Finite Group Databasebeta

← #121368

Subgroups — #121368

Order 512.

868 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 868 conjugacy classes of subgroups — too many to draw. Only its 162 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#21942C211yesmaximal, characteristic
2C2×#1852C212yesmaximal
3#10727C211yesmaximal, characteristic
4C2×#1298C211yesmaximal, characteristic
5#7915C212yesmaximal
6C23×C4⋊C4C2211yescharacteristic
7C22×#107—22no—
8C2×#407C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C22×C22⋊C8C2212yes—
11C2×#380C2212yes—
12#1853—22no—
13#1852—24no—
14#1852—24no—
15#1852—22no—
16#1809—22no—
17C2×#343—22no—
18C2×#340—22no—
19#1298C414yescomplemented
20#1280—24no—
21C24×C4—21no—
22C24×C4C2311yescharacteristic
23C22×M16—21no—
24C22×M16D412yes—
25C23×C8—22no—
26C22×C4⋊C4—22no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4—22no—
29C22×C4⋊C4—44no—
30C22×C4⋊C4D412yes—
31C22×C4⋊C4D412yes—
32C22×C4⋊C4D412yes—
33C22×C22⋊C4—22no—
34C2×#107—28no—
35C2×#107—22no—
36C2×#107—48no—
37C2×#106—22no—
38#407—22no—
39#407—22no—
40#407—21no—
41#407C2×C412yes—
42#407Q812yes—
43#407Q812yes—
44C2×C22⋊C8—22no—
45C2×C22⋊C8—22no—
46C2×C22⋊C8—22no—
47C2×C22⋊C8—22no—
48C2×C22⋊C8—22no—
49C2×C22⋊C8C2×C414yes—
50C2×C22⋊C8D414yes—
51C2×C22⋊C8Q814yes—
52#380—22no—
53#380—22no—
54#380—22no—
55#380—22no—
56C2×#95—22no—
57#343—44no—
58#340—24no—
59#340—28no—
60C25#4111yescharacteristic
61C23×C4—22no—
62C23×C4—22no—
63C23×C4—22no—
64C23×C4—42no—
65C23×C4—42no—
66C23×C4—44no—
67C23×C4—42no—
68C23×C4—84no—
69C23×C4C22×C411yescharacteristic
70C23×C4C2×D411yescharacteristic
71C23×C4C2×D412yes—
72C23×C4C2×D412yes—
73C23×C4C2×D411yescharacteristic
74C23×C4C2×Q812yes—
75C23×C4C2×Q811yescharacteristic
76C23×C4C2×Q811yescharacteristic
77C23×C4#4111yescharacteristic
78C23×C4#4112yes—
79C2×M16—22no—
80C2×M16—22no—
81C2×M16—41no—
82C2×M16—42no—
83C2×M16—42no—
84C2×M16—42no—
85C2×M16—42no—
86C2×M16C4⋊C414yes—
87C22×C8—24no—
88C22×C8—24no—
89C22×C8—22no—
90C22×C8—24no—
91C22×C8—44no—
92C22×C8—44no—
93C22×C8—41no—
94C2×C4⋊C4—28no—
95C2×C4⋊C4—24no—
96C2×C4⋊C4—24no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—44no—
102C2×C4⋊C4—48no—
103C2×C4⋊C4—44no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4—44no—
108C2×C4⋊C4—44no—
109C2×C22⋊C4—42no—
110C2×C22⋊C4—42no—
111C2×C22⋊C4—84no—
112#107—416no—
113#107—82no—
114#106—82no—
115C22⋊C8—24no—
116C22⋊C8—24no—
117C22⋊C8—24no—
118C22⋊C8—24no—
119C22⋊C8—24no—
120C22⋊C8—24no—
121C22⋊C8—42no—
122C22⋊C8C4⋊C418yes—
123#95—42no—
124#95—42no—
125C24—24no—
126C24—24no—
127C24—41no—
128C24—41no—
129C24C4×Q811yescharacteristic
130C24#12111yescharacteristic
131C24#12112yes—
132C24#12511yescharacteristic
133C24#12512yes—
134C22×C4—24no—
135C22×C4—22no—
136C22×C4—24no—
137C22×C4—22no—
138C22×C4—24no—
139C22×C4—42no—
140C22×C4—44no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—44no—
145C22×C4—41no—
146C22×C4—42no—
147C22×C4—41no—
148C22×C4—42no—
