Finite Group Databasebeta

← #123299

Subgroups — #123299

Order 512.

232 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 52 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#20006C211yesmaximal, characteristic
2#12510C211yesmaximal, characteristic, complemented
3#12446C211yesmaximal, characteristic, complemented
4#12426C211yesmaximal, characteristic
5#7098C211yesmaximal, characteristic, complemented
6#7069C211yesmaximal, characteristic, complemented
7#7068C211yesmaximal, characteristic
8#3380—21no—
9#3338—21no—
10#2241—22no—
11#2225—21no—
12#2208—21no—
13#1713C2211yescharacteristic
14#1713C2211yescharacteristic
15#1676—21no—
16#1675—21no—
17#1632—22no—
18C8×Q16C2211yescharacteristic
19C8×D8C2211yescharacteristic
20#1370C2211yescharacteristic
21#1370C2211yescharacteristic
22#1364—21no—
23#1362—21no—
24C8×C16C2211yescharacteristic
25#554—22no—
26C2×SD32—41no—
27#500—21no—
28#500—21no—
29#500—21no—
30#500—21no—
31#499—41no—
32#494—21no—
33#494D412yes—
34#484—42no—
35#478—42no—
36#477—22no—
37#476—22no—
38#467—41no—
39C8×Q8—21no—
40C4×Q16D411yescharacteristic
41C4×D8—41no—
42C4×D8D411yescharacteristic
43C8×D4—21no—
44C2×#107—41no—
45#367—41no—
46C4⋊C16—21no—
47C4⋊C16—21no—
48#358—44no—
49#358—41no—
50#358—41no—
51C4×C16—21no—
52#335D411yescharacteristic
53#335D411yescharacteristic
54#333—22no—
55Q8⋊C8—21no—
56D4⋊C8—21no—
57C82C2311yescharacteristic
58C2×Q16—42no—
59C2×Q16—41no—
60C2×Q16—41no—
61C2×Q16D811yescharacteristic
62C2×D8D811yescharacteristic
63C22×C8—41no—
64C4⋊Q8—22no—
65C4⋊Q8—41no—
66#125—42no—
67C22⋊Q8—42no—
68C22⋊Q8—82no—
69C4×Q8—21no—
70C4×D4—21no—
71C4×D4—42no—
72C2×C4⋊C4—42no—
73SD32—82no—
74C2×C16—22no—
75C2×C16—41no—
76C2×C16—41no—
77#107—42no—
78#107—41no—
79#107—41no—
80#107—41no—
81#107—41no—
82#107—42no—
83#107—41no—
84#107—41no—
85#107D811yescharacteristic
86#107D811yescharacteristic
87#106—42no—
88C4⋊C8—21no—
89C4⋊C8—21no—
90C4⋊C8—41no—
91#103—21no—
92#103—42no—
93#102—21no—
94#102—42no—
95C22⋊C8—41no—
96C4×C8—41no—
97C4×C8C2×D411yescharacteristic
98C4×C8C2×D411yescharacteristic
99C4×C8C2×D411yescharacteristic
100C2×Q8—21no—
101C2×Q8—42no—
102C2×Q8—81no—
103C2×D4—21no—
104C22×C4—41no—
105C22×C4—82no—
106Q16—21no—
107Q16—21no—
108Q16—84no—
109Q16—81no—
110Q16—81no—
111Q16—161no—
112D8—21no—
113D8SD3212yes—
114C2×C8—21no—
115C2×C8—41no—
116C2×C8—41no—
117C2×C8—81no—
118C2×C8—81no—
119C2×C8C2×D811yescharacteristic
120C2×C8C2×D811yescharacteristic
121C2×C8AGL(1,ℤ/8)11yescharacteristic
122C2×C8AGL(1,ℤ/8)11yescharacteristic
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—41no—
126C4⋊C4—42no—
127C4⋊C4—42no—
128C4⋊C4—82no—
129C4⋊C4—82no—
130C4⋊C4—81no—
131C4⋊C4—81no—
132C22⋊C4—41no—
133C22⋊C4—82no—
134C42—41no—
135C42#12011yescharacteristic
136C16—42no—
137C16—81no—
138C16—81no—
139C23—41no—
140Q8—41no—
141Q8—41no—
142Q8—162no—
143Q8—161no—
144Q8—161no—
145D4—22no—
146D4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—81no—
151C2×C4—81no—
152C2×C4—82no—
153C2×C4—161no—
154C2×C4—322no—
155C2×C4#44711yescharacteristic
156C2×C4#44711yescharacteristic
157C2×C4#45311yescharacteristic
158C8—22no—
159C8—21no—
160C8—21no—
161C8—81no—
162C8—81no—
163C8C2×SD3211yescharacteristic
164C8#50811yescharacteristic
165C8#51011yescharacteristic
166C8#51011yescharacteristic
167C22—42no—
168C22—81no—
169C22—81no—
170C22#161911yescharacteristic
171C4—21no—
172C4—81no—
173C4—81no—
174C4—81no—
175C4—162no—
176C4—321no—
177C4#218511yescharacteristic
178C4#218911yescharacteristic
179C4#219311yescharacteristic
180C4#219511yescharacteristic
181C2—82no—
182C2#1231211yescharacteristic
183C2#1231811yescharacteristic
184C2#1232311yescharacteristic
subgroupG/H
center, socleC22#1619
commutator, Frattini, cosocleC82C23
prime radicalC2×D8D8
LCSG ▷ C82 ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ C82 ◁ G
LECSG ▷ C82 ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12510C2semidirect
3. #12446C2semidirect
5. #7098C2semidirect
6. #7069C2semidirect

Maximal subgroups

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

subgroupindex
#200062
#125102
#124462
#124262
#70982
#70692
#70682