Finite Group Databasebeta

← #10724473

Subgroups — #10724473

Order 768.

1406 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4946C211yesmaximal, characteristic
2C3×#1437C211yesmaximal, characteristic
3#71549C212yesmaximal
4#71545C211yesmaximal, characteristic
5#71544C212yesmaximal
6#71534C212yesmaximal, complemented
7#69421C212yesmaximal, complemented
8#69417C212yesmaximal, complemented
9#69405C212yesmaximal, complemented
10#14515—31nomaximal, Sylow
11C24×Dic3C2211yescharacteristic
12C22×C3×C22⋊C4C2212yes—
13C22×C3×C22⋊C4C2211yescharacteristic
14C22×C3×C42C2211yescharacteristic
15C22×#217C2212yes—
16C22×#217C2211yescharacteristic
17C22×Dic3⋊C4C2212yes—
18C22×C4×Dic3C2211yescharacteristic
19C2×C3×#95C2212yes—
20C2×C3×#95C2211yescharacteristic
21C2×C3⋊#95C2212yes—
22C2×C3⋊#95C2212yes—
23C2×C3⋊#95C2212yes—
24C2×C3⋊#95C2212yes—
25C2×C3⋊#95C2212yes—
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2211yescharacteristic
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32#4946C418yescomplemented
33C2×#385—31no—
34#1445—32no—
35#1443—32no—
36#1440—32no—
37#1437—31no—
38#1437D311yescharacteristic
39#1436—32no—
40#1434—32no—
41C22⋊#95—32no—
42C25×C3C2311yescharacteristic
43C23×C3×C4C2312yes—
44C23×C3×C4C2311yescharacteristic
45C23×C3×C4C2311yescharacteristic
46C23×C3×C4C2312yes—
47C23×Dic3—41no—
48C23×Dic3—41no—
49C23×Dic3—42no—
50C23×Dic3—42no—
51C23×Dic3C2×C414yes—
52C23×Dic3C2312yes—
53C23×Dic3C2312yes—
54C23×Dic3C2312yes—
55C23×Dic3C2312yes—
56C2×C3×C22⋊C4—24no—
57C2×C3×C22⋊C4—24no—
58C2×C3×C22⋊C4—41no—
59C2×C3×C22⋊C4—41no—
60C2×C3×C22⋊C4—42no—
61C2×C3×C22⋊C4C2×C414yes—
62C2×C3×C42—24no—
63C2×C3×C42—41no—
64C2×#217—24no—
65C2×#217—24no—
66C2×#217—24no—
67C2×#217—42no—
68C2×#217C2×C414yes—
69C2×Dic3⋊C4—42no—
70C2×Dic3⋊C4—42no—
71C2×Dic3⋊C4C2×C418yes—
72C2×C4×Dic3—41no—
73C2×C4×Dic3—42no—
74C3×#95—42no—
75C3×#95—41no—
76C3⋊#95—24no—
77C3⋊#95—24no—
78C3⋊#95—24no—
79C3⋊#95—42no—
80C3⋊#95—42no—
81C3⋊#95—42no—
82C3⋊#95—42no—
83C3⋊#95—42no—
84C3⋊#95—42no—
85C3⋊#95—41no—
86C3⋊#95C2×C418yes—
87C24×C4—31no—
88C22×C4⋊C4—32no—
89C22×C22⋊C4—31no—
90C22×C22⋊C4—32no—
91C22×C22⋊C4C2×D312yes—
92C22×C22⋊C4C2×D311yescharacteristic
93C22×C42—31no—
94C22×C42C2×D311yescharacteristic
95#385—38no—
96C2×#95—32no—
97C2×#95—32no—
98C2×#95—32no—
99C2×#95—32no—
100C2×#95—32no—
101C2×#95—32no—
102C2×#95—32no—
103C2×#95—32no—
104C2×#95—32no—
105C2×#95—31no—
106C2×#95—32no—
107C2×#95C2×D312yes—
108C2×#95C2×D311yescharacteristic
109C24×C3—22no—
110C24×C3—22no—
111C24×C3—22no—
112C24×C3—21no—
113C24×C3—21no—
114C24×C3—22no—
115C24×C3—22no—
116C24×C3C22×C412yes—
117C24×C3#4114yes—
