Finite Group Databasebeta

← #6437289

Subgroups — #6437289

Order 512.

1512 conjugacy classes of subgroups.

The subgroup lattice has 357 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36645C211yesmaximal, characteristic
2#36640C211yesmaximal, characteristic
3#30307C214yesmaximal, complemented
4#30305C212yesmaximal, complemented
5#30304C212yesmaximal, complemented
6#30295C218yesmaximal
7#23841C218yesmaximal, complemented
8#20824C214yesmaximal
9#20364C211yesmaximal, characteristic
10C2×#533—21no—
11C2×C4×M16C2212yes—
12#2868C2212yes—
13#2864—22no—
14#2862—21no—
15#2859—21no—
16#2859C2211yescharacteristic
17#2859C2211yescharacteristic
18#2859C2212yes—
19#2859C2212yes—
20#2857—22no—
21#2857C2214yes—
22#2851—21no—
23#2851—21no—
24#2851C2214yes—
25C4×C4⋊Q8—22no—
26C4×C4⋊Q8C2214yes—
27C4×#125—21no—
28C4×#125—21no—
29C4×#125C2214yes—
30#2296—22no—
31#2285—21no—
32#2285C2212yes—
33#2285C2212yes—
34C2×C43C2211yescharacteristic
35#2085C22116yes—
36#2084C2218yes—
37#2083C2218yes—
38#2059C22116yes—
39#2058C2218yes—
40#2057C2218yes—
41#1845C2214yes—
42#1844C2214yes—
43#1834C2214yes—
44#1834C2214yes—
45#1809C22116yes—
46#1776C22116yes—
47#1745C2214yes—
48#1744C2212yes—
49#1744C2212yes—
50#1728C2214yes—
51C22×M16C2314yes—
52#533—48no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8C2314yes—
56C2×#125—21no—
57C2×#125—21no—
58C2×#125—22no—
59C2×#125C2314yes—
60C2×C22⋊Q8—24no—
61C2×C4×Q8—24no—
62C2×#117—22no—
63C2×#117—22no—
64C22×C42D411yescharacteristic
65C22×C42D411yescharacteristic
66C22×C42D412yes—
67C22×C42C2312yes—
68C22×C42C2311yescharacteristic
69C2×#107C23116yes—
70C2×#106C23116yes—
71C4×M16Q8116yescomplemented
72C2×#97C2314yes—
73C2×C4×C8C2314yes—
74#391—24no—
75#391—24no—
76#389—22no—
77#389—22no—
78#389—24no—
79#389—24no—
80#389—24no—
81#389—22no—
82#389—22no—
83#389—22no—
84#389—22no—
85#389C2314yes—
86#389C2314yes—
87#389C2318yes—
88#387—24no—
89#387—24no—
90#387—24no—
91#387—24no—
92#385—24no—
93#385—24no—
94#384—22no—
95#384—22no—
96#384C2314yes—
97#384C2312yes—
98#384C2312yes—
99#384C2314yes—
100#382—21no—
101#382—21no—
102#382—22no—
103#382—22no—
104#382—22no—
105#382C2314yes—
106#380—22no—
107#380—22no—
108#380—24no—
109#380C2314yes—
110#380C2314yes—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—24no—
113C4×C4⋊C4—24no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—24no—
116C4×C4⋊C4C2314yes—
117C4×C4⋊C4C2314yes—
118C43D411yescharacteristic
119C43D411yescharacteristic
120C43D412yes—
121C43C2312yes—
122C43C2311yescharacteristic
123C43C2311yescharacteristic
124#340C23116yes—
125#340C23116yes—
126#336C2314yes—
127#336C2314yes—
128#336C2318yes—
129C22×Q8—24no—
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D412yes—
133C23×C4C2×D412yes—
134C23×C4C2411yescharacteristic
135C2×M16—48no—
136C2×M16C2×Q8116yes—
137C22×C8C2418yes—
138C4⋊Q8—48no—
139C4⋊Q8—48no—
140C4⋊Q8—48no—
141#125—48no—
142#125—48no—
143#125—44no—
144#125—44no—
145C22⋊Q8—416no—
146C4×Q8—416no—
147#117—48no—
148#117—48no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—28no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4C2414yes—
164C2×C4⋊C4C2414yes—
165C2×C4⋊C4C2418yes—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—24no—
168C2×C42—24no—
169C2×C42—24no—
170C2×C42—24no—
171C2×C42—24no—
172C2×C42—44no—
173C2×C42—44no—
174C2×C42C2×D412yes—
