Finite Group Databasebeta

← #116562

Subgroups — #116562

Order 512.

606 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 606 conjugacy classes of subgroups — too many to draw. Only its 170 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#21301C211yesmaximal, characteristic
2#20762C211yesmaximal, characteristic
3#17575C212yesmaximal
4#8095C212yesmaximal
5C2×#1275C211yesmaximal, characteristic
6C2×#533C2211yescharacteristic
7C2×#431C2212yes—
8C2×#423C2211yescharacteristic
9C22×#97C2211yescharacteristic
10#2285C2212yes—
11#1985—22no—
12#1985C414yescomplemented
13#1836—24no—
14#1735—22no—
15#1729—22no—
16#1520—24no—
17#1518—24no—
18#1503—24no—
19#1303—24no—
20#1275—22no—
21C22×M16—21no—
22C23×C8D412yes—
23#533—22no—
24#533—22no—
25#533—22no—
26#533—22no—
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—
35C2×#117—22no—
36C22×C42C2311yescharacteristic
37#431—22no—
38#431—22no—
39#431—22no—
40#431D414yes—
41#423—22no—
42#423—22no—
43C2×C4⋊C8—22no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8C2×C412yes—
46C2×C22⋊C8—22no—
47C2×#97—22no—
48C2×#97—22no—
49C2×#97C2×C412yes—
50C2×#97D412yes—
51C2×C4×C8—22no—
52#384—22no—
53#382D414yes—
54#380—22no—
55#380—22no—
56C4×C4⋊C4—22no—
57C4×C4⋊C4—22no—
58#340—28no—
59#336—28no—
60#328—216no—
61C22×Q8—21no—
62C22×Q8C4⋊C412yes—
63C23×C4C2×D411yescharacteristic
64C23×C4C2×D411yescharacteristic
65C23×C4#4111yescharacteristic
66C2×M16—41no—
67C2×M16—41no—
68C2×M16—44no—
69C22×C8—22no—
70C22×C8—22no—
71C22×C8—24no—
72C22×C8—24no—
73C22×C8—22no—
74C22×C8—42no—
75C22×C8C22⋊C414yes—
76C4⋊Q8—22no—
77C4⋊Q8—22no—
78C4⋊Q8—22no—
79C4⋊Q8—22no—
80#125—42no—
81#125—42no—
82C22⋊Q8—24no—
83C22⋊Q8—24no—
84C22⋊Q8—42no—
85C22⋊Q8—42no—
86C4×Q8—24no—
87C4×Q8—24no—
88C4×Q8—42no—
89C4×Q8—42no—
90#117—41no—
91#117—41no—
92#117—41no—
93#117—41no—
94#117—44no—
95#117—44no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4C4⋊C412yes—
105C2×C22⋊C4—22no—
106C2×C22⋊C4—42no—
107C2×C42—22no—
108C2×C42—42no—
109C2×C42C22×C411yescharacteristic
110C2×C42C2×D412yes—
111C2×C42C2×D412yes—
112C2×C42C2×D411yescharacteristic
113C2×C42C2×Q811yescharacteristic
114C2×C42#4112yes—
115C2×C42#4112yes—
116C2×C42#4111yescharacteristic
117C4⋊C8—24no—
118C4⋊C8—24no—
119C4⋊C8—42no—
120C4⋊C8—42no—
121C22⋊C8—24no—
122C22⋊C8—42no—
123#97—24no—
124#97—24no—
125C4×C8—24no—
126C4×C8—42no—
127#95—24no—
128#95—24no—
129C24#12111yescharacteristic
130C2×Q8—24no—
131C2×Q8—42no—
132C2×Q8—82no—
133C2×Q8—82no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—44no—
137C22×C4—81no—
138C22×C4—81no—
139C22×C4C2×C22⋊C411yescharacteristic
140C22×C4C2×C4⋊C411yescharacteristic
141C22×C4#11711yescharacteristic
142C22×C4#12012yes—
143C22×C4#12011yescharacteristic
144C22×C4#12112yes—
145C22×C4#12111yescharacteristic
146C22×C4C22⋊Q811yescharacteristic
147C22×C4C22⋊Q811yescharacteristic
148C22×C4#12312yes—
149C22×C4#12311yescharacteristic
150C22×C4#12411yescharacteristic
151C22×C4#12612yes—
152C22×C4#12711yescharacteristic
153C22×C4#13514yes—
154C22×C4#13714yes—
155M16—82no—
156M16—82no—
157C2×C8—24no—
158C2×C8—24no—
159C2×C8—24no—
160C2×C8—44no—
161C2×C8—44no—
162C2×C8—42no—
163C2×C8—44no—
164C2×C8—44no—
165C2×C8—84no—
166C4⋊C4—24no—
167C4⋊C4—44no—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—44no—
171C4⋊C4—42no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—48no—
175C4⋊C4—84no—
176C4⋊C4—84no—
177C22⋊C4—42no—
178C22⋊C4—42no—
179C22⋊C4—84no—
180C42—42no—
181C42—42no—
182C42—82no—
183C42—82no—
184C42C4×D412yes—
185C42C4×D412yes—
186C42#12112yes—
187C42#12112yes—
188C42C22⋊Q812yes—
189C42C22⋊Q812yes—
190C42#12312yes—
191C42#12412yes—
192C23—41no—
193C23—41no—
194C23#38011yescharacteristic
195C23#39211yescharacteristic
196C23#39711yescharacteristic
197C23#46414yes—
198Q8—84no—
199Q8—84no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—44no—
209C2×C4—44no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—84no—
213C2×C4—84no—
214C2×C4—84no—
215C2×C4#38512yes—
216C2×C4#38611yescharacteristic
217C2×C4#38812yes—
218C2×C4#39011yescharacteristic
219C2×C4#39311yescharacteristic
220C2×C4#39412yes—
221C2×C4#39612yes—
222C2×C4#39911yescharacteristic
223C2×C4#41612yes—
224C2×C4#41912yes—
225C2×C4#45214yes—
226C2×C4#47914yes—
227C2×C4#48514yes—
228C2×C4#48812yes—
229C2×C4#49112yes—
230C2×C4#49512yes—
231C2×C4#49712yes—
232C8—44no—
233C8—84no—
234C8—82no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—21no—
240C22—21no—
241C22—44no—
242C22#143911yescharacteristic
243C22#180512yes—
244C22#201512yes—
245C22#207412yes—
246C4—22no—
247C4—22no—
248C4—22no—
249C4—22no—
250C4—82no—
251C4—82no—
252C4—164no—
253C4#187312yes—
254C4#195811yescharacteristic
255C4#196011yescharacteristic
256C4#202612yes—
257C4#209712yes—
258C2—22no—
259C2—22no—
260C2—22no—
261C2#1105211yescharacteristic
262C2#1110211yescharacteristic
263C2#1110911yescharacteristic
subgroupG/H
centerC2×C4#390
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1439
prime radicalC24#121
LCSG ▷ C2×C42 ▷ 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
12. #1985C4semidirect

Maximal subgroups

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

subgroupindex
#213012
#207622
#175752
#80952
C2×#12752