Finite Group Databasebeta

← #10723391

Subgroups — #10723391

Order 768.

1302 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#384C211yesmaximal, characteristic
2C2×#4980C214yesmaximal
3#71569C214yesmaximal
4#71565C212yesmaximal
5#71564C212yesmaximal
6#71534C212yesmaximal
7#14424—31nomaximal, Sylow
8C22×C3×C4⋊C4C2214yes—
9C22×C3×C4⋊C4C2211yescharacteristic
10C22×C3×C4⋊C4C2211yescharacteristic
11C22×C3×C42C2211yescharacteristic
12C22×C4⋊Dic3C2212yes—
13C22×C4×Dic3C2212yes—
14C3×#384C418yescomplemented
15C2×C3⋊#95C2214yes—
16C2×C3⋊#95C2212yes—
17C2×C3⋊#95C2214yes—
18C2×C3⋊#95C2214yes—
19C2×C3⋊#95C2214yes—
20C2×C3⋊#95C2212yes—
21C2×C3⋊#95C2212yes—
22C2×C3⋊#95C2212yes—
23#4980—216no—
24C2×#387—34no—
25C2×#384D311yescharacteristic
26#1445—34no—
27#1441—32no—
28#1436—32no—
29#1434—32no—
30C23×C3×C4C2312yes—
31C23×C3×C4C2312yes—
32C23×C3×C4C2312yes—
33C23×C3×C4C2311yescharacteristic
34C23×Dic3C2314yes—
35C23×Dic3C2314yes—
36C2×C3×C4⋊C4—28no—
37C2×C3×C4⋊C4—28no—
38C2×C3×C4⋊C4—41no—
39C2×C3×C4⋊C4—41no—
40C2×C3×C4⋊C4—41no—
41C2×C3×C4⋊C4—41no—
42C2×C3×C4⋊C4C2×C4116yes—
43C2×C3×C4⋊C4C2×C414yes—
44C2×C3×C4⋊C4C2×C414yes—
45C2×C3×C42—24no—
46C2×C3×C42C2×C414yes—
47C2×C4⋊Dic3—28no—
48C2×C4⋊Dic3—42no—
49C2×C4×Dic3—28no—
50C2×C4×Dic3—42no—
51C3⋊#95—28no—
52C3⋊#95—28no—
53C3⋊#95—24no—
54C3⋊#95—28no—
55C3⋊#95—28no—
56C3⋊#95—42no—
57C3⋊#95—42no—
58C3⋊#95D418yes—
59C22×C4⋊C4—32no—
60C22×C4⋊C4C2×D314yes—
61C22×C4⋊C4C2×D311yescharacteristic
62C22×C4⋊C4C2×D311yescharacteristic
63C22×C42—32no—
64C22×C42C2×D311yescharacteristic
65#387—616no—
66#384Dic318yescomplemented
67C2×#95—32no—
68C2×#95—34no—
69C2×#95—32no—
70C2×#95—34no—
71C2×#95—34no—
72C2×#95—34no—
73C2×#95—32no—
74C2×#95—32no—
75C24×C3C2411yescharacteristic
76C22×C3×C4—24no—
77C22×C3×C4—21no—
78C22×C3×C4—21no—
79C22×C3×C4—22no—
80C22×C3×C4—24no—
81C22×C3×C4—24no—
82C22×C3×C4—24no—
83C22×C3×C4—24no—
84C22×C3×C4—24no—
85C22×C3×C4—22no—
86C22×C3×C4—22no—
87C22×C3×C4—22no—
88C22×C3×C4—22no—
89C22×C3×C4C22×C414yes—
90C22×C3×C4C22×C414yes—
91C22×C3×C4C22×C412yes—
92C22×C3×C4C22×C414yes—
93C22×C3×C4C2×D414yes—
94C22×C3×C4#4114yes—
95C22×C3×C4#4114yes—
96C22×Dic3—24no—
97C22×Dic3—24no—
98C22×Dic3—24no—
99C22×Dic3—24no—
100C22×Dic3—24no—
101C22×Dic3—24no—
102C22×Dic3—24no—
103C22×Dic3—24no—
104C22×Dic3—24no—
105C22×Dic3—24no—
106C22×Dic3—24no—
107C22×Dic3—24no—
108C22×Dic3—24no—
109C22×Dic3C2×D418yes—
110C3×C4⋊C4—232no—
111C3×C4⋊C4—44no—
112C3×C4⋊C4—44no—
113C3×C42—28no—
114C4⋊Dic3—48no—
115C4×Dic3—48no—
116C23×C4—34no—
117C23×C4—34no—
118C23×C4C22×D312yes—
119C23×C4C22×D312yes—
120C23×C4C22×D311yescharacteristic
121C23×C4C22×D312yes—
122C2×C4⋊C4—28no—
123C2×C4⋊C4—28no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—68no—
129C2×C4⋊C4—122no—
130C2×C4⋊C4C2×Dic3116yes—
131C2×C4⋊C4C2×Dic314yes—
132C2×C4⋊C4C2×Dic314yes—
133C2×C42—24no—
134C2×C42—68no—
135C2×C42—122no—
136C2×C42C2×Dic314yes—
137#95—38no—
138#95—68no—
139#95—68no—
140#95—68no—
141#95—68no—
142#95—64no—
143#95—122no—
144#95—122no—
145C23×C3C23×C411yescharacteristic
146C23×C3C22×D412yes—
147C23×C3C2×#4112yes—
148C23×C3C2×#4112yes—
149C23×C3H(5,𝔽2)11yescharacteristic
150C23×C3H(5,𝔽2)11yescharacteristic
151C23×C32−1+412yes—
152C23×C32−1+414yes—
153C2×C3×C4—24no—
154C2×C3×C4—22no—
155C2×C3×C4—22no—
156C2×C3×C4—24no—
157C2×C3×C4—24no—
158C2×C3×C4—24no—
159C2×C3×C4—28no—
160C2×C3×C4—28no—
161C2×C3×C4—28no—
162C2×C3×C4—28no—
163C2×C3×C4—24no—
164C2×C3×C4—24no—
165C2×C3×C4—42no—
166C2×C3×C4—42no—
167C2×C3×C4—44no—
168C2×C3×C4—44no—
