Finite Group Databasebeta

← #10732663

Subgroups — #10732663

Order 768.

1542 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#380C211yesmaximal, characteristic
2C2×#5212C211yesmaximal, characteristic
3C2×C4×#217C211yesmaximal, characteristic
4C2×#4977C211yesmaximal, characteristic, complemented
5#71563C211yesmaximal, characteristic, complemented
6#71563C211yesmaximal, characteristic, complemented
7#71563C211yesmaximal, characteristic, complemented
8#71548C214yesmaximal
9#69418C214yesmaximal, complemented
10#15233—31nomaximal, Sylow
11C24×C3×C4C2211yescharacteristic
12C22×C3×C4⋊C4C2211yescharacteristic
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C22⋊C4C2212yes—
15C22×#217C2212yes—
16C22×#217C2212yes—
17C22×C4⋊Dic3C2211yescharacteristic
18C22×C4⋊Dic3C2211yescharacteristic
19C22×C4×Dic3C2211yescharacteristic
20C22×C4×Dic3C2211yescharacteristic
21C3×#380—24no—
22C2×C3×#95C2212yes—
23#5212—24no—
24C4×#217—24no—
25C2×C3⋊#95C2212yes—
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2214yes—
28C2×C3⋊#95C2214yes—
29C2×C3⋊#95C2214yes—
30C2×C3⋊#95C2214yes—
31#4977—24no—
32C2×#382—31no—
33C2×#380—31no—
34C2×#380D311yescharacteristic
35C2×C4×C22⋊C4—31no—
36#1443—34no—
37#1440—34no—
38C4⋊#95—31no—
39C4⋊#95—31no—
40C4⋊#95—31no—
41C25×C3C2311yescharacteristic
42C23×C3×C4—22no—
43C23×C3×C4—22no—
44C23×C3×C4—22no—
45C23×C3×C4—22no—
46C23×C3×C4—22no—
47C23×C3×C4—22no—
48C23×C3×C4D414yes—
49C23×C3×C4C2312yes—
50C23×C3×C4C2312yes—
51C23×C3×C4C2311yescharacteristic
52C23×C3×C4C2311yescharacteristic
53C23×Dic3C2312yes—
54C23×Dic3C2312yes—
55C23×Dic3C2312yes—
56C23×Dic3C2312yes—
57C2×C3×C4⋊C4—22no—
58C2×C3×C4⋊C4—22no—
59C2×C3×C4⋊C4—22no—
60C2×C3×C4⋊C4—22no—
61C2×C3×C4⋊C4—22no—
62C2×C3×C4⋊C4—22no—
63C2×C3×C22⋊C4—24no—
64C2×C3×C22⋊C4—28no—
65C2×#217—24no—
66C2×#217—24no—
67C2×#217—24no—
68C2×#217—24no—
69C2×#217—44no—
70C2×C4⋊Dic3—22no—
71C2×C4⋊Dic3—22no—
72C2×C4⋊Dic3—22no—
73C2×C4⋊Dic3—22no—
74C2×C4⋊Dic3—22no—
75C2×C4⋊Dic3—22no—
76C2×C4×Dic3—22no—
77C2×C4×Dic3—22no—
78C2×C4×Dic3—22no—
79C2×C4×Dic3—22no—
80C2×C4×Dic3—22no—
81C2×C4×Dic3D414yes—
82C3×#95—24no—
83C3⋊#95—24no—
84C3⋊#95—24no—
85C3⋊#95—28no—
86C3⋊#95—28no—
87C3⋊#95—44no—
88C3⋊#95—44no—
89C24×C4C2×D311yescharacteristic
90C22×C4⋊C4—31no—
91C22×C4⋊C4—31no—
92C22×C4⋊C4C2×D311yescharacteristic
93C22×C4⋊C4C2×D311yescharacteristic
94C22×C22⋊C4—32no—
95C22×C22⋊C4—32no—
96C22×C22⋊C4C2×D312yes—
97C22×C42—31no—
98C22×C42—31no—
99#382—64no—
100#380—24no—
101#380—64no—
102C4×C22⋊C4—64no—
103C2×#95—34no—
104C2×#95—34no—
105C2×#95—32no—
106C2×#95—32no—
107C2×#95—34no—
108C2×#95—34no—
109C2×#95C2×D312yes—
110C24×C3—21no—
111C24×C3—22no—
112C24×C3—21no—
113C24×C3—21no—
114C24×C3—21no—
115C24×C3—21no—
116C24×C3—21no—
117C24×C3—22no—
118C24×C3—22no—
119C24×C3C2×D412yes—
120C24×C3#4114yes—
121C24×C3C2411yescharacteristic
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—28no—
129C22×C3×C4—28no—
130C22×C3×C4—28no—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—24no—
135C22×C3×C4—24no—
136C22×C3×C4—24no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—24no—
141C22×C3×C4—44no—
142C22×C3×C4—44no—
143C22×C3×C4—44no—
144C22×C3×C4—44no—
145C22×C3×C4C2×D412yes—
146C22×C3×C4C2×D412yes—
147C22×C3×C4C2×D412yes—
148C22×C3×C4#4112yes—
149C22×C3×C4#4112yes—
150C22×C3×C4#4112yes—
151C22×C3×C4#4112yes—
152C22×C3×C4#4112yes—
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—24no—
169C22×Dic3—22no—
170C22×Dic3—22no—
171C22×Dic3—22no—
172C22×Dic3—24no—
173C22×Dic3—24no—
174C22×Dic3—24no—
175C22×Dic3C2×D414yes—
176C22×Dic3#4114yes—
177C3×C4⋊C4—44no—
178C3×C4⋊C4—44no—
