Finite Group Databasebeta

← #150596

Subgroups — #150596

Order 512.

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

normal subgroups only

This group has 340 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#29726C211yesmaximal, characteristic
2#12859C211yesmaximal, characteristic
3#11508C211yesmaximal, characteristic
4#7412C212yesmaximal, complemented
5#7403C212yesmaximal, complemented
6C2×#452—21no—
7#2621C411yescharacteristic, complemented
8#2311C2211yescharacteristic
9#2074C411yescharacteristic, complemented
10#2073C2211yescharacteristic
11#2056—21no—
12#2055—21no—
13#2016C411yescharacteristic, complemented
14#2013C2211yescharacteristic
15#1872—21no—
16#1798C411yescharacteristic, complemented
17#1782—21no—
18#1463—21no—
19#1393—21no—
20#1393C2212yes—
21#1319—21no—
22#1318C2212yes—
23C22×#41D411yescharacteristic
24C22×Q16—21no—
25C22×SD16—21no—
26C2×C4⋊Q8C2×C411yescharacteristic
27C2×C22⋊Q8—21no—
28C2×#117C2×C411yescharacteristic
29C22×C4⋊C4D411yescharacteristic
30#452—44no—
31C2×C4⋊C8—21no—
32C2×#103—21no—
33C2×#103—21no—
34C2×#103—21no—
35C2×#102—21no—
36C2×#99—22no—
37C2×#99—41no—
38#407—41no—
39C2×C22⋊C8—21no—
40#398—41no—
41#397—41no—
42#391—21no—
43#391C2×C411yescharacteristic
44#391D411yescharacteristic
45#390D411yescharacteristic
46#390C2311yescharacteristic
47#386—41no—
48#382C2×C411yescharacteristic
49#381—41no—
50#380—41no—
51#340—21no—
52#340—21no—
53#336C2×C411yescharacteristic
54#336C2×C411yescharacteristic
55#324—22no—
56#324—41no—
57C2×#41—22no—
58C2×#41—41no—
59C2×#41—81no—
60C2×#41—81no—
61C22×Q8—21no—
62C22×Q8C22⋊C411yescharacteristic
63C22×D4—21no—
64C22×D4C2×D411yescharacteristic
65C23×C4—21no—
66C23×C4—21no—
67C23×C4C22⋊C411yescharacteristic
68C2×Q16—42no—
69C2×Q16—81no—
70C2×Q16—81no—
71C2×SD16—42no—
72C2×SD16—81no—
73C2×SD16—81no—
74C2×M16—81no—
75C22×C8—21no—
76C22×C8—21no—
77C4⋊Q8—81no—
78C22⋊Q8—42no—
79C22⋊Q8—42no—
80#117—81no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—81no—
87C2×C4⋊C4C22⋊C411yescharacteristic
88C2×C4⋊C4C2×D411yescharacteristic
89C2×C22⋊C4—21no—
90C2×C22⋊C4—41no—
91C2×C22⋊C4—42no—
92C2×C22⋊C4—41no—
93C2×C22⋊C4—41no—
94C2×C42C22⋊C411yescharacteristic
95C2×C42C22×C411yescharacteristic
96C4⋊C8—81no—
97#103—42no—
98#103—81no—
99#103—81no—
100#102—42no—
101#99—42no—
102#99—42no—
103#99—82no—
104C22⋊C8—42no—
105C22⋊C8—42no—
106C22⋊C8—81no—
107#95—41no—
108#95—41no—
109#95—41no—
110#95—42no—
111#95—82no—
112C24—21no—
113C24#9911yescharacteristic
114#41—84no—
115#41—84no—
116C2×Q8—21no—
117C2×Q8—21no—
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×D4—21no—
127C2×D4—41no—
128C2×D4—41no—
129C2×D4—42no—
130C2×D4—42no—
131C2×D4—81no—
132C2×D4AGL(1,ℤ/8)12yes—
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—41no—
147C22×C4—41no—
148C22×C4—81no—
149C22×C4—82no—
150C22×C4—161no—
151C22×C4#9911yescharacteristic
152C22×C4C2×C22⋊C411yescharacteristic
153Q16—84no—
154SD16—84no—
155M16—162no—
156C2×C8—41no—
157C2×C8—41no—
158C2×C8—41no—
159C2×C8—41no—
160C2×C8—41no—
161C2×C8—41no—
162C2×C8—82no—
163C4⋊C4—41no—
164C4⋊C4—41no—
165C4⋊C4—41no—
166C4⋊C4—41no—
167C4⋊C4—81no—
168C4⋊C4—81no—
169C4⋊C4—82no—
170C4⋊C4—82no—
171C22⋊C4—81no—
172C22⋊C4—82no—
173C22⋊C4—82no—
174C22⋊C4—82no—
175C22⋊C4—161no—
176C22⋊C4—161no—
177C42—41no—
178C42—41no—
179C23—21no—
180C23—21no—
181C23—21no—
182C23—22no—
183C23—41no—
184C23—41no—
185C23—41no—
186C23—41no—
187C23—81no—
188C23—81no—
189C23C2×#9911yescharacteristic
190Q8—42no—
191Q8—81no—
192Q8—81no—
193Q8—81no—
194Q8—81no—
195Q8—81no—
196D4—42no—
197D4—81no—
198D4—84no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—84no—
218C2×C4—82no—
219C2×C4—81no—
220C2×C4—81no—
221C2×C4—81no—
222C2×C4—81no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4—81no—
227C2×C4—161no—
228C2×C4—162no—
229C2×C4—162no—
230C2×C4—162no—
231C2×C4#41811yescharacteristic
232C2×C4#42111yescharacteristic
233C8—81no—
234C8—81no—
235C8—162no—
236C22—21no—
237C22—21no—
238C22—42no—
239C22—41no—
240C22—42no—
241C22—41no—
242C22—41no—
243C22—81no—
244C22—82no—
245C22—81no—
246C22—81no—
247C22—161no—
248C22#151011yescharacteristic
249C22#212111yescharacteristic
250C22#212711yescharacteristic
251C4—21no—
252C4—21no—
253C4—41no—
254C4—81no—
255C4—81no—
256C4—81no—
257C4—81no—
258C4—81no—
259C4—161no—
260C4—161no—
261C4—322no—
262C2—21no—
263C2—21no—
264C2—82no—
265C2—81no—
266C2#1179911yescharacteristic
267C2#1360811yescharacteristic
268C2#1361011yescharacteristic
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. #7412C2semidirect
5. #7403C2semidirect
7. #2621C4semidirect
9. #2074C4semidirect
13. #2016C4semidirect
16. #1798C4semidirect

Maximal subgroups

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

subgroupindex
#297262
#128592
#115082
#74122
#74032