Finite Group Databasebeta

← #6434802

Subgroups — #6434802

Order 512.

1390 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2647C211yesmaximal, characteristic
2#34074C211yesmaximal, characteristic
3#30104C218yesmaximal, complemented
4#30100C218yesmaximal, complemented
5#23747C212yesmaximal
6#23745C212yesmaximal
7#23659C212yesmaximal
8#23554C212yesmaximal
9#21034C214yesmaximal
10#20773C211yesmaximal, characteristic
11C2×#533C2211yescharacteristic
12C2×#428C2214yes—
13#2647—24no—
14#2647C2214yes—
15#2647C2214yes—
16C2×#400—22no—
17C2×#398—22no—
18#2331C2214yes—
19#2330C2214yes—
20#2329—22no—
21#2329—22no—
22#2296C2212yes—
23#2295—24no—
24C2×#389C2212yes—
25C2×#389C2212yes—
26#2285C2211yescharacteristic
27C2×#384C2211yescharacteristic
28C2×#382C2211yescharacteristic
29C4×#117C2212yes—
30C2×#376C2211yescharacteristic
31#2101C2218yes—
32#2100C2218yes—
33#2100C2214yes—
34#2099C2218yes—
35#2099C2214yes—
36#2085C2218yes—
37#2085C2218yes—
38#2059C2218yes—
39#2058C2214yes—
40#2057C2214yes—
41#1921C2218yes—
42#1920C2218yes—
43#1919C2218yes—
44#1918C2218yes—
45#1840C2214yes—
46#1840C2214yes—
47#1736C2214yes—
48#1736C2214yes—
49#1736C2214yes—
50#1728C2212yes—
51#1728C2212yes—
52#1728C2212yes—
53C22×M16C2314yes—
54#533—28no—
55C2×C4⋊Q8C2312yes—
56C2×#125C2312yes—
57C2×C22⋊Q8—22no—
58C2×C4×Q8—22no—
59C2×#117C2312yes—
60C2×#117C2312yes—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C42C2311yescharacteristic
67C22×C42C2311yescharacteristic
68C22×C42C2311yescharacteristic
69#428—216no—
70C2×#107C2318yes—
71C2×#106C2318yes—
72#400—28no—
73#400—28no—
74#398—28no—
75#398—28no—
76#391—24no—
77#391—24no—
78#389—22no—
79#389—22no—
80#389—24no—
81#389—24no—
82#389—24no—
83#389C2314yes—
84#389C2314yes—
85#389C2314yes—
86#389C2314yes—
87#389C2318yes—
88#387—24no—
89#387—24no—
90#387—24no—
91#387—24no—
92#385—24no—
93#384—22no—
94#384C2312yes—
95#384C2314yes—
96#382—22no—
97#382C2312yes—
98#382C2314yes—
99#380—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—24no—
102C4×C4⋊C4C2314yes—
103C4×C4⋊C4C2314yes—
104C4×C22⋊C4—22no—
105C4×C22⋊C4—22no—
106#376—22no—
107#376D414yes—
108#376C2312yes—
109#376C2312yes—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95D412yes—
114C2×#95D412yes—
115C43—22no—
116#340C2318yes—
117#340C2318yes—
118#340C2318yes—
119#340C2318yes—
120#340C2318yes—
121#340C2318yes—
122#336C2314yes—
123#336C2314yes—
124#336C2314yes—
125#336C2314yes—
126#336C2318yes—
127C22×Q8—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D412yes—
132C23×C4C2×D412yes—
133C23×C4C2×D411yescharacteristic
134C23×C4C2411yescharacteristic
135C2×M16—28no—
136C2×M16—28no—
137C2×M16—28no—
138C22×C8C2418yes—
139C4⋊Q8—28no—
140#125—28no—
141C22⋊Q8—216no—
142C4×Q8—216no—
143#117—24no—
144#117—28no—
145#117#4118yes—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—28no—
149C2×C4⋊C4—28no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—28no—
153C2×C4⋊C4—28no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—84no—
161C2×C4⋊C4—84no—
162C2×C4⋊C4—84no—
163C2×C4⋊C4C2414yes—
164C2×C4⋊C4C2414yes—
