Finite Group Databasebeta

← #145232

Subgroups — #145232

Order 512.

261 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 261 conjugacy classes of subgroups — too many to draw. Only its 62 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×#1690C211yesmaximal, characteristic
2#12028C211yesmaximal, characteristic
3#11965C211yesmaximal, characteristic
4#11965C211yesmaximal, characteristic
5#11644C211yesmaximal, characteristic
6#8625C211yesmaximal, characteristic
7#7294C211yesmaximal, characteristic
8C2×#494C2211yescharacteristic
9C2×#475—21no—
10C2×#446C2211yescharacteristic
11#2163C2211yescharacteristic
12#2163C2211yescharacteristic
13#1690—22no—
14#1690—22no—
15#1568C2211yescharacteristic
16#1568C2211yescharacteristic
17C2×#334C2211yescharacteristic
18C2×#332—21no—
19#1467—21no—
20#1368—22no—
21#1362—22no—
22#1362—22no—
23#1338—22no—
24#1290—21no—
25C22×Q16—21no—
26C22×Q16—21no—
27C2×C4⋊Q8—21no—
28C2×C4×Q8D411yescharacteristic
29#494—21no—
30#494—21no—
31#494—21no—
32#494D412yes—
33#475—24no—
34#475—42no—
35#446—21no—
36#446—21no—
37#446—21no—
38#446—21no—
39#432—21no—
40#432—21no—
41C2×#106—41no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8D411yescharacteristic
45C2×#103—41no—
46C2×#97—21no—
47C2×C4×C8C2311yescharacteristic
48C4⋊C16—21no—
49C4⋊C16D412yes—
50C22⋊C16—21no—
51C22⋊C16—21no—
52#346—21no—
53#346—21no—
54#335—21no—
55#335—21no—
56#334—21no—
57#334—21no—
58#334—21no—
59#334D412yes—
60#332—24no—
61#324—42no—
62C22×Q8—41no—
63C22×Q8SD1611yescharacteristic
64C2×Q16—22no—
65C2×Q16—22no—
66C2×Q16—41no—
67C2×Q16—41no—
68C2×Q16—82no—
69C2×Q16—82no—
70C22×C8—21no—
71C22×C8—41no—
72C22×C8#4111yescharacteristic
73C22×C8#4111yescharacteristic
74C4⋊Q8—22no—
75C4⋊Q8—41no—
76C4⋊Q8—41no—
77C4⋊Q8—41no—
78C4×Q8—21no—
79C4×Q8—21no—
80C4×Q8—21no—
81C4×Q8—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4SD1611yescharacteristic
86C2×C42—21no—
87C2×C42C2×D411yescharacteristic
88C2×C16—22no—
89C2×C16—41no—
90#106—44no—
91C4⋊C8—21no—
92C4⋊C8—22no—
93C4⋊C8—21no—
94C4⋊C8—41no—
95C4⋊C8—41no—
96C4⋊C8—41no—
97C4⋊C8—41no—
98C4⋊C8—41no—
99#103—44no—
100C22⋊C8—41no—
101C22⋊C8—41no—
102#97—41no—
103#97—41no—
104#97—41no—
105#97—41no—
106C4×C8C2×D411yescharacteristic
107C4×C8C2×D411yescharacteristic
108C4×C8C2×D411yescharacteristic
109C4×C8#4111yescharacteristic
110C2×Q8—21no—
111C2×Q8—21no—
112C2×Q8—42no—
113C2×Q8—42no—
114C2×Q8—82no—
115C2×Q8—82no—
116C2×Q8—81no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—81no—
120C22×C4#12311yescharacteristic
121C22×C4C2×SD1611yescharacteristic
122C22×C4AGL(1,ℤ/8)11yescharacteristic
123Q16—82no—
124Q16—82no—
125Q16—82no—
126Q16—82no—
127C2×C8—22no—
128C2×C8—41no—
129C2×C8—41no—
130C2×C8—42no—
131C2×C8—82no—
132C2×C8—81no—
133C2×C8—81no—
134C2×C8—81no—
135C2×C8#12111yescharacteristic
136C2×C8#12111yescharacteristic
137C2×C8#12311yescharacteristic
138C2×C8#12411yescharacteristic
139C4⋊C4—21no—
140C4⋊C4—21no—
141C4⋊C4—41no—
142C4⋊C4—41no—
143C4⋊C4—42no—
144C4⋊C4—42no—
145C4⋊C4—81no—
146C4⋊C4—81no—
147C42—41no—
148C42—41no—
149C42#12011yescharacteristic
150C42#12111yescharacteristic
151C42#13511yescharacteristic
152C42AGL(1,ℤ/8)11yescharacteristic
153C16—42no—
154C16—81no—
155C23#48111yescharacteristic
156Q8—42no—
157Q8—81no—
158Q8—82no—
159Q8—82no—
160Q8—162no—
161C2×C4—22no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—41no—
165C2×C4—42no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—81no—
169C2×C4—82no—
170C2×C4—162no—
171C2×C4#39411yescharacteristic
172C2×C4#44911yescharacteristic
173C2×C4#45011yescharacteristic
174C2×C4#46511yescharacteristic
175C2×C4#46911yescharacteristic
176C2×C4#48211yescharacteristic
177C8—21no—
178C8—21no—
179C8—21no—
180C8—82no—
181C8—81no—
182C8—161no—
183C8—161no—
184C8#47112yes—
185C22—21no—
186C22—21no—
187C22#203411yescharacteristic
188C22#223912yes—
189C4—21no—
190C4—21no—
191C4—21no—
192C4—21no—
193C4—21no—
194C4—81no—
195C4—81no—
196C4—162no—
197C4#204611yescharacteristic
198C4#204811yescharacteristic
199C2—22no—
200C2#1044311yescharacteristic
201C2#1339911yescharacteristic
202C2#1340511yescharacteristic
subgroupG/H
center, socleC22#2034
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC23#481
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#446 ◁ 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×#16902
#120282
#119652
#116442
#86252
#72942