Finite Group Databasebeta

← #517954

Subgroups — #517954

Order 512.

2290 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
1C22×#401C211yesmaximal, characteristic
2#14349C211yesmaximal, characteristic
3#14333C212yesmaximal, complemented
4#14251C216yesmaximal, complemented
5#14251C216yesmaximal, complemented
6#13883C213yesmaximal
7#13874C213yesmaximal
8C4×#401C212yesmaximal, complemented
9#13802C213yesmaximal
10#13799C213yesmaximal
11#13749C211yesmaximal, characteristic
12C22×#117C2211yescharacteristic
13C2×#401C4116yescomplemented
14C2×#401C2211yescharacteristic
15C2×#401C2213yes—
16C2×#401C2213yes—
17C2×#401C2211yescharacteristic
18C2×#381C2213yes—
19C2×#380C2213yes—
20C2×C4×C22⋊C4C2213yes—
21C2×C4×C22⋊C4C2213yes—
22C2×C4×C22⋊C4C2213yes—
23C22×#95C2211yescharacteristic
24C22×#95C2213yes—
25C22×#95C2213yes—
26#1443C2212yes—
27#1443C2216yes—
28#1443C2216yes—
29#1443C2216yes—
30#1442C2216yes—
31#1442C2216yes—
32#1436C2216yes—
33#1436C2216yes—
34#1436C2216yes—
35#1436C2216yes—
36#1436C2216yes—
37#1436C2216yes—
38#1434C2216yes—
39#1434C2216yes—
40#1434C2216yes—
41#1434C2216yes—
42C22⋊#95C2213yes—
43C22⋊#95C2213yes—
44C22⋊#95C2213yes—
45C22⋊#95C2213yes—
46C4×#95C2216yes—
47C4×#95C2216yes—
48C4×#95C2212yes—
49C4×#95C2216yes—
50C24×C4C2311yescharacteristic
51C24×C4C2313yes—
52C24×C4C2313yes—
53C2×#117—212no—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2316yes—
56C22×C22⋊C4C2311yescharacteristic
57C22×C22⋊C4C2313yes—
58C22×C22⋊C4C2313yes—
59C22×C22⋊C4C2311yescharacteristic
60C22×C42C2312yes—
61C22×C42C2316yes—
62C22×C42C2316yes—
63C22×C42C2316yes—
64#401C2×C418yes—
65#401C2×C4124yes—
66#401C2×C4124yes—
67#401C2×C418yes—
68#381—212no—
69#380—212no—
70C4×C22⋊C4—212no—
71C4×C22⋊C4—212no—
72C4×C22⋊C4—212no—
73C2×#95C2×C4124yes—
74C2×#95C2×C418yes—
75C2×#95C2×C4124yes—
76C2×#95C2316yes—
77C2×#95C2316yes—
78C2×#95C2316yes—
79C2×#95C2316yes—
80C2×#95C2316yes—
81C2×#95C2316yes—
82C2×#95C2316yes—
83C2×#95C2316yes—
84C2×#95C2316yes—
85C2×#95C2316yes—
86C2×#95C2316yes—
87C2×#95C2316yes—
88C2×#95C2316yes—
89C2×#95C2316yes—
90C2×#95C2313yes—
91C2×#95C2313yes—
92C2×#95C2311yescharacteristic
93C2×#95C2313yes—
94C2×#95C2313yes—
95C2×#95C2313yes—
96C2×#95C2313yes—
97C2×#95C2313yes—
98C2×#95C2313yes—
99C2×#95C2313yes—
100C25C2411yescharacteristic
101C23×C4—26no—
102C23×C4—26no—
103C23×C4—26no—
104C23×C4—26no—
105C23×C4—26no—
106C23×C4—26no—
107C23×C4—46no—
108C23×C4—46no—
109C23×C4—46no—
110C23×C4—46no—
111C23×C4C22×C414yes—
112C23×C4C22×C4112yes—
113C23×C4C22×C4112yes—
114C23×C4C2416yes—
115C23×C4C2412yes—
116C23×C4C2411yescharacteristic
117C23×C4C2413yes—
118C23×C4C2411yescharacteristic
119C23×C4C2413yes—
120C23×C4C2413yes—
121C23×C4C2413yes—
122C23×C4C2416yes—
123C23×C4C2412yes—
124#117—416no—
125C2×C4⋊C4—212no—
126C2×C4⋊C4—212no—
127C2×C4⋊C4—46no—
128C2×C4⋊C4—46no—
129C2×C22⋊C4—26no—
130C2×C22⋊C4—26no—
131C2×C22⋊C4—26no—
132C2×C22⋊C4—26no—
133C2×C22⋊C4—26no—
134C2×C22⋊C4—26no—
135C2×C22⋊C4#41112yes—
136C2×C22⋊C4#41112yes—
137C2×C42—212no—
138C2×C42—212no—
139C2×C42—212no—
140C2×C42—212no—
141C2×C42—46no—
142C2×C42—46no—
143C2×C42—46no—
144C2×C42—46no—
145C2×C42—46no—
146C2×C42—46no—
147#95—212no—
148#95—212no—
149#95—212no—
150#95—212no—
151#95—212no—
152#95—212no—
153#95—46no—
154#95—46no—
155#95—46no—
156#95—46no—
157#95—46no—
158#95—46no—
159#95—46no—
160#95—46no—