149C22×C4—48no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—44no—
155C22×C4—44no—
156C22×C4—84no—
157C22×C4—816no—
158C22×C4—84no—
159C22×C4—84no—
160C22×C4—82no—
161C22×C4—82no—
162C22×C4—82no—
163C22×C4—82no—
164C22×C4C2×C4⋊C411yescharacteristic
165C22×C4C2×C4⋊C412yes—
166C22×C4C4×D412yes—
167C22×C4C4×D411yescharacteristic
168C22×C4#12012yes—
169C22×C4#12011yescharacteristic
170C22×C4C22⋊Q812yes—
171C22×C4C22⋊Q812yes—
172C22×C4C22⋊Q811yescharacteristic
173C22×C4C22⋊Q812yes—
174C22×C4C22⋊Q812yes—
175C22×C4#12311yescharacteristic
176C22×C4#12312yes—
177C22×C4#12711yescharacteristic
178C22×C4C4⋊Q811yescharacteristic
179C22×C4C4⋊Q812yes—
180C22×C4C4⋊Q812yes—
181C22×C4C4⋊Q811yescharacteristic
182M16—44no—
183M16—44no—
184M16—82no—
185M16—82no—
186C2×C8—24no—
187C2×C8—28no—
188C2×C8—44no—
189C2×C8—44no—
190C2×C8—48no—
191C2×C8—48no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—42no—
195C2×C8—81no—
196C2×C8—81no—
197C4⋊C4—28no—
198C4⋊C4—44no—
199C4⋊C4—44no—
200C4⋊C4—44no—
201C4⋊C4—44no—
202C4⋊C4—44no—
203C4⋊C4—44no—
204C4⋊C4—44no—
205C22⋊C4—84no—
206C22⋊C4—84no—
207C23—22no—
208C23—22no—
209C23—22no—
210C23—22no—
211C23—24no—
212C23—24no—
213C23—24no—
214C23—24no—
215C23—22no—
216C23—22no—
217C23—21no—
218C23—21no—
219C23—41no—
220C23—42no—
221C23—41no—
222C23—44no—
223C23—44no—
224C23—48no—
225C23#38911yescharacteristic
226C23#38912yes—
227C23#39211yescharacteristic
228C23#39811yescharacteristic
229C23#39812yes—
230C23#47214yes—
231C23#47314yes—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—24no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—24no—
240C2×C4—24no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—24no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—44no—
252C2×C4—816no—
253C2×C4—84no—
254C2×C4—84no—
255C2×C4—84no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—164no—
263C2×C4—164no—
264C2×C4#38411yescharacteristic
265C2×C4#38512yes—
266C2×C4#38511yescharacteristic
267C2×C4#39512yes—
268C2×C4#39511yescharacteristic
269C2×C4#39911yescharacteristic
270C2×C4B(4,𝔽2)12yes—
271C2×C4#45812yes—
272C8—48no—
273C8—44no—
274C8—82no—
275C22—24no—
276C22—24no—
277C22—24no—
278C22—24no—
279C22—22no—
280C22—22no—
281C22—22no—
282C22—22no—
283C22—22no—
284C22—22no—
285C22—21no—
286C22—21no—
287C22—44no—
288C22—48no—
289C22—44no—
290C22—41no—
291C22—41no—
292C22—42no—
293C22#144411yescharacteristic
294C22#195214yes—
295C22#201812yes—
296C22#202012yes—
297C22#202111yescharacteristic
298C22#202211yescharacteristic
299C22#208914yes—
300C4—22no—
301C4—21no—
302C4—21no—
303C4—22no—
304C4—84no—
305C4—162no—
306C4—162no—
307C4#186712yes—
308C4#206212yes—
309C2—24no—
310C2—24no—
311C2—41no—
312C2—41no—
313C2#998611yescharacteristic
314C2#999111yescharacteristic
315C2#999412yes—
316C2#1133112yes—
317C2#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
19. #1298C4semidirect

Maximal subgroups

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

subgroupindex
#219422
C2×#18522
#107272
C2×#12982
#79152