Finite Group Databasebeta

← #6532184

Subgroups — #6532184

Order 512.

1478 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2584C211yesmaximal, characteristic
2#34187C211yesmaximal, characteristic
3#27617C214yesmaximal
4#27616C214yesmaximal
5#26291C218yesmaximal, complemented
6#26289C218yesmaximal, complemented
7#21838C212yesmaximal
8#21822C211yesmaximal, characteristic
9#21365C212yesmaximal
10C22×#125C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C2×#432C2212yes—
13#2584—24no—
14#2584C2218yes—
15C2×#398—21no—
16C2×#395—21no—
17#2356—24no—
18#2355C2214yes—
19#2355C2214yes—
20#2313—22no—
21#2310—21no—
22C2×#389—22no—
23C2×#389C2212yes—
24C2×#389C2212yes—
25#2285C2212yes—
26#2283C2214yes—
27#2274—21no—
28C2×C4×C4⋊C4C2211yescharacteristic
29#2101C2218yes—
30#2101C2218yes—
31#2085C2218yes—
32#2085C2218yes—
33#1992C2214yes—
34#1992C2214yes—
35#1989C2218yes—
36#1985C2218yes—
37#1948C2214yes—
38#1947C2214yes—
39#1921C2218yes—
40#1920C2218yes—
41#1919C2218yes—
42#1918C2218yes—
43#1853C2214yes—
44#1852C2214yes—
45#1824C2214yes—
46#1823C2214yes—
47#1817C2214yes—
48#1776C2218yes—
49C23×Q8D411yescharacteristic
50C2×#125—22no—
51C2×#125—28no—
52C2×#125C2314yes—
53C2×C4×Q8—28no—
54C2×C4×Q8D414yes—
55C2×C4×Q8C2314yes—
56C2×#117—22no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4D412yes—
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C42—21no—
67C22×C42C2312yes—
68C22×C42C2311yescharacteristic
69#432—28no—
70C2×#107C2318yes—
71C2×#106C2318yes—
72C2×C4⋊C8C2314yes—
73C2×C22⋊C8C2314yes—
74C2×#97C2314yes—
75#398—28no—
76#395—28no—
77#391—24no—
78#391—24no—
79#389—28no—
80#389—28no—
81#389—24no—
82#389—24no—
83#389C2318yes—
84#389C2318yes—
85#387—24no—
86#387—24no—
87#387—24no—
88#387—24no—
89#384C2312yes—
90#384C2312yes—
91#383—24no—
92#382—22no—
93#382C2314yes—
94#382C2314yes—
95#381—24no—
96#380—22no—
97#380C2314yes—
98C4×C4⋊C4—24no—
99C4×C4⋊C4—24no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4C2314yes—
103C4×C4⋊C4C2314yes—
104C4×C4⋊C4C2314yes—
105C4×C4⋊C4C2314yes—
106C4×C22⋊C4—24no—
107#376—24no—
108C2×#95—21no—
109C2×#95—22no—
110C2×#95—22no—
111#340C2318yes—
112#340C2318yes—
113#340C2318yes—
114#340C2318yes—
115#340C2318yes—
116#340C2318yes—
117#336C2318yes—
118#336C2318yes—
119C22×Q8—24no—
120C22×Q8—44no—
121C22×Q8C2×D414yes—
122C23×C4—22no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4C2×D412yes—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D412yes—
128C23×C4C2×D411yescharacteristic
129C23×C4C2411yescharacteristic
130C22×C8C2418yes—
131#125—28no—
132#125—28no—
133#125—48no—
134C4×Q8—216no—
135C4×Q8—28no—
136C4×Q8—28no—
137#117—48no—
138C2×C4⋊C4—28no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—28no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—28no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—28no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—48no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—84no—
158C2×C4⋊C4C2×D414yes—
159C2×C4⋊C4C2×D414yes—
