Finite Group Databasebeta

← #108213

Subgroups — #108213

Order 512.

837 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 837 conjugacy classes of subgroups — too many to draw. Only its 190 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#34059C211yesmaximal, characteristic
2#21820C211yesmaximal, characteristic
3#20727C211yesmaximal, characteristic
4#7948C212yesmaximal
5#7870C212yesmaximal
6C22×C4×Q8C2211yescharacteristic
7#2920C2212yes—
8C2×#432C2211yescharacteristic
9C2×#431C2211yescharacteristic
10#2320—24no—
11#2316—21no—
12#2313C414yes—
13#2310—21no—
14#2295—24no—
15#2283—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
27C2×C4×Q8—22no—
28C2×C4×Q8—22no—
29C2×C4×Q8—22no—
30C2×C4×Q8—22no—
31C2×C4×Q8C2×C412yes—
32C2×C4×Q8C2×C412yes—
33C2×C4×Q8D412yes—
34C2×C4×Q8Q812yes—
35C2×#117—22no—
36C22×C4⋊C4C2×C412yes—
37C22×C4⋊C4D411yescharacteristic
38C22×C42D412yes—
39C22×C42C2311yescharacteristic
40#432—22no—
41#432—22no—
42#432—22no—
43#432—22no—
44#431—22no—
45#431—22no—
46#431—22no—
47#431—22no—
48#423C2×C414yes—
49C2×C4⋊C8—21no—
50C2×C4⋊C8C2×C412yes—
51C2×#101—24no—
52#407—22no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—21no—
55C4×M16—22no—
56C2×#97C2×C412yes—
57C2×C4×C8—21no—
58#391—28no—
59#389—28no—
60#387—216no—
61#383—24no—
62#382—24no—
63#381—24no—
64#380—24no—
65C4×C4⋊C4—24no—
66C4×C4⋊C4—22no—
67C4×C4⋊C4D414yes—
68C4×C22⋊C4—22no—
69C4×C22⋊C4C2×C414yes—
70#376—24no—
71#376—22no—
72C43D414yes—
73#343—24no—
74#336—28no—
75C22×Q8—22no—
76C22×Q8—22no—
77C22×Q8—44no—
78C22×Q8C22⋊C412yes—
79C22×Q8C4⋊C412yes—
80C23×C4C22⋊C412yes—
81C23×C4C22⋊C412yes—
82C23×C4C22×C411yescharacteristic
83C23×C4C2×D411yescharacteristic
84C23×C4C2×D411yescharacteristic
85C2×M16—24no—
86C22×C8—22no—
87C22×C8—22no—
88C4×Q8—24no—
89C4×Q8—24no—
90C4×Q8—24no—
91C4×Q8—24no—
92C4×Q8—24no—
93C4×Q8—24no—
94C4×Q8—24no—
95C4×Q8—24no—
96#117—48no—
97C2×C4⋊C4—24no—
98C2×C4⋊C4—24no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—28no—
103C2×C4⋊C4C4214yes—
104C2×C4⋊C4C22⋊C412yes—
105C2×C4⋊C4C4⋊C414yes—
106C2×C4⋊C4C4⋊C412yes—
107C2×C22⋊C4—24no—
108C2×C42—22no—
109C2×C42—22no—
110C2×C42—22no—
111C2×C42—22no—
112C2×C42—24no—
113C2×C42—24no—
114C2×C42C22⋊C414yes—
115C2×C42C22⋊C418yes—
116C2×C42C4⋊C414yes—
117C2×C42C22×C411yescharacteristic
118C2×C42C22×C411yescharacteristic
119C2×C42C2×D412yes—
120C2×C42C2×D412yes—
121C2×C42C2×D411yescharacteristic
122C2×C42C2×Q811yescharacteristic
123C2×C42#4112yes—
124C2×C42#4112yes—
125C4⋊C8—24no—
126C4⋊C8—24no—
127C4⋊C8—24no—
128C4⋊C8—42no—
129#101—48no—
130C22⋊C8—41no—
131C22⋊C8—41no—
132C22⋊C8—42no—
133C22⋊C8—41no—
134C22⋊C8—41no—
135C22⋊C8—42no—