169C2×C3×C4—42no—
170C2×C3×C4—42no—
171C2×C3×C4#11714yes—
172C2×C3×C4C4×D418yes—
173C2×Dic3—28no—
174C2×Dic3—28no—
175C2×Dic3—28no—
176C2×Dic3—44no—
177C2×Dic3—44no—
178C2×Dic3—44no—
179C2×Dic3—44no—
180C2×Dic3—44no—
181C2×Dic3—44no—
182C2×Dic3—44no—
183C2×Dic3—44no—
184C2×Dic3—44no—
185C2×Dic3—44no—
186C2×Dic3—44no—
187C24C23×D311yescharacteristic
188C22×C4—24no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—24no—
192C22×C4—22no—
193C22×C4—24no—
194C22×C4—24no—
195C22×C4—24no—
196C22×C4—24no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—38no—
202C22×C4—64no—
203C22×C4—64no—
204C22×C4—64no—
205C22×C4—64no—
206C22×C4—64no—
207C22×C4—64no—
208C22×C4—64no—
209C22×C4—64no—
210C22×C4—64no—
211C22×C4—64no—
212C22×C4—64no—
213C22×C4—64no—
214C22×C4—64no—
215C22×C4#23514yes—
216C22×C4D3×D414yes—
217C22×C4#23714yes—
218C22×C4C22×Dic314yes—
219C22×C4C22×Dic314yes—
220C22×C4C22×Dic312yes—
221C22×C4C22×Dic314yes—
222C4⋊C4—232no—
223C4⋊C4—44no—
224C4⋊C4—44no—
225C4⋊C4—128no—
226C42—28no—
227C42—128no—
228C22×C3C2×#11711yescharacteristic
229C22×C3C2×C4×D412yes—
230C22×C3#51811yescharacteristic
231C22×C3#51911yescharacteristic
232C22×C3#51912yes—
233C22×C3#53614yes—
234C22×C3#53912yes—
235C22×C3#53912yes—
236C22×C3#54114yes—
237C22×C3D4211yescharacteristic
238C22×C3#54714yes—
239C22×C3#55012yes—
240C22×C3#55214yes—
241C22×C3#55911yescharacteristic
242C22×C3#56312yes—
243C22×C3#56312yes—
244C3×C4—24no—
245C3×C4—28no—
246C3×C4—44no—
247C3×C4—44no—
248Dic3—48no—
249Dic3—84no—
250C23C2×#23512yes—
251C23C2×D3×D412yes—
252C23C2×#23712yes—
253C23#100411yescharacteristic
254C23#100411yescharacteristic
255C23#100712yes—
256C23#100714yes—
257C23C23×Dic311yescharacteristic
258C2×C4—28no—
259C2×C4—28no—
260C2×C4—24no—
261C2×C4—24no—
262C2×C4—24no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—24no—
266C2×C4—28no—
267C2×C4—28no—
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—44no—
275C2×C4—44no—
276C2×C4—68no—
277C2×C4—68no—
278C2×C4—68no—
279C2×C4—124no—
280C2×C4—124no—
281C2×C4—124no—
282C2×C4—124no—
283C2×C4—124no—
284C2×C4—124no—
285C2×C4—124no—
286C2×C4—124no—
287C2×C4—124no—
288C2×C4—124no—
289C2×C4—124no—
290C2×C4#92614yes—
291C2×C4D4×Dic318yes—
292C2×C3#231311yescharacteristic
293C2×C3#232712yes—
294C2×C3#234412yes—
295C2×C3#236611yescharacteristic
296C2×C3#238111yescharacteristic
297C2×C3#268012yes—
298C2×C3#272114yes—
299C2×C3#279412yes—
300C22#550714yes—
301C22#551314yes—
302C22#551512yes—
303C22#551512yes—
304C22#571311yescharacteristic
305C22#571711yescharacteristic
306C22#571812yes—
307C22#572414yes—
308C22#573114yes—
309C22#573512yes—
310C22#573512yes—
311C22C2×#92611yescharacteristic
312C22C2×D4×Dic312yes—
313C22#579811yescharacteristic
314C22#581211yescharacteristic
315C22#581212yes—
316C4—24no—
317C4—28no—
318C4—44no—
319C4—44no—
320C4—128no—
321C4—244no—
322C3#1442411yescharacteristic, complemented, Sylow
323C2#7434814yes—
324C2#7435212yes—
325C2#7437112yes—
326C2#8215511yescharacteristic
327C2#8215912yes—
328C2#8317311yescharacteristic
329C2#8322712yes—
330C2#8323111yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC2×C3×#384C2
socle, prime radical, cosocleC24×C3C24
hypercenterC2×#384D3
hypocenterC3#14424
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#384

Complements

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

NH ≅ G/Ntype
14. C3×#384C4semidirect
66. #384Dic3semidirect
322. C3#14424semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3842
C2×#49802
#715692
#715652
#715642
#715342
#144243