Finite Group Databasebeta

← #4417855

Subgroups — #4417855

Order 512.

1770 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16570C216yesmaximal
2#16523C216yesmaximal
3#15231C213yesmaximal
4#15228C213yesmaximal
5#15146C213yesmaximal
6#15137C213yesmaximal
7#14692C211yesmaximal, characteristic
8#14688C211yesmaximal, characteristic
9#14678C212yesmaximal
10#14676C212yesmaximal
11C4×#383C211yesmaximal, characteristic
12C22×#125C2211yescharacteristic
13C2×#400C2216yes—
14C2×#398C2216yes—
15C2×#389C2213yes—
16C2×#389C2213yes—
17C2×#384C2212yes—
18C2×#384C2213yes—
19C2×#384C2213yes—
20C2×#384C2216yes—
21C2×#383C2211yescharacteristic
22C2×#383C2212yes—
23C2×#383C2211yescharacteristic
24C2×#382C2213yes—
25C2×C43C2211yescharacteristic
26#1444C2216yes—
27#1444C2216yes—
28#1444C2216yes—
29#1442C2216yes—
30#1442C2216yes—
31#1442C2216yes—
32#1442C2216yes—
33#1442C2216yes—
34#1441C2216yes—
35#1441C2216yes—
36#1441C2216yes—
37#1441C2216yes—
38#1441C2216yes—
39#1441C2216yes—
40C4⋊#95C2213yes—
41C4⋊#95C2213yes—
42C4⋊#95C2213yes—
43C4⋊#95C2213yes—
44#1434C2216yes—
45#1434C2213yes—
46#1434C2213yes—
47C4×#95C2213yes—
48C4×#95C2213yes—
49C2×#125—46no—
50C22×C4⋊C4C2311yescharacteristic
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2313yes—
53C22×C4⋊C4C2313yes—
54C22×C4⋊C4C2316yes—
55C22×C4⋊C4C2316yes—
56C22×C4⋊C4C2316yes—
57C22×C4⋊C4C2316yes—
58C22×C4⋊C4C2316yes—
59C22×C4⋊C4C2316yes—
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2312yes—
62C22×C42C2311yescharacteristic
63C22×C42C2313yes—
64C22×C42C2313yes—
65C22×C42C2313yes—
66C22×C42C2311yescharacteristic
67#400—412no—
68#398—412no—
69#389—46no—
70#389—46no—
71#384—28no—
72#384—46no—
73#384—46no—
74#384—412no—
75#383—24no—
76#383—28no—
77#383—24no—
78#382—46no—
79C2×#95C2313yes—
80C2×#95C2316yes—
81C2×#95C2313yes—
82C2×#95C2313yes—
83C2×#95C2313yes—
84C2×#95C2313yes—
85C2×#95C2313yes—
86C2×#95C2313yes—
87C2×#95C2313yes—
88C2×#95C2316yes—
89C2×#95C2316yes—
90C2×#95C2316yes—
91C2×#95C2316yes—
92C2×#95C2316yes—
93C2×#95C2316yes—
94C2×#95C2316yes—
95C2×#95C2316yes—
96C2×#95C2316yes—
97C2×#95C2316yes—
98C2×#95C2316yes—
99C43Q818yes—
100C23×C4C2412yes—
101C23×C4C2416yes—
102C23×C4C2411yescharacteristic
103C23×C4C2416yes—
104C23×C4C2412yes—
105C23×C4C2413yes—
106C23×C4C2411yescharacteristic
107C23×C4C2413yes—
108C23×C4C2413yes—
109C23×C4C2413yes—
110C23×C4C2411yescharacteristic
111#125—416no—
112C2×C4⋊C4—26no—
113C2×C4⋊C4—26no—
114C2×C4⋊C4—26no—
115C2×C4⋊C4—26no—
116C2×C4⋊C4—26no—
117C2×C4⋊C4—26no—
118C2×C4⋊C4—26no—
119C2×C4⋊C4—26no—
120C2×C4⋊C4—212no—
121C2×C4⋊C4—212no—
122C2×C4⋊C4—212no—
123C2×C4⋊C4—212no—
124C2×C4⋊C4—46no—
125C2×C4⋊C4—46no—
126C2×C4⋊C4—46no—
127C2×C4⋊C4—46no—
128C2×C4⋊C4—46no—
129C2×C4⋊C4—46no—
130C2×C4⋊C4—46no—
131C2×C4⋊C4—46no—
132C2×C4⋊C4—46no—
133C2×C4⋊C4—46no—
134C2×C4⋊C4—46no—
135C2×C4⋊C4—46no—
136C2×C4⋊C4—46no—
137C2×C4⋊C4—46no—
138C2×C4⋊C4—46no—
139C2×C4⋊C4—46no—
140C2×C42—26no—
141C2×C42—26no—
142C2×C42—26no—
143C2×C42—26no—
144C2×C42—26no—
145C2×C42—26no—
146C2×C42—26no—
147C2×C42—26no—
148C2×C42C2×Q8112yes—
149C2×C42C2×Q8112yes—
150C2×C42C2×Q814yes—
151C2×C42#4114yes—
152C2×C42#4114yes—
153#95—26no—
154#95—26no—
155#95—212no—
156#95—212no—
157#95—26no—
158#95—26no—
159#95—26no—
160#95—212no—
161#95—26no—
162#95—26no—
163#95—26no—
