Finite Group Databasebeta

← #108644

Subgroups — #108644

Order 512.

467 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 467 conjugacy classes of subgroups — too many to draw. Only its 150 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#20546C211yesmaximal, characteristic
2#8121C212yesmaximal
3#8097C212yesmaximal
4C2×#1292C211yesmaximal, characteristic
5#7988C211yesmaximal, characteristic
6#2920C2212yes—
7C2×#432C2211yescharacteristic
8C2×#423C2212yes—
9C22×C4⋊C8C2211yescharacteristic
10C4×#117C2211yescharacteristic
11#1771—22no—
12#1767—21no—
13C4×C4⋊C8—21no—
14#1752C412yes—
15C8×C22⋊C4C412yes—
16#1307C418yescomplemented
17#1303—22no—
18#1303—22no—
19#1292—21no—
20#1292—21no—
21#1290—22no—
22C22×M16—22no—
23C23×C8C2×C412yes—
24C2×#117—21no—
25C22×C42C2311yescharacteristic
26#432—22no—
27#432—22no—
28#432C2×C414yes—
29#432C2×C414yes—
30#431—22no—
31#431—22no—
32#423—22no—
33#423—22no—
34#423—22no—
35#423—22no—
36#423C2×C414yes—
37C2×C4⋊C8—21no—
38C2×C4⋊C8—22no—
39C2×C4⋊C8—22no—
40C2×C4⋊C8—22no—
41C2×C4⋊C8—22no—
42C2×C4⋊C8D412yes—
43C2×C4⋊C8D411yescharacteristic
44C2×C4⋊C8D411yescharacteristic
45#407—22no—
46C2×C22⋊C8C2×C412yes—
47C4×M16—22no—
48C2×#97—21no—
49C2×#97—22no—
50C2×C4×C8—22no—
51C4×C4⋊C4—21no—
52C4×C4⋊C4D411yescharacteristic
53C4×C4⋊C4D411yescharacteristic
54C4×C22⋊C4—21no—
55C4×C22⋊C4C2×C411yescharacteristic
56C4×C22⋊C4C2×C411yescharacteristic
57#376Q811yescharacteristic
58#376Q811yescharacteristic
59C43—21no—
60#336—22no—
61#336—22no—
62C23×C4C22×C411yescharacteristic
63C23×C4#4111yescharacteristic
64C23×C4#4111yescharacteristic
65C2×M16—24no—
66C2×M16—42no—
67C2×M16—42no—
68C2×M16—82no—
69C2×M16—82no—
70C22×C8—22no—
71C22×C8—22no—
72C22×C8—22no—
73C22×C8—42no—
74C22×C8—82no—
75C22×C8C22⋊C412yes—
76C22×C8C22⋊C412yes—
77#117—44no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C22⋊C4—21no—
81C2×C22⋊C4—21no—
82C2×C42—21no—
83C2×C42—21no—
84C2×C42—21no—
85C2×C42—21no—
86C2×C42—21no—
87C2×C42—21no—
88C2×C42C22×C412yes—
89C2×C42C2×D411yescharacteristic
90C2×C42C2×D411yescharacteristic
91C2×C42C2×D411yescharacteristic
92C2×C42C2×Q811yescharacteristic
93C2×C42#4112yes—
94C2×C42#4112yes—
95C2×C42#4112yes—
96C4⋊C8—24no—
97C4⋊C8—24no—
98C4⋊C8—24no—
99C4⋊C8—24no—
100C4⋊C8—24no—
101C4⋊C8—22no—
102C4⋊C8—22no—
103C4⋊C8—42no—
104C4⋊C8—42no—
105C4⋊C8—42no—
106C22⋊C8—22no—
107C22⋊C8—24no—
108C22⋊C8—42no—
109C22⋊C8C4214yes—
110#97—24no—
111#97—24no—
112#97—82no—
113C4×C8—42no—
114C4×C8—82no—
115#95—21no—
116#95—21no—
117#95C4⋊C411yescharacteristic
118#95C4⋊C411yescharacteristic
119#95C4⋊C411yescharacteristic
120#95C4⋊C411yescharacteristic
121C24#11711yescharacteristic
122C22×C4—22no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—41no—
130C22×C4—41no—
131C22×C4—42no—
132C22×C4C2×C4211yescharacteristic
133C22×C4C2×C22⋊C411yescharacteristic
134C22×C4C2×C4⋊C411yescharacteristic
135C22×C4#11711yescharacteristic
136C22×C4#11711yescharacteristic
137C22×C4#11712yes—
138C22×C4#11711yescharacteristic
139C22×C4#11712yes—
140C22×C4#12111yescharacteristic
141C22×C4#12111yescharacteristic
142C22×C4C22⋊Q811yescharacteristic
143C22×C4C22⋊Q811yescharacteristic
144C22×C4#12612yes—
145C22×C4#12612yes—
146M16—84no—
147M16—82no—
148M16—82no—
149M16—84no—
150C2×C8—24no—
151C2×C8—22no—
152C2×C8—22no—
153C2×C8—42no—
154C2×C8—42no—
155C2×C8—42no—
156C2×C8—42no—
157C2×C8—44no—
158C2×C8—44no—
159C2×C8—82no—
160C2×C8—82no—
161C2×C8—84no—
162C4⋊C4—42no—
163C4⋊C4—42no—
164C22⋊C4—24no—
165C22⋊C4—42no—
166C42—41no—
167C42—41no—
168C42—41no—
169C42—41no—
170C42—42no—
171C42—42no—
172C42—42no—
173C42—41no—
174C42—41no—
175C42C4×D412yes—
176C42C4×D412yes—
177C42C4×D412yes—
178C42C4×Q812yes—
179C42#12312yes—
180C42#12312yes—
181C42#12412yes—
182C42#12512yes—
183C23—21no—
184C23—21no—
185C23—22no—
186C23—41no—
187C23#37611yescharacteristic
188C23#38011yescharacteristic
189C23#38311yescharacteristic
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—24no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—42no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4C4×C22⋊C411yescharacteristic
222C2×C4C4×C4⋊C411yescharacteristic
223C2×C4#38111yescharacteristic
224C2×C4#38211yescharacteristic
225C2×C4#38712yes—
226C2×C4#38712yes—
227C2×C4#38812yes—
228C2×C4#38812yes—
229C2×C4#47112yes—
230C2×C4#47211yescharacteristic
231C2×C4#47311yescharacteristic
232C8—44no—
233C8—42no—
234C8—42no—
235C8—84no—
236C8—84no—
237C22—21no—
238C22—21no—
239C22—21no—
240C22—21no—
241C22—21no—
242C22—21no—
243C22—21no—
244C22—42no—
245C22—42no—
246C22#143411yescharacteristic
247C22#181511yescharacteristic
248C22#181611yescharacteristic
249C22#184112yes—
250C4—22no—
251C4—22no—
252C4—22no—
253C4—21no—
254C4—21no—
255C4—21no—
256C4—42no—
257C4—42no—
258C4—81no—
259C4—81no—
260C4#182911yescharacteristic
261C4#183011yescharacteristic
262C4#193012yes—
263C4#193112yes—
264C2—21no—
265C2—21no—
266C2—22no—
267C2—41no—
268C2#1080511yescharacteristic
269C2#1082711yescharacteristic
270C2#1083011yescharacteristic
subgroupG/H
center, socleC22#1434
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24#117
LCSG ▷ C22×C4 ▷ 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
16. #1307C4semidirect

Maximal subgroups

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

subgroupindex
#205462
#81212
#80972
C2×#12922
#79882