Finite Group Databasebeta

← #150603

Subgroups — #150603

Order 512.

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

normal subgroups only

This group has 266 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#29820C211yesmaximal, characteristic
2#12864C211yesmaximal, characteristic
3#11524C211yesmaximal, characteristic
4#7401C212yesmaximal
5#7399C212yesmaximal
6C2×#494C411yescharacteristic, complemented
7#3039—21no—
8#2922C2211yescharacteristic
9#2165—21no—
10#2047—21no—
11#2047C412yescomplemented
12#2020—21no—
13#2020C2212yes—
14#1966—21no—
15#1837C411yescharacteristic, complemented
16#1827—21no—
17#1395—21no—
18#1392C2211yescharacteristic
19#1391C2211yescharacteristic
20#1328—21no—
21#1321C2212yes—
22C2×#137—22no—
23C22×Q16—21no—
24C22×Q16C2×C412yes—
25C2×C4⋊Q8—21no—
26C4×#41D411yescharacteristic
27#494—41no—
28#494—41no—
29#462—42no—
30#461—42no—
31#433—41no—
32#431—41no—
33#424C2×C411yescharacteristic
34#424D411yescharacteristic
35#423—21no—
36#419—22no—
37#413—21no—
38C2×#101—22no—
39C2×#101—22no—
40#410—21no—
41#408D411yescharacteristic
42#408C2311yescharacteristic
43C4×M16D411yescharacteristic
44C2×C4×C8C2×C411yescharacteristic
45#364—41no—
46#349—22no—
47#347—41no—
48#341—21no—
49#341C2×C412yes—
50#339—41no—
51H(3,ℤ/4)—41no—
52C2×#41C2×D411yescharacteristic
53C22×Q8—22no—
54#137—82no—
55#137—82no—
56C2×Q16—42no—
57C2×Q16—42no—
58C2×Q16—42no—
59C2×Q16—82no—
60C2×Q16—82no—
61C2×SD16—42no—
62C2×M16—22no—
63C2×M16—41no—
64C2×M16C22⋊C411yescharacteristic
65C2×M16C2×D411yescharacteristic
66C22×C8C22⋊C411yescharacteristic
67C22×C8C22×C411yescharacteristic
68C4⋊Q8—42no—
69C4⋊Q8—41no—
70C4⋊Q8—41no—
71C4×Q8—21no—
72C4×D4—21no—
73C4×D4—41no—
74#117—21no—
75#117—41no—
76#117C22⋊C411yescharacteristic
77C2×C4⋊C4—42no—
78C2×C42—21no—
79C2×C42C22⋊C411yescharacteristic
80M32—42no—
81C4⋊C8—41no—
82C4⋊C8—41no—
83C4⋊C8—42no—
84#103—44no—
85#101—41no—
86#101—82no—
87#101—82no—
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—42no—
100C2×Q8—42no—
101C2×Q8—42no—
102C2×Q8—42no—
103C2×Q8—42no—
104C2×Q8—42no—
105C2×Q8AGL(1,ℤ/8)11yescharacteristic
106C2×D4—21no—
107C2×D4AGL(1,ℤ/8)11yescharacteristic
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—41no—
111C22×C4—42no—
112C22×C4C2×C22⋊C411yescharacteristic
113Q16—84no—
114Q16—84no—
115Q16—84no—
116SD16—84no—
117M16—41no—
118M16—41no—
119M16—42no—
120M16—42no—
121M16—81no—
122M16—81no—
123C2×C8—21no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—41no—
128C2×C8—41no—
129C2×C8—42no—
130C2×C8—81no—
131C4⋊C4—21no—
132C4⋊C4—41no—
133C4⋊C4—82no—
134C4⋊C4—82no—
135C4⋊C4—81no—
136C22⋊C4—21no—
137C22⋊C4—41no—
138C22⋊C4—81no—
139C42—21no—
140C42—21no—
141C42—41no—
142C42—81no—
143C42#9911yescharacteristic
144C42#9911yescharacteristic
145C16—82no—
146C23—21no—
147C23#42111yescharacteristic
148Q8—41no—
149Q8—82no—
150Q8—82no—
151Q8—82no—
152Q8—82no—
153D4—41no—
154D4—81no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—42no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—82no—
165C2×C4—82no—
166C2×C4—82no—
167C2×C4—82no—
168C2×C4—161no—
169C2×C4C2×#9911yescharacteristic
170C2×C4#41811yescharacteristic
171C8—41no—
172C8—41no—
173C8—41no—
174C8—41no—
175C8—82no—
176C8—161no—
177C22—21no—
178C22—41no—
179C22—41no—
180C22—81no—
181C22#151011yescharacteristic
182C4—22no—
183C4—21no—
184C4—41no—
185C4—41no—
186C4—81no—
187C4—162no—
188C4—321no—
189C4#212211yescharacteristic
190C4#212811yescharacteristic
191C2—21no—
192C2—41no—
193C2—81no—
194C2#1180011yescharacteristic
subgroupG/H
center, socleC2#11800
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×#494C4semidirect
11. #2047C4semidirect
15. #1837C4semidirect

Maximal subgroups

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

subgroupindex
#298202
#128642
#115242
#74012
#73992