Finite Group Databasebeta

← #6435613

Subgroups — #6435613

Order 512.

1672 conjugacy classes of subgroups.

The subgroup lattice has 387 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34638C211yesmaximal, characteristic, complemented
2C2×#2319C211yesmaximal, characteristic
3#30003C218yesmaximal, complemented
4#30002C218yesmaximal, complemented
5#23670C214yesmaximal, complemented
6#23618C214yesmaximal
7#21170C212yesmaximal, complemented
8#20792C211yesmaximal, characteristic, complemented
9#20646C212yesmaximal
10C22×C4×D4C2211yescharacteristic
11C2×#428C2214yes—
12C2×#423C2212yes—
13C2×#423C2214yes—
14C2×#418C2214yes—
15#2647C2214yescomplemented
16#2647C2214yescomplemented
17#2646—21no—
18#2646—21no—
19#2646—22no—
20#2402—24no—
21#2319—24no—
22#2319C2218yes—
23#2296C2214yes—
24C2×#390C2212yes—
25C2×#389C2212yes—
26C2×#388—22no—
27#2285C2212yes—
28C2×#381—21no—
29C2×#380—21no—
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×#376C2211yescharacteristic
32#2088C2218yescomplemented
33#2088C2218yescomplemented
34#2087C2218yescomplemented
35#2086C2218yescomplemented
36#2067C2218yes—
37#2067C2218yes—
38#2065C2218yes—
39#2065C2218yes—
40#1949C2218yescomplemented
41#1948C2214yescomplemented
42#1947C2214yescomplemented
43#1843C2218yescomplemented
44#1804C2218yes—
45#1802C2214yes—
46#1802C2214yes—
47#1736C2214yes—
48#1736C2214yes—
49C23×D4C2311yescharacteristic
50C24×C4—21no—
51C22×M16C2314yes—
52C2×C4×D4—22no—
53C2×C4×D4—24no—
54C2×C4×D4Q818yes—
55C2×C4×D4C2314yes—
56C2×#117C2314yes—
57C22×C4⋊C4—22no—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2311yescharacteristic
60C22×C22⋊C4—21no—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—22no—
63C22×C42—21no—
64C22×C42C2312yes—
65C22×C42C2311yescharacteristic
66#428—48no—
67C2×#107C2318yes—
68C2×#106C2318yes—
69#423—28no—
70#423—48no—
71C2×C4⋊C8C2314yes—
72C2×C4⋊C8C2314yes—
73C2×C4⋊C8C2318yes—
74C2×C4⋊C8C2318yes—
75#418—48no—
76C2×#102C2318yes—
77C2×#102C2318yes—
78#400—24no—
79#400—24no—
80#400—24no—
81#400—24no—
82#398—24no—
83#398—24no—
84#398—24no—
85#398—24no—
86#390—24no—
87#390C2318yes—
88#389—24no—
89#389C2318yes—
90#389C2314yes—
91#389C2314yes—
92#389C2314yes—
93#389C2314yes—
94#388—28no—
95#388—28no—
96#385—24no—
97#385—24no—
98#384C2312yes—
99#384C2312yes—
100#382C2314yes—
101#381—28no—
102#380—22no—
103#380—22no—
104#380—28no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—24no—
109C4×C4⋊C4—24no—
110C4×C4⋊C4C2314yes—
111#376D414yes—
112#376C2314yes—
113C2×#95—21no—
114C2×#95—22no—
115C2×#95D411yescharacteristic
116C2×#95D411yescharacteristic
117C2×#95D412yes—
118#340C2318yes—
119#340C2318yes—
120#340C2318yes—
121#340C2318yes—
122C25—21no—
123C22×D4—21no—
124C22×D4—22no—
125C22×D4—24no—
126C22×D4—24no—
127C22×D4C2×Q814yes—
128C22×D4C2412yes—
129C23×C4—21no—
130C23×C4—28no—
131C23×C4—24no—
132C23×C4—22no—
133C23×C4—84no—
134C23×C4C2×D412yes—
135C23×C4C2×D412yes—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2411yescharacteristic
139C2×M16—28no—
140C2×M16—28no—
141C2×M16—28no—
142C22×C8C2418yes—
143C4×D4—216no—
144C4×D4—28no—
145C4×D4C2×Q8116yes—
146#117—28no—
147#117—28no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—28no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—84no—
163C2×C4⋊C4C2×Q814yes—
164C2×C4⋊C4C2414yes—
165C2×C4⋊C4C2414yes—
166C2×C4⋊C4C2414yes—
167C2×C4⋊C4C2412yes—
168C2×C22⋊C4—28no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—28no—
172C2×C22⋊C4—28no—
173C2×C22⋊C4—84no—
174C2×C22⋊C4—84no—
175C2×C42—24no—
176C2×C42—24no—
177C2×C42—24no—
178C2×C42—24no—
179C2×C42—82no—
180C2×C42C2×D414yes—
181C2×C42C2×D412yes—
182C2×C42C2×Q814yes—
183C2×C42#4114yes—