179C3×C22⋊C4—48no—
180#217—216no—
181#217—48no—
182C4⋊Dic3—44no—
183C4⋊Dic3—44no—
184C4×Dic3—28no—
185C4×Dic3—44no—
186C25C22×D311yescharacteristic
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—32no—
194C23×C4—32no—
195C23×C4—32no—
196C23×C4—32no—
197C23×C4#6714yes—
198C23×C4C22×D312yes—
199C23×C4C22×D312yes—
200C23×C4C22×D311yescharacteristic
201C23×C4C22×D311yescharacteristic
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—62no—
209C2×C4⋊C4—62no—
210C2×C4⋊C4—62no—
211C2×C4⋊C4—62no—
212C2×C4⋊C4—62no—
213C2×C4⋊C4—62no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—64no—
217C2×C22⋊C4—64no—
218C2×C22⋊C4—64no—
219C2×C22⋊C4—64no—
220C2×C22⋊C4—124no—
221C2×C42—34no—
222C2×C42—62no—
223C2×C42—62no—
224C2×C42—62no—
225C2×C42—62no—
226C2×C42—62no—
227#95—24no—
228#95—68no—
229#95—64no—
230#95—68no—
231#95—64no—
232#95—124no—
233#95—124no—
234C23×C3—22no—
235C23×C3—22no—
236C23×C3—22no—
237C23×C3—24no—
238C23×C3—24no—
239C23×C3—22no—
240C23×C3—22no—
241C23×C3—22no—
242C23×C3—24no—
243C23×C3—24no—
244C23×C3—24no—
245C23×C3—24no—
246C23×C3—41no—
247C23×C3—41no—
248C23×C3—41no—
249C23×C3—41no—
250C23×C3—42no—
251C23×C3—42no—
252C23×C3—42no—
253C23×C3—42no—
254C23×C3—42no—
255C23×C3—42no—
256C23×C3#12414yes—
257C23×C3C22×D411yescharacteristic
258C23×C3C22×D411yescharacteristic
259C23×C3C2×#4111yescharacteristic
260C23×C3C2×#4111yescharacteristic
261C23×C3C2×#4111yescharacteristic
262C23×C3C2×#4111yescharacteristic
263C23×C3C2×#4111yescharacteristic
264C23×C3C2×#4112yes—
265C23×C3C2×#4112yes—
266C23×C3H(5,𝔽2)12yes—
267C23×C32−1+412yes—
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—22no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—41no—
274C2×C3×C4—41no—
275C2×C3×C4—416no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—44no—
279C2×C3×C4—42no—
280C2×C3×C4—42no—
281C2×C3×C4—44no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4—44no—
285C2×C3×C4—44no—
286C2×C3×C4#12114yes—
287C2×C3×C4#12314yes—
288C2×C3×C4#12314yes—
289C2×C3×C4#12314yes—
290C2×C3×C4#12414yes—
291C2×C3×C4#12614yes—
292C2×C3×C4#12614yes—
293C2×Dic3—24no—
294C2×Dic3—24no—
295C2×Dic3—24no—
296C2×Dic3—24no—
297C2×Dic3—24no—
298C2×Dic3—24no—
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—
311C2×Dic3—42no—
312C2×Dic3—42no—
313C2×Dic3—44no—
314C2×Dic3—44no—
315C2×Dic3—44no—
316C2×Dic3—44no—
317C24—21no—
318C24—21no—
319C24—21no—
320C24—21no—
321C24—21no—
322C24—21no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24#23714yes—
327C24C2×#6712yes—
328C24C23×D311yescharacteristic
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—28no—
336C22×C4—28no—
337C22×C4—28no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—34no—
349C22×C4—34no—
350C22×C4—44no—
351C22×C4—44no—
352C22×C4—44no—
353C22×C4—44no—
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—64no—
371C22×C4—64no—
372C22×C4—64no—
373C22×C4—64no—
374C22×C4—62no—
375C22×C4—62no—
376C22×C4#23512yes—
377C22×C4#23512yes—
378C22×C4D3×D412yes—
379C22×C4#23712yes—
380C22×C4#23712yes—
381C22×C4#23712yes—
382C22×C4C2×#6712yes—
383C22×C4C2×#6712yes—
384C4⋊C4—44no—
385C4⋊C4—44no—
386C4⋊C4—124no—
387C4⋊C4—124no—
388C22⋊C4—48no—
389C22⋊C4—616no—
390C22⋊C4—128no—
391C42—68no—
392C42—124no—
393C22×C3—24no—
394C22×C3—24no—
395C22×C3—24no—
396C22×C3—44no—
397C22×C3—42no—
398C22×C3—42no—
399C22×C3—44no—
400C22×C3—44no—
401C22×C3—44no—
402C22×C3—42no—
403C22×C3—42no—
404C22×C3C2×#12111yescharacteristic
405C22×C3C2×#12311yescharacteristic
406C22×C3C2×#12311yescharacteristic
407C22×C3C2×#12311yescharacteristic
408C22×C3#52512yes—
409C22×C3C2×#12411yescharacteristic
