Finite Group Databasebeta

← #150348

Subgroups — #150348

Order 512.

337 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 337 conjugacy classes of subgroups — too many to draw. Only its 56 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#23824C211yesmaximal, characteristic
2C4×#353C211yesmaximal, characteristic
3#11475C211yesmaximal, characteristic
4#7428C212yesmaximal, complemented
5#7412C212yesmaximal, complemented
6C4×#127C2211yescharacteristic
7C4×#125C411yescharacteristic, complemented
8C2×#353—21no—
9#2099—21no—
10#2073C411yescharacteristic, complemented
11#2073C2211yescharacteristic
12#2041—21no—
13#2025C411yescharacteristic, complemented
14#2025C2211yescharacteristic
15#1762—21no—
16#1762—21no—
17C4×#99—21no—
18#1745C411yescharacteristic, complemented
19#1459—21no—
20#1393C2212yes—
21#1325—21no—
22#1318C2212yes—
23C2×#127D411yescharacteristic
24C2×#125C2×C411yescharacteristic
25C2×C4×D4—41no—
26C2×C4×D4D411yescharacteristic
27C2×C4×D4D411yescharacteristic
28C2×#102—21no—
29C2×#102—21no—
30C2×#102—21no—
31C2×#102—21no—
32C2×#99—22no—
33C2×C22⋊C8—41no—
34C2×#97—21no—
35#390—41no—
36#390C2×C411yescharacteristic
37#390C2311yescharacteristic
38#389—21no—
39#387—41no—
40#384D411yescharacteristic
41#382C2×C411yescharacteristic
42C4×C4⋊C4—21no—
43C4×C4⋊C4—41no—
44C4×C22⋊C4—41no—
45C43C2×C411yescharacteristic
46#353—24no—
47#342—41no—
48#340—21no—
49#340—21no—
50#340—21no—
51#340—21no—
52#336—21no—
53#336C2×C411yescharacteristic
54#336C2×C411yescharacteristic
55#324—22no—
56#323—41no—
57C22×D4—41no—
58C22×D4C22⋊C411yescharacteristic
59C22×D4C2×D411yescharacteristic
60C23×C4—21no—
61C23×C4—21no—
62C22×C8—21no—
63C22×C8—21no—
64C22×C8—81no—
65#127—22no—
66#127—22no—
67#125—42no—
68C4×D4—41no—
69C4×D4—41no—
70C4×D4—41no—
71C4×D4—41no—
72C4×D4—84no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4C22⋊C411yescharacteristic
79C2×C4⋊C4C2×D411yescharacteristic
80C2×C22⋊C4—21no—
81C2×C22⋊C4—21no—
82C2×C22⋊C4—42no—
83C2×C22⋊C4—41no—
84C2×C22⋊C4—41no—
85C2×C42—21no—
86C2×C42—21no—
87C2×C42—41no—
88C2×C42—81no—
89C2×C42C22⋊C411yescharacteristic
90C2×C42C22⋊C411yescharacteristic
91C2×C42C22×C411yescharacteristic
92#102—42no—
93#102—42no—
94#102—42no—
95#102—42no—
96#99—24no—
97#99—42no—
98C22⋊C8—42no—
99C22⋊C8—81no—
100#97—41no—
101#97—41no—
102#95—41no—
103#95—41no—
104#95—41no—
105#95—81no—
106C24—21no—
107C24—21no—
108C2×D4—21no—
109C2×D4—21no—
110C2×D4—41no—
111C2×D4—41no—
112C2×D4—41no—
113C2×D4—41no—
114C2×D4—44no—
115C2×D4—82no—
116C2×D4—82no—
117C2×D4#10412yes—
118C2×D4#13512yes—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—41no—
124C22×C4—41no—
125C22×C4—41no—
126C22×C4—41no—
127C22×C4—41no—
128C22×C4—41no—
129C22×C4—82no—
130C22×C4—82no—
131C22×C4—82no—
132C22×C4#9911yescharacteristic
133C22×C4#9911yescharacteristic
134C22×C4C2×C22⋊C411yescharacteristic
135C2×C8—41no—
136C2×C8—41no—
137C2×C8—41no—
138C2×C8—41no—
139C2×C8—41no—
140C2×C8—41no—
141C2×C8—82no—
142C2×C8—162no—
143C4⋊C4—22no—
144C4⋊C4—41no—
145C4⋊C4—41no—
146C4⋊C4—41no—
147C4⋊C4—42no—
148C4⋊C4—42no—
149C4⋊C4—41no—
150C4⋊C4—41no—
151C4⋊C4—82no—
152C4⋊C4—82no—
153C22⋊C4—44no—
154C22⋊C4—81no—
155C22⋊C4—81no—
156C22⋊C4—82no—
157C22⋊C4—82no—
158C22⋊C4—82no—
159C42—21no—
160C42—21no—
161C42—21no—
162C42—21no—
163C42—41no—
164C42—42no—
165C42—81no—
166C42—82no—
167C42—162no—
168C23—22no—
169C23—22no—
170C23—41no—
171C23—41no—
172C23—41no—
173C23—41no—
174C23—81no—
175C23—81no—
176C23—81no—
177C23—81no—
178C23C2×#9911yescharacteristic
179D4—42no—
180D4—42no—
181D4—81no—
182D4—81no—
183D4—88no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—21no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—42no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4—81no—
201C2×C4—81no—
202C2×C4—81no—
203C2×C4—81no—
204C2×C4—81no—
205C2×C4—81no—
206C2×C4—84no—
207C2×C4—81no—
208C2×C4—161no—
209C2×C4—161no—
210C2×C4—162no—
211C2×C4—162no—
212C2×C4#35411yescharacteristic
213C2×C4#35411yescharacteristic
214C2×C4#41611yescharacteristic
215C2×C4C2×#10411yescharacteristic
216C8—81no—
217C8—81no—
218C8—162no—
219C22—21no—
220C22—21no—
221C22—42no—
222C22—42no—
223C22—82no—
224C22—82no—
225C22—81no—
226C22—81no—
227C22—81no—
228C22—81no—
229C22—161no—
230C22—161no—
231C22#150111yescharacteristic
232C22#212011yescharacteristic
233C22C2×#35411yescharacteristic
234C4—21no—
235C4—21no—
236C4—21no—
237C4—21no—
238C4—42no—
239C4—41no—
240C4—81no—
241C4—82no—
242C4—81no—
243C4—81no—
244C4—161no—
245C4—164no—
246C2—21no—
247C2—21no—
248C2—82no—
249C2—82no—
250C2#1172811yescharacteristic
251C2#1359511yescharacteristic
252C2#1360411yescharacteristic
subgroupG/H
center, socleC22C2×#354
commutatorC2×C42C22×C4
Frattini, cosocle#390C23
prime radicalC2×#127D4
LCSG ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×#127 ◁ G
LECSG ▷ #390 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7428C2semidirect
5. #7412C2semidirect
7. C4×#125C4semidirect
10. #2073C4semidirect
13. #2025C4semidirect
18. #1745C4semidirect

Maximal subgroups

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

subgroupindex
#238242
C4×#3532
#114752
#74282
#74122