Finite Group Databasebeta

← #10726599

Subgroups — #10726599

Order 768.

2982 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C4⋊C4C211yesmaximal, characteristic, complemented
2C22×#930C211yesmaximal, characteristic
3C2×#5213C212yesmaximal, complemented
4C2×#5213C212yesmaximal, complemented
5C2×#4991C212yesmaximal, complemented
6#72127C211yesmaximal, characteristic, complemented
7#71570C214yesmaximal, complemented
8#71565C212yesmaximal, complemented
9#14719—31nomaximal, Sylow
10C24×C3×C4C2211yescharacteristic
11C24×C3×C4C2211yescharacteristic
12C24×C3×C4C2211yescharacteristic
13C23×#67C2212yes—
14C23×D12C2211yescharacteristic
15C22×C3×C4⋊C4—24no—
16C22×C3×C4⋊C4—28no—
17C22×C3×C4⋊C4C2212yes—
18C22×C3×C4⋊C4C2212yes—
19C22×#217C2212yes—
20C2×#930—24no—
21C2×#930—28no—
22C22×(C2×D3)⋊C4C2214yes—
23C22×(C2×D3)⋊C4C2212yes—
24C22×(C2×D3)⋊C4C2214yes—
25C22×C4⋊Dic3C2211yescharacteristic
26#5213—28no—
27#5213—28no—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2214yes—
30C2×C3⋊#95C2214yes—
31C2×C3⋊#95C2212yes—
32#4991—28no—
33C22×#121—31no—
34C23×C4⋊C4D311yescharacteristic, complemented
35C2×#394—32no—
36C2×#386—32no—
37C2×#386—32no—
38#1441—32no—
39#1439—34no—
40C22⋊#95—31no—
41C25×C3C2311yescharacteristic
42C24×D3C2312yes—
43C23×C3×C4—22no—
44C23×C3×C4—22no—
45C23×C3×C4—24no—
46C23×C3×C4—24no—
47C23×C3×C4—24no—
48C23×C3×C4—22no—
49C23×C3×C4—24no—
50C23×C3×C4—24no—
51C23×C3×C4—24no—
52C23×C3×C4—42no—
53C23×C3×C4—42no—
54C23×C3×C4D414yes—
55C23×C3×C4D414yescomplemented
56C23×C3×C4C2312yes—
57C23×C3×C4C2311yescharacteristic
58C23×C3×C4C2311yescharacteristic
59C23×C3×C4C2312yes—
60C22×#67—24no—
61C22×#67—24no—
62C22×#67—24no—
63C22×#67—24no—
64C22×#67—24no—
65C22×#67—24no—
66C22×#67D418yes—
67C23×Dic3C2314yes—
68C23×Dic3C2312yes—
69C22×D12—24no—
70C22×D12—22no—
71C22×D12—24no—
72C22×D12—42no—
73C2×C3×C4⋊C4—232no—
74C2×C3×C4⋊C4—24no—
75C2×C3×C4⋊C4—24no—
76C2×C3×C4⋊C4—24no—
77C2×C3×C4⋊C4—24no—
78C2×C3×C4⋊C4—24no—
79C2×C3×C4⋊C4D418yescomplemented
80C2×#217—24no—
81C2×#217—24no—
82C2×#217—24no—
83#930—232no—
84C2×(C2×D3)⋊C4—24no—
85C2×(C2×D3)⋊C4—24no—
86C2×(C2×D3)⋊C4—24no—
87C2×(C2×D3)⋊C4—28no—
88C2×(C2×D3)⋊C4—28no—
89C2×(C2×D3)⋊C4—28no—
90C2×(C2×D3)⋊C4—28no—
91C2×(C2×D3)⋊C4—28no—
92C2×(C2×D3)⋊C4—44no—
93C2×C4⋊Dic3—22no—
94C2×C4⋊Dic3—24no—
95C3⋊#95—28no—
96C3⋊#95—24no—
97C3⋊#95—28no—
98C3⋊#95—42no—
99C23×D4—32no—
100C23×D4—31no—
101C24×C4C2×D311yescharacteristic
102C24×C4C2×D311yescharacteristic
103C24×C4C2×D311yescharacteristic
104C2×#121—68no—
105C2×#121—64no—
106C22×C4⋊C4—24no—
107C22×C4⋊C4—28no—
108C22×C4⋊C4—31no—
109C22×C4⋊C4C2×D312yes—
110C22×C4⋊C4C2×D312yes—
111C22×C22⋊C4—32no—
112C22×C22⋊C4—34no—
113C22×C22⋊C4—32no—
114C22×C22⋊C4—34no—
115#394—68no—
116#386—68no—
117#386—68no—
118C2×#95—32no—
119C2×#95—34no—
120C2×#95—34no—
121C2×#95—32no—
122C24×C3—22no—
123C24×C3—22no—
124C24×C3—22no—
125C24×C3—22no—
126C24×C3C2×D412yes—
127C24×C3C2×D414yes—
128C24×C3C2×D412yes—
129C24×C3#4112yes—
130C24×C3#4114yes—
131C24×C3C2411yescharacteristic
132C23×D3—24no—
133C23×D3—22no—
134C23×D3—22no—
135C23×D3—22no—
136C23×D3—24no—
137C23×D3—22no—
138C23×D3—22no—
139C23×D3—22no—
140C23×D3—24no—
141C23×D3—24no—
