Finite Group Databasebeta

← #194798

Subgroups — #194798

Order 512.

1464 conjugacy classes of subgroups; only the 266 normal subgroups are shown.

normal subgroups only

This group has 1464 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×#2398C211yesmaximal, characteristic
2#34186C211yesmaximal, characteristic
3#20740C212yesmaximal, complemented
4C2×#1466C211yesmaximal, characteristic
5#9028C214yesmaximal, complemented
6#9016C214yesmaximal, complemented
7#8022C212yesmaximal, complemented
8C22×C4×Q8C2211yescharacteristic
9C22×C4×D4C2211yescharacteristic
10C2×#432C2212yes—
11#2398—24no—
12#2398C418yescomplemented
13C2×#400—21no—
14C2×#397C2212yes—
15C2×#394—21no—
16C2×#393C2212yes—
17#2357—24no—
18#2354—24no—
19#2312C2214yes—
20#2309—21no—
21#2285C2212yes—
22#2280—22no—
23#2274—21no—
24C2×C4×C22⋊C4—22no—
25C2×#376C2211yescharacteristic
26#1823C2214yes—
27#1803—24no—
28#1787C2214yes—
29#1763—24no—
30#1466—24no—
31C2×#323C2214yes—
32#1294C2218yes—
33C23×Q8C2311yescharacteristic
34C23×D4D411yescharacteristic
35C24×C4D412yes—
36C2×C4×Q8—22no—
37C2×C4×Q8—22no—
38C2×C4×Q8—22no—
39C2×C4×Q8—24no—
40C2×C4×Q8C2×C414yes—
41C2×C4×D4—22no—
42C2×C4×D4—22no—
43C2×C4×D4—24no—
44C2×C4×D4C2×C414yes—
45C2×C4×D4D414yes—
46C2×#117—22no—
47C2×#117—24no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4D411yescharacteristic
50C22×C4⋊C4C2311yescharacteristic
51C22×C22⋊C4—22no—
52C22×C22⋊C4—22no—
53C22×C22⋊C4C2312yes—
54C22×C42—21no—
55C22×C42—21no—
56C22×C42C2311yescharacteristic
57#432—24no—
58#432—24no—
59C2×C4⋊C8—22no—
60C2×#103—28no—
61C2×C22⋊C8C2314yes—
62C2×#97—22no—
63#400—28no—
64#397—24no—
65#397C2×C418yes—
66#394—28no—
67#393—24no—
68#393C2×C418yes—
69#390—28no—
70#388—216no—
71#384D414yes—
72#382—22no—
73#381—24no—
74#380—24no—
75#380—22no—
76#380C2314yes—
77#379—24no—
78C4×C4⋊C4—28no—
79C4×C4⋊C4—22no—
80C4×C4⋊C4D414yes—
81C4×C22⋊C4—24no—
82C4×C22⋊C4—216no—
83C4×C22⋊C4—24no—
84#376—22no—
85#376C2×C414yes—
86C2×#95—22no—
87C2×#95—21no—
88C2×#95—22no—
89C2×#95C2312yes—
90#340—28no—
91#323—28no—
92C25C2×D412yes—
93C22×Q8—21no—
94C22×Q8—22no—
95C22×Q8—22no—
96C22×Q8—44no—
97C22×Q8C22×C412yes—
98C22×D4—22no—
99C22×D4—22no—
100C22×D4—44no—
101C22×D4C22⋊C414yes—
102C23×C4—21no—
103C23×C4—22no—
104C23×C4—22no—
105C23×C4—24no—
106C23×C4—21no—
107C23×C4—22no—
108C23×C4—48no—
109C23×C4C22⋊C418yes—
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D412yes—
113C23×C4C2411yescharacteristic
114C22×C8—24no—
115C4×Q8—28no—
116C4×Q8—28no—
117C4×Q8—44no—
118C4×Q8—44no—
119C4×D4—28no—
120C4×D4—28no—
121C4×D4—28no—
122C4×D4—28no—
123#117—48no—
124#117—88no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4C22⋊C414yes—
136C2×C4⋊C4C22×C412yes—
137C2×C4⋊C4C2×D418yes—
138C2×C22⋊C4—24no—
139C2×C22⋊C4—24no—
140C2×C22⋊C4—28no—
141C2×C22⋊C4—28no—
142C2×C22⋊C4—24no—
143C2×C22⋊C4—28no—
144C2×C22⋊C4—28no—
145C2×C22⋊C4—48no—
146C2×C22⋊C4—48no—
147C2×C22⋊C4C22×C414yes—
148C2×C22⋊C4C2×D414yes—
149C2×C42—24no—
150C2×C42—24no—
151C2×C42—24no—
152C2×C42—28no—
153C2×C42—44no—
154C2×C42—41no—
155C2×C42—41no—
156C2×C42—42no—
157C2×C42C22×C412yes—
158C2×C42C2×D412yes—
159C2×C42C2×D412yes—
160C2×C42C2×D412yes—
161C2×C42#4114yes—
162C4⋊C8—44no—
163#103—416no—
164C22⋊C8—24no—
165C22⋊C8—24no—
166#97—44no—
167#95—28no—
168#95—24no—
169#95—28no—
170#95—28no—