164#95—212no—
165#95—46no—
166#95—46no—
167#95—46no—
168#95—46no—
169#95—46no—
170#95—46no—
171#95—46no—
172#95—46no—
173C24C2511yescharacteristic
174C22×C4—23no—
175C22×C4—23no—
176C22×C4—23no—
177C22×C4—23no—
178C22×C4—23no—
179C22×C4—23no—
180C22×C4—23no—
181C22×C4—23no—
182C22×C4—21no—
183C22×C4—23no—
184C22×C4—26no—
185C22×C4—22no—
186C22×C4—23no—
187C22×C4—21no—
188C22×C4—22no—
189C22×C4—26no—
190C22×C4—23no—
191C22×C4—23no—
192C22×C4—26no—
193C22×C4—26no—
194C22×C4—23no—
195C22×C4—23no—
196C22×C4—23no—
197C22×C4—23no—
198C22×C4—23no—
199C22×C4—26no—
200C22×C4—26no—
201C22×C4—26no—
202C22×C4—26no—
203C22×C4—26no—
204C22×C4—26no—
205C22×C4—26no—
206C22×C4—26no—
207C22×C4—23no—
208C22×C4—26no—
209C22×C4—26no—
210C22×C4—23no—
211C22×C4—23no—
212C22×C4—26no—
213C22×C4—23no—
214C22×C4—26no—
215C22×C4—23no—
216C22×C4—23no—
217C22×C4—23no—
218C22×C4—23no—
219C22×C4—23no—
220C22×C4—26no—
221C22×C4—26no—
222C22×C4C22×Q816yes—
223C22×C4C22×Q816yes—
224C22×C4C22×Q812yes—
225C22×C4C2×#4116yes—
226C22×C4C2×#4116yes—
227C22×C4H(5,𝔽2)16yes—
228C22×C4H(5,𝔽2)16yes—
229C22×C42−1+416yes—
230C22×C42−1+416yes—
231C4⋊C4—48no—
232C4⋊C4—48no—
233C4⋊C4—412no—
234C4⋊C4—44no—
235C4⋊C4—44no—
236C4⋊C4—412no—
237C4⋊C4—812no—
238C4⋊C4—812no—
239C4⋊C4—812no—
240C4⋊C4—812no—
241C4⋊C4—812no—
242C4⋊C4—812no—
243C42—224no—
244C42—224no—
245C42—44no—
246C42—412no—
247C42#125116yescomplemented
248C23C23×Q811yescharacteristic
249C23C22×#4111yescharacteristic
250C23C22×#4111yescharacteristic
251C23C2×H(5,𝔽2)13yes—
252C23C2×H(5,𝔽2)13yes—
253C23C2×2−1+413yes—
254C23C2×2−1+413yes—
255C2×C4—26no—
256C2×C4—26no—
257C2×C4—26no—
258C2×C4—26no—
259C2×C4—26no—
260C2×C4—26no—
261C2×C4—26no—
262C2×C4—26no—
263C2×C4—26no—
264C2×C4—26no—
265C2×C4—26no—
266C2×C4—26no—
267C2×C4—46no—
268C2×C4—46no—
269C2×C4—46no—
270C2×C4—46no—
271C2×C4—43no—
272C2×C4—43no—
273C2×C4—43no—
274C2×C4—43no—
275C2×C4—43no—
276C2×C4—43no—
277C2×C4—46no—
278C2×C4—46no—
279C2×C4—43no—
280C2×C4—46no—
281C2×C4—43no—
282C2×C4—46no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—46no—
288C2×C4—43no—
289C2×C4—43no—
290C2×C4—46no—
291C2×C4—46no—
292C2×C4—46no—
293C2×C4—43no—
294C2×C4—43no—
295C2×C4—46no—
296C2×C4—46no—
297C2×C4—46no—
298C2×C4—43no—
299C2×C4—43no—
300C2×C4—46no—
301C2×C4—43no—
302C2×C4—43no—
303C2×C4—43no—
304C2×C4—46no—
305C2×C4—43no—
306C2×C4—46no—
307C2×C4C2×#125112yes—
308C2×C4#544112yes—
309C2×C4#544112yes—
310C2×C4#56014yes—
311C2×C4#563112yes—
312C22C22×#12511yescharacteristic
313C22C2×#54413yes—
314C22C2×#54413yes—
315C22#350813yes—
316C22#350913yes—
317C22#351113yes—
318C22#351113yes—
319C22C2×#56011yescharacteristic
320C22C2×#56313yes—
321C22#354313yes—
322C22#354413yes—
323C22#355213yes—
324C22#355213yes—
325C4—212no—
326C4—212no—
327C4—82no—
328C4—86no—
329C4—81no—
330C4—83no—
331C4—83no—
332C4—86no—
333C4—81no—
334C4—82no—
335C4#284318yes—
336C2#3528613yes—
337C2C2×#284311yescharacteristic
338C2#3762313yes—
339C2#4601011yescharacteristic
340C2#4601113yes—
341C2#4601913yes—
342C2#4602011yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
247. C42#125semidirect

Maximal subgroups

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

subgroupindex
#165702
#165232
#152312
#152282
#151462
#151372
#146922
#146882
#146782
#146762
C4×#3832