Finite Group Databasebeta

← #121608

Subgroups — #121608

Order 512.

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

normal subgroups only

This group has 992 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#13881C212yesmaximal
3#10727C211yesmaximal, characteristic
4C2×#1297C211yesmaximal, characteristic
5#8019C212yesmaximal
6C23×C4⋊C4C2211yescharacteristic
7C2×#407C2211yescharacteristic
8C2×#407C2211yescharacteristic
9C2×#382—22no—
10C2×#380C2212yes—
11C2×C4×C22⋊C4C2212yes—
12#1853—22no—
13#1852—22no—
14#1809—22no—
15C2×#343—22no—
16#1442—24no—
17C4⋊#95—22no—
18C22⋊#95—24no—
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—44no—
30C22×C4⋊C4D412yes—
31C22×C4⋊C4D412yes—
32C22×C22⋊C4—22no—
33C22×C22⋊C4—22no—
34C22×C42—22no—
35C2×#107—22no—
36C2×#106—22no—
37#407—22no—
38#407—22no—
39#407—21no—
40#407C2×C412yes—
41#407Q812yes—
42#407Q812yes—
43C2×C22⋊C8—22no—
44C2×C22⋊C8—22no—
45#382—48no—
46#380—22no—
47#380—22no—
48#380—22no—
49#380—22no—
50C4×C22⋊C4—22no—
51C4×C22⋊C4—22no—
52C4×C22⋊C4C2×C414yes—
53C4×C22⋊C4Q814yes—
54C2×#95—28no—
55C2×#95—22no—
56C2×#95—28no—
57C2×#95D414yes—
58#343—44no—
59#340—24no—
60C25C2×D411yescharacteristic
61C23×C4—24no—
62C23×C4—24no—
63C23×C4—22no—
64C23×C4—22no—
65C23×C4—22no—
66C23×C4—42no—
67C23×C4—42no—
68C23×C4—44no—
69C23×C4—48no—
70C23×C4C22×C411yescharacteristic
71C23×C4C2×D411yescharacteristic
72C23×C4C2×D412yes—
73C23×C4C2×D412yes—
74C23×C4C2×Q811yescharacteristic
75C23×C4C2×Q812yes—
76C23×C4C2×Q811yescharacteristic
77C23×C4#4112yes—
78C23×C4#4111yescharacteristic
79C23×C4#4111yescharacteristic
80C2×M16—22no—
81C2×M16—22no—
82C2×M16—41no—
83C2×M16—42no—
84C2×M16—42no—
85C2×M16—42no—
86C2×M16—42no—
87C2×M16C4⋊C414yes—
88C22×C8—22no—
89C22×C8—41no—
90C2×C4⋊C4—24no—
91C2×C4⋊C4—28no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—48no—
97C2×C4⋊C4—44no—
98C2×C4⋊C4—44no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—44no—
104C2×C4⋊C4—44no—
105C2×C22⋊C4—24no—
106C2×C22⋊C4—24no—
107C2×C22⋊C4—44no—
108C2×C22⋊C4—44no—
109C2×C22⋊C4—48no—
110C2×C22⋊C4—42no—
111C2×C22⋊C4—42no—
112C2×C42—24no—
113C2×C42—24no—
114C2×C42—44no—
115C2×C42—44no—
116#107—82no—
117#106—82no—
118C22⋊C8—24no—
119C22⋊C8—24no—
120C22⋊C8—24no—
121C22⋊C8—42no—
122#95—24no—
123#95—416no—
124#95—42no—
125#95—42no—
126#95—88no—
127#95C4⋊C418yes—
128C24—24no—
129C24—24no—
130C24—41no—
131C24—41no—
132C24C4×D411yescharacteristic
133C24#12012yes—
134C24#12111yescharacteristic
135C24C22⋊Q812yes—
136C24C4⋊Q811yescharacteristic
137C22×C4—24no—
138C22×C4—22no—
139C22×C4—24no—
140C22×C4—22no—
141C22×C4—28no—
142C22×C4—28no—
143C22×C4—24no—
144C22×C4—42no—
145C22×C4—44no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—44no—
150C22×C4—42no—
151C22×C4—41no—
152C22×C4—42no—
153C22×C4—41no—
154C22×C4—44no—
155C22×C4—44no—
156C22×C4—44no—
157C22×C4—44no—
158C22×C4—44no—
159C22×C4—42no—
160C22×C4—48no—
161C22×C4—42no—
162C22×C4—44no—
163C22×C4—44no—
164C22×C4—48no—
165C22×C4—44no—
166C22×C4—44no—
