Finite Group Databasebeta

← #6477791

Subgroups — #6477791

Order 512.

1550 conjugacy classes of subgroups.

The subgroup lattice has 377 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35020C211yesmaximal, characteristic
2C2×#2347C211yesmaximal, characteristic
3#33150C214yesmaximal, complemented
4#29972C214yesmaximal, complemented
5#26015C218yesmaximal, complemented
6#26013C218yesmaximal, complemented
7#21942C212yesmaximal
8#21934C211yesmaximal, characteristic
9#20702C212yesmaximal
10C22×#117C2211yescharacteristic
11C2×#407C2212yes—
12#2851C2214yes—
13#2646—21no—
14#2646—21no—
15#2569—24no—
16#2442C2214yes—
17#2442C2214yes—
18#2347—24no—
19#2347C2218yes—
20#2315—22no—
21C2×#389C2212yes—
22C2×#388—22no—
23C2×#383—21no—
24C2×#380C2211yescharacteristic
25C2×#380C2212yes—
26C2×C4×C4⋊C4—21no—
27C2×C4×C4⋊C4C2212yes—
28C2×C4×C22⋊C4C2211yescharacteristic
29#2101C2218yescomplemented
30#2088C2218yescomplemented
31#2087C2218yescomplemented
32#2059C2218yescomplemented
33#1989C2218yescomplemented
34#1949C2218yescomplemented
35#1920C2218yescomplemented
36#1918C2218yescomplemented
37#1853C2214yes—
38#1853C2214yes—
39#1852C2214yes—
40#1852C2214yes—
41#1851C2218yes—
42#1825C2218yes—
43#1824C2214yes—
44#1823C2214yes—
45#1809C2218yes—
46#1797C2214yes—
47#1736C2214yes—
48#1736C2214yes—
49C24×C4C2311yescharacteristic
50C22×M16C2314yes—
51C2×#125C2314yes—
52C2×#117—22no—
53C2×#117—24no—
54C2×#117Q818yes—
55C2×#117C2314yes—
56C22×C4⋊C4—22no—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2312yes—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—22no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4D412yes—
66C22×C42—21no—
67C22×C42—22no—
68C22×C42C2312yes—
69C2×#107C2318yes—
70C2×#106C2318yes—
71C2×C4⋊C8C2318yes—
72#407—28no—
73C2×C22⋊C8C2314yes—
74C2×C22⋊C8C2314yes—
75#400—24no—
76#400—24no—
77#398—24no—
78#398—24no—
79#389—24no—
80#389C2318yes—
81#389C2314yes—
82#389C2314yes—
83#388—28no—
84#388—28no—
85#387—24no—
86#387—24no—
87#385—24no—
88#385—24no—
89#385—24no—
90#385—24no—
91#383—28no—
92#382C2314yes—
93#380—22no—
94#380—22no—
95#380—22no—
96#380—24no—
97#380—24no—
98#380C2314yes—
99#380C2314yes—
100#380C2314yes—
101C4×C4⋊C4—24no—
102C4×C4⋊C4—28no—
103C4×C4⋊C4C2318yes—
104C4×C22⋊C4D414yes—
105C4×C22⋊C4C2314yes—
106C2×#95—22no—
107C2×#95—21no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95D411yescharacteristic
111C2×#95D411yescharacteristic
112#340C2318yes—
113#340C2318yes—
114#340C2318yes—
115#340C2318yes—
116#340C2318yes—
117#340C2318yes—
118#336C2318yes—
119C25C2×D411yescharacteristic
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—44no—
126C23×C4C2×D412yes—
127C23×C4C2×D412yes—
128C23×C4C2×D412yes—
129C23×C4C2×D411yescharacteristic
130C23×C4C2×Q814yes—
131C23×C4#4114yes—
132C23×C4C2411yescharacteristic
133C23×C4C2412yes—
134C2×M16—28no—
135C2×M16—44no—
136C2×M16—44no—
137C22×C8C2418yes—
138#125—48no—
139#117—216no—
140#117—28no—
141#117C2×Q8116yes—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—28no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—48no—
158C2×C4⋊C4C2×Q814yes—
159C2×C4⋊C4C2×Q814yes—
160C2×C4⋊C4C2414yes—
161C2×C4⋊C4C2414yes—
162C2×C4⋊C4C2414yes—
163C2×C4⋊C4C2412yes—
164C2×C4⋊C4C2412yes—
165C2×C22⋊C4—28no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—44no—
171C2×C22⋊C4—48no—
172C2×C22⋊C4—82no—
173C2×C22⋊C4C2×D414yes—
174C2×C22⋊C4C2×D414yes—