142C23×D3C2×D414yes—
143C22×C3×C4—28no—
144C22×C3×C4—216no—
145C22×C3×C4—28no—
146C22×C3×C4—28no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—216no—
154C22×C3×C4—28no—
155C22×C3×C4—28no—
156C22×C3×C4—24no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—24no—
163C22×C3×C4—28no—
164C22×C3×C4—44no—
165C22×C3×C4—416no—
166C22×C3×C4—416no—
167C22×C3×C4—44no—
168C22×C3×C4—44no—
169C22×C3×C4C2×D414yes—
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2×D414yes—
172C22×C3×C4C2×D414yes—
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2×D414yes—
175C22×C3×C4#4114yes—
176C2×#67—216no—
177C2×#67—216no—
178C2×#67—216no—
179C2×#67—48no—
180C2×#67—48no—
181C2×#67—48no—
182C2×#67—48no—
183C22×Dic3—22no—
184C22×Dic3—22no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—22no—
188C22×Dic3—22no—
189C22×Dic3—24no—
190C22×Dic3—24no—
191C22×Dic3—24no—
192C22×Dic3—24no—
193C22×Dic3—24no—
194C22×Dic3—24no—
195C22×Dic3—24no—
196C22×Dic3C2×D414yes—
197C2×D12—28no—
198C2×D12—416no—
199C2×D12—44no—
200C2×D12—44no—
201C3×C4⋊C4—216no—
202C3×C4⋊C4—216no—
203#217—48no—
204(C2×D3)⋊C4—216no—
205(C2×D3)⋊C4—416no—
206(C2×D3)⋊C4—416no—
207C4⋊Dic3—28no—
208C25—32no—
209C25C22×D311yescharacteristic
210C22×D4—38no—
211C22×D4—64no—
212C22×D4—64no—
213C22×D4—64no—
214C22×D4—64no—
215C22×D4—64no—
216C22×D4—64no—
217C22×D4—64no—
218C22×D4—62no—
219C22×D4—64no—
220C22×D4—122no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—22no—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—24no—
230C23×C4—32no—
231C23×C4—34no—
232C23×C4—42no—
233C23×C4—42no—
234C23×C4D1214yescomplemented
235C23×C4#6714yes—
236C23×C4C22×D312yes—
237C23×C4C22×D311yescharacteristic
238C23×C4C22×D312yes—
239C23×C4C22×D311yescharacteristic
240#121—632no—
241C2×C4⋊C4—232no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—64no—
248C2×C4⋊C4—62no—
249C2×C4⋊C4#6718yescomplemented
250C2×C22⋊C4—64no—
251C2×C22⋊C4—64no—
252C2×C22⋊C4—64no—
253C2×C22⋊C4—68no—
254C2×C22⋊C4—64no—
255C2×C22⋊C4—64no—
256C2×C22⋊C4—64no—
257C2×C22⋊C4—68no—
258C2×C22⋊C4—68no—
259C2×C22⋊C4—68no—
260C2×C22⋊C4—68no—
261C2×C22⋊C4—124no—
262#95—68no—
263#95—68no—
264#95—64no—
265#95—122no—
266C23×C3—22no—
267C23×C3—24no—
268C23×C3—22no—
269C23×C3—24no—
270C23×C3—22no—
271C23×C3—22no—
272C23×C3—24no—
273C23×C3—24no—
274C23×C3—28no—
275C23×C3—28no—
276C23×C3—28no—
277C23×C3—28no—
278C23×C3#12014yes—
279C23×C3#12118yes—
280C23×C3#12114yes—
281C23×C3#12318yes—
282C23×C3#12414yes—
283C23×C3C22×D411yescharacteristic
284C23×C3C22×D412yes—
285C23×C3C22×D411yescharacteristic
286C23×C3C22×D412yes—
287C23×C3C2×#4111yescharacteristic
288C23×C3C2×#4112yes—
289C23×C3C2×#4112yes—
290C23×C3H(5,𝔽2)12yes—
291C23×C32−1+412yes—
292C22×D3—24no—
293C22×D3—24no—
294C22×D3—24no—
295C22×D3—28no—
296C22×D3—28no—
297C22×D3—42no—
298C22×D3—44no—
299C22×D3—42no—
300C22×D3—42no—
301C22×D3—42no—
302C22×D3—42no—
303C22×D3—44no—
304C22×D3—42no—
305C22×D3—42no—
306C22×D3—42no—
307C22×D3—44no—
308C22×D3—44no—
309C22×D3—44no—
310C22×D3—44no—
311C22×D3—44no—
312C22×D3—44no—
313C22×D3—44no—
314C22×D3—44no—
315C2×C3×C4—232no—
316C2×C3×C4—22no—
317C2×C3×C4—24no—
318C2×C3×C4—24no—
319C2×C3×C4—24no—
320C2×C3×C4—24no—
