Finite Group Databasebeta

← #141311

Subgroups — #141311

Order 512.

334 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 98 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
1C2×#1707C211yesmaximal, characteristic
2#13279C211yesmaximal, characteristic
3#13256C211yesmaximal, characteristic
4#12004C211yesmaximal, characteristic
5#11941C211yesmaximal, characteristic
6#11932C211yesmaximal, characteristic
7#8293C211yesmaximal, characteristic
8C2×#500C2211yescharacteristic
9C2×#498—21no—
10C2×#494C2211yescharacteristic
11C2×#487—21no—
12#2150C2211yescharacteristic
13#2150C2211yescharacteristic
14C2×#362—22no—
15C4×#108C2211yescharacteristic
16#1707—22no—
17#1707C414yes—
18#1564C2211yescharacteristic
19#1451—21no—
20C2×C82C2211yescharacteristic
21#1387—22no—
22#1370—22no—
23#1365—22no—
24#1362—22no—
25#1278—21no—
26C22×Q16—21no—
27C22×Q16—21no—
28C2×C4⋊Q8—21no—
29C2×C4⋊Q8—21no—
30C2×M32—21no—
31C2×M32—21no—
32#500—21no—
33#500—21no—
34#500—21no—
35#500C2×C412yes—
36#498—24no—
37#498—42no—
38#494—21no—
39#494—21no—
40#494—21no—
41#494C2×C412yes—
42#487—24no—
43#487—42no—
44#431—21no—
45C2×#108D412yes—
46C2×#107—21no—
47C2×#107—21no—
48C2×#106—42no—
49C2×#103—42no—
50C4×M16—21no—
51C2×C4×C8D411yescharacteristic
52C2×C4×C8D411yescharacteristic
53C2×C4×C8C2311yescharacteristic
54C4⋊C16—21no—
55C4⋊C16—21no—
56C4⋊C16—21no—
57C4⋊C16—21no—
58#362—44no—
59#341—21no—
60#335D412yes—
61#334—21no—
62#322—42no—
63C82C2×C412yes—
64C82D412yes—
65C22×Q8—41no—
66C22×Q8—41no—
67C2×Q16—22no—
68C2×Q16—22no—
69C2×Q16—41no—
70C2×Q16—41no—
71C2×Q16—42no—
72C2×Q16—42no—
73C2×Q16—82no—
74C2×M16—22no—
75C22×C8—21no—
76C22×C8SD1612yes—
77C22×C8C2×D411yescharacteristic
78C22×C8#4111yescharacteristic
79C4⋊Q8—22no—
80C4⋊Q8—22no—
81C4⋊Q8—41no—
82C4⋊Q8—41no—
83C4⋊Q8—41no—
84C4⋊Q8—41no—
85C4⋊Q8—41no—
86C4⋊Q8—41no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C42C2×D411yescharacteristic
92M32—42no—
93M32—42no—
94C2×C16—41no—
95C2×C16—41no—
96#108—22no—
97#108—22no—
98#107—22no—
99#107—22no—
100#107—41no—
101#107—41no—
102#106—48no—
103C4⋊C8—22no—
104#103—48no—
105C22⋊C8—41no—
106#97—41no—
107C4×C8—21no—
108C4×C8—41no—
109C4×C8—41no—
110C4×C8C22⋊C412yes—
111C4×C8C22⋊C412yes—
112C4×C8D812yes—
113C4×C8C22×C411yescharacteristic
114C4×C8C2×D411yescharacteristic
115C4×C8C2×D411yescharacteristic
116C4×C8#4111yescharacteristic
117C2×Q8—42no—
118C2×Q8—42no—
119C2×Q8—84no—
120C2×Q8—81no—
121C2×Q8—81no—
122C2×Q8—82no—
123C2×Q8—82no—
124C22×C4—81no—
125C22×C4—81no—
126C22×C4#12111yescharacteristic
127C22×C4C2×SD1611yescharacteristic
128C22×C4AGL(1,ℤ/8)11yescharacteristic
129Q16—44no—
130Q16—82no—
131Q16—82no—
132Q16—81no—
133Q16—81no—
134M16—42no—
135M16—42no—
136C2×C8—24no—
137C2×C8—22no—
138C2×C8—22no—
139C2×C8—42no—
140C2×C8—42no—
141C2×C8—42no—
142C2×C8#10214yes—
143C2×C8C4×D411yescharacteristic
144C2×C8C4×D411yescharacteristic
145C2×C8#12411yescharacteristic
146C2×C8#12711yescharacteristic
147C2×C8AGL(1,ℤ/8)12yes—
148C2×C8#13712yes—
149C4⋊C4—42no—
150C4⋊C4—42no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—81no—
154C4⋊C4—81no—
155C4⋊C4—81no—
156C4⋊C4—81no—
157C42C2×C22⋊C411yescharacteristic
158C42#12111yescharacteristic
159C42C2×D811yescharacteristic
160C42AGL(1,ℤ/8)11yescharacteristic
161C16—81no—
162C16—81no—
163C23#46011yescharacteristic
164Q8—84no—
165Q8—82no—
166Q8—82no—
167Q8—162no—
168Q8—162no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—82no—
172C2×C4—82no—
173C2×C4—162no—
174C2×C4—162no—
175C2×C4#39011yescharacteristic
176C2×C4C2×#10211yescharacteristic
177C2×C4#41811yescharacteristic
178C2×C4#45911yescharacteristic
179C2×C4#45911yescharacteristic
180C2×C4#46011yescharacteristic
181C2×C4#46812yes—
182C2×C4#47112yes—
183C8—24no—
184C8—24no—
185C8—21no—
186C8—21no—
187C8—82no—
188C8—82no—
189C8#44012yes—
190C8#49611yescharacteristic
191C8#49711yescharacteristic
192C22—21no—
193C22—21no—
194C22#168612yes—
195C22#198211yescharacteristic
196C4—21no—
197C4—21no—
198C4—162no—
199C4—162no—
200C4#168512yes—
201C4#197311yescharacteristic
202C4#197711yescharacteristic
203C4#214812yes—
204C4#221412yes—
205C2—22no—
206C2#974411yescharacteristic
207C2#1325811yescharacteristic
208C2#1326211yescharacteristic
subgroupG/H
center, socleC22#1982
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC23#460
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ 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
C2×#17072
#132792
#132562
#120042
#119412
#119322
#82932