Finite Group Databasebeta

← #106201

Subgroups — #106201

Order 512.

695 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 695 conjugacy classes of subgroups — too many to draw. Only its 180 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
1C2×#2985C211yesmaximal, characteristic
2#17590C212yesmaximal
3#8000C211yesmaximal, characteristic
4#7980C211yesmaximal, characteristic
5#7954C212yesmaximal
6C2×#533C2211yescharacteristic
7#2985—24no—
8#2985—24no—
9C2×#446—22no—
10C2×#446C412yescomplemented
11C2×C8×Q8C412yescomplemented
12#2920C2212yes—
13C2×#433—22no—
14C2×#432C2211yescharacteristic
15C2×#431C2211yescharacteristic
16#2283C2212yes—
17#1518—24no—
18#1511—24no—
19#1506C418yescomplemented
20#1292—22no—
21#1289—22no—
22#1285—24no—
23#533—22no—
24#533—22no—
25#533C2×C414yes—
26#533D414yes—
27C2×C4⋊Q8Q811yescharacteristic
28C2×C4⋊Q8Q811yescharacteristic
29C2×#125—21no—
30C2×C22⋊Q8—21no—
31C2×C22⋊Q8D412yes—
32C2×C4×Q8—21no—
33C2×C4×Q8C2×C412yes—
34C2×#117—21no—
35C22×C42C2311yescharacteristic
36#446—216no—
37#446—24no—
38#446—24no—
39C8×Q8—24no—
40C8×Q8—24no—
41#433—216no—
42#432—22no—
43#432—22no—
44#432C2×C414yes—
45#432D414yes—
46#431—22no—
47#431—22no—
48#431—22no—
49#431—22no—
50#423—22no—
51C2×C4⋊C8—22no—
52C2×C4⋊C8—22no—
53C2×C4⋊C8C2×C412yes—
54C2×C4⋊C8D412yes—
55#407—22no—
56C2×C22⋊C8—21no—
57C2×C22⋊C8D412yes—
58C4×M16—22no—
59C2×#97—22no—
60C2×#97—21no—
61C2×C4×C8—22no—
62C2×C4×C8C2×C411yescharacteristic
63C2×C4×C8C2×C411yescharacteristic
64#383—22no—
65#382—22no—
66#381—22no—
67#380—22no—
68C4×C4⋊C4—22no—
69C4×C22⋊C4—22no—
70#376C2×C414yes—
71#328—216no—
72C22×Q8—21no—
73C22×Q8C4⋊C412yes—
74C23×C4C2×D411yescharacteristic
75C23×C4C2×D411yescharacteristic
76C23×C4#4111yescharacteristic
77C2×M16—42no—
78C22×C8—22no—
79C22×C8C22⋊C412yes—
80C22×C8C22⋊C412yes—
81C4⋊Q8—22no—
82C4⋊Q8—22no—
83C4⋊Q8C4⋊C414yes—
84C4⋊Q8C4⋊C414yes—
85#125—24no—
86#125—24no—
87C22⋊Q8—24no—
88C22⋊Q8—24no—
89C22⋊Q8—24no—
90C22⋊Q8C22⋊C418yes—
91C4×Q8—22no—
92C4×Q8—22no—
93C4×Q8—24no—
94C4×Q8—24no—
95C4×Q8C4214yes—
96C4×Q8C22⋊C414yes—
97#117—24no—
98#117—24no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4C4⋊C412yes—
106C2×C22⋊C4—22no—
107C2×C22⋊C4—42no—
108C2×C42—22no—
109C2×C42—24no—
110C2×C42C22×C411yescharacteristic
111C2×C42C22×C412yes—
112C2×C42C2×D412yes—
113C2×C42C2×D412yes—
114C2×C42C2×D411yescharacteristic
115C2×C42C2×Q811yescharacteristic
116C2×C42#4112yes—
117C2×C42#4111yescharacteristic
118C4⋊C8—28no—
119C4⋊C8—28no—
120C4⋊C8—22no—
121C4⋊C8—24no—
122C4⋊C8—24no—
123C4⋊C8—42no—
124C4⋊C8C22⋊C414yes—
125C22⋊C8—24no—
126C22⋊C8—24no—
127C22⋊C8—24no—
