Finite Group Databasebeta

← #194924

Subgroups — #194924

Order 512.

1470 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2632C211yesmaximal, characteristic
2#34182C211yesmaximal, characteristic, complemented
3#21808C212yesmaximal
4C2×#1467C211yesmaximal, characteristic, complemented
5#9019C214yesmaximal
6#9015C214yesmaximal, complemented
7#8018C212yesmaximal, complemented
8C22×#124C2211yescharacteristic
9C2×#519—21no—
10C22×C4×Q8C2211yescharacteristic
11C2×#432C2212yes—
12#2632—24no—
13#2632C418yescomplemented
14C2×#399—21no—
15C2×#395C2212yes—
16C2×#394C2212yes—
17C2×#393—21no—
18#2351—24no—
19Q8×C4⋊C4—24no—
20#2316—21no—
21#2316C2214yes—
22C2×#388—22no—
23#2285—22no—
24#2285C2212yes—
25C2×#383C2211yescharacteristic
26#1983—24no—
27#1959—24no—
28#1787C2214yes—
29#1786C2214yes—
30#1467—24no—
31C2×#324C2214yes—
32#1294C2218yes—
33C23×Q8D411yescharacteristic
34C23×Q8C2311yescharacteristic
35C23×D4C2311yescharacteristic
36C2×#124—22no—
37C2×#124—22no—
38C2×#124—22no—
39C2×#124—24no—
40C2×#124C2×C414yes—
41#519—48no—
42C2×C4×Q8—28no—
43C2×C4×Q8—22no—
44C2×C4×Q8—22no—
45C2×C4×Q8—24no—
46C2×C4×Q8C2×C414yes—
47C2×C4×Q8D414yes—
48C2×#117—22no—
49C2×#117—24no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4D411yescharacteristic
52C22×C4⋊C4C2312yes—
53C22×C22⋊C4—22no—
54C22×C22⋊C4—21no—
55C22×C22⋊C4—21no—
56C22×C22⋊C4—22no—
57C22×C42D412yes—
58C22×C42C2311yescharacteristic
59#432—24no—
60#432—24no—
61C2×C4⋊C8—22no—
62C2×#103—28no—
63C2×#102—28no—
64C2×C22⋊C8C2314yes—
65C2×#97—22no—
66#399—28no—
67#395—24no—
68#395C2×C418yes—
69#394—24no—
70#394C2×C418yes—
71#393—28no—
72#389—28no—
73#389—216no—
74#388—216no—
75#384—24no—
76#384D414yes—
77#383—22no—
78#383C2×C414yes—
79#382—24no—
80#382—22no—
81#380—24no—
82#380—22no—
83#380C2314yes—
84C4×C4⋊C4—24no—
85C4×C4⋊C4—24no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4D414yes—
88C2×#95—22no—
89C2×#95—21no—
90C2×#95—21no—
91C2×#95C2312yes—
92#324—28no—
93C25—21no—
94C22×Q8—22no—
95C22×Q8—22no—
96C22×Q8—24no—
97C22×Q8—24no—
98C22×Q8—21no—
99C22×Q8—22no—
100C22×Q8—22no—
101C22×Q8—82no—
102C22×Q8C22⋊C414yes—
103C22×Q8C22×C412yes—
104C22×D4—22no—
105C22×D4—21no—
106C22×D4—22no—
107C22×D4—82no—
108C22×D4C22×C412yes—
109C23×C4—21no—
110C23×C4—22no—
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D412yes—
113C23×C4C2×D412yes—
114C23×C4C2×D411yescharacteristic
115C23×C4C2411yescharacteristic
116C22×C8—24no—
117#124—28no—
118#124—28no—
119#124—44no—
120#124—44no—
121C4×Q8—28no—
122C4×Q8—28no—
123C4×Q8—28no—
124C4×Q8—28no—
125C4×Q8—432no—
126#117—48no—
127#117—48no—
128#117—48no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—28no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—28no—
137C2×C4⋊C4—28no—
138C2×C4⋊C4—84no—
139C2×C4⋊C4C22⋊C414yes—
140C2×C4⋊C4C22×C414yes—
141C2×C4⋊C4C2×D414yes—
142C2×C4⋊C4C2×D418yes—
143C2×C22⋊C4—24no—
144C2×C22⋊C4—24no—
145C2×C22⋊C4—24no—
146C2×C22⋊C4—28no—
147C2×C22⋊C4—28no—
148C2×C22⋊C4—41no—
149C2×C22⋊C4—41no—
150C2×C22⋊C4—42no—
151C2×C22⋊C4—42no—
152C2×C22⋊C4—42no—
153C2×C22⋊C4—44no—
154C2×C22⋊C4—44no—
155C2×C22⋊C4—84no—
156C2×C42—22no—
157C2×C42—22no—
158C2×C42—24no—
159C2×C42—28no—
160C2×C42—24no—
161C2×C42C22⋊C418yes—
162C2×C42C22×C412yes—
163C2×C42C2×D412yes—
164C2×C42C2×D412yes—
165C2×C42C2×D412yes—
166C2×C42#4114yes—
167C4⋊C8—44no—
168#103—88no—
