Finite Group Databasebeta

← #194797

Subgroups — #194797

Order 512.

1572 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2398C211yesmaximal, characteristic
2#34182C211yesmaximal, characteristic, complemented
3#20738C212yesmaximal
4C2×#1467C211yesmaximal, characteristic, complemented
5#9023C214yesmaximal, complemented
6#9019C214yesmaximal
7#8022C212yesmaximal, complemented
8C2×#519—21no—
9C22×C4×Q8C2211yescharacteristic
10C22×C4×D4C2211yescharacteristic
11C2×#432C2212yes—
12#2398—24no—
13#2398C418yescomplemented
14C2×#400C2212yes—
15C2×#397—21no—
16C2×#394C2212yes—
17C2×#393—21no—
18#2351—24no—
19Q8×C4⋊C4—24no—
20#2316—21no—
21#2316C2214yes—
22#2285—22no—
23#2285C2212yes—
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—
36#519—48no—
37C2×C4×Q8—28no—
38C2×C4×Q8—22no—
39C2×C4×Q8—22no—
40C2×C4×Q8—24no—
41C2×C4×Q8C2×C414yes—
42C2×C4×Q8D414yes—
43C2×C4×D4—22no—
44C2×C4×D4—22no—
45C2×C4×D4—22no—
46C2×C4×D4—24no—
47C2×C4×D4C2×C414yes—
48C2×#117—22no—
49C2×#117—24no—
50C22×C4⋊C4D411yescharacteristic
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2312yes—
53C22×C22⋊C4—22no—
54C22×C22⋊C4—21no—
55C22×C22⋊C4—22no—
56C22×C42—21no—
57C22×C42D412yes—
58C22×C42C2311yescharacteristic
59#432—24no—
60#432—24no—
61C2×C4⋊C8—22no—
62C2×#102—28no—
63C2×C22⋊C8C2314yes—
64C2×#97—22no—
65#400—24no—
66#400C2×C418yes—
67#397—28no—
68#394—24no—
69#394C2×C418yes—
70#393—28no—
71#389—28no—
72#389—216no—
73#384—24no—
74#384D414yes—
75#382—24no—
76#382—22no—
77#380—24no—
78#380—22no—
79#380C2314yes—
80C4×C4⋊C4—24no—
81C4×C4⋊C4—24no—
82C4×C4⋊C4—22no—
83C4×C4⋊C4D414yes—
84C4×C22⋊C4—216no—
85#376—22no—
86#376C2×C414yes—
87C2×#95—21no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95C2312yes—
91#340—28no—
92#324—28no—
93C25—21no—
94C22×Q8—22no—
95C22×Q8—22no—
96C22×Q8—24no—
97C22×Q8—24no—
98C22×Q8C22⋊C414yes—
99C22×D4—22no—
100C22×D4—21no—
101C22×D4—22no—
102C22×D4—44no—
103C22×D4C22×C412yes—
104C23×C4—22no—
105C23×C4—21no—
106C23×C4—24no—
107C23×C4—48no—
108C23×C4—41no—
109C23×C4—41no—
110C23×C4—42no—
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D412yes—
114C23×C4C2×D412yes—
115C23×C4C2411yescharacteristic
116C22×C8—24no—
117C4×Q8—28no—
118C4×Q8—28no—
119C4×Q8—28no—
120C4×Q8—28no—
121C4×Q8—432no—
122C4×D4—28no—
123C4×D4—28no—
124C4×D4—44no—
125C4×D4—44no—
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—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—28no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—28no—
140C2×C4⋊C4C22⋊C414yes—
141C2×C4⋊C4C22×C412yes—
142C2×C4⋊C4C22×C414yes—
143C2×C4⋊C4C2×D414yes—
144C2×C4⋊C4C2×D418yes—
145C2×C22⋊C4—24no—
146C2×C22⋊C4—24no—
147C2×C22⋊C4—28no—
148C2×C22⋊C4—28no—
149C2×C22⋊C4—48no—
150C2×C22⋊C4—42no—
151C2×C22⋊C4—42no—
152C2×C22⋊C4—44no—
153C2×C22⋊C4—44no—
154C2×C42—22no—
155C2×C42—22no—
156C2×C42—24no—
157C2×C42—28no—
158C2×C42—24no—
159C2×C42—24no—
160C2×C42—44no—
161C2×C42C22⋊C418yes—
162C2×C42C22×C412yes—
163C2×C42C2×D412yes—
164C2×C42C2×D412yes—
165C2×C42C2×D412yes—
166C2×C42#4114yes—
167C4⋊C8—44no—
168#102—416no—
169C22⋊C8—24no—
170C22⋊C8—24no—
171#97—44no—
172#95—28no—
173#95—24no—
174#95—28no—
175#95—28no—
176#95—24no—
177#95—22no—
