Finite Group Databasebeta

← #121370

Subgroups — #121370

Order 512.

800 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 154 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#21938C211yesmaximal, characteristic
2#20846C212yesmaximal
3#10713C211yesmaximal, characteristic
4C2×#1293C211yesmaximal, characteristic
5#7915C212yesmaximal
6C22×#117C2211yescharacteristic
7C2×#428—22no—
8C2×#407C2212yes—
9C22×C22⋊C8C2211yescharacteristic
10C22×C22⋊C8C2211yescharacteristic
11C2×#380C2212yes—
12#1854—24no—
13#1853—22no—
14#1853—22no—
15#1853—22no—
16#1852—22no—
17#1807—22no—
18#1736—22no—
19C2×#336—22no—
20#1293C414yescomplemented
21#1280—24no—
22C24×C4C2311yescharacteristic
23C22×M16—22no—
24C23×C8—21no—
25C23×C8D412yes—
26C2×#117—22no—
27C2×#117—22no—
28C2×#117—44no—
29C22×C4⋊C4D412yes—
30C22×C4⋊C4D412yes—
31C22×C4⋊C4D412yes—
32C22×C22⋊C4—21no—
33C22×C22⋊C4—22no—
34C22×C42—21no—
35#428—48no—
36C2×#107—24no—
37C2×#106—24no—
38C2×#106—22no—
39C2×#106—22no—
40#407—22no—
41#407—22no—
42#407C2×C414yes—
43#407Q814yes—
44C2×C22⋊C8—22no—
45C2×C22⋊C8—22no—
46C2×C22⋊C8—22no—
47C2×C22⋊C8—22no—
48C2×C22⋊C8—22no—
49C2×C22⋊C8—21no—
50C2×C22⋊C8C2×C412yes—
51C2×C22⋊C8D414yes—
52C2×C22⋊C8Q812yes—
53C2×C22⋊C8Q812yes—
54#380—22no—
55#380—22no—
56#380—22no—
57#380—22no—
58C2×#95—22no—
59#340—24no—
60#340—24no—
61#340—24no—
62#336—44no—
63C25#4111yescharacteristic
64C23×C4—22no—
65C23×C4—22no—
66C23×C4—22no—
67C23×C4—42no—
68C23×C4—42no—
69C23×C4C22×C411yescharacteristic
70C23×C4C2×D411yescharacteristic
71C23×C4C2×D412yes—
72C23×C4C2×D412yes—
73C23×C4C2×D411yescharacteristic
74C23×C4C2×Q811yescharacteristic
75C23×C4C2×Q812yes—
76C23×C4C2×Q811yescharacteristic
77C23×C4#4112yes—
78C23×C4#4111yescharacteristic
79C2×M16—24no—
80C2×M16—24no—
81C2×M16—44no—
82C2×M16—44no—
83C22×C8—22no—
84C22×C8—24no—
85C22×C8—22no—
86C22×C8—22no—
87C22×C8—42no—
88C22×C8—42no—
89C22×C8—42no—
90C22×C8—42no—
91C22×C8—41no—
92C22×C8—41no—
93C22×C8C4⋊C414yes—
94#117—28no—
95#117—44no—
96#117—44no—
97#117—44no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—44no—
105C2×C4⋊C4—44no—
106C2×C4⋊C4—44no—
107C2×C22⋊C4—44no—
108C2×C22⋊C4—42no—
109C2×C22⋊C4—42no—
110C2×C22⋊C4—42no—
111C2×C22⋊C4—84no—
112C2×C42—24no—
113C2×C42—42no—
114C2×C42—41no—
115C2×C42—41no—
116#107—48no—
117#106—48no—
118#106—82no—
119#106—82no—
120C22⋊C8—24no—
121C22⋊C8—24no—
122C22⋊C8—24no—
123C22⋊C8—24no—
124C22⋊C8—24no—
125C22⋊C8—24no—
126C22⋊C8—42no—
127C22⋊C8C4⋊C418yes—
128#95—42no—
129#95—42no—
130C24—24no—
131C24—41no—
132C24—41no—
133C24—42no—
134C24C4×Q811yescharacteristic
135C24#12112yes—
136C24#12111yescharacteristic
137C24#12512yes—
138C24#12511yescharacteristic
139C22×C4—22no—
140C22×C4—24no—
141C22×C4—24no—
142C22×C4—22no—
143C22×C4—24no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—44no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—44no—
150C22×C4—41no—
151C22×C4—42no—
152C22×C4—41no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—44no—
156C22×C4—42no—
