Finite Group Databasebeta

← #6528422

Subgroups — #6528422

Order 512.

1594 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#35167C211yesmaximal, characteristic
2C2×#2552C211yesmaximal, characteristic
3#29297C214yesmaximal
4#28829C218yesmaximal, complemented
5#28791C218yesmaximal, complemented
6#27948C214yesmaximal
7#21981C212yesmaximal
8#21359C212yesmaximal
9#20533C211yesmaximal, characteristic
10C2×#544—21no—
11C22×#125—21no—
12C2×#431C2212yes—
13#2857C2214yes—
14#2659—22no—
15#2552—24no—
16#2552C2214yes—
17#2552C2218yes—
18#2551C2214yes—
19#2550—24no—
20C2×#398—22no—
21#2316—21no—
22C2×#389—21no—
23C2×#389C2212yes—
24#2285C2212yes—
25C2×#384C2212yes—
26C2×#384C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29#2084C22116yes—
30#2084C2218yes—
31#2058C2218yes—
32#1990C2218yes—
33#1948C2214yes—
34#1948C2214yes—
35#1948C2218yes—
36#1941C22116yes—
37#1853C2214yes—
38#1853C2218yes—
39#1853C2214yes—
40#1844C2218yes—
41#1844C2214yes—
42#1834C2214yes—
43C4×#106C2218yes—
44C8×C4⋊C4C2218yes—
45C4×C22⋊C8C2214yes—
46#544—48no—
47C2×C4⋊Q8—24no—
48C2×C4⋊Q8C2314yes—
49C2×#125—28no—
50C2×#125—24no—
51C2×#125—48no—
52C2×C22⋊Q8—24no—
53C2×#117—22no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C42—21no—
65C22×C42C2312yes—
66C22×C42C2311yescharacteristic
67#431Q8116yescomplemented
68C2×#106C2318yes—
69C2×#106C23116yes—
70C2×#106C2318yes—
71C2×C4⋊C8C2314yes—
72C2×C22⋊C8C2314yes—
73C2×C4×C8C2318yes—
74C2×C4×C8C2314yes—
75#398—216no—
76#398—28no—
77#395—28no—
78#391—24no—
79#389—24no—
80#389—24no—
81#389—28no—
82#389—24no—
83#389—24no—
84#389C2318yes—
85#389C2318yes—
86#384—24no—
87#384—22no—
88#384C2314yes—
89#384C2318yes—
90#384C2312yes—
91#384C2314yes—
92#384C2312yes—
93#382—22no—
94#380—24no—
95#380—22no—
96#380C2314yes—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4D414yes—
100C4×C4⋊C4C2314yes—
101C4×C4⋊C4C2314yes—
102C4×C4⋊C4C2314yes—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95D412yes—
106#340C23116yes—
107#340C2318yes—
108#340C2318yes—
109#336C2318yes—
110C22×Q8—24no—
111C23×C4—22no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D412yes—
116C23×C4C2×D412yes—
117C23×C4C2×D411yescharacteristic
118C23×C4C2411yescharacteristic
119C22×C8C2418yes—
120C4⋊Q8—48no—
121C4⋊Q8—416no—
122#125—432no—
123#125—416no—
124C22⋊Q8—416no—
125#117—48no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—28no—
128C2×C4⋊C4—28no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—28no—
136C2×C4⋊C4—28no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4—44no—
145C2×C4⋊C4—48no—
146C2×C4⋊C4—48no—
147C2×C4⋊C4—44no—
148C2×C4⋊C4—82no—
149C2×C4⋊C4—84no—
150C2×C4⋊C4SD1618yes—
151C2×C4⋊C4SD1618yes—
152C2×C4⋊C4C2×D414yes—
153C2×C4⋊C4C2×D414yes—
154C2×C4⋊C4C2412yes—
155C2×C4⋊C4C2412yes—
156C2×C4⋊C4C2414yes—
157C2×C4⋊C4C2414yes—
158C2×C4⋊C4C2414yes—
159C2×C22⋊C4—24no—
160C2×C42—24no—
161C2×C42—24no—
162C2×C42—44no—
163C2×C42—44no—
164C2×C42C2×D414yes—
165C2×C42C2×D412yes—
166C2×C42C2×Q818yes—
167C2×C42C2414yes—
168C2×C42C2412yes—
169#106—416no—
170#106—48no—
171#106—48no—
172C4⋊C8C2×Q8116yes—
173C22⋊C8C2×Q8116yes—
174C4×C8—48no—
