Finite Group Databasebeta

← #146186

Subgroups — #146186

Order 512.

213 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 213 conjugacy classes of subgroups — too many to draw. Only its 56 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#12034C211yesmaximal, characteristic, complemented
2#12031C211yesmaximal, characteristic, complemented
3#11667C211yesmaximal, characteristic
4#11661C211yesmaximal, characteristic
5#9670C211yesmaximal, characteristic
6#8798C211yesmaximal, characteristic, complemented
7#7147C211yesmaximal, characteristic, complemented
8C2×C4×D8C2211yescharacteristic
9#2163C2211yescharacteristic
10#2162C2211yescharacteristic
11#1568C2211yescharacteristic
12#1568C2211yescharacteristic
13#1564C2211yescharacteristic
14#1478—21no—
15#1460—21no—
16#1452C2211yescharacteristic
17#1372—21no—
18#1371—21no—
19#1368—22no—
20#1333—22no—
21#1331—21no—
22#1331—21no—
23#1287—21no—
24#1287—21no—
25C22×D8Q811yescharacteristic
26C2×C4×D4—21no—
27C4×D8—21no—
28C4×D8—21no—
29C4×D8—21no—
30C4×D8—21no—
31#432—21no—
32#432—21no—
33#431—21no—
34#431D411yescharacteristic
35#431D411yescharacteristic
36C2×#107Q811yescharacteristic
37C2×#102—21no—
38C2×C4×C8C2311yescharacteristic
39C4⋊C16—21no—
40C4⋊C16—21no—
41C22⋊C16—21no—
42C22⋊C16—21no—
43#346D412yes—
44C4×C16D411yescharacteristic
45C4×C16D411yescharacteristic
46#335—21no—
47#335—21no—
48#335—21no—
49#334—21no—
50#334—21no—
51#334—21no—
52D4⋊C8—42no—
53#323—42no—
54#322—21no—
55#322—21no—
56C22×D4—21no—
57C23×C4—41no—
58C2×D8—21no—
59C2×D8—21no—
60C2×D8—42no—
61C22×C8C2×Q811yescharacteristic
62C22×C8#4111yescharacteristic
63C4×D4—41no—
64C4×D4—41no—
65C4×D4—41no—
66C4×D4—41no—
67C2×C4⋊C4—21no—
68C2×C22⋊C4—41no—
69C2×C42C2×D411yescharacteristic
70C2×C16—21no—
71C2×C16—21no—
72C2×C16—22no—
73#107—21no—
74#107—21no—
75C4⋊C8—21no—
76C4⋊C8—21no—
77C4⋊C8—41no—
78C4⋊C8—41no—
79C4⋊C8—41no—
80#102—41no—
81#102—41no—
82C22⋊C8—21no—
83C22⋊C8—21no—
84C22⋊C8—41no—
85C22⋊C8—41no—
86C22⋊C8—41no—
87#97—41no—
88#97—41no—
89C4×C8—21no—
90C4×C8—21no—
91C4×C8—41no—
92C4×C8C2×D411yescharacteristic
93C4×C8C2×D411yescharacteristic
94C4×C8#4111yescharacteristic
95C4×C8#4111yescharacteristic
96C24—41no—
97C2×D4—22no—
98C2×D4—41no—
99C2×D4—41no—
100C2×D4—41no—
101C2×D4—41no—
102C2×D4—41no—
103C22×C4—41no—
104C22×C4—81no—
105C22×C4—81no—
106C22×C4—162no—
107C22×C4C22⋊Q811yescharacteristic
108C22×C4#13511yescharacteristic
109C22×C4AGL(1,ℤ/8)11yescharacteristic
110D8—22no—
111D8—22no—
112D8—41no—
113D8—41no—
114C2×C8—41no—
115C2×C8—41no—
116C2×C8—41no—
117C2×C8—41no—
118C2×C8—81no—
119C2×C8—81no—
120C2×C8—81no—
121C2×C8C22⋊Q811yescharacteristic
122C2×C8C22⋊Q811yescharacteristic
123C2×C8#12311yescharacteristic
124C2×C8#12311yescharacteristic
125C4⋊C4—41no—
126C4⋊C4—41no—
127C22⋊C4—81no—
128C22⋊C4—81no—
129C42#12311yescharacteristic
130C42#12711yescharacteristic
131C42#13511yescharacteristic
132C42AGL(1,ℤ/8)11yescharacteristic
133C16—41no—
134C16—41no—
135C16—42no—
136C23—42no—
137C23—81no—
138C23—81no—
139C23—161no—
140C23—161no—
141C23#47811yescharacteristic
142D4—42no—
143D4—42no—
144D4—81no—
145D4—81no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—81no—
151C2×C4—162no—
152C2×C4—162no—
153C2×C4—161no—
154C2×C4#39911yescharacteristic
155C2×C4#47811yescharacteristic
156C2×C4#48211yescharacteristic
157C2×C4#48211yescharacteristic
158C2×C4#49511yescharacteristic
159C2×C4#49611yescharacteristic
160C8—21no—
161C8—21no—
162C8—21no—
163C8—21no—
164C8—81no—
165C8—81no—
166C8—161no—
167C8—161no—
168C8—161no—
169C22—21no—
170C22—21no—
171C22—21no—
172C22—82no—
173C22—162no—
174C22—162no—
175C22—161no—
176C22—161no—
177C22#209211yescharacteristic
178C4—21no—
179C4—21no—
180C4—21no—
181C4—21no—
182C4—21no—
183C4—161no—
184C4#209111yescharacteristic
185C4#209811yescharacteristic
186C2—41no—
187C2—162no—
188C2#1065911yescharacteristic
189C2#1352311yescharacteristic
190C2#1352511yescharacteristic
subgroupG/H
center, socleC22#2092
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC22×D8Q8
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1452 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #12034C2semidirect
2. #12031C2semidirect
6. #8798C2semidirect
7. #7147C2semidirect

Maximal subgroups

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

subgroupindex
#120342
#120312
#116672
#116612
#96702
#87982
#71472