Finite Group Databasebeta

← #150888

Subgroups — #150888

Order 512.

277 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 277 conjugacy classes of subgroups — too many to draw. Only its 60 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#29268C211yesmaximal, characteristic
2#12803C211yesmaximal, characteristic
3#11481C211yesmaximal, characteristic
4#7436C212yesmaximal, complemented
5#7429C212yesmaximal, complemented
6#2330C411yescharacteristic
7#2326C2211yescharacteristic
8C2×#355—21no—
9#2095C2211yescharacteristic
10#2093C411yescharacteristic
11#2026C2211yescharacteristic
12#2025C411yescharacteristic
13#1983—21no—
14#1947—21no—
15#1863—21no—
16#1862—21no—
17#1832C411yescharacteristic
18#1801—21no—
19#1459—21no—
20#1394C2212yes—
21#1325—21no—
22#1319C2212yes—
23C2×#125C2×C411yescharacteristic
24C2×#124D411yescharacteristic
25C2×C4×Q8D411yescharacteristic
26C2×C4×D4D411yescharacteristic
27C2×#107—21no—
28C2×#107—21no—
29#423—41no—
30C2×C4⋊C8—21no—
31C2×#103—21no—
32C2×#103—21no—
33C2×#102—21no—
34C2×#102—21no—
35C2×#101—22no—
36C2×C22⋊C8—41no—
37#391C2311yescharacteristic
38#390C2×C411yescharacteristic
39#389—21no—
40#389—41no—
41#388—41no—
42#387—41no—
43#385—41no—
44#384C2×C411yescharacteristic
45#384D411yescharacteristic
46C4×C4⋊C4—21no—
47#376C2×C411yescharacteristic
48#355—42no—
49#343—41no—
50#340—21no—
51#340—21no—
52#336—21no—
53#336C2×C411yescharacteristic
54#336C2×C411yescharacteristic
55#324—22no—
56#323—41no—
57C22×Q8C2×D411yescharacteristic
58C22×D4C22⋊C411yescharacteristic
59C23×C4—21no—
60C2×M16—42no—
61C22×C8—21no—
62C22×C8—21no—
63C22×C8—81no—
64#125—42no—
65#124—41no—
66#124SD1612yes—
67#124Q1612yes—
68C4×Q8—81no—
69C4×D4—81no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—41no—
74C2×C4⋊C4—41no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4C22⋊C411yescharacteristic
78C2×C4⋊C4C2×D411yescharacteristic
79C2×C22⋊C4—21no—
80C2×C22⋊C4—41no—
81C2×C22⋊C4—41no—
82C2×C42—21no—
83C2×C42—41no—
84C2×C42C22⋊C411yescharacteristic
85C2×C42C22⋊C411yescharacteristic
86C2×C42C22×C411yescharacteristic
87#107—81no—
88#107—81no—
89C4⋊C8—42no—
90C4⋊C8—82no—
91#103—42no—
92#103—81no—
93#102—42no—
94#102—81no—
95#101—42no—
96#101—42no—
97C22⋊C8—42no—
98C22⋊C8—81no—
99#95—21no—
100#95—21no—
101#95—41no—
102#95—41no—
103#95—41no—
104C24—21no—
105C2×Q8—21no—
106C2×Q8—41no—
107C2×Q8—41no—
108C2×Q8C2×SD1611yescharacteristic
109C2×Q8C2×Q1611yescharacteristic
110C2×D4—21no—
111C2×D4—41no—
112C2×D4—41no—
113C2×D4#10312yes—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—41no—
120C22×C4—41no—
121C22×C4—41no—
122C22×C4—41no—
123C22×C4—161no—
124C22×C4#9911yescharacteristic
125C22×C4#9911yescharacteristic
126C22×C4C2×C22⋊C411yescharacteristic
127M16—82no—
128M16—82no—
129C2×C8—41no—
130C2×C8—41no—
131C2×C8—41no—
132C2×C8—41no—
133C2×C8—41no—
134C2×C8—41no—
135C2×C8—82no—
136C2×C8—82no—
137C2×C8—162no—
138C4⋊C4—22no—
139C4⋊C4—41no—
140C4⋊C4—41no—
141C4⋊C4—41no—
142C4⋊C4—41no—
143C4⋊C4—41no—
144C4⋊C4—42no—
145C4⋊C4—42no—
146C4⋊C4—81no—
147C4⋊C4—161no—
148C4⋊C4—161no—
149C4⋊C4—161no—
150C22⋊C4—42no—
151C22⋊C4—42no—
152C22⋊C4—81no—
153C22⋊C4—161no—
154C42—21no—
155C42—41no—
156C42—41no—
157C42—42no—
158C42—161no—
159C42#10312yes—
160C23—22no—
161C23—41no—
162C23—41no—
163C23—81no—
164C23—81no—
165C23C2×#9911yescharacteristic
166Q8—41no—
167Q8—41no—
168Q8—81no—
169D4—42no—
170D4—81no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—42no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—82no—
191C2×C4—81no—
192C2×C4—81no—
193C2×C4—81no—
194C2×C4—81no—
195C2×C4—81no—
196C2×C4—81no—
197C2×C4—81no—
198C2×C4—161no—
199C2×C4C2×#10311yescharacteristic
200C2×C4#42111yescharacteristic
201C8—81no—
202C8—81no—
203C8—162no—
204C8—162no—
205C22—21no—
206C22—21no—
207C22—42no—
208C22—82no—
209C22—81no—
210C22—81no—
211C22—161no—
212C22#152011yescharacteristic
213C22#212011yescharacteristic
214C22#212711yescharacteristic
215C4—21no—
216C4—21no—
217C4—42no—
218C4—41no—
219C4—41no—
220C4—41no—
221C4—41no—
222C4—81no—
223C4—82no—
224C4—81no—
225C4—81no—
226C4—161no—
227C2—21no—
228C2—21no—
229C2—82no—
230C2#1189711yescharacteristic
231C2#1360511yescharacteristic
232C2#1361211yescharacteristic
subgroupG/H
center, socleC22#2127
commutatorC2×C42C22×C4
Frattini, cosocle#391C23
prime radicalC22×D4C22⋊C4
LCSG ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C2×C42 ◁ #2326 ◁ G
LECSG ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7436C2semidirect
5. #7429C2semidirect

Maximal subgroups

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

subgroupindex
#292682
#128032
#114812
#74362
#74292