118C24×C3C2411yescharacteristic
119C22×C3×C4—22no—
120C22×C3×C4—22no—
121C22×C3×C4—22no—
122C22×C3×C4—22no—
123C22×C3×C4—22no—
124C22×C3×C4—22no—
125C22×C3×C4—21no—
126C22×C3×C4—22no—
127C22×C3×C4—21no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4C22×C412yes—
141C22×C3×C4C22×C412yes—
142C22×C3×C4#4114yes—
143C22×Dic3—22no—
144C22×Dic3—22no—
145C22×Dic3—22no—
146C22×Dic3—22no—
147C22×Dic3—22no—
148C22×Dic3—22no—
149C22×Dic3—22no—
150C22×Dic3—22no—
151C22×Dic3—22no—
152C22×Dic3—22no—
153C22×Dic3—22no—
154C22×Dic3—22no—
155C22×Dic3—22no—
156C22×Dic3—22no—
157C22×Dic3—22no—
158C22×Dic3—22no—
159C22×Dic3—22no—
160C22×Dic3—22no—
161C22×Dic3—22no—
162C22×Dic3—22no—
163C22×Dic3—22no—
164C22×Dic3—22no—
165C22×Dic3—22no—
166C22×Dic3—22no—
167C22×Dic3—22no—
168C22×Dic3—48no—
169C22×Dic3—44no—
170C22×Dic3—44no—
171C22×Dic3—48no—
172C22×Dic3—44no—
173C22×Dic3C22×C414yes—
174C22×Dic3C22×C414yes—
175C22×Dic3#4114yes—
176C3×C22⋊C4—44no—
177C3×C22⋊C4—48no—
178C3×C42—44no—
179#217—28no—
180#217—48no—
181Dic3⋊C4—48no—
182C4×Dic3—44no—
183C25C22×D311yescharacteristic
184C23×C4—32no—
185C23×C4—32no—
186C23×C4—32no—
187C23×C4—34no—
188C23×C4—32no—
189C23×C4—121no—
190C23×C4—121no—
191C23×C4—122no—
192C23×C4—122no—
193C23×C4C22×D312yes—
194C23×C4C22×D311yescharacteristic
195C23×C4C22×D311yescharacteristic
196C23×C4C22×D312yes—
197C2×C4⋊C4—38no—
198C2×C4⋊C4—122no—
199C2×C4⋊C4—122no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—34no—
203C2×C22⋊C4—41no—
204C2×C22⋊C4—41no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—64no—
207C2×C22⋊C4—64no—
208C2×C22⋊C4—64no—
209C2×C22⋊C4—122no—
210C2×C22⋊C4C4×D314yes—
211C2×C42—24no—
212C2×C42—41no—
213C2×C42—121no—
214C2×C42—122no—
215#95—38no—
216#95—42no—
217#95—41no—
218#95—64no—
219#95—64no—
220#95—64no—
221#95—122no—
222#95—121no—
223#95—122no—
224#95—122no—
225#95—122no—
226#95—122no—
227#95—122no—
228C23×C3—24no—
229C23×C3—22no—
230C23×C3—24no—
231C23×C3—22no—
232C23×C3—24no—
233C23×C3—24no—
234C23×C3—24no—
235C23×C3—24no—
236C23×C3—24no—
237C23×C3—24no—
238C23×C3—42no—
239C23×C3—42no—
240C23×C3—42no—
241C23×C3—42no—
242C23×C3—41no—
243C23×C3—42no—
244C23×C3—41no—
245C23×C3—41no—
246C23×C3—42no—
247C23×C3—41no—
248C23×C3#11714yes—
249C23×C3C23×C411yescharacteristic
250C23×C3C2×#4112yes—
251C23×C3C2×#4112yes—
252C23×C3C2×#4112yes—
253C23×C3H(5,𝔽2)11yescharacteristic
254C23×C3H(5,𝔽2)11yescharacteristic
255C23×C3H(5,𝔽2)12yes—
256C23×C3H(5,𝔽2)12yes—
257C23×C32−1+412yes—
258C2×C3×C4—24no—
259C2×C3×C4—24no—
260C2×C3×C4—22no—
261C2×C3×C4—22no—
262C2×C3×C4—24no—
263C2×C3×C4—24no—
264C2×C3×C4—24no—
265C2×C3×C4—42no—