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D412yes—
177C2×C42C2×D412yes—
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D412yes—
180C2×C42C2×D412yes—
181C2×C42C2×D412yes—
182C2×C42C2×D412yes—
183C2×C42C2×D412yes—
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2×Q818yes—
189C2×C42C2412yes—
190C2×C42C2412yes—
191C2×C42C2411yescharacteristic
192C2×C42C2411yescharacteristic
193#107—416no—
194#106—416no—
195#97C2×Q8116yes—
196C4×C8C2×Q8116yes—
197#95—28no—
198#95—24no—
199#95—24no—
200#95—24no—
201#95—24no—
202#95—24no—
203#95—24no—
204#95—24no—
205#95—24no—
206C24C22×D411yescharacteristic
207C2×Q8—48no—
208C2×Q8—48no—
209C2×Q8—48no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—28no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4C4⋊Q814yes—
224C22×C4C4⋊Q814yes—
225C22×C4C22×D412yes—
226C22×C4C22×D412yes—
227C22×C4C22×D411yescharacteristic
228C22×C4C22×D411yescharacteristic
229C22×C4C22×D411yescharacteristic
230C22×C4C22×D412yes—
231C22×C4C22×D412yes—
232C22×C4C22×D411yescharacteristic
233C22×C4C22×D411yescharacteristic
234C22×C4C22×Q814yes—
235C22×C4C2511yescharacteristic
236M16—416no—
237C2×C8—28no—
238C2×C8—28no—
239C2×C8C22×Q8116yes—
240C4⋊C4—28no—
241C4⋊C4—216no—
242C4⋊C4—28no—
243C4⋊C4—416no—
244C4⋊C4—48no—
245C4⋊C4—48no—
246C4⋊C4—48no—
247C4⋊C4—48no—
248C4⋊C4—48no—
249C4⋊C4—48no—
250C4⋊C4—48no—
251C4⋊C4—44no—
252C4⋊C4—48no—
253C4⋊C4—44no—
254C4⋊C4—44no—
255C4⋊C4—44no—
256C4⋊C4—48no—
257C22⋊C4—48no—
258C22⋊C4—48no—
259C42—24no—
260C42—22no—
261C42—24no—
262C42—22no—
263C42—24no—
264C42—24no—
265C42—24no—
266C42—24no—
267C42—42no—
268C42—42no—
269C42—42no—
270C42—42no—
271C42—48no—
272C42—48no—
273C42C4⋊Q814yes—
274C42C4⋊Q814yes—
275C42AGL(1,ℤ/8)14yes—
276C42#13714yes—
277C42C22×Q814yes—
278C42C22×Q814yes—
279C23—22no—
280C23—21no—
281C23—21no—
282C23—22no—
283C23C2×C4⋊Q812yes—
284C23C23×D411yescharacteristic
285Q8—816no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—28no—
295C2×C4—28no—
296C2×C4—28no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—48no—
304C2×C4—48no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—48no—
308C2×C4—48no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—48no—
314C2×C4C2×C4⋊Q812yes—
315C2×C4C2×C4⋊Q812yes—
316C2×C4C2×C4⋊Q812yes—
317C2×C4C2×C4⋊Q812yes—
318C2×C4C2×C4⋊Q812yes—
319C2×C4C2×AGL(1,ℤ/8)14yes—
320C2×C4C2×AGL(1,ℤ/8)12yes—
321C2×C4C2×#13714yes—
322C2×C4C2×#13712yes—
323C2×C4#57514yes—
324C2×C4C23×Q812yes—
325C2×C4#58514yes—
326C8—216no—
327C22—22no—
328C22—22no—
329C22—22no—
330C22—42no—
331C22—42no—
332C22C22×C4⋊Q811yescharacteristic
333C22#344912yes—
334C22C22×AGL(1,ℤ/8)11yescharacteristic
335C22C22×#13711yescharacteristic
336C22C2×#57512yes—
337C4—24no—
338C4—24no—
339C4—22no—
340C4—22no—
341C4—22no—
342C4—42no—
343C4—42no—
344C4—84no—
345C4—88no—
346C4—84no—
347C4#316514yes—
348C4#316614yes—
349C4#345814yes—
350C2—42no—
351C2#3464211yescharacteristic
352C2C2×#316511yescharacteristic
353C2C2×#316611yescharacteristic
354C2#6096512yes—
355C2#6096612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #30307C2semidirect
4. #30305C2semidirect
5. #30304C2semidirect
7. #23841C2semidirect
71. C4×M16Q8semidirect

Maximal subgroups

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

subgroupindex
#366452
#366402
#303072
#303052
#303042
#302952
#238412
#208242
#203642