Finite Group Databasebeta

← #152247

Subgroups — #152247

Order 512.

297 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 297 conjugacy classes of subgroups — too many to draw. Only its 68 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#21661C211yesmaximal, characteristic
2#13594C212yesmaximal
3#12807C212yesmaximal
4#7506C212yesmaximal
5#3135C2211yescharacteristic
6#2960C411yescharacteristic, complemented
7#2931C2211yescharacteristic
8#2126C2214yes—
9#1995C411yescharacteristic, complemented
10#1963C2211yescharacteristic
11#1938—22no—
12#1838C411yescharacteristic, complemented
13#1812—22no—
14#1764—22no—
15#1726C411yescharacteristic, complemented
16#1412—24no—
17#1391—22no—
18C2×#131D412yes—
19#533C2×C411yescharacteristic
20#532C2311yescharacteristic
21#491Q812yes—
22#490D412yes—
23#489—22no—
24#446—41no—
25C8×Q8—41no—
26#436—41no—
27C8×D4—41no—
28#427—42no—
29#424—21no—
30#423C2×C411yescharacteristic
31#423C2×C411yescharacteristic
32C2×#104—22no—
33#416—22no—
34#410—24no—
35#408D412yes—
36#405—21no—
37C2×#97—21no—
38C2×#97C2×C411yescharacteristic
39C2×C4×C8—21no—
40C2×C4×C8C2×C411yescharacteristic
41#376C2×C411yescharacteristic
42#354—24no—
43#353—24no—
44#341—22no—
45#339—22no—
46H(3,ℤ/4)—42no—
47C2×#41C2×D412yes—
48#131—42no—
49#131—42no—
50C2×M16—22no—
51C2×M16—21no—
52C2×M16C22⋊C412yes—
53C22×C8—21no—
54C22×C8—22no—
55C22×C8C22⋊C411yescharacteristic
56C22×C8C22⋊C411yescharacteristic
57C4⋊Q8—21no—
58C4⋊Q8C2×Q811yescharacteristic
59#127C2×D411yescharacteristic
60#125C4⋊C412yes—
61#124#4112yes—
62C22⋊Q8—41no—
63#121—41no—
64C4×Q8—41no—
65C4×D4—41no—
66#117—21no—
67#117—44no—
68C2×C42—21no—
69C2×C42C22×C411yescharacteristic
70#107—82no—
71#106—82no—
72C4⋊C8—41no—
73C4⋊C8—41no—
74C4⋊C8—42no—
75#104—44no—
76#103—24no—
77#102—24no—
78#99—44no—
79#99—44no—
80C22⋊C8—42no—
81#97—21no—
82#97—42no—
83#97—41no—
84#97C4⋊C412yes—
85C4×C8—21no—
86C4×C8—21no—
87C4×C8—41no—
88C4×C8—41no—
89C4×C8—41no—
90#95—41no—
91#41—42no—
92#41—42no—
93C2×Q8—41no—
94C2×Q8C22⋊Q812yes—
95C2×D4—22no—
96C2×D4—41no—
97C2×D4#12112yes—
98C22×C4—21no—
99C22×C4—22no—
100C22×C4—41no—
101C22×C4C2×C22⋊C411yescharacteristic
102M16—42no—
103M16—42no—
104M16—82no—
105C2×C8—21no—
106C2×C8—21no—
107C2×C8—22no—
108C2×C8—21no—
109C2×C8—22no—
110C2×C8—41no—
111C2×C8—41no—
112C2×C8—42no—
113C2×C8—42no—
114C2×C8—41no—
115C2×C8—41no—
116C2×C8#9911yescharacteristic
117C2×C8#9911yescharacteristic
118C4⋊C4—22no—
119C4⋊C4—41no—
120C4⋊C4—42no—
121C4⋊C4—41no—
122C4⋊C4—84no—
123C22⋊C4—42no—
124C22⋊C4—41no—
125C22⋊C4—84no—
126C42—21no—
127C42—22no—
128C42—41no—
129C42—41no—
130C42—84no—
131C42C2×C4⋊C411yescharacteristic
132C42#11711yescharacteristic
133C23—22no—
134C23#41011yescharacteristic
135Q8—42no—
136Q8—82no—
137D4—42no—
138D4—82no—
139D4—82no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—22no—
145C2×C4—42no—
146C2×C4—42no—
147C2×C4—42no—
148C2×C4—42no—
149C2×C4—81no—
150C2×C4—82no—
151C2×C4—164no—
152C2×C4#38011yescharacteristic
153C2×C4C2×#9911yescharacteristic
154C8—21no—
155C8—42no—
156C8—42no—
157C8—41no—
158C8—81no—
159C8#35411yescharacteristic
160C8#35411yescharacteristic
161C22—21no—
162C22—41no—
163C22—42no—
164C22—82no—
165C22#179911yescharacteristic
166C4—21no—
167C4—41no—
168C4—41no—
169C4—42no—
170C4—82no—
171C4—82no—
172C4—324no—
173C4C2×#35411yescharacteristic
174C4#212611yescharacteristic
175C2—21no—
176C2—41no—
177C2—82no—
178C2#1279811yescharacteristic
subgroupG/H
centerC4#2126
commutatorC2×C42C22×C4
Frattini, prime radical, cosocle#532C23
socleC2#12798
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C2×C8 ◁ C22×C8 ◁ #2931 ◁ G
LECSG ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #2960C4semidirect
9. #1995C4semidirect
12. #1838C4semidirect
15. #1726C4semidirect

Maximal subgroups

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

subgroupindex
#216612
#135942
#128072
#75062