Finite Group Databasebeta

← #135071

Subgroups — #135071

Order 512.

240 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 76 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#17897C211yesmaximal, characteristic
2#12014C211yesmaximal, characteristic
3#11668C211yesmaximal, characteristic
4#11662C211yesmaximal, characteristic
5#8797C211yesmaximal, characteristic
6#7293C211yesmaximal, characteristic
7#7289C211yesmaximal, characteristic
8#2983—21no—
9C2×C4×Q16C2211yescharacteristic
10#2163C2211yescharacteristic
11#2162C2211yescharacteristic
12#1593C412yes—
13#1593C412yes—
14#1591—21no—
15#1591—21no—
16#1568C2211yescharacteristic
17#1568C2211yescharacteristic
18#1564C2211yescharacteristic
19#1479—21no—
20#1450C2211yescharacteristic
21#1370—21no—
22#1367—21no—
23#1334—22no—
24#1332—22no—
25#1287—21no—
26#1287—21no—
27C22×Q16Q811yescharacteristic
28C2×C4×Q8—21no—
29#446—22no—
30#446—22no—
31#446—42no—
32C4×Q16—21no—
33C4×Q16—21no—
34C4×Q16C2×C412yes—
35C4×Q16D412yes—
36#432—21no—
37#432—21no—
38#432—42no—
39#431—21no—
40#431D411yescharacteristic
41#431D411yescharacteristic
42C2×#107Q811yescharacteristic
43C2×#103—21no—
44C2×C4×C8C2311yescharacteristic
45C4⋊C16—21no—
46C4⋊C16—21no—
47C22⋊C16—21no—
48C22⋊C16—21no—
49#346—21no—
50C4×C16—21no—
51#335—21no—
52#335—21no—
53#335—21no—
54#334—21no—
55#334—21no—
56#334C2×C412yes—
57Q8⋊C8—22no—
58Q8⋊C8—22no—
59#322C2×C411yescharacteristic
60#322C2×C411yescharacteristic
61C82D411yescharacteristic
62C82D411yescharacteristic
63C22×Q8—21no—
64C2×Q16—21no—
65C2×Q16—42no—
66C2×Q16C4⋊C412yes—
67C22×C8C2×Q811yescharacteristic
68C22×C8#4111yescharacteristic
69C4×Q8—22no—
70C4×Q8—22no—
71C4×Q8—41no—
72C4×Q8—41no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—41no—
75C2×C42—41no—
76C2×C42C2×D411yescharacteristic
77C2×C16—41no—
78C2×C16—41no—
79#107—21no—
80#107C4⋊C412yes—
81C4⋊C8—22no—
82C4⋊C8—21no—
83C4⋊C8—21no—
84C4⋊C8—42no—
85C4⋊C8—42no—
86C4⋊C8—41no—
87C4⋊C8—41no—
88#103—22no—
89#103—22no—
90C22⋊C8—21no—
91C22⋊C8—21no—
92C22⋊C8—41no—
93C22⋊C8—41no—
94C22⋊C8—41no—
95#97—42no—
96#97—42no—
97#97—41no—
98#97—41no—
99C4×C8—41no—
100C4×C8C22⋊C411yescharacteristic
101C4×C8C22⋊C411yescharacteristic
102C4×C8C22⋊C411yescharacteristic
103C4×C8C22⋊C411yescharacteristic
104C4×C8C22×C411yescharacteristic
105C4×C8C2×D411yescharacteristic
106C4×C8C2×D411yescharacteristic
107C4×C8#4111yescharacteristic
108C2×Q8—22no—
109C2×Q8—42no—
110C2×Q8—42no—
111C2×Q8—41no—
112C22×C4—41no—
113C22×C4—41no—
114C22×C4C22⋊Q811yescharacteristic
115C22×C4#13511yescharacteristic
116C22×C4AGL(1,ℤ/8)11yescharacteristic
117Q16—24no—
118Q16—42no—
119C2×C8—22no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—21no—
124C2×C8—41no—
125C2×C8—42no—
126C2×C8—81no—
127C2×C8—81no—
128C2×C8C2×C4⋊C411yescharacteristic
129C2×C8#11711yescharacteristic
130C2×C8#12111yescharacteristic
131C2×C8C22⋊Q811yescharacteristic
132C4⋊C4—22no—
133C4⋊C4—42no—
134C4⋊C4—42no—
135C4⋊C4—41no—
136C42—42no—
137C42—42no—
138C42C2×C22⋊C411yescharacteristic
139C42#12111yescharacteristic
140C42#13511yescharacteristic
141C42AGL(1,ℤ/8)11yescharacteristic
142C16—81no—
143C16—81no—
144C23#47811yescharacteristic
145Q8—42no—
146Q8—42no—
147Q8—82no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—42no—
153C2×C4—42no—
154C2×C4—82no—
155C2×C4—161no—
156C2×C4#38011yescharacteristic
157C2×C4#41611yescharacteristic
158C2×C4#41811yescharacteristic
159C2×C4#46311yescharacteristic
160C2×C4#46311yescharacteristic
161C2×C4#47811yescharacteristic
162C8—21no—
163C8—21no—
164C8—21no—
165C8—41no—
166C8—41no—
167C8—41no—
168C8—41no—
169C8—82no—
170C8—161no—
171C8—161no—
172C8#43012yes—
173C22—21no—
174C22—21no—
175C22—21no—
176C22#180411yescharacteristic
177C4—21no—
178C4—21no—
179C4—21no—
180C4—21no—
181C4—21no—
182C4—82no—
183C4—82no—
184C4#181111yescharacteristic
185C4#181411yescharacteristic
186C2—41no—
187C2#866411yescharacteristic
188C2#1282711yescharacteristic
189C2#1283011yescharacteristic
subgroupG/H
center, socleC22#1804
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC23#478
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1450 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#178972
#120142
#116682
#116622
#87972
#72932
#72892