Finite Group Databasebeta

← #121372

Subgroups — #121372

Order 512.

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

normal subgroups only

This group has 760 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#21948C211yesmaximal, characteristic
2#20851C212yesmaximal
3#10715C211yesmaximal, characteristic
4#8008C211yesmaximal, characteristic
5#7929C212yesmaximal
6C23×M16C2211yescharacteristic
7C2×#430—22no—
8C2×#407C2212yes—
9C2×#407C2212yes—
10C22×C22⋊C8C2211yescharacteristic
11C2×C4×C22⋊C4C2211yescharacteristic
12#1855—24no—
13#1855—24no—
14#1855—22no—
15#1855—22no—
16#1810—22no—
17#1738—22no—
18C2×#336—22no—
19#1293C414yescomplemented
20#1282—24no—
21C24×C4C2311yescharacteristic
22C22×M16—22no—
23C22×M16—21no—
24C22×M16—22no—
25C22×M16—22no—
26C22×M16—44no—
27C22×M16D412yes—
28C22×M16D412yes—
29C22×M16D412yes—
30C23×C8—21no—
31C23×C8D412yes—
32C22×C22⋊C4Q812yes—
33C22×C42—21no—
34#430—48no—
35C2×#108—28no—
36C2×#108—22no—
37C2×#108—22no—
38#407—22no—
39#407—22no—
40#407—22no—
41#407—22no—
42#407—22no—
43#407—22no—
44#407C2×C414yes—
45#407Q814yes—
46C2×C22⋊C8—22no—
47C2×C22⋊C8—22no—
48C2×C22⋊C8—22no—
49C2×C22⋊C8—22no—
50C2×C22⋊C8—22no—
51C2×C22⋊C8C2×C412yes—
52C2×C22⋊C8D414yes—
53C2×C22⋊C8Q812yes—
54C4×C22⋊C4—22no—
55C4×C22⋊C4—22no—
56C2×#95—21no—
57#341—24no—
58#341—28no—
59#336—44no—
60C25C2×Q811yescharacteristic
61C23×C4—22no—
62C23×C4—42no—
63C23×C4—42no—
64C23×C4C22×C411yescharacteristic
65C23×C4C2×D411yescharacteristic
66C23×C4C2×D412yes—
67C23×C4C2×D412yes—
68C23×C4C2×D411yescharacteristic
69C23×C4C2×Q811yescharacteristic
70C23×C4C2×Q812yes—
71C23×C4#4111yescharacteristic
72C23×C4#4112yes—
73C23×C4#4111yescharacteristic
74C2×M16—24no—
75C2×M16—24no—
76C2×M16—28no—
77C2×M16—24no—
78C2×M16—22no—
79C2×M16—22no—
80C2×M16—22no—
81C2×M16—22no—
82C2×M16—44no—
83C2×M16—44no—
84C2×M16—48no—
85C2×M16—44no—
86C2×M16—44no—
87C2×M16—44no—
88C2×M16—42no—
89C2×M16—44no—
90C2×M16—44no—
91C22×C8—24no—
92C22×C8—22no—
93C22×C8—24no—
94C22×C8—22no—
95C22×C8—22no—
96C22×C8—22no—
97C22×C8—22no—
98C22×C8—42no—
99C22×C8—41no—
100C22×C8—41no—
101C22×C8—42no—
102C22×C8—42no—
103C22×C8C4⋊C414yes—
104C2×C22⋊C4—24no—
105C2×C22⋊C4—44no—
106C2×C42—42no—
107C2×C42—42no—
108C2×C42—41no—
109C2×C42—41no—
110#108—416no—
111#108—82no—
112#108—82no—
113C22⋊C8—24no—
114C22⋊C8—24no—
115C22⋊C8—24no—
116C22⋊C8—24no—
117C22⋊C8—24no—
118C22⋊C8—42no—
119C22⋊C8—42no—
120C22⋊C8—42no—
121C22⋊C8—42no—
122C22⋊C8C4⋊C418yes—
123#95—42no—
124C24—42no—
125C24—42no—
126C24—42no—
127C24C4×Q811yescharacteristic
128C24C22⋊Q811yescharacteristic
129C24C22⋊Q812yes—
130C24C4⋊Q812yes—
131C24C4⋊Q811yescharacteristic
132C22×C4—22no—
133C22×C4—24no—
134C22×C4—22no—
135C22×C4—24no—
136C22×C4—24no—
137C22×C4—42no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—41no—
143C22×C4—42no—
144C22×C4—44no—
145C22×C4—41no—
146C22×C4—44no—
147C22×C4—42no—
