Finite Group Databasebeta

← #120281

Subgroups — #120281

Order 512.

522 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 132 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#21960C211yesmaximal, characteristic
2#20793C211yesmaximal, characteristic
3#8002C211yesmaximal, characteristic
4#7962C212yesmaximal
5#7940C212yesmaximal
6#2920C2212yes—
7C2×#424—21no—
8C2×#424—21no—
9C2×#423C2211yescharacteristic
10C22×C4⋊C8C2211yescharacteristic
11#2283C2212yes—
12C4×#117C2211yescharacteristic
13#1949—24no—
14#1843—22no—
15#1842C414yescomplemented
16#1808—21no—
17#1807—21no—
18#1292—22no—
19#1286—24no—
20#1284—24no—
21C22×M16—21no—
22C23×C8D411yescharacteristic
23C23×C8D411yescharacteristic
24C2×#117—21no—
25C22×C42C2311yescharacteristic
26#432—22no—
27#432—22no—
28#431—22no—
29#431Q814yes—
30C2×#107—22no—
31C2×#106—22no—
32#424—44no—
33#424—44no—
34#423—22no—
35#423—22no—
36#423—22no—
37C2×C4⋊C8—22no—
38C2×C4⋊C8—22no—
39C2×C4⋊C8—24no—
40C2×C4⋊C8—24no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8C2×C412yes—
43C2×C4⋊C8C2×C412yes—
44C2×C4⋊C8D412yes—
45#407—22no—
46C4×M16—22no—
47#383—22no—
48#382D414yes—
49#381—22no—
50#380—22no—
51C4×C4⋊C4—21no—
52C4×C4⋊C4—22no—
53C4×C4⋊C4Q812yes—
54C4×C22⋊C4—21no—
55C4×C22⋊C4—22no—
56C4×C22⋊C4D412yes—
57#376—21no—
58#376—22no—
59C43C2×C412yes—
60#340—22no—
61#340—22no—
62C23×C4C2×D411yescharacteristic
63C23×C4C2×D411yescharacteristic
64C23×C4#4111yescharacteristic
65C2×M16—42no—
66C2×M16—42no—
67C2×M16—41no—
68C2×M16—42no—
69C2×M16—41no—
70C22×C8—21no—
71C22×C8—21no—
72C22×C8—21no—
73C22×C8—21no—
74C22×C8—22no—
75C22×C8—42no—
76C22×C8—42no—
77C22×C8C22⋊C412yes—
78C22×C8C22⋊C412yes—
79#117—24no—
80#117—24no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—24no—
83C2×C22⋊C4—22no—
84C2×C22⋊C4—42no—
85C2×C42—22no—
86C2×C42—22no—
87C2×C42—42no—
88C2×C42C4⋊C414yes—
89C2×C42C22×C411yescharacteristic
90C2×C42C2×D412yes—
91C2×C42C2×D411yescharacteristic
92C2×C42C2×Q812yes—
93C2×C42C2×Q811yescharacteristic
94C2×C42#4112yes—
95C2×C42#4112yes—
96C2×C42#4111yescharacteristic
97#107—82no—
98#106—82no—
99C4⋊C8—24no—
100C4⋊C8—24no—
101C4⋊C8—24no—
102C4⋊C8—24no—
103C4⋊C8—48no—
104C4⋊C8—48no—
105C4⋊C8—42no—
106C4⋊C8—42no—
107C22⋊C8—24no—
108C22⋊C8—42no—
109#97—42no—
110C4×C8—24no—
111#95—22no—
112#95—22no—
113#95—24no—
114#95—24no—
115#95—42no—
116#95—42no—
117C24#12111yescharacteristic
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—24no—
121C22×C4—42no—
122C22×C4—44no—
123C22×C4—81no—
124C22×C4C2×C22⋊C411yescharacteristic
125C22×C4C2×C4⋊C411yescharacteristic
126C22×C4#11711yescharacteristic
127C22×C4#12011yescharacteristic
128C22×C4#12111yescharacteristic
129C22×C4#12112yes—
130C22×C4C22⋊Q811yescharacteristic
131C22×C4C22⋊Q812yes—
132C22×C4C22⋊Q811yescharacteristic
133C22×C4#12311yescharacteristic
134C22×C4#12312yes—
135C22×C4#12411yescharacteristic
136C22×C4#12512yes—
137C22×C4C4⋊Q811yescharacteristic
138M16—84no—
139M16—82no—
140M16—82no—
141C2×C8—22no—
142C2×C8—22no—
143C2×C8—42no—
144C2×C8—42no—
145C2×C8—42no—
146C2×C8—42no—
147C2×C8—42no—
148C2×C8—42no—
149C2×C8—42no—
150C2×C8—44no—
151C2×C8—81no—
152C2×C8—81no—
153C4⋊C4—28no—
154C4⋊C4—44no—
155C4⋊C4—44no—
156C22⋊C4—28no—
157C22⋊C4—84no—
158C42—24no—
159C42—24no—
160C42—44no—
161C42—82no—
162C42—84no—
163C42C4×D412yes—
164C42C4×Q812yes—
165C42#12112yes—
166C42C22⋊Q812yes—
167C42C22⋊Q812yes—
168C42#12512yes—
169C42#12612yes—
170C42C4⋊Q812yes—
171C23—22no—
172C23—22no—
173C23—41no—
174C23#38011yescharacteristic
175C23#39411yescharacteristic
176C23#39811yescharacteristic
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—28no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—44no—
194C2×C4—44no—
195C2×C4—84no—
196C2×C4—84no—
197C2×C4—84no—
198C2×C4#38611yescharacteristic
199C2×C4#38712yes—
200C2×C4#38912yes—
201C2×C4#39111yescharacteristic
202C2×C4#39511yescharacteristic
203C2×C4#39612yes—
204C2×C4#39812yes—
205C2×C4#40011yescharacteristic
206C2×C4#45312yes—
207C2×C4#45612yes—
208C8—42no—
209C8—42no—
210C8—84no—
211C8—82no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—21no—
217C22—21no—
218C22—21no—
219C22—44no—
220C22#144111yescharacteristic
221C22#181212yes—
222C22#203912yes—
223C4—21no—
224C4—21no—
225C4—22no—
226C4—22no—
227C4—22no—
228C4—22no—
229C4—48no—
230C4—164no—
231C4#188611yescharacteristic
232C4#188711yescharacteristic
233C4#206012yes—
234C2—22no—
235C2—22no—
236C2—41no—
237C2#1121011yescharacteristic
238C2#1122911yescharacteristic
239C2#1124511yescharacteristic
subgroupG/H
center, socleC22#1441
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C2×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
15. #1842C4semidirect

Maximal subgroups

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

subgroupindex
#219602
#207932
#80022
#79622
#79402