Finite Group Databasebeta

← #10731723

Subgroups — #10731723

Order 768.

3458 conjugacy classes of subgroups.

The subgroup lattice has 726 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#938C212yesmaximal, complemented
2C22×C4×D12C211yesmaximal, characteristic, complemented
3C2×C3×#390C211yesmaximal, characteristic, complemented
4C2×#4987C212yesmaximal, complemented
5C2×#4960C212yesmaximal, complemented
6#71550C214yesmaximal, complemented
7#71546C212yesmaximal, complemented
8#71535C211yesmaximal, characteristic, complemented
9#15123—31nomaximal, Sylow
10C23×C3×D4C2211yescharacteristic
11C23×#67C2214yes—
12C23×D12C2211yescharacteristic
13C23×C4×D3C2212yes—
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C22⋊C4C2212yes—
16C22×C3×C22⋊C4C2212yes—
17C22×C3×C42C2211yescharacteristic
18C22×#217C2214yes—
19C2×#938—28no—
20C2×#938—28no—
21C2×#938—28no—
22C22×(C2×D3)⋊C4C2212yes—
23C22×(C2×D3)⋊C4C2214yes—
24C22×C4⋊Dic3C2211yescharacteristic
25C22×C4⋊Dic3C2212yes—
26C2×C4×D12—28no—
27C2×C4×D12—24no—
28C3×#390—24no—
29C2×C3⋊#95C2214yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32#4987—28no—
33#4960—28no—
34C22×#121—32no—
35C22×C4×D4—31no—
36C2×#392—32no—
37C2×#390D311yescharacteristic, complemented
38C2×#380—32no—
39#1439—32no—
40#1438—34no—
41C4⋊#95—31no—
42C25×C3C2312yes—
43C24×D3C2312yes—
44C22×C3×D4—22no—
45C22×C3×D4—24no—
46C22×C3×D4—24no—
47C22×C3×D4—22no—
48C22×C3×D4D414yescomplemented
49C23×C3×C4C2312yes—
50C23×C3×C4C2311yescharacteristic
51C23×C3×C4C2311yescharacteristic
52C23×C3×C4C2311yescharacteristic
53C22×#67—28no—
54C22×#67—28no—
55C22×#67—28no—
56C22×#67—28no—
57C22×#67—28no—
58C22×#67—44no—
59C22×#67—44no—
60C23×Dic3C2312yes—
61C23×Dic3C2314yes—
62C22×D12—24no—
63C22×D12—24no—
64C22×D12—22no—
65C22×D12—22no—
66C22×D12D414yescomplemented
67C22×C4×D3—24no—
68C22×C4×D3—24no—
69C22×C4×D3—24no—
70C22×C4×D3—24no—
71C22×C4×D3—24no—
72C22×C4×D3—24no—
73C22×C4×D3D418yescomplemented
74C2×C3×C4⋊C4—24no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C22⋊C4—24no—
77C2×C3×C22⋊C4—24no—
78C2×C3×C22⋊C4—24no—
79C2×C3×C22⋊C4—24no—
80C2×C3×C22⋊C4—44no—
81C2×C3×C42—24no—
82C2×C3×C42D414yescomplemented
83C2×#217—28no—
84C2×#217—28no—
85C2×#217—28no—
86#938—264no—
87C2×(C2×D3)⋊C4—24no—
88C2×(C2×D3)⋊C4—24no—
89C2×(C2×D3)⋊C4—24no—
90C2×(C2×D3)⋊C4—28no—
91C2×(C2×D3)⋊C4—28no—
92C2×(C2×D3)⋊C4—44no—
93C2×C4⋊Dic3—22no—
94C2×C4⋊Dic3—24no—
95C2×C4⋊Dic3—24no—
96C2×C4⋊Dic3—24no—
97C2×C4⋊Dic3D418yescomplemented
98C4×D12—232no—
99C3⋊#95—24no—
100C3⋊#95—28no—
101C3⋊#95—42no—
102C23×D4—34no—
103C23×D4—31no—
104C23×D4C2×D311yescharacteristic
105C24×C4—32no—
106C2×#121—68no—
107C2×#121—68no—
108C2×#121—68no—
109C2×C4×D4—68no—
110C2×C4×D4—64no—
111C22×C4⋊C4—32no—
112C22×C4⋊C4—31no—
113C22×C4⋊C4C2×D311yescharacteristic
114C22×C22⋊C4—34no—
115C22×C22⋊C4—32no—
116C22×C22⋊C4—34no—
117C22×C22⋊C4C2×D312yes—
