Finite Group Databasebeta

← #6510705

Subgroups — #6510705

Order 512.

1804 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35002C211yesmaximal, characteristic
2C2×#2502C211yesmaximal, characteristic
3#29105C214yesmaximal, complemented
4#27961C214yesmaximal, complemented
5#27697C2116yesmaximal, complemented
6#21759C212yesmaximal
7C2×#1824C212yesmaximal
8#20523C211yesmaximal, characteristic
9C2×#543—21no—
10C22×C22⋊Q8C2211yescharacteristic
11C23×C4⋊C4C2211yescharacteristic
12C22×C22⋊C8C2212yes—
13#2857C2214yes—
14#2659—22no—
15#2549—24no—
16#2502—24no—
17#2502C2218yes—
18#2440C2218yes—
19C2×#398—22no—
20#2315—21no—
21C2×#391C2212yes—
22C2×#385—21no—
23C2×#385—21no—
24C2×#384C2212yes—
25C2×#380C2212yes—
26C2×C4×C22⋊C4C2211yescharacteristic
27#2100C22116yescomplemented
28#2072C2218yescomplemented
29#2024C2218yescomplemented
30#1990C2218yescomplemented
31#1933C22116yescomplemented
32#1892C2218yes—
33#1872C2218yes—
34#1834C2218yescomplemented
35#1824C2218yes—
36#1824C2214yes—
37#1824C2218yes—
38#1787C2214yes—
39C8×C22⋊C4C2218yes—
40C4×C22⋊C8C2214yes—
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⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2314yes—
62C22×C4⋊C4C2314yes—
63C22×C22⋊C4—22no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4D411yescharacteristic
66C22×C22⋊C4D411yescharacteristic
67C22×C42C2312yes—
68C2×#103C23116yes—
69C2×C22⋊C8—28no—
70C2×C22⋊C8C2314yes—
71C2×C22⋊C8C2314yes—
72C2×C4×C8C2318yes—
73#398—216no—
74#398—28no—
75#395—28no—
76#391—24no—
77#391—24no—
78#391C2318yes—
79#389—24no—
80#389C2318yes—
81#385—28no—
82#385—28no—
83#385—28no—
84#384—24no—
85#384C2314yes—
86#384C2318yes—
87#380—24no—
88#380—22no—
89#380—22no—
90#380C2314yes—
91#380C2314yes—
92C4×C22⋊C4D414yes—
93C4×C22⋊C4C2314yes—
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—84no—
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D412yes—
118C23×C4C2×D412yes—
119C23×C4C2×D412yes—
120C23×C4#4118yes—
121C23×C4C2412yes—
122C23×C4C2411yescharacteristic
123C22×C8—28no—
124C22×C8—48no—
125C22×C8C2418yes—
126C4⋊Q8—48no—
127C22⋊Q8—28no—
128C22⋊Q8—28no—
129C22⋊Q8—48no—
130C22⋊Q8—416no—
131C22⋊Q8—432no—
132C2×C4⋊C4—28no—
133C2×C4⋊C4—28no—
134C2×C4⋊C4—28no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—28no—
142C2×C4⋊C4—28no—
143C2×C4⋊C4—28no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—48no—
148C2×C4⋊C4—44no—
149C2×C4⋊C4—48no—
150C2×C4⋊C4—44no—
151C2×C4⋊C4—84no—
152C2×C4⋊C4—84no—
153C2×C4⋊C4C2414yes—
154C2×C4⋊C4C2412yes—
155C2×C4⋊C4C2418yes—
156C2×C22⋊C4—24no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—44no—
160C2×C22⋊C4—48no—
161C2×C22⋊C4SD1618yes—
162C2×C22⋊C4Q1618yes—
163C2×C22⋊C4C2×D414yes—
164C2×C22⋊C4C2×D414yes—
165C2×C42—44no—
166C2×C42C2×D414yes—
167C2×C42C2414yes—
168#103—416no—
169C22⋊C8—28no—
170C22⋊C8—28no—
171C4×C8—48no—
172#95—24no—
173#95—24no—
174#95—28no—
175#95—28no—
176#95—28no—
177#95C2×D418yes—
178C24—42no—
179C24—42no—
180C24#12414yes—
181C24C2×SD1614yes—
