Finite Group Databasebeta

← #121996

Subgroups — #121996

Order 512.

756 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 146 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#34552C211yesmaximal, characteristic
2C4×H(3,ℤ/4)C211yesmaximal, characteristic
3#8068C212yesmaximal
4#8048C212yesmaximal
5#7868C211yesmaximal, characteristic
6#2920C2212yes—
7C2×C4×M16C2211yescharacteristic
8#2402—22no—
9#2402C412yescomplemented
10#2399—22no—
11#2399C412yescomplemented
12#2396—24no—
13#2396—22no—
14C42×Q8—22no—
15C4×#117C2212yes—
16C4×#117C2211yescharacteristic
17C4×#117C2211yescharacteristic
18C2×H(3,ℤ/4)—22no—
19#1300—22no—
20#1300—22no—
21#1297—24no—
22#1276—22no—
23C22×M16—21no—
24C2×C4×Q8—24no—
25C2×C4×Q8—22no—
26C2×#117—22no—
27C2×#117—21no—
28C2×#117D412yes—
29C22×C42C2311yescharacteristic
30#432—22no—
31#432—22no—
32#431—22no—
33#431—22no—
34#423Q814yescomplemented
35#407—22no—
36C4×M16—22no—
37C4×M16—22no—
38C4×M16—22no—
39C2×#97Q812yescomplemented
40C2×C4×C8—21no—
41#400—22no—
42#400—24no—
43#399—22no—
44#398—24no—
45#398—24no—
46#398—24no—
47#396—24no—
48#395—22no—
49#395—24no—
50#393—22no—
51C4×C4⋊C4—24no—
52C4×C4⋊C4—24no—
53C4×C4⋊C4—24no—
54C4×C4⋊C4—22no—
55C4×C4⋊C4—24no—
56C4×C4⋊C4—22no—
57C4×C4⋊C4—22no—
58C4×C4⋊C4—22no—
59C4×C4⋊C4C2×C414yes—
60C4×C22⋊C4—22no—
61C4×C22⋊C4—22no—
62C4×C22⋊C4—22no—
63C4×C22⋊C4D414yes—
64#376C2×C411yescharacteristic
65#376C2×C411yescharacteristic
66#376D412yes—
67#376D412yes—
68#376Q812yescomplemented
69C43—22no—
70C43—21no—
71C43D412yes—
72H(3,ℤ/4)—28no—
73C22×Q8—22no—
74C23×C4C2×D411yescharacteristic
75C23×C4C2×D412yes—
76C2×M16—24no—
77C2×M16—24no—
78C2×M16—42no—
79C2×M16—41no—
80C2×M16—41no—
81C22×C8—22no—
82C4×Q8—416no—
83C4×Q8—44no—
84C4×Q8—44no—
85#117—24no—
86#117—24no—
87#117—42no—
88#117—48no—
89#117—42no—
90C2×C4⋊C4—24no—
91C2×C4⋊C4—24no—
92C2×C4⋊C4—24no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—24no—
95C2×C4⋊C4—24no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—41no—
98C2×C22⋊C4—24no—
99C2×C22⋊C4—22no—
100C2×C22⋊C4—41no—
101C2×C42—22no—
102C2×C42—22no—
103C2×C42—24no—
104C2×C42—22no—
105C2×C42—24no—
106C2×C42—24no—
107C2×C42—42no—
108C2×C42C22⋊C412yes—
109C2×C42C22⋊C412yes—
110C2×C42C22×C411yescharacteristic
111C2×C42C2×D412yes—
112C2×C42C2×D411yescharacteristic
113C2×C42C2×Q811yescharacteristic
114C2×C42C2×Q812yes—
115C2×C42C2×Q811yescharacteristic
116C2×C42#4112yes—
117C2×C42#4111yescharacteristic
118C2×C42#4111yescharacteristic
119C4⋊C8—24no—
120C4⋊C8—24no—
121C22⋊C8—42no—
122C22⋊C8—42no—
123#97—24no—
124#97—42no—
125C4×C8—42no—
126C4×C8—42no—
127#95—22no—
128#95—24no—
129#95—22no—
130#95—24no—
131#95—22no—
132#95—22no—
133#95—22no—
134#95—22no—
135#95—24no—
136#95C22⋊C414yes—
137#95C22⋊C414yes—
138C24#12011yescharacteristic
