Finite Group Databasebeta

← #194793

Subgroups — #194793

Order 512.

1420 conjugacy classes of subgroups; only the 266 normal subgroups are shown.

normal subgroups only

This group has 1420 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#34187C211yesmaximal, characteristic, complemented
3#20738C212yesmaximal
4C2×#1467C211yesmaximal, characteristic, complemented
5#9022C214yesmaximal, complemented
6#9018C214yesmaximal
7#8022C212yesmaximal, complemented
8C22×C4×Q8C2211yescharacteristic
9C22×C4×D4C2211yescharacteristic
10C2×#432C2212yes—
11#2398—24no—
12#2398C418yescomplemented
13C2×#400C2212yes—
14C2×#397—21no—
15C2×#394C2212yes—
16C2×#393—21no—
17#2356—24no—
18#2355—24no—
19#2313C2214yes—
20#2310—21no—
21#2285C2212yes—
22#2283—22no—
23#2274—21no—
24C2×C4×C22⋊C4—22no—
25C2×#376C2211yescharacteristic
26#1824C2214yes—
27#1802—24no—
28#1786C2214yes—
29#1762—24no—
30#1467—24no—
31C2×#324C2214yes—
32#1294C2218yes—
33C23×Q8D411yescharacteristic
34C23×D4C2311yescharacteristic
35C24×C4—21no—
36C2×C4×Q8—22no—
37C2×C4×Q8—22no—
38C2×C4×Q8—24no—
39C2×C4×Q8C2×C414yes—
40C2×C4×Q8D414yes—
41C2×C4×D4—22no—
42C2×C4×D4—22no—
43C2×C4×D4—22no—
44C2×C4×D4—24no—
45C2×C4×D4C2×C414yes—
46C2×#117—22no—
47C22×C4⋊C4D411yescharacteristic
48C22×C4⋊C4C2311yescharacteristic
49C22×C4⋊C4C2312yes—
50C22×C22⋊C4—22no—
51C22×C22⋊C4—21no—
52C22×C22⋊C4—22no—
53C22×C42—21no—
54C22×C42D412yes—
55C22×C42C2311yescharacteristic
56#432—24no—
57#432—24no—
58C2×C4⋊C8—22no—
59C2×#102—28no—
60C2×C22⋊C8C2314yes—
61C2×#97—22no—
62#400—24no—
63#400C2×C418yes—
64#397—28no—
65#394—24no—
66#394C2×C418yes—
67#393—28no—
68#391—28no—
69#387—216no—
70#384D414yes—
71#383—24no—
72#382—24no—
73#382—22no—
74#381—24no—
75#380—22no—
76#380C2314yes—
77C4×C4⋊C4—24no—
78C4×C4⋊C4—28no—
79C4×C4⋊C4—22no—
80C4×C4⋊C4D414yes—
81C4×C22⋊C4—24no—
82C4×C22⋊C4—216no—
83#376—24no—
84#376—22no—
85#376C2×C414yes—
86C2×#95—21no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95C2312yes—
90#340—28no—
91#324—28no—
92C25—21no—
93C22×Q8—22no—
94C22×Q8—22no—
95C22×Q8—44no—
96C22×Q8C22⋊C414yes—
97C22×D4—22no—
98C22×D4—21no—
99C22×D4—22no—
100C22×D4—44no—
101C22×D4C22×C412yes—
102C23×C4—22no—
103C23×C4—21no—
104C23×C4—24no—
105C23×C4—48no—
106C23×C4—41no—
107C23×C4—41no—
108C23×C4—42no—
109C23×C4C2×D412yes—
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D412yes—
113C23×C4C2411yescharacteristic
114C22×C8—24no—
115C4×Q8—28no—
116C4×Q8—28no—
117C4×Q8—28no—
118C4×Q8—28no—
119C4×D4—28no—
120C4×D4—28no—
121C4×D4—44no—
122C4×D4—44no—
123#117—48no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4C22⋊C414yes—
134C2×C4⋊C4C22×C412yes—
135C2×C4⋊C4C22×C414yes—
136C2×C4⋊C4C2×D414yes—
137C2×C4⋊C4C2×D418yes—
138C2×C22⋊C4—24no—
139C2×C22⋊C4—24no—
140C2×C22⋊C4—28no—
141C2×C22⋊C4—28no—
142C2×C22⋊C4—48no—
143C2×C22⋊C4—48no—
144C2×C22⋊C4—42no—
145C2×C22⋊C4—42no—
146C2×C42—22no—
147C2×C42—22no—
148C2×C42—24no—
149C2×C42—24no—
150C2×C42—24no—
151C2×C42—28no—
152C2×C42—44no—
153C2×C42C22⋊C418yes—
154C2×C42C22×C412yes—
155C2×C42C2×D412yes—
156C2×C42C2×D412yes—
157C2×C42C2×D412yes—
158C2×C42#4114yes—
159C4⋊C8—44no—
160#102—416no—
161C22⋊C8—24no—
162C22⋊C8—24no—
163#97—44no—
164#95—28no—
165#95—28no—
166#95—28no—
167#95—24no—
