Finite Group Databasebeta

← #121609

Subgroups — #121609

Order 512.

1136 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 1136 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
2#13880C212yesmaximal
3#10727C211yesmaximal, characteristic
4C2×#1297C211yesmaximal, characteristic
5#8019C212yesmaximal
6C23×C4⋊C4C2211yescharacteristic
7C2×#407C2211yescharacteristic
8C2×#407C2211yescharacteristic
9C2×#385—24no—
10C2×#384—22no—
11C2×#380C2212yes—
12C2×C4×C22⋊C4C2212yes—
13#1853—22no—
14#1852—22no—
15#1809—22no—
16C2×#343—22no—
17#1444—24no—
18C4⋊#95—22no—
19#1297C414yescomplemented
20#1294—24no—
21C24×C4D412yes—
22C24×C4C2311yescharacteristic
23C22×M16—21no—
24C22×M16D412yes—
25C22×C4⋊C4—22no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4—22no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4—28no—
30C22×C4⋊C4—44no—
31C22×C4⋊C4D412yes—
32C22×C4⋊C4D412yes—
33C22×C22⋊C4—22no—
34C22×C22⋊C4—22no—
35C22×C42—22no—
36C2×#107—22no—
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—
46#385—416no—
47#384—48no—
48#380—22no—
49#380—22no—
50#380—22no—
51#380—22no—
52C4×C22⋊C4—22no—
53C4×C22⋊C4—22no—
54C4×C22⋊C4C2×C414yes—
55C4×C22⋊C4Q814yes—
56C2×#95—28no—
57C2×#95—22no—
58C2×#95D414yes—
59#343—44no—
60#340—24no—
61C25C2×D411yescharacteristic
62C23×C4—24no—
63C23×C4—24no—
64C23×C4—22no—
65C23×C4—22no—
66C23×C4—22no—
67C23×C4—42no—
68C23×C4—42no—
69C23×C4—44no—
70C23×C4—48no—
71C23×C4C22×C411yescharacteristic
72C23×C4C2×D411yescharacteristic
73C23×C4C2×D412yes—
74C23×C4C2×D412yes—
75C23×C4C2×Q811yescharacteristic
76C23×C4C2×Q812yes—
77C23×C4C2×Q811yescharacteristic
78C23×C4#4112yes—
79C23×C4#4111yescharacteristic
80C23×C4#4111yescharacteristic
81C2×M16—22no—
82C2×M16—22no—
83C2×M16—41no—
84C2×M16—42no—
85C2×M16—42no—
86C2×M16—42no—
87C2×M16—42no—
88C2×M16C4⋊C414yescomplemented
89C22×C8—22no—
90C22×C8—41no—
91C2×C4⋊C4—24no—
92C2×C4⋊C4—28no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—48no—
98C2×C4⋊C4—44no—
99C2×C4⋊C4—44no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—44no—
105C2×C4⋊C4—44no—
106C2×C4⋊C4—416no—
107C2×C4⋊C4—416no—
108C2×C4⋊C4—88no—
109C2×C22⋊C4—28no—
110C2×C22⋊C4—28no—
111C2×C22⋊C4—24no—
112C2×C22⋊C4—24no—
113C2×C22⋊C4—48no—
114C2×C22⋊C4—42no—
115C2×C22⋊C4—42no—
116C2×C42—24no—
117C2×C42—24no—
118C2×C42—44no—
119C2×C42—44no—
120#107—82no—
121#106—82no—
122C22⋊C8—24no—
123C22⋊C8—24no—
124C22⋊C8—24no—
125C22⋊C8—42no—
126#95—24no—
127#95—416no—
128#95—42no—
129#95—42no—
130#95C4⋊C418yescomplemented
131C24—24no—
132C24—24no—
133C24—41no—
134C24—41no—
135C24C4×D411yescharacteristic
136C24#12012yes—
137C24#12111yescharacteristic
138C24C22⋊Q812yes—
139C24C4⋊Q811yescharacteristic
140C22×C4—24no—
141C22×C4—22no—
142C22×C4—24no—
143C22×C4—22no—
144C22×C4—28no—
145C22×C4—28no—
146C22×C4—28no—
147C22×C4—28no—
148C22×C4—24no—
149C22×C4—42no—
150C22×C4—44no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—44no—
155C22×C4—42no—
156C22×C4—41no—
157C22×C4—42no—
158C22×C4—41no—
159C22×C4—44no—
160C22×C4—44no—
161C22×C4—44no—
162C22×C4—44no—
163C22×C4—48no—
164C22×C4—44no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—48no—
