Finite Group Databasebeta

← #10823329

Subgroups — #10823329

Order 768.

1118 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85725C211yesmaximal, characteristic
2C3×#2317C211yesmaximal, characteristic
3#82157C212yesmaximal
4#71593C212yesmaximal
5#71592C211yesmaximal, characteristic
6#69594C218yesmaximal
7#23566—31nomaximal, Sylow
8C2×C3×#117C2211yescharacteristic
9C2×C3×#117C2211yescharacteristic
10C22×C3×C4⋊C4C2211yescharacteristic
11C2×#926C2212yes—
12C2×#926C2212yes—
13#5519—22no—
14#5519—22no—
15#5517—24no—
16#5500—21no—
17#5500—21no—
18#5483C2214yes—
19#5470—22no—
20C3×#389—22no—
21C3×#385C2214yes—
22C3×#384—21no—
23C3×#382—21no—
24#5210C2218yes—
25#4977—22no—
26#4975—22no—
27#4972—24no—
28#4527C2218yes—
29#4526C2214yes—
30#3458—31no—
31#2317—32no—
32#2317D311yescharacteristic
33#2062—38no—
34#1740—31no—
35#1740—32no—
36C23×C3×C4C2312yes—
37C23×C3×C4C2311yescharacteristic
38C3×#117—22no—
39C3×#117—22no—
40C3×#117—22no—
41C3×#117Q814yescomplemented
42C2×C3×C4⋊C4—21no—
43C2×C3×C4⋊C4—22no—
44C2×C3×C4⋊C4—22no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—24no—
48C2×C3×C4⋊C4D411yescharacteristic
49C2×C3×C4⋊C4D411yescharacteristic
50C2×C3×C22⋊C4C2312yes—
51C2×C3×C22⋊C4C2312yes—
52C2×C3×C42—21no—
53C2×C3×C42D412yes—
54C2×#217C2314yes—
55C2×#217C2314yes—
56#926—22no—
57#926—22no—
58#926—22no—
59#926—22no—
60#926—24no—
61#926Q818yescomplemented
62C2×C4⋊Dic3—22no—
63C2×C4⋊Dic3D412yes—
64C2×C4⋊Dic3D412yes—
65#924—24no—
66#924—24no—
67C2×Dic3⋊C4—28no—
68C2×C4×Dic3—22no—
69C2×C4×Dic3D414yes—
70#890—24no—
71#890—24no—
72#888—24no—
73#888—24no—
74#878—28no—
75#878—28no—
76#870—24no—
77#869—22no—
78#869—22no—
79C4×Dic6—24no—
80C4×Dic6—24no—
81C3×#95—24no—
82C3⋊#95—28no—
83#544—62no—
84#544—62no—
85#544—61no—
86#544—61no—
87#543—34no—
88#533—64no—
89#533—62no—
90C2×#117—32no—
91C2×#117—32no—
92C2×#117C2×D311yescharacteristic
93C2×#117C2×D311yescharacteristic
94C22×C4⋊C4C2×D311yescharacteristic
95#389—22no—
96#389—64no—
97#385—38no—
98#385C2×D314yes—
99#384—21no—
100#384—62no—
101#382—21no—
102#382—62no—
103#342—34no—
104#342—38no—
105C24×C3C2411yescharacteristic
106C22×C3×C4—22no—
107C22×C3×C4—24no—
108C22×C3×C4—22no—
109C22×C3×C4—82no—
110C22×C3×C4C2×D412yes—
111C22×C3×C4C2×D412yes—
112C22×C3×C4C2×D412yes—
113C22×C3×C4C2×D412yes—
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×Q812yes—
117C22×C3×C4C2×Q814yes—
118C22×C3×C4#4112yes—
119C22×Dic3—24no—
120C22×Dic3C2×D414yes—
121C22×Dic3C2×D414yes—
122C2×Dic6—24no—
123C2×Dic6—24no—
124C3×C4⋊C4—22no—
125C3×C4⋊C4—22no—
126C3×C4⋊C4—24no—
127C3×C4⋊C4—41no—
128C3×C4⋊C4—42no—
