Finite Group Databasebeta

← #150650

Subgroups — #150650

Order 512.

307 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 307 conjugacy classes of subgroups — too many to draw. Only its 52 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#29964C211yesmaximal, characteristic
2#12859C211yesmaximal, characteristic
3#11508C211yesmaximal, characteristic
4#7434C212yesmaximal, complemented
5#7428C212yesmaximal, complemented
6#2629C411yescharacteristic, complemented
7#2311C2211yescharacteristic
8#2095C411yescharacteristic, complemented
9#2093C2211yescharacteristic
10#2068—21no—
11#2067—21no—
12#2026C411yescharacteristic, complemented
13#2025C2211yescharacteristic
14#1852—21no—
15#1798C411yescharacteristic, complemented
16#1788—21no—
17#1782—21no—
18#1463—21no—
19#1393—21no—
20#1393C2212yes—
21#1319—21no—
22#1318C2212yes—
23C22×#41D411yescharacteristic
24C2×#125C2×C411yescharacteristic
25C2×#117C2×C411yescharacteristic
26C22×C4⋊C4D411yescharacteristic
27C2×#107—21no—
28C2×#107—21no—
29C2×C4⋊C8—21no—
30C2×#103—21no—
31C2×#103—21no—
32C2×#102—21no—
33C2×#102—21no—
34C2×#99—22no—
35C2×#99—41no—
36#407—41no—
37C2×C22⋊C8—21no—
38#398—41no—
39#396—41no—
40#391C2×C411yescharacteristic
41#391D411yescharacteristic
42#390D411yescharacteristic
43#390C2311yescharacteristic
44#389—21no—
45#386—41no—
46#384C2×C411yescharacteristic
47#381—41no—
48#380—21no—
49#380—41no—
50#340—21no—
51#340—21no—
52#336C2×C411yescharacteristic
53#336C2×C411yescharacteristic
54#324—22no—
55#324—41no—
56C2×#41—22no—
57C2×#41—41no—
58C2×#41—81no—
59C2×#41—81no—
60C22×Q8C22⋊C411yescharacteristic
61C22×D4—21no—
62C22×D4C2×D411yescharacteristic
63C23×C4—21no—
64C23×C4—21no—
65C23×C4C22⋊C411yescharacteristic
66C2×M16—81no—
67C22×C8—21no—
68C22×C8—21no—
69#125—81no—
70#117—81no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—81no—
79C2×C4⋊C4C22⋊C411yescharacteristic
80C2×C4⋊C4C2×D411yescharacteristic
81C2×C22⋊C4—21no—
82C2×C22⋊C4—42no—
83C2×C22⋊C4—41no—
84C2×C22⋊C4—41no—
85C2×C22⋊C4—41no—
86C2×C42C22⋊C411yescharacteristic
87C2×C42C22×C411yescharacteristic
88#107—81no—
89#107—81no—
90C4⋊C8—81no—
91#103—81no—
92#103—81no—
93#102—81no—
94#102—81no—
95#99—42no—
96#99—42no—
97#99—82no—
98C22⋊C8—42no—
99C22⋊C8—42no—
100C22⋊C8—81no—
101#95—41no—
102#95—41no—
103#95—41no—
104#95—41no—
105#95—82no—
106C24—21no—
107C24#9911yescharacteristic
108#41—84no—
109#41—84no—
110C2×Q8—21no—
111C2×Q8—21no—
112C2×Q8—41no—
113C2×Q8—41no—
114C2×D4—21no—
115C2×D4—41no—
116C2×D4—41no—
117C2×D4—42no—
118C2×D4—42no—
119C2×D4—81no—
120C2×D4#13712yes—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—41no—
129C22×C4—41no—
130C22×C4—42no—
131C22×C4—42no—
132C22×C4—41no—
133C22×C4—41no—
134C22×C4—41no—
135C22×C4—41no—
136C22×C4—81no—
137C22×C4—82no—
138C22×C4—161no—
139C22×C4#9911yescharacteristic
140C22×C4C2×C22⋊C411yescharacteristic
141M16—162no—
142C2×C8—41no—
143C2×C8—41no—
144C2×C8—41no—
145C2×C8—41no—
146C2×C8—41no—
147C2×C8—41no—
148C2×C8—82no—
149C4⋊C4—41no—
150C4⋊C4—41no—
151C4⋊C4—41no—
152C4⋊C4—41no—
153C4⋊C4—41no—
154C4⋊C4—41no—
155C4⋊C4—81no—
156C4⋊C4—81no—
157C4⋊C4—82no—
158C4⋊C4—161no—
159C4⋊C4—161no—
160C22⋊C4—81no—
161C22⋊C4—82no—
162C22⋊C4—82no—
163C22⋊C4—161no—
164C22⋊C4—161no—
165C22⋊C4—161no—
166C42—41no—
167C42—41no—
168C23—21no—
169C23—21no—
170C23—21no—
171C23—22no—
172C23—41no—
173C23—41no—
174C23—41no—
175C23—41no—
176C23—81no—
177C23—81no—
178C23C2×#9911yescharacteristic
179Q8—42no—
180Q8—81no—
181D4—42no—
182D4—81no—
183D4—84no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—84no—
203C2×C4—82no—
204C2×C4—81no—
205C2×C4—81no—
206C2×C4—81no—
207C2×C4—81no—
208C2×C4—81no—
209C2×C4—81no—
210C2×C4—81no—
211C2×C4—81no—
212C2×C4—161no—
213C2×C4—162no—
214C2×C4—162no—
215C2×C4—162no—
216C2×C4#41911yescharacteristic
217C2×C4#42111yescharacteristic
218C8—81no—
219C8—81no—
220C8—162no—
221C22—21no—
222C22—21no—
223C22—42no—
224C22—41no—
225C22—42no—
226C22—41no—
227C22—41no—
228C22—81no—
229C22—82no—
230C22—81no—
231C22—81no—
232C22—161no—
233C22#151111yescharacteristic
234C22#212111yescharacteristic
235C22#212711yescharacteristic
236C4—21no—
237C4—21no—
238C4—41no—
239C4—81no—
240C4—81no—
241C4—81no—
242C4—81no—
243C4—81no—
244C4—161no—
245C4—161no—
246C4—322no—
247C2—21no—
248C2—21no—
249C2—82no—
250C2—81no—
251C2#1181111yescharacteristic
252C2#1360811yescharacteristic
253C2#1360911yescharacteristic
subgroupG/H
center, socleC22#2127
commutatorC2×C42C22×C4
Frattini, cosocle#390C23
prime radicalC22×#41D4
LCSG ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C23×C4 ◁ #2311 ◁ G
LECSG ▷ #390 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7434C2semidirect
5. #7428C2semidirect
6. #2629C4semidirect
8. #2095C4semidirect
12. #2026C4semidirect
15. #1798C4semidirect

Maximal subgroups

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

subgroupindex
#299642
#128592
#115082
#74342
#74282