Finite Group Databasebeta

← #10722363

Subgroups — #10722363

Order 768.

2142 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4⋊Dic3C211yesmaximal, characteristic
2C2×C3×#381C211yesmaximal, characteristic
3C2×#5211C211yesmaximal, characteristic
4C2×#4942C212yesmaximal
5#71544C212yesmaximal
6#69411C218yesmaximal, complemented
7#14339—31nomaximal, Sylow
8C24×C3×C4C2211yescharacteristic
9C24×Dic3C2211yescharacteristic
10C24×Dic3C2211yescharacteristic
11C22×C3×C22⋊C4C2212yes—
12C22×#217C2212yes—
13C22×C4⋊Dic3—28no—
14C22×C4⋊Dic3C418yescomplemented
15C22×C4⋊Dic3C2214yes—
16C22×Dic3⋊C4C2214yes—
17C3×#381—24no—
18C2×C3×#95C2214yes—
19#5211—24no—
20C2×C3⋊#95C2214yes—
21C2×C3⋊#95C2214yes—
22C2×C3⋊#95C2218yes—
23#4942—28no—
24C23×C4⋊C4—31no—
25C2×#381—31no—
26C2×#381D311yescharacteristic
27C2×#380—32no—
28#1441—38no—
29C22⋊#95—32no—
30C25×C3C2311yescharacteristic
31C23×C3×C4—24no—
32C23×C3×C4—24no—
33C23×C3×C4C2×C414yes—
34C23×C3×C4D414yes—
35C23×C3×C4Q814yes—
36C23×C3×C4C2314yes—
37C23×C3×C4C2312yes—
38C23×Dic3—24no—
39C23×Dic3—44no—
40C23×Dic3—44no—
41C23×Dic3C2×C414yes—
42C23×Dic3C2×C414yes—
43C23×Dic3D418yes—
44C23×Dic3C2314yes—
45C23×Dic3C2312yes—
46C23×Dic3C2312yes—
47C2×C3×C22⋊C4—24no—
48C2×C3×C22⋊C4—24no—
49C2×C3×C22⋊C4—24no—
50C2×#217—24no—
51C2×#217—24no—
52C2×#217—24no—
53C2×C4⋊Dic3—232no—
54C2×C4⋊Dic3—28no—
55C2×C4⋊Dic3—28no—
56C2×C4⋊Dic3C2×C4116yes—
57C2×Dic3⋊C4—28no—
58C2×Dic3⋊C4—48no—
59C3×#95—28no—
60C3⋊#95—28no—
61C3⋊#95—28no—
62C3⋊#95—48no—
63C24×C4—31no—
64C24×C4—31no—
65C24×C4C2×D311yescharacteristic
66C22×C4⋊C4—38no—
67C22×C4⋊C4—34no—
68C22×C4⋊C4—34no—
69C22×C4⋊C4—68no—
70C22×C22⋊C4—32no—
71C22×C22⋊C4C2×D312yes—
72#381—24no—
73#381—64no—
74#380—68no—
75C2×#95—34no—
76C2×#95—34no—
77C2×#95—38no—
78C2×#95C2×D314yes—
79C24×C3—24no—
80C24×C3—24no—
81C24×C3C22×C412yes—
82C24×C3C2×D414yes—
83C24×C3C2×D412yes—
84C24×C3C2×Q812yes—
85C24×C3#4114yes—
86C24×C3C2411yescharacteristic
87C22×C3×C4—28no—
88C22×C3×C4—28no—
89C22×C3×C4—216no—
90C22×C3×C4—216no—
91C22×C3×C4—24no—
92C22×C3×C4—24no—
93C22×C3×C4—24no—
94C22×C3×C4—24no—
95C22×C3×C4—24no—
96C22×C3×C4—28no—
97C22×C3×C4—28no—
98C22×C3×C4C4⋊C4116yescomplemented
99C22×C3×C4C22×C414yes—
100C22×C3×C4C2×D414yes—
101C22×C3×C4C2×Q814yes—
102C22×Dic3—28no—
103C22×Dic3—28no—
104C22×Dic3—24no—
105C22×Dic3—28no—
106C22×Dic3—24no—
107C22×Dic3—24no—
108C22×Dic3—24no—
109C22×Dic3—28no—
110C22×Dic3—28no—
111C22×Dic3—48no—
112C22×Dic3—48no—
113C22×Dic3—416no—
114C22×Dic3—416no—
115C22×Dic3C22⋊C4116yes—
116C22×Dic3C22×C414yes—
117C22×Dic3C22×C414yes—
