Finite Group Databasebeta

← #135117

Subgroups — #135117

Order 512.

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

normal subgroups only

This group has 253 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#17970C211yesmaximal, characteristic
2#12014C211yesmaximal, characteristic, complemented
3#11862C211yesmaximal, characteristic
4#11858C211yesmaximal, characteristic
5#8792C211yesmaximal, characteristic, complemented
6#7224C211yesmaximal, characteristic, complemented
7#7218C211yesmaximal, characteristic, complemented
8#3169C2211yescharacteristic
9#2981—21no—
10#2163C2211yescharacteristic
11#2162C2211yescharacteristic
12#1837C2211yescharacteristic
13#1835C2211yescharacteristic
14#1594C412yescomplemented
15#1589—21no—
16#1580C412yescomplemented
17C8×D8—21no—
18#1568C2211yescharacteristic
19#1496—21no—
20#1450C2211yescharacteristic
21#1370—21no—
22#1367—21no—
23#1349—22no—
24#1347—22no—
25#1286—21no—
26#1286—21no—
27#532—21no—
28#500Q811yescharacteristic
29#493Q811yescharacteristic
30#486—21no—
31#466—21no—
32#466D412yes—
33C4×D8—21no—
34C4×D8C2×C412yes—
35#436—22no—
36#435—42no—
37C8×D4—22no—
38#432—21no—
39#431D411yescharacteristic
40#431D411yescharacteristic
41#423—21no—
42#408—42no—
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—
57D4⋊C8—22no—
58D4⋊C8—22no—
59#322C2×C411yescharacteristic
60#322C2×C411yescharacteristic
61C82D411yescharacteristic
62C82D411yescharacteristic
63C2×#41—41no—
64C2×D8—21no—
65C2×D8C4⋊C412yes—
66C2×M16—42no—
67C2×M16—41no—
68C22×C8—42no—
69C22×C8C2×D411yescharacteristic
70C22×C8#4111yescharacteristic
71C4⋊Q8—21no—
72#127—21no—
73#124—41no—
74#121—22no—
75#121—41no—
76C4×D4—22no—
77C4×D4—41no—
78C2×C4⋊C4—41no—
79C2×C42C2×D411yescharacteristic
80C2×C16—41no—
81C2×C16—41no—
82#107—21no—
83#107C4⋊C412yes—
84C4⋊C8—22no—
85C4⋊C8—21no—
86C4⋊C8—21no—
87C4⋊C8—41no—
88C4⋊C8—41no—
89#102—22no—
90#102—22no—
91C22⋊C8—21no—
92C22⋊C8—21no—
93C22⋊C8—42no—
94C22⋊C8—42no—
95C22⋊C8—41no—
96#97—41no—
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—
107#41—162no—
108C2×Q8—41no—
109C2×D4—22no—
110C2×D4—42no—
111C2×D4—41no—
112C2×D4—41no—
113C22×C4—42no—
114C22×C4—81no—
115C22×C4#12111yescharacteristic
116C22×C4#13511yescharacteristic
117C22×C4AGL(1,ℤ/8)11yescharacteristic
118D8—42no—
119D8—42no—
120M16—81no—
121M16—81no—
122M16—162no—
123C2×C8—21no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—41no—
128C2×C8—41no—
129C2×C8—42no—
130C2×C8—81no—
131C2×C8—162no—
132C2×C8C2×C4⋊C411yescharacteristic
133C2×C8#11711yescharacteristic
134C2×C8#12111yescharacteristic
135C2×C8C22⋊Q811yescharacteristic
136C4⋊C4—22no—
137C4⋊C4—41no—
138C4⋊C4—81no—
139C4⋊C4—81no—
140C22⋊C4—42no—
141C22⋊C4—42no—
142C42C2×C22⋊C411yescharacteristic
143C42C22⋊Q811yescharacteristic
144C42#13511yescharacteristic
145C42AGL(1,ℤ/8)11yescharacteristic
146C16—81no—
147C16—81no—
148C23—42no—
149C23#46311yescharacteristic
150Q8—161no—
151D4—42no—
152D4—42no—
153D4—161no—
154D4—161no—
155D4—162no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—42no—
161C2×C4—82no—
162C2×C4—82no—
163C2×C4—161no—
164C2×C4—161no—
165C2×C4—161no—
166C2×C4#38011yescharacteristic
167C2×C4#41611yescharacteristic
168C2×C4#41811yescharacteristic
169C2×C4#46311yescharacteristic
170C2×C4#47811yescharacteristic
171C2×C4#47811yescharacteristic
172C8—21no—
173C8—21no—
174C8—21no—
175C8—21no—
176C8—41no—
177C8—41no—
178C8—41no—
179C8—41no—
180C8—82no—
181C8—161no—
182C22—21no—
183C22—21no—
184C22—21no—
185C22—82no—
186C22—82no—
187C22—82no—
188C22#180411yescharacteristic
189C4—21no—
190C4—21no—
191C4—21no—
192C4—21no—
193C4—21no—
194C4—82no—
195C4—321no—
196C4#181211yescharacteristic
197C4#181411yescharacteristic
198C2—41no—
199C2—162no—
200C2#867811yescharacteristic
201C2#1282611yescharacteristic
202C2#1283011yescharacteristic
subgroupG/H
center, socleC22#1804
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radical#3169C22
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
2. #12014C2semidirect
5. #8792C2semidirect
6. #7224C2semidirect
7. #7218C2semidirect
14. #1594C4semidirect
16. #1580C4semidirect

Maximal subgroups

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

subgroupindex
#179702
#120142
#118622
#118582
#87922
#72242
#72182