Finite Group Databasebeta

← #136769

Subgroups — #136769

Order 512.

339 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 339 conjugacy classes of subgroups — too many to draw. Only its 92 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#33493C211yesmaximal, characteristic
2#21658C211yesmaximal, characteristic
3#11638C212yesmaximal, complemented
4#11490C211yesmaximal, characteristic
5#7148C212yesmaximal, complemented
6#3131C411yescharacteristic
7#2960C2211yescharacteristic
8#2931C2211yescharacteristic
9#2169—22no—
10D4×C16—22no—
11#2162C2212yes—
12#2162C2212yes—
13#2113—22no—
14#1994C411yescharacteristic
15#1962C411yescharacteristic
16#1937—22no—
17#1837C411yescharacteristic
18#1764—22no—
19C42×C8C2211yescharacteristic
20Q8⋊C16—22no—
21Q8⋊C16—22no—
22#1315—22no—
23C2×#131D412yes—
24#533C2×C411yescharacteristic
25#532C2×C411yescharacteristic
26C2×M32—22no—
27C22×C16—22no—
28#488—22no—
29#487C812yes—
30#486C812yes—
31#446—22no—
32C8×Q8—22no—
33#436—22no—
34C8×D4—22no—
35#424—21no—
36#423C2×C411yescharacteristic
37#423C2×C411yescharacteristic
38C2×#104—22no—
39#416—22no—
40#408D412yes—
41#405—21no—
42C2×C4×C8—21no—
43C2×C4×C8—21no—
44C2×C4×C8C2×C411yescharacteristic
45C2×C4×C8C2311yescharacteristic
46C43C2×C411yescharacteristic
47C4⋊C16—22no—
48C4⋊C16D412yes—
49C4⋊C16D412yes—
50C22⋊C16—24no—
51C22⋊C16—22no—
52C22⋊C16—22no—
53C4×C16—22no—
54C4×C16—22no—
55#341—22no—
56#339—22no—
57C2×#41C22⋊C412yes—
58#131—42no—
59#131—42no—
60C2×M16—22no—
61C2×M16—21no—
62C2×M16C22⋊C412yes—
63C22×C8—21no—
64C22×C8—22no—
65C22×C8C2×D411yescharacteristic
66C22×C8C2×D411yescharacteristic
67C4⋊Q8C2×C812yes—
68C4⋊Q8C2×C811yescharacteristic
69#127C2×C811yescharacteristic
70#125—21no—
71#124M1612yes—
72C22⋊Q8—22no—
73#121—22no—
74C4×Q8—22no—
75C4×D4—22no—
76#117—21no—
77C2×C42—21no—
78C2×C42—22no—
79C2×C42C22×C411yescharacteristic
80M32—82no—
81C2×C16—22no—
82C2×C16—22no—
83C2×C16—42no—
84C2×C16—82no—
85C4⋊C8—22no—
86C4⋊C8—22no—
87C4⋊C8—42no—
88#104—44no—
89#103—24no—
90#102—24no—
91C22⋊C8—24no—
92#97—41no—
93C4×C8—21no—
94C4×C8—42no—
95C4×C8—41no—
96C4×C8—41no—
97C4×C8—41no—
98C4×C8C2×C812yes—
99C4×C8C2×D412yes—
100C4×C8#4112yes—
101#41—42no—
102#41—42no—
103C2×Q8—22no—
104C2×Q8C22⋊C812yes—
105C2×D4—22no—
106C2×D4—22no—
107C2×D4C22⋊C812yes—
108C22×C4—22no—
109C22×C4—41no—
110C22×C4C2×C22⋊C411yescharacteristic
111C22×C4#13112yes—
112M16—42no—
113M16—42no—
114M16—82no—
115C2×C8—21no—
116C2×C8—21no—
117C2×C8—22no—
118C2×C8—22no—
119C2×C8—41no—
120C2×C8—41no—
121C2×C8—41no—
122C2×C8—42no—
123C2×C8—42no—
124C2×C8—41no—
125C2×C8#12011yescharacteristic
126C2×C8#12112yes—
127C2×C8#12311yescharacteristic
128C4⋊C4—22no—
129C4⋊C4—22no—
130C4⋊C4—22no—
131C4⋊C4—42no—
132C22⋊C4—24no—
133C22⋊C4—41no—
134C42—22no—
135C42—24no—
136C42—41no—
137C42—42no—
138C42—41no—
139C42C4×D412yes—
140C42C22×C811yescharacteristic
141C42C2×M1611yescharacteristic
142C16—42no—
143C16—42no—
144C16—82no—
145C23—22no—
146C23#40811yescharacteristic
147Q8—42no—
148Q8—42no—
149Q8—42no—
150D4—42no—
151D4—42no—
152D4—42no—
153D4—82no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—44no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4#38611yescharacteristic
165C2×C4C2×C22⋊C811yescharacteristic
166C2×C4C8×D412yes—
167C2×C4#43512yes—
168C8—22no—
169C8—21no—
170C8—42no—
171C8—42no—
172C8—81no—
173C8#45511yescharacteristic
174C8#45511yescharacteristic
175C22—22no—
176C22—21no—
177C22—44no—
178C22—42no—
179C22#186911yescharacteristic
180C4—22no—
181C4—21no—
182C4—44no—
183C4—42no—
184C4—82no—
185C4#188411yescharacteristic
186C4#188511yescharacteristic
187C4#217012yes—
188C4#217112yes—
189C2—41no—
190C2—82no—
191C2#895711yescharacteristic
192C2#1298012yes—
subgroupG/H
centerC2×C8#120
commutatorC42C22×C8
Frattini, cosocleC2×C4×C8C23
socleC22#1869
prime radical#532C2×C4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11638C2semidirect
5. #7148C2semidirect

Maximal subgroups

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

subgroupindex
#334932
#216582
#116382
#114902
#71482