Finite Group Databasebeta

← #6434800

Subgroups — #6434800

Order 512.

1394 conjugacy classes of subgroups.

The subgroup lattice has 349 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34638C211yesmaximal, characteristic
2C2×#2330C211yesmaximal, characteristic
3#30104C218yesmaximal, complemented
4#30103C218yesmaximal, complemented
5#23747C212yesmaximal
6#23745C212yesmaximal
7#23657C212yesmaximal
8#23553C212yesmaximal
9#21033C214yesmaximal
10#20773C211yesmaximal, characteristic
11C22×#125C2211yescharacteristic
12C2×#428C2214yes—
13#2647C2214yes—
14#2647C2214yes—
15#2646—22no—
16#2646—22no—
17#2402—22no—
18#2402—22no—
19#2330—22no—
20#2330—22no—
21#2330C2214yes—
22#2330C2214yes—
23#2296C2212yes—
24#2296C2212yes—
25C2×#389—22no—
26C2×#389C2212yes—
27C2×#387—22no—
28#2285C2211yescharacteristic
29#2285C2211yescharacteristic
30C2×#384C2211yescharacteristic
31C2×C4×C4⋊C4C2212yes—
32C2×#376C2211yescharacteristic
33#2101C2218yes—
34#2100C2218yes—
35#2100C2214yes—
36#2099C2218yes—
37#2099C2214yes—
38#2085C2218yes—
39#2084C2214yes—
40#2083C2214yes—
41#2059C2218yes—
42#2059C2218yes—
43#1921C2218yes—
44#1920C2218yes—
45#1919C2218yes—
46#1918C2218yes—
47#1840C2214yes—
48#1840C2214yes—
49#1736C2214yes—
50#1736C2214yes—
51#1736C2214yes—
52#1728C2212yes—
53#1728C2212yes—
54#1728C2212yes—
55C22×M16C2314yes—
56C2×#125—28no—
57C2×#125—22no—
58C2×#125C2314yes—
59C2×#117C2312yes—
60C2×#117C2312yes—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C42—22no—
67C22×C42C2311yescharacteristic
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70#428—216no—
71C2×#107C2318yes—
72C2×#106C2318yes—
73#400—24no—
74#400—24no—
75#400—24no—
76#400—24no—
77#398—24no—
78#398—24no—
79#398—24no—
80#398—24no—
81#389—22no—
82#389—22no—
83#389—28no—
84#389—28no—
85#389C2314yes—
86#389C2314yes—
87#389C2314yes—
88#389C2314yes—
89#389C2314yes—
90#389C2314yes—
91#387—28no—
92#387—28no—
93#385—24no—
94#385—24no—
95#384—22no—
96#384C2314yes—
97#384C2312yes—
98#384C2312yes—
99#382C2312yes—
100#382C2312yes—
101#380—21no—
102#380—21no—
103#380—21no—
104#380—21no—
105C4×C4⋊C4—24no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—24no—
111C4×C4⋊C4—24no—
112C4×C4⋊C4C2318yes—
113#376D412yes—
114#376D412yes—
115#376C2312yes—
116#376C2312yes—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95D412yes—
120C2×#95D412yes—
121#340C2318yes—
122#340C2318yes—
123#340C2318yes—
124#340C2318yes—
125#340C2318yes—
126#340C2318yes—
127#336C2314yes—
128#336C2314yes—
129#336C2314yes—
130#336C2314yes—
131#336C2318yes—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D412yes—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D412yes—
138C23×C4C2411yescharacteristic
139C2×M16—28no—
140C2×M16—28no—
141C2×M16—28no—
142C22×C8C2418yes—
143#125—216no—
144#125—216no—
145#117—28no—
146#117—28no—
147C2×C4⋊C4—216no—
148C2×C4⋊C4—28no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—28no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—28no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—84no—
165C2×C4⋊C4—84no—