139C2×Q8—48no—
140C2×Q8—44no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—24no—
145C22×C4—22no—
146C22×C4—24no—
147C22×C4—44no—
148C22×C4—42no—
149C22×C4C2×C22⋊C412yes—
150C22×C4C2×C22⋊C411yescharacteristic
151C22×C4#12011yescharacteristic
152C22×C4#12012yes—
153C22×C4C22⋊Q811yescharacteristic
154C22×C4C22⋊Q812yes—
155C22×C4C22⋊Q811yescharacteristic
156C22×C4C22⋊Q812yes—
157C22×C4#12412yes—
158C22×C4#12411yescharacteristic
159C22×C4C4⋊Q811yescharacteristic
160C22×C4C4⋊Q812yes—
161M16—44no—
162M16—44no—
163M16—84no—
164C2×C8—24no—
165C2×C8—42no—
166C2×C8—42no—
167C2×C8—44no—
168C4⋊C4—44no—
169C4⋊C4—48no—
170C4⋊C4—48no—
171C4⋊C4—48no—
172C4⋊C4—48no—
173C4⋊C4—48no—
174C4⋊C4—44no—
175C4⋊C4—44no—
176C22⋊C4—44no—
177C22⋊C4—48no—
178C22⋊C4—44no—
179C42—28no—
180C42—22no—
181C42—24no—
182C42—22no—
183C42—22no—
184C42—42no—
185C42—44no—
186C42—48no—
187C42—48no—
188C42—44no—
189C42—44no—
190C42—84no—
191C42#10414yes—
192C42C4×Q811yescharacteristic
193C42C4×Q812yes—
194C42C4×Q811yescharacteristic
195C42C22⋊Q812yes—
196C42C22⋊Q812yes—
197C42C22⋊Q812yes—
198C42#12412yes—
199C42#12411yescharacteristic
200C42C4⋊Q812yes—
201C42C4⋊Q811yescharacteristic
202C23—22no—
203C23—42no—
204C23#37911yescharacteristic
205C23#39311yescharacteristic
206C23#39511yescharacteristic
207Q8—416no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—24no—
216C2×C4—24no—
217C2×C4—24no—
218C2×C4—28no—
219C2×C4—24no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—44no—
224C2×C4—44no—
225C2×C4—44no—
226C2×C4—44no—
227C2×C4—44no—
228C2×C4—44no—
229C2×C4—44no—
230C2×C4—42no—
231C2×C4—44no—
232C2×C4—84no—
233C2×C4#39112yes—
234C2×C4#39112yes—
235C2×C4#39111yescharacteristic
236C2×C4#39111yescharacteristic
237C2×C4#39311yescharacteristic
238C2×C4#39312yes—
239C2×C4#39511yescharacteristic
240C2×C4#39512yes—
241C2×C4C2×#10412yes—
242C2×C4#42112yes—
243C8—44no—
244C8—84no—
245C22—22no—
246C22—21no—
247C22—21no—
248C22—22no—
249C22—22no—
250C22—82no—
251C22#144611yescharacteristic
252C22#178912yes—
253C4—21no—
254C4—21no—
255C4—22no—
256C4—22no—
257C4—22no—
258C4—48no—
259C4—44no—
260C4—84no—
261C4—82no—
262C4—84no—
263C4#198612yes—
264C4#199411yescharacteristic
265C4#199511yescharacteristic
266C4#203114yes—
267C2—22no—
268C2—41no—
269C2—41no—
270C2#1138611yescharacteristic
271C2#1138712yes—
subgroupG/H
centerC42#124
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1446
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C42 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
9. #2402C4semidirect
11. #2399C4semidirect
34. #423Q8semidirect
39. C2×#97Q8semidirect
68. #376Q8semidirect

Maximal subgroups

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

subgroupindex
#345522
C4×H(3,ℤ/4)2
#80682
#80482
#78682