175C4×C8C2×Q8116yes—
176#95—24no—
177#95—28no—
178#95—24no—
179#95—24no—
180#95—28no—
181#95C2×D418yes—
182C24C22×D411yescharacteristic
183C2×Q8—48no—
184C2×Q8—84no—
185C2×Q8—84no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—24no—
190C22×C4—24no—
191C22×C4—22no—
192C22×C4—24no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—41no—
196C22×C4—42no—
197C22×C4—41no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—44no—
201C22×C4—42no—
202C22×C4—44no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—44no—
208C22×C4C4⋊Q814yes—
209C22×C4C4⋊Q814yes—
210C22×C4C2×SD1618yes—
211C22×C4C2×SD1618yes—
212C22×C4C2×SD1614yes—
213C22×C4C2×SD1614yes—
214C22×C4C22×D412yes—
215C22×C4C22×D411yescharacteristic
216C22×C4C22×D414yes—
217C22×C4C22×D414yes—
218C22×C4C22×D412yes—
219C22×C4C22×Q814yes—
220C22×C4C2511yescharacteristic
221C2×C8—28no—
222C2×C8—28no—
223C2×C8C22×Q8116yes—
224C4⋊C4—28no—
225C4⋊C4—24no—
226C4⋊C4—24no—
227C4⋊C4—28no—
228C4⋊C4—28no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—44no—
232C4⋊C4—416no—
233C4⋊C4—416no—
234C4⋊C4—48no—
235C4⋊C4—44no—
236C4⋊C4—44no—
237C4⋊C4—48no—
238C4⋊C4—416no—
239C4⋊C4—82no—
240C4⋊C4—82no—
241C4⋊C4—84no—
242C4⋊C4—84no—
243C4⋊C4—88no—
244C4⋊C4—84no—
245C4⋊C4C2×SD16116yes—
246C4⋊C4C2×SD16116yes—
247C22⋊C4—48no—
248C42—28no—
249C42—44no—
250C42—48no—
251C42—48no—
252C42C4⋊Q818yes—
253C42C22×Q814yes—
254C42C22×Q814yes—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23C2×C4⋊Q812yes—
260C23C22×SD1612yes—
261C23C22×SD1612yes—
262C23C23×D411yescharacteristic
263Q8—168no—
264C2×C4—22no—
265C2×C4—24no—
266C2×C4—22no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—28no—
273C2×C4—44no—
274C2×C4—48no—
275C2×C4—48no—
276C2×C4—42no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—44no—
283C2×C4—42no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—84no—
292C2×C4—88no—
293C2×C4—84no—
294C2×C4#498116yes—
295C2×C4C2×C4⋊Q812yes—
296C2×C4C2×C4⋊Q814yes—
297C2×C4C2×C4⋊Q814yes—
298C2×C4C22×SD1618yes—
299C2×C4C22×SD1618yes—
300C2×C4C22×SD1614yes—
301C2×C4C22×SD1614yes—
302C2×C4C23×Q812yes—
303C2×C4#58512yes—
304C2×C4#58512yes—
305C8—216no—
306C22—22no—
307C22—22no—
308C22—22no—
309C22—22no—
310C22—22no—
311C22—22no—
312C22C2×#49814yes—
313C22C22×C4⋊Q811yescharacteristic
314C22#348511yescharacteristic
315C22#348711yescharacteristic
316C22C23×SD1611yescharacteristic
317C22C23×SD1611yescharacteristic
318C22#358511yescharacteristic
319C22#358611yescharacteristic
320C4—28no—
321C4—22no—
322C4—24no—
323C4—28no—
324C4—84no—
325C4—82no—
326C4—84no—
327C4—84no—
328C4—82no—
329C4C2×#49818yes—
330C4#350414yes—
331C2—24no—
332C2#3564011yescharacteristic
333C2C22×#49811yescharacteristic
334C2#6066411yescharacteristic
335C2#6169511yescharacteristic
336C2#6169711yescharacteristic
337C2#6188111yescharacteristic
338C2#6188311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #28829C2semidirect
5. #28791C2semidirect
67. #431Q8semidirect

Maximal subgroups

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

subgroupindex
#351672
C2×#25522
#292972
#288292
#287912
#279482
#219812
#213592
#205332