
167C22⋊C4—42no—
168C22⋊C4—82no—
169C22⋊C4—82no—
170C42—22no—
171C42C22×D411yescharacteristic
172C16—22no—
173C16—21no—
174C16—42no—
175C16C2×Q1612yes—
176C23—42no—
177Q8—22no—
178Q8—22no—
179Q8—82no—
180Q8—82no—
181Q8—162no—
182Q8—164no—
183D4—24no—
184D4—42no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—42no—
188C2×C4—82no—
189C2×C4—82no—
190C2×C4—82no—
191C2×C4—82no—
192C2×C4—164no—
193C2×C4—322no—
194C2×C4D4211yescharacteristic
195C2×C4C22×D811yescharacteristic
196C2×C4#57611yescharacteristic
197C8—21no—
198C8—21no—
199C8—21no—
200C8—41no—
201C8—42no—
202C8—82no—
203C8C22×Q1611yescharacteristic
204C8#57811yescharacteristic
205C22—44no—
206C22—82no—
207C22—82no—
208C22D4×D811yescharacteristic
209C4—21no—
210C4—42no—
211C4—42no—
212C4—82no—
213C4—162no—
214C4—162no—
215C4D4×Q1611yescharacteristic
216C4#329311yescharacteristic
217C4#341411yescharacteristic
218C4#341511yescharacteristic
219C2—84no—
220C2D8×Q1611yescharacteristic
221C2#3383911yescharacteristic
222C2#3385711yescharacteristic
subgroupG/H
center, socleC22D4×D8
commutator, Frattini, cosocleC4×C8C24
prime radicalC2×D16D4
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
2. #19993C2semidirect
3. #12522C2semidirect
4. #12462C2semidirect
5. #12449C2semidirect
6. #12411C2semidirect
7. #12404C2semidirect
8. #12054C2semidirect
104. D16Q16semidirect

Maximal subgroups

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

subgroupindex
#339512
#199932
#125222
#124622
#124492
#124112
#124042
#120542
C8×D162