Finite Group Databasebeta

← #150646

Subgroups — #150646

Order 512.

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

normal subgroups only

This group has 391 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#23734C211yesmaximal, characteristic
2#12861C211yesmaximal, characteristic
3#11507C211yesmaximal, characteristic
4#7437C214yesmaximal, complemented
5C2×#428—21no—
6C2×#417—21no—
7#2297C2211yescharacteristic
8C2×#384C411yescharacteristic
9#2096C2212yes—
10#2095—21no—
11#2094—21no—
12#2093C412yes—
13#1785—21no—
14#1728C411yescharacteristic
15#1464—21no—
16#1394—21no—
17#1394C2212yes—
18#1319C2212yes—
19#1318—21no—
20C22×#41D411yescharacteristic
21C22×M16—21no—
22C2×#117D411yescharacteristic
23C22×C4⋊C4—21no—
24C22×C4⋊C4—41no—
25C22×C42C2×C411yescharacteristic
26#432—41no—
27#428—42no—
28#428—42no—
29C2×#107—22no—
30C2×#106—22no—
31#417—42no—
32#417—42no—
33C2×#103—22no—
34C2×#102—22no—
35C2×#101—22no—
36C2×#100—41no—
37#407—41no—
38#391D411yescharacteristic
39#391C2311yescharacteristic
40#390C2×C411yescharacteristic
41#390D411yescharacteristic
42#386—41no—
43#384—21no—
44#384—41no—
45#384C2×C412yes—
46C4×C22⋊C4—41no—
47#336—21no—
48#336C2×C412yes—
49#324—22no—
50#323—41no—
51C2×#41—22no—
52C2×#41—41no—
53C2×#41—41no—
54C2×#41—41no—
55C2×#41—42no—
56C22×Q8C2×D411yescharacteristic
57C22×D4—21no—
58C22×D4C22⋊C411yescharacteristic
59C23×C4—21no—
60C23×C4—21no—
61C23×C4—41no—
62C23×C4C22⋊C411yescharacteristic
63C2×M16—22no—
64C2×M16—22no—
65C2×M16—42no—
66C2×M16—42no—
67C2×M16—42no—
68C22×C8—22no—
69#117—41no—
70#117—41no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—41no—
73C2×C4⋊C4—41no—
74C2×C4⋊C4—41no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4—42no—
77C2×C4⋊C4—42no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—84no—
80C2×C4⋊C4C22⋊C411yescharacteristic
81C2×C4⋊C4C2×D411yescharacteristic
82C2×C22⋊C4—21no—
83C2×C22⋊C4—41no—
84C2×C22⋊C4—41no—
85C2×C42—21no—
86C2×C42—21no—
87C2×C42—42no—
88C2×C42C22⋊C411yescharacteristic
89C2×C42C22×C411yescharacteristic
90#107—44no—
91#106—44no—
92C4⋊C8—81no—
93#103—44no—
94#102—44no—
95#101—42no—
96#101—42no—
97#100—82no—
98C22⋊C8—42no—
99C22⋊C8—81no—
100C22⋊C8—81no—
101#97—81no—
102#95—41no—
103#95—41no—
104C24—21no—
105C24#9911yescharacteristic
106#41—44no—
107#41—44no—
108#41—82no—
109#41—82no—
110C2×Q8—21no—
111C2×Q8—41no—
112C2×Q8—41no—
113C2×Q8#13712yes—
114C2×D4—21no—
115C2×D4—21no—
116C2×D4—41no—
117C2×D4—41no—
118C2×D4—42no—
119C2×D4—42no—
120C2×D4—81no—
121C22×C4—22no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—42no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—42no—
138C22×C4—81no—
139C22×C4—81no—
140C22×C4—81no—
141C22×C4—84no—
142C22×C4#9911yescharacteristic
143C22×C4C2×C22⋊C411yescharacteristic
144M16—44no—
145M16—44no—
146M16—82no—
147M16—82no—
148C2×C8—24no—
149C2×C8—42no—
150C2×C8—42no—
151C2×C8—82no—
152C2×C8—82no—
153C4⋊C4—41no—
154C4⋊C4—41no—
155C4⋊C4—42no—
156C4⋊C4—42no—
157C4⋊C4—84no—
158C4⋊C4—84no—
159C4⋊C4—81no—
160C4⋊C4—81no—
161C22⋊C4—82no—
162C22⋊C4—82no—
163C22⋊C4—81no—
164C42—41no—
165C42—41no—
166C42—42no—
167C42—41no—
168C42—41no—
169C23—21no—
170C23—21no—
171C23—41no—
172C23—41no—
173C23—41no—
174C23—41no—
175C23—41no—
176C23—81no—
177C23—81no—
178C23#35212yes—
179C23C2×#9911yescharacteristic
180Q8—42no—
181Q8—81no—
182D4—42no—
183D4—81no—
184D4—84no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—48no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—82no—
206C2×C4—82no—
207C2×C4—81no—
208C2×C4—81no—
209C2×C4—84no—
210C2×C4—82no—
211C2×C4—162no—
212C2×C4—162no—
213C2×C4#35311yescharacteristic
214C2×C4#35311yescharacteristic
215C2×C4#41911yescharacteristic
216C2×C4#42111yescharacteristic
217C8—44no—
218C8—162no—
219C8—162no—
220C22—21no—
221C22—21no—
222C22—22no—
223C22—42no—
224C22—42no—
225C22—41no—
226C22—41no—
227C22—82no—
228C22—81no—
229C22—81no—
230C22—81no—
231C22—161no—
232C22#151111yescharacteristic
233C22C2×#35211yescharacteristic
234C22C2×#35311yescharacteristic
235C4—21no—
236C4—21no—
237C4—21no—
238C4—21no—
239C4—42no—
240C4—42no—
241C4—42no—
242C4—81no—
243C4—81no—
244C4—162no—
245C2—21no—
246C2—21no—
247C2—42no—
248C2—161no—
249C2#1180911yescharacteristic
250C2#1358011yescharacteristic
251C2#1359111yescharacteristic
subgroupG/H
center, socleC22C2×#353
commutatorC2×C42C22×C4
Frattini, cosocle#391C23
prime radicalC22×#41D4
LCSG ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ #2297 ◁ G
LECSG ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7437C2semidirect

Maximal subgroups

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

subgroupindex
#237342
#128612
#115072
#74372