166C2×C4⋊C4#4118yes—
167C2×C4⋊C4C2414yes—
168C2×C4⋊C4C2414yes—
169C2×C4⋊C4C2414yes—
170C2×C4⋊C4C2414yes—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C42—24no—
174C2×C42—24no—
175C2×C42—24no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—28no—
179C2×C42—84no—
180C2×C42C2×D412yes—
181C2×C42C2×D412yes—
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2412yes—
185C2×C42C2412yes—
186C2×C42C2411yescharacteristic
187C2×C42C2411yescharacteristic
188#107—216no—
189#106—216no—
190#95—28no—
191#95—24no—
192#95—24no—
193#95—22no—
194#95—22no—
195#95—22no—
196#95—22no—
197#95—24no—
198#95—24no—
199#95—24no—
200#95—24no—
201#95C2×D414yes—
202#95C2×D414yes—
203#95C2×D414yes—
204#95C2×D414yes—
205C24C22×D411yescharacteristic
206C22×C4—24no—
207C22×C4—28no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—44no—
222C22×C4—44no—
223C22×C4—44no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4C22×D411yescharacteristic
227C22×C4C22×D411yescharacteristic
228C22×C4C22×D414yes—
229C22×C4C22×D411yescharacteristic
230C22×C4C22×D411yescharacteristic
231C22×C4C22×D414yes—
232C22×C4C22×D411yescharacteristic
233C22×C4C2×#4114yes—
234C22×C42−1+414yes—
235C22×C42−1+414yes—
236C22×C4C2511yescharacteristic
237M16—416no—
238C2×C8—28no—
239C2×C8—28no—
240C2×C8—28no—
241C4⋊C4—232no—
242C4⋊C4—28no—
243C4⋊C4—24no—
244C4⋊C4—24no—
245C4⋊C4—24no—
246C4⋊C4—28no—
247C4⋊C4—28no—
248C4⋊C4—44no—
249C4⋊C4—84no—
250C4⋊C4—82no—
251C4⋊C4—82no—
252C4⋊C4—84no—
253C4⋊C4—82no—
254C4⋊C4—82no—
255C4⋊C4—84no—
256C4⋊C4—84no—
257C4⋊C4—84no—
258C4⋊C4—84no—
259C4⋊C4—88no—
260C4⋊C4C2×#4118yes—
261C22⋊C4—44no—
262C22⋊C4—44no—
263C42—24no—
264C42—24no—
265C42—22no—
266C42—22no—
267C42—24no—
268C42—24no—
269C42—22no—
270C42—22no—
271C42—44no—
272C42—44no—
273C42—88no—
274C23—22no—
275C23—22no—
276C23#52512yes—
277C23#53612yes—
278C23#53612yes—
279C23C23×D411yescharacteristic
280C2×C4—24no—
281C2×C4—24no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—24no—
289C2×C4—28no—
290C2×C4—24no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—48no—
296C2×C4—416no—
297C2×C4—48no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4—88no—
312C2×C4#52512yes—
313C2×C4#53612yes—
314C2×C4#53612yes—
315C2×C4#541116yes—
316C2×C4C22×#4112yes—
317C2×C4C2×2−1+412yes—
318C2×C4C2×2−1+412yes—
319C8—48no—
320C22—24no—
321C22—24no—
322C22—22no—
323C22—22no—
324C22#263414yes—
325C22C2×#52511yescharacteristic
326C22C2×#53611yescharacteristic
327C22C2×#53611yescharacteristic
328C22#358512yes—
329C22#358612yes—
330C4—44no—
331C4—48no—
332C4—44no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—84no—
338C4—84no—
339C4—84no—
340C4#263414yes—
341C4C2×#54114yes—
342C2—24no—
343C2C2×#263411yescharacteristic
344C2#6056512yes—
345C2#6086712yes—
346C2#6086811yescharacteristic
347C2#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. #30103C2semidirect

Maximal subgroups

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

subgroupindex
#346382
C2×#23302
#301042
#301032
#237472
#237452
#236572
#235532
#210332
#207732