Finite Group Databasebeta

← #124682

Subgroups — #124682

Order 512.

283 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 283 conjugacy classes of subgroups — too many to draw. Only its 68 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#21287C211yesmaximal, characteristic
2#12595C211yesmaximal, characteristic
3#8800C211yesmaximal, characteristic
4#7127C211yesmaximal, characteristic
5#7126C211yesmaximal, characteristic
6#7111C211yesmaximal, characteristic
7#7111C211yesmaximal, characteristic
8C2×#441C411yescharacteristic
9#2295—21no—
10C2×#347—21no—
11#1971C2211yescharacteristic
12#1969C411yescharacteristic
13#1944C2211yescharacteristic
14#1941C411yescharacteristic
15#1935—21no—
16#1933—21no—
17#1776C411yescharacteristic
18#1763—21no—
19C4×#97C2211yescharacteristic
20#1377C2211yescharacteristic
21#1377C2211yescharacteristic
22#1376—21no—
23#1281—21no—
24#1276—21no—
25#1275C2211yescharacteristic
26#1275C2211yescharacteristic
27C22×Q16C2×C411yescharacteristic
28C2×C4×Q8—21no—
29C2×M32—21no—
30C2×M32—21no—
31#441—22no—
32#441—22no—
33#431—21no—
34#431—21no—
35C2×#107C2×C411yescharacteristic
36C2×#106C2×C411yescharacteristic
37C2×#106C2×C411yescharacteristic
38C2×#103—21no—
39C2×#103—21no—
40C2×#103—21no—
41C2×#97C2×C411yescharacteristic
42C2×#97D411yescharacteristic
43C2×#97D411yescharacteristic
44C2×#97C2311yescharacteristic
45C2×C4×C8C2×C411yescharacteristic
46C2×C4×C8D411yescharacteristic
47#391—21no—
48#391—21no—
49#389—21no—
50#389—21no—
51#387—41no—
52#387—41no—
53C4×C4⋊C4—21no—
54C4×C4⋊C4—41no—
55#376—41no—
56C43D411yescharacteristic
57#347—24no—
58#340—21no—
59#340—21no—
60#340—21no—
61C22×Q8—21no—
62C2×Q16—21no—
63C2×Q16—41no—
64C2×Q16—41no—
65C2×Q16M1612yes—
66C22×C8C22⋊C411yescharacteristic
67C22×C8C22⋊C411yescharacteristic
68C22×C8C22×C411yescharacteristic
69C22×C8C2×D411yescharacteristic
70C4×Q8—24no—
71C4×Q8—42no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—41no—
78C2×C42—21no—
79C2×C42—21no—
80C2×C42—41no—
81C2×C42—41no—
82C2×C42C22⋊C411yescharacteristic
83C2×C42C22⋊C411yescharacteristic
84C2×C42C2×D411yescharacteristic
85M32—22no—
86M32—22no—
87M32—41no—
88M32—41no—
89C2×C16—41no—
90C2×C16—41no—
91#107—41no—
92#106—21no—
93#106—41no—
94#106M1612yes—
95C4⋊C8—41no—
96C4⋊C8—41no—
97#103—24no—
98#103—42no—
99#103—41no—
100#103—41no—
101C22⋊C8—41no—
102C22⋊C8—41no—
103#97—21no—
104#97—21no—
105#97—22no—
106#97—41no—
107#97—41no—
108C4×C8—21no—
109C4×C8—21no—
110C4×C8—41no—
111C4×C8—41no—
112C4×C8—41no—
113#95—41no—
114#95—41no—
115#95—41no—
116#95—41no—
117#95—41no—
118#95—41no—
119#95—41no—
120C2×Q8—22no—
121C2×Q8—41no—
122C2×Q8—41no—
123C2×Q8—41no—
124C2×Q8—41no—
125C2×Q8—41no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—41no—
129C22×C4—41no—
130C22×C4—41no—
131C22×C4—41no—
132C22×C4C2×C22⋊C411yescharacteristic
133C22×C4AGL(1,ℤ/8)11yescharacteristic
134C22×C4AGL(1,ℤ/8)11yescharacteristic
135Q16—42no—
136Q16—42no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—81no—
148C2×C8—81no—
149C2×C8C2×M1611yescharacteristic
150C2×C8C2×M1611yescharacteristic
151C2×C8#13511yescharacteristic
152C2×C8#13511yescharacteristic
153C4⋊C4—22no—
154C4⋊C4—44no—
155C4⋊C4—41no—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4—41no—
159C4⋊C4—41no—
160C4⋊C4—82no—
161C4⋊C4—82no—
162C42—21no—
163C42—21no—
164C42—21no—
165C42—21no—
166C42—21no—
167C42—42no—
168C42—44no—
169C42—81no—
170C42—82no—
171C42#10412yes—
172C16—42no—
173C16—42no—
174C23#41811yescharacteristic
175Q8—42no—
176Q8—42no—
177Q8—81no—
178Q8—81no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—81no—
192C2×C4—81no—
193C2×C4—81no—
194C2×C4—81no—
195C2×C4—81no—
196C2×C4—81no—
197C2×C4—81no—
198C2×C4—81no—
199C2×C4—81no—
200C2×C4—81no—
201C2×C4#40711yescharacteristic
202C2×C4#41011yescharacteristic
203C2×C4#41311yescharacteristic
204C2×C4#41611yescharacteristic
205C2×C4C2×#10411yescharacteristic
206C2×C4#42111yescharacteristic
207C8—21no—
208C8—21no—
209C8—21no—
210C8—21no—
211C8—41no—
212C8—41no—
213C8—161no—
214C8—161no—
215C22—21no—
216C22—21no—
217C22#149011yescharacteristic
218C22#151411yescharacteristic
219C22#155311yescharacteristic
220C4—42no—
221C4—42no—
222C4—41no—
223C4—41no—
224C4—82no—
225C4—82no—
226C4—161no—
227C4—161no—
228C4#146411yescharacteristic
229C4#148811yescharacteristic
230C4#150011yescharacteristic
231C4#152911yescharacteristic
232C2—21no—
233C2—21no—
234C2#760811yescharacteristic
235C2#1183611yescharacteristic
236C2#1183811yescharacteristic
subgroupG/H
center, socleC22#1514
commutatorC22×C8C22×C4
Frattini, cosocleC2×#97C23
prime radicalC23#418
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C4×#97 ◁ G
LECSG ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#212872
#125952
#88002
#71272
#71262
#71112