Finite Group Databasebeta

← #121610

Subgroups — #121610

Order 512.

810 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 810 conjugacy classes of subgroups — too many to draw. Only its 142 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#21949C211yesmaximal, characteristic
2#20837C212yesmaximal
3#10729C211yesmaximal, characteristic
4#8051C212yesmaximal
5C2×#1297C211yesmaximal, characteristic
6C23×M16C2211yescharacteristic
7C2×#423—22no—
8C2×#407C2212yes—
9C2×#407C2211yescharacteristic
10#2280C2211yescharacteristic
11C2×C4×C22⋊C4C2212yes—
12#1855—22no—
13#1855—22no—
14#1851—24no—
15#1850—24no—
16#1811—22no—
17C2×#344—22no—
18#1728—22no—
19#1297—24no—
20#1297C411yescharacteristic, complemented
21#1297C412yescomplemented
22#1297C411yescharacteristic, complemented
23C24×C4C2311yescharacteristic
24C22×M16—22no—
25C22×M16—21no—
26C22×M16—22no—
27C22×M16—21no—
28C22×M16—44no—
29C22×M16D412yes—
30C22×M16D412yes—
31C22×M16D412yes—
32C23×C8D412yes—
33C2×#117—21no—
34C22×C22⋊C4—22no—
35C22×C42—22no—
36C2×#108—22no—
37C2×#108—22no—
38#423—48no—
39C2×C4⋊C8—28no—
40#407—22no—
41#407—22no—
42#407—22no—
43#407—22no—
44#407—22no—
45#407C2×C411yescharacteristic
46#407C2×C411yescharacteristic
47#407Q812yes—
48C2×C22⋊C8—22no—
49C2×C22⋊C8—22no—
50C2×C22⋊C8—22no—
51#381Q812yes—
52#380—22no—
53#379—21no—
54C4×C22⋊C4—22no—
55C4×C22⋊C4—22no—
56C4×C22⋊C4—22no—
57C4×C22⋊C4C2×C412yes—
58C4×C22⋊C4C2×C412yes—
59C4×C22⋊C4D414yes—
60C2×#95Q814yes—
61#344—44no—
62#341—24no—
63#336—28no—
64C25#4111yescharacteristic
65C23×C4—24no—
66C23×C4—42no—
67C23×C4—42no—
68C23×C4C22×C411yescharacteristic
69C23×C4C2×D411yescharacteristic
70C23×C4C2×D412yes—
71C23×C4C2×D412yes—
72C23×C4C2×D411yescharacteristic
73C23×C4C2×Q811yescharacteristic
74C23×C4C2×Q812yes—
75C23×C4C2×Q811yescharacteristic
76C23×C4#4112yes—
77C23×C4#4111yescharacteristic
78C2×M16—24no—
79C2×M16—28no—
80C2×M16—22no—
81C2×M16—22no—
82C2×M16—22no—
83C2×M16—22no—
84C2×M16—22no—
85C2×M16—48no—
86C2×M16—44no—
87C2×M16—44no—
88C2×M16—42no—
89C2×M16—41no—
90C2×M16—42no—
91C2×M16—41no—
92C2×M16—44no—
93C2×M16—44no—
94C2×M16—42no—
95C2×M16—42no—
96C2×M16C4⋊C412yes—
97C2×M16C4⋊C412yes—
98C22×C8—24no—
99C22×C8—22no—
100C22×C8—22no—
101C22×C8—22no—
102C22×C8—22no—
103C22×C8—44no—
104#117—42no—
105#117—42no—
106C2×C4⋊C4—41no—
107C2×C22⋊C4—22no—
108C2×C22⋊C4—22no—
109C2×C22⋊C4—24no—
110C2×C22⋊C4—22no—
111C2×C22⋊C4—48no—
112C2×C22⋊C4—42no—
113C2×C42—22no—
114C2×C42—22no—
115C2×C42—24no—
116C2×C42—44no—
117C2×C42—44no—
118C2×C42—41no—
119#108—82no—
120#108—82no—
121C4⋊C8—416no—
122C22⋊C8—22no—
123C22⋊C8—22no—
124C22⋊C8—24no—
125C22⋊C8—42no—
126C22⋊C8—42no—
127C22⋊C8—42no—
128C22⋊C8—42no—
129#95—24no—
130#95—24no—
131#95C4⋊C414yes—
132#95C4⋊C414yes—
133C24—44no—
134C24—42no—
135C24C4×D411yescharacteristic
136C24C22⋊Q812yes—
137C24C22⋊Q811yescharacteristic
138C24#12312yes—
139C24#12511yescharacteristic
140C22×C4—24no—
141C22×C4—22no—
142C22×C4—24no—
143C22×C4—22no—
144C22×C4—24no—
145C22×C4—24no—
146C22×C4—41no—
