Finite Group Databasebeta

← #6561546

Subgroups — #6561546

Order 512.

1408 conjugacy classes of subgroups.

The subgroup lattice has 337 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36593C211yesmaximal, characteristic
2#36447C211yesmaximal, characteristic
3#30304C213yesmaximal
4#29319C213yesmaximal
5#29213C213yesmaximal
6#28835C216yesmaximal
7#28798C216yesmaximal
8#23838C213yesmaximal
9#22738C212yesmaximal
10Q8×#107C212yesmaximal
11#20572C211yesmaximal, characteristic
12C2×#557—21no—
13C2×C8×Q8C2212yes—
14#2868C2213yes—
15#2859C2213yes—
16#2571C2213yes—
17#2552—23no—
18#2552C2216yes—
19#2552C2213yes—
20#2546—23no—
21#2508—23no—
22#2506—23no—
23#2351—21no—
24Q8×C4⋊C4—21no—
25Q8×C4⋊C4—21no—
26Q8×C4⋊C4C2211yescharacteristic
27Q8×C4⋊C4C2211yescharacteristic
28Q8×C4⋊C4C2212yes—
29C4×C4⋊Q8C2213yes—
30C4×#125C2213yes—
31C42×Q8C2211yescharacteristic
32#2085C2216yes—
33#2084C2213yes—
34#2083C2211yescharacteristic
35#2083C22112yes—
36#2059C2216yes—
37#2058C2213yes—
38#2057C2216yes—
39#2057C2213yes—
40#1992C2216yes—
41#1948C2216yes—
42#1948C2216yes—
43#1947C2216yes—
44#1947C2216yes—
45#1943C22112yes—
46#1847C2216yes—
47#1845C2216yes—
48#1834C2216yes—
49C4×#107C2216yes—
50C4×#106C2216yes—
51#1770C2216yes—
52#1767C2213yes—
53#1744C2213yes—
54#557—48no—
55C2×C4⋊Q8—23no—
56C2×C4⋊Q8—23no—
57C2×C4⋊Q8C2313yes—
58C2×#125—26no—
59C2×#125—23no—
60C2×#125C2313yes—
61C2×C4×Q8—21no—
62C2×C4×Q8—21no—
63C2×C4×Q8—22no—
64C2×C4×Q8—23no—
65C2×C4×Q8D411yescharacteristic
66C2×C4×Q8D411yescharacteristic
67C2×C4×Q8C2311yescharacteristic
68C8×Q8Q8116yescomplemented
69C2×#107C2312yes—
70C2×#107C2316yes—
71C2×#107C23112yes—
72C2×#106C2316yes—
73C2×#106C2316yes—
74C2×C4⋊C8C2316yes—
75C2×#97C2316yes—
76C2×C4×C8C2316yes—
77#400—26no—
78#398—26no—
79#398—212no—
80#391—23no—
81#389—23no—
82#389—26no—
83#389—23no—
84#389—23no—
85#389—26no—
86#389—23no—
87#389—23no—
88#389—23no—
89#389—23no—
90#389C2311yescharacteristic
91#389C2313yes—
92#389C2316yes—
93#389C2316yes—
94#387—23no—
95#387—23no—
96#384—23no—
97#384—23no—
98#384—23no—
99#384C2311yescharacteristic
100#384C2313yes—
101#384C2316yes—
102#384C2313yes—
103#384C2316yes—
104#382—23no—
105#382—23no—
106#382C2313yes—
107C4×C4⋊C4—23no—
108C4×C4⋊C4—23no—
109C4×C4⋊C4—23no—
110C4×C4⋊C4D413yes—
111C4×C4⋊C4D413yes—
112C4×C4⋊C4C2313yes—
113C4×C4⋊C4C2313yes—
114C4×C4⋊C4C2316yes—
115C4×C4⋊C4C2313yes—
116C43C2313yes—
117#340C2316yes—
118#340C23112yes—
119#340C2316yes—
120#340C2312yes—
121#340C2316yes—
122#336C2311yescharacteristic
123#336C2313yes—
124#336C2316yes—
125C22×Q8—23no—
126C22×Q8C2×D411yescharacteristic
127C22×C8C2412yes—
128C22×C8C2416yes—
129C4⋊Q8—46no—
130C4⋊Q8—412no—
131C4⋊Q8—86no—
132#125—26no—
133#125—43no—
134#125—424no—
135#125—86no—
136C4×Q8—24no—
137C4×Q8—24no—
138C4×Q8—412no—
139C4×Q8—44no—
140C4×Q8—48no—
141C4×Q8—44no—
142C4×Q8C2×Q818yes—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—26no—
145C2×C4⋊C4—26no—
146C2×C4⋊C4—23no—
147C2×C4⋊C4—23no—
148C2×C4⋊C4—26no—
149C2×C4⋊C4—23no—
150C2×C4⋊C4—23no—
151C2×C4⋊C4—23no—
152C2×C4⋊C4—23no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—23no—
156C2×C4⋊C4—23no—
157C2×C4⋊C4—26no—
158C2×C4⋊C4—26no—
159C2×C4⋊C4—26no—
160C2×C4⋊C4C2×D413yes—
161C2×C4⋊C4C2×D413yes—
162C2×C4⋊C4C2×D413yes—
163C2×C4⋊C4C2413yes—
164C2×C4⋊C4C2416yes—
165C2×C4⋊C4C2412yes—
166C2×C4⋊C4C2411yescharacteristic