169#102—88no—
170C22⋊C8—24no—
171C22⋊C8—24no—
172#97—44no—
173#95—28no—
174#95—24no—
175#95—24no—
176#95—28no—
177#95—24no—
178#95—22no—
179#95—24no—
180#95C22×C414yes—
181C24—22no—
182C24—44no—
183C24—44no—
184C24—41no—
185C24—41no—
186C24—42no—
187C24—42no—
188C24C22×D411yescharacteristic
189C2×Q8—28no—
190C2×Q8—216no—
191C2×Q8—28no—
192C2×Q8—28no—
193C2×Q8—216no—
194C2×Q8—24no—
195C2×Q8—48no—
196C2×Q8—42no—
197C2×Q8—82no—
198C2×Q8—82no—
199C2×Q8—84no—
200C2×D4—24no—
201C2×D4—48no—
202C2×D4—42no—
203C2×D4—82no—
204C2×D4—84no—
205C2×D4—82no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—24no—
215C22×C4—22no—
216C22×C4—28no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—44no—
220C22×C4—44no—
221C22×C4—44no—
222C22×C4—44no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4C2×C22⋊C412yes—
228C22×C4C2×C22⋊C414yes—
229C22×C4C2×C22⋊C414yes—
230C22×C4C2×C22⋊C412yes—
231C22×C4#12011yescharacteristic
232C22×C4#12011yescharacteristic
233C22×C4#12012yes—
234C22×C4#12312yes—
235C22×C4#12312yes—
236C22×C4AGL(1,ℤ/8)14yes—
237C22×C4#13714yes—
238C22×C4C23×C411yescharacteristic
239C22×C4C22×D412yes—
240C22×C4C22×D411yescharacteristic
241C22×C4C2×#4112yes—
242C2×C8—48no—
243C2×C8—44no—
244C4⋊C4—28no—
245C4⋊C4—28no—
246C4⋊C4—28no—
247C4⋊C4—48no—
248C4⋊C4—48no—
249C4⋊C4—48no—
250C4⋊C4—416no—
251C4⋊C4—88no—
252C4⋊C4—88no—
253C22⋊C4—48no—
254C22⋊C4—44no—
255C22⋊C4—44no—
256C22⋊C4—48no—
257C22⋊C4—416no—
258C22⋊C4—88no—
259C42—28no—
260C42—28no—
261C42—48no—
262C42—416no—
263C42C4×D414yes—
264C42C4×D414yes—
265C42#12114yes—
266C42#12114yes—
267C23—22no—
268C23—22no—
269C23—44no—
270C23—44no—
271C23—44no—
272C23—42no—
273C23—82no—
274C23—88no—
275C23—88no—
276C23—82no—
277C23—82no—
278C23—81no—
279C23—81no—
280C23—82no—
281C23—82no—
282C23C22×C22⋊C411yescharacteristic
283C23C2×#12011yescharacteristic
284C23C2×#12311yescharacteristic
285C23C2×AGL(1,ℤ/8)12yes—
286C23C2×#13712yes—
287Q8—232no—
288Q8—48no—
289Q8—84no—
290D4—48no—
291D4—84no—
292C2×C4—24no—
293C2×C4—24no—
294C2×C4—24no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—22no—
298C2×C4—24no—
299C2×C4—22no—
300C2×C4—24no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—48no—
318C2×C4—48no—
319C2×C4—48no—
320C2×C4—42no—
321C2×C4—82no—
322C2×C4—84no—
323C2×C4—84no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4#38612yes—
327C2×C4#38612yes—
328C2×C4#38614yes—
329C2×C4#41718yes—
330C2×C4C2×C4×D412yes—
331C2×C4C2×#12112yes—
332C8—84no—
333C22—24no—
334C22—24no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—88no—
340C22—82no—
341C22—82no—
342C22—82no—
343C22—164no—
344C22—162no—
345C22—162no—
346C22C2×#38611yescharacteristic
347C22C2×#41712yes—
348C22#300311yescharacteristic
349C22#300411yescharacteristic
350C22#302012yes—
351C22#302712yes—
352C22#309212yes—
353C4—24no—
354C4—28no—
355C4—22no—
356C4—22no—
357C4—22no—
358C4—22no—
359C4—44no—
360C4—84no—
361C4—88no—
362C4—82no—
363C2—22no—
364C2—22no—
365C2—162no—
366C2#1836812yes—
367C2#1836912yes—
368C2#2090911yescharacteristic
369C2#2095411yescharacteristic
370C2#2096111yescharacteristic
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. #34182C2semidirect
4. C2×#1467C2semidirect
6. #9015C2semidirect
7. #8018C2semidirect
13. #2632C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26322
#341822
#218082
C2×#14672
#90192
#90152
#80182