Finite Group Databasebeta

← #6510735

Subgroups — #6510735

Order 512.

1736 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35002C211yesmaximal, characteristic
2C2×#2503C211yesmaximal, characteristic
3#29111C214yesmaximal, complemented
4#27969C214yesmaximal, complemented
5#27713C2116yesmaximal, complemented
6#21759C212yesmaximal
7C2×#1823C212yesmaximal
8#20526C211yesmaximal, characteristic
9C2×#543—21no—
10C22×C22⋊Q8C2211yescharacteristic
11C23×C4⋊C4C2211yescharacteristic
12C22×C22⋊C8C2212yes—
13#2857C2214yes—
14#2659—22no—
15#2549—24no—
16#2503—24no—
17#2503C2218yes—
18#2440C2218yes—
19C2×#400—22no—
20#2315—21no—
21C2×#391C2212yes—
22C2×#382C2212yes—
23C2×#381—21no—
24C2×#381—21no—
25C2×#380C2212yes—
26C2×C4×C22⋊C4C2211yescharacteristic
27#2101C22116yescomplemented
28#2072C2218yescomplemented
29#2024C2218yescomplemented
30#1993C2218yescomplemented
31#1935C22116yescomplemented
32#1892C2218yes—
33#1872C2218yes—
34#1836C2218yescomplemented
35#1824C2214yes—
36#1823C2218yes—
37#1823C2218yes—
38#1787C2214yes—
39#1752C2218yes—
40#1749C2214yes—
41C2×#340C2218yes—
42C23×Q8C2311yescharacteristic
43C24×C4—21no—
44C24×C4C2311yescharacteristic
45C23×C8C2314yes—
46#543—48no—
47C2×C4⋊Q8C2314yes—
48C2×C22⋊Q8—24no—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—28no—
51C2×C22⋊Q8—28no—
52C2×C22⋊Q8C2314yes—
53C22×C4⋊C4—28no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2314yes—
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2314yes—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4D411yescharacteristic
64C22×C22⋊C4D411yescharacteristic
65C22×C42C2312yes—
66C2×#103C23116yes—
67C2×C22⋊C8—28no—
68C2×C22⋊C8C2314yes—
69C2×C22⋊C8C2314yes—
70C2×#97C2318yes—
71#400—216no—
72#398—28no—
73#395—28no—
74#391—24no—
75#391—24no—
76#391C2318yes—
77#389—24no—
78#389C2318yes—
79#385—28no—
80#384C2314yes—
81#382—24no—
82#382C2318yes—
83#381—28no—
84#381—28no—
85#380—24no—
86#380—22no—
87#380—22no—
88#380C2314yes—
89#380C2314yes—
90C4×C22⋊C4D414yes—
91C4×C22⋊C4C2314yes—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95D412yes—
98#340C23116yes—
99#340C23116yes—
100#340C23116yes—
101#336C2318yes—
102C25C2×D411yescharacteristic
103C22×Q8—21no—
104C22×Q8—24no—
105C22×Q8—24no—
106C22×Q8—44no—
107C22×Q8C2412yes—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—22no—
111C23×C4—24no—
112C23×C4—24no—
113C23×C4—44no—
114C23×C4—44no—
115C23×C4—44no—
116C23×C4—44no—
117C23×C4C2×D412yes—
118C23×C4C2×D412yes—
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D412yes—
121C23×C4#4118yes—
122C23×C4C2412yes—
123C23×C4C2411yescharacteristic
124C22×C8—28no—
125C22×C8—48no—
126C22×C8C2418yes—
127C4⋊Q8—48no—
128C22⋊Q8—28no—
129C22⋊Q8—28no—
130C22⋊Q8—432no—
131C22⋊Q8—416no—
132C22⋊Q8—48no—
133C2×C4⋊C4—28no—
134C2×C4⋊C4—28no—
135C2×C4⋊C4—28no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—28no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—48no—
147C2×C4⋊C4—44no—
148C2×C4⋊C4—48no—
149C2×C4⋊C4—44no—
150C2×C4⋊C4C2414yes—
151C2×C4⋊C4C2412yes—
152C2×C4⋊C4C2418yes—
153C2×C22⋊C4—24no—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—24no—
156C2×C22⋊C4—24no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—44no—
159C2×C22⋊C4—48no—
160C2×C22⋊C4C2×D414yes—
161C2×C22⋊C4C2×D414yes—
162C2×C42—44no—
163C2×C42C2×D414yes—
164C2×C42C2414yes—
165#103—416no—