160C2×C4⋊C4C2×D414yes—
161C2×C4⋊C4C2412yes—
162C2×C4⋊C4C2412yes—
163C2×C4⋊C4C2414yes—
164C2×C4⋊C4C2414yes—
165C2×C4⋊C4C2414yes—
166C2×C22⋊C4—24no—
167C2×C42—28no—
168C2×C42—24no—
169C2×C42—24no—
170C2×C42—24no—
171C2×C42—24no—
172C2×C42—82no—
173C2×C42C2×D414yes—
174C2×C42C2×D412yes—
175C2×C42#4118yes—
176C2×C42C2414yes—
177C2×C42C2412yes—
178#107—216no—
179#106—216no—
180C4⋊C8—28no—
181C22⋊C8—28no—
182#97—28no—
183#95—24no—
184#95—28no—
185#95—28no—
186#95—24no—
187#95—24no—
188#95—24no—
189#95—24no—
190#95—24no—
191#95—24no—
192#95—24no—
193#95—24no—
194C24C22×D411yescharacteristic
195C2×Q8—28no—
196C2×Q8—44no—
197C2×Q8—44no—
198C2×Q8—416no—
199C22×C4—24no—
200C22×C4—24no—
201C22×C4—28no—
202C22×C4—24no—
203C22×C4—24no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—44no—
216C22×C4—44no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—44no—
222C22×C4#12314yes—
223C22×C4#12314yes—
224C22×C4C22×D414yes—
225C22×C4C22×D414yes—
226C22×C4C22×D411yescharacteristic
227C22×C4C22×D412yes—
228C22×C4C22×D412yes—
229C22×C4C2×#4114yes—
230C22×C4C2511yescharacteristic
231C2×C8—28no—
232C2×C8—28no—
233C2×C8—28no—
234C4⋊C4—28no—
235C4⋊C4—28no—
236C4⋊C4—28no—
237C4⋊C4—28no—
238C4⋊C4—24no—
239C4⋊C4—24no—
240C4⋊C4—24no—
241C4⋊C4—28no—
242C4⋊C4—28no—
243C4⋊C4—416no—
244C4⋊C4—44no—
245C4⋊C4—44no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—48no—
249C4⋊C4—48no—
250C4⋊C4—82no—
251C4⋊C4—82no—
252C4⋊C4—88no—
253C4⋊C42−1+418yes—
254C22⋊C4—48no—
255C42—28no—
256C42—28no—
257C42—48no—
258C42—84no—
259C42—84no—
260C42—84no—
261C42#12318yes—
262C42C2×#4114yes—
263C42C2×#4114yes—
264C23—21no—
265C23—21no—
266C23—22no—
267C23C2×#12312yes—
268C23#57514yes—
269C23C23×D411yescharacteristic
270Q8—416no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—28no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—28no—
278C2×C4—24no—
279C2×C4—24no—
280C2×C4—24no—
281C2×C4—24no—
282C2×C4—28no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—44no—
286C2×C4—416no—
287C2×C4—44no—
288C2×C4—48no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—42no—
292C2×C4—44no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—84no—
302C2×C4—84no—
303C2×C4—84no—
304C2×C4C2×#12312yes—
305C2×C4C2×#12314yes—
306C2×C4C2×#12314yes—
307C2×C4C22×#4112yes—
308C2×C4C2×2−1+414yes—
309C8—48no—
310C22—22no—
311C22—22no—
312C22—24no—
313C22—24no—
314C22#309314yes—
315C22C22×#12311yescharacteristic
316C22#348612yes—
317C22C2×#57512yes—
318C22#358511yescharacteristic
319C22#358611yescharacteristic
320C4—22no—
321C4—24no—
322C4—24no—
323C4—44no—
324C4—44no—
325C4—48no—
326C4—82no—
327C4—82no—
328C4—84no—
329C4—84no—
330C4#350914yes—
331C2—24no—
332C2#3567411yescharacteristic
333C2C2×#309311yescharacteristic
334C2#6052011yescharacteristic
335C2#6183112yes—
336C2#6183212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #26291C2semidirect
6. #26289C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25842
#341872
#276172
#276162
#262912
#262892
#218382
#218222
#213652