118C22×Dic3C2×D414yes—
119C22×Dic3C2×D414yes—
120C3×C22⋊C4—48no—
121#217—48no—
122C4⋊Dic3—232no—
123Dic3⋊C4—416no—
124C25C22×D311yescharacteristic
125C23×C4—24no—
126C23×C4—24no—
127C23×C4—32no—
128C23×C4—34no—
129C23×C4—38no—
130C23×C4—32no—
131C23×C4—34no—
132C23×C4—34no—
133C23×C4—64no—
134C23×C4—124no—
135C23×C4—124no—
136C23×C4Dic614yes—
137C23×C4C4×D314yes—
138C23×C4#6714yes—
139C23×C4C22×D314yes—
140C23×C4C22×D312yes—
141C2×C4⋊C4—316no—
142C2×C4⋊C4—632no—
143C2×C4⋊C4—68no—
144C2×C4⋊C4—68no—
145C2×C4⋊C4—68no—
146C2×C4⋊C4—128no—
147C2×C22⋊C4—24no—
148C2×C22⋊C4—24no—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—64no—
151C2×C22⋊C4—64no—
152C2×C22⋊C4—64no—
153#95—28no—
154#95—68no—
155#95—68no—
156#95—128no—
157C23×C3—24no—
158C23×C3—24no—
159C23×C3—28no—
160C23×C3—28no—
161C23×C3—28no—
162C23×C3—216no—
163C23×C3—28no—
164C23×C3C2×C22⋊C414yes—
165C23×C3C2×C4⋊C414yes—
166C23×C3#11714yes—
167C23×C3#12118yes—
168C23×C3C22⋊Q818yes—
169C23×C3C23×C411yescharacteristic
170C23×C3C22×D412yes—
171C23×C3C22×D411yescharacteristic
172C23×C3C22×Q811yescharacteristic
173C23×C3C2×#4112yes—
174C23×C3H(5,𝔽2)14yes—
175C23×C32−1+414yes—
176C2×C3×C4—24no—
177C2×C3×C4—24no—
178C2×C3×C4—28no—
179C2×C3×C4—28no—
180C2×C3×C4—232no—
181C2×C3×C4—44no—
182C2×C3×C4—44no—
183C2×C3×C4—44no—
184C2×C3×C4—48no—
185C2×C3×C4—48no—
186C2×C3×C4C2×C4⋊C418yes—
187C2×Dic3—28no—
188C2×Dic3—28no—
189C2×Dic3—28no—
190C2×Dic3—24no—
191C2×Dic3—28no—
192C2×Dic3—416no—
193C2×Dic3—416no—
194C2×Dic3—48no—
195C2×Dic3—44no—
196C2×Dic3—44no—
197C2×Dic3—44no—
198C2×Dic3—48no—
199C2×Dic3—48no—
200C2×Dic3C2×C22⋊C418yes—
201C24—24no—
202C24—24no—
203C24C2×Dic612yes—
204C24C2×C4×D312yes—
205C24D3×D414yes—
206C24#23714yes—
207C24C2×#6712yes—
208C24C23×D311yescharacteristic
209C22×C4—28no—
210C22×C4—216no—
211C22×C4—28no—
212C22×C4—216no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—28no—
218C22×C4—28no—
219C22×C4—24no—
220C22×C4—34no—
221C22×C4—34no—
222C22×C4—34no—
223C22×C4—316no—
224C22×C4—34no—
225C22×C4—68no—
226C22×C4—68no—
227C22×C4—64no—
228C22×C4—68no—
229C22×C4—64no—
230C22×C4—64no—
231C22×C4—68no—
232C22×C4—68no—
233C22×C4—64no—
234C22×C4—1216no—
235C22×C4—128no—
236C22×C4—128no—
237C22×C4—1216no—
238C22×C4Dic3⋊C4116yescomplemented
239C22×C4C2×Dic614yes—
240C22×C4C2×C4×D314yes—
241C22×C4C2×#6714yes—
242C4⋊C4—632no—
243C4⋊C4—1216no—
244C22⋊C4—48no—
245C22⋊C4—128no—
246C22×C3—24no—
247C22×C3—24no—
248C22×C3—28no—
249C22×C3—28no—
250C22×C3—216no—
251C22×C3—216no—
252C22×C3#38018yes—
253C22×C3C22×C22⋊C411yescharacteristic