161#95C22×C4112yes—
162#95C22×C414yes—
163#95C22×C4112yes—
164#95C22×C4112yes—
165#95C22×C4112yes—
166#95C22×C4112yes—
167#95C22×C4112yes—
168#95C22×C4112yes—
169#95C22×C4112yes—
170#95C22×C4112yes—
171C24—22no—
172C24—26no—
173C24C23×C412yes—
174C24C2×#4116yes—
175C24C2×#4116yes—
176C24C2511yescharacteristic
177C22×C4—22no—
178C22×C4—26no—
179C22×C4—224no—
180C22×C4—23no—
181C22×C4—23no—
182C22×C4—23no—
183C22×C4—23no—
184C22×C4—23no—
185C22×C4—23no—
186C22×C4—26no—
187C22×C4—26no—
188C22×C4—23no—
189C22×C4—23no—
190C22×C4—23no—
191C22×C4—23no—
192C22×C4—23no—
193C22×C4—23no—
194C22×C4—26no—
195C22×C4—26no—
196C22×C4—26no—
197C22×C4—26no—
198C22×C4—26no—
199C22×C4—26no—
200C22×C4—26no—
201C22×C4—26no—
202C22×C4—26no—
203C22×C4—26no—
204C22×C4—26no—
205C22×C4—26no—
206C22×C4—224no—
207C22×C4—224no—
208C22×C4—26no—
209C22×C4—26no—
210C22×C4—26no—
211C22×C4—26no—
212C22×C4—22no—
213C22×C4—26no—
214C22×C4—26no—
215C22×C4—26no—
216C22×C4—26no—
217C22×C4—26no—
218C22×C4—412no—
219C22×C4—412no—
220C22×C4—412no—
221C22×C4—412no—
222C22×C4—44no—
223C22×C4—412no—
224C22×C4—812no—
225C22×C4—812no—
226C22×C4—812no—
227C22×C4—812no—
228C22×C4C23×C412yes—
229C22×C4C23×C412yes—
230C22×C4C23×C416yes—
231C22×C4C23×C416yes—
232C22×C4C23×C416yes—
233C22×C4C23×C416yes—
234C22×C4C2×#41112yes—
235C22×C4C2×#41112yes—
236C4⋊C4—48no—
237C4⋊C4—812no—
238C22⋊C4—224no—
239C22⋊C4—44no—
240C22⋊C4—412no—
241C22⋊C4#126116yes—
242C42—48no—
243C42—812no—
244C42—812no—
245C42—812no—
246C23—26no—
247C23—22no—
248C23—212no—
249C23—212no—
250C23—212no—
251C23—212no—
252C23C4×#41112yes—
253C23C2×#12614yes—
254C23#529112yes—
255C23C24×C411yescharacteristic
256C23C22×#4113yes—
257C23C22×#4113yes—
258C23#58512yes—
259C23#58516yes—
260C2×C4—212no—
261C2×C4—212no—
262C2×C4—26no—
263C2×C4—26no—
264C2×C4—26no—
265C2×C4—26no—
266C2×C4—26no—
267C2×C4—26no—
268C2×C4—26no—
269C2×C4—26no—
270C2×C4—26no—
271C2×C4—26no—
272C2×C4—26no—
273C2×C4—26no—
274C2×C4—26no—
275C2×C4—26no—
276C2×C4—212no—
277C2×C4—212no—
278C2×C4—212no—
279C2×C4—416no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—43no—
283C2×C4—43no—
284C2×C4—46no—
285C2×C4—46no—
286C2×C4—46no—
287C2×C4—46no—
288C2×C4—46no—
289C2×C4—46no—
290C2×C4—46no—
291C2×C4—46no—
292C2×C4—46no—
293C2×C4—46no—
294C2×C4—46no—
295C2×C4—46no—
296C2×C4—42no—
297C2×C4—46no—
298C2×C4—46no—
299C2×C4—46no—
300C2×C4—46no—
301C2×C4—46no—
302C2×C4—46no—
303C2×C4—46no—
304C2×C4—46no—
305C2×C4—46no—
306C2×C4—824no—
307C2×C4—824no—
308C2×C4C2×#12618yes—
309C22—212no—
310C22—212no—
311C22—28no—
312C22—224no—
313C22C4×#12618yes—
314C22C2×C4×#4113yes—
315C22#342813yes—
316C22#342811yescharacteristic
317C22C22×#12611yescharacteristic
318C22C2×#52913yes—
319C22#352116yes—
320C22#352116yes—
321C22#352116yes—
322C22#352116yes—
323C4—28no—
324C4—46no—
325C4—46no—
326C4—46no—
327C4—46no—
328C4—412no—
329C4—42no—
330C4—42no—
331C4—82no—
332C4—86no—
333C2—28no—
334C2C2×C4×#12611yescharacteristic
335C2#3424713yes—
336C2#3424713yes—
337C2#3605312yes—
338C2#3605416yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC25C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #14333C2semidirect
4. #14251C2semidirect
5. #14251C2semidirect
8. C4×#401C2semidirect
13. C2×#401C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4012
#143492
#143332
#142512
#138832
#138742
C4×#4012
#138022
#137992
#137492