Finite Group Databasebeta

← #134261

Subgroups — #134261

Order 512.

394 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 114 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#20438C211yesmaximal, characteristic
2#11447C212yesmaximal
3#8601C212yesmaximal
4#7188C212yesmaximal
5C2×#423C411yescharacteristic
6C2×#405C411yescharacteristic
7C2×#405C2211yescharacteristic
8C22×#97C411yescharacteristic
9#2285C2211yescharacteristic
10#1771—22no—
11#1771—22no—
12#1729C411yescharacteristic
13#1729C2211yescharacteristic
14#1564—22no—
15#1458C2214yes—
16#1325—22no—
17#1324—22no—
18#1275—22no—
19C22×M16C2×C411yescharacteristic
20C22×M16C2×C411yescharacteristic
21C23×C8C2×C411yescharacteristic
22C23×C8C2×C411yescharacteristic
23C2×#117C2×C411yescharacteristic
24C2×#117C2311yescharacteristic
25C22×C42C2×C411yescharacteristic
26#431—24no—
27#423—22no—
28#423—22no—
29C2×C4⋊C8—22no—
30#405—42no—
31#405—42no—
32C2×#97—21no—
33C2×#97—21no—
34C2×#97—22no—
35C2×#97—21no—
36C2×#97C814yescomplemented
37C2×#97D412yes—
38C2×C4×C8—21no—
39C2×C4×C8D412yes—
40#384D411yescharacteristic
41#384Q811yescharacteristic
42#382D411yescharacteristic
43#382Q811yescharacteristic
44#380—22no—
45C4×C4⋊C4—22no—
46#336—22no—
47#336—22no—
48#335—42no—
49#334—42no—
50#324—24no—
51#323—24no—
52C23×C4—21no—
53C23×C4C22×C411yescharacteristic
54C2×M16—21no—
55C2×M16—21no—
56C2×M16—21no—
57C2×M16—41no—
58C2×M16—41no—
59C2×M16—41no—
60C2×M16—41no—
61C2×M16M1612yes—
62C22×C8—22no—
63C22×C8—22no—
64C22×C8—21no—
65C22×C8—21no—
66C22×C8—21no—
67C22×C8—41no—
68C22×C8—41no—
69C22×C8C22⋊C412yes—
70C22×C8C22⋊C411yescharacteristic
71C22×C8C22⋊C411yescharacteristic
72C22×C8C2×C812yes—
73C22×C8C2×C812yes—
74C22×C8M1612yes—
75#117—41no—
76#117—41no—
77#117—41no—
78#117—41no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4C4⋊C411yescharacteristic
82C2×C4⋊C4C4⋊C411yescharacteristic
83C2×C4⋊C4C2×D411yescharacteristic
84C2×C4⋊C4C2×Q811yescharacteristic
85C2×C22⋊C4—21no—
86C2×C22⋊C4#4112yes—
87C2×C42—21no—
88C2×C42—21no—
89C2×C42—21no—
90C2×C42—21no—
91C2×C42C4⋊C411yescharacteristic
92C2×C42C4⋊C411yescharacteristic
93C2×C42C2×C812yes—
94C2×C42M1612yes—
95C2×C42C2×D412yes—
96C4⋊C8—28no—
97C4⋊C8—44no—
98C22⋊C8—44no—
99#97—24no—
100#97—24no—
101#97—42no—
102#97—42no—
103C4×C8—22no—
104C4×C8—22no—
105C4×C8—44no—
106C4×C8—42no—
107#95—22no—
108#95—22no—
109C24#11711yescharacteristic
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—81no—
121C22×C4—81no—
122C22×C4C2×C22⋊C411yescharacteristic
123C22×C4C2×C4⋊C411yescharacteristic
124C22×C4#12112yes—
125C22×C4C22⋊Q812yes—
126C22×C4C22×C811yescharacteristic
127C22×C4C2×M1611yescharacteristic
128C22×C4C2×M1611yescharacteristic
129C22×C4C2×M1611yescharacteristic
130M16—42no—
131M16—42no—
132M16—42no—
133M16—42no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—24no—
137C2×C8—24no—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—22no—
142C2×C8—22no—
143C2×C8—42no—
144C2×C8—41no—
145C2×C8—41no—
146C2×C8—41no—
147C2×C8—41no—
148C2×C8—42no—
149C2×C8—84no—
150C2×C8C22⋊C812yes—
151C2×C8C22⋊C812yes—
152C2×C8#9912yes—
153C2×C8#10112yes—
154C4⋊C4—41no—
155C4⋊C4—41no—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4—82no—
159C4⋊C4—82no—
160C22⋊C4—42no—
161C22⋊C4—42no—
162C42—22no—
163C42—22no—
164C42—22no—
165C42—22no—
166C42—21no—
167C42—21no—
168C42—41no—
169C42—41no—
170C42C4⋊C812yes—
171C42C4⋊C812yes—
172C23—21no—
173C23—21no—
174C23—41no—
175C23—41no—
176C23#38011yescharacteristic
177C23#43111yescharacteristic
178C23#43211yescharacteristic
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—82no—
204C2×C4—82no—
205C2×C4C2×C22⋊C811yescharacteristic
206C2×C4#40711yescharacteristic
207C2×C4C2×#9911yescharacteristic
208C2×C4#41011yescharacteristic
209C2×C4C2×#10111yescharacteristic
210C2×C4#41311yescharacteristic
211C2×C4C2×C4⋊C811yescharacteristic
212C2×C4#42311yescharacteristic
213C8—22no—
214C8—42no—
215C8—42no—
216C8—42no—
217C8—164no—
218C8#32414yes—
219C22—21no—
220C22—21no—
221C22—21no—
222C22—21no—
223C22—41no—
224C22—41no—
225C22—42no—
226C22—41no—
227C22—41no—
228C22#179711yescharacteristic
229C22#179911yescharacteristic
230C22#180111yescharacteristic
231C4—21no—
232C4—21no—
233C4—42no—
234C4—42no—
235C4—42no—
236C4—42no—
237C4—81no—
238C4—81no—
239C4C2×#32411yescharacteristic
240C4#145811yescharacteristic
241C4#146511yescharacteristic
242C4#146711yescharacteristic
243C2—21no—
244C2—21no—
245C2—41no—
246C2—41no—
247C2#855811yescharacteristic
248C2#1280811yescharacteristic
249C2#1281011yescharacteristic
subgroupG/H
centerC2×C4#410
commutatorC22×C4C22×C8
Frattini, cosocleC2×#117C23
socleC22#1799
prime radicalC24#117
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#405 ◁ G
LECSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
36. C2×#97C8semidirect

Maximal subgroups

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

subgroupindex
#204382
#114472
#86012
#71882