157C22×C4—44no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—44no—
161C22×C4—44no—
162C22×C4—82no—
163C22×C4—82no—
164C22×C4—82no—
165C22×C4—82no—
166C22×C4C2×C4⋊C411yescharacteristic
167C22×C4C2×C4⋊C412yes—
168C22×C4C4×D412yes—
169C22×C4C4×D411yescharacteristic
170C22×C4#12012yes—
171C22×C4#12011yescharacteristic
172C22×C4C22⋊Q812yes—
173C22×C4C22⋊Q812yes—
174C22×C4C22⋊Q811yescharacteristic
175C22×C4C22⋊Q812yes—
176C22×C4C22⋊Q812yes—
177C22×C4#12311yescharacteristic
178C22×C4#12312yes—
179C22×C4#12711yescharacteristic
180C22×C4C4⋊Q811yescharacteristic
181C22×C4C4⋊Q812yes—
182C22×C4C4⋊Q812yes—
183C22×C4C4⋊Q811yescharacteristic
184M16—48no—
185M16—48no—
186C2×C8—24no—
187C2×C8—28no—
188C2×C8—44no—
189C2×C8—44no—
190C2×C8—42no—
191C2×C8—42no—
192C2×C8—44no—
193C2×C8—44no—
194C2×C8—42no—
195C2×C8—82no—
196C2×C8—81no—
197C2×C8—81no—
198C2×C8—82no—
199C4⋊C4—24no—
200C4⋊C4—24no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—44no—
204C4⋊C4—44no—
205C4⋊C4—44no—
206C4⋊C4—44no—
207C22⋊C4—48no—
208C22⋊C4—84no—
209C22⋊C4—84no—
210C42—44no—
211C42—44no—
212C23—22no—
213C23—21no—
214C23—21no—
215C23—22no—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—22no—
220C23—22no—
221C23—24no—
222C23—22no—
223C23—22no—
224C23—41no—
225C23—41no—
226C23—42no—
227C23—41no—
228C23—41no—
229C23—42no—
230C23—84no—
231C23—84no—
232C23#38912yes—
233C23#38911yescharacteristic
234C23#39211yescharacteristic
235C23#39812yes—
236C23#39811yescharacteristic
237C23#47114yes—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—24no—
244C2×C4—22no—
245C2×C4—24no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—24no—
249C2×C4—24no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—48no—
258C2×C4—44no—
259C2×C4—84no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—164no—
269C2×C4—164no—
270C2×C4#38411yescharacteristic
271C2×C4#38512yes—
272C2×C4#38511yescharacteristic
273C2×C4#39512yes—
274C2×C4#39511yescharacteristic
275C2×C4#39911yescharacteristic
276C2×C4B(4,𝔽2)12yes—
277C2×C4#45812yes—
278C8—48no—
279C8—44no—
280C8—82no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—24no—
286C22—24no—
287C22—22no—
288C22—22no—
289C22—22no—
290C22—22no—
291C22—22no—
292C22—22no—
293C22—42no—
294C22—42no—
295C22—42no—
296C22—42no—
297C22—84no—
298C22—82no—
299C22—82no—
300C22#144411yescharacteristic
301C22#195112yes—
302C22#201712yes—
303C22#201912yes—
304C22#202111yescharacteristic
305C22#202211yescharacteristic
306C22#209012yes—
307C4—22no—
308C4—21no—
309C4—21no—
310C4—22no—
311C4—84no—
312C4—162no—
313C4—162no—
314C4#186712yes—
315C4#206212yes—
316C2—22no—
317C2—22no—
318C2—42no—
319C2—42no—
320C2#997512yes—
321C2#997812yes—
322C2#1133011yescharacteristic
323C2#1133711yescharacteristic
324C2#1134411yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25#41
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
20. #1293C4semidirect

Maximal subgroups

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

subgroupindex
#219382
#208462
#107132
C2×#12932
#79152