Finite Group Databasebeta

← #108689

Subgroups — #108689

Order 512.

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

normal subgroups only

This group has 455 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#20470C211yesmaximal, characteristic
2#8097C212yesmaximal
3#8006C211yesmaximal, characteristic
4#8001C211yesmaximal, characteristic
5#7990C212yesmaximal
6#2920C2212yes—
7C2×#432C2211yescharacteristic
8C2×#431C2211yescharacteristic
9C2×#423C2212yes—
10C4×#117C2211yescharacteristic
11#1767—21no—
12C4×C4⋊C8—21no—
13#1750—22no—
14#1746C412yes—
15#1726C412yes—
16#1303—22no—
17#1303—22no—
18#1293—22no—
19#1292—21no—
20#1292—21no—
21#1290C418yescomplemented
22C22×M16—22no—
23C2×#117—21no—
24C22×C42C2311yescharacteristic
25#432—22no—
26#432—22no—
27#432C2×C414yes—
28#432C2×C414yes—
29#431—22no—
30#431—22no—
31#431—22no—
32#431—22no—
33#423—22no—
34#423—22no—
35#423—22no—
36#423—22no—
37#423—22no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8—22no—
40C2×C4⋊C8C2×C414yes—
41C2×C4⋊C8D411yescharacteristic
42C2×C4⋊C8D411yescharacteristic
43#407—22no—
44C2×C22⋊C8—21no—
45C2×C22⋊C8D412yes—
46C4×M16—22no—
47C2×#97—22no—
48C2×#97C2×C412yes—
49C2×C4×C8—22no—
50C2×C4×C8C2×C412yes—
51C4×C4⋊C4—21no—
52C4×C4⋊C4D411yescharacteristic
53C4×C4⋊C4D411yescharacteristic
54C4×C22⋊C4—21no—
55C4×C22⋊C4Q811yescharacteristic
56C4×C22⋊C4Q811yescharacteristic
57#376C2×C411yescharacteristic
58#376C2×C411yescharacteristic
59C43—21no—
60#336—22no—
61#336—22no—
62C23×C4C2×D411yescharacteristic
63C23×C4C2×Q811yescharacteristic
64C23×C4#4111yescharacteristic
65C2×M16—42no—
66C2×M16—42no—
67C2×M16—42no—
68C2×M16—82no—
69C2×M16—82no—
70C22×C8—22no—
71C22×C8—24no—
72C22×C8C22⋊C412yes—
73C22×C8C22⋊C412yes—
74#117—42no—
75#117—42no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C22⋊C4—21no—
79C2×C22⋊C4—21no—
80C2×C42—21no—
81C2×C42—21no—
82C2×C42—21no—
83C2×C42—21no—
84C2×C42—21no—
85C2×C42—21no—
86C2×C42C22×C411yescharacteristic
87C2×C42C22×C412yes—
88C2×C42C2×D411yescharacteristic
89C2×C42C2×D411yescharacteristic
90C2×C42#4112yes—
91C2×C42#4111yescharacteristic
92C2×C42#4112yes—
93C2×C42#4112yes—
94C4⋊C8—24no—
95C4⋊C8—24no—
96C4⋊C8—24no—
97C4⋊C8—24no—
98C4⋊C8—22no—
99C4⋊C8—22no—
100C4⋊C8—42no—
101C4⋊C8—42no—
102C4⋊C8—42no—
103C22⋊C8—24no—
104C22⋊C8—24no—
105C22⋊C8—24no—
106C22⋊C8—42no—
107#97—22no—
108#97—24no—
109#97—44no—
110#97C4214yes—
111C4×C8—22no—
112C4×C8—22no—
113C4×C8—44no—
114C4×C8—42no—
115#95—21no—
116#95—21no—
117#95C4⋊C411yescharacteristic
118#95C4⋊C411yescharacteristic
119#95C4⋊C411yescharacteristic
120#95C4⋊C411yescharacteristic
121C24C22⋊Q811yescharacteristic
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—81no—
132C22×C4C2×C22⋊C411yescharacteristic
133C22×C4C2×C4⋊C411yescharacteristic
134C22×C4#11711yescharacteristic
135C22×C4#11712yes—
136C22×C4C4×D412yes—
137C22×C4C4×Q812yes—
138C22×C4#12111yescharacteristic
139C22×C4#12111yescharacteristic
140C22×C4C22⋊Q811yescharacteristic
141C22×C4#12312yes—
142C22×C4#12512yes—
143C22×C4#12612yes—
144M16—82no—
145M16—82no—
146M16—84no—
147M16—82no—
148M16—82no—
149C2×C8—24no—
150C2×C8—22no—
151C2×C8—22no—
152C2×C8—42no—
153C2×C8—42no—
154C2×C8—42no—
155C2×C8—42no—
156C2×C8—44no—
157C2×C8—44no—
158C2×C8—84no—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C22⋊C4—42no—
162C22⋊C4—42no—
163C42—22no—
164C42—22no—
165C42—41no—
166C42—41no—
167C42—42no—
168C42—41no—
169C42—41no—
170C42—81no—
171C42—81no—
172C42C2×C4211yescharacteristic
173C42#11711yescharacteristic
174C42#11712yes—
175C42#11711yescharacteristic
176C42#11711yescharacteristic
177C42C4×D412yes—
178C42C4×D412yes—
179C42#12312yes—
180C42#12412yes—
181C42#12612yes—
182C23—21no—
183C23—21no—
184C23—41no—
185C23—41no—
186C23#38011yescharacteristic
187C23#38712yes—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—42no—
203C2×C4—41no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4#37611yescharacteristic
217C2×C4C4×C22⋊C411yescharacteristic
218C2×C4C4×C4⋊C411yescharacteristic
219C2×C4#38111yescharacteristic
220C2×C4#38211yescharacteristic
221C2×C4#38311yescharacteristic
222C2×C4#38712yes—
223C2×C4#38812yes—
224C2×C4#38812yes—
225C2×C4#47112yes—
226C2×C4#47211yescharacteristic
227C2×C4#47311yescharacteristic
228C8—44no—
229C8—42no—
230C8—42no—
231C8—84no—
232C8—84no—
233C22—21no—
234C22—21no—
235C22—22no—
236C22—22no—
237C22—22no—
238C22—81no—
239C22—81no—
240C22#143411yescharacteristic
241C22#181511yescharacteristic
242C22#181611yescharacteristic
243C4—21no—
244C4—21no—
245C4—22no—
246C4—21no—
247C4—22no—
248C4—21no—
249C4—21no—
250C4—42no—
251C4—42no—
252C4—81no—
253C4—81no—
254C4#182711yescharacteristic
255C4#182711yescharacteristic
256C4#193012yes—
257C4#193112yes—
258C2—21no—
259C2—21no—
260C2—42no—
261C2#1081211yescharacteristic
262C2#1081611yescharacteristic
263C2#1082711yescharacteristic
subgroupG/H
center, socleC22#1434
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24C22⋊Q8
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
21. #1290C4semidirect

Maximal subgroups

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

subgroupindex
#204702
#80972
#80062
#80012
#79902