136#97—24no—
137#97—42no—
138C4×C8—42no—
139C4×C8—42no—
140#95—28no—
141#95—28no—
142#95—28no—
143#95—24no—
144#95—24no—
145C24C2×C22⋊C411yescharacteristic
146C2×Q8—24no—
147C2×Q8—24no—
148C2×Q8—24no—
149C2×Q8—24no—
150C2×Q8—44no—
151C2×Q8—48no—
152C2×Q8—48no—
153C22×C4—24no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—28no—
159C22×C4—42no—
160C22×C4—44no—
161C22×C4—42no—
162C22×C4—44no—
163C22×C4#9514yes—
164C22×C4#9514yes—
165C22×C4C2×C4211yescharacteristic
166C22×C4C2×C22⋊C412yes—
167C22×C4C2×C22⋊C411yescharacteristic
168C22×C4C2×C22⋊C411yescharacteristic
169C22×C4C2×C4⋊C411yescharacteristic
170C22×C4C2×C4⋊C411yescharacteristic
171C22×C4C2×C4⋊C411yescharacteristic
172C22×C4#11712yes—
173C22×C4#12012yes—
174C22×C4#12011yescharacteristic
175C22×C4#12011yescharacteristic
176C22×C4#12312yes—
177C22×C4#12311yescharacteristic
178C22×C4#12311yescharacteristic
179M16—48no—
180C2×C8—44no—
181C2×C8—44no—
182C2×C8—44no—
183C2×C8—42no—
184C2×C8—42no—
185C4⋊C4—28no—
186C4⋊C4—28no—
187C4⋊C4—24no—
188C4⋊C4—24no—
189C4⋊C4—48no—
190C4⋊C4—88no—
191C22⋊C4—48no—
192C42—24no—
193C42—24no—
194C42—24no—
195C42—24no—
196C42—48no—
197C42—48no—
198C42—48no—
199C42C4×D412yes—
200C42C4×D412yes—
201C42C4×D412yes—
202C42C4×D412yes—
203C42#12112yes—
204C42#12112yes—
205C42C22⋊Q812yes—
206C42C22⋊Q812yes—
207C23—22no—
208C23—22no—
209C23—41no—
210C23C2×#9511yescharacteristic
211C23#37911yescharacteristic
212C23#38111yescharacteristic
213Q8—48no—
214Q8—48no—
215C2×C4—22no—
216C2×C4—24no—
217C2×C4—24no—
218C2×C4—24no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—24no—
223C2×C4—24no—
224C2×C4—24no—
225C2×C4—24no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—42no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—44no—
233C2×C4—44no—
234C2×C4—44no—
235C2×C4—44no—
236C2×C4—48no—
237C2×C4—48no—
238C2×C4—48no—
239C2×C4—82no—
240C2×C4C4×C22⋊C412yes—
241C2×C4#38012yes—
242C2×C4#38511yescharacteristic
243C2×C4#38511yescharacteristic
244C2×C4#38512yes—
245C2×C4#38611yescharacteristic
246C2×C4#38612yes—
247C2×C4#38611yescharacteristic
248C2×C4#42114yes—
249C8—82no—
250C8—84no—
251C8—82no—
252C22—21no—
253C22—21no—
254C22—21no—
255C22—21no—
256C22—21no—
257C22—21no—
258C22—21no—
259C22—44no—
260C22C22⋊#9511yescharacteristic
261C22#173312yes—
262C22#179112yes—
263C4—22no—
264C4—21no—
265C4—21no—
266C4—21no—
267C4—22no—
268C4—22no—
269C4—44no—
270C4—44no—
271C4—88no—
272C4#189714yes—
273C4#191212yes—
274C2—22no—
275C2—22no—
276C2—41no—
277C2#1072011yescharacteristic
278C2#1074111yescharacteristic
279C2#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
#340592
#218202
#207272
#79482
#78702