Finite Group Databasebeta

← #135121

Subgroups — #135121

Order 512.

233 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 233 conjugacy classes of subgroups — too many to draw. Only its 74 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#17972C211yesmaximal, characteristic
2#12014C211yesmaximal, characteristic
3#11863C211yesmaximal, characteristic
4#11859C211yesmaximal, characteristic
5#8791C211yesmaximal, characteristic
6#7224C211yesmaximal, characteristic
7#7218C211yesmaximal, characteristic
8#3170C2211yescharacteristic
9#2985—21no—
10#2163C2211yescharacteristic
11#2162C2211yescharacteristic
12#1837C2211yescharacteristic
13#1835C2211yescharacteristic
14#1593C412yescomplemented
15#1591—21no—
16#1579C412yescomplemented
17C8×Q16—21no—
18#1568C2211yescharacteristic
19#1497—21no—
20#1450C2211yescharacteristic
21#1370—21no—
22#1367—21no—
23#1350—22no—
24#1348—22no—
25#1286—21no—
26#1286—21no—
27#533—21no—
28#500Q811yescharacteristic
29#494Q811yescharacteristic
30#487—21no—
31#467—21no—
32#467D412yes—
33#446—22no—
34#446—42no—
35C8×Q8—22no—
36C4×Q16—21no—
37C4×Q16C2×C412yes—
38#433—42no—
39#432—21no—
40#431D411yescharacteristic
41#431D411yescharacteristic
42#423—21no—
43C2×C4×C8C2311yescharacteristic
44#382—21no—
45C4⋊C16—21no—
46C4⋊C16—21no—
47C22⋊C16—21no—
48C22⋊C16—21no—
49#346—21no—
50C4×C16—21no—
51#341—21no—
52#341—21no—
53#340—21no—
54#340—21no—
55#335—21no—
56#334C2×C412yes—
57Q8⋊C8—22no—
58Q8⋊C8—22no—
59#322C2×C411yescharacteristic
60#322C2×C411yescharacteristic
61C82D411yescharacteristic
62C82D411yescharacteristic
63C2×Q16—21no—
64C2×Q16C4⋊C412yes—
65C2×M16—41no—
66C22×C8C2×D411yescharacteristic
67C22×C8#4111yescharacteristic
68C4⋊Q8—21no—
69C4⋊Q8—21no—
70#125—41no—
71C22⋊Q8—22no—
72C22⋊Q8—41no—
73C4×Q8—22no—
74C4×Q8—41no—
75#117—41no—
76C2×C4⋊C4—41no—
77C2×C42C2×D411yescharacteristic
78C2×C16—41no—
79C2×C16—41no—
80#107—21no—
81#107C4⋊C412yes—
82C4⋊C8—22no—
83C4⋊C8—21no—
84C4⋊C8—21no—
85C4⋊C8—42no—
86C4⋊C8—42no—
87C4⋊C8—41no—
88C4⋊C8—41no—
89#103—22no—
90#103—22no—
91C22⋊C8—21no—
92C22⋊C8—21no—
93C22⋊C8—41no—
94#97—42no—
95#97—41no—
96C4×C8—42no—
97C4×C8C22⋊C411yescharacteristic
98C4×C8C22⋊C411yescharacteristic
99C4×C8C22⋊C411yescharacteristic
100C4×C8C22⋊C411yescharacteristic
101C4×C8C22×C411yescharacteristic
102C4×C8C2×D411yescharacteristic
103C4×C8C2×Q811yescharacteristic
104C4×C8#4111yescharacteristic
105#95—41no—
106#95—41no—
107C2×Q8—22no—
108C2×Q8—41no—
109C22×C4—81no—
110C22×C4#12111yescharacteristic
111C22×C4#13511yescharacteristic
112C22×C4AGL(1,ℤ/8)11yescharacteristic
113Q16—42no—
114Q16—42no—
115M16—81no—
116M16—81no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—21no—
120C2×C8—21no—
121C2×C8—41no—
122C2×C8—41no—
123C2×C8—42no—
124C2×C8—81no—
125C2×C8C2×C4⋊C411yescharacteristic
126C2×C8#11711yescharacteristic
127C2×C8#12111yescharacteristic
128C2×C8C22⋊Q811yescharacteristic
129C4⋊C4—22no—
130C4⋊C4—42no—
131C4⋊C4—42no—
132C4⋊C4—41no—
133C4⋊C4—41no—
134C4⋊C4—41no—
135C4⋊C4—81no—
136C4⋊C4—81no—
137C22⋊C4—42no—
138C42—42no—
139C42C2×C22⋊C411yescharacteristic
140C42C22⋊Q811yescharacteristic
141C42#13511yescharacteristic
142C42AGL(1,ℤ/8)11yescharacteristic
143C16—81no—
144C16—81no—
145C23#46311yescharacteristic
146Q8—42no—
147Q8—42no—
148Q8—161no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—161no—
156C2×C4—161no—
157C2×C4—161no—
158C2×C4#38011yescharacteristic
159C2×C4#41611yescharacteristic
160C2×C4#41811yescharacteristic
161C2×C4#46311yescharacteristic
162C2×C4#47811yescharacteristic
163C2×C4#47811yescharacteristic
164C8—21no—
165C8—21no—
166C8—21no—
167C8—21no—
168C8—41no—
169C8—41no—
170C8—41no—
171C8—41no—
172C8—82no—
173C8—161no—
174C22—21no—
175C22—21no—
176C22—21no—
177C22#180411yescharacteristic
178C4—21no—
179C4—21no—
180C4—21no—
181C4—21no—
182C4—21no—
183C4—82no—
184C4—82no—
185C4—321no—
186C4#181211yescharacteristic
187C4#181411yescharacteristic
188C2—41no—
189C2#868011yescharacteristic
190C2#1282711yescharacteristic
191C2#1283011yescharacteristic
subgroupG/H
center, socleC22#1804
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC23#463
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1450 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
14. #1593C4semidirect
16. #1579C4semidirect

Maximal subgroups

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

subgroupindex
#179722
#120142
#118632
#118592
#87912
#72242
#72182