Finite Group Databasebeta

← #10731725

Subgroups — #10731725

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×C3×C4×D4C211yesmaximal, characteristic, complemented
2C22×#930C212yesmaximal, complemented
3C2×#5223C212yesmaximal, complemented
4C2×#5212C212yesmaximal, complemented
5C2×#4939C211yesmaximal, characteristic, complemented
6#71570C212yesmaximal, complemented
7#71550C214yesmaximal, complemented
8#71535C211yesmaximal, characteristic, complemented
9#15123—31nomaximal, Sylow
10C23×C3×D4C2211yescharacteristic
11C24×C3×C4C2212yes—
12C23×#67C2214yes—
13C23×D12C2211yescharacteristic
14C2×C3×C4×D4—28no—
15C2×C3×C4×D4—24no—
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×C3×C22⋊C4C2212yes—
18C22×C3×C42C2211yescharacteristic
19C22×#217C2214yes—
20C2×#930—28no—
21C2×#930—28no—
22C2×#930—28no—
23C22×(C2×D3)⋊C4C2212yes—
24C22×(C2×D3)⋊C4C2214yes—
25C22×(C2×D3)⋊C4C2212yes—
26C22×C4⋊Dic3C2211yescharacteristic
27C22×C4⋊Dic3C2212yes—
28#5223—28no—
29#5212—28no—
30C2×C3⋊#95C2214yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33#4939—24no—
34C22×#121—32no—
35C22×C4×D4D311yescharacteristic, complemented
36C2×#392—32no—
37C2×#390—31no—
38C2×#380—32no—
39#1439—32no—
40#1438—34no—
41C4⋊#95—31no—
42C25×C3C2312yes—
43C24×D3C2312yes—
44C22×C3×D4—24no—
45C22×C3×D4—24no—
46C22×C3×D4—22no—
47C22×C3×D4—22no—
48C22×C3×D4D414yescomplemented
49C23×C3×C4—24no—
50C23×C3×C4—24no—
51C23×C3×C4—24no—
52C23×C3×C4—24no—
53C23×C3×C4—24no—
54C23×C3×C4—24no—
55C23×C3×C4D418yescomplemented
56C23×C3×C4C2312yes—
57C23×C3×C4C2311yescharacteristic
58C23×C3×C4C2311yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C22×#67—28no—
61C22×#67—28no—
62C22×#67—28no—
63C22×#67—28no—
64C22×#67—28no—
65C22×#67—44no—
66C22×#67—44no—
67C23×Dic3C2312yes—
68C23×Dic3C2314yes—
69C22×D12—22no—
70C22×D12—24no—
71C22×D12—24no—
72C22×D12—22no—
73C22×D12D414yescomplemented
74C3×C4×D4—232no—
75C2×C3×C4⋊C4—24no—
76C2×C3×C4⋊C4—22no—
77C2×C3×C22⋊C4—24no—
78C2×C3×C22⋊C4—24no—
79C2×C3×C22⋊C4—24no—
80C2×C3×C42—24no—
81C2×C3×C42D414yescomplemented
82C2×#217—28no—
83C2×#217—28no—
84C2×#217—44no—
85#930—264no—
86C2×(C2×D3)⋊C4—28no—
87C2×(C2×D3)⋊C4—28no—
88C2×(C2×D3)⋊C4—28no—
89C2×(C2×D3)⋊C4—24no—
90C2×(C2×D3)⋊C4—24no—
91C2×(C2×D3)⋊C4—24no—
92C2×(C2×D3)⋊C4—24no—
93C2×(C2×D3)⋊C4—44no—
94C2×C4⋊Dic3—24no—
95C2×C4⋊Dic3—24no—
96C2×C4⋊Dic3—22no—
97C2×C4⋊Dic3—24no—
98C2×C4⋊Dic3D418yescomplemented
99C3⋊#95—24no—
100C3⋊#95—28no—
101C3⋊#95—42no—
102C23×D4—34no—
103C23×D4—31no—
104C23×D4C2×D311yescharacteristic
105C24×C4C2×D312yes—
106C2×#121—68no—
107C2×#121—68no—
108C2×#121—68no—
109C2×C4×D4—28no—
110C2×C4×D4—24no—
111C22×C4⋊C4—32no—
112C22×C4⋊C4—31no—
113C22×C4⋊C4C2×D311yescharacteristic
114C22×C22⋊C4—32no—
115C22×C22⋊C4—34no—