129C3×C4⋊C4—41no—
130C3×C4⋊C4—84no—
131C3×C22⋊C4—24no—
132C3×C22⋊C4—22no—
133C3×C22⋊C4C2×Q814yes—
134C3×C42—22no—
135C3×C42—22no—
136C3×C42—24no—
137#217—24no—
138#217—24no—
139#217—24no—
140#217C2×Q818yes—
141C4⋊Dic3—24no—
142C4⋊Dic3—24no—
143C4⋊Dic3—22no—
144C4⋊Dic3—22no—
145C4⋊Dic3—22no—
146C4⋊Dic3—22no—
147C4⋊Dic3—28no—
148Dic3⋊C4—28no—
149Dic3⋊C4—28no—
150Dic3⋊C4—88no—
151C4×Dic3—24no—
152C4×Dic3—24no—
153C4×Dic3—28no—
154C23×C4C22×D312yes—
155C23×C4C22×D311yescharacteristic
156C4⋊Q8—62no—
157C4⋊Q8—62no—
158C4⋊Q8—64no—
159C4⋊Q8—64no—
160#125—64no—
161#125—64no—
162#125—64no—
163C22⋊Q8—68no—
164C22⋊Q8—68no—
165C22⋊Q8—64no—
166C22⋊Q8—64no—
167C4×Q8—64no—
168C4×Q8—64no—
169#117—22no—
170#117—22no—
171#117—22no—
172#117—38no—
173#117—64no—
174#117—62no—
175#117—62no—
176#117—62no—
177#117—62no—
178#117Dic614yescomplemented
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—32no—
186C2×C4⋊C4—32no—
187C2×C4⋊C4—62no—
188C2×C4⋊C4—68no—
189C2×C4⋊C4#6711yescharacteristic
190C2×C4⋊C4#6711yescharacteristic
191C2×C22⋊C4—34no—
192C2×C22⋊C4—34no—
193C2×C22⋊C4C22×D312yes—
194C2×C22⋊C4C22×D312yes—
195C2×C42—21no—
196C2×C42—34no—
197C2×C42—62no—
198C2×C42#6712yes—
199#95—24no—
200#95—68no—
201C23×C3—41no—
202C23×C3—41no—
203C23×C3C22×D412yes—
204C23×C3C22×D411yescharacteristic
205C23×C3C22×Q811yescharacteristic
206C23×C3C22×Q812yes—
207C23×C3C2×#4111yescharacteristic
208C2×C3×C4—22no—
209C2×C3×C4—22no—
210C2×C3×C4—24no—
211C2×C3×C4—42no—
212C2×C3×C4—42no—
213C2×C3×C4—42no—
214C2×C3×C4—42no—
215C2×C3×C4—48no—
216C2×C3×C4—44no—
217C2×C3×C4—42no—
218C2×C3×C4#12011yescharacteristic
219C2×C3×C4#12012yes—
220C2×C3×C4#12011yescharacteristic
221C2×C3×C4C22⋊Q814yes—
222C2×C3×C4C22⋊Q814yes—
223C2×C3×C4C22⋊Q812yes—
224C2×C3×C4C22⋊Q812yes—
225C2×C3×C4C4⋊Q814yes—
226C2×C3×C4C4⋊Q814yes—
227C2×C3×C4C4⋊Q812yes—
228C2×C3×C4C4⋊Q812yes—
229C2×Dic3—24no—
230C2×Dic3—24no—
231C2×Dic3—28no—
232C2×Dic3—44no—
233C2×Dic3—44no—
234C2×Dic3—44no—
235C2×Dic3—44no—
236Dic6—84no—
237Dic6—84no—
238C24C23×D311yescharacteristic
239C2×Q8—64no—
240C2×Q8—64no—
241C22×C4—22no—
242C22×C4—24no—
243C22×C4—22no—
244C22×C4—34no—
245C22×C4—34no—
246C22×C4—64no—
247C22×C4—82no—
248C22×C4C2×Dic612yes—
249C22×C4#23512yes—
250C22×C4D3×D412yes—
251C22×C4D3×D412yes—
252C22×C4D3×Q814yes—
253C22×C4C2×#6712yes—
254C22×C4C2×#6712yes—
255C22×C4C2×#6711yescharacteristic
256C22×C4C2×#6711yescharacteristic
257C4⋊C4—24no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—42no—
263C4⋊C4—68no—
264C4⋊C4—68no—
265C4⋊C4—64no—
266C4⋊C4—64no—
267C4⋊C4—68no—
268C4⋊C4—62no—
269C4⋊C4—62no—
270C4⋊C4—62no—
