Finite Group Databasebeta

← #108686

Subgroups — #108686

Order 512.

453 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 453 conjugacy classes of subgroups — too many to draw. Only its 140 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#20470C211yesmaximal, characteristic
2#8052C212yesmaximal
3#7987C211yesmaximal, characteristic
4#7980C211yesmaximal, characteristic
5#7977C212yesmaximal
6#2920C2212yes—
7C2×#432C2211yescharacteristic
8C2×#431C2211yescharacteristic
9#2283C2212yes—
10C4×#117C2211yescharacteristic
11#1767—21no—
12C4×C4⋊C8—21no—
13#1750—22no—
14#1746C412yescomplemented
15#1726C412yescomplemented
16#1297—24no—
17#1290—21no—
18#1290—21no—
19#1289—22no—
20#1288C418yescomplemented
21C2×#117—21no—
22C22×C42C2311yescharacteristic
23#432—22no—
24#432—22no—
25#432—22no—
26#432C2×C414yes—
27#431—22no—
28#431—22no—
29#431—22no—
30#431—22no—
31#423—22no—
32C2×C4⋊C8—21no—
33C2×C4⋊C8D411yescharacteristic
34C2×C4⋊C8D411yescharacteristic
35#407—22no—
36C2×C22⋊C8—21no—
37C2×C22⋊C8D412yes—
38C4×M16C2×C414yes—
39C2×#97—22no—
40C2×#97C2×C412yes—
41C2×C4×C8—22no—
42C2×C4×C8C2×C412yes—
43#383C2×C414yes—
44#382—22no—
45#381—22no—
46#380—22no—
47C4×C4⋊C4—21no—
48C4×C4⋊C4—22no—
49C4×C4⋊C4Q812yes—
50C4×C22⋊C4—21no—
51C4×C22⋊C4—22no—
52C4×C22⋊C4D412yes—
53#376—22no—
54#376C2×C411yescharacteristic
55#376C2×C411yescharacteristic
56C43—21no—
57#336—22no—
58#336—22no—
59C23×C4C2×D411yescharacteristic
60C23×C4C2×D411yescharacteristic
61C23×C4#4111yescharacteristic
62C2×M16—24no—
63C22×C8—22no—
64C22×C8C22⋊C412yes—
65C22×C8C22⋊C412yes—
66#117—42no—
67#117—42no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—42no—
70C2×C22⋊C4—22no—
71C2×C22⋊C4—42no—
72C2×C42—21no—
73C2×C42—21no—
74C2×C42—22no—
75C2×C42—22no—
76C2×C42—42no—
77C2×C42C22×C411yescharacteristic
78C2×C42C22×C412yes—
79C2×C42C2×D411yescharacteristic
80C2×C42C2×Q811yescharacteristic
81C2×C42#4112yes—
82C2×C42#4111yescharacteristic
83C2×C42#4112yes—
84C2×C42#4112yes—
85C4⋊C8—24no—
86C4⋊C8—22no—
87C4⋊C8—22no—
88C4⋊C8—42no—
89C4⋊C8—42no—
90C22⋊C8—24no—
91C22⋊C8—24no—
92C22⋊C8—42no—
93C22⋊C8—42no—
94#97—22no—
95#97—24no—
96#97—44no—
97#97C4214yes—
98C4×C8—22no—
99C4×C8—22no—
100C4×C8—24no—
101C4×C8—44no—
102#95—21no—
103#95—21no—
104#95—24no—
105#95—24no—
106#95—24no—
107#95—42no—
108#95C4⋊C412yes—
109#95C4⋊C412yes—
110C24#12111yescharacteristic
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—41no—
114C22×C4—41no—
115C22×C4—42no—
116C22×C4—44no—
117C22×C4—81no—
118C22×C4C2×C22⋊C411yescharacteristic
119C22×C4C2×C4⋊C411yescharacteristic
120C22×C4#11711yescharacteristic
121C22×C4#11712yes—
122C22×C4C4×D412yes—
123C22×C4C4×D412yes—
124C22×C4#12111yescharacteristic
125C22×C4C22⋊Q811yescharacteristic
126C22×C4C22⋊Q811yescharacteristic
127C22×C4#12312yes—
128C22×C4#12412yes—
129C22×C4#12612yes—
130M16—84no—
131C2×C8—24no—
132C2×C8—22no—
133C2×C8—22no—
134C2×C8—42no—
135C2×C8—42no—
136C2×C8—42no—
137C2×C8—42no—
138C2×C8—44no—
139C4⋊C4—44no—
140C4⋊C4—82no—
141C4⋊C4—82no—
142C22⋊C4—44no—
143C22⋊C4—84no—
144C42—22no—
145C42—22no—
146C42—44no—
147C42—41no—
148C42—41no—
149C42—82no—
150C42—84no—
151C42C2×C4211yescharacteristic
152C42#11711yescharacteristic
153C42#11712yes—
154C42#11711yescharacteristic
155C42#11711yescharacteristic
156C42C4×D412yes—
157C42C4×Q812yes—
158C42#12312yes—
159C42#12512yes—
160C42#12612yes—
161C23—22no—
162C23—41no—
163C23—41no—
164C23#38011yescharacteristic
165C23#38812yes—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—24no—
173C2×C4—41no—
174C2×C4—42no—
175C2×C4—41no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—41no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—84no—
187C2×C4—84no—
188C2×C4—84no—
189C2×C4#37611yescharacteristic
190C2×C4C4×C22⋊C411yescharacteristic
191C2×C4C4×C4⋊C411yescharacteristic
192C2×C4#38111yescharacteristic
193C2×C4#38211yescharacteristic
194C2×C4#38311yescharacteristic
195C2×C4#38712yes—
196C2×C4#38712yes—
197C2×C4#38812yes—
198C8—44no—
199C8—42no—
200C8—42no—
201C8—84no—
202C22—21no—
203C22—21no—
204C22—22no—
205C22—22no—
206C22—22no—
207C22—82no—
208C22#143411yescharacteristic
209C22#181412yes—
210C4—22no—
211C4—21no—
212C4—21no—
213C4—21no—
214C4—22no—
215C4—21no—
216C4—21no—
217C4—22no—
218C4—44no—
219C4—82no—
220C4—164no—
221C4#182711yescharacteristic
222C4#182711yescharacteristic
223C2—22no—
224C2—42no—
225C2#1081111yescharacteristic
226C2#1081611yescharacteristic
227C2#1082811yescharacteristic
subgroupG/H
center, socleC22#1434
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
14. #1746C4semidirect
15. #1726C4semidirect
20. #1288C4semidirect

Maximal subgroups

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

subgroupindex
#204702
#80522
#79872
#79802
#79772