Finite Group Databasebeta

← #10732472

Subgroups — #10732472

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×C4×#95C211yesmaximal, characteristic
2#71536C213yesmaximal
3#71536C213yesmaximal
4#69421C213yesmaximal
5#69416C211yesmaximal, characteristic
6#69416C211yesmaximal, characteristic
7#69414C213yesmaximal
8#15210—31nomaximal, Sylow
9C22×C3×C42C2213yes—
10C22×Dic3⋊C4C2211yescharacteristic
11C22×Dic3⋊C4C2213yes—
12C2×C3×#95C2213yes—
13C2×C3×#95C2211yescharacteristic
14C2×C3⋊#95C2213yes—
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#1443—31no—
25#1442—33no—
26#1436—33no—
27#1436—33no—
28C4×#95D311yescharacteristic
29C23×C3×C4C2313yes—
30C23×C3×C4C2313yes—
31C23×C3×C4C2311yescharacteristic
32C23×Dic3C2313yes—
33C23×Dic3C2311yescharacteristic
34C23×Dic3C2311yescharacteristic
35C23×Dic3C2313yes—
36C2×C3×C42—26no—
37C2×C3×C42—43no—
38C2×C3×C42—43no—
39C2×Dic3⋊C4—26no—
40C2×Dic3⋊C4—26no—
41C2×Dic3⋊C4—26no—
42C2×Dic3⋊C4—43no—
43C3×#95—26no—
44C3×#95Q814yes—
45C3⋊#95—26no—
46C3⋊#95—26no—
47C3⋊#95—26no—
48C3⋊#95—26no—
49C3⋊#95—26no—
50C3⋊#95—43no—
51C3⋊#95—43no—
52C3⋊#95—43no—
53C22×C4⋊C4—31no—
54C22×C4⋊C4—33no—
55C22×C42C2×D313yes—
56C2×#95—33no—
57C2×#95—33no—
58C2×#95—33no—
59C2×#95—33no—
60C2×#95—33no—
61C2×#95—33no—
62C2×#95—33no—
63C2×#95—33no—
64C2×#95C2×D313yes—
65C2×#95C2×D311yescharacteristic
66C24×C3C2411yescharacteristic
67C22×C3×C4—23no—
68C22×C3×C4—21no—
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×C4C2×Q816yes—
82C22×C3×C4#4116yes—
83C22×C3×C4#4116yes—
84C22×Dic3—23no—
85C22×Dic3—23no—
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—23no—
101C22×Dic3—21no—
102C22×Dic3#4112yes—
103C22×Dic3#4116yes—
104C3×C42—412no—
105Dic3⋊C4—412no—
106Dic3⋊C4—44no—
107C23×C4—31no—
108C23×C4—33no—
109C23×C4—33no—
110C23×C4—31no—
111C23×C4C22×D313yes—
112C23×C4C22×D313yes—
113C23×C4C22×D311yescharacteristic
114C2×C4⋊C4—66no—
115C2×C4⋊C4—66no—
116C2×C4⋊C4—66no—
117C2×C4⋊C4—123no—
118C2×C42—26no—
119C2×C42—43no—
120C2×C42—43no—
121#95—26no—
122#95—66no—
123#95—66no—
124#95—66no—
125#95—66no—
126#95—66no—
127#95—123no—
128#95—123no—
129#95—123no—
130#95Dic614yes—
131C23×C3C22×Q811yescharacteristic
132C23×C3C2×#4113yes—
133C23×C3C2×#4113yes—
134C23×C3C2×#4113yes—
135C23×C3C2×#4111yescharacteristic
136C23×C3H(5,𝔽2)13yes—
137C23×C32−1+411yescharacteristic
138C2×C3×C4—26no—
139C2×C3×C4—26no—
140C2×C3×C4—26no—
141C2×C3×C4—26no—
142C2×C3×C4—26no—
143C2×C3×C4—26no—
144C2×C3×C4—26no—
145C2×C3×C4—26no—
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#12614yes—
155C2×Dic3—26no—
156C2×Dic3—26no—
157C2×Dic3—26no—
158C2×Dic3—26no—
159C2×Dic3—41no—
160C2×Dic3—43no—
161C2×Dic3—43no—
162C2×Dic3—43no—
163C2×Dic3—43no—
164C2×Dic3—43no—
165C2×Dic3—41no—
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—43no—
175C24C23×D311yescharacteristic
176C22×C4—23no—
177C22×C4—21no—
178C22×C4—23no—
179C22×C4—23no—
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—32no—
191C22×C4—36no—
192C22×C4—63no—
193C22×C4—63no—
194C22×C4—63no—
195C22×C4—63no—
196C22×C4—63no—
197C22×C4—63no—
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—61no—
208C22×C4—63no—
209C22×C4—63no—
210C22×C4C2×Dic616yes—
211C22×C4#23516yes—
212C22×C4#23516yes—
213C4⋊C4—1212no—
214C4⋊C4—124no—
215C42—412no—
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—24no—
230C3×C4—46no—
231C3×C4—46no—
232Dic3—46no—
233Dic3—42no—
234Dic3—83no—
235Dic3—81no—
236C23C22×Dic611yescharacteristic
237C23C2×#23513yes—
238C23C2×#23513yes—
239C23#100413yes—
240C23#100711yescharacteristic
241C23D3×#4113yes—
242C23D3×#4111yescharacteristic
243C2×C4—26no—
244C2×C4—26no—
245C2×C4—26no—
246C2×C4—26no—
247C2×C4—26no—
248C2×C4—26no—
249C2×C4—26no—
250C2×C4—26no—
251C2×C4—43no—
252C2×C4—43no—
253C2×C4—43no—
254C2×C4—43no—
255C2×C4—43no—
256C2×C4—43no—
257C2×C4—43no—
258C2×C4—43no—
259C2×C4—66no—
260C2×C4—66no—
261C2×C4—66no—
262C2×C4—66no—
263C2×C4—121no—
264C2×C4—123no—
265C2×C4—123no—
266C2×C4—123no—
267C2×C4—123no—
268C2×C4—123no—
269C2×C4—121no—
270C2×C4—123no—
271C2×C4—123no—
272C2×C4—123no—
273C2×C4—123no—
274C2×C4—123no—
275C2×C4—123no—
276C2×C4—123no—
277C2×C4—123no—
278C2×C4—123no—
279C2×C4#87614yes—
280C2×C3#240111yescharacteristic
281C2×C3#257513yes—
282C2×C3#258913yes—
283C2×C3#259013yes—
284C2×C3#259513yes—
285C2×C3#264311yescharacteristic
286C2×C3#286311yescharacteristic
287C22#547013yes—
288C22C2×#87611yescharacteristic
289C22#547913yes—
290C22#549413yes—
291C22#551513yes—
292C22#552813yes—
293C22#552813yes—
294C22#553913yes—
295C22#554013yes—
296C22#555613yes—
297C22#555913yes—
298C22#556911yescharacteristic
299C22#558113yes—
300C4—24no—
301C4—46no—
302C4—46no—
303C4—126no—
304C4—122no—
305C4—243no—
306C4—241no—
307C3#1521011yescharacteristic, complemented, Sylow
308C2#7330911yescharacteristic
309C2#7338011yescharacteristic
310C2#7342011yescharacteristic
311C2#7354913yes—
312C2#7355113yes—
313C2#7373413yes—
314C2#7380913yes—
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC3×C4×#95C2
hypercenterC4×#95D3
hypocenterC3#15210
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C4×#95

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×C4×#952
#715362
#694212
#694162
#694142
#152103