Finite Group Databasebeta

← #122843

Subgroups — #122843

Order 512.

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

normal subgroups only

This group has 317 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#27661C211yesmaximal, characteristic
2#11466C211yesmaximal, characteristic
3#8614C211yesmaximal, characteristic
4#7091C214yesmaximal, complemented
5C2×C8×D4C2211yescharacteristic
6C22×#107C411yescharacteristic
7#2486—21no—
8#2112—21no—
9#2086C2212yes—
10#2075C412yes—
11#1942—21no—
12#1823—21no—
13#1808—21no—
14#1808C411yescharacteristic
15#1800—21no—
16#1566—21no—
17#1322C2212yes—
18#1314—21no—
19#1292C2212yes—
20C23×C8C2×C411yescharacteristic
21C23×C8D411yescharacteristic
22C2×#123—41no—
23C2×C4×D4D411yescharacteristic
24C22×C4⋊C4—21no—
25C2×M32—41no—
26C8×D4—41no—
27C8×D4—41no—
28C2×#107—21no—
29C2×#107—21no—
30C2×#107—41no—
31C2×#107—41no—
32C2×#107C2×C412yes—
33#423—22no—
34C2×C4⋊C8C2311yescharacteristic
35C2×#100—41no—
36C2×C22⋊C8C2×C411yescharacteristic
37C2×C22⋊C8D411yescharacteristic
38C2×C4×C8D411yescharacteristic
39#398—41no—
40#397—41no—
41#389—22no—
42#381—41no—
43#380—22no—
44#380—21no—
45C22⋊C16—22no—
46C22⋊C16—41no—
47#343—41no—
48#340—22no—
49#340—22no—
50#340C2×C412yes—
51#334—41no—
52#334—41no—
53C22×D4D811yescharacteristic
54C23×C4—41no—
55C23×C4C22⋊C411yescharacteristic
56C23×C4C22⋊C411yescharacteristic
57C2×M16—42no—
58C22×C8—21no—
59C22×C8—21no—
60C22×C8—21no—
61C22×C8—21no—
62C22×C8—42no—
63C22×C8—42no—
64C22×C8C22⋊C411yescharacteristic
65C22×C8C22⋊C411yescharacteristic
66C22×C8C22×C411yescharacteristic
67C22×C8C2×D411yescharacteristic
68#123—84no—
69C4×D4—41no—
70C4×D4—41no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—42no—
74C2×C4⋊C4—42no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4—42no—
77C2×C4⋊C4D811yescharacteristic
78C2×C22⋊C4—41no—
79C2×C22⋊C4—42no—
80C2×C22⋊C4SD1611yescharacteristic
81C2×C22⋊C4SD1611yescharacteristic
82C2×C42C2×D411yescharacteristic
83M32—82no—
84C2×C16—42no—
85#107—42no—
86#107—41no—
87#107—41no—
88#107—42no—
89C4⋊C8—41no—
90C4⋊C8—42no—
91C4⋊C8—42no—
92#100—82no—
93C22⋊C8—41no—
94C22⋊C8—41no—
95C4×C8—21no—
96C4×C8—21no—
97#95—42no—
98#95—41no—
99#95—42no—
100C24#10211yescharacteristic
101C24#10211yescharacteristic
102C2×D4—22no—
103C2×D4—22no—
104C2×D4—41no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—41no—
110C22×C4—41no—
111C22×C4—41no—
112C22×C4—41no—
113C22×C4—42no—
114C22×C4—42no—
115C22×C4—82no—
116C22×C4—82no—
117C22×C4—82no—
118C22×C4#10211yescharacteristic
119C22×C4#10211yescharacteristic
120C22×C4C2×C22⋊C411yescharacteristic
121C22×C4C2×D811yescharacteristic
122C22×C4C2×SD1611yescharacteristic
123M16—82no—
124M16—82no—
125C2×C8—22no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—42no—
130C2×C8—42no—
131C2×C8—42no—
132C2×C8—41no—
133C2×C8—41no—
134C2×C8—41no—
135C2×C8—81no—
136C2×C8—82no—
137C2×C8#9911yescharacteristic
138C2×C8#9911yescharacteristic
139C4⋊C4—41no—
140C4⋊C4—42no—
141C4⋊C4—42no—
142C4⋊C4—42no—
143C4⋊C4—42no—
144C4⋊C4—82no—
145C4⋊C4—84no—
146C22⋊C4—41no—
147C22⋊C4—41no—
148C22⋊C4—82no—
149C22⋊C4—84no—
150C42—21no—
151C42—21no—
152C16—82no—
153C23—21no—
154C23—21no—
155C23—21no—
156C23—21no—
157C23—21no—
158C23—41no—
159C23—41no—
160C23—41no—
161C23—41no—
162C23#36012yes—
163C23C2×#10211yescharacteristic
164D4—44no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—42no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—81no—
181C2×C4—82no—
182C2×C4—82no—
183C2×C4—82no—
184C2×C4—82no—
185C2×C4—82no—
186C2×C4—82no—
187C2×C4—164no—
188C2×C4C2×#9911yescharacteristic
189C2×C4#42111yescharacteristic
190C2×C4#50912yes—
191C8—21no—
192C8—41no—
193C8—41no—
194C8—41no—
195C8—162no—
196C8#35212yes—
197C22—21no—
198C22—21no—
199C22—22no—
200C22—42no—
201C22—42no—
202C22—42no—
203C22—41no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22—81no—
209C22#151911yescharacteristic
210C22#214011yescharacteristic
211C22C2×#36011yescharacteristic
212C4—21no—
213C4—22no—
214C4—41no—
215C4—41no—
216C4—41no—
217C4—162no—
218C4—162no—
219C4C2×#35211yescharacteristic
220C4#213511yescharacteristic
221C2—21no—
222C2—21no—
223C2—42no—
224C2—81no—
225C2#1187111yescharacteristic
226C2#1188111yescharacteristic
227C2#1188811yescharacteristic
subgroupG/H
center, socleC22#1519
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4⋊C8C23
prime radicalC22×D4D8
LCSG ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #7091C2semidirect

Maximal subgroups

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

subgroupindex
#276612
#114662
#86142
#70912