175C2×C42—28no—
176C2×C42—24no—
177C2×C42—28no—
178C2×C42—44no—
179C2×C42—84no—
180C2×C42C2×D414yes—
181C2×C42C2414yes—
182#107—48no—
183#106—48no—
184C4⋊C8—48no—
185C22⋊C8—28no—
186C22⋊C8—28no—
187#95—24no—
188#95—28no—
189#95—24no—
190#95—24no—
191#95—28no—
192#95—24no—
193#95—24no—
194#95—24no—
195#95—24no—
196#95C2×D414yes—
197#95C2×D414yes—
198C24—24no—
199C24—41no—
200C24—42no—
201C24—41no—
202C24C22⋊Q814yes—
203C24C22×D412yes—
204C24C22×D411yescharacteristic
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—48no—
218C22×C4—48no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—44no—
229C22×C4—44no—
230C22×C4—44no—
231C22×C4—44no—
232C22×C4—44no—
233C22×C4—44no—
234C22×C4—82no—
235C22×C4—82no—
236C22×C4—82no—
237C22×C4—82no—
238C22×C4C22⋊Q814yes—
239C22×C4C22⋊Q818yes—
240C22×C4C22×D414yes—
241C22×C4C22×D414yes—
242C22×C4C22×D412yes—
243C22×C4C22×D411yescharacteristic
244C22×C4C22×Q812yes—
245C22×C4C22×Q814yes—
246C22×C4C2×#4112yes—
247C22×C4C2×#4114yes—
248C22×C4C2511yescharacteristic
249M16—88no—
250C2×C8—28no—
251C2×C8—28no—
252C2×C8—28no—
253C4⋊C4—24no—
254C4⋊C4—24no—
255C4⋊C4—22no—
256C4⋊C4—22no—
257C4⋊C4—28no—
258C4⋊C4—28no—
259C4⋊C4—28no—
260C4⋊C4—44no—
261C4⋊C4—416no—
262C4⋊C4—84no—
263C4⋊C4—84no—
264C4⋊C4—82no—
265C4⋊C4—82no—
266C4⋊C4—84no—
267C4⋊C4—84no—
268C4⋊C4C22×Q814yes—
269C4⋊C4C22×Q814yes—
270C22⋊C4—216no—
271C22⋊C4—44no—
272C22⋊C4—44no—
273C22⋊C4—48no—
274C22⋊C4—48no—
275C22⋊C4—84no—
276C42—216no—
277C42—28no—
278C42—44no—
279C42—88no—
280C23—21no—
281C23—21no—
282C23—22no—
283C23—22no—
284C23—22no—
285C23—22no—
286C23—22no—
287C23—22no—
288C23—22no—
289C23—28no—
290C23—44no—
291C23—44no—
292C23—84no—
293C23—84no—
294C23C2×C22⋊Q814yes—
295C23C2×C22⋊Q812yes—
296C23#57514yes—
297C23C23×D411yescharacteristic
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—24no—
302C2×C4—28no—
303C2×C4—28no—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—44no—
314C2×C4—48no—
315C2×C4—48no—
316C2×C4—48no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—48no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—84no—
331C2×C4—88no—
332C2×C4—164no—
333C2×C4—164no—
334C2×C4C2×C22⋊Q814yes—
335C2×C4C2×C22⋊Q812yes—
336C2×C4C23×Q811yescharacteristic
337C2×C4C22×#4111yescharacteristic
338C2×C4#58514yes—
339C8—48no—
340C22—24no—
341C22—22no—
342C22—24no—
343C22—22no—
344C22—24no—
345C22—24no—
346C22—24no—
347C22—42no—
348C22—42no—
349C22—42no—
350C22—42no—
351C22—88no—
352C22#308014yes—
353C22C22×C22⋊Q811yescharacteristic
354C22#348512yes—
355C22C2×#57512yes—
356C22#358511yescharacteristic
357C22#358611yescharacteristic
358C4—22no—
359C4—24no—
360C4—44no—
361C4—44no—
362C4—44no—
363C4—44no—
364C4—84no—
365C4—84no—
366C4—84no—
367C4#345812yes—
368C4#349112yes—
369C2—24no—
370C2—44no—
371C2#3509011yescharacteristic
372C2C2×#308011yescharacteristic
373C2#6049211yescharacteristic
374C2#6171212yes—
375C2#6171812yes—
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. #33150C2semidirect
4. #29972C2semidirect
5. #26015C2semidirect
6. #26013C2semidirect
29. #2101C22semidirect
30. #2088C22semidirect
31. #2087C22semidirect
32. #2059C22semidirect
33. #1989C22semidirect
34. #1949C22semidirect
35. #1920C22semidirect
36. #1918C22semidirect

Maximal subgroups

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

subgroupindex
#350202
C2×#23472
#331502
#299722
#260152
#260132
#219422
#219342
#207022