Finite Group Databasebeta

← #11660563

Subgroups — #11660563

Order 768.

2038 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×#41C211yesmaximal, characteristic
2#82523C218yesmaximal, complemented
3#82519C218yesmaximal, complemented
4#82435C212yesmaximal, complemented
5#82304C216yesmaximal, complemented
6#82049C216yesmaximal, complemented
7#60284—31nomaximal, Sylow
8C22×C3×#41C411yescharacteristic
9C3×C4×#41C414yes—
10C3×C4×#41C414yes—
11C3×C4×#41C2214yes—
12C3×C4×#41C2214yes—
13C2×C3×C4×Q8C411yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C2×C3×C4×D4C413yes—
16C2×C3×C4×D4C2213yes—
17C2×C3×#117C413yes—
18C22×C3×C42C2213yes—
19#5207C22112yes—
20Q8×C3⋊C8C22116yescomplemented
21#5012C22148yescomplemented
22#4926C2214yes—
23#4926C22112yescomplemented
24#4925C22112yes—
25#4925C22112yescomplemented
26#4924C22112yescomplemented
27C4×#208C22112yescomplemented
28C2×C4×#41D311yescharacteristic
29#2991—38no—
30#2982—32no—
31#2975—36no—
32#2971—38no—
33#2920—36no—
34C2×C3×#41—28no—
35C2×C3×#41C2×C414yes—
36C2×C3×#41C2×C414yes—
37C22×C3×Q8C2×C411yescharacteristic
38C22×C3×D4C2×C413yes—
39C23×C3×C4C2×C413yes—
40C23×C3×C4C2×C413yes—
41C23×C3×C4C2313yes—
42C3×C4×Q8C2×C414yes—
43C3×C4×Q8C2×C414yes—
44C3×C4×Q8C2×C414yes—
45C3×C4×Q8C2314yes—
46C3×C4×D4C2×C4112yes—
47C3×C4×D4C2×C4112yes—
48C3×C4×D4C2×C4112yes—
49C3×C4×D4C23112yes—
50C3×#117C2×C4112yes—
51C3×#117C2×C4112yes—
52C2×C3×C4⋊C4C2×C413yes—
53C2×C3×C4⋊C4C2×C413yes—
54C2×C3×C22⋊C4C2×C416yes—
55C2×C3×C42C2×C413yes—
56C2×C3×C42C2×C413yes—
57C2×C3×C42C2×C416yes—
58C2×C3×C42C2×C416yes—
59C2×C3×C42C2316yes—
60C2×C3×C42C2316yes—
61C2×C3×C42C2313yes—
62C2×C3×C42C2311yescharacteristic
63C2×#208C23124yes—
64#830C2318yes—
65#830C23124yes—
66C4⋊(C3⋊C8)C2318yes—
67C4⋊(C3⋊C8)C23124yes—
68#803C2318yes—
69C4×C3⋊C8C23124yes—
70C22×#41Dic311yescharacteristic
71C4×#41Dic314yes—
72C4×#41Dic314yes—
73C4×#41C2×D314yes—
74C4×#41C2×D314yes—
75C2×C4×Q8Dic311yescharacteristic
76C2×C4×Q8C2×D311yescharacteristic
77C2×C4×D4Dic313yes—
78C2×C4×D4C2×D313yes—
79C2×#117Dic313yes—
80C22×C42C2×D313yes—
81C8×Q8—316no—
82#436—348no—
83#432—34no—
84#432—312no—
85#431—312no—
86#431—312no—
87#423—312no—
88#407—312no—
89C4×M16—312no—
90C24×C3C22×C413yes—
91C3×#41—216no—
92C3×#41—216no—
93C2×C3×Q8M1618yescomplemented
94C2×C3×Q8C22×C414yes—
95C2×C3×D4—212no—
96C2×C3×D4C22×C4112yes—
97C22×C3×C4—212no—
98C22×C3×C4—46no—
99C22×C3×C4—46no—
100C22×C3×C4C22×C411yescharacteristic
101C22×C3×C4C22×C411yescharacteristic
102C22×C3×C4C22×C416yes—
103C22×C3×C4C22×C416yes—
104C22×C3×C4C22×C416yes—
105C22×C3×C4C22×C413yes—
