Finite Group Databasebeta

← #112806

Subgroups — #112806

Order 512.

792 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 792 conjugacy classes of subgroups — too many to draw. Only its 142 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×#3160C211yesmaximal, characteristic
2#21826C211yesmaximal, characteristic
3#17676C211yesmaximal, characteristic
4#17596C212yesmaximal
5#7898C212yesmaximal
6C2×#533C2211yescharacteristic
7C2×#533C2211yescharacteristic
8#3160—24no—
9#3160—24no—
10C2×#500C412yescomplemented
11C2×#499—22no—
12C2×#498C412yescomplemented
13C2×#432C2212yes—
14C2×#427—22no—
15C22×C4×C8C2211yescharacteristic
16#2283C2212yes—
17#1996—24no—
18#1820—22no—
19#1534—22no—
20#1531—24no—
21#1520—24no—
22#1515—24no—
23#1506—24no—
24#1279—24no—
25C23×C8—21no—
26#533—22no—
27#533—22no—
28#533—22no—
29#533—22no—
30#533—22no—
31#533—22no—
32#533—22no—
33#533D414yes—
34C2×C4⋊Q8C2×C412yes—
35C2×C4⋊Q8C2×C412yes—
36C2×#125—21no—
37C2×#125D412yes—
38C2×C22⋊Q8—22no—
39C2×C22⋊Q8—22no—
40C2×C4×Q8—22no—
41C2×C4×Q8—22no—
42C2×#117—21no—
43C2×#117D412yes—
44C22×C42C2311yescharacteristic
45#500—24no—
46#500—24no—
47#499—216no—
48#498—24no—
49#498—24no—
50#432—22no—
51#432—22no—
52#432—22no—
53#432D414yes—
54#427—216no—
55C2×#107—24no—
56C2×#106—24no—
57C2×C4⋊C8—22no—
58C2×C22⋊C8—22no—
59C2×#97—22no—
60C2×C4×C8—21no—
61C2×C4×C8—22no—
62C2×C4×C8C2×C411yescharacteristic
63C2×C4×C8C2×C411yescharacteristic
64C2×C4×C8D414yes—
65#383—22no—
66#382—22no—
67#381—22no—
68#380D414yes—
69C4×C4⋊C4—22no—
70C4×C22⋊C4—22no—
71#376—22no—
72#342—28no—
73#330—28no—
74#328—216no—
75C22×Q8—22no—
76C22×Q8—22no—
77C23×C4C2×D411yescharacteristic
78C23×C4C2×D412yes—
79C22×C8—28no—
80C22×C8—21no—
81C22×C8—21no—
82C22×C8—24no—
83C22×C8—42no—
84C4⋊Q8—22no—
85C4⋊Q8—22no—
86C4⋊Q8—22no—
87C4⋊Q8—22no—
88C4⋊Q8—22no—
89C4⋊Q8—22no—
90C4⋊Q8—22no—
91C4⋊Q8C22⋊C414yes—
92#125—24no—
93#125—22no—
94#125—24no—
95#125—22no—
96#125—41no—
97#125—41no—
98C22⋊Q8—24no—
99C22⋊Q8—24no—
100C22⋊Q8—42no—
101C22⋊Q8—42no—
102C22⋊Q8—42no—
103C22⋊Q8—42no—
104C22⋊Q8—42no—
105C22⋊Q8—42no—
106C4×Q8—24no—
107C4×Q8—24no—
108C4×Q8—42no—
109C4×Q8—42no—
110C4×Q8—42no—
111C4×Q8—42no—
112C4×Q8—42no—
113C4×Q8—42no—
114#117—24no—
115#117—24no—
116#117—24no—
117#117—42no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4C22⋊C414yes—
125C2×C22⋊C4—22no—
126C2×C22⋊C4—24no—
127C2×C22⋊C4—41no—
128C2×C42—22no—
129C2×C42—41no—
130C2×C42—42no—
131C2×C42C22×C411yescharacteristic
132C2×C42C2×D412yes—
133C2×C42C2×D412yes—
134C2×C42C2×D412yes—
