Finite Group Databasebeta

← #10732473

Subgroups — #10732473

Order 768.

1066 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#1443C211yesmaximal, characteristic
2#69421C213yesmaximal
3#69416C211yesmaximal, characteristic
4#69414C213yesmaximal
5#69406C213yesmaximal
6#69406C213yesmaximal
7Dic3×#95C211yesmaximal, characteristic
8#15210—31nomaximal, Sylow
9C22×Dic3⋊C4C2211yescharacteristic
10C22×Dic3⋊C4C2213yes—
11C22×C4×Dic3C2213yes—
12C2×C3×#95C2213yes—
13C2×C3×#95C2213yes—
14C2×C3×#95C2211yescharacteristic
15C2×C3⋊#95C2213yes—
16C2×C3⋊#95C2213yes—
17C2×C3⋊#95C2213yes—
18C2×C3⋊#95C2213yes—
19C2×C3⋊#95C2213yes—
20C2×C3⋊#95C2213yes—
21C2×C3⋊#95C2213yes—
22#1445—33no—
23#1443—31no—
24#1443D311yescharacteristic
25#1442—33no—
26#1436—33no—
27#1436—33no—
28C4×#95—31no—
29C23×C3×C4C2313yes—
30C23×C3×C4C2311yescharacteristic
31C23×C3×C4C2313yes—
32C23×Dic3C2311yescharacteristic
33C23×Dic3C2313yes—
34C23×Dic3C2313yes—
35C23×Dic3C2311yescharacteristic
36C2×Dic3⋊C4—26no—
37C2×Dic3⋊C4—26no—
38C2×Dic3⋊C4—26no—
39C2×Dic3⋊C4—43no—
40C2×C4×Dic3—26no—
41C2×C4×Dic3—43no—
42C2×C4×Dic3—43no—
43C3×#95—43no—
44C3×#95—43no—
45C3×#95Q814yes—
46C3⋊#95—26no—
47C3⋊#95—26no—
48C3⋊#95—26no—
49C3⋊#95—26no—
50C3⋊#95—26no—
51C3⋊#95—26no—
52C3⋊#95—43no—
53C22×C4⋊C4—31no—
54C22×C4⋊C4—33no—
55C22×C42—33no—
56C2×#95—33no—
57C2×#95—33no—
58C2×#95—33no—
59C2×#95—33no—
60C2×#95—33no—
61C2×#95—33no—
62C2×#95—33no—
63C2×#95C2×D313yes—
64C2×#95C2×D313yes—
65C2×#95C2×D311yescharacteristic
66C24×C3C2411yescharacteristic
67C22×C3×C4—23no—
68C22×C3×C4—23no—
69C22×C3×C4—23no—
70C22×C3×C4—23no—
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—23no—
78C22×C3×C4—23no—
79C22×C3×C4—23no—
80C22×C3×C4—23no—
81C22×C3×C4—23no—
82C22×C3×C4—21no—
83C22×C3×C4C2×Q816yes—
84C22×Dic3—23no—
85C22×Dic3—21no—
86C22×Dic3—23no—
87C22×Dic3—23no—
88C22×Dic3—23no—
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#4112yes—
101C22×Dic3#4116yes—
102C22×Dic3#4116yes—
103C22×Dic3#4116yes—
104Dic3⋊C4—412no—
105Dic3⋊C4—44no—
106C4×Dic3—412no—
107C23×C4—31no—
108C23×C4—31no—
109C23×C4—33no—
110C23×C4—33no—
111C23×C4C22×D313yes—
112C23×C4C22×D311yescharacteristic
113C23×C4C22×D313yes—
114C2×C4⋊C4—66no—
115C2×C4⋊C4—66no—
116C2×C4⋊C4—66no—
117C2×C4⋊C4—123no—
118C2×C42—66no—
119C2×C42—123no—
120C2×C42—123no—
121#95—43no—
122#95—43no—
123#95—66no—
124#95—66no—
125#95—66no—
126#95—66no—
127#95—66no—
128#95—66no—
129#95—123no—
130#95Dic614yes—
131C23×C3C22×Q811yescharacteristic
132C23×C3C2×#4113yes—
133C23×C3C2×#4113yes—
134C23×C3C2×#4111yescharacteristic
135C23×C3C2×#4113yes—
136C23×C3H(5,𝔽2)13yes—
137C23×C32−1+411yescharacteristic
138C2×C3×C4—26no—
139C2×C3×C4—26no—
140C2×C3×C4—26no—
141C2×C3×C4—43no—
142C2×C3×C4—43no—
143C2×C3×C4—43no—
144C2×C3×C4—43no—
145C2×C3×C4—41no—
146C2×C3×C4—43no—
147C2×C3×C4—43no—
148C2×C3×C4—43no—
149C2×C3×C4—43no—
150C2×C3×C4—43no—
151C2×C3×C4—43no—
152C2×C3×C4—43no—
153C2×C3×C4—43no—
154C2×C3×C4—43no—
155C2×Dic3—26no—
156C2×Dic3—26no—
157C2×Dic3—26no—
158C2×Dic3—26no—
159C2×Dic3—26no—
160C2×Dic3—26no—
161C2×Dic3—26no—
162C2×Dic3—26no—