171#95—24no—
172#95—22no—
173#95—24no—
174#95C22×C414yes—
175C24—22no—
176C24—24no—
177C24—22no—
178C24—24no—
179C24—44no—
180C24—44no—
181C24C2×C22⋊C414yes—
182C24C22×D411yescharacteristic
183C2×Q8—28no—
184C2×Q8—24no—
185C2×Q8—44no—
186C2×Q8—44no—
187C2×Q8—44no—
188C2×Q8—48no—
189C2×Q8—42no—
190C2×D4—28no—
191C2×D4—48no—
192C2×D4—48no—
193C2×D4—44no—
194C2×D4—44no—
195C22×C4—24no—
196C22×C4—24no—
197C22×C4—24no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—28no—
202C22×C4—28no—
203C22×C4—24no—
204C22×C4—22no—
205C22×C4—28no—
206C22×C4—42no—
207C22×C4—416no—
208C22×C4—42no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—48no—
212C22×C4—44no—
213C22×C4—44no—
214C22×C4—44no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—44no—
218C22×C4—44no—
219C22×C4—88no—
220C22×C4—84no—
221C22×C4C2×C22⋊C412yes—
222C22×C4C2×C22⋊C412yes—
223C22×C4C2×C22⋊C414yes—
224C22×C4#12011yescharacteristic
225C22×C4#12012yes—
226C22×C4#12011yescharacteristic
227C22×C4#12312yes—
228C22×C4#12312yes—
229C22×C4C23×C411yescharacteristic
230C22×C4C22×D412yes—
231C22×C4C22×D411yescharacteristic
232C22×C4C2×#4112yes—
233C2×C8—48no—
234C2×C8—44no—
235C4⋊C4—28no—
236C4⋊C4—24no—
237C4⋊C4—48no—
238C4⋊C4—48no—
239C4⋊C4—42no—
240C4⋊C4—48no—
241C4⋊C4—48no—
242C22⋊C4—28no—
243C22⋊C4—28no—
244C22⋊C4—416no—
245C22⋊C4—416no—
246C22⋊C4—48no—
247C22⋊C4—88no—
248C22⋊C4—88no—
249C42—48no—
250C42—48no—
251C42—44no—
252C42—44no—
253C42—88no—
254C42C4×D414yes—
255C42C4×D414yes—
256C42#12114yes—
257C42#12114yes—
258C23—22no—
259C23—22no—
260C23—24no—
261C23—24no—
262C23—24no—
263C23—44no—
264C23—44no—
265C23—416no—
266C23—44no—
267C23—44no—
268C23—44no—
269C23—44no—
270C23—82no—
271C23—82no—
272C23—88no—
273C23C22×C22⋊C411yescharacteristic
274C23C2×#12011yescharacteristic
275C23C2×#12311yescharacteristic
276C23#57514yes—
277Q8—28no—
278Q8—48no—
279Q8—44no—
280D4—416no—
281C2×C4—24no—
282C2×C4—24no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—24no—
287C2×C4—24no—
288C2×C4—24no—
289C2×C4—24no—
290C2×C4—24no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—44no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—42no—
299C2×C4—416no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—44no—
305C2×C4—48no—
306C2×C4—48no—
307C2×C4—48no—
308C2×C4—42no—
309C2×C4—88no—
310C2×C4—82no—
311C2×C4—88no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4#38612yes—
315C2×C4#38614yes—
316C2×C4#38612yes—
317C2×C4#45514yes—
318C2×C4C2×C4×D412yes—
319C2×C4C2×#12112yes—
320C8—84no—
321C22—28no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—48no—
327C22—44no—
328C22—44no—
329C22—44no—
330C22—84no—
331C22—88no—
332C22—88no—
333C22C2×#38611yescharacteristic
334C22#289612yes—
335C22#300512yes—
336C22C2×#45512yes—
337C22#302512yes—
338C22#309312yes—
339C4—24no—
340C4—22no—
341C4—22no—
342C4—22no—
343C4—44no—
344C4—44no—
345C4—84no—
346C4—88no—
347C4—82no—
348C4#188514yes—
349C2—22no—
350C2—22no—
351C2—84no—
352C2#1819114yes—
353C2#2091011yescharacteristic
354C2C2×#188511yescharacteristic
355C2#2095911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #20740C2semidirect
5. #9028C2semidirect
6. #9016C2semidirect
7. #8022C2semidirect
12. #2398C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23982
#341862
#207402
C2×#14662
#90282
#90162
#80222