106C22×C3×C4C22×C416yes—
107C22×C3×C4C22×C413yes—
108C22×C3×C4C22×C416yes—
109C22×C3×C4C22×C413yes—
110C22×C3×C4C2411yescharacteristic
111C22×C3×C4C2416yes—
112C3×C4⋊C4C22×C4112yes—
113C3×C4⋊C4C22×C4112yes—
114C3×C22⋊C4C22×C4124yes—
115C3×C42C22×C411yescharacteristic
116C3×C42C22×C411yescharacteristic
117C3×C42C22×C416yes—
118C3×C42C22×C416yes—
119C3×C42C22×C416yes—
120C3×C42C22×C416yes—
121C3×C42C22×C416yes—
122C3×C42C2416yes—
123C3×C42C2411yescharacteristic
124C3×C42C2411yescharacteristic
125#208—424no—
126C2×C3⋊C8C24116yes—
127C2×#41—28no—
128C2×#41C2×Dic314yes—
129C2×#41C2×Dic314yes—
130C22×Q8C2×Dic311yescharacteristic
131C22×D4C2×Dic313yes—
132C23×C4C2×Dic313yes—
133C23×C4C2×Dic313yes—
134C23×C4C22×D313yes—
135C2×M16—324no—
136C4×Q8C2×Dic314yes—
137C4×Q8C2×Dic314yes—
138C4×Q8C2×Dic314yes—
139C4×Q8C22×D314yes—
140C4×D4C2×Dic3112yes—
141C4×D4C2×Dic3112yes—
142C4×D4C2×Dic3112yes—
143C4×D4C22×D3112yes—
144#117C2×Dic3112yes—
145#117C2×Dic3112yes—
146C2×C4⋊C4C2×Dic313yes—
147C2×C4⋊C4C2×Dic313yes—
148C2×C22⋊C4C2×Dic316yes—
149C2×C42C2×Dic313yes—
150C2×C42C2×Dic313yes—
151C2×C42C2×Dic316yes—
152C2×C42C2×Dic316yes—
153C2×C42C22×D313yes—
154C2×C42C22×D316yes—
155C2×C42C22×D316yes—
156C2×C42C22×D311yescharacteristic
157C4⋊C8—38no—
158C4⋊C8—324no—
159C22⋊C8—38no—
160C22⋊C8—324no—
161#97—38no—
162C4×C8—324no—
163C23×C3—26no—
164C23×C3—43no—
165C23×C3—43no—
166C23×C3C23×C416yes—
167C23×C3C23×C411yescharacteristic
168C3×Q8C2×M16116yes—
169C3×D4—224no—
170C2×C3×C4—22no—
171C2×C3×C4—212no—
172C2×C3×C4—212no—
173C2×C3×C4—22no—
174C2×C3×C4—212no—
175C2×C3×C4—212no—
176C2×C3×C4—22no—
177C2×C3×C4—46no—
178C2×C3×C4—43no—
179C2×C3×C4—46no—
180C2×C3×C4—43no—
181C2×C3×C4C2×M16112yes—
182C2×C3×C4C23×C411yescharacteristic
183C2×C3×C4C23×C411yescharacteristic
184C2×C3×C4C23×C411yescharacteristic
185C2×C3×C4C23×C411yescharacteristic
186C2×C3×C4C23×C416yes—
187C2×C3×C4C23×C416yes—
188C2×C3×C4C23×C416yes—
189C2×C3×C4C23×C411yescharacteristic
190C2×C3×C4C2511yescharacteristic
191C3⋊C8—216no—
192C24C22×Dic313yes—
193#41—216no—
194#41—216no—
195C2×Q8#20818yescomplemented
196C2×Q8C22×Dic314yes—
197C2×D4—212no—
198C2×D4C22×Dic3112yes—
199C22×C4—212no—
200C22×C4—46no—
201C22×C4—46no—
202C22×C4C22×Dic311yescharacteristic
203C22×C4C22×Dic311yescharacteristic
204C22×C4C22×Dic316yes—
205C22×C4C22×Dic316yes—
206C22×C4C22×Dic316yes—
207C22×C4C22×Dic313yes—
208C22×C4C22×Dic316yes—
209C22×C4C22×Dic313yes—
210C22×C4C22×Dic316yes—
211C22×C4C22×Dic313yes—
212C22×C4C23×D311yescharacteristic
213C22×C4C23×D316yes—
214M16—1224no—
215C2×C8—316no—