118C22×C22⋊C4C2×D312yes—
119C22×C42C2×D311yescharacteristic
120#392—68no—
121#390—24no—
122#380—68no—
123C2×#95—32no—
124C2×#95—34no—
125C2×#95—32no—
126C24×C3—24no—
127C24×C3—24no—
128C24×C3—22no—
129C24×C3—22no—
130C24×C3—22no—
131C24×C3—24no—
132C24×C3—22no—
133C24×C3—24no—
134C24×C3—22no—
135C24×C3—22no—
136C24×C3C2×D414yes—
137C24×C3C2411yescharacteristic
138C23×D3—22no—
139C23×D3—22no—
140C23×D3—22no—
141C23×D3—22no—
142C23×D3—22no—
143C23×D3—22no—
144C23×D3—22no—
145C23×D3—22no—
146C23×D3—22no—
147C23×D3—22no—
148C23×D3—22no—
149C23×D3—22no—
150C23×D3C2×D414yes—
151C23×D3C2×D414yes—
152C23×D3#4114yes—
153C2×C3×D4—216no—
154C2×C3×D4—28no—
155C2×C3×D4—44no—
156C2×C3×D4—48no—
157C2×C3×D4—44no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—21no—
162C22×C3×C4—22no—
163C22×C3×C4—21no—
164C22×C3×C4—24no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—24no—
169C22×C3×C4C2×D412yes—
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2×D414yes—
172C22×C3×C4C2×D414yes—
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2×D412yes—
176C22×C3×C4#4112yes—
177C22×C3×C4#4112yes—
178C2×#67—232no—
179C2×#67—416no—
180C2×#67—416no—
181C2×#67—416no—
182C2×#67—416no—
183C2×#67—416no—
184C2×#67—416no—
185C22×Dic3—22no—
186C22×Dic3—24no—
187C22×Dic3—22no—
188C22×Dic3—22no—
189C22×Dic3—24no—
190C22×Dic3—24no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—24no—
195C22×Dic3—24no—
196C22×Dic3—24no—
197C22×Dic3C2×D418yes—
198C22×Dic3C2×D414yes—
199C2×D12—28no—
200C2×D12—28no—
201C2×D12—216no—
202C2×D12—216no—
203C2×D12—28no—
204C2×C4×D3—216no—
205C2×C4×D3—216no—
206C2×C4×D3—216no—
207C2×C4×D3—48no—
208C2×C4×D3—48no—
209C2×C4×D3—48no—
210C2×C4×D3—48no—
211C3×C4⋊C4—44no—
212C3×C22⋊C4—48no—
213C3×C22⋊C4—48no—
214C3×C42—28no—
215#217—232no—
216(C2×D3)⋊C4—216no—
217(C2×D3)⋊C4—416no—
218C4⋊Dic3—28no—
219C4⋊Dic3—216no—
220C25—32no—
221C25C22×D312yes—
222C22×D4—22no—
223C22×D4—24no—
224C22×D4—24no—
225C22×D4—22no—
226C22×D4—34no—
227C22×D4—64no—
228C22×D4—64no—
229C22×D4—68no—
230C22×D4—68no—
231C22×D4—68no—
232C22×D4—68no—
233C22×D4—68no—
234C22×D4—62no—
235C22×D4—62no—
236C22×D4—124no—
237C22×D4—124no—
238C22×D4D1214yescomplemented
239C23×C4—32no—
240C23×C4—34no—
241C23×C4—38no—
242C23×C4—64no—
243C23×C4—64no—
244C23×C4—64no—
245C23×C4—64no—
246C23×C4—64no—
247C23×C4—64no—
248C23×C4C22×D312yes—
249C23×C4C22×D311yescharacteristic
250C23×C4C22×D311yescharacteristic
251C23×C4C22×D311yescharacteristic
252#121—664no—
253C4×D4—632no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—38no—
257C2×C4⋊C4—64no—
258C2×C4⋊C4—64no—
259C2×C4⋊C4—64no—
260C2×C4⋊C4—62no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—64no—
267C2×C22⋊C4—64no—
268C2×C22⋊C4—68no—
269C2×C22⋊C4—68no—
270C2×C22⋊C4—68no—
271C2×C22⋊C4—64no—
272C2×C22⋊C4—68no—
273C2×C22⋊C4—68no—
274C2×C22⋊C4—124no—
275C2×C42—24no—
