Finite Group Databasebeta

← #108739

Subgroups — #108739

Order 512.

477 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 477 conjugacy classes of subgroups — too many to draw. Only its 148 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#20481C211yesmaximal, characteristic
2#8034C212yesmaximal
3#7986C211yesmaximal, characteristic
4#7984C212yesmaximal
5#7981C211yesmaximal, characteristic
6C2×#432C2212yes—
7C2×#431C2211yescharacteristic
8C22×#97C2211yescharacteristic
9#2285C2212yes—
10#2283C2211yescharacteristic
11#1771—22no—
12#1752—21no—
13C8×C22⋊C4—21no—
14#1746C412yescomplemented
15#1726C412yescomplemented
16#1295C418yescomplemented
17#1290—22no—
18#1289—21no—
19#1289—21no—
20#1289—22no—
21#1289—22no—
22C23×C8—21no—
23C2×#117—22no—
24C22×C42C2311yescharacteristic
25#432—22no—
26#432—22no—
27#432—22no—
28#432—22no—
29#431—22no—
30#431—22no—
31C2×C4⋊C8—22no—
32C2×C4⋊C8D412yes—
33C2×C22⋊C8—22no—
34C2×C22⋊C8D411yescharacteristic
35C2×C22⋊C8D411yescharacteristic
36C2×#97—21no—
37C2×#97—22no—
38C2×#97—22no—
39C2×#97C2×C414yes—
40C2×#97C2×C414yes—
41C2×#97C2×C411yescharacteristic
42C2×#97C2×C411yescharacteristic
43C2×C4×C8—22no—
44C2×C4×C8C2×C412yes—
45#384—22no—
46#383Q812yes—
47#382—21no—
48#382C2×C414yes—
49#381—21no—
50#380—22no—
51#380—22no—
52#380D411yescharacteristic
53#380D411yescharacteristic
54C4×C4⋊C4—21no—
55C4×C4⋊C4—22no—
56C4×C4⋊C4—22no—
57C4×C22⋊C4—21no—
58#376C2×C412yes—
59#336—22no—
60#336—22no—
61C23×C4C2×D411yescharacteristic
62C23×C4C2×D411yescharacteristic
63C23×C4#4111yescharacteristic
64C22×C8—21no—
65C22×C8—21no—
66C22×C8—24no—
67C22×C8—24no—
68C22×C8—44no—
69C22×C8C22⋊C412yes—
70C22×C8C22⋊C412yes—
71#117—82no—
72#117—82no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—24no—
76C2×C4⋊C4—42no—
77C2×C4⋊C4—42no—
78C2×C22⋊C4—21no—
79C2×C22⋊C4—21no—
80C2×C22⋊C4—42no—
81C2×C42—22no—
82C2×C42—42no—
83C2×C42C22×C411yescharacteristic
84C2×C42C22×C412yes—
85C2×C42C2×D411yescharacteristic
86C2×C42C2×Q811yescharacteristic
87C2×C42#4112yes—
88C2×C42#4111yescharacteristic
89C2×C42#4112yes—
90C2×C42#4112yes—
91C4⋊C8—24no—
92C4⋊C8—42no—
93C4⋊C8—42no—
94C22⋊C8—22no—
95C22⋊C8—22no—
96C22⋊C8—42no—
97C22⋊C8—42no—
98#97—21no—
99#97—21no—
100#97—24no—
101#97—24no—
102#97—24no—
103#97—44no—
104#97C4212yes—
105#97C4212yes—
106C4×C8—22no—
107C4×C8—22no—
108C4×C8—44no—
109#95—22no—
110#95—22no—
111#95—22no—
112#95—24no—
113#95—24no—
114#95C4⋊C412yes—
115#95C4⋊C412yes—
116C24#12111yescharacteristic
117C22×C4—22no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—42no—
123C22×C4—44no—
124C22×C4—81no—
125C22×C4C2×C22⋊C411yescharacteristic
126C22×C4C2×C4⋊C411yescharacteristic
127C22×C4#11711yescharacteristic
128C22×C4#11712yes—
129C22×C4C4×D412yes—
130C22×C4C4×D412yes—
131C22×C4#12111yescharacteristic
132C22×C4C22⋊Q811yescharacteristic
133C22×C4C22⋊Q811yescharacteristic
134C22×C4#12312yes—
135C22×C4#12412yes—
136C22×C4#12612yes—
137C2×C8—24no—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—22no—
142C2×C8—48no—
143C2×C8—42no—
144C2×C8—42no—
145C2×C8—42no—
146C2×C8—44no—
147C2×C8—84no—
148C4⋊C4—84no—
149C4⋊C4—84no—
150C4⋊C4—81no—
151C4⋊C4—81no—
152C4⋊C4—84no—
153C22⋊C4—42no—
154C22⋊C4—41no—
155C22⋊C4—41no—
156C22⋊C4—82no—
157C22⋊C4—82no—
158C42—42no—
159C42—42no—
160C42—84no—
161C42C2×C4211yescharacteristic
162C42#11711yescharacteristic
163C42#11712yes—
164C42#11711yescharacteristic
165C42#11711yescharacteristic
166C42C4×D412yes—
167C42C4×Q812yes—
168C42#12312yes—
169C42#12512yes—
170C42#12612yes—
171C23—41no—
172C23—41no—
173C23#38011yescharacteristic
174C23#38812yes—
175C23#47112yes—
176C23#47211yescharacteristic
177C23#47311yescharacteristic
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—24no—
188C2×C4—41no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—84no—
201C2×C4—84no—
202C2×C4—84no—
203C2×C4#37611yescharacteristic
204C2×C4C4×C22⋊C411yescharacteristic
205C2×C4C4×C4⋊C411yescharacteristic
206C2×C4#38111yescharacteristic
207C2×C4#38211yescharacteristic
208C2×C4#38311yescharacteristic
209C2×C4#38712yes—
210C2×C4#38712yes—
211C2×C4#38812yes—
212C8—24no—
213C8—42no—
214C8—42no—
215C8—42no—
216C8—84no—
217C22—21no—
218C22—22no—
219C22—21no—
220C22—22no—
221C22—42no—
222C22—41no—
223C22—41no—
224C22#143411yescharacteristic
225C22#181511yescharacteristic
226C22#181611yescharacteristic
227C22#193012yes—
228C22#193112yes—
229C4—22no—
230C4—21no—
231C4—21no—
232C4—21no—
233C4—22no—
234C4—21no—
235C4—21no—
236C4—22no—
237C4—82no—
238C4—82no—
239C4—164no—
240C4#182712yes—
241C2—21no—
242C2—21no—
243C2—22no—
244C2—22no—
245C2#1081111yescharacteristic
246C2#1082411yescharacteristic
247C2#1082711yescharacteristic
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
16. #1295C4semidirect

Maximal subgroups

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

subgroupindex
#204812
#80342
#79862
#79842
#79812