135C2×C42C2×D411yescharacteristic
136C2×C42#4112yes—
137C2×C42#4112yes—
138#107—216no—
139#106—216no—
140C4⋊C8—24no—
141C4⋊C8—42no—
142C22⋊C8—24no—
143C22⋊C8—42no—
144#97—24no—
145#97—42no—
146C4×C8—21no—
147C4×C8—21no—
148C4×C8—24no—
149C4×C8C22⋊C412yes—
150C4×C8C22⋊C412yes—
151#95—24no—
152#95—42no—
153#95—42no—
154#95—42no—
155C24#12011yescharacteristic
156C2×Q8—24no—
157C2×Q8—42no—
158C2×Q8—42no—
159C2×Q8—42no—
160C2×Q8—84no—
161C2×Q8—84no—
162C22×C4—24no—
163C22×C4—24no—
164C22×C4—44no—
165C22×C4—42no—
166C22×C4—44no—
167C22×C4C2×C22⋊C411yescharacteristic
168C22×C4C4×D412yes—
169C22×C4#12012yes—
170C22×C4#12012yes—
171C22×C4#12112yes—
172C22×C4#12112yes—
173C22×C4#12112yes—
174C22×C4#12112yes—
175C22×C4#12311yescharacteristic
176C22×C4#12712yes—
177C2×C8—24no—
178C2×C8—24no—
179C2×C8—216no—
180C2×C8—22no—
181C2×C8—22no—
182C2×C8—44no—
183C2×C8—84no—
184C4⋊C4—24no—
185C4⋊C4—24no—
186C4⋊C4—24no—
187C4⋊C4—24no—
188C4⋊C4—48no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—44no—
194C4⋊C4—44no—
195C4⋊C4—44no—
196C4⋊C4—44no—
197C22⋊C4—44no—
198C22⋊C4—44no—
199C22⋊C4—48no—
200C42—44no—
201C42—44no—
202C42—84no—
203C42C2×C22⋊C411yescharacteristic
204C42C2×C22⋊C411yescharacteristic
205C42C4×D412yes—
206C42#12012yes—
207C42#12011yescharacteristic
208C42#12112yes—
209C42#12112yes—
210C42#12312yes—
211C42#12311yescharacteristic
212C42#12412yes—
213C23—42no—
214C23#38611yescharacteristic
215C23#39212yes—
216C23#45614yes—
217Q8—84no—
218Q8—84no—
219Q8—84no—
220Q8—84no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—24no—
224C2×C4—24no—
225C2×C4—24no—
226C2×C4—24no—
227C2×C4—24no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—44no—
232C2×C4—44no—
233C2×C4—84no—
234C2×C4—84no—
235C2×C4—84no—
236C2×C4—84no—
237C2×C4—84no—
238C2×C4#37911yescharacteristic
239C2×C4#38612yes—
240C2×C4#38611yescharacteristic
241C2×C4#39012yes—
242C2×C4#39212yes—
243C2×C4#39412yes—
244C2×C4#39412yes—
245C8—24no—
246C8—24no—
247C8—84no—
248C22—22no—
249C22—22no—
250C22—22no—
251C22—44no—
252C22#143811yescharacteristic
253C22#188612yes—
254C22#200514yes—
255C4—22no—
256C4—22no—
257C4—22no—
258C4—21no—
259C4—21no—
260C4—22no—
261C4—84no—
262C4—84no—
263C4—164no—
264C4#197614yes—
265C2—24no—
266C2—22no—
267C2#1097811yescharacteristic
268C2#1102612yes—
subgroupG/H
center, socleC22#1438
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ C2×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×#500C4semidirect
12. C2×#498C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31602
#218262
#176762
#175962
#78982