168C22×C4—44no—
169C22×C4—44no—
170C22×C4—44no—
171C22×C4—44no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—44no—
175C22×C4—44no—
176C22×C4—816no—
177C22×C4—84no—
178C22×C4C2×C4⋊C411yescharacteristic
179C22×C4C2×C4⋊C412yes—
180C22×C4C4×D412yes—
181C22×C4C4×Q811yescharacteristic
182C22×C4#12011yescharacteristic
183C22×C4#12112yes—
184C22×C4C22⋊Q812yes—
185C22×C4C22⋊Q812yes—
186C22×C4C22⋊Q811yescharacteristic
187C22×C4C22⋊Q812yes—
188C22×C4#12311yescharacteristic
189C22×C4#12312yes—
190C22×C4#12512yes—
191C22×C4#12511yescharacteristic
192C22×C4#12711yescharacteristic
193C22×C4C4⋊Q811yescharacteristic
194C22×C4C4⋊Q812yes—
195C22×C4C4⋊Q812yes—
196M16—44no—
197M16—44no—
198M16—82no—
199M16—82no—
200C2×C8—24no—
201C2×C8—42no—
202C2×C8—42no—
203C2×C8—42no—
204C2×C8—81no—
205C2×C8—81no—
206C4⋊C4—48no—
207C4⋊C4—44no—
208C4⋊C4—44no—
209C4⋊C4—44no—
210C4⋊C4—44no—
211C4⋊C4—82no—
212C4⋊C4—82no—
213C4⋊C4—832no—
214C22⋊C4—216no—
215C22⋊C4—216no—
216C22⋊C4—84no—
217C22⋊C4—84no—
218C42—48no—
219C42—48no—
220C23—22no—
221C23—22no—
222C23—22no—
223C23—22no—
224C23—24no—
225C23—24no—
226C23—24no—
227C23—24no—
228C23—22no—
229C23—22no—
230C23—21no—
231C23—21no—
232C23—41no—
233C23—42no—
234C23—41no—
235C23—44no—
236C23—48no—
237C23—44no—
238C23#38512yes—
239C23#38911yescharacteristic
240C23#39211yescharacteristic
241C23#39512yes—
242C23#39811yescharacteristic
243C23#45314yes—
244C23#45614yes—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—24no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—24no—
253C2×C4—24no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—24no—
257C2×C4—216no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—42no—
265C2×C4—48no—
266C2×C4—48no—
267C2×C4—48no—
268C2×C4—48no—
269C2×C4—48no—
270C2×C4—48no—
271C2×C4—416no—
272C2×C4—44no—
273C2×C4—88no—
274C2×C4—84no—
275C2×C4—84no—
276C2×C4—84no—
277C2×C4—88no—
278C2×C4—88no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4#38411yescharacteristic
286C2×C4#38511yescharacteristic
287C2×C4#38912yes—
288C2×C4#39511yescharacteristic
289C2×C4#39812yes—
290C2×C4#39911yescharacteristic
291C2×C4B(4,𝔽2)12yes—
292C2×C4#45812yes—
293C8—44no—
294C8—82no—
295C22—22no—
296C22—22no—
297C22—22no—
298C22—22no—
299C22—24no—
300C22—24no—
301C22—24no—
302C22—24no—
303C22—22no—
304C22—22no—
305C22—21no—
306C22—21no—
307C22—44no—
308C22—48no—
309C22—44no—
310C22—41no—
311C22—42no—
312C22—41no—
313C22#144411yescharacteristic
314C22#185914yes—
315C22#200212yes—
316C22#200512yes—
317C22#202111yescharacteristic
318C22#202211yescharacteristic
319C22#206014yes—
320C4—21no—
321C4—21no—
322C4—22no—
323C4—22no—
324C4—416no—
325C4—84no—
326C4—162no—
327C4—162no—
328C4#186712yes—
329C4#206212yes—
330C2—24no—
331C2—24no—
332C2—41no—
333C2—41no—
334C2#1032611yescharacteristic
335C2#1033112yes—
336C2#1033411yescharacteristic
337C2#1134012yes—
338C2#1134411yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
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. #1297C4semidirect
88. C2×M16C4⋊C4semidirect
130. #95C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#219422
#138802
#107272
C2×#12972
#80192