167C22×C4—44no—
168C22×C4—44no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—44no—
172C22×C4—44no—
173C22×C4—816no—
174C22×C4—84no—
175C22×C4C2×C4⋊C411yescharacteristic
176C22×C4C2×C4⋊C412yes—
177C22×C4C4×D412yes—
178C22×C4C4×Q811yescharacteristic
179C22×C4#12011yescharacteristic
180C22×C4#12112yes—
181C22×C4C22⋊Q812yes—
182C22×C4C22⋊Q812yes—
183C22×C4C22⋊Q811yescharacteristic
184C22×C4C22⋊Q812yes—
185C22×C4#12311yescharacteristic
186C22×C4#12312yes—
187C22×C4#12512yes—
188C22×C4#12511yescharacteristic
189C22×C4#12711yescharacteristic
190C22×C4C4⋊Q811yescharacteristic
191C22×C4C4⋊Q812yes—
192C22×C4C4⋊Q812yes—
193M16—44no—
194M16—44no—
195M16—82no—
196M16—82no—
197C2×C8—24no—
198C2×C8—42no—
199C2×C8—42no—
200C2×C8—42no—
201C2×C8—81no—
202C2×C8—81no—
203C4⋊C4—48no—
204C4⋊C4—44no—
205C4⋊C4—44no—
206C4⋊C4—44no—
207C4⋊C4—44no—
208C4⋊C4—82no—
209C4⋊C4—82no—
210C22⋊C4—48no—
211C22⋊C4—48no—
212C22⋊C4—84no—
213C22⋊C4—84no—
214C42—48no—
215C42—48no—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—22no—
220C23—24no—
221C23—24no—
222C23—24no—
223C23—24no—
224C23—22no—
225C23—22no—
226C23—21no—
227C23—21no—
228C23—41no—
229C23—42no—
230C23—41no—
231C23—44no—
232C23—44no—
233C23—48no—
234C23#38512yes—
235C23#38911yescharacteristic
236C23#39211yescharacteristic
237C23#39512yes—
238C23#39811yescharacteristic
239C23#45414yes—
240C23#45514yes—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—24no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—24no—
249C2×C4—24no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—24no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—48no—
261C2×C4—48no—
262C2×C4—416no—
263C2×C4—48no—
264C2×C4—44no—
265C2×C4—84no—
266C2×C4—84no—
267C2×C4—84no—
268C2×C4—84no—
269C2×C4—88no—
270C2×C4—84no—
271C2×C4—84no—
272C2×C4—84no—
273C2×C4—88no—
274C2×C4—88no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4#38411yescharacteristic
282C2×C4#38511yescharacteristic
283C2×C4#38912yes—
284C2×C4#39511yescharacteristic
285C2×C4#39812yes—
286C2×C4#39911yescharacteristic
287C2×C4B(4,𝔽2)12yes—
288C2×C4#45812yes—
289C8—44no—
290C8—82no—
291C22—22no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—24no—
296C22—24no—
297C22—24no—
298C22—24no—
299C22—22no—
300C22—22no—
301C22—21no—
302C22—21no—
303C22—44no—
304C22—48no—
305C22—44no—
306C22—41no—
307C22—42no—
308C22—41no—
309C22#144411yescharacteristic
310C22#186014yes—
311C22#200312yes—
312C22#200412yes—
313C22#202111yescharacteristic
314C22#202211yescharacteristic
315C22#206114yes—
316C4—21no—
317C4—21no—
318C4—22no—
319C4—22no—
320C4—88no—
321C4—84no—
322C4—162no—
323C4—162no—
324C4#186712yes—
325C4#206212yes—
326C2—24no—
327C2—24no—
328C2—41no—
329C2—41no—
330C2#1032711yescharacteristic
331C2#1033212yes—
332C2#1033311yescharacteristic
333C2#1133912yes—
334C2#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

Maximal subgroups

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

subgroupindex
#219422
#138812
#107272
C2×#12972
#80192