165C2×C4⋊C4C2414yes—
166C2×C4⋊C4C2414yes—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C42—22no—
170C2×C42—24no—
171C2×C42—24no—
172C2×C42—24no—
173C2×C42—22no—
174C2×C42—24no—
175C2×C42—24no—
176C2×C42C2×D412yes—
177C2×C42C2×D412yes—
178C2×C42C2×D412yes—
179C2×C42C2411yescharacteristic
180C2×C42C2411yescharacteristic
181C2×C42C2412yes—
182C2×C42C2412yes—
183#107—216no—
184#106—216no—
185#95—28no—
186#95—24no—
187#95—24no—
188#95—28no—
189#95—24no—
190#95—24no—
191#95—24no—
192#95—24no—
193#95—24no—
194#95C2×D414yes—
195#95C2×D414yes—
196#95C2×D414yes—
197#95C2×D414yes—
198C24C22×D411yescharacteristic
199C2×Q8—28no—
200C2×Q8—82no—
201C2×Q8—82no—
202C22×C4—24no—
203C22×C4—24no—
204C22×C4—24no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—44no—
217C22×C4—44no—
218C22×C4—44no—
219C22×C4—44no—
220C22×C4—44no—
221C22×C4—44no—
222C22×C4C22×D412yes—
223C22×C4C22×D414yes—
224C22×C4C22×D412yes—
225C22×C4C22×D412yes—
226C22×C4C22×D412yes—
227C22×C4C22×D411yescharacteristic
228C22×C4C2×#4114yes—
229C22×C42−1+414yes—
230C22×C42−1+414yes—
231C22×C4C2511yescharacteristic
232M16—416no—
233C2×C8—28no—
234C2×C8—28no—
235C2×C8—28no—
236C4⋊C4—216no—
237C4⋊C4—28no—
238C4⋊C4—24no—
239C4⋊C4—28no—
240C4⋊C4—28no—
241C4⋊C4—28no—
242C4⋊C4—44no—
243C4⋊C4—44no—
244C4⋊C4—84no—
245C4⋊C4—82no—
246C4⋊C4—84no—
247C4⋊C4—82no—
248C4⋊C4—88no—
249C4⋊C4—88no—
250C4⋊C4—84no—
251C4⋊C4—84no—
252C4⋊C4C2×#4118yes—
253C22⋊C4—28no—
254C22⋊C4—44no—
255C42—24no—
256C42—28no—
257C42—24no—
258C42—24no—
259C42—22no—
260C42—22no—
261C42—24no—
262C42—24no—
263C42—44no—
264C42—84no—
265C42—84no—
266C23—22no—
267C23—22no—
268C23#52512yes—
269C23#53612yes—
270C23#53612yes—
271C23C23×D411yescharacteristic
272Q8—88no—
273C2×C4—24no—
274C2×C4—22no—
275C2×C4—24no—
276C2×C4—22no—
277C2×C4—24no—
278C2×C4—22no—
279C2×C4—24no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—28no—
284C2×C4—24no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—48no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—44no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—84no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—88no—
303C2×C4—88no—
304C2×C4#52512yes—
305C2×C4#53612yes—
306C2×C4#53612yes—
307C2×C4#541116yes—
308C2×C4C22×#4112yes—
309C2×C4C2×2−1+412yes—
310C2×C4C2×2−1+412yes—
311C8—48no—
312C22—24no—
313C22—24no—
314C22—22no—
315C22—22no—
316C22#263414yes—
317C22C2×#52511yescharacteristic
318C22C2×#53611yescharacteristic
319C22C2×#53611yescharacteristic
320C22#358512yes—
321C22#358612yes—
322C4—42no—
323C4—42no—
324C4—48no—
325C4—44no—
326C4—42no—
327C4—42no—
328C4—84no—
329C4—84no—
330C4—84no—
331C4#263414yes—
332C4C2×#54114yes—
333C2—24no—
334C2C2×#263411yescharacteristic
335C2#6056712yes—
336C2#6086712yes—
337C2#6086811yescharacteristic
338C2#6086911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
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
3. #30104C2semidirect
4. #30100C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26472
#340742
#301042
#301002
#237472
#237452
#236592
#235542
#210342
#207732