Finite Group Databasebeta

← #10729002

Subgroups — #10729002

Order 768.

1130 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#401C211yesmaximal, characteristic
2C2×#4648C211yesmaximal, characteristic
3#69421C213yesmaximal
4#69416C211yesmaximal, characteristic
5#69413C213yesmaximal
6#69406C213yesmaximal
7#69406C213yesmaximal
8#14909—31nomaximal, Sylow
9C22×Dic3⋊C4C2211yescharacteristic
10C22×Dic3⋊C4C2213yes—
11C22×C4×Dic3C2213yes—
12C3×#401—24no—
13C2×C3×#95C2213yes—
14C2×C3×#95C2213yes—
15C2×C3×#95C2211yescharacteristic
16C2×C3⋊#95C2213yes—
17C2×C3⋊#95C2213yes—
18C2×C3⋊#95C2213yes—
19C2×C3⋊#95C2213yes—
20C2×C3⋊#95C2213yes—
21C2×C3⋊#95C2213yes—
22C2×C3⋊#95C2213yes—
23#4648—24no—
24C2×#401D311yescharacteristic
25C2×#376—31no—
26#1445—33no—
27#1443—31no—
28#1442—33no—
29#1436—33no—
30#1436—33no—
31C23×C3×C4C2311yescharacteristic
32C23×C3×C4C2313yes—
33C23×C3×C4C2313yes—
34C23×Dic3C2311yescharacteristic
35C23×Dic3C2313yes—
36C23×Dic3C2313yes—
37C23×Dic3C2311yescharacteristic
38C2×Dic3⋊C4—26no—
39C2×Dic3⋊C4—26no—
40C2×Dic3⋊C4—26no—
41C2×Dic3⋊C4—43no—
42C2×C4×Dic3—26no—
43C2×C4×Dic3—26no—
44C2×C4×Dic3—26no—
45C3×#95—26no—
46C3×#95—26no—
47C3×#95D414yes—
48C3⋊#95—26no—
49C3⋊#95—26no—
50C3⋊#95—26no—
51C3⋊#95—26no—
52C3⋊#95—26no—
53C3⋊#95—26no—
54C3⋊#95—43no—
55C22×C4⋊C4—31no—
56C22×C4⋊C4—33no—
57C22×C42—33no—
58#401—24no—
59#376—64no—
60C2×#95—33no—
61C2×#95—33no—
62C2×#95—33no—
63C2×#95—33no—
64C2×#95—33no—
65C2×#95—33no—
66C2×#95—33no—
67C2×#95C2×D313yes—
68C2×#95C2×D313yes—
69C2×#95C2×D311yescharacteristic
70C24×C3C2411yescharacteristic
71C22×C3×C4—23no—
72C22×C3×C4—23no—
73C22×C3×C4—23no—
74C22×C3×C4—23no—
75C22×C3×C4—23no—
76C22×C3×C4—23no—
77C22×C3×C4—21no—
78C22×C3×C4—23no—
79C22×C3×C4—23no—
80C22×C3×C4—23no—
81C22×C3×C4—23no—
82C22×C3×C4—23no—
83C22×C3×C4—23no—
84C22×C3×C4—23no—
85C22×C3×C4—23no—
86C22×C3×C4—23no—
87C22×C3×C4C2×D416yes—
88C22×Dic3—21no—
89C22×Dic3—23no—
90C22×Dic3—23no—
91C22×Dic3—23no—
92C22×Dic3—23no—
93C22×Dic3—23no—
94C22×Dic3—23no—
95C22×Dic3—23no—
96C22×Dic3—23no—
97C22×Dic3—23no—
98C22×Dic3—23no—
99C22×Dic3—23no—
100C22×Dic3—23no—
101C22×Dic3—23no—
102C22×Dic3—23no—
103C22×Dic3—23no—
104C22×Dic3#4112yes—
105C22×Dic3#4116yes—
106C22×Dic3#4116yes—
107C22×Dic3#4116yes—
108Dic3⋊C4—412no—
109Dic3⋊C4—44no—
110C4×Dic3—224no—
111C23×C4—31no—
112C23×C4—31no—
113C23×C4—33no—
114C23×C4—33no—
115C23×C4C22×D311yescharacteristic
116C23×C4C22×D313yes—
117C23×C4C22×D313yes—
118C2×C4⋊C4—66no—
119C2×C4⋊C4—66no—
120C2×C4⋊C4—66no—
121C2×C4⋊C4—123no—
122C2×C42—66no—
123C2×C42—66no—
124C2×C42—66no—
125#95—26no—
126#95—26no—
127#95—66no—
128#95—66no—
129#95—66no—
130#95—66no—
131#95—66no—
132#95—66no—
133#95—123no—
134#95#6714yes—
135C23×C3C22×D411yescharacteristic
136C23×C3C2×#4113yes—
137C23×C3C2×#4113yes—
138C23×C3C2×#4113yes—
139C23×C3C2×#4111yescharacteristic
140C23×C3H(5,𝔽2)11yescharacteristic
141C23×C32−1+413yes—
142C2×C3×C4—26no—
143C2×C3×C4—26no—
144C2×C3×C4—26no—
145C2×C3×C4—43no—
146C2×C3×C4—43no—
147C2×C3×C4—43no—
148C2×C3×C4—43no—
149C2×C3×C4—43no—
150C2×C3×C4—43no—
151C2×C3×C4—41no—
152C2×C3×C4—43no—
153C2×C3×C4—43no—
154C2×C3×C4—43no—
155C2×C3×C4—43no—
156C2×C3×C4—43no—
157C2×C3×C4—43no—
158C2×C3×C4—43no—
159C2×Dic3—26no—
160C2×Dic3—26no—