216C4⋊C4C22×Dic3112yes—
217C4⋊C4C22×Dic3112yes—
218C22⋊C4C22×Dic3124yes—
219C42C22×Dic311yescharacteristic
220C42C22×Dic311yescharacteristic
221C42C22×Dic316yes—
222C42C22×Dic316yes—
223C42C22×Dic316yes—
224C42C22×Dic316yes—
225C42C22×Dic316yes—
226C42C23×D316yes—
227C42C23×D311yescharacteristic
228C42C23×D311yescharacteristic
229C22×C3—21no—
230C22×C3—21no—
231C22×C3—26no—
232C22×C3—26no—
233C22×C3—26no—
234C22×C3—412no—
235C22×C3C22×M1612yes—
236C22×C3C24×C411yescharacteristic
237C3×C4—21no—
238C3×C4—26no—
239C3×C4—21no—
240C3×C4—21no—
241C3×C4—21no—
242C3×C4—26no—
243C3×C4C22×M16112yes—
244C3×C4#56812yes—
245C3×C4#58512yes—
246C23—26no—
247C23—43no—
248C23—43no—
249C23C23×Dic316yes—
250C23C23×Dic311yescharacteristic
251Q8C2×#208116yes—
252D4—224no—
253C2×C4—22no—
254C2×C4—212no—
255C2×C4—22no—
256C2×C4—212no—
257C2×C4—212no—
258C2×C4—22no—
259C2×C4—212no—
260C2×C4—46no—
261C2×C4—43no—
262C2×C4—46no—
263C2×C4—43no—
264C2×C4C2×#208112yes—
265C2×C4C23×Dic311yescharacteristic
266C2×C4C23×Dic311yescharacteristic
267C2×C4C23×Dic311yescharacteristic
268C2×C4C23×Dic311yescharacteristic
269C2×C4C23×Dic316yes—
270C2×C4C23×Dic316yes—
271C2×C4C23×Dic316yes—
272C2×C4C23×Dic311yescharacteristic
273C2×C4C24×D311yescharacteristic
274C8—616no—
275C2×C3—22no—
276C2×C3—46no—
277C2×C3#342811yescharacteristic
278C2×C3C23×M1611yescharacteristic
279C2×C3C2×#56811yescharacteristic
280C22—21no—
281C22—21no—
282C22—26no—
283C22—26no—
284C22—26no—
285C22—412no—
286C22C22×#20812yes—
287C22C24×Dic311yescharacteristic
288C4—21no—
289C4—21no—
290C4—21no—
291C4—26no—
292C4—21no—
293C4—26no—
294C4C22×#208112yes—
295C4#581912yes—
296C4#596412yes—
297C3#6028411yescharacteristic, complemented, Sylow
298C2—22no—
299C2—46no—
300C2C23×#20811yescharacteristic
301C2C2×#581911yescharacteristic
302C2#8726011yescharacteristic
subgroupG/H
center, FrattiniC2×C4C24×D3
commutator, socleC22×C3C24×C4
FittingC2×C3×C4×#41C2
prime radicalC22×C3×#41C4
cosocleC2×C3×C4C25
hypercenterC2×C4×#41D3
hypocenterC3#60284
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C4×#41

Complements

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

NH ≅ G/Ntype
2. #82523C2semidirect
3. #82519C2semidirect
4. #82435C2semidirect
5. #82304C2semidirect
6. #82049C2semidirect
20. Q8×C3⋊C8C22semidirect
21. #5012C22semidirect
23. #4926C22semidirect
25. #4925C22semidirect
26. #4924C22semidirect
27. C4×#208C22semidirect
93. C2×C3×Q8M16semidirect
195. C2×Q8#208semidirect
297. C3#60284semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×#412
#825232
#825192
#824352
#823042
#820492
#602843