Finite Group Databasebeta

← #4963789

Subgroups — #4963789

Order 512.

2360 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16385C216yesmaximal, complemented
2#15994C216yesmaximal, complemented
3#15695C216yesmaximal, complemented
4#14829C216yesmaximal, complemented
5#14501C213yesmaximal, complemented
6#13937C213yesmaximal
7#13834C211yesmaximal, characteristic, complemented
8C23×#41C2211yescharacteristic
9C2×#397C2216yes—
10C2×#391C2213yes—
11C2×#391C2213yes—
12C2×#390C2213yes—
13C2×#390C2213yes—
14C2×#383C2212yes—
15C2×#380C2213yes—
16C2×C4×C22⋊C4C2213yes—
17#1443C2216yes—
18#1443C2212yes—
19#1442C2216yes—
20#1442C2216yes—
21#1442C2216yes—
22#1441C2216yes—
23#1440C2216yes—
24#1440C2216yes—
25#1440C2216yes—
26#1440C2216yes—
27#1439C2216yes—
28#1439C2216yes—
29#1438C2216yes—
30#1438C2216yes—
31#1437C2216yes—
32#1436C2216yes—
33#1436C2216yes—
34#1436C2216yes—
35#1434C2216yes—
36#1434C2216yes—
37C22⋊#95C2216yes—
38C22⋊#95C2216yes—
39C22×#41—212no—
40C22×#41—42no—
41C22×#41—46no—
42C23×Q8C2311yescharacteristic
43C23×D4C2313yes—
44C24×C4C2313yes—
45C22×C4⋊C4C2313yes—
46C22×C4⋊C4C2313yes—
47C22×C4⋊C4C2316yes—
48C22×C22⋊C4C2316yes—
49C22×C22⋊C4C2316yes—
50C22×C22⋊C4C2316yes—
51C22×C22⋊C4C2316yes—
52C22×C22⋊C4C2316yes—
53C22×C22⋊C4C2316yes—
54C22×C42C2313yes—
55C22×C42C2313yes—
56C22×C42C2312yes—
57#397—412no—
58#391—212no—
59#391—212no—
60#390—46no—
61#390—46no—
62#383—44no—
63#380—46no—
64C4×C22⋊C4—46no—
65C2×#95C2316yes—
66C2×#95C2316yes—
67C2×#95C2316yes—
68C2×#95C2316yes—
69C2×#95C2316yes—
70C2×#95C2316yes—
71C2×#95C2316yes—
72C2×#95C2316yes—
73C2×#95C2316yes—
74C2×#95C2312yes—
75C2×#95C2316yes—
76C2×#95C2316yes—
77C2×#95C2316yes—
78C2×#95C2316yes—
79C2×#95C2316yes—
80C2×#95C2316yes—
81C25C2413yes—
82C2×#41—448no—
83C2×#41—448no—
84C2×#41—416no—
85C22×Q8—43no—
86C22×Q8—41no—
87C22×Q8#41112yes—
88C22×D4—26no—
89C22×D4—26no—
90C22×D4—26no—
91C22×D4—26no—
92C22×D4—26no—
93C22×D4—43no—
94C22×D4—43no—
95C23×C4—26no—
96C23×C4—26no—
97C23×C4—26no—
98C23×C4—26no—
99C23×C4—26no—
100C23×C4—43no—
101C23×C4—43no—
102C23×C4C2416yes—
103C23×C4C2416yes—
104C23×C4C2416yes—
105C23×C4C2416yes—
106C23×C4C2411yescharacteristic
107C23×C4C2413yes—
108C2×C4⋊C4—26no—
109C2×C4⋊C4—26no—
110C2×C4⋊C4—46no—
111C2×C4⋊C4—46no—
112C2×C4⋊C4—43no—
113C2×C4⋊C4—43no—
114C2×C4⋊C4—43no—
115C2×C4⋊C4—43no—
116C2×C4⋊C4—46no—
117C2×C22⋊C4—212no—
118C2×C22⋊C4—212no—
119C2×C22⋊C4—212no—
120C2×C22⋊C4—46no—
121C2×C22⋊C4—46no—
122C2×C22⋊C4—46no—
123C2×C22⋊C4—46no—
124C2×C22⋊C4—46no—
125C2×C22⋊C4—46no—
126C2×C22⋊C4—46no—
127C2×C22⋊C4—46no—
128C2×C22⋊C4—46no—
129C2×C22⋊C4—46no—
130C2×C22⋊C4—46no—
131C2×C22⋊C4—46no—
132C2×C22⋊C4—46no—
133C2×C22⋊C4—46no—
134C2×C22⋊C4—46no—
135C2×C42—26no—
136C2×C42—26no—
137C2×C42—42no—
138C2×C42—42no—
139C2×C42—42no—
140C2×C42—43no—
141C2×C42—43no—
142C2×C42—43no—
143C2×C42—43no—
144#95—212no—
145#95—212no—
146#95—212no—
147#95—212no—
148#95—212no—