266C2×C3×C4—42no—
267C2×C3×C4—41no—
268C2×C3×C4—42no—
269C2×C3×C4—42no—
270C2×C3×C4—41no—
271C2×C3×C4—41no—
272C2×C3×C4—42no—
273C2×C3×C4—42no—
274C2×C3×C4—41no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—42no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×Dic3—24no—
283C2×Dic3—24no—
284C2×Dic3—24no—
285C2×Dic3—24no—
286C2×Dic3—24no—
287C2×Dic3—24no—
288C2×Dic3—24no—
289C2×Dic3—24no—
290C2×Dic3—24no—
291C2×Dic3—416no—
292C2×Dic3—42no—
293C2×Dic3—42no—
294C2×Dic3—42no—
295C2×Dic3—42no—
296C2×Dic3—42no—
297C2×Dic3—42no—
298C2×Dic3—42no—
299C2×Dic3—42no—
300C2×Dic3—42no—
301C2×Dic3—42no—
302C2×Dic3—42no—
303C2×Dic3—42no—
304C2×Dic3—42no—
305C2×Dic3—42no—
306C2×Dic3—42no—
307C2×Dic3—42no—
308C2×Dic3—42no—
309C2×Dic3—42no—
310C2×Dic3—42no—
311C24—22no—
312C24—21no—
313C24—21no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24C2×C4×D312yes—
319C24#23514yes—
320C24C23×D311yescharacteristic
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—34no—
343C22×C4—34no—
344C22×C4—34no—
345C22×C4—62no—
346C22×C4—62no—
347C22×C4—62no—
348C22×C4—62no—
349C22×C4—62no—
350C22×C4—62no—
351C22×C4—62no—
352C22×C4—62no—
353C22×C4—62no—
354C22×C4—62no—
355C22×C4—62no—
356C22×C4—62no—
357C22×C4—62no—
358C22×C4—62no—
359C22×C4—62no—
360C22×C4—62no—
361C22×C4—62no—
362C22×C4—62no—
363C22×C4—62no—
364C22×C4—62no—
365C22×C4—62no—
366C22×C4—62no—
367C22×C4—62no—
368C22×C4—62no—
369C22×C4—62no—
370C22×C4—124no—
371C22×C4—128no—
372C22×C4—124no—
373C22×C4—124no—
374C22×C4—128no—
375C22×C4C2×C4×D312yes—
376C22×C4C2×C4×D312yes—
377C22×C4#23714yes—
378C4⋊C4—128no—
379C22⋊C4—44no—
380C22⋊C4—48no—
381C22⋊C4—68no—
382C22⋊C4—128no—
383C42—44no—
384C42—124no—
385C22×C3—24no—
386C22×C3—28no—
387C22×C3—44no—
388C22×C3—42no—
389C22×C3—42no—
390C22×C3—44no—
391C22×C3—44no—
392C22×C3—44no—
393C22×C3—44no—
394C22×C3C2×#11711yescharacteristic
395C22×C3C4×#4112yes—
396C22×C3#51811yescharacteristic
397C22×C3#51812yes—
398C22×C3#51911yescharacteristic
399C22×C3#53912yes—
400C22×C3#53912yes—
401C22×C3#53912yes—
402C22×C3#53912yes—
403C22×C3#54212yes—
404C22×C3#54212yes—
405C22×C3#55112yes—
406C22×C3#55112yes—
407C22×C3#55111yescharacteristic
408C22×C3#55312yes—
409C22×C3#55312yes—
410C22×C3#55312yes—
411C22×C3#55911yescharacteristic
412C22×C3#56011yescharacteristic
413C22×C3#56012yes—
414C22×C3H(3,𝔽4)11yescharacteristic
415C3×C4—42no—
416C3×C4—42no—
417C3×C4—44no—
418C3×C4—82no—
419Dic3—44no—
420Dic3—44no—
421Dic3—44no—
422Dic3—82no—
423C23—22no—
424C23—24no—
425C23—22no—
426C23—24no—
427C23—24no—
428C23—24no—
429C23—24no—
430C23—24no—
431C23—24no—
432C23—24no—
433C23—41no—
434C23—42no—