184C2×C42C2414yes—
185C2×C42C2412yes—
186#107—48no—
187#106—48no—
188C4⋊C8—28no—
189C4⋊C8—28no—
190C4⋊C8—48no—
191C4⋊C8—48no—
192#102—48no—
193#102—48no—
194#95—24no—
195#95—28no—
196#95—24no—
197#95—24no—
198#95—24no—
199#95—24no—
200#95—24no—
201#95—24no—
202#95C2×D414yes—
203#95C2×D414yes—
204#95C2×D414yes—
205#95C2×D414yes—
206C24—24no—
207C24—22no—
208C24—42no—
209C24—44no—
210C24—42no—
211C24—44no—
212C24C22×D411yescharacteristic
213C2×D4—24no—
214C2×D4—216no—
215C2×D4—22no—
216C2×D4—24no—
217C2×D4—24no—
218C2×D4—44no—
219C2×D4—44no—
220C2×D4—44no—
221C2×D4C22×Q814yes—
222C22×C4—24no—
223C22×C4—22no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—216no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—41no—
231C22×C4—42no—
232C22×C4—41no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—44no—
238C22×C4—44no—
239C22×C4—44no—
240C22×C4—44no—
241C22×C4—84no—
242C22×C4—84no—
243C22×C4—816no—
244C22×C4C22⋊Q814yes—
245C22×C4C22⋊Q814yes—
246C22×C4C22×D414yes—
247C22×C4C22×D414yes—
248C22×C4C22×D412yes—
249C22×C4C22×D412yes—
250C22×C4C22×D411yescharacteristic
251C22×C4C22×Q812yes—
252C22×C4C2×#4112yes—
253C22×C4H(5,𝔽2)14yes—
254C22×C42−1+414yes—
255C22×C4C2511yescharacteristic
256M16—416no—
257C2×C8—28no—
258C2×C8—28no—
259C2×C8—28no—
260C4⋊C4—24no—
261C4⋊C4—22no—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C4⋊C4—44no—
269C4⋊C4—82no—
270C4⋊C4—82no—
271C4⋊C4—82no—
272C4⋊C4—82no—
273C4⋊C4—84no—
274C4⋊C4—84no—
275C4⋊C4—84no—
276C4⋊C4—84no—
277C4⋊C4—88no—
278C4⋊C4C22×Q814yes—
279C22⋊C4—216no—
280C22⋊C4—48no—
281C22⋊C4—88no—
282C22⋊C4—88no—
283C42—28no—
284C42—24no—
285C42—24no—
286C42—48no—
287C42—84no—
288C42C22⋊Q818yes—
289C42C22×Q814yes—
290C42C2×#4114yes—
291C23—22no—
292C23—21no—
293C23—21no—
294C23—24no—
295C23—48no—
296C23—48no—
297C23—42no—
298C23—44no—
299C23—42no—
300C23—48no—
301C23—44no—
302C23—88no—
303C23—88no—
304C23C2×C22⋊Q812yes—
305C23#53714yes—
306C23C23×D411yescharacteristic
307D4—28no—
308D4—48no—
309D4—42no—
310D4—42no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—48no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—816no—
336C2×C4—84no—
337C2×C4—84no—
338C2×C4—88no—
339C2×C4C2×C22⋊Q814yes—
340C2×C4C2×C22⋊Q814yes—
341C2×C4C2×C22⋊Q812yes—
342C2×C4#53714yes—
343C2×C4#54014yes—
344C2×C4#54114yes—
345C2×C4#54418yes—
346C2×C4C23×Q811yescharacteristic
347C2×C4C22×#4111yescharacteristic
348C2×C4C2×H(5,𝔽2)12yes—
349C2×C4C2×2−1+412yes—
350C8—48no—
351C22—22no—
352C22—22no—
353C22—24no—
354C22—24no—
355C22—44no—
356C22—44no—
357C22—44no—
358C22—816no—
359C22—84no—
360C22—84no—
361C22#262914yes—
362C22C22×C22⋊Q811yescharacteristic
363C22C2×#53712yes—
364C22#358512yes—
365C22#358511yescharacteristic
366C22#358611yescharacteristic
367C4—24no—
368C4—24no—
369C4—44no—
370C4—44no—
371C4—44no—
372C4—84no—
373C4—84no—
374C4—84no—
375C4#262914yes—
376C4C2×#54011yescharacteristic
377C4C2×#54111yescharacteristic
378C4C2×#54412yes—
379C2—24no—
380C2—84no—
381C2C2×#262911yescharacteristic
382C2#6049011yescharacteristic
383C2#6049211yescharacteristic
384C2#6091912yes—
385C2#6092012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #34638C2semidirect
3. #30003C2semidirect
4. #30002C2semidirect
5. #23670C2semidirect
7. #21170C2semidirect
8. #20792C2semidirect
15. #2647C22semidirect
16. #2647C22semidirect
32. #2088C22semidirect
33. #2088C22semidirect
34. #2087C22semidirect
35. #2086C22semidirect
40. #1949C22semidirect
41. #1948C22semidirect
42. #1947C22semidirect
43. #1843C22semidirect

Maximal subgroups

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

subgroupindex
#346382
C2×#23192
#300032
#300022
#236702
#236182
#211702
#207922
#206462