178#95—24no—
179#95C22×C414yes—
180C24—24no—
181C24—22no—
182C24—44no—
183C24—42no—
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×D4—28no—
195C2×D4—24no—
196C2×D4—44no—
197C2×D4—44no—
198C2×D4—48no—
199C2×D4—44no—
200C2×D4—42no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—24no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—24no—
212C22×C4—28no—
213C22×C4—42no—
214C22×C4—416no—
215C22×C4—44no—
216C22×C4—44no—
217C22×C4—44no—
218C22×C4—44no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—44no—
224C22×C4—44no—
225C22×C4—84no—
226C22×C4—88no—
227C22×C4—84no—
228C22×C4C2×C22⋊C412yes—
229C22×C4C2×C22⋊C412yes—
230C22×C4C2×C22⋊C414yes—
231C22×C4C2×C22⋊C414yes—
232C22×C4#12011yescharacteristic
233C22×C4#12012yes—
234C22×C4#12011yescharacteristic
235C22×C4#12312yes—
236C22×C4#12312yes—
237C22×C4AGL(1,ℤ/8)14yes—
238C22×C4#13714yes—
239C22×C4C23×C411yescharacteristic
240C22×C4C22×D412yes—
241C22×C4C22×D411yescharacteristic
242C22×C4C2×#4112yes—
243C2×C8—48no—
244C2×C8—44no—
245C4⋊C4—28no—
246C4⋊C4—28no—
247C4⋊C4—28no—
248C4⋊C4—24no—
249C4⋊C4—48no—
250C4⋊C4—48no—
251C4⋊C4—416no—
252C4⋊C4—42no—
253C4⋊C4—48no—
254C4⋊C4—88no—
255C22⋊C4—416no—
256C22⋊C4—48no—
257C22⋊C4—48no—
258C22⋊C4—416no—
259C42—28no—
260C42—28no—
261C42—48no—
262C42—48no—
263C42—416no—
264C42C4×D414yes—
265C42C4×D414yes—
266C42#12114yes—
267C42#12114yes—
268C23—22no—
269C23—22no—
270C23—24no—
271C23—44no—
272C23—42no—
273C23—44no—
274C23—48no—
275C23—48no—
276C23—44no—
277C23—44no—
278C23—42no—
279C23—82no—
280C23—84no—
281C23—84no—
282C23—82no—
283C23—81no—
284C23—81no—
285C23—82no—
286C23C22×C22⋊C411yescharacteristic
287C23C2×#12011yescharacteristic
288C23C2×#12311yescharacteristic
289C23C2×AGL(1,ℤ/8)12yes—
290C23C2×#13712yes—
291Q8—232no—
292D4—28no—
293D4—48no—
294D4—44no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—22no—
301C2×C4—24no—
302C2×C4—22no—
303C2×C4—24no—
304C2×C4—22no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—48no—
320C2×C4—48no—
321C2×C4—48no—
322C2×C4—42no—
323C2×C4—88no—
324C2×C4—82no—
325C2×C4—84no—
326C2×C4—84no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—88no—
330C2×C4#38612yes—
331C2×C4#38614yes—
332C2×C4#38612yes—
333C2×C4#41718yes—
334C2×C4#45414yes—
335C2×C4C2×C4×D412yes—
336C2×C4C2×#12112yes—
337C8—84no—
338C22—24no—
339C22—24no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—48no—
345C22—44no—
346C22—84no—
347C22—88no—
348C22—84no—
349C22—84no—
350C22—82no—
351C22—82no—
352C22C2×#38611yescharacteristic
353C22C2×#41712yes—
354C22#300311yescharacteristic
355C22#300411yescharacteristic
356C22C2×#45412yes—
357C22#302512yes—
358C22#309212yes—
359C4—24no—
360C4—28no—
361C4—22no—
362C4—22no—
363C4—22no—
364C4—44no—
365C4—84no—
366C4—88no—
367C4—82no—
368C4#188814yes—
369C2—22no—
370C2—22no—
371C2—84no—
372C2#1818612yes—
373C2#1818812yes—
374C2#2090911yescharacteristic
375C2C2×#188811yescharacteristic
376C2#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. #34182C2semidirect
4. C2×#1467C2semidirect
5. #9023C2semidirect
7. #8022C2semidirect
13. #2398C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23982
#341822
#207382
C2×#14672
#90232
#90192
#80222