Finite Group Databasebeta

← #150601

Subgroups — #150601

Order 512.

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

normal subgroups only

This group has 317 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#29724C211yesmaximal, characteristic
2#12862C211yesmaximal, characteristic
3#11510C211yesmaximal, characteristic
4#7413C212yesmaximal, complemented
5#7405C212yesmaximal, complemented
6C2×#451—21no—
7#2621C411yescharacteristic
8#2310C2211yescharacteristic
9#2074C411yescharacteristic
10#2074C2211yescharacteristic
11#2056—21no—
12#2056—21no—
13#2016C411yescharacteristic
14#2016C2211yescharacteristic
15#1873—21no—
16#1798C411yescharacteristic
17#1784—21no—
18#1464—21no—
19#1394—21no—
20#1394C2212yes—
21#1319—21no—
22#1319C2212yes—
23C23×Q8D411yescharacteristic
24C22×Q16—21no—
25C22×Q16—21no—
26C2×C4⋊Q8C2×C411yescharacteristic
27C2×C22⋊Q8—21no—
28C2×#117C2×C411yescharacteristic
29C2×#117D411yescharacteristic
30#451—44no—
31#432—41no—
32C2×C4⋊C8—21no—
33C2×#103—21no—
34C2×#103—21no—
35C2×#103—21no—
36C2×#103—21no—
37C2×#101—22no—
38C2×#101—41no—
39#407—41no—
40C2×C22⋊C8—21no—
41#398—41no—
42#397—41no—
43#391—21no—
44#391—41no—
45#391C2×C411yescharacteristic
46#391D411yescharacteristic
47#391D411yescharacteristic
48#391C2311yescharacteristic
49#382C2×C411yescharacteristic
50#381—41no—
51#340—21no—
52#340—21no—
53#336C2×C411yescharacteristic
54#336C2×C411yescharacteristic
55#324—22no—
56#323—41no—
57C22×Q8—21no—
58C22×Q8—22no—
59C22×Q8—21no—
60C22×Q8—41no—
61C22×Q8—81no—
62C22×Q8—81no—
63C22×Q8C22⋊C411yescharacteristic
64C22×Q8C2×D411yescharacteristic
65C23×C4—21no—
66C23×C4C22⋊C411yescharacteristic
67C2×Q16—42no—
68C2×Q16—42no—
69C2×Q16—81no—
70C2×Q16—81no—
71C2×Q16—81no—
72C2×Q16—81no—
73C2×M16—42no—
74C22×C8—21no—
75C22×C8—21no—
76C4⋊Q8—81no—
77C22⋊Q8—42no—
78C22⋊Q8—42no—
79#117—81no—
80#117—81no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4C22⋊C411yescharacteristic
86C2×C4⋊C4C2×D411yescharacteristic
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—41no—
90C2×C42—21no—
91C2×C42C22⋊C411yescharacteristic
92C2×C42C22×C411yescharacteristic
93C4⋊C8—81no—
94C4⋊C8—81no—
95#103—42no—
96#103—42no—
97#103—81no—
98#103—81no—
99#101—42no—
100#101—42no—
101#101—82no—
102C22⋊C8—42no—
103C22⋊C8—42no—
104C22⋊C8—82no—
105#97—81no—
106#95—41no—
107#95—41no—
108#95—41no—
109#95—41no—
110#95—41no—
111#95—42no—
112C24#9911yescharacteristic
113C2×Q8—21no—
114C2×Q8—21no—
115C2×Q8—21no—
116C2×Q8—42no—
117C2×Q8—42no—
118C2×Q8—41no—
119C2×Q8—41no—
120C2×Q8—41no—
121C2×Q8—41no—
122C2×Q8—41no—
123C2×Q8—41no—
124C2×Q8—41no—
125C2×Q8—41no—
126C2×Q8—41no—
127C2×Q8—41no—
128C2×Q8—81no—
129C2×Q8—84no—
130C2×Q8—84no—
131C2×Q8AGL(1,ℤ/8)12yes—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—81no—
147C22×C4#9911yescharacteristic
148C22×C4C2×C22⋊C411yescharacteristic
149Q16—84no—
150Q16—84no—
151M16—82no—
152M16—82no—
153C2×C8—41no—
154C2×C8—41no—
155C2×C8—41no—
156C2×C8—41no—
157C2×C8—41no—
158C2×C8—41no—
159C2×C8—82no—
160C2×C8—82no—
161C4⋊C4—41no—
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4—41no—
165C4⋊C4—81no—
166C4⋊C4—82no—
167C4⋊C4—82no—
168C22⋊C4—81no—
169C22⋊C4—81no—
170C22⋊C4—82no—
171C42—41no—
172C42—41no—
173C42—41no—
174C42—41no—
175C23—21no—
176C23—21no—
177C23—21no—
178C23—41no—
179C23—41no—
180C23C2×#9911yescharacteristic
181Q8—42no—
182Q8—42no—
183Q8—81no—
184Q8—81no—
185Q8—81no—
186Q8—81no—
187Q8—81no—
188Q8—81no—
189Q8—84no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—84no—
212C2×C4—81no—
213C2×C4—81no—
214C2×C4—81no—
215C2×C4—81no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4#41811yescharacteristic
221C2×C4#42111yescharacteristic
222C8—81no—
223C8—81no—
224C8—162no—
225C8—162no—
226C22—21no—
227C22—21no—
228C22—42no—
229C22—41no—
230C22—41no—
231C22—41no—
232C22—81no—
233C22#151011yescharacteristic
234C22#212111yescharacteristic
235C22#212711yescharacteristic
236C4—21no—
237C4—21no—
238C4—42no—
239C4—41no—
240C4—81no—
241C4—81no—
242C4—81no—
243C4—81no—
244C4—81no—
245C4—161no—
246C4—161no—
247C2—21no—
248C2—21no—
249C2—81no—
250C2#1179911yescharacteristic
251C2#1360811yescharacteristic
252C2#1361211yescharacteristic
subgroupG/H
center, socleC22#2127
commutatorC2×C42C22×C4
Frattini, cosocle#391C23
prime radicalC24#99
LCSG ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C23×C4 ◁ #2310 ◁ G
LECSG ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7413C2semidirect
5. #7405C2semidirect

Maximal subgroups

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

subgroupindex
#297242
#128622
#115102
#74132
#74052