182C24C2×Q1614yes—
183C24C22×D412yes—
184C24C22×D411yescharacteristic
185C2×Q8—24no—
186C2×Q8—28no—
187C2×Q8—44no—
188C2×Q8—42no—
189C2×Q8—48no—
190C2×Q8—84no—
191C2×Q8—84no—
192C2×Q8—88no—
193C22×C4—24no—
194C22×C4—28no—
195C22×C4—22no—
196C22×C4—28no—
197C22×C4—24no—
198C22×C4—22no—
199C22×C4—416no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—44no—
205C22×C4—42no—
206C22×C4—44no—
207C22×C4—44no—
208C22×C4—44no—
209C22×C4—44no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—44no—
213C22×C4—48no—
214C22×C4—48no—
215C22×C4—88no—
216C22×C4—84no—
217C22×C4—84no—
218C22×C4—816no—
219C22×C4#12414yes—
220C22×C4#12418yes—
221C22×C4C2×SD1618yes—
222C22×C4C2×Q1618yes—
223C22×C4C22×D411yescharacteristic
224C22×C4C22×D412yes—
225C22×C4C22×D414yes—
226C22×C4C22×D414yes—
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—44no—
239C4⋊C4—42no—
240C4⋊C4—48no—
241C4⋊C4—48no—
242C4⋊C4—416no—
243C4⋊C4—84no—
244C4⋊C4—84no—
245C4⋊C4—88no—
246C4⋊C4—88no—
247C4⋊C4—84no—
248C22⋊C4—48no—
249C22⋊C4—48no—
250C22⋊C4—48no—
251C22⋊C4C2×SD16116yes—
252C22⋊C4C2×Q16116yes—
253C42—28no—
254C42—44no—
255C23—22no—
256C23—24no—
257C23—22no—
258C23—24no—
259C23—44no—
260C23—44no—
261C23—48no—
262C23—48no—
263C23#487116yes—
264C23C2×#12414yes—
265C23C2×#12412yes—
266C23C22×SD1612yes—
267C23C22×SD1614yes—
268C23C22×Q1614yes—
269C23C22×Q1612yes—
270C23C23×D411yescharacteristic
271Q8—48no—
272Q8—84no—
273Q8—168no—
274C2×C4—24no—
275C2×C4—24no—
276C2×C4—24no—
277C2×C4—24no—
278C2×C4—22no—
279C2×C4—24no—
280C2×C4—28no—
281C2×C4—28no—
282C2×C4—28no—
283C2×C4—44no—
284C2×C4—48no—
285C2×C4—48no—
286C2×C4—48no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—48no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—44no—
294C2×C4—48no—
295C2×C4—44no—
296C2×C4—48no—
297C2×C4—84no—
298C2×C4—88no—
299C2×C4—84no—
300C2×C4—816no—
301C2×C4—168no—
302C2×C4C2×#12414yes—
303C2×C4C2×#12412yes—
304C2×C4C22×SD1618yes—
305C2×C4C22×Q1618yes—
306C2×C4C22×#4112yes—
307C2×C4#58514yes—
308C8—48no—
309C22—24no—
310C22—28no—
311C22—24no—
312C22—28no—
313C22—22no—
314C22—24no—
315C22—22no—
316C22—24no—
317C22—416no—
318C22C2×#48718yes—
319C22C2×#48714yes—
320C22C22×#12411yescharacteristic
321C22#348512yes—
322C22C23×SD1611yescharacteristic
323C22C23×Q1611yescharacteristic
324C22#358512yes—
325C4—28no—
326C4—28no—
327C4—24no—
328C4—22no—
329C4—88no—
330C4—82no—
331C4—82no—
332C4—84no—
333C4#349114yes—
334C2—24no—
335C2—28no—
336C2#3545511yescharacteristic
337C2C22×#48711yescharacteristic
338C2#6061911yescharacteristic
339C2#6169712yes—
340C2#6169812yes—
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. #29105C2semidirect
4. #27961C2semidirect
5. #27697C2semidirect
27. #2100C22semidirect
28. #2072C22semidirect
29. #2024C22semidirect
30. #1990C22semidirect
31. #1933C22semidirect
34. #1834C22semidirect

Maximal subgroups

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

subgroupindex
#350022
C2×#25022
#291052
#279612
#276972
#217592
C2×#18242
#205232