147C22×C4—42no—
148C22×C4—41no—
149C22×C4—42no—
150C22×C4—41no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—44no—
154C22×C4—41no—
155C22×C4—44no—
156C22×C4—42no—
157C22×C4—48no—
158C22×C4—48no—
159C22×C4—42no—
160C22×C4—84no—
161C22×C4C2×C4⋊C411yescharacteristic
162C22×C4C2×C4⋊C412yes—
163C22×C4C4×D412yes—
164C22×C4C4×Q811yescharacteristic
165C22×C4#12012yes—
166C22×C4#12011yescharacteristic
167C22×C4#12112yes—
168C22×C4#12111yescharacteristic
169C22×C4C22⋊Q812yes—
170C22×C4C22⋊Q812yes—
171C22×C4C22⋊Q812yes—
172C22×C4#12311yescharacteristic
173C22×C4#12512yes—
174C22×C4#12711yescharacteristic
175C22×C4C4⋊Q811yescharacteristic
176C22×C4C4⋊Q812yes—
177C22×C4C4⋊Q812yes—
178C22×C4C4⋊Q811yescharacteristic
179M16—44no—
180M16—48no—
181M16—44no—
182M16—42no—
183M16—42no—
184M16—44no—
185M16—44no—
186M16—82no—
187M16—82no—
188M16—82no—
189C2×C8—28no—
190C2×C8—24no—
191C2×C8—22no—
192C2×C8—22no—
193C2×C8—44no—
194C2×C8—42no—
195C2×C8—42no—
196C2×C8—42no—
197C2×C8—42no—
198C2×C8—44no—
199C2×C8—44no—
200C2×C8—42no—
201C2×C8—42no—
202C4⋊C4—82no—
203C22⋊C4—28no—
204C22⋊C4—48no—
205C22⋊C4—44no—
206C22⋊C4—44no—
207C22⋊C4—84no—
208C42—44no—
209C42—44no—
210C42—44no—
211C42—44no—
212C42—81no—
213C42—81no—
214C23—24no—
215C23—24no—
216C23—24no—
217C23—22no—
218C23—24no—
219C23—41no—
220C23—42no—
221C23—42no—
222C23—42no—
223C23—41no—
224C23—42no—
225C23—42no—
226C23—88no—
227C23#38512yes—
228C23#38911yescharacteristic
229C23#39511yescharacteristic
230C23#39812yes—
231C23#39911yescharacteristic
232C2×C4—22no—
233C2×C4—24no—
234C2×C4—24no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—24no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—28no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—44no—
255C2×C4—44no—
256C2×C4—44no—
257C2×C4—44no—
258C2×C4—44no—
259C2×C4—816no—
260C2×C4—82no—
261C2×C4—84no—
262C2×C4#38411yescharacteristic
263C2×C4#38511yescharacteristic
264C2×C4#38912yes—
265C2×C4#39211yescharacteristic
266C2×C4#39512yes—
267C2×C4#39811yescharacteristic
268C2×C4B(4,𝔽2)12yes—
269C2×C4#45812yes—
270C8—42no—
271C8—42no—
272C8—44no—
273C8—82no—
274C8—82no—
275C22—22no—
276C22—22no—
277C22—22no—
278C22—22no—
279C22—22no—
280C22—22no—
281C22—22no—
282C22—21no—
283C22—21no—
284C22—22no—
285C22—48no—
286C22—44no—
287C22—84no—
288C22—82no—
289C22—82no—
290C22#144411yescharacteristic
291C22#186114yes—
292C22#206212yes—
293C4—21no—
294C4—22no—
295C4—22no—
296C4—21no—
297C4—48no—
298C4—84no—
299C4—162no—
300C4#186712yes—
301C4#202111yescharacteristic
302C4#202211yescharacteristic
303C2—24no—
304C2—22no—
305C2—41no—
306C2—41no—
307C2—42no—
308C2#1134112yes—
309C2#1134411yescharacteristic
subgroupG/H
centerC2×C4#392
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
socleC22#1444
prime radicalC25#41
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
20. #1297C4semidirect
21. #1297C4semidirect
22. #1297C4semidirect

Maximal subgroups

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

subgroupindex
#219492
#208372
#107292
#80512
C2×#12972