Finite Group Databasebeta

← #194733

Subgroups — #194733

Order 512.

1402 conjugacy classes of subgroups; only the 266 normal subgroups are shown.

normal subgroups only

This group has 1402 conjugacy classes of subgroups — too many to draw. Only its 266 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
1C2×#2484C211yesmaximal, characteristic
2C2×#2313C211yesmaximal, characteristic
3#20702C212yesmaximal
4C2×#1463C211yesmaximal, characteristic
5#9026C214yesmaximal, complemented
6#9022C214yesmaximal, complemented
7#8015C212yesmaximal
8C22×#117C2211yescharacteristic
9C23×C4⋊C4C2211yescharacteristic
10C2×#407C2212yes—
11#2484—24no—
12#2484C418yescomplemented
13C2×#400C2212yes—
14C2×#398C2212yes—
15#2313—24no—
16#2313—22no—
17#2313C2214yes—
18C2×#387—22no—
19C2×#387—24no—
20C2×#385—21no—
21C2×#381—21no—
22C2×#381—21no—
23C2×#381C2211yescharacteristic
24C2×#380C2212yes—
25C2×C4×C22⋊C4—21no—
26#1825—24no—
27#1824C2214yes—
28#1823C2214yes—
29#1736—24no—
30#1463—24no—
31C2×#324C2214yes—
32#1294C2218yes—
33C24×C4D411yescharacteristic
34C24×C4D411yescharacteristic
35C24×C4C2311yescharacteristic
36C22×M16—22no—
37C2×#117—22no—
38C2×#117—22no—
39C2×#117—22no—
40C2×#117—24no—
41C2×#117C2×C414yes—
42C22×C4⋊C4—22no—
43C22×C4⋊C4—22no—
44C22×C4⋊C4—24no—
45C22×C4⋊C4—22no—
46C22×C4⋊C4C2×C414yes—
47C22×C4⋊C4D412yes—
48C22×C4⋊C4D414yes—
49C22×C4⋊C4D414yes—
50C22×C4⋊C4C2311yescharacteristic
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2312yes—
53C22×C22⋊C4—21no—
54C22×C22⋊C4—22no—
55C22×C22⋊C4—21no—
56C22×C42—21no—
57C22×C42—22no—
58#407—24no—
59#407—24no—
60C2×C22⋊C8—22no—
61C2×C22⋊C8C2314yes—
62#400—24no—
63#400C2×C418yescomplemented
64#398—24no—
65#398C2×C418yescomplemented
66#387—216no—
67#387—216no—
68#387—216no—
69#385—28no—
70#381—24no—
71#381—28no—
72#381—24no—
73#381—22no—
74#381C2×C414yes—
75#380—22no—
76#380—22no—
77#380D414yes—
78#380C2314yes—
79C4×C22⋊C4—24no—
80C4×C22⋊C4—24no—
81C2×#95—22no—
82C2×#95—24no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—21no—
88C2×#95C2312yes—
89#340—28no—
90#340—28no—
91#324—28no—
92C25C2×D411yescharacteristic
93C23×C4—21no—
94C23×C4—21no—
95C23×C4—21no—
96C23×C4—22no—
97C23×C4—21no—
98C23×C4—22no—
99C23×C4—24no—
100C23×C4—21no—
101C23×C4—22no—
102C23×C4—21no—
103C23×C4—44no—
104C23×C4—44no—
105C23×C4—44no—
106C23×C4C22⋊C414yes—
107C23×C4C22⋊C414yes—
108C23×C4C22×C412yes—
109C23×C4C2×D412yes—
110C23×C4C2×D412yes—
111C23×C4C2×D412yes—
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D412yes—
114C23×C4C2×D412yes—
115C23×C4#4114yes—
116C23×C4C2411yescharacteristic
117C2×M16—48no—
118C2×M16—44no—
119C22×C8—24no—
120#117—28no—
121#117—28no—
122#117—44no—
123#117—44no—
124C2×C4⋊C4—28no—
125C2×C4⋊C4—28no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—28no—
128C2×C4⋊C4—28no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—28no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—48no—
141C2×C4⋊C4—48no—
142C2×C4⋊C4C22⋊C418yes—
143C2×C4⋊C4C22×C412yes—
144C2×C4⋊C4C22×C412yes—
145C2×C4⋊C4C22×C414yes—
146C2×C4⋊C4C2×D414yes—
147C2×C4⋊C4C2×D418yes—
148C2×C22⋊C4—24no—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—24no—
151C2×C22⋊C4—24no—
152C2×C22⋊C4—44no—
153C2×C22⋊C4—48no—
154C2×C22⋊C4—41no—
155C2×C22⋊C4—41no—
156C2×C22⋊C4—42no—
157C2×C42—24no—
158C2×C42—28no—
159C2×C42—42no—
160C2×C42—42no—
161C2×C42—84no—
162C22⋊C8—24no—
163C22⋊C8—24no—
164C22⋊C8—44no—
165#95—28no—
