Finite Group Databasebeta

← #139178

Subgroups — #139178

Order 512.

277 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 277 conjugacy classes of subgroups — too many to draw. Only its 80 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#33499C211yesmaximal, characteristic
2#19350C211yesmaximal, characteristic
3#11865C211yesmaximal, characteristic
4#11646C211yesmaximal, characteristic
5#8689C211yesmaximal, characteristic
6#7292C211yesmaximal, characteristic
7#7219C211yesmaximal, characteristic
8#3255—21no—
9#3170C2211yescharacteristic
10#2960C2211yescharacteristic
11#2183—22no—
12#2183—21no—
13Q8×C16—21no—
14#2164—22no—
15#2163C2211yescharacteristic
16#2162C2211yescharacteristic
17#1837C2211yescharacteristic
18#1694C412yes—
19#1690C412yes—
20#1680—22no—
21#1566C2211yescharacteristic
22C2×#335C2211yescharacteristic
23#1561—21no—
24#1550—21no—
25#1350—22no—
26#1332—21no—
27#1332—21no—
28#1292—21no—
29#533—21no—
30#533D411yescharacteristic
31#500C2×C411yescharacteristic
32#494C2×C411yescharacteristic
33#487—21no—
34#475—22no—
35#475—41no—
36#470—42no—
37#467—21no—
38#467—21no—
39#462—22no—
40#462—41no—
41#446—21no—
42#446C2×C412yes—
43C8×Q8—21no—
44C8×Q8D412yes—
45C4×Q16—21no—
46C4×Q16—21no—
47#424—21no—
48#423—21no—
49#423—21no—
50#423D411yescharacteristic
51C2×C4⋊C8—21no—
52#405—21no—
53C2×C4×C8C2311yescharacteristic
54C4⋊C16—22no—
55C4⋊C16—22no—
56C4⋊C16—21no—
57C4⋊C16—21no—
58C22⋊C16—21no—
59C22⋊C16D412yes—
60#346—22no—
61#346—21no—
62C4×C16—22no—
63C4×C16D411yescharacteristic
64C4×C16D411yescharacteristic
65#341—21no—
66#341—21no—
67#335—21no—
68#335—21no—
69#334—21no—
70#334C2×C412yes—
71#333—41no—
72#332—22no—
73#332—22no—
74#332—41no—
75C2×Q16—21no—
76C2×Q16—21no—
77C2×Q16—42no—
78C2×M16—21no—
79C2×M16—41no—
80C2×M16—41no—
81C22×C8—41no—
82C22×C8C2×D411yescharacteristic
83C22×C8#4111yescharacteristic
84C4⋊Q8—21no—
85C4⋊Q8—21no—
86C4⋊Q8D811yescharacteristic
87C4⋊Q8D811yescharacteristic
88#125—21no—
89#125—41no—
90C22⋊Q8—21no—
91C22⋊Q8—41no—
92C22⋊Q8—41no—
93C22⋊Q8SD1612yes—
94C4×Q8—21no—
95C4×Q8—41no—
96C4×Q8—41no—
97C4×Q8C22⋊C412yes—
98#117—21no—
99#117—41no—
100C2×C42C2×D411yescharacteristic
101C2×C16—21no—
102C2×C16—22no—
103C2×C16—21no—
104#107—21no—
105#107—21no—
106#106—42no—
107C4⋊C8—22no—
108C4⋊C8—22no—
109C4⋊C8—22no—
110C4⋊C8—21no—
111C4⋊C8—41no—
112C4⋊C8—41no—
113C4⋊C8—41no—
114C4⋊C8—41no—
115C4⋊C8—41no—
116C4⋊C8C22⋊C412yes—
117#103—42no—
118#103—41no—
119#103—41no—
120#103—42no—
121#97—22no—
122C4×C8—22no—
123C4×C8C22×C411yescharacteristic
124C4×C8C2×D411yescharacteristic
125C4×C8C2×D411yescharacteristic
126C4×C8#4111yescharacteristic
127C2×Q8—21no—
128C2×Q8—41no—
129C2×Q8—41no—
130C2×Q8#10212yes—
131C22×C4#12111yescharacteristic
132C22×C4C2×SD1611yescharacteristic
133C22×C4#13511yescharacteristic
134Q16—42no—
135Q16—81no—
136Q16—162no—
137M16—81no—
138M16—81no—
139M16—81no—
140M16—161no—
141C2×C8—22no—
142C2×C8—41no—
143C2×C8—41no—
144C2×C8—42no—
145C2×C8—81no—
146C2×C8—81no—
147C2×C8—161no—
148C2×C8C4×D411yescharacteristic
149C2×C8C4×D411yescharacteristic
150C2×C8#12011yescharacteristic
151C2×C8#12311yescharacteristic
152C4⋊C4—22no—
153C4⋊C4—22no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—41no—
158C4⋊C4—41no—
159C4⋊C4—41no—
160C4⋊C4—41no—
161C4⋊C4—81no—
162C4⋊C4—81no—
163C4⋊C4#10212yes—
164C22⋊C4—22no—
165C22⋊C4—81no—
166C42—22no—
167C42—81no—
168C42C2×C22⋊C411yescharacteristic
169C42#12111yescharacteristic
170C42C2×D811yescharacteristic
171C42#13511yescharacteristic
172C16—41no—
173C16—42no—
174C16—41no—
175C23#46511yescharacteristic
176Q8—42no—
177Q8—41no—
178Q8—41no—
179Q8—81no—
180Q8—81no—
181Q8—81no—
182Q8—81no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—41no—
190C2×C4—81no—
191C2×C4—81no—
192C2×C4#38611yescharacteristic
193C2×C4C2×#10211yescharacteristic
194C2×C4#41611yescharacteristic
195C2×C4#46511yescharacteristic
196C2×C4#46611yescharacteristic
197C2×C4#46611yescharacteristic
198C8—21no—
199C8—21no—
200C8—21no—
201C8—21no—
202C8—42no—
203C8—82no—
204C8—161no—
205C8—161no—
206C22—21no—
207C22—21no—
208C22—21no—
209C22#189311yescharacteristic
210C4—21no—
211C4—21no—
212C4—21no—
213C4—21no—
214C4—42no—
215C4—42no—
216C4—161no—
217C4—161no—
218C4#188111yescharacteristic
219C4#190511yescharacteristic
220C4#222211yescharacteristic
221C4#222311yescharacteristic
222C2—41no—
223C2#930611yescharacteristic
224C2#1307211yescharacteristic
225C2#1307611yescharacteristic
subgroupG/H
center, socleC22#1893
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC23#465
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2960 ◁ 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
#334992
#193502
#118652
#116462
#86892
#72922
#72192