Finite Group Databasebeta

← #150586

Subgroups — #150586

Order 512.

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

normal subgroups only

This group has 353 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#23627C211yesmaximal, characteristic
2#12857C211yesmaximal, characteristic
3#11507C211yesmaximal, characteristic
4#7412C214yesmaximal, complemented
5C2×#416—21no—
6#2297C2211yescharacteristic
7#2285C411yescharacteristic, complemented
8#2074C412yescomplemented
9#2073C2212yes—
10#2072—21no—
11#2071—21no—
12#1782—21no—
13#1735—21no—
14C2×#336C411yescharacteristic, complemented
15#1464—21no—
16#1393—21no—
17#1393C2212yes—
18#1319—21no—
19#1318C2212yes—
20C22×#41D411yescharacteristic
21C23×C8—21no—
22C2×#117—21no—
23C2×#117D411yescharacteristic
24C22×C42C2×C411yescharacteristic
25#432—41no—
26#416—44no—
27C2×#103—22no—
28C2×#102—22no—
29C2×#99—22no—
30C2×#99—41no—
31#391C2×C411yescharacteristic
32#391D411yescharacteristic
33#390D411yescharacteristic
34#390C2311yescharacteristic
35#386—41no—
36#384—21no—
37#382C2×C412yes—
38#381—41no—
39#380—41no—
40C4×C4⋊C4—41no—
41C4×C22⋊C4—41no—
42#340—22no—
43#340—22no—
44#336—21no—
45#336C2×C412yes—
46#324—22no—
47#323—41no—
48C2×#41—22no—
49C2×#41—41no—
50C2×#41—41no—
51C2×#41—41no—
52C2×#41—42no—
53C22×Q8C22⋊C411yescharacteristic
54C22×D4—21no—
55C22×D4C2×D411yescharacteristic
56C23×C4—21no—
57C23×C4—21no—
58C23×C4C22⋊C411yescharacteristic
59C22×C8—22no—
60C22×C8—42no—
61C22×C8—41no—
62C22×C8—41no—
63C22×C8—42no—
64#117—41no—
65#117—41no—
66#117—42no—
67#117—42no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—41no—
70C2×C4⋊C4—41no—
71C2×C4⋊C4C22⋊C411yescharacteristic
72C2×C4⋊C4C2×D411yescharacteristic
73C2×C22⋊C4—21no—
74C2×C22⋊C4—41no—
75C2×C22⋊C4—41no—
76C2×C22⋊C4—41no—
77C2×C22⋊C4—42no—
78C2×C42—21no—
79C2×C42—21no—
80C2×C42—42no—
81C2×C42—41no—
82C2×C42C22⋊C411yescharacteristic
83C2×C42C22×C411yescharacteristic
84C4⋊C8—81no—
85#103—44no—
86#102—44no—
87#99—42no—
88#99—42no—
89#99—82no—
90C22⋊C8—42no—
91#97—81no—
92#95—41no—
93#95—41no—
94#95—42no—
95#95—81no—
96#95—81no—
97C24—21no—
98C24#9911yescharacteristic
99#41—44no—
100#41—44no—
101#41—82no—
102#41—82no—
103C2×Q8—21no—
104C2×Q8—21no—
105C2×Q8—41no—
106C2×Q8—41no—
107C2×D4—21no—
108C2×D4—41no—
109C2×D4—41no—
110C2×D4—42no—
111C2×D4—42no—
112C2×D4—81no—
113C2×D4AGL(1,ℤ/8)12yes—
114C22×C4—22no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—42no—
125C22×C4—42no—
126C22×C4—42no—
127C22×C4—41no—
128C22×C4—41no—
129C22×C4—42no—
130C22×C4—81no—
131C22×C4—81no—
132C22×C4—82no—
133C22×C4#9911yescharacteristic
134C22×C4C2×C22⋊C411yescharacteristic
135C2×C8—42no—
136C2×C8—42no—
137C2×C8—42no—
138C2×C8—82no—
139C2×C8—82no—
140C2×C8—82no—
141C4⋊C4—41no—
142C4⋊C4—41no—
143C4⋊C4—42no—
144C4⋊C4—42no—
145C4⋊C4—82no—
146C4⋊C4—82no—
147C22⋊C4—82no—
148C22⋊C4—81no—
149C22⋊C4—82no—
150C22⋊C4—82no—
151C22⋊C4—82no—
152C22⋊C4—82no—
153C42—41no—
154C42—41no—
155C42—42no—
156C42—41no—
157C42—41no—
158C42—82no—
159C23—21no—
160C23—21no—
161C23—22no—
162C23—41no—
163C23—41no—
164C23—41no—
165C23—41no—
166C23—81no—
167C23—81no—
168C23#35212yes—
169C23C2×#9911yescharacteristic
170Q8—42no—
171Q8—81no—
172D4—42no—
173D4—81no—
174D4—84no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—82no—
195C2×C4—82no—
196C2×C4—84no—
197C2×C4—81no—
198C2×C4—81no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4—162no—
203C2×C4—162no—
204C2×C4—162no—
205C2×C4#35411yescharacteristic
206C2×C4#35411yescharacteristic
207C2×C4#41811yescharacteristic
208C2×C4#42111yescharacteristic
209C8—82no—
210C8—162no—
211C22—21no—
212C22—21no—
213C22—22no—
214C22—42no—
215C22—42no—
216C22—41no—
217C22—41no—
218C22—42no—
219C22—82no—
220C22—81no—
221C22—81no—
222C22—161no—
223C22#151011yescharacteristic
224C22C2×#35211yescharacteristic
225C22C2×#35411yescharacteristic
226C4—21no—
227C4—21no—
228C4—21no—
229C4—21no—
230C4—42no—
231C4—81no—
232C4—81no—
233C4—81no—
234C4—81no—
235C4—162no—
236C4—322no—
237C2—21no—
238C2—21no—
239C2—42no—
240C2—82no—
241C2#1179711yescharacteristic
242C2#1359411yescharacteristic
243C2#1360211yescharacteristic
subgroupG/H
center, socleC22C2×#354
commutatorC2×C42C22×C4
Frattini, cosocle#390C23
prime radicalC22×#41D4
LCSG ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ #2297 ◁ G
LECSG ▷ #390 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7412C2semidirect
7. #2285C4semidirect
8. #2074C4semidirect
14. C2×#336C4semidirect

Maximal subgroups

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

subgroupindex
#236272
#128572
#115072
#74122