116C22×C22⋊C4—34no—
117C22×C22⋊C4—32no—
118C22×C22⋊C4C2×D312yes—
119C22×C42C2×D311yescharacteristic
120#392—68no—
121#390—64no—
122#380—68no—
123C2×#95—32no—
124C2×#95—34no—
125C2×#95—32no—
126C24×C3—22no—
127C24×C3—22no—
128C24×C3—22no—
129C24×C3—22no—
130C24×C3—22no—
131C24×C3—22no—
132C24×C3—22no—
133C24×C3—22no—
134C24×C3—22no—
135C24×C3—22no—
136C24×C3—22no—
137C24×C3—22no—
138C24×C3C2×D414yes—
139C24×C3C2×D414yes—
140C24×C3#4114yes—
141C24×C3C2411yescharacteristic
142C23×D3—22no—
143C23×D3—24no—
144C23×D3—24no—
145C23×D3—22no—
146C23×D3—22no—
147C23×D3—22no—
148C23×D3—24no—
149C23×D3—24no—
150C23×D3—22no—
151C23×D3—22no—
152C23×D3C2×D414yes—
153C2×C3×D4—28no—
154C2×C3×D4—28no—
155C2×C3×D4—216no—
156C2×C3×D4—216no—
157C2×C3×D4—28no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—21no—
168C22×C3×C4—22no—
169C22×C3×C4—21no—
170C22×C3×C4—216no—
171C22×C3×C4—216no—
172C22×C3×C4—216no—
173C22×C3×C4—48no—
174C22×C3×C4—48no—
175C22×C3×C4—48no—
176C22×C3×C4—48no—
177C22×C3×C4C2×D412yes—
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2×D414yes—
180C22×C3×C4C2×D412yes—
181C22×C3×C4C2×D414yes—
182C22×C3×C4C2×D412yes—
183C22×C3×C4C2×D414yes—
184C22×C3×C4C2×D412yes—
185C22×C3×C4#4112yes—
186C22×C3×C4#4112yes—
187C2×#67—232no—
188C2×#67—416no—
189C2×#67—416no—
190C2×#67—416no—
191C2×#67—416no—
192C2×#67—416no—
193C2×#67—416no—
194C22×Dic3—24no—
195C22×Dic3—24no—
196C22×Dic3—24no—
197C22×Dic3—24no—
198C22×Dic3—24no—
199C22×Dic3—24no—
200C22×Dic3—24no—
201C22×Dic3—22no—
202C22×Dic3—22no—
203C22×Dic3—22no—
204C22×Dic3—24no—
205C22×Dic3C2×D418yes—
206C2×D12—216no—
207C2×D12—28no—
208C2×D12—44no—
209C2×D12—48no—
210C2×D12—44no—
211C3×C4⋊C4—28no—
212C3×C22⋊C4—216no—
213C3×C42—28no—
214#217—416no—
215(C2×D3)⋊C4—232no—
216(C2×D3)⋊C4—48no—
217(C2×D3)⋊C4—48no—
218C4⋊Dic3—216no—
219C4⋊Dic3—44no—
220C25—32no—
221C25C22×D312yes—
222C22×D4—24no—
223C22×D4—24no—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—34no—
227C22×D4—68no—
228C22×D4—68no—
229C22×D4—68no—
230C22×D4—68no—
231C22×D4—68no—
232C22×D4—64no—
233C22×D4—62no—
234C22×D4—64no—
235C22×D4—62no—
236C22×D4—124no—
237C22×D4—124no—
238C22×D4D1214yescomplemented
239C23×C4—24no—
240C23×C4—24no—
241C23×C4—24no—
242C23×C4—24no—
243C23×C4—24no—
244C23×C4—24no—
245C23×C4—34no—
246C23×C4—32no—
247C23×C4#6718yescomplemented
248C23×C4C22×D311yescharacteristic
249C23×C4C22×D312yes—
250C23×C4C22×D311yescharacteristic
251C23×C4C22×D311yescharacteristic
252#121—664no—
253C4×D4—232no—
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—68no—
265C2×C22⋊C4—68no—
266C2×C22⋊C4—68no—
267C2×C22⋊C4—68no—
268C2×C22⋊C4—68no—
269C2×C22⋊C4—64no—
270C2×C22⋊C4—64no—
271C2×C22⋊C4—64no—
272C2×C22⋊C4—64no—
273C2×C22⋊C4—124no—
274C2×C22⋊C4—124no—
275C2×C42—24no—