148C22×C4—44no—
149C22×C4—42no—
150C22×C4—44no—
151C22×C4—42no—
152C22×C4C2×C4⋊C411yescharacteristic
153C22×C4C2×C4⋊C412yes—
154C22×C4C4×D412yes—
155C22×C4C4×D411yescharacteristic
156C22×C4#12012yes—
157C22×C4#12011yescharacteristic
158C22×C4#12111yescharacteristic
159C22×C4#12112yes—
160C22×C4C22⋊Q812yes—
161C22×C4C22⋊Q812yes—
162C22×C4C22⋊Q812yes—
163C22×C4#12311yescharacteristic
164C22×C4#12312yes—
165C22×C4#12512yes—
166C22×C4#12511yescharacteristic
167C22×C4#12711yescharacteristic
168C22×C4C4⋊Q811yescharacteristic
169C22×C4C4⋊Q812yes—
170M16—28no—
171M16—48no—
172M16—48no—
173M16—44no—
174M16—48no—
175M16—44no—
176M16—44no—
177M16—44no—
178M16—44no—
179C2×C8—24no—
180C2×C8—24no—
181C2×C8—28no—
182C2×C8—44no—
183C2×C8—44no—
184C2×C8—42no—
185C2×C8—44no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8—44no—
189C2×C8—42no—
190C2×C8—44no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—82no—
194C2×C8—82no—
195C2×C8—82no—
196C22⋊C4—28no—
197C22⋊C4—44no—
198C42—82no—
199C42—81no—
200C42—81no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—24no—
205C23—24no—
206C23—22no—
207C23—22no—
208C23—41no—
209C23—42no—
210C23—41no—
211C23—41no—
212C23—42no—
213C23—42no—
214C23—41no—
215C23—42no—
216C23—84no—
217C23—84no—
218C23#38912yes—
219C23#38911yescharacteristic
220C23#39512yes—
221C23#39511yescharacteristic
222C23#39911yescharacteristic
223C2×C4—24no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—24no—
228C2×C4—24no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—42no—
238C2×C4—41no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—41no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—48no—
245C2×C4—44no—
246C2×C4—82no—
247C2×C4—84no—
248C2×C4—82no—
249C2×C4#38411yescharacteristic
250C2×C4#38512yes—
251C2×C4#38511yescharacteristic
252C2×C4#39211yescharacteristic
253C2×C4#39811yescharacteristic
254C2×C4#39812yes—
255C2×C4B(4,𝔽2)12yes—
256C2×C4#45812yes—
257C8—48no—
258C8—44no—
259C8—44no—
260C8—82no—
261C8—82no—
262C22—24no—
263C22—24no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—22no—
271C22—22no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—84no—
277C22—84no—
278C22—82no—
279C22—82no—
280C22#144411yescharacteristic
281C22#195312yes—
282C22#206212yes—
283C22#209112yes—
284C4—22no—
285C4—21no—
286C4—22no—
287C4—21no—
288C4—84no—
289C4#186712yes—
290C4#202111yescharacteristic
291C4#202211yescharacteristic
292C2—22no—
293C2—22no—
294C2—22no—
295C2—42no—
296C2—42no—
297C2#1133211yescharacteristic
298C2#1133811yescharacteristic
299C2#1134411yescharacteristic
subgroupG/H
centerC2×C4#392
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
socleC22#1444
prime radicalC25C2×Q8
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
19. #1293C4semidirect

Maximal subgroups

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

subgroupindex
#219482
#208512
#107152
#80082
#79292