321C2×C3×C4—24no—
322C2×C3×C4—24no—
323C2×C3×C4—22no—
324C2×C3×C4—28no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—24no—
328C2×C3×C4—22no—
329C2×C3×C4—28no—
330C2×C3×C4—416no—
331C2×C3×C4—416no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—44no—
335C2×C3×C4—44no—
336C2×C3×C4#12014yes—
337C2×C3×C4#12118yescomplemented
338#67—432no—
339C2×Dic3—24no—
340C2×Dic3—24no—
341C2×Dic3—24no—
342C2×Dic3—42no—
343C2×Dic3—42no—
344C2×Dic3—42no—
345C2×Dic3—42no—
346C2×Dic3—44no—
347C2×Dic3—44no—
348C2×Dic3—44no—
349C2×Dic3—44no—
350C2×Dic3—44no—
351C2×Dic3—44no—
352C2×Dic3—44no—
353D12—416no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—34no—
359C24—62no—
360C24—64no—
361C24—62no—
362C24—62no—
363C24—64no—
364C24—62no—
365C24—62no—
366C24—64no—
367C24—64no—
368C24—62no—
369C24C2×D1212yes—
370C24#23512yes—
371C24D3×D414yes—
372C24#23914yes—
373C24C2×#6712yes—
374C24C23×D311yescharacteristic
375C2×D4—68no—
376C2×D4—616no—
377C2×D4—616no—
378C2×D4—616no—
379C2×D4—1216no—
380C2×D4—124no—
381C2×D4—128no—
382C2×D4—128no—
383C2×D4—128no—
384C2×D4—128no—
385C2×D4—124no—
386C22×C4—28no—
387C22×C4—28no—
388C22×C4—216no—
389C22×C4—28no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—216no—
397C22×C4—28no—
398C22×C4—28no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—28no—
407C22×C4—34no—
408C22×C4—44no—
409C22×C4—416no—
410C22×C4—416no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—62no—
414C22×C4—62no—
415C22×C4—62no—
416C22×C4—62no—
417C22×C4—62no—
418C22×C4—62no—
419C22×C4—64no—
420C22×C4—64no—
421C22×C4—64no—
422C22×C4—64no—
423C22×C4—64no—
424C22×C4—64no—
425C22×C4—64no—
426C22×C4C2×D1214yes—
427C22×C4#23514yes—
428C22×C4C2×#6712yes—
429C22×C4C2×#6714yes—
430C22×C4C2×#6714yes—
431C22×C4C2×#6714yes—
432C22×C4C2×#6712yes—
433C4⋊C4—216no—
434C4⋊C4—216no—
435C4⋊C4—68no—
436C22⋊C4—616no—
437C22⋊C4—1216no—
438C22⋊C4—128no—
439C22⋊C4—1216no—
440C22×C3—24no—
441C22×C3—24no—
442C22×C3—24no—
443C22×C3—24no—
444C22×C3—24no—
445C22×C3—216no—
446C22×C3—24no—
447C22×C3—216no—
448C22×C3#39418yescomplemented
449C22×C3C2×#12011yescharacteristic
450C22×C3C2×#12011yescharacteristic
451C22×C3C2×#12112yes—
452C22×C3C2×#12111yescharacteristic
453C22×C3C2×#12112yes—
454C22×C3C2×#12312yes—
455C22×C3#52511yescharacteristic
456C22×C3#52512yes—
457C22×C3C2×#12411yescharacteristic
458C22×C3#53411yescharacteristic
459C22×C3#53611yescharacteristic
460C22×C3#53712yes—
461C22×C3#53912yes—
462C22×C3D4214yes—
463C22×C3#54714yes—
464C22×C3#54814yes—
465C22×C3#55314yes—
466C2×D3—44no—
467C2×D3—44no—
468C2×D3—44no—
469C2×D3—48no—
470C2×D3—48no—
471C2×D3—82no—
472C2×D3—82no—
473C2×D3—82no—
474C2×D3—84no—
475C2×D3—82no—
476C2×D3—84no—
477C3×C4—28no—
478C3×C4—24no—
479C3×C4—42no—
480C3×C4—44no—
481Dic3—44no—
482Dic3—84no—
483C23—28no—
484C23—24no—
485C23—28no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—22no—
490C23—28no—
491C23—22no—
492C23—22no—
493C23—28no—
494C23—22no—
495C23—64no—
496C23—64no—
497C23—68no—
498C23—64no—
499C23—68no—
500C23—122no—
501C23—122no—
502C23—124no—
503C23—124no—
504C23—124no—
505C23—124no—
506C23—124no—
507C23—124no—
508C23—122no—
509C23—122no—
510C23—124no—
511C23—124no—