276C2×C42#6714yescomplemented
277#95—68no—
278#95—64no—
279#95—122no—
280C23×C3—24no—
281C23×C3—24no—
282C23×C3—24no—
283C23×C3—24no—
284C23×C3—24no—
285C23×C3—24no—
286C23×C3—24no—
287C23×C3—24no—
288C23×C3—24no—
289C23×C3—24no—
290C23×C3—24no—
291C23×C3—24no—
292C23×C3—24no—
293C23×C3—24no—
294C23×C3—24no—
295C23×C3—24no—
296C23×C3—24no—
297C23×C3—24no—
298C23×C3—42no—
299C23×C3—42no—
300C23×C3—42no—
301C23×C3—42no—
302C23×C3—42no—
303C23×C3—42no—
304C23×C3—42no—
305C23×C3—42no—
306C23×C3—42no—
307C23×C3—42no—
308C23×C3—42no—
309C23×C3—42no—
310C23×C3—42no—
311C23×C3—42no—
312C23×C3—42no—
313C23×C3—42no—
314C23×C3#12118yescomplemented
315C23×C3C22×D411yescharacteristic
316C23×C3C22×D411yescharacteristic
317C23×C3C22×D412yes—
318C23×C3C22×D412yes—
319C23×C3C22×D411yescharacteristic
320C23×C3C2×#4112yes—
321C23×C3C2×#4111yescharacteristic
322C23×C3C2×#4111yescharacteristic
323C23×C3H(5,𝔽2)11yescharacteristic
324C23×C3H(5,𝔽2)12yes—
325C23×C32−1+411yescharacteristic
326C22×D3—28no—
327C22×D3—24no—
328C22×D3—24no—
329C22×D3—28no—
330C22×D3—24no—
331C22×D3—42no—
332C22×D3—42no—
333C22×D3—44no—
334C22×D3—44no—
335C22×D3—44no—
336C22×D3—42no—
337C22×D3—42no—
338C22×D3—42no—
339C22×D3—44no—
340C22×D3—44no—
341C22×D3—42no—
342C22×D3—44no—
343C22×D3—44no—
344C22×D3—42no—
345C22×D3—44no—
346C22×D3—44no—
347C22×D3—44no—
348C22×D3—42no—
349C3×D4—232no—
350C2×C3×C4—22no—
351C2×C3×C4—24no—
352C2×C3×C4—22no—
353C2×C3×C4—24no—
354C2×C3×C4—24no—
355C2×C3×C4—22no—
356C2×C3×C4—24no—
357C2×C3×C4—24no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—42no—
361C2×C3×C4—42no—
362C2×C3×C4—42no—
363C2×C3×C4—42no—
364C2×C3×C4—42no—
365C2×C3×C4—42no—
366C2×C3×C4—432no—
367C2×C3×C4#12018yes—
368C2×C3×C4#12014yes—
369C2×C3×C4#12114yes—
370C2×C3×C4#12118yes—
371C2×C3×C4#12114yescomplemented
372C2×C3×C4#12714yes—
373#67—464no—
374C2×Dic3—28no—
375C2×Dic3—28no—
376C2×Dic3—28no—
377C2×Dic3—44no—
378C2×Dic3—44no—
379C2×Dic3—44no—
380C2×Dic3—44no—
381C2×Dic3—42no—
382C2×Dic3—42no—
383C2×Dic3—42no—
384C2×Dic3—44no—
385C2×Dic3—44no—
386D12—416no—
387C24—22no—
388C24—22no—
389C24—22no—
390C24—22no—
391C24—22no—
392C24—22no—
393C24—22no—
394C24—22no—
395C24—22no—
396C24—22no—
397C24—22no—
398C24—22no—
399C24—34no—
400C24—62no—
401C24—62no—
402C24—64no—
403C24—64no—
404C24—62no—
405C24—64no—
406C24—62no—
407C24—62no—
408C24—62no—
409C24—64no—
410C24C2×D1214yes—
411C24#23514yes—
412C24C2×#6714yes—
413C24C23×D311yescharacteristic
414C2×D4—28no—
415C2×D4—216no—
416C2×D4—28no—
417C2×D4—216no—
418C2×D4—28no—
419C2×D4—632no—
420C2×D4—616no—
421C2×D4—68no—
422C2×D4—1216no—
423C2×D4—1216no—
424C2×D4—1216no—
425C2×D4—1216no—
426C2×D4—1216no—
427C2×D4—1216no—
428C2×D4—128no—
429C2×D4—124no—