161C2×Dic3—26no—
162C2×Dic3—26no—
163C2×Dic3—26no—
164C2×Dic3—26no—
165C2×Dic3—26no—
166C2×Dic3—26no—
167C2×Dic3—26no—
168C2×Dic3—41no—
169C2×Dic3—43no—
170C2×Dic3—43no—
171C2×Dic3—43no—
172C2×Dic3—43no—
173C2×Dic3—43no—
174C2×Dic3—43no—
175C2×Dic3—43no—
176C2×Dic3—43no—
177C2×Dic3—43no—
178C2×Dic3#12614yes—
179C24C23×D311yescharacteristic
180C22×C4—23no—
181C22×C4—23no—
182C22×C4—23no—
183C22×C4—23no—
184C22×C4—23no—
185C22×C4—23no—
186C22×C4—23no—
187C22×C4—23no—
188C22×C4—23no—
189C22×C4—23no—
190C22×C4—23no—
191C22×C4—23no—
192C22×C4—23no—
193C22×C4—23no—
194C22×C4—23no—
195C22×C4—21no—
196C22×C4—36no—
197C22×C4—32no—
198C22×C4—36no—
199C22×C4—36no—
200C22×C4—63no—
201C22×C4—61no—
202C22×C4—63no—
203C22×C4—63no—
204C22×C4—63no—
205C22×C4—63no—
206C22×C4—63no—
207C22×C4—63no—
208C22×C4—63no—
209C22×C4—63no—
210C22×C4—63no—
211C22×C4—63no—
212C22×C4—63no—
213C22×C4—63no—
214C22×C4—63no—
215C22×C4—63no—
216C22×C4C2×#6716yes—
217C4⋊C4—124no—
218C4⋊C4—1212no—
219C42—624no—
220C22×C3#52513yes—
221C22×C3C2×#12611yescharacteristic
222C22×C3#52913yes—
223C22×C3#52913yes—
224C22×C3#52913yes—
225C22×C3#54113yes—
226C22×C3#54213yes—
227C22×C3#54713yes—
228C22×C3#55011yescharacteristic
229C22×C3#55113yes—
230C22×C3#55213yes—
231C22×C3#55213yes—
232C22×C3#55613yes—
233C3×C4—46no—
234C3×C4—81no—
235C3×C4—83no—
236Dic3—24no—
237Dic3—42no—
238Dic3—46no—
239Dic3—46no—
240C23D3×#4113yes—
241C23D3×#4113yes—
242C23D3×#4113yes—
243C23D3×#4111yescharacteristic
244C23#100911yescharacteristic
245C23#101013yes—
246C23C22×#6711yescharacteristic
247C2×C4—26no—
248C2×C4—26no—
249C2×C4—26no—
250C2×C4—34no—
251C2×C4—43no—
252C2×C4—43no—
253C2×C4—43no—
254C2×C4—43no—
255C2×C4—41no—
256C2×C4—43no—
257C2×C4—43no—
258C2×C4—43no—
259C2×C4—43no—
260C2×C4—43no—
261C2×C4—43no—
262C2×C4—43no—
263C2×C4—43no—
264C2×C4—43no—
265C2×C4—66no—
266C2×C4—66no—
267C2×C4—66no—
268C2×C4—66no—
269C2×C4—66no—
270C2×C4—66no—
271C2×C4—66no—
272C2×C4—66no—
273C2×C4—66no—
274C2×C4—123no—
275C2×C4—121no—
276C2×C4—123no—
277C2×C4—123no—
278C2×C4—123no—
279C2×C4—123no—
280C2×C4—123no—
281C2×C4—123no—
282C2×C4—123no—
283C2×C4—123no—
284C2×C3#240111yescharacteristic
285C2×C3#247813yes—
286C2×C3#247913yes—
287C2×C3#252713yes—
288C2×C3#252813yes—
289C2×C3#264511yescharacteristic
290C2×C3#285211yescharacteristic
291C22#570113yes—
292C22#570213yes—
293C22D3×#12611yescharacteristic
294C22#570513yes—
295C22#570613yes—
296C22#570613yes—
297C22#570613yes—
298C22#570813yes—
299C22#570813yes—
300C22#571013yes—
301C22#582413yes—
302C22#582813yes—
303C22#583411yescharacteristic
304C4—46no—
305C4—64no—
306C4—83no—
307C4—81no—
308C4—122no—
309C4—126no—
310C4—126no—
311C3#1490911yescharacteristic, complemented, Sylow
312C2#7840911yescharacteristic
313C2#7841213yes—
314C2#7841313yes—
315C2#7841611yescharacteristic
316C2#8307913yes—
317C2#8308013yes—
318C2#8312011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC2×C3×#401C2
hypercenterC2×#401D3
hypocenterC3#14909
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#401

Complements

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

NH ≅ G/Ntype
311. C3#14909semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4012
C2×#46482
#694212
#694162
#694132
#694062
#149093