Finite Group Databasebeta

← #150605

Subgroups — #150605

Order 512.

226 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 50 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#29887C211yesmaximal, characteristic
2#12864C211yesmaximal, characteristic
3#11524C211yesmaximal, characteristic
4#7411C212yesmaximal
5#7410C212yesmaximal
6C2×#498C411yescharacteristic, complemented
7#3079—21no—
8#2922C2211yescharacteristic
9#2165—21no—
10#2090—21no—
11#2090C412yescomplemented
12#2079—21no—
13#2079C2212yes—
14#1997—21no—
15#1837C411yescharacteristic, complemented
16#1827—21no—
17#1395—21no—
18#1392C2211yescharacteristic
19#1391C2211yescharacteristic
20#1328—21no—
21#1321C2212yes—
22C2×C4⋊Q8—21no—
23C4×#41D411yescharacteristic
24#498—41no—
25#498—41no—
26#477—41no—
27#476—41no—
28#475—41no—
29#474—41no—
30#433—41no—
31#431—41no—
32#428—22no—
33C2×#106—21no—
34C2×#106C2×C412yes—
35#424C2×C411yescharacteristic
36#424D411yescharacteristic
37#423—21no—
38#417—22no—
39#413—21no—
40#410—21no—
41#408D411yescharacteristic
42#408C2311yescharacteristic
43C4×M16D411yescharacteristic
44C2×C4×C8C2×C411yescharacteristic
45#364—41no—
46#349—22no—
47#347—41no—
48#343—22no—
49#343—22no—
50#341—21no—
51#341C2×C412yes—
52#339—41no—
53H(3,ℤ/4)—41no—
54C2×#41C2×D411yescharacteristic
55C22×Q8—41no—
56C2×M16—22no—
57C2×M16—41no—
58C2×M16C22⋊C411yescharacteristic
59C2×M16C2×D411yescharacteristic
60C22×C8C22⋊C411yescharacteristic
61C22×C8C22×C411yescharacteristic
62C4⋊Q8—41no—
63C4⋊Q8—41no—
64C4⋊Q8—41no—
65C4⋊Q8—41no—
66C4×Q8—21no—
67C4×D4—21no—
68C4×D4—41no—
69#117—21no—
70#117—41no—
71#117C22⋊C411yescharacteristic
72C2×C4⋊C4—22no—
73C2×C4⋊C4—41no—
74C2×C42—21no—
75C2×C42C22⋊C411yescharacteristic
76M32—42no—
77#107—42no—
78#106—42no—
79#106—42no—
80#106—42no—
81C4⋊C8—41no—
82C4⋊C8—41no—
83C4⋊C8—41no—
84C4⋊C8—41no—
85#103—42no—
86#102—42no—
87#101—41no—
88#100—41no—
89#99—41no—
90#99—41no—
91C22⋊C8—41no—
92C22⋊C8—41no—
93#97—21no—
94C4×C8—21no—
95C4×C8—21no—
96C4×C8—21no—
97#41—41no—
98#41—41no—
99C2×Q8—81no—
100C2×Q8—81no—
101C2×Q8—162no—
102C2×Q8AGL(1,ℤ/8)11yescharacteristic
103C2×D4—21no—
104C2×D4AGL(1,ℤ/8)11yescharacteristic
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—41no—
108C22×C4—42no—
109C22×C4C2×C22⋊C411yescharacteristic
110M16—41no—
111M16—41no—
112M16—42no—
113M16—42no—
114M16—81no—
115M16—81no—
116C2×C8—21no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—21no—
120C2×C8—41no—
121C2×C8—41no—
122C2×C8—42no—
123C2×C8—81no—
124C4⋊C4—21no—
125C4⋊C4—41no—
126C4⋊C4—42no—
127C4⋊C4—42no—
128C4⋊C4—81no—
129C4⋊C4—81no—
130C4⋊C4—81no—
131C22⋊C4—21no—
132C22⋊C4—41no—
133C22⋊C4—81no—
134C42—21no—
135C42—21no—
136C42—41no—
137C42—81no—
138C42#9911yescharacteristic
139C42#9911yescharacteristic
140C16—82no—
141C23—21no—
142C23#42111yescharacteristic
143Q8—41no—
144Q8—162no—
145Q8—162no—
146D4—41no—
147D4—81no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—42no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—82no—
158C2×C4—82no—
159C2×C4—162no—
160C2×C4—161no—
161C2×C4C2×#9911yescharacteristic
162C2×C4#41811yescharacteristic
163C8—41no—
164C8—41no—
165C8—41no—
166C8—41no—
167C8—82no—
168C8—161no—
169C22—21no—
170C22—41no—
171C22—41no—
172C22—81no—
173C22#151011yescharacteristic
174C4—22no—
175C4—21no—
176C4—41no—
177C4—41no—
178C4—81no—
179C4—162no—
180C4—321no—
181C4#212311yescharacteristic
182C4#212911yescharacteristic
183C2—21no—
184C2—41no—
185C2—81no—
186C2#1180111yescharacteristic
subgroupG/H
center, socleC2#11801
commutatorC22×C8C22×C4
Frattini, cosocle#408C23
prime radicalC2×#41C2×D4
LCSG ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C42 ◁ #2922 ◁ G
LECSG ▷ #408 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. C2×#498C4semidirect
11. #2090C4semidirect
15. #1837C4semidirect

Maximal subgroups

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

subgroupindex
#298872
#128642
#115242
#74112
#74102