Finite Group Databasebeta

← #108211

Subgroups — #108211

Order 512.

1125 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 1125 conjugacy classes of subgroups — too many to draw. Only its 206 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
1C4×#519C211yesmaximal, characteristic
2#21820C211yesmaximal, characteristic
3#20727C211yesmaximal, characteristic
4#7948C212yesmaximal
5#7870C212yesmaximal
6C2×#519—21no—
7C22×C4×Q8C2211yescharacteristic
8#2920C2212yes—
9C2×#432C2211yescharacteristic
10C2×#431C2211yescharacteristic
11#2276—24no—
12#2274—21no—
13#2274C414yes—
14C42×Q8—24no—
15C4×#117—22no—
16C4×#117C2212yes—
17#1985C414yescomplemented
18#1984—22no—
19#1821—21no—
20#1817—22no—
21#1801C414yescomplemented
22#1784—21no—
23C4×#101—22no—
24#1285—24no—
25#1276—24no—
26C23×Q8D411yescharacteristic
27#519—216no—
28C2×C4×Q8—28no—
29C2×C4×Q8—22no—
30C2×C4×Q8—22no—
31C2×C4×Q8—22no—
32C2×C4×Q8—22no—
33C2×C4×Q8C2×C412yes—
34C2×C4×Q8C2×C412yes—
35C2×C4×Q8D412yes—
36C2×C4×Q8Q812yes—
37C2×#117—24no—
38C2×#117—22no—
39C22×C4⋊C4C2×C412yes—
40C22×C4⋊C4D411yescharacteristic
41C22×C42D412yes—
42C22×C42C2311yescharacteristic
43#432—22no—
44#432—22no—
45#432—22no—
46#432—22no—
47#431—22no—
48#431—22no—
49#431—22no—
50#431—22no—
51#423C2×C414yes—
52C2×C4⋊C8—21no—
53C2×C4⋊C8C2×C412yes—
54C2×#101—24no—
55#407—22no—
56C2×C22⋊C8—21no—
57C2×C22⋊C8—21no—
58C4×M16—22no—
59C2×#97C2×C412yes—
60C2×C4×C8—21no—
61C4×C4⋊C4—24no—
62C4×C4⋊C4—24no—
63C4×C4⋊C4—28no—
64C4×C4⋊C4—216no—
65C4×C4⋊C4—22no—
66C4×C4⋊C4D414yes—
67C4×C22⋊C4—24no—
68C4×C22⋊C4—22no—
69C4×C22⋊C4C2×C414yes—
70#376—24no—
71#376—22no—
72C43—24no—
73C43D414yes—
74#343—24no—
75#336—28no—
76C22×Q8—28no—
77C22×Q8—22no—
78C22×Q8—22no—
79C22×Q8C22⋊C412yes—
80C22×Q8C4⋊C412yes—
81C23×C4C22⋊C412yes—
82C23×C4C22⋊C412yes—
83C23×C4C22×C411yescharacteristic
84C23×C4C2×D411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C2×M16—24no—
87C22×C8—22no—
88C22×C8—22no—
89C4×Q8—264no—
90C4×Q8—24no—
91C4×Q8—24no—
92C4×Q8—24no—
93C4×Q8—24no—
94C4×Q8—24no—
95C4×Q8—24no—
96C4×Q8—24no—
97C4×Q8—24no—
98#117—232no—
99#117—216no—
100C2×C4⋊C4—28no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4C4214yes—
107C2×C4⋊C4C22⋊C412yes—
108C2×C4⋊C4C4⋊C414yes—
109C2×C4⋊C4C4⋊C412yes—
110C2×C22⋊C4—24no—
111C2×C42—28no—
112C2×C42—24no—
113C2×C42—22no—
114C2×C42—22no—
115C2×C42—22no—
116C2×C42—22no—
117C2×C42—24no—
118C2×C42—28no—
119C2×C42C22⋊C414yes—
120C2×C42C22⋊C418yes—
121C2×C42C4⋊C414yes—
122C2×C42C22×C411yescharacteristic
123C2×C42C22×C411yescharacteristic
124C2×C42C2×D412yes—
125C2×C42C2×D412yes—
126C2×C42C2×D411yescharacteristic
127C2×C42C2×Q811yescharacteristic
128C2×C42#4112yes—
129C2×C42#4112yes—
130C4⋊C8—24no—
131C4⋊C8—24no—
132C4⋊C8—24no—
133C4⋊C8—42no—
134#101—48no—
135C22⋊C8—41no—
136C22⋊C8—41no—
137C22⋊C8—42no—
138C22⋊C8—41no—
139C22⋊C8—41no—
140C22⋊C8—42no—