128C22⋊C8—42no—
129#97—28no—
130#97—24no—
131#97—24no—
132C4×C8—28no—
133C4×C8—21no—
134C4×C8—21no—
135C4×C8—21no—
136C4×C8—21no—
137C4×C8—42no—
138#95—24no—
139#95—24no—
140#95—42no—
141#95—42no—
142C24#12111yescharacteristic
143C2×Q8—22no—
144C2×Q8—24no—
145C2×Q8—44no—
146C2×Q8—82no—
147C2×Q8#9514yes—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—44no—
153C22×C4—81no—
154C22×C4C2×C22⋊C411yescharacteristic
155C22×C4C2×C22⋊C412yes—
156C22×C4C2×C4⋊C411yescharacteristic
157C22×C4#11711yescharacteristic
158C22×C4C4×D412yes—
159C22×C4C4×D412yes—
160C22×C4#12012yes—
161C22×C4#12111yescharacteristic
162C22×C4#12112yes—
163C22×C4C22⋊Q811yescharacteristic
164C22×C4C22⋊Q811yescharacteristic
165C22×C4#12312yes—
166M16—84no—
167C2×C8—22no—
168C2×C8—22no—
169C2×C8—24no—
170C2×C8—44no—
171C2×C8—42no—
172C2×C8—42no—
173C2×C8—44no—
174C4⋊C4—24no—
175C4⋊C4—22no—
176C4⋊C4—24no—
177C4⋊C4—24no—
178C4⋊C4—24no—
179C4⋊C4—24no—
180C4⋊C4—28no—
181C4⋊C4—84no—
182C4⋊C4#9514yes—
183C22⋊C4—28no—
184C22⋊C4—84no—
185C42—24no—
186C42—24no—
187C42—84no—
188C42C2×C4211yescharacteristic
189C42C2×C22⋊C411yescharacteristic
190C42C2×C22⋊C411yescharacteristic
191C42C2×C4⋊C412yes—
192C42#11711yescharacteristic
193C42C4×D412yes—
194C42C4×D412yes—
195C42#12012yes—
196C42C22⋊Q812yes—
197C42#12312yes—
198C23—22no—
199C23—41no—
200C23—41no—
201C23#38011yescharacteristic
202C23#38612yes—
203Q8—44no—
204Q8—44no—
205Q8—84no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—24no—
209C2×C4—22no—
210C2×C4—24no—
211C2×C4—24no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—42no—
220C2×C4—44no—
221C2×C4—44no—
222C2×C4—84no—
223C2×C4—84no—
224C2×C4—84no—
225C2×C4C2×#9511yescharacteristic
226C2×C4C4×C22⋊C411yescharacteristic
227C2×C4C4×C22⋊C411yescharacteristic
228C2×C4#37911yescharacteristic
229C2×C4#38011yescharacteristic
230C2×C4#38111yescharacteristic
231C2×C4#38512yes—
232C2×C4#38512yes—
233C2×C4#38612yes—
234C8—42no—
235C8—42no—
236C8—44no—
237C8—84no—
238C22—21no—
239C22—21no—
240C22—22no—
241C22—22no—
242C22—22no—
243C22—82no—
244C22C22⋊#9511yescharacteristic
245C22#181212yes—
246C4—22no—
247C4—21no—
248C4—21no—
249C4—21no—
250C4—22no—
251C4—21no—
252C4—21no—
253C4—22no—
254C4—44no—
255C4—44no—
256C4—164no—
257C4#179612yes—
258C2—22no—
259C2—42no—
260C2#1075911yescharacteristic
261C2#1077711yescharacteristic
262C2#1079411yescharacteristic
subgroupG/H
center, socleC22C22⋊#95
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
10. C2×#446C4semidirect
11. C2×C8×Q8C4semidirect
19. #1506C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29852
#175902
#80002
#79802
#79542