166#95—216no—
167#95—28no—
168#95—28no—
169#95—24no—
170#95—28no—
171#95—24no—
172#95—24no—
173#95—22no—
174#95C22×C414yes—
175C24—24no—
176C24—24no—
177C24—42no—
178C24C2×C22⋊C412yes—
179C24#12011yescharacteristic
180C24#12011yescharacteristic
181C24#12312yes—
182C24C22×D411yescharacteristic
183C22×C4—28no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—24no—
188C22×C4—24no—
189C22×C4—24no—
190C22×C4—24no—
191C22×C4—24no—
192C22×C4—22no—
193C22×C4—24no—
194C22×C4—24no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—28no—
198C22×C4—48no—
199C22×C4—44no—
200C22×C4—44no—
201C22×C4—44no—
202C22×C4—42no—
203C22×C4—48no—
204C22×C4—42no—
205C22×C4—48no—
206C22×C4—44no—
207C22×C4—44no—
208C22×C4—44no—
209C22×C4—48no—
210C22×C4—44no—
211C22×C4—44no—
212C22×C4—48no—
213C22×C4—44no—
214C22×C4—44no—
215C22×C4—48no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—44no—
221C22×C4—44no—
222C22×C4—48no—
223C22×C4—84no—
224C22×C4—82no—
225C22×C4C2×C22⋊C414yes—
226C22×C4C2×C22⋊C414yes—
227C22×C4C2×C22⋊C412yes—
228C22×C4C4×D414yes—
229C22×C4C4×D414yes—
230C22×C4#12012yes—
231C22×C4#12114yes—
232C22×C4#12114yes—
233C22×C4#12312yes—
234C22×C4C23×C411yescharacteristic
235C22×C4C22×D412yes—
236C22×C4C22×D411yescharacteristic
237C22×C4C2×#4112yes—
238M16—88no—
239C2×C8—48no—
240C2×C8—44no—
241C4⋊C4—28no—
242C4⋊C4—28no—
243C4⋊C4—216no—
244C4⋊C4—24no—
245C4⋊C4—24no—
246C4⋊C4—48no—
247C4⋊C4—416no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—84no—
251C22⋊C4—48no—
252C22⋊C4—48no—
253C22⋊C4—48no—
254C22⋊C4—44no—
255C22⋊C4—44no—
256C42—48no—
257C42—88no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—22no—
262C23—24no—
263C23—24no—
264C23—24no—
265C23—24no—
266C23—22no—
267C23—22no—
268C23—22no—
269C23—42no—
270C23—42no—
271C23—416no—
272C23#38612yes—
273C23#38612yes—
274C23#45814yes—
275C23C22×C22⋊C411yescharacteristic
276C23C2×#12011yescharacteristic
277C23C2×#12311yescharacteristic
278C23#57514yes—
279C2×C4—28no—
280C2×C4—28no—
281C2×C4—24no—
282C2×C4—24no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—42no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—42no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—48no—
295C2×C4—48no—
296C2×C4—44no—
297C2×C4—48no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—48no—
307C2×C4—48no—
308C2×C4—48no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—82no—
312C2×C4—84no—
313C2×C4—88no—
314C2×C4—82no—
315C2×C4—816no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4#38614yes—
320C2×C4C2×C4×D412yes—
321C2×C4C2×#12112yes—
322C8—84no—
323C22—22no—
324C22—21no—
325C22—21no—
326C22—22no—
327C22—28no—
328C22—28no—
329C22—21no—
330C22—21no—
331C22—22no—
332C22—22no—
333C22—22no—
334C22—48no—
335C22—48no—
336C22—44no—
337C22#189014yes—
338C22C2×#38611yescharacteristic
339C22#289612yes—
340C22#300512yes—
341C22#302012yes—
342C22C2×#45812yes—
343C22#309312yes—
344C4—22no—
345C4—22no—
346C4—24no—
347C4—44no—
348C4—44no—
349C4—84no—
350C4—82no—
351C4—82no—
352C4—88no—
353C2—22no—
354C2—22no—
355C2—24no—
356C2—42no—
357C2#1813914yes—
358C2#2091011yescharacteristic
359C2#2095411yescharacteristic
360C2C2×#189011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #9026C2semidirect
6. #9022C2semidirect
12. #2484C4semidirect
63. #400C2×C4semidirect
65. #398C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#24842
C2×#23132
#207022
C2×#14632
#90262
#90222
#80152