Finite Group Databasebeta

← #10822592

Subgroups — #10822592

Order 768.

1378 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85725C211yesmaximal, characteristic, complemented
2C3×#2280C211yesmaximal, characteristic
3#84027C212yesmaximal
4#71593C212yesmaximal, complemented
5#71592C211yesmaximal, characteristic, complemented
6#69536C214yesmaximal, complemented
7#69497C214yesmaximal
8#23467—31nomaximal, Sylow
9C24×C3×C4C2211yescharacteristic
10C2×C3×#117C2211yescharacteristic
11C2×C3×#117C2211yescharacteristic
12C2×#926C2212yes—
13C2×#926C2212yes—
14#5519—24no—
15#5517—24no—
16#5500—22no—
17#5483C2214yes—
18#5470—22no—
19C3×#381C2212yes—
20C3×#380—22no—
21C3×#379C2212yes—
22C3×C4×C22⋊C4—22no—
23#5247C2214yes—
24#5212—24no—
25#5211C2214yes—
26C4×#217—24no—
27#4527C2218yes—
28#4526C2214yes—
29#3458—31no—
30#2280—32no—
31#2280D311yescharacteristic
32#2030—34no—
33#2029—34no—
34#1740—31no—
35#1740—32no—
36C25×C3C2311yescharacteristic
37C23×C3×C4—22no—
38C23×C3×C4—22no—
39C23×C3×C4—21no—
40C23×C3×C4—21no—
41C23×C3×C4—22no—
42C23×C3×C4—24no—
43C23×C3×C4Q814yescomplemented
44C23×C3×C4C2311yescharacteristic
45C23×C3×C4C2311yescharacteristic
46C3×#117—22no—
47C3×#117—22no—
48C3×#117—24no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4D412yes—
51C2×C3×C22⋊C4—24no—
52C2×C3×C22⋊C4C2312yes—
53C2×C3×C22⋊C4C2312yes—
54C2×C3×C42—21no—
55C2×C3×C42D412yes—
56C2×#217—28no—
57C2×#217C2314yes—
58C2×#217C2314yes—
59#926—22no—
60#926—22no—
61#926—22no—
62#926—22no—
63#926—24no—
64#926—24no—
65C2×C4⋊Dic3—22no—
66C2×C4⋊Dic3D414yes—
67#924—28no—
68C2×C4×Dic3—22no—
69C2×C4×Dic3D414yes—
70#890—28no—
71#888—28no—
72#878—216no—
73#870—24no—
74#869—24no—
75C4×Dic6—28no—
76C3×#95—24no—
77C3⋊#95—28no—
78C24×C4C2×D311yescharacteristic
79#544—64no—
80#544—62no—
81#543—34no—
82#533—64no—
83#533—62no—
84C2×#117—32no—
85C2×#117—32no—
86C2×#117C2×D311yescharacteristic
87C2×#117C2×D311yescharacteristic
88#381—34no—
89#381C2×D312yes—
90#380—22no—
91#380—64no—
92#379—34no—
93#379C2×D312yes—
94C4×C22⋊C4—22no—
95C4×C22⋊C4—64no—
96#342—34no—
97#342—38no—
98C24×C3—22no—
99C24×C3—21no—
100C24×C3—21no—
101C24×C3—22no—
102C24×C3—44no—
103C24×C3C2×Q812yes—
104C24×C3C2411yescharacteristic
105C22×C3×C4—22no—
106C22×C3×C4—21no—
107C22×C3×C4—21no—
108C22×C3×C4—22no—
109C22×C3×C4—28no—
110C22×C3×C4—28no—
111C22×C3×C4—24no—
112C22×C3×C4—24no—
113C22×C3×C4—22no—
114C22×C3×C4—42no—
115C22×C3×C4—41no—
116C22×C3×C4—41no—
117C22×C3×C4—44no—
118C22×C3×C4—44no—
119C22×C3×C4—42no—
120C22×C3×C4—42no—
121C22×C3×C4C2×D412yes—
122C22×C3×C4C2×D412yes—
123C22×C3×C4C2×D414yes—
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×Q812yes—
127C22×C3×C4C2×Q812yes—
128C22×C3×C4#4112yes—
129C22×C3×C4#4114yes—
130C22×Dic3—24no—
131C22×Dic3C2×D418yes—
132C2×Dic6—28no—
133C3×C4⋊C4—24no—
134C3×C4⋊C4—42no—