163C2×Dic3—26no—
164C2×Dic3—41no—
165C2×Dic3—43no—
166C2×Dic3—43no—
167C2×Dic3—43no—
168C2×Dic3—43no—
169C2×Dic3—43no—
170C2×Dic3—43no—
171C2×Dic3—43no—
172C2×Dic3—43no—
173C2×Dic3—43no—
174C2×Dic3#12614yes—
175C24C23×D311yescharacteristic
176C22×C4—23no—
177C22×C4—23no—
178C22×C4—23no—
179C22×C4—23no—
180C22×C4—23no—
181C22×C4—23no—
182C22×C4—23no—
183C22×C4—23no—
184C22×C4—23no—
185C22×C4—21no—
186C22×C4—23no—
187C22×C4—23no—
188C22×C4—23no—
189C22×C4—23no—
190C22×C4—23no—
191C22×C4—23no—
192C22×C4—36no—
193C22×C4—32no—
194C22×C4—36no—
195C22×C4—36no—
196C22×C4—63no—
197C22×C4—61no—
198C22×C4—63no—
199C22×C4—63no—
200C22×C4—63no—
201C22×C4—63no—
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×C4C2×Dic616yes—
213C4⋊C4—124no—
214C4⋊C4—1212no—
215C42—1212no—
216C22×C3C2×#12611yescharacteristic
217C22×C3#52913yes—
218C22×C3#52913yes—
219C22×C3#52913yes—
220C22×C3#53313yes—
221C22×C3#53813yes—
222C22×C3#53913yes—
223C22×C3#55113yes—
224C22×C3#55213yes—
225C22×C3#55313yes—
226C22×C3#55413yes—
227C22×C3#55711yescharacteristic
228C22×C3#55913yes—
229C3×C4—46no—
230C3×C4—83no—
231C3×C4—81no—
232Dic3—24no—
233Dic3—46no—
234Dic3—46no—
235Dic3—42no—
236C23C22×Dic611yescharacteristic
237C23#100413yes—
238C23#100711yescharacteristic
239C23D3×#4113yes—
240C23D3×#4111yescharacteristic
241C23D3×#4113yes—
242C23D3×#4113yes—
243C2×C4—26no—
244C2×C4—26no—
245C2×C4—26no—
246C2×C4—34no—
247C2×C4—43no—
248C2×C4—43no—
249C2×C4—43no—
250C2×C4—43no—
251C2×C4—43no—
252C2×C4—43no—
253C2×C4—43no—
254C2×C4—43no—
255C2×C4—43no—
256C2×C4—41no—
257C2×C4—43no—
258C2×C4—43no—
259C2×C4—43no—
260C2×C4—43no—
261C2×C4—66no—
262C2×C4—66no—
263C2×C4—66no—
264C2×C4—66no—
265C2×C4—66no—
266C2×C4—66no—
267C2×C4—66no—
268C2×C4—66no—
269C2×C4—66no—
270C2×C4—123no—
271C2×C4—123no—
272C2×C4—123no—
273C2×C4—123no—
274C2×C4—123no—
275C2×C4—121no—
276C2×C4—123no—
277C2×C4—123no—
278C2×C4—123no—
279C2×C4—123no—
280C2×C3#240111yescharacteristic
281C2×C3#257513yes—
282C2×C3#258913yes—
283C2×C3#259013yes—
284C2×C3#259513yes—
285C2×C3#264311yescharacteristic
286C2×C3#286311yescharacteristic
287C22#551713yes—
288C22#553913yes—
289C22#556911yescharacteristic
290C22#561313yes—
291C22#563813yes—
292C22#566113yes—
293C22#566413yes—
294C22#567913yes—
295C22#569313yes—
296C22D3×#12611yescharacteristic
297C22#570613yes—
298C22#570613yes—
299C22#570613yes—
300C4—46no—
301C4—64no—
302C4—81no—
303C4—83no—
304C4—122no—
305C4—126no—
306C4—126no—
307C3#1521011yescharacteristic, complemented, Sylow
308C2#7429311yescharacteristic
309C2#7511113yes—
310C2#7561813yes—
311C2#7742511yescharacteristic
312C2#7752813yes—
313C2#7818713yes—
314C2#7840911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC3×#1443C2
hypercenter#1443D3
hypocenterC3#15210
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1443

Complements

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

NH ≅ G/Ntype
307. C3#15210semidirect

Maximal subgroups

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

subgroupindex
C3×#14432
#694212
#694162
#694142
#694062
Dic3×#952
#152103