Finite Group Databasebeta

← #145987

Subgroups — #145987

Order 512.

227 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 227 conjugacy classes of subgroups — too many to draw. Only its 72 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#12033C211yesmaximal, characteristic
2#12015C211yesmaximal, characteristic
3#11962C211yesmaximal, characteristic
4C2×#1362C211yesmaximal, characteristic
5#9798C211yesmaximal, characteristic
6#9624C211yesmaximal, characteristic
7#7241C211yesmaximal, characteristic
8C2×#494C2211yescharacteristic
9#2162C2211yescharacteristic
10C2×C4⋊C16C2211yescharacteristic
11#1997—21no—
12#1950—21no—
13#1567C2211yescharacteristic
14#1567C2211yescharacteristic
15C2×#335C2211yescharacteristic
16#1548—21no—
17C8×M16C2211yescharacteristic
18#1372—22no—
19#1370—21no—
20#1367—21no—
21#1362—21no—
22#1362—21no—
23#1362—22no—
24#1301—21no—
25C22×Q16—21no—
26C22×Q16Q812yes—
27C2×C4⋊Q8—21no—
28C22×C16—21no—
29#494—21no—
30#494—21no—
31#494—21no—
32#494—21no—
33#431D411yescharacteristic
34C2×#108—41no—
35#423—21no—
36#423—21no—
37C2×C4⋊C8—21no—
38C4×M16D411yescharacteristic
39C2×C4×C8C2311yescharacteristic
40C4⋊C16—21no—
41C4⋊C16—21no—
42C4⋊C16D411yescharacteristic
43C4⋊C16D411yescharacteristic
44C22⋊C16—21no—
45C4×C16—21no—
46#343—41no—
47#343—41no—
48#341—41no—
49#335—21no—
50#335—21no—
51#335—21no—
52#335—21no—
53#335—21no—
54#335D412yes—
55#333—42no—
56#322—21no—
57C82—21no—
58C22×Q8—22no—
59C2×Q16—22no—
60C2×Q16—22no—
61C2×Q16—41no—
62C2×Q16—41no—
63C2×Q16—81no—
64C2×Q16—81no—
65C2×Q16—82no—
66C2×M16—41no—
67C2×M16—41no—
68C2×M16D811yescharacteristic
69C2×M16D811yescharacteristic
70C22×C8—41no—
71C22×C8C2×Q811yescharacteristic
72C22×C8#4111yescharacteristic
73C4⋊Q8—41no—
74C4⋊Q8—41no—
75C4⋊Q8—41no—
76C4⋊Q8—41no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—41no—
79C2×C42C2×D411yescharacteristic
80C2×C16—21no—
81C2×C16—21no—
82C2×C16—41no—
83C2×C16—81no—
84#108—82no—
85C4⋊C8—22no—
86C4⋊C8—21no—
87C4⋊C8—41no—
88C4⋊C8—41no—
89C4⋊C8—41no—
90C4⋊C8—41no—
91C4⋊C8—41no—
92C22⋊C8—21no—
93#97—21no—
94C4×C8—21no—
95C4×C8—21no—
96C4×C8C2×D411yescharacteristic
97C4×C8C2×D411yescharacteristic
98C4×C8#4111yescharacteristic
99C4×C8#4111yescharacteristic
100C2×Q8—42no—
101C2×Q8—42no—
102C2×Q8—82no—
103C2×Q8—82no—
104C22×C4—42no—
105C22×C4C22⋊Q811yescharacteristic
106C22×C4C2×D811yescharacteristic
107C22×C4#13711yescharacteristic
108Q16—82no—
109Q16—82no—
110Q16—82no—
111Q16—81no—
112Q16—81no—
113M16—21no—
114M16—21no—
115M16—21no—
116M16—81no—
117M16—81no—
118M16—81no—
119M16—81no—
120M16SD3212yes—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—21no—
124C2×C8—21no—
125C2×C8—41no—
126C2×C8—42no—
127C2×C8—41no—
128C2×C8—81no—
129C2×C8—81no—
130C2×C8—81no—
131C2×C8—81no—
132C2×C8#12111yescharacteristic
133C2×C8C22⋊Q811yescharacteristic
134C2×C8C22⋊Q811yescharacteristic
135C2×C8#12611yescharacteristic
136C4⋊C4—81no—
137C4⋊C4—81no—
138C4⋊C4—81no—
139C4⋊C4—81no—
140C42#12111yescharacteristic
141C42#12111yescharacteristic
142C42#13511yescharacteristic
143C42AGL(1,ℤ/8)11yescharacteristic
144C16—41no—
145C16—41no—
146C16—81no—
147C23#47411yescharacteristic
148Q8—82no—
149Q8—82no—
150Q8—162no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—41no—
154C2×C4—82no—
155C2×C4—162no—
156C2×C4#39611yescharacteristic
157C2×C4#45911yescharacteristic
158C2×C4#46411yescharacteristic
159C2×C4#46611yescharacteristic
160C2×C4#46811yescharacteristic
161C2×C4#47811yescharacteristic
162C2×C4C2×SD3211yescharacteristic
163C2×C4#50811yescharacteristic
164C2×C4#51011yescharacteristic
165C2×C4#51011yescharacteristic
166C8—21no—
167C8—21no—
168C8—21no—
169C8—22no—
170C8—41no—
171C8—81no—
172C8—82no—
173C8—161no—
174C8—161no—
175C8#47111yescharacteristic
176C8#47211yescharacteristic
177C22—21no—
178C22—21no—
179C22#206711yescharacteristic
180C22#222511yescharacteristic
181C22#222911yescharacteristic
182C4—21no—
183C4—21no—
184C4—21no—
185C4—162no—
186C4#206311yescharacteristic
187C4#206911yescharacteristic
188C4#220811yescharacteristic
189C4#221111yescharacteristic
190C4#221711yescharacteristic
191C4#221911yescharacteristic
192C2—21no—
193C2—21no—
194C2#1060311yescharacteristic
195C2#1347311yescharacteristic
196C2#1347711yescharacteristic
subgroupG/H
center, socleC22#2067
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC23#474
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C8×M16 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#120332
#120152
#119622
C2×#13622
#97982
#96242
#72412