271C4⋊C4—62no—
272C4⋊C4—84no—
273C4⋊C4—248no—
274C22⋊C4—22no—
275C22⋊C4—24no—
276C22⋊C4—38no—
277C22⋊C4—64no—
278C22⋊C4—64no—
279C22⋊C4—64no—
280C22⋊C4C2×Dic614yes—
281C42—24no—
282C42—22no—
283C42—22no—
284C42—68no—
285C42—64no—
286C42—64no—
287C22×C3—22no—
288C22×C3—21no—
289C22×C3—22no—
290C22×C3—21no—
291C22×C3—41no—
292C22×C3—42no—
293C22×C3—41no—
294C22×C3C2×#12011yescharacteristic
295C22×C3C2×C22⋊Q812yes—
296C22×C3C2×C22⋊Q811yescharacteristic
297C22×C3C2×C4⋊Q812yes—
298C22×C3C2×C4⋊Q811yescharacteristic
299C3×C4—44no—
300C3×C4—42no—
301C3×C4—42no—
302C3×C4—82no—
303C3×C4#39512yes—
304C3×C4#39514yes—
305C3×C4#39512yes—
306Dic3—48no—
307Dic3—44no—
308Dic3—44no—
309C23—41no—
310C23—41no—
311C23C22×Dic611yescharacteristic
312C23C2×#23511yescharacteristic
313C23C2×D3×D412yes—
314C23C2×D3×Q812yes—
315C23C22×#6711yescharacteristic
316Q8—244no—
317Q8—244no—
318C2×C4—24no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—44no—
326C2×C4—48no—
327C2×C4—42no—
328C2×C4—68no—
329C2×C4—64no—
330C2×C4—64no—
331C2×C4—124no—
332C2×C4—124no—
333C2×C4—124no—
334C2×C4—124no—
335C2×C4#88814yes—
336C2×C4#88814yes—
337C2×C4#89614yes—
338C2×C4#89614yes—
339C2×C4#92412yes—
340C2×C4#92412yes—
341C2×C4#93711yescharacteristic
342C2×C4#93711yescharacteristic
343C2×C4#93712yes—
344C2×C4#94412yes—
345C2×C4#94412yes—
346C2×C3—22no—
347C2×C3—21no—
348C2×C3—22no—
349C2×C3—21no—
350C2×C3C2×#39511yescharacteristic
351C2×C3#302711yescharacteristic
352C2×C3#302811yescharacteristic
353C22—21no—
354C22—22no—
355C22—21no—
356C22—22no—
357C22—41no—
358C22—41no—
359C22—42no—
360C22C2×#88812yes—
361C22C2×#89612yes—
362C22C2×#92411yescharacteristic
363C22C2×#93711yescharacteristic
364C22C2×#94411yescharacteristic
365C4—44no—
366C4—42no—
367C4—42no—
368C4—82no—
369C4—128no—
370C4—124no—
371C4—124no—
372C4#497912yes—
373C4#497914yes—
374C4#497912yes—
375C3#2356611yescharacteristic, complemented, Sylow
376C2—21no—
377C2—22no—
378C2—22no—
379C2—21no—
380C2C2×#497911yescharacteristic
381C2#8339611yescharacteristic
382C2#8341711yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, prime radical, cosocleC24×C3C24
FrattiniC24C23×D3
FittingC3×#2317C2
socleC22×C3C2×#120
hypercenter#2317D3
hypocenterC3#23566
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2317

Complements

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

NH ≅ G/Ntype
41. C3×#117Q8semidirect
61. #926Q8semidirect
178. #117Dic6semidirect
375. C3#23566semidirect

Maximal subgroups

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

subgroupindex
#857252
C3×#23172
#821572
#715932
#715922
#695942
#235663