Finite Group Databasebeta

← #120577

Subgroups — #120577

Order 512.

556 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 136 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#21355C211yesmaximal, characteristic
2#20451C211yesmaximal, characteristic
3#8048C212yesmaximal
4#7949C212yesmaximal
5#7861C211yesmaximal, characteristic
6#2920C2212yes—
7C2×#432C2211yescharacteristic
8C2×#427—21no—
9C22×#97C2211yescharacteristic
10C4×#117C2212yes—
11C4×#117C2211yescharacteristic
12#1940C412yescomplemented
13#1940C412yescomplemented
14#1850—21no—
15#1850—21no—
16#1776—24no—
17#1767—22no—
18#1735—21no—
19#1297—22no—
20#1297—22no—
21#1285—24no—
22#1275—22no—
23C23×C8D411yescharacteristic
24C23×C8D411yescharacteristic
25C2×#117—21no—
26C2×#117—22no—
27C22×C42C2311yescharacteristic
28#432—22no—
29#432—22no—
30#432—22no—
31#432—22no—
32#431—22no—
33#431—22no—
34#427—28no—
35C2×#107—22no—
36C2×#106—22no—
37#423—22no—
38C2×C4⋊C8—22no—
39C2×C4⋊C8—22no—
40C2×C4⋊C8C2×C411yescharacteristic
41C2×C4⋊C8C2×C411yescharacteristic
42#407—22no—
43C2×C22⋊C8—21no—
44C4×M16Q814yes—
45C2×#97—22no—
46C2×#97—22no—
47C2×#97—21no—
48C2×#97C2×C412yes—
49C2×#97D412yes—
50C4×C4⋊C4—21no—
51C4×C4⋊C4—22no—
52C4×C4⋊C4C2×C412yes—
53C4×C4⋊C4D414yes—
54C4×C22⋊C4—21no—
55C4×C22⋊C4—22no—
56C4×C22⋊C4—22no—
57C4×C22⋊C4D411yescharacteristic
58C4×C22⋊C4D411yescharacteristic
59#376—22no—
60#376Q812yes—
61C43—21no—
62C43—22no—
63#340—24no—
64#336—22no—
65#336—22no—
66C23×C4C2×D411yescharacteristic
67C23×C4C2×D411yescharacteristic
68C23×C4#4111yescharacteristic
69C2×M16—24no—
70C22×C8—21no—
71C22×C8—21no—
72C22×C8—24no—
73C22×C8—21no—
74C22×C8—21no—
75C22×C8—21no—
76C22×C8—21no—
77C22×C8—42no—
78C22×C8C22⋊C412yes—
79C22×C8C22⋊C412yes—
80#117—28no—
81#117—44no—
82#117—44no—
83C2×C4⋊C4—24no—
84C2×C4⋊C4—22no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4—21no—
87C2×C22⋊C4—42no—
88C2×C42—22no—
89C2×C42—22no—
90C2×C42—21no—
91C2×C42—21no—
92C2×C42—24no—
93C2×C42—42no—
94C2×C42—42no—
95C2×C42C22×C411yescharacteristic
96C2×C42C2×D412yes—
97C2×C42C2×D411yescharacteristic
98C2×C42C2×Q812yes—
99C2×C42C2×Q811yescharacteristic
100C2×C42#4112yes—
101C2×C42#4112yes—
102C2×C42#4111yescharacteristic
103#107—28no—
104#106—28no—
105C4⋊C8—22no—
106C4⋊C8—22no—
107C4⋊C8—44no—
108C4⋊C8—44no—
109C4⋊C8—42no—
110C4⋊C8—42no—
111C22⋊C8—41no—
112C22⋊C8—42no—
113C22⋊C8—42no—
114C22⋊C8—41no—
115#97—24no—
116#97—24no—
117#97—24no—
118#97—42no—
119C4×C8—24no—
120#95—22no—
121#95—24no—
122#95—42no—
123#95C4⋊C412yes—
124#95C4⋊C412yes—
125C24#12111yescharacteristic
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—42no—
131C22×C4—44no—
132C22×C4—81no—
133C22×C4C2×C22⋊C411yescharacteristic
134C22×C4C2×C4⋊C411yescharacteristic
135C22×C4#11711yescharacteristic
136C22×C4#12011yescharacteristic
137C22×C4#12112yes—
138C22×C4#12111yescharacteristic
139C22×C4C22⋊Q811yescharacteristic
140C22×C4C22⋊Q812yes—
141C22×C4C22⋊Q811yescharacteristic
142C22×C4#12311yescharacteristic
143C22×C4#12312yes—
144C22×C4#12411yescharacteristic
145C22×C4#12512yes—
146C22×C4C4⋊Q811yescharacteristic
147M16—84no—
148C2×C8—22no—
149C2×C8—22no—
150C2×C8—24no—
151C2×C8—22no—
152C2×C8—22no—
153C2×C8—22no—
154C2×C8—22no—
155C2×C8—22no—
156C2×C8—42no—
157C2×C8—42no—
158C2×C8—41no—
159C2×C8—41no—
160C2×C8—44no—
161C2×C8—42no—
162C2×C8—42no—
163C2×C8#10414yes—
164C4⋊C4—28no—
165C4⋊C4—44no—
166C4⋊C4—44no—
167C22⋊C4—24no—
168C22⋊C4—42no—
169C22⋊C4—48no—
170C42—44no—
171C42—48no—
172C42—42no—
173C42—42no—
174C42—84no—
175C42—81no—
176C42—81no—
177C42—84no—
178C42C4×D412yes—
179C42C4×Q812yes—
180C42#12112yes—
181C42C22⋊Q812yes—
182C42C22⋊Q812yes—
183C42#12512yes—
184C42#12612yes—
185C42C4⋊Q812yes—
186C23—22no—
187C23—22no—
188C23—41no—
189C23#38011yescharacteristic
190C23#39411yescharacteristic
191C23#39811yescharacteristic
192C2×C4—24no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—24no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—41no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—48no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—84no—
215C2×C4—84no—
216C2×C4#38611yescharacteristic
217C2×C4#38712yes—
218C2×C4#38912yes—
219C2×C4#39111yescharacteristic
220C2×C4#39511yescharacteristic
221C2×C4#39612yes—
222C2×C4#39812yes—
223C2×C4#40011yescharacteristic
224C2×C4C2×#10412yes—
225C2×C4#42112yes—
226C8—24no—
227C8—42no—
228C8—84no—
229C8—81no—
230C8—81no—
231C22—21no—
232C22—21no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—44no—
239C22#144111yescharacteristic
240C22#180612yes—
241C22#205012yes—
242C4—21no—
243C4—21no—
244C4—22no—
245C4—22no—
246C4—22no—
247C4—22no—
248C4—44no—
249C4—82no—
250C4—88no—
251C4#189611yescharacteristic
252C4#189711yescharacteristic
253C4#198612yes—
254C2—22no—
255C2—22no—
256C2—41no—
257C2#1122611yescharacteristic
258C2#1123411yescharacteristic
259C2#1124311yescharacteristic
subgroupG/H
centerC2×C4#391
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1441
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
12. #1940C4semidirect
13. #1940C4semidirect

Maximal subgroups

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

subgroupindex
#213552
#204512
#80482
#79492
#78612