168#95—28no—
169#95—28no—
170#95—24no—
171#95—22no—
172#95—24no—
173#95C22×C414yes—
174C24—24no—
175C24—22no—
176C24—44no—
177C24—42no—
178C24—41no—
179C24—41no—
180C24—44no—
181C24C22×D411yescharacteristic
182C2×Q8—28no—
183C2×Q8—48no—
184C2×Q8—48no—
185C2×Q8—44no—
186C2×Q8—44no—
187C2×D4—28no—
188C2×D4—24no—
189C2×D4—44no—
190C2×D4—44no—
191C2×D4—48no—
192C2×D4—44no—
193C2×D4—42no—
194C22×C4—24no—
195C22×C4—24no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—24no—
200C22×C4—24no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—24no—
204C22×C4—28no—
205C22×C4—42no—
206C22×C4—416no—
207C22×C4—44no—
208C22×C4—44no—
209C22×C4—44no—
210C22×C4—44no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—44no—
214C22×C4—44no—
215C22×C4—44no—
216C22×C4—84no—
217C22×C4—88no—
218C22×C4—84no—
219C22×C4C2×C22⋊C414yes—
220C22×C4C2×C22⋊C412yes—
221C22×C4C2×C22⋊C412yes—
222C22×C4C2×C22⋊C414yes—
223C22×C4#12011yescharacteristic
224C22×C4#12012yes—
225C22×C4#12011yescharacteristic
226C22×C4#12312yes—
227C22×C4#12312yes—
228C22×C4C23×C411yescharacteristic
229C22×C4C22×D412yes—
230C22×C4C22×D411yescharacteristic
231C22×C4C2×#4112yes—
232C2×C8—48no—
233C2×C8—44no—
234C4⋊C4—28no—
235C4⋊C4—28no—
236C4⋊C4—28no—
237C4⋊C4—24no—
238C4⋊C4—48no—
239C4⋊C4—48no—
240C4⋊C4—42no—
241C4⋊C4—48no—
242C22⋊C4—416no—
243C22⋊C4—416no—
244C22⋊C4—48no—
245C22⋊C4—48no—
246C42—28no—
247C42—28no—
248C42—48no—
249C42—48no—
250C42—88no—
251C42C4×D414yes—
252C42C4×D414yes—
253C42#12114yes—
254C42#12114yes—
255C23—22no—
256C23—22no—
257C23—24no—
258C23—44no—
259C23—42no—
260C23—44no—
261C23—416no—
262C23—44no—
263C23—44no—
264C23—42no—
265C23—82no—
266C23—88no—
267C23—82no—
268C23—82no—
269C23—82no—
270C23C22×C22⋊C411yescharacteristic
271C23C2×#12011yescharacteristic
272C23C2×#12311yescharacteristic
273C23#57514yes—
274Q8—416no—
275D4—28no—
276D4—48no—
277D4—44no—
278C2×C4—24no—
279C2×C4—24no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—24no—
288C2×C4—24no—
289C2×C4—24no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—44no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—48no—
302C2×C4—48no—
303C2×C4—48no—
304C2×C4—42no—
305C2×C4—88no—
306C2×C4—82no—
307C2×C4—88no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—88no—
311C2×C4#38612yes—
312C2×C4#38614yes—
313C2×C4#38612yes—
314C2×C4#45414yes—
315C2×C4C2×C4×D412yes—
316C2×C4C2×#12112yes—
317C8—84no—
318C22—28no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—48no—
324C22—44no—
325C22—84no—
326C22—88no—
327C22—88no—
328C22—82no—
329C22—82no—
330C22C2×#38611yescharacteristic
331C22#289612yes—
332C22#300512yes—
333C22C2×#45412yes—
334C22#302512yes—
335C22#309312yes—
336C4—24no—
337C4—22no—
338C4—22no—
339C4—22no—
340C4—44no—
341C4—44no—
342C4—84no—
343C4—88no—
344C4—82no—
345C4#188814yes—
346C2—22no—
347C2—22no—
348C2—84no—
349C2#1818214yes—
350C2#2091011yescharacteristic
351C2C2×#188811yescharacteristic
352C2#2095911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
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
2. #34187C2semidirect
4. C2×#1467C2semidirect
5. #9022C2semidirect
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
#341872
#207382
C2×#14672
#90222
#90182
#80222