167C2×C4⋊C4C2413yes—
168C2×C4⋊C4C2411yescharacteristic
169C2×C42—26no—
170C2×C42—23no—
171C2×C42—23no—
172C2×C42C2×D413yes—
173C2×C42C2×D413yes—
174C2×C42C2×D413yes—
175C2×C42C2×D413yes—
176C2×C42C2413yes—
177C2×C42C2411yescharacteristic
178C2×C42C2413yes—
179#107—24no—
180#107—46no—
181#107—412no—
182#106—46no—
183#106—46no—
184C4⋊C8C2×Q8124yes—
185#97—46no—
186C4×C8C2×Q8124yes—
187#95—21no—
188#95—23no—
189#95—23no—
190#95—23no—
191#95—26no—
192#95—23no—
193#95—23no—
194#95—26no—
195#95—26no—
196#95C2×D416yes—
197C2×Q8—46no—
198C2×Q8—46no—
199C2×Q8—46no—
200C2×Q8C4⋊Q814yes—
201C2×Q8AGL(1,ℤ/8)14yes—
202C2×Q8#13714yes—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—22no—
206C22×C4—23no—
207C22×C4—23no—
208C22×C4—26no—
209C22×C4C22×D413yes—
210C22×C4C22×D413yes—
211C22×C4C22×D413yes—
212C22×C4C22×D411yescharacteristic
213C22×C4C22×D411yescharacteristic
214C22×C4C22×D413yes—
215C22×C4C2511yescharacteristic
216C2×C8—22no—
217C2×C8—26no—
218C2×C8—26no—
219C2×C8—22no—
220C2×C8C22×Q814yes—
221C2×C8C22×Q8112yes—
222C4⋊C4—26no—
223C4⋊C4—26no—
224C4⋊C4—21no—
225C4⋊C4—22no—
226C4⋊C4—23no—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—23no—
230C4⋊C4—26no—
231C4⋊C4—26no—
232C4⋊C4—26no—
233C4⋊C4—46no—
234C4⋊C4—46no—
235C4⋊C4—42no—
236C4⋊C4—412no—
237C4⋊C4—412no—
238C4⋊C4—46no—
239C4⋊C4—46no—
240C4⋊C4—412no—
241C4⋊C4—46no—
242C4⋊C4—46no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—83no—
246C4⋊C4—83no—
247C4⋊C4—86no—
248C4⋊C4—86no—
249C4⋊C4—86no—
250C4⋊C4C4⋊Q8112yes—
251C4⋊C42−1+412yes—
252C42—26no—
253C42—26no—
254C42—26no—
255C42—26no—
256C42—23no—
257C42—23no—
258C42—22no—
259C42—46no—
260C42—412no—
261C42—46no—
262C42C22×Q8112yes—
263C23C23×D411yescharacteristic
264Q8—812no—
265Q8#501116yes—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—23no—
271C2×C4—23no—
272C2×C4—23no—
273C2×C4—23no—
274C2×C4—22no—
275C2×C4—23no—
276C2×C4—23no—
277C2×C4—23no—
278C2×C4—23no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—46no—
282C2×C4—41no—
283C2×C4—43no—
284C2×C4—43no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—46no—
288C2×C4—43no—
289C2×C4—41no—
290C2×C4—43no—
291C2×C4—43no—
292C2×C4—43no—
293C2×C4—46no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4C2×C4⋊Q816yes—
297C2×C4C2×C4⋊Q816yes—
298C2×C4C2×AGL(1,ℤ/8)16yes—
299C2×C4C2×#13716yes—
300C2×C4#57612yes—
301C2×C4#57812yes—
302C2×C4C23×Q812yes—
303C2×C4C2×2−1+411yescharacteristic
304C2×C4#58512yes—
305C2×C4#58511yescharacteristic
306C8—24no—
307C8—212no—
308C22C22×C4⋊Q811yescharacteristic
309C22#348711yescharacteristic
310C22#348911yescharacteristic
311C22C22×AGL(1,ℤ/8)11yescharacteristic
312C22C22×#13711yescharacteristic
313C22C2×#57611yescharacteristic
314C22C2×#57811yescharacteristic
315C4—22no—
316C4—22no—
317C4—26no—
318C4—43no—
319C4—42no—
320C4—43no—
321C4—82no—
322C4—83no—
323C4—81no—
324C4—83no—
325C4—86no—
326C4C2×#501112yes—
327C4#350212yes—
328C4#350412yes—
329C2#3595711yescharacteristic
330C2C22×#50111yescharacteristic
331C2#6066711yescharacteristic
332C2#6190011yescharacteristic
333C2#6190711yescharacteristic
334C2#6214811yescharacteristic
335C2#6214911yescharacteristic
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
68. C8×Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#365932
#364472
#303042
#293192
#292132
#288352
#287982
#238382
#227382
Q8×#1072
#205722