254C22×C3C22×C4⋊C411yescharacteristic
255C22×C3C2×#11711yescharacteristic
256C22×C3#51812yes—
257C22×C3#51912yes—
258C22×C3C2×#12112yes—
259C22×C3C2×C22⋊Q812yes—
260C22×C3#53414yes—
261C22×C3#53614yes—
262C22×C3#53714yes—
263C22×C3#53918yes—
264C22×C3#54414yes—
265C3×C4—28no—
266C3×C4—84no—
267Dic3—28no—
268Dic3—44no—
269Dic3—84no—
270C23—28no—
271C23—28no—
272C23—216no—
273C23—24no—
274C23—28no—
275C23—24no—
276C23—28no—
277C23#87714yes—
278C23#87818yes—
279C23D3×C22⋊C414yes—
280C23C2×Dic3⋊C414yes—
281C23#93918yes—
282C23C22×Dic611yescharacteristic
283C23C22×C4×D311yescharacteristic
284C23C2×D3×D412yes—
285C23C2×#23712yes—
286C23#100914yes—
287C23#101014yes—
288C23C22×#6711yescharacteristic
289C2×C4—232no—
290C2×C4—24no—
291C2×C4—28no—
292C2×C4—24no—
293C2×C4—28no—
294C2×C4—38no—
295C2×C4—44no—
296C2×C4—44no—
297C2×C4—48no—
298C2×C4—44no—
299C2×C4—48no—
300C2×C4—68no—
301C2×C4—68no—
302C2×C4—68no—
303C2×C4—64no—
304C2×C4—68no—
305C2×C4—1216no—
306C2×C4—1216no—
307C2×C4—128no—
308C2×C4—124no—
309C2×C4—124no—
310C2×C4—128no—
311C2×C4—124no—
312C2×C4—128no—
313C2×C4C2×Dic3⋊C418yes—
314C2×C3—28no—
315C2×C3C2×#38011yescharacteristic
316C2×C3#230811yescharacteristic
317C2×C3#231011yescharacteristic
318C2×C3#231312yes—
319C2×C3#231612yes—
320C2×C3#261814yes—
321C2×C3#262714yes—
322C22—28no—
323C22—24no—
324C22—28no—
325C22—216no—
326C22—24no—
327C22—216no—
328C22#494218yes—
329C22C2×#87711yescharacteristic
330C22C2×#87812yes—
331C22C2×D3×C22⋊C411yescharacteristic
332C22#551612yes—
333C22#551914yes—
334C22#552112yes—
335C22#564814yes—
336C22#565314yes—
337C22#566518yes—
338C22C22×Dic3⋊C411yescharacteristic
339C22C2×#93912yes—
340C22#583314yes—
341C4—28no—
342C4—68no—
343C4—84no—
344C4—124no—
345C4—244no—
346C3#1433911yescharacteristic, complemented, Sylow
347C2—28no—
348C2#7405311yescharacteristic
349C2#7405412yes—
350C2#7409014yes—
351C2#7411911yescharacteristic
352C2C2×#494211yescharacteristic
353C2#8223212yes—
354C2#8286514yes—
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC2×C3×#381C2
socle, cosocleC24×C3C24
prime radicalC25×C3C23
hypercenterC2×#381D3
hypocenterC3#14339
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#381

Complements

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

NH ≅ G/Ntype
6. #69411C2semidirect
14. C22×C4⋊Dic3C4semidirect
98. C22×C3×C4C4⋊C4semidirect
238. C22×C4Dic3⋊C4semidirect
346. C3#14339semidirect

Maximal subgroups

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

subgroupindex
C23×C4⋊Dic32
C2×C3×#3812
C2×#52112
C2×#49422
#715442
#694112
#143393