135C3×C22⋊C4—24no—
136C3×C22⋊C4—24no—
137C3×C22⋊C4—48no—
138C3×C42—22no—
139C3×C42—22no—
140C3×C42—42no—
141#217—24no—
142#217—24no—
143#217—24no—
144#217—24no—
145#217—416no—
146C4⋊Dic3—28no—
147C4⋊Dic3—24no—
148C4⋊Dic3—24no—
149C4⋊Dic3—44no—
150Dic3⋊C4—216no—
151C4×Dic3—24no—
152C4×Dic3—24no—
153C4×Dic3—44no—
154C25C22×D311yescharacteristic
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—24no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4Dic614yescomplemented
162C23×C4C22×D311yescharacteristic
163C23×C4C22×D311yescharacteristic
164C4⋊Q8—64no—
165C4⋊Q8—68no—
166#125—64no—
167#125—68no—
168C22⋊Q8—616no—
169C22⋊Q8—68no—
170C4×Q8—68no—
171#117—22no—
172#117—22no—
173#117—24no—
174#117—62no—
175#117—62no—
176#117—62no—
177#117—62no—
178#117—64no—
179#117—64no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—34no—
182C2×C4⋊C4—62no—
183C2×C4⋊C4#6712yes—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—34no—
186C2×C22⋊C4—34no—
187C2×C22⋊C4—68no—
188C2×C22⋊C4C22×D312yes—
189C2×C22⋊C4C22×D312yes—
190C2×C42—21no—
191C2×C42—34no—
192C2×C42—62no—
193C2×C42#6712yes—
194#95—24no—
195#95—68no—
196C23×C3—28no—
197C23×C3—24no—
198C23×C3—24no—
199C23×C3—22no—
200C23×C3—44no—
201C23×C3—42no—
202C23×C3—44no—
203C23×C3—42no—
204C23×C3—44no—
205C23×C3—48no—
206C23×C3—41no—
207C23×C3—42no—
208C23×C3—41no—
209C23×C3C22×D412yes—
210C23×C3C22×D411yescharacteristic
211C23×C3C22×Q811yescharacteristic
212C23×C3C2×#4111yescharacteristic
213C23×C3C2×#4112yes—
214C2×C3×C4—24no—
215C2×C3×C4—24no—
216C2×C3×C4—22no—
217C2×C3×C4—41no—
218C2×C3×C4—42no—
219C2×C3×C4—41no—
220C2×C3×C4—48no—
221C2×C3×C4—44no—
222C2×C3×C4—44no—
223C2×C3×C4—44no—
224C2×C3×C4—44no—
225C2×C3×C4—42no—
226C2×C3×C4#12012yes—
227C2×C3×C4#12011yescharacteristic
228C2×C3×C4#12011yescharacteristic
229C2×C3×C4C22⋊Q814yes—
230C2×C3×C4C22⋊Q814yes—
231C2×C3×C4C22⋊Q812yes—
232C2×C3×C4C22⋊Q812yes—
233C2×C3×C4#12414yes—
234C2×C3×C4#12414yes—
235C2×C3×C4#12412yes—
236C2×C3×C4#12412yes—
237C2×Dic3—28no—
238C2×Dic3—48no—
239C2×Dic3—48no—
240C2×Dic3—44no—
241Dic6—88no—
242C24—22no—
243C24—21no—
244C24—21no—
245C24—22no—
246C24—44no—
247C24C2×Dic612yes—
248C24C23×D311yescharacteristic
249C2×Q8—68no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—28no—
255C22×C4—28no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4—38no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—41no—
265C22×C4—42no—
266C22×C4—41no—
267C22×C4—64no—
268C22×C4C2×Dic612yes—
269C22×C4C2×Dic612yes—
270C22×C4#23512yes—
271C22×C4D3×D412yes—
272C22×C4D3×D412yes—
273C22×C4#23714yes—
274C22×C4C2×#6714yes—
275C22×C4C2×#6711yescharacteristic
276C22×C4C2×#6711yescharacteristic
277C4⋊C4—24no—
278C4⋊C4—42no—
279C4⋊C4—616no—
280C4⋊C4—68no—
281C4⋊C4—64no—
282C4⋊C4—64no—
283C4⋊C4—124no—
284C22⋊C4—24no—
285C22⋊C4—24no—
286C22⋊C4—48no—