430C2×D4—124no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—216no—
444C22×C4—216no—
445C22×C4—216no—
446C22×C4—38no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—64no—
452C22×C4—64no—
453C22×C4—64no—
454C22×C4—64no—
455C22×C4—64no—
456C22×C4—64no—
457C22×C4—62no—
458C22×C4—62no—
459C22×C4—64no—
460C22×C4—64no—
461C22×C4—62no—
462C22×C4C2×D1212yes—
463C22×C4#23512yes—
464C22×C4D3×D412yes—
465C22×C4D3×D414yes—
466C22×C4#23712yes—
467C22×C4C2×#6712yes—
468C22×C4C2×#6714yes—
469C22×C4C2×#6712yes—
470C22×C4C2×#6714yes—
471C22×C4C2×#6712yes—
472C4⋊C4—28no—
473C4⋊C4—616no—
474C4⋊C4—124no—
475C22⋊C4—216no—
476C22⋊C4—632no—
477C22⋊C4—128no—
478C22⋊C4—1216no—
479C22⋊C4—128no—
480C42—28no—
481C22×C3—28no—
482C22×C3—28no—
483C22×C3—28no—
484C22×C3—44no—
485C22×C3—44no—
486C22×C3—44no—
487C22×C3—44no—
488C22×C3—44no—
489C22×C3—44no—
490C22×C3—44no—
491C22×C3—44no—
492C22×C3—44no—
493C22×C3—44no—
494C22×C3—44no—
495C22×C3—44no—
496C22×C3C2×#12012yes—
497C22×C3C2×#12011yescharacteristic
498C22×C3C2×#12112yes—
499C22×C3C2×#12111yescharacteristic
500C22×C3C2×#12111yescharacteristic
501C22×C3C2×#12112yes—
502C22×C3#52512yes—
503C22×C3#52511yescharacteristic
504C22×C3C2×#12711yescharacteristic
505C22×C3#53512yes—
506C22×C3#53511yescharacteristic
507C22×C3#53712yes—
508C22×C3#53711yescharacteristic
509C22×C3D4211yescharacteristic
510C22×C3D4212yes—
511C22×C3#54612yes—
512C22×C3#54612yes—
513C22×C3#54711yescharacteristic
514C22×C3#54711yescharacteristic
515C22×C3#54812yes—
516C22×C3#55012yes—
517C22×C3#55312yes—
518C22×C3#55511yescharacteristic
519C2×D3—44no—
520C2×D3—48no—
521C2×D3—48no—
522C2×D3—44no—
523C2×D3—44no—
524C2×D3—82no—
525C2×D3—82no—
526C2×D3—84no—
527C2×D3—84no—
528C2×D3—82no—
529C2×D3—82no—
530C3×C4—24no—
531C3×C4—42no—
532C3×C4—44no—
533C3×C4#39218yescomplemented
534Dic3—48no—
535Dic3—82no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—24no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—68no—
571C23—64no—
572C23—64no—
573C23—68no—
574C23—64no—
575C23—122no—
576C23—124no—
577C23—124no—
578C23—122no—
579C23—124no—
580C23—122no—
581C23—124no—
582C23—124no—
583C23—124no—
584C23—124no—
585C23—122no—
586C23—122no—
587C23—124no—
588C23—122no—
589C23—124no—
590C23—124no—
591C23—122no—
592C23—122no—
593C23#93018yescomplemented
594C23C22×D1211yescharacteristic
595C23C2×#23512yes—
596C23C2×#23511yescharacteristic
597C23C2×D3×D411yescharacteristic
598C23C2×D3×D412yes—
599C23C2×#23711yescharacteristic
600C23#100911yescharacteristic
601C23#100912yes—
602C23#101011yescharacteristic
603C23C22×#6712yes—
604C23C22×#6711yescharacteristic
605D4—232no—
606D4—1264no—
607D4—1216no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—24no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—432no—