276C2×C42#6714yescomplemented
277#95—68no—
278#95—64no—
279#95—122no—
280C23×C3—28no—
281C23×C3—24no—
282C23×C3—24no—
283C23×C3—28no—
284C23×C3—24no—
285C23×C3—42no—
286C23×C3—42no—
287C23×C3—44no—
288C23×C3—44no—
289C23×C3—42no—
290C23×C3—44no—
291C23×C3—42no—
292C23×C3—42no—
293C23×C3—44no—
294C23×C3—42no—
295C23×C3—44no—
296C23×C3—44no—
297C23×C3—44no—
298C23×C3—42no—
299C23×C3—44no—
300C23×C3—44no—
301C23×C3—44no—
302C23×C3—42no—
303C23×C3C22×D411yescharacteristic
304C23×C3C22×D412yes—
305C23×C3C22×D412yes—
306C23×C3C22×D411yescharacteristic
307C23×C3C22×D411yescharacteristic
308C23×C3C2×#4111yescharacteristic
309C23×C3C2×#4111yescharacteristic
310C23×C3C2×#4112yes—
311C23×C3H(5,𝔽2)12yes—
312C23×C3H(5,𝔽2)11yescharacteristic
313C23×C32−1+411yescharacteristic
314C22×D3—24no—
315C22×D3—24no—
316C22×D3—24no—
317C22×D3—24no—
318C22×D3—24no—
319C22×D3—24no—
320C22×D3—24no—
321C22×D3—24no—
322C22×D3—24no—
323C22×D3—24no—
324C22×D3—24no—
325C22×D3—24no—
326C22×D3—24no—
327C22×D3—24no—
328C22×D3—24no—
329C22×D3—24no—
330C22×D3—24no—
331C22×D3—24no—
332C22×D3—42no—
333C22×D3—42no—
334C22×D3—42no—
335C22×D3—42no—
336C22×D3—42no—
337C22×D3—42no—
338C22×D3—42no—
339C22×D3—42no—
340C22×D3—42no—
341C22×D3—42no—
342C22×D3—42no—
343C22×D3—42no—
344C22×D3—42no—
345C22×D3—42no—
346C22×D3—42no—
347C22×D3—42no—
348C22×D3#12118yescomplemented
349C3×D4—416no—
350C2×C3×C4—22no—
351C2×C3×C4—24no—
352C2×C3×C4—22no—
353C2×C3×C4—24no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—42no—
358C2×C3×C4—42no—
359C2×C3×C4—42no—
360C2×C3×C4—42no—
361C2×C3×C4—42no—
362C2×C3×C4—44no—
363C2×C3×C4—44no—
364C2×C3×C4#12018yes—
365C2×C3×C4#12014yes—
366C2×C3×C4#12118yes—
367C2×C3×C4#12114yes—
368C2×C3×C4#12114yescomplemented
369C2×C3×C4#12714yes—
370#67—464no—
371C2×Dic3—24no—
372C2×Dic3—28no—
373C2×Dic3—24no—
374C2×Dic3—24no—
375C2×Dic3—28no—
376C2×Dic3—28no—
377C2×Dic3—42no—
378C2×Dic3—42no—
379C2×Dic3—42no—
380C2×Dic3—44no—
381C2×Dic3—42no—
382C2×Dic3—44no—
383C2×Dic3—44no—
384C2×Dic3—44no—
385D12—232no—
386C4×D3—432no—
387C24—22no—
388C24—22no—
389C24—24no—
390C24—24no—
391C24—22no—
392C24—24no—
393C24—24no—
394C24—22no—
395C24—22no—
396C24—22no—
397C24—34no—
398C24—34no—
399C24—34no—
400C24—62no—
401C24—62no—
402C24—62no—
403C24—62no—
404C24—62no—
405C24—62no—
406C24—62no—
407C24—62no—
408C24—62no—
409C24—62no—
410C24—62no—
411C24—62no—
412C24C2×D1214yes—
413C24C23×D311yescharacteristic
414C2×D4—216no—
415C2×D4—28no—
416C2×D4—48no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4—616no—
420C2×D4—68no—
421C2×D4—68no—
422C2×D4—616no—
423C2×D4—632no—
424C2×D4—68no—
425C2×D4—1216no—
426C2×D4—1216no—
427C2×D4—1216no—
428C2×D4—1216no—
429C2×D4—1216no—
430C2×D4—1216no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—38no—
443C22×C4—34no—
444C22×C4—62no—
445C22×C4—64no—
446C22×C4—62no—
447C22×C4—62no—
448C22×C4—64no—
