Finite Group Databasebeta

← #101778

Subgroups — #101778

Order 512.

416 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 92 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#28751C211yesmaximal, characteristic
2#23263C211yesmaximal, characteristic, complemented
3#8584C211yesmaximal, characteristic, complemented
4#7679C212yesmaximal
5#7027C212yesmaximal, complemented
6C2×#488—21no—
7#2858C2211yescharacteristic
8C4×#125C2211yescharacteristic
9#2043C412yes—
10#2041C412yes—
11#1983—22no—
12#1958—21no—
13#1916—22no—
14#1916—22no—
15C8×C4⋊C4—22no—
16#1767C2212yes—
17C4×C4⋊C8C2211yescharacteristic
18#1744C2212yes—
19C2×#330—21no—
20#1295—21no—
21#1270—22no—
22#1270—22no—
23C2×C4⋊Q8C2×C411yescharacteristic
24C2×#127C2×C411yescharacteristic
25C2×#125D411yescharacteristic
26C2×#124—21no—
27C2×#124—21no—
28#488—42no—
29#488—42no—
30C2×C4⋊C8—22no—
31C2×C4⋊C8—22no—
32C2×C4⋊C8—21no—
33C2×#103—22no—
34C2×#103—22no—
35C2×#102—22no—
36C2×#102—22no—
37C2×#97—22no—
38C2×C4×C8—22no—
39C2×C4×C8—21no—
40C2×C4×C8—22no—
41#390—22no—
42#390—22no—
43#389C2×C412yes—
44#387—22no—
45#382D411yescharacteristic
46C4×C4⋊C4—22no—
47C4×C4⋊C4D412yes—
48C43C2311yescharacteristic
49#336—22no—
50#336—22no—
51#336—22no—
52#336—22no—
53#336C2×C412yes—
54#336D412yes—
55#336D412yes—
56#330—42no—
57C22×Q8—21no—
58C22×D4—21no—
59C22×D4—22no—
60C22×C8—22no—
61C22×C8—22no—
62C22×C8—22no—
63C22×C8—42no—
64C4⋊Q8—81no—
65#127—81no—
66#125—22no—
67#125D812yes—
68#125SD1612yes—
69#124—44no—
70#124—41no—
71#124—41no—
72#124—42no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4C22⋊C412yes—
77C2×C4⋊C4C22⋊C412yes—
78C2×C22⋊C4—41no—
79C2×C22⋊C4—42no—
80C2×C22⋊C4—41no—
81C2×C42—22no—
82C2×C42C22×C411yescharacteristic
83C2×C42C2×D411yescharacteristic
84C2×C42C2×D412yes—
85C2×C42C2×D411yescharacteristic
86C2×C42#4112yes—
87C4⋊C8—24no—
88C4⋊C8—42no—
89C4⋊C8—42no—
90C4⋊C8—44no—
91#103—44no—
92#103—82no—
93#102—44no—
94#102—82no—
95#97—42no—
96#97—42no—
97C4×C8—44no—
98C4×C8—44no—
99C4×C8—41no—
100C4×C8—41no—
101#95—22no—
102#95—22no—
103#95—22no—
104C24—41no—
105C2×Q8—42no—
106C2×Q8—42no—
107C2×Q8—41no—
108C2×Q8—41no—
109C2×D4—42no—
110C2×D4—42no—
111C2×D4—41no—
112C2×D4—41no—
113C2×D4—42no—
114C2×D4—42no—
115C2×D4—82no—
116C2×D4—82no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—41no—
120C22×C4C2×C22⋊C411yescharacteristic
121C22×C4C4×D412yes—
122C22×C4#12011yescharacteristic
123C22×C4#12112yes—
124C22×C4#12311yescharacteristic
125C2×C8—24no—
126C2×C8—42no—
127C2×C8—42no—
128C2×C8—42no—
129C2×C8—42no—
130C2×C8—44no—
131C2×C8—42no—
132C2×C8—42no—
133C2×C8—42no—
134C2×C8—42no—
135C2×C8—82no—
136C2×C8—82no—
137C4⋊C4—22no—
138C4⋊C4—24no—
139C4⋊C4—24no—
140C4⋊C4—42no—
141C4⋊C4—42no—
142C4⋊C4—42no—
143C4⋊C4—82no—
144C4⋊C4#10214yes—
145C22⋊C4—82no—
146C22⋊C4—82no—
147C22⋊C4—82no—
148C22⋊C4—82no—
149C42—22no—
150C42—22no—
151C42—22no—
152C42—42no—
153C42—41no—
154C42—42no—
155C42—42no—
156C42C2×D811yescharacteristic
157C42C2×SD1611yescharacteristic
158C42#13512yes—
159C23—82no—
160C23—81no—
161C23—81no—
162C23—82no—
163C23—81no—
164C23#38611yescharacteristic
165Q8—82no—
166Q8—82no—
167D4—82no—
168D4—82no—
169D4—162no—
170D4—162no—
171C2×C4—22no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—21no—
180C2×C4—24no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—81no—
187C2×C4—81no—
188C2×C4—81no—
189C2×C4C2×#10211yescharacteristic
190C2×C4#41611yescharacteristic
191C2×C4#44312yes—
192C2×C4#44412yes—
193C2×C4#45412yes—
194C2×C4#46512yes—
195C2×C4#46612yes—
196C8—44no—
197C8—82no—
198C8—82no—
199C8—84no—
200C22—162no—
201C22—161no—
202C22—161no—
203C22—161no—
204C22—162no—
205C22#187612yes—
206C22#188811yescharacteristic
207C22#189311yescharacteristic
208C22#190612yes—
209C22#191411yescharacteristic
210C4—22no—
211C4—22no—
212C4—21no—
213C4—21no—
214C4—22no—
215C4—22no—
216C4—22no—
217C4—41no—
218C4—44no—
219C4—82no—
220C4—161no—
221C2—321no—
222C2#907011yescharacteristic
223C2#920812yes—
224C2#926512yes—
225C2#939811yescharacteristic
226C2#944211yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
prime radicalC2×#127C2×C4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #23263C2semidirect
3. #8584C2semidirect
5. #7027C2semidirect

Maximal subgroups

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

subgroupindex
#287512
#232632
#85842
#76792
#70272