435C23—42no—
436C23—41no—
437C23—42no—
438C23—41no—
439C23—41no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23#87214yes—
444C23C22×C4×D311yescharacteristic
445C23C2×#23512yes—
446C23C2×#23712yes—
447C23#100411yescharacteristic
448C23#100412yes—
449C23#100411yescharacteristic
450C23D3×#4112yes—
451C23#100912yes—
452C23#101012yes—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—64no—
478C2×C4—64no—
479C2×C4—64no—
480C2×C4—64no—
481C2×C4—64no—
482C2×C4—64no—
483C2×C4—64no—
484C2×C4—64no—
485C2×C4—64no—
486C2×C4—1216no—
487C2×C4—122no—
488C2×C4—122no—
489C2×C4—122no—
490C2×C4—122no—
491C2×C4—122no—
492C2×C4—122no—
493C2×C4—122no—
494C2×C4—122no—
495C2×C4—122no—
496C2×C4—122no—
497C2×C4—122no—
498C2×C4—122no—
499C2×C4—122no—
500C2×C4—122no—
501C2×C4—122no—
502C2×C4—122no—
503C2×C4—122no—
504C2×C4—122no—
505C2×C4—122no—
506C2×C3—44no—
507C2×C3#231311yescharacteristic
508C2×C3#233212yes—
509C2×C3#235312yes—
510C2×C3#237411yescharacteristic
511C2×C3#237511yescharacteristic
512C2×C3#274612yes—
513C2×C3#275712yes—
514C2×C3#278212yes—
515C2×C3#279612yes—
516C22—24no—
517C22—28no—
518C22—44no—
519C22—44no—
520C22—44no—
521C22—44no—
522C22—42no—
523C22—42no—
524C22—44no—
525C22C2×#87211yescharacteristic
526C22#548611yescharacteristic
527C22#549512yes—
528C22#551611yescharacteristic
529C22#552712yes—
530C22C4×#23712yes—
531C22#554012yes—
532C22#554512yes—
533C22#556412yes—
534C22#556412yes—
535C22#556512yes—
536C22#558711yescharacteristic
537C22#560312yes—
538C22#561412yes—
539C22#562012yes—
540C22#566411yescharacteristic
541C22#566512yes—
542C22#566912yes—
543C22#568311yescharacteristic
544C22#570712yes—
545C22#572011yescharacteristic
546C4—44no—
547C4—42no—
548C4—42no—
549C4—82no—
550C4—124no—
551C4—124no—
552C4—124no—
553C4—242no—
554C3#1451511yescharacteristic, complemented, Sylow
555C2—44no—
556C2#7344511yescharacteristic
557C2#7346212yes—
558C2#7468111yescharacteristic
559C2#7483612yes—
560C2#7546412yes—
561C2#7577712yes—
562C2#7579512yes—
563C2#7641112yes—
564C2#7643411yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC3×#1437C2
socle, cosocleC24×C3C24
prime radicalC25×C3C23
hypercenter#1437D3
hypocenterC3#14515
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1437

Complements

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

NH ≅ G/Ntype
6. #71534C2semidirect
7. #69421C2semidirect
8. #69417C2semidirect
9. #69405C2semidirect
32. #4946C4semidirect
554. C3#14515semidirect

Maximal subgroups

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

subgroupindex
C2×#49462
C3×#14372
#715492
#715452
#715442
#715342
#694212
#694172
#694052
#145153