Finite Group Databasebeta

← #120531

Subgroups — #120531

Order 512.

458 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 126 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#21974C211yesmaximal, characteristic
2#17815C211yesmaximal, characteristic
3#7980C211yesmaximal, characteristic
4#7962C212yesmaximal
5#7941C212yesmaximal
6#2920C2212yes—
7C2×#431C2211yescharacteristic
8C2×#423C2211yescharacteristic
9#2283C2212yes—
10C4×#117C2211yescharacteristic
11#1939—21no—
12#1939—21no—
13#1851—21no—
14#1851—21no—
15#1842C412yescomplemented
16#1831C412yescomplemented
17#1572—24no—
18#1567—21no—
19#1566—21no—
20#1289—21no—
21#1289—21no—
22#1286—24no—
23#1284—24no—
24C22×M16—21no—
25C2×#117—21no—
26C22×C42C2311yescharacteristic
27#432—22no—
28#432—22no—
29#431—22no—
30#431—22no—
31#431—22no—
32#431D414yes—
33#423—22no—
34#423—22no—
35#423—22no—
36C2×C4⋊C8—21no—
37C2×C4⋊C8—22no—
38C2×C4⋊C8—22no—
39C2×C4⋊C8C2×C412yes—
40C2×C4⋊C8C2×C411yescharacteristic
41C2×C4⋊C8C2×C411yescharacteristic
42#407—22no—
43C2×C22⋊C8D411yescharacteristic
44C2×C22⋊C8D411yescharacteristic
45C4×M16—22no—
46C2×C4×C8D411yescharacteristic
47C2×C4×C8D411yescharacteristic
48#383Q814yes—
49#382—22no—
50#381—22no—
51#380—22no—
52C4×C4⋊C4—21no—
53C4×C4⋊C4—22no—
54C4×C4⋊C4Q811yescharacteristic
55C4×C4⋊C4Q811yescharacteristic
56C4×C22⋊C4—21no—
57C4×C22⋊C4—22no—
58C4×C22⋊C4D412yes—
59#376—21no—
60#376—22no—
61C43C2×C412yes—
62#336—22no—
63#336—22no—
64#335—24no—
65#334—24no—
66C23×C4C2×D411yescharacteristic
67C23×C4C2×D411yescharacteristic
68C23×C4#4111yescharacteristic
69C2×M16—42no—
70C2×M16—42no—
71C2×M16—82no—
72C22×C8—22no—
73C22×C8C22⋊C411yescharacteristic
74C22×C8C22⋊C411yescharacteristic
75C22×C8C22⋊C411yescharacteristic
76C22×C8C22⋊C411yescharacteristic
77#117—44no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—42no—
81C2×C22⋊C4—22no—
82C2×C22⋊C4—42no—
83C2×C42—22no—
84C2×C42—22no—
85C2×C42—42no—
86C2×C42C4⋊C412yes—
87C2×C42C4⋊C412yes—
88C2×C42C22×C411yescharacteristic
89C2×C42C2×D412yes—
90C2×C42C2×D411yescharacteristic
91C2×C42C2×Q812yes—
92C2×C42C2×Q811yescharacteristic
93C2×C42#4112yes—
94C2×C42#4112yes—
95C2×C42#4111yescharacteristic
96C4⋊C8—22no—
97C4⋊C8—22no—
98C4⋊C8—24no—
99C4⋊C8—24no—
100C4⋊C8—41no—
101C4⋊C8—41no—
102C4⋊C8—42no—
103C4⋊C8—82no—
104C4⋊C8—82no—
105C22⋊C8—22no—
106C22⋊C8—22no—
107C22⋊C8—24no—
108C22⋊C8—42no—
109#97—42no—
110C4×C8—21no—
111C4×C8—21no—
112C4×C8—21no—
113C4×C8—21no—
114C4×C8—24no—
115#95—22no—
116#95—21no—
117#95—21no—
118#95—24no—
119#95—24no—
120#95—24no—
121#95—42no—
122C24#12111yescharacteristic
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—42no—
126C22×C4—41no—
127C22×C4—41no—
128C22×C4—41no—
129C22×C4—41no—
130C22×C4—44no—
131C22×C4C2×C22⋊C411yescharacteristic
132C22×C4C2×C4⋊C411yescharacteristic
133C22×C4#11711yescharacteristic
134C22×C4#12112yes—
135C22×C4#12111yescharacteristic
136C22×C4#12112yes—
137C22×C4C22⋊Q811yescharacteristic
138C22×C4C22⋊Q812yes—
139C22×C4C22⋊Q812yes—
140C22×C4C22⋊Q812yes—
141C22×C4C22⋊Q811yescharacteristic
142C22×C4#12612yes—
143M16—82no—
144M16—82no—
145M16—84no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—41no—
151C2×C8—41no—
152C2×C8—41no—
153C2×C8—41no—
154C2×C8—42no—
155C2×C8—42no—
156C2×C8—42no—
157C2×C8—44no—
158C4⋊C4—24no—
159C4⋊C4—42no—
160C4⋊C4—82no—
161C4⋊C4—82no—
162C22⋊C4—44no—
163C22⋊C4—82no—
164C22⋊C4—82no—
165C42—42no—
166C42—42no—
167C42—42no—
168C42—42no—
169C42—44no—
170C42—84no—
171C42C4×D412yes—
172C42C4×Q812yes—
173C42#12011yescharacteristic
174C42#12311yescharacteristic
175C42#12312yes—
176C42#12411yescharacteristic
177C42#12512yes—
178C42#12512yes—
179C42C4⋊Q812yes—
180C42C4⋊Q811yescharacteristic
181C23—21no—
182C23—21no—
183C23—41no—
184C23—41no—
185C23#38011yescharacteristic
186C23#39612yes—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—24no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—24no—
198C2×C4—41no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—41no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—84no—
210C2×C4—84no—
211C2×C4—84no—
212C2×C4#38611yescharacteristic
213C2×C4#38712yes—
214C2×C4#38912yes—
215C2×C4#39111yescharacteristic
216C2×C4#39411yescharacteristic
217C2×C4#39511yescharacteristic
218C2×C4#39812yes—
219C2×C4#39811yescharacteristic
220C2×C4#40011yescharacteristic
221C8—41no—
222C8—41no—
223C8—41no—
224C8—41no—
225C8—84no—
226C8—82no—
227C22—21no—
228C22—21no—
229C22—22no—
230C22—22no—
231C22—22no—
232C22—81no—
233C22—81no—
234C22#144111yescharacteristic
235C22#181211yescharacteristic
236C22#181211yescharacteristic
237C4—21no—
238C4—22no—
239C4—21no—
240C4—22no—
241C4—21no—
242C4—22no—
243C4—21no—
244C4—44no—
245C4—82no—
246C4—164no—
247C4#190911yescharacteristic
248C4#191011yescharacteristic
249C4#210212yes—
250C2—21no—
251C2—21no—
252C2—42no—
253C2#1122511yescharacteristic
254C2#1122911yescharacteristic
255C2#1124411yescharacteristic
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
16. #1831C4semidirect

Maximal subgroups

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

subgroupindex
#219742
#178152
#79802
#79622
#79412