449C22×C4—62no—
450C22×C4—62no—
451C22×C4—64no—
452C22×C4—64no—
453C22×C4—64no—
454C22×C4—64no—
455C22×C4—62no—
456C22×C4—616no—
457C22×C4—616no—
458C22×C4—616no—
459C22×C4—128no—
460C22×C4—128no—
461C22×C4—128no—
462C22×C4—128no—
463C22×C4C2×D1212yes—
464C22×C4#23512yes—
465C22×C4D3×D412yes—
466C22×C4D3×D414yes—
467C22×C4D3×D414yes—
468C22×C4#23712yes—
469C22×C4C2×#6712yes—
470C22×C4C2×#6712yes—
471C22×C4C2×#6712yes—
472C4⋊C4—44no—
473C4⋊C4—68no—
474C4⋊C4—616no—
475C22⋊C4—48no—
476C22⋊C4—48no—
477C22⋊C4—616no—
478C22⋊C4—632no—
479C22⋊C4—1216no—
480C42—28no—
481C22×C3—44no—
482C22×C3—48no—
483C22×C3—48no—
484C22×C3—44no—
485C22×C3—44no—
486C22×C3—82no—
487C22×C3—82no—
488C22×C3—84no—
489C22×C3—84no—
490C22×C3—82no—
491C22×C3—82no—
492C22×C3C2×#12012yes—
493C22×C3C2×#12011yescharacteristic
494C22×C3C2×#12112yes—
495C22×C3C2×#12111yescharacteristic
496C22×C3C2×#12111yescharacteristic
497C22×C3C2×#12112yes—
498C22×C3#52511yescharacteristic
499C22×C3#52512yes—
500C22×C3C2×#12711yescharacteristic
501C22×C3#53511yescharacteristic
502C22×C3#53512yes—
503C22×C3#53711yescharacteristic
504C22×C3#53712yes—
505C22×C3D4212yes—
506C22×C3D4211yescharacteristic
507C22×C3#54612yes—
508C22×C3#54612yes—
509C22×C3#54711yescharacteristic
510C22×C3#54711yescharacteristic
511C22×C3#54812yes—
512C22×C3#55012yes—
513C22×C3#55312yes—
514C22×C3#55511yescharacteristic
515C2×D3—28no—
516C2×D3—28no—
517C2×D3—28no—
518C2×D3—44no—
519C2×D3—44no—
520C2×D3—44no—
521C2×D3—44no—
522C2×D3—44no—
523C2×D3—44no—
524C2×D3—44no—
525C2×D3—44no—
526C2×D3—44no—
527C2×D3—44no—
528C2×D3—44no—
529C2×D3—44no—
530C3×C4—24no—
531C3×C4—42no—
532C3×C4—82no—
533C3×C4#39218yescomplemented
534Dic3—44no—
535Dic3—48no—
536C23—28no—
537C23—24no—
538C23—24no—
539C23—28no—
540C23—24no—
541C23—38no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—44no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—64no—
561C23—64no—
562C23—64no—
563C23—64no—
564C23—64no—
565C23—64no—
566C23—64no—
567C23—64no—
568C23—64no—
569C23—64no—
570C23—64no—
571C23—64no—
572C23—64no—
573C23—64no—
574C23—64no—
575C23—64no—
576C23—64no—
577C23—64no—
578C23—122no—
579C23—122no—
580C23—122no—
581C23—122no—
582C23—122no—
583C23—122no—
584C23—122no—
585C23—122no—
586C23—122no—
587C23—122no—
588C23—122no—
589C23—122no—
590C23—122no—
591C23—122no—
592C23—122no—
593C23—122no—
594C23C22×D1211yescharacteristic
595C23C2×#23511yescharacteristic
596C23C2×D3×D412yes—
597C23C2×D3×D411yescharacteristic
598C23C2×D3×D412yes—
599C23C2×#23711yescharacteristic
600C23#100412yes—
601C23D3×#4112yes—
602C23#100911yescharacteristic
603C23#101011yescharacteristic
604C23C22×#6711yescharacteristic
605D4—416no—
606D4—632no—
607D4—1264no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—68no—
623C2×C4—68no—
624C2×C4—64no—
625C2×C4—68no—
626C2×C4—64no—
627C2×C4—64no—
628C2×C4—122no—
629C2×C4—122no—