141#97—24no—
142#97—42no—
143C4×C8—42no—
144C4×C8—42no—
145#95—28no—
146#95—24no—
147#95—24no—
148C24C2×C22⋊C411yescharacteristic
149C2×Q8—216no—
150C2×Q8—216no—
151C2×Q8—28no—
152C2×Q8—24no—
153C2×Q8—24no—
154C2×Q8—24no—
155C2×Q8#10418yes—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—28no—
159C22×C4—24no—
160C22×C4—28no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—44no—
166C22×C4#9514yes—
167C22×C4#9514yes—
168C22×C4C2×C4211yescharacteristic
169C22×C4C2×C22⋊C412yes—
170C22×C4C2×C22⋊C411yescharacteristic
171C22×C4C2×C22⋊C411yescharacteristic
172C22×C4C2×C4⋊C411yescharacteristic
173C22×C4C2×C4⋊C411yescharacteristic
174C22×C4C2×C4⋊C411yescharacteristic
175C22×C4#11712yes—
176C22×C4#12012yes—
177C22×C4#12011yescharacteristic
178C22×C4#12011yescharacteristic
179C22×C4#12312yes—
180C22×C4#12311yescharacteristic
181C22×C4#12311yescharacteristic
182M16—48no—
183C2×C8—44no—
184C2×C8—44no—
185C2×C8—44no—
186C2×C8—42no—
187C2×C8—42no—
188C4⋊C4—232no—
189C4⋊C4—216no—
190C4⋊C4—28no—
191C4⋊C4—28no—
192C4⋊C4—24no—
193C4⋊C4—24no—
194C22⋊C4—216no—
195C42—232no—
196C42—216no—
197C42—24no—
198C42—24no—
199C42—24no—
200C42—24no—
201C42—48no—
202C42—48no—
203C42—88no—
204C42C4×D412yes—
205C42C4×D412yes—
206C42C4×D412yes—
207C42C4×D412yes—
208C42#12112yes—
209C42#12112yes—
210C42C22⋊Q812yes—
211C42C22⋊Q812yes—
212C23—22no—
213C23—22no—
214C23—41no—
215C23C2×#9511yescharacteristic
216C23#37911yescharacteristic
217C23#38111yescharacteristic
218Q8—216no—
219Q8—216no—
220C2×C4—28no—
221C2×C4—28no—
222C2×C4—24no—
223C2×C4—24no—
224C2×C4—22no—
225C2×C4—24no—
226C2×C4—24no—
227C2×C4—24no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—216no—
232C2×C4—24no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—42no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—48no—
240C2×C4—48no—
241C2×C4—44no—
242C2×C4—44no—
243C2×C4—82no—
244C2×C4#33918yes—
245C2×C4C4×C22⋊C412yes—
246C2×C4#38012yes—
247C2×C4#38511yescharacteristic
248C2×C4#38511yescharacteristic
249C2×C4#38512yes—
250C2×C4#38611yescharacteristic
251C2×C4#38612yes—
252C2×C4#38611yescharacteristic
253C2×C4C2×#10414yes—
254C8—82no—
255C8—84no—
256C8—82no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—44no—
265C22C22⋊#9511yescharacteristic
266C22C2×#33912yes—
267C22#179112yes—
268C4—28no—
269C4—22no—
270C4—21no—
271C4—21no—
272C4—21no—
273C4—22no—
274C4—22no—
275C4—416no—
276C4—44no—
277C4#189614yes—
278C4#191212yes—
279C2—22no—
280C2—22no—
281C2—41no—
282C2#1071911yescharacteristic
283C2#1074111yescharacteristic
284C2#1074711yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
socleC22C22⋊#95
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
17. #1985C4semidirect
21. #1801C4semidirect

Maximal subgroups

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

subgroupindex
C4×#5192
#218202
#207272
#79482
#78702