512C23—122no—
513C23—122no—
514C23—122no—
515C23—124no—
516C23—124no—
517C23—122no—
518C23#89418yes—
519C23#89518yes—
520C23#93014yes—
521C23#93714yes—
522C23#94714yes—
523C23C22×D1211yescharacteristic
524C23C2×#23511yescharacteristic
525C23C2×#23512yes—
526C23C2×D3×D412yes—
527C23#100412yes—
528C23C2×#23912yes—
529C23#100712yes—
530C23C22×#6711yescharacteristic
531C23C22×#6712yes—
532D4—1216no—
533D4—1232no—
534C2×C4—232no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—28no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—28no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—416no—
550C2×C4—416no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—64no—
556C2×C4—64no—
557C2×C4—64no—
558C2×C4—122no—
559C2×C4—122no—
560C2×C4—122no—
561C2×C4—122no—
562C2×C4—124no—
563C2×C4—124no—
564C2×C4—124no—
565C2×C4—124no—
566C2×C4—124no—
567C2×C4—124no—
568C2×C4—124no—
569C2×C4#93018yescomplemented
570C2×C4#95314yes—
571C2×C3—28no—
572C2×C3C2×#39411yescharacteristic
573C2×C3#240311yescharacteristic
574C2×C3#240611yescharacteristic
575C2×C3#241614yes—
576C2×C3#243611yescharacteristic
577C2×C3#244314yes—
578C2×C3#259111yescharacteristic
579C2×C3#261811yescharacteristic
580C2×C3#284711yescharacteristic
581D3—84no—
582C22—24no—
583C22—24no—
584C22—216no—
585C22—216no—
586C22—24no—
587C22—24no—
588C22—24no—
589C22—24no—
590C22—124no—
591C22—124no—
592C22—128no—
593C22—128no—
594C22—124no—
595C22—242no—
596C22—244no—
597C22—244no—
598C22—242no—
599C22—242no—
600C22—242no—
601C22#499118yescomplemented
602C22C2×#89412yes—
603C22C2×#89512yes—
604C22#550714yes—
605C22#551012yes—
606C22#551414yes—
607C22#551512yes—
608C22C2×#93011yescharacteristic
609C22C2×#93012yes—
610C22#579111yescharacteristic
611C22#579112yes—
612C22C2×#93711yescharacteristic
613C22D4×#6714yes—
614C22#580411yescharacteristic
615C22C2×#94711yescharacteristic
616C22#581611yescharacteristic
617C22#581714yes—
618C22C2×#95311yescharacteristic
619C4—28no—
620C4—24no—
621C4—42no—
622C4—44no—
623C4—124no—
624C4—244no—
625C3#1471911yescharacteristic, complemented, Sylow
626C2—28no—
627C2—244no—
628C2#8218211yescharacteristic
629C2C2×#499111yescharacteristic
630C2#8219211yescharacteristic
631C2#8219314yes—
632C2#8219514yes—
633C2#8219911yescharacteristic
634C2#8403211yescharacteristic
635C2#8405211yescharacteristic
636C2#8406611yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC23×C3×C4⋊C4C2
prime radicalC22×#930C2
hypercenterC23×C4⋊C4D3
hypocenterC3#14719
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C23×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C23×C3×C4⋊C4C2semidirect
3. C2×#5213C2semidirect
4. C2×#5213C2semidirect
5. C2×#4991C2semidirect
6. #72127C2semidirect
7. #71570C2semidirect
8. #71565C2semidirect
34. C23×C4⋊C4D3semidirect
55. C23×C3×C4D4semidirect
79. C2×C3×C4⋊C4D4semidirect
234. C23×C4D12semidirect
249. C2×C4⋊C4#67semidirect
337. C2×C3×C4#121semidirect
448. C22×C3#394semidirect
569. C2×C4#930semidirect
601. C22#4991semidirect
625. C3#14719semidirect

Maximal subgroups

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

subgroupindex
C23×C3×C4⋊C42
C22×#9302
C2×#52132
C2×#49912
#721272
#715702
#715652
#147193