149#95—42no—
150#95—46no—
151#95—46no—
152#95—46no—
153#95—46no—
154#95—46no—
155#95—46no—
156#95—46no—
157#95—46no—
158#95—46no—
159#95—46no—
160C24—23no—
161C24—23no—
162C24—23no—
163C24—23no—
164C24—23no—
165C24—23no—
166C24—26no—
167C24—26no—
168C24—26no—
169C24—26no—
170C24H(5,𝔽2)16yes—
171C24C2511yescharacteristic
172#41—4128no—
173C2×Q8—412no—
174C2×Q8—412no—
175C2×Q8#126116yes—
176C2×D4—224no—
177C2×D4—412no—
178C2×D4—412no—
179C2×D4—412no—
180C2×D4—412no—
181C2×D4—412no—
182C2×D4—412no—
183C22×C4—23no—
184C22×C4—23no—
185C22×C4—23no—
186C22×C4—23no—
187C22×C4—224no—
188C22×C4—23no—
189C22×C4—23no—
190C22×C4—26no—
191C22×C4—26no—
192C22×C4—26no—
193C22×C4—26no—
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—26no—
207C22×C4—26no—
208C22×C4—26no—
209C22×C4—26no—
210C22×C4—26no—
211C22×C4—26no—
212C22×C4—26no—
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—412no—
223C22×C4—412no—
224C22×C4C2×#4116yes—
225C22×C4C2×#4116yes—
226C22×C4C2×#4116yes—
227C22×C4H(5,𝔽2)12yes—
228C4⋊C4—86no—
229C4⋊C4—86no—
230C4⋊C4—812no—
231C22⋊C4—424no—
232C22⋊C4—812no—
233C22⋊C4—812no—
234C22⋊C4—812no—
235C22⋊C4—812no—
236C22⋊C4—812no—
237C42—48no—
238C42—86no—
239C42—86no—
240C23—26no—
241C23—26no—
242C23—26no—
243C23—212no—
244C23—212no—
245C23—43no—
246C23—43no—
247C23—43no—
248C23—43no—
249C23—46no—
250C23—46no—
251C23—46no—
252C23—46no—
253C23—46no—
254C23—46no—
255C23—43no—
256C23—46no—
257C23—46no—
258C23—43no—
259C23—43no—
260C23—43no—
261C23—46no—
262C23—46no—
263C23C22×#4113yes—
264C23C2×H(5,𝔽2)11yescharacteristic
265C23C2×H(5,𝔽2)13yes—
266C23#58512yes—
267C23#58516yes—
268Q8—416no—
269D4—448no—
270C2×C4—26no—
271C2×C4—26no—
272C2×C4—26no—
273C2×C4—26no—
274C2×C4—26no—
275C2×C4—26no—
276C2×C4—26no—
277C2×C4—448no—
278C2×C4—41no—
279C2×C4—43no—
280C2×C4—46no—
281C2×C4—46no—
282C2×C4—46no—
283C2×C4—46no—
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—46no—
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—46no—
307C2×C4—46no—
308C2×C4C2×#126112yes—
309C22—46no—
310C22—412no—
311C22—412no—
312C22—46no—
313C22—46no—
314C22—83no—
315C22—83no—
316C22—86no—
317C22—83no—
318C22—83no—
319C22—86no—
320C22C22×#12611yescharacteristic
321C22#349013yes—
322C22#349013yes—
323C22#352416yes—
324C22#352416yes—
325C22#352911yescharacteristic
326C22#353316yes—
327C22#355113yes—
328C22#356016yes—
329C4—212no—
330C4—42no—
331C4—86no—
332C4—86no—
333C4—86no—
334C4—86no—
335C2—86no—
336C2#3536911yescharacteristic
337C2#3605912yes—
338C2#4627513yes—
339C2#4867316yes—
340C2#5198413yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×#41C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #16385C2semidirect
2. #15994C2semidirect
3. #15695C2semidirect
4. #14829C2semidirect
5. #14501C2semidirect
7. #13834C2semidirect

Maximal subgroups

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

subgroupindex
#163852
#159942
#156952
#148292
#145012
#139372
#138342