166C22⋊C8—28no—
167C22⋊C8—28no—
168#97—48no—
169#95—24no—
170#95—24no—
171#95—28no—
172#95—28no—
173#95—28no—
174#95—28no—
175#95—28no—
176#95C2×D418yes—
177C24—42no—
178C24—42no—
179C24#12414yes—
180C24AGL(1,ℤ/8)14yes—
181C24#13714yes—
182C24C22×D412yes—
183C24C22×D411yescharacteristic
184C2×Q8—24no—
185C2×Q8—28no—
186C2×Q8—44no—
187C2×Q8—48no—
188C2×Q8—42no—
189C2×Q8—84no—
190C2×Q8—84no—
191C2×Q8—88no—
192C22×C4—24no—
193C22×C4—28no—
194C22×C4—24no—
195C22×C4—24no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—24no—
199C22×C4—28no—
200C22×C4—416no—
201C22×C4—44no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—44no—
208C22×C4—44no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—48no—
212C22×C4—44no—
213C22×C4—48no—
214C22×C4—44no—
215C22×C4—48no—
216C22×C4—44no—
217C22×C4—48no—
218C22×C4—84no—
219C22×C4—84no—
220C22×C4—816no—
221C22×C4#12414yes—
222C22×C4#12418yes—
223C22×C4C22×D414yes—
224C22×C4C22×D414yes—
225C22×C4C22×D411yescharacteristic
226C22×C4C22×D412yes—
227C22×C4C2×#4114yes—
228C22×C4C2×#4114yes—
229C22×C4C2×#4114yes—
230C22×C4C2511yescharacteristic
231C2×C8—28no—
232C2×C8—216no—
233C2×C8—88no—
234C4⋊C4—28no—
235C4⋊C4—24no—
236C4⋊C4—216no—
237C4⋊C4—48no—
238C4⋊C4—416no—
239C4⋊C4—48no—
240C4⋊C4—44no—
241C4⋊C4—42no—
242C4⋊C4—48no—
243C4⋊C4—84no—
244C4⋊C4—84no—
245C4⋊C4—84no—
246C22⋊C4—28no—
247C22⋊C4—28no—
248C22⋊C4—48no—
249C22⋊C4—48no—
250C22⋊C4—48no—
251C42—28no—
252C42—44no—
253C23—22no—
254C23—24no—
255C23—22no—
256C23—24no—
257C23—44no—
258C23—44no—
259C23—48no—
260C23—48no—
261C23#489116yes—
262C23C2×#12414yes—
263C23C2×#12412yes—
264C23C2×AGL(1,ℤ/8)14yes—
265C23C2×AGL(1,ℤ/8)12yes—
266C23C2×#13714yes—
267C23C2×#13712yes—
268C23C23×D411yescharacteristic
269Q8—48no—
270Q8—84no—
271Q8—168no—
272C2×C4—22no—
273C2×C4—24no—
274C2×C4—28no—
275C2×C4—28no—
276C2×C4—28no—
277C2×C4—24no—
278C2×C4—24no—
279C2×C4—24no—
280C2×C4—24no—
281C2×C4—24no—
282C2×C4—24no—
283C2×C4—44no—
284C2×C4—48no—
285C2×C4—48no—
286C2×C4—44no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—44no—
290C2×C4—48no—
291C2×C4—44no—
292C2×C4—48no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—84no—
296C2×C4—84no—
297C2×C4—84no—
298C2×C4—84no—
299C2×C4—816no—
300C2×C4—88no—
301C2×C4—168no—
302C2×C4C2×#12414yes—
303C2×C4C2×#12412yes—
304C2×C4C22×#4112yes—
305C2×C4#58514yes—
306C8—48no—
307C22—24no—
308C22—28no—
309C22—24no—
310C22—28no—
311C22—22no—
312C22—24no—
313C22—22no—
314C22—24no—
315C22—416no—
316C22C2×#48918yes—
317C22C2×#48914yes—
318C22C22×#12411yescharacteristic
319C22#348512yes—
320C22C22×AGL(1,ℤ/8)11yescharacteristic
321C22C22×#13711yescharacteristic
322C22#358612yes—
323C4—24no—
324C4—22no—
325C4—44no—
326C4—44no—
327C4—88no—
328C4—82no—
329C4—84no—
330C4—82no—
331C4#349114yes—
332C2—24no—
333C2—28no—
334C2#3545511yescharacteristic
335C2C22×#48911yescharacteristic
336C2#6062011yescharacteristic
337C2#6170412yes—
338C2#6170512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29111C2semidirect
4. #27969C2semidirect
5. #27713C2semidirect
27. #2101C22semidirect
28. #2072C22semidirect
29. #2024C22semidirect
30. #1993C22semidirect
31. #1935C22semidirect
34. #1836C22semidirect

Maximal subgroups

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

subgroupindex
#350022
C2×#25032
#291112
#279692
#277132
#217592
C2×#18232
#205262