Finite Group Databasebeta

← #150362

Subgroups — #150362

Order 512.

278 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 54 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#30089C211yesmaximal, characteristic
2#12627C211yesmaximal, characteristic
3#11480C211yesmaximal, characteristic
4#7434C212yesmaximal
5#7412C212yesmaximal
6#2647C411yescharacteristic, complemented
7#2326C2211yescharacteristic
8C2×#354—21no—
9#2095C411yescharacteristic, complemented
10#2093C2211yescharacteristic
11#2083—21no—
12#2074C411yescharacteristic, complemented
13#2073C2211yescharacteristic
14#2042—21no—
15#1863—21no—
16#1862—21no—
17#1840C411yescharacteristic, complemented
18C4×#99—21no—
19#1461—21no—
20#1393C2212yes—
21#1325—21no—
22#1318C2212yes—
23C2×#124D411yescharacteristic
24C2×C4×Q8D411yescharacteristic
25C2×C4×D4D411yescharacteristic
26#431—41no—
27C2×#107—21no—
28C2×#107—21no—
29C2×#103—21no—
30C2×#103—21no—
31C2×#102—21no—
32C2×#102—21no—
33C2×#99—22no—
34#400—41no—
35#398—41no—
36#391C2×C411yescharacteristic
37#390C2311yescharacteristic
38#389—21no—
39#389—21no—
40#388—41no—
41#385—41no—
42#384C2×C411yescharacteristic
43#384D411yescharacteristic
44#382C2×C411yescharacteristic
45C4×C22⋊C4—41no—
46#376C2×C411yescharacteristic
47#354—24no—
48#342—41no—
49#340—21no—
50#340—21no—
51#336—21no—
52#336—21no—
53#336C2×C411yescharacteristic
54#336C2×C411yescharacteristic
55#324—22no—
56#324—41no—
57C22×Q8C22⋊C411yescharacteristic
58C22×D4C2×D411yescharacteristic
59C23×C4—21no—
60C22×C8—21no—
61C22×C8—21no—
62#124—22no—
63#124—22no—
64C4×Q8—81no—
65C4×D4—81no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—41no—
70C2×C4⋊C4—41no—
71C2×C4⋊C4—41no—
72C2×C4⋊C4—41no—
73C2×C4⋊C4C22⋊C411yescharacteristic
74C2×C4⋊C4C2×D411yescharacteristic
75C2×C22⋊C4—21no—
76C2×C22⋊C4—41no—
77C2×C22⋊C4—42no—
78C2×C22⋊C4—41no—
79C2×C42—21no—
80C2×C42—81no—
81C2×C42C22⋊C411yescharacteristic
82C2×C42C22⋊C411yescharacteristic
83C2×C42C22×C411yescharacteristic
84#107—81no—
85#107—81no—
86C4⋊C8—81no—
87#103—42no—
88#103—81no—
89#102—42no—
90#102—81no—
91#99—24no—
92#99—42no—
93C22⋊C8—42no—
94C4×C8—81no—
95#95—21no—
96#95—21no—
97#95—41no—
98#95—41no—
99#95—41no—
100#95—41no—
101#95—81no—
102C24—21no—
103C2×Q8—21no—
104C2×Q8—41no—
105C2×Q8—41no—
106C2×Q8#10412yes—
107C2×D4—21no—
108C2×D4—41no—
109C2×D4—41no—
110C2×D4#13512yes—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—41no—
117C22×C4—41no—
118C22×C4—41no—
119C22×C4—41no—
120C22×C4—82no—
121C22×C4—161no—
122C22×C4#9911yescharacteristic
123C22×C4#9911yescharacteristic
124C22×C4C2×C22⋊C411yescharacteristic
125C2×C8—41no—
126C2×C8—41no—
127C2×C8—41no—
128C2×C8—41no—
129C2×C8—41no—
130C2×C8—41no—
131C2×C8—82no—
132C4⋊C4—41no—
133C4⋊C4—41no—
134C4⋊C4—41no—
135C4⋊C4—41no—
136C4⋊C4—41no—
137C4⋊C4—41no—
138C4⋊C4—81no—
139C4⋊C4—161no—
140C4⋊C4—161no—
141C4⋊C4—161no—
142C4⋊C4—161no—
143C22⋊C4—44no—
144C22⋊C4—44no—
145C22⋊C4—81no—
146C22⋊C4—82no—
147C22⋊C4—161no—
148C42—21no—
149C42—21no—
150C42—41no—
151C42—41no—
152C42—41no—
153C42—41no—
154C42—162no—
155C23—22no—
156C23—41no—
157C23—41no—
158C23—81no—
159C23—81no—
160C23C2×#9911yescharacteristic
161Q8—42no—
162Q8—81no—
163D4—42no—
164D4—81no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—82no—
183C2×C4—81no—
184C2×C4—81no—
185C2×C4—81no—
186C2×C4—81no—
187C2×C4—81no—
188C2×C4—81no—
189C2×C4—84no—
190C2×C4—81no—
191C2×C4—81no—
192C2×C4—161no—
193C2×C4—162no—
194C2×C4—162no—
195C2×C4#41611yescharacteristic
196C2×C4C2×#10411yescharacteristic
197C8—81no—
198C8—81no—
199C8—162no—
200C22—21no—
201C22—21no—
202C22—42no—
203C22—82no—
204C22—81no—
205C22—81no—
206C22—161no—
207C22#150111yescharacteristic
208C22#212311yescharacteristic
209C22#212711yescharacteristic
210C4—21no—
211C4—21no—
212C4—41no—
213C4—41no—
214C4—42no—
215C4—81no—
216C4—81no—
217C4—81no—
218C4—81no—
219C4—161no—
220C4—164no—
221C4—161no—
222C2—21no—
223C2—21no—
224C2—82no—
225C2#1173211yescharacteristic
226C2#1360811yescharacteristic
227C2#1360911yescharacteristic
subgroupG/H
center, socleC22#2127
commutatorC2×C42C22×C4
Frattini, cosocle#390C23
prime radicalC22×D4C2×D4
LCSG ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C2×C42 ◁ #2326 ◁ G
LECSG ▷ #390 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #2647C4semidirect
9. #2095C4semidirect
12. #2074C4semidirect
17. #1840C4semidirect

Maximal subgroups

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

subgroupindex
#300892
#126272
#114802
#74342
#74122