630C2×C4—122no—
631C2×C4—122no—
632C2×C4—124no—
633C2×C4—124no—
634C2×C4—124no—
635C2×C4—124no—
636C2×C4—1232no—
637C2×C4#88218yes—
638C2×C4#88418yes—
639C2×C4#93014yes—
640C2×C4#93714yes—
641C2×C4#93814yescomplemented
642C2×C4#94014yes—
643C2×C3—84no—
644C2×C3C2×#39211yescharacteristic
645C2×C3#240712yes—
646C2×C3#240812yes—
647C2×C3#241412yes—
648C2×C3#241612yes—
649C2×C3#253511yescharacteristic
650C2×C3#253711yescharacteristic
651C2×C3#254011yescharacteristic
652C2×C3#254111yescharacteristic
653C2×C3#261511yescharacteristic
654C2×C3#284711yescharacteristic
655D3—48no—
656C22—48no—
657C22—48no—
658C22—44no—
659C22—44no—
660C22—44no—
661C22—68no—
662C22—68no—
663C22—68no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22—84no—
668C22—84no—
669C22—82no—
670C22—124no—
671C22—124no—
672C22—124no—
673C22—124no—
674C22—124no—
675C22—124no—
676C22—124no—
677C22—124no—
678C22—124no—
679C22—124no—
680C22—124no—
681C22—124no—
682C22C2×#88212yes—
683C22C2×#88412yes—
684C22D4×D1211yescharacteristic
685C22#555111yescharacteristic
686C22#555412yes—
687C22#555511yescharacteristic
688C22#555812yes—
689C22#556311yescharacteristic
690C22D3×#12112yes—
691C22#560612yes—
692C22#560712yes—
693C22#560812yes—
694C22#567612yes—
695C22#567712yes—
696C22#571712yes—
697C22#571812yes—
698C22C2×#93011yescharacteristic
699C22C2×#93711yescharacteristic
700C22C2×#93811yescharacteristic
701C22C2×#94011yescharacteristic
702C22#580511yescharacteristic
703C22#582811yescharacteristic
704C22#583311yescharacteristic
705C4—24no—
706C4—42no—
707C4—82no—
708C4—128no—
709C4—124no—
710C4#496018yescomplemented
711C3#1512311yescharacteristic, complemented, Sylow
712C2—84no—
713C2—128no—
714C2#7643512yes—
715C2#7643612yes—
716C2#7643812yes—
717C2#7643912yes—
718C2C2×#496011yescharacteristic
719C2#8241611yescharacteristic
720C2#8241711yescharacteristic
721C2#8270911yescharacteristic
722C2#8272311yescharacteristic
723C2#8299811yescharacteristic
724C2#8318811yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC2×C3×#390C2
hypercenterC2×#390D3
hypocenterC3#15123
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
1. C22×#938C2semidirect
2. C22×C4×D12C2semidirect
3. C2×C3×#390C2semidirect
4. C2×#4987C2semidirect
5. C2×#4960C2semidirect
6. #71550C2semidirect
7. #71546C2semidirect
8. #71535C2semidirect
37. C2×#390D3semidirect
48. C22×C3×D4D4semidirect
66. C22×D12D4semidirect
73. C22×C4×D3D4semidirect
82. C2×C3×C42D4semidirect
97. C2×C4⋊Dic3D4semidirect
238. C22×D4D12semidirect
276. C2×C42#67semidirect
348. C22×D3#121semidirect
368. C2×C3×C4#121semidirect
533. C3×C4#392semidirect
641. C2×C4#938semidirect
710. C4#4960semidirect
711. C3#15123semidirect

Maximal subgroups

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

subgroupindex
C22×#9382
C22×C4×D122
C2×C3×#3902
C2×#49872
C2×#49602
#715502
#715462
#715352
#151233