287C22⋊C4—64no—
288C22⋊C4—64no—
289C22⋊C4—64no—
290C22⋊C4—64no—
291C22⋊C4—1216no—
292C42—22no—
293C42—22no—
294C42—42no—
295C42—64no—
296C42—64no—
297C42—124no—
298C22×C3—24no—
299C22×C3—22no—
300C22×C3—28no—
301C22×C3—24no—
302C22×C3—41no—
303C22×C3—41no—
304C22×C3—42no—
305C22×C3—48no—
306C22×C3—44no—
307C22×C3—44no—
308C22×C3—42no—
309C22×C3—42no—
310C22×C3—44no—
311C22×C3C2×#12011yescharacteristic
312C22×C3C2×C22⋊Q812yes—
313C22×C3C2×C22⋊Q811yescharacteristic
314C22×C3C2×#12412yes—
315C22×C3C2×#12411yescharacteristic
316C3×C4—42no—
317C3×C4—44no—
318C3×C4—82no—
319C3×C4#39314yes—
320C3×C4#39312yes—
321C3×C4#39312yes—
322Dic3—48no—
323Dic3—84no—
324C23—24no—
325C23—28no—
326C23—24no—
327C23—22no—
328C23—44no—
329C23—42no—
330C23—42no—
331C23—44no—
332C23—44no—
333C23—48no—
334C23—42no—
335C23—41no—
336C23—41no—
337C23C22×Dic611yescharacteristic
338C23C2×#23511yescharacteristic
339C23C2×D3×D412yes—
340C23C2×#23712yes—
341C23C22×#6711yescharacteristic
342Q8—248no—
343C2×C4—24no—
344C2×C4—22no—
345C2×C4—24no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—48no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—42no—
355C2×C4—68no—
356C2×C4—128no—
357C2×C4—128no—
358C2×C4—124no—
359C2×C4#87814yes—
360C2×C4#87814yes—
361C2×C4#88514yes—
362C2×C4#88514yes—
363C2×C4#92412yes—
364C2×C4#92412yes—
365C2×C4#93612yes—
366C2×C4#93612yes—
367C2×C4#93711yescharacteristic
368C2×C4#93712yes—
369C2×C4#93711yescharacteristic
370C2×C3—22no—
371C2×C3—21no—
372C2×C3—22no—
373C2×C3—21no—
374C2×C3—44no—
375C2×C3C2×#39311yescharacteristic
376C2×C3#302512yes—
377C22—22no—
378C22—24no—
379C22—28no—
380C22—24no—
381C22—41no—
382C22—41no—
383C22—42no—
384C22—44no—
385C22—48no—
386C22—44no—
387C22—42no—
388C22—42no—
389C22—44no—
390C22C2×#87812yes—
391C22C2×#88512yes—
392C22C2×#92411yescharacteristic
393C22C2×#93611yescharacteristic
394C22C2×#93711yescharacteristic
395C4—42no—
396C4—44no—
397C4—82no—
398C4—128no—
399C4—244no—
400C4#494714yes—
401C4#494712yes—
402C4#494712yes—
403C3#2346711yescharacteristic, complemented, Sylow
404C2—21no—
405C2—22no—
406C2—22no—
407C2—21no—
408C2—44no—
409C2C2×#494711yescharacteristic
410C2#8293612yes—
subgroupG/H
centerC2×C4#937
commutator, cosocleC24×C3C24
FrattiniC24C23×D3
FittingC3×#2280C2
socleC22×C3C2×#120
prime radicalC25×C3C23
hypercenter#2280D3
hypocenterC3#23467
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #2280

Complements

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

NH ≅ G/Ntype
1. #85725C2semidirect
4. #71593C2semidirect
5. #71592C2semidirect
6. #69536C2semidirect
43. C23×C3×C4Q8semidirect
161. C23×C4Dic6semidirect
403. C3#23467semidirect

Maximal subgroups

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

subgroupindex
#857252
C3×#22802
#840272
#715932
#715922
#695362
#694972
#234673