410C22×C3C2×#12411yescharacteristic
411C22×C3C2×#12611yescharacteristic
412C22×C3C2×#12611yescharacteristic
413C22×C3#52912yes—
414C22×C3#53211yescharacteristic
415C22×C3#53511yescharacteristic
416C22×C3#53611yescharacteristic
417C22×C3#53712yes—
418C22×C3#54212yes—
419C22×C3#55212yes—
420C22×C3#55212yes—
421C22×C3#55212yes—
422C22×C3#55312yes—
423C22×C3#55312yes—
424C22×C3#55312yes—
425C22×C3#55512yes—
426C22×C3#55612yes—
427C3×C4—42no—
428C3×C4—82no—
429C3×C4—82no—
430C3×C4#39718yes—
431Dic3—44no—
432Dic3—44no—
433Dic3—82no—
434Dic3—82no—
435C23—24no—
436C23—24no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—24no—
444C23—24no—
445C23—24no—
446C23—24no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23—41no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23#93614yes—
458C23C2×#23511yescharacteristic
459C23C2×#23511yescharacteristic
460C23C2×D3×D411yescharacteristic
461C23C2×#23711yescharacteristic
462C23C2×#23711yescharacteristic
463C23C2×#23711yescharacteristic
464C23C2×#23712yes—
465C23#100712yes—
466C23D3×#4112yes—
467C23#100912yes—
468C23C22×#6711yescharacteristic
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—416no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—64no—
488C2×C4—64no—
489C2×C4—64no—
490C2×C4—64no—
491C2×C4—64no—
492C2×C4—64no—
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—124no—
502C2×C4—124no—
503C2×C4—124no—
504C2×C4—122no—
505C2×C4—122no—
506C2×C4—122no—
507C2×C4—122no—
508C2×C4—122no—
509C2×C4—122no—
510C2×C4—124no—
511C2×C4#87914yes—
512C2×C4#87914yes—
513C2×C4#88314yes—
514C2×C4#88514yes—
515C2×C4#92914yes—
516C2×C4#93514yes—
517C2×C4#93914yes—
518C2×C3—44no—
519C2×C3C2×#39711yescharacteristic
520C2×C3#248512yes—
521C2×C3#249412yes—
522C2×C3#249512yes—
523C2×C3#249911yescharacteristic
524C2×C3#250011yescharacteristic
525C2×C3#250311yescharacteristic
526C2×C3#250411yescharacteristic
527C2×C3#256212yes—
528C2×C3#262011yescharacteristic
529C2×C3#285011yescharacteristic
530C22—24no—
531C22—24no—
532C22—24no—
533C22—44no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—44no—
539C22—44no—
540C22—44no—
541C22C2×#87911yescharacteristic
542C22C2×#87911yescharacteristic
543C22C2×#88311yescharacteristic
544C22C2×#88511yescharacteristic
545C22#551312yes—
546C22#551312yes—
547C22#553012yes—
548C22#553212yes—
549C22#560012yes—
550C22#560612yes—
551C22#561812yes—
552C22#561812yes—
553C22#562912yes—
554C22#563912yes—
555C22#564112yes—
556C22#564312yes—
557C22C2×#92911yescharacteristic
558C22C2×#93511yescharacteristic
559C22C2×#93611yescharacteristic
560C22C2×#93911yescharacteristic
561C22#581611yescharacteristic
562C22#583011yescharacteristic
563C22#583211yescharacteristic
564C4—42no—
565C4—82no—
566C4—82no—
567C4—124no—
568C4—124no—
569C4—242no—
570C4—242no—
571C4#495218yes—
572C3#1523311yescharacteristic, complemented, Sylow
573C2—44no—
574C2#7391812yes—
575C2#7392112yes—
576C2#7394012yes—
577C2#7402412yes—
578C2C2×#495211yescharacteristic
579C2#8217911yescharacteristic
580C2#8225711yescharacteristic
581C2#8266511yescharacteristic
582C2#8269811yescharacteristic
583C2#8277411yescharacteristic
584C2#8282111yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC2×C3×#380C2
prime radicalC25×C3C23
hypercenterC2×#380D3
hypocenterC3#15233
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
4. C2×#4977C2semidirect
5. #71563C2semidirect
6. #71563C2semidirect
7. #71563C2semidirect
9. #69418C2semidirect
572. C3#15233semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3802
C2×#52122
C2×C4×#2172
C2×#49772
#715632
#715482
#694182
#152333