625C2×C4—68no—
626C2×C4—68no—
627C2×C4—68no—
628C2×C4—124no—
629C2×C4—124no—
630C2×C4—124no—
631C2×C4—124no—
632C2×C4—122no—
633C2×C4—122no—
634C2×C4—124no—
635C2×C4—124no—
636C2×C4—122no—
637C2×C4#93014yescomplemented
638C2×C4#93718yes—
639C2×C4#93814yes—
640C2×C4#93918yes—
641C2×C4#94014yes—
642C2×C4#95314yes—
643C2×C3—48no—
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—84no—
656C22—28no—
657C22—28no—
658C22—28no—
659C22—44no—
660C22—44no—
661C22—44no—
662C22—44no—
663C22—44no—
664C22—44no—
665C22—44no—
666C22—44no—
667C22—44no—
668C22—44no—
669C22—44no—
670C22—44no—
671C22—124no—
672C22—128no—
673C22—128no—
674C22—124no—
675C22—124no—
676C22—242no—
677C22—242no—
678C22—244no—
679C22—244no—
680C22—242no—
681C22—242no—
682C22D4×D1212yes—
683C22D4×D1211yescharacteristic
684C22#555012yes—
685C22#555012yes—
686C22#555111yescharacteristic
687C22#555212yes—
688C22#555312yes—
689C22#555511yescharacteristic
690C22#556311yescharacteristic
691C22#556512yes—
692C22C2×#93012yes—
693C22C2×#93011yescharacteristic
694C22#579112yes—
695C22#579111yescharacteristic
696C22C2×#93712yes—
697C22C2×#93811yescharacteristic
698C22C2×#93912yes—
699C22C2×#94011yescharacteristic
700C22#583012yes—
701C22#583011yescharacteristic
702C22#583312yes—
703C22#583311yescharacteristic
704C22C2×#95311yescharacteristic
705C4—24no—
706C4—44no—
707C4—42no—
708C4—128no—
709C4—242no—
710C4#522318yescomplemented
711C3#1512311yescharacteristic, complemented, Sylow
712C2—48no—
713C2—244no—
714C2#8235012yes—
715C2#8235212yes—
716C2#8237711yescharacteristic
717C2#8237811yescharacteristic
718C2#8240212yes—
719C2#8240312yes—
720C2#8241511yescharacteristic
721C2#8241711yescharacteristic
722C2#8403211yescharacteristic
723C2C2×#522311yescharacteristic
724C2#8409811yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC22×C3×C4×D4C2
hypercenterC22×C4×D4D3
hypocenterC3#15123
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×C4×D4

Complements

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

NH ≅ G/Ntype
1. C22×C3×C4×D4C2semidirect
2. C22×#930C2semidirect
3. C2×#5223C2semidirect
4. C2×#5212C2semidirect
5. C2×#4939C2semidirect
6. #71570C2semidirect
7. #71550C2semidirect
8. #71535C2semidirect
35. C22×C4×D4D3semidirect
48. C22×C3×D4D4semidirect
55. C23×C3×C4D4semidirect
73. C22×D12D4semidirect
81. C2×C3×C42D4semidirect
98. C2×C4⋊Dic3D4semidirect
238. C22×D4D12semidirect
247. C23×C4#67semidirect
276. C2×C42#67semidirect
314. C23×C3#121semidirect
371. C2×C3×C4#121semidirect
533. C3×C4#392semidirect
593. C23#930semidirect
637. C2×C4#930semidirect
710. C4#5223semidirect
711. C3#15123semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4×D42
C22×#9302
C2×#52232
C2×#52122
C2×#49392
#715702
#715502
#715352
#151233