Finite Group Databasebeta

← #10718675

Subgroups — #10718675

Order 768.

2574 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×D4C211yesmaximal, characteristic
2C22×#936C211yesmaximal, characteristic
3C2×#5221C211yesmaximal, characteristic
4C2×#4973C211yesmaximal, characteristic, complemented
5C2×#4929C211yesmaximal, characteristic, complemented
6#72127C214yesmaximal, complemented
7#71563C211yesmaximal, characteristic, complemented
8#71549C214yesmaximal, complemented
9C4×C3⋊#95C211yesmaximal, characteristic, complemented
10#14027—31nomaximal, Sylow
11C23×C3×D4C2211yescharacteristic
12C24×C3×C4C2212yes—
13C23×Dic6C2211yescharacteristic
14C2×C3×C4×D4—24no—
15C2×C3×C4×D4—24no—
16C2×C3×C4×D4—24no—
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C22⋊C4C2212yes—
19C22×C3×C42C2211yescharacteristic
20C22×#217C2214yes—
21C22×#217C2214yes—
22C2×#936—24no—
23C2×#936—24no—
24C2×#936C418yescomplemented
25C22×C4⋊Dic3C2211yescharacteristic
26C22×C4×Dic3C2211yescharacteristic
27C22×C4×Dic3C2211yescharacteristic
28#5221—24no—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2214yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2214yes—
35#4973—24no—
36#4929—24no—
37C22×#124—31no—
38C22×C4×D4D311yescharacteristic
39C2×#391—31no—
40C2×#391—31no—
41C2×#390—31no—
42#1440—34no—
43C4⋊#95—31no—
44C22⋊#95—34no—
45C4×#95—31no—
46C25×C3C2312yes—
47C22×C3×D4—22no—
48C22×C3×D4—22no—
49C22×C3×D4—22no—
50C22×C3×D4—22no—
51C22×C3×D4C2×C414yes—
52C22×C3×D4D414yes—
53C22×C3×D4D414yes—
54C23×C3×C4—24no—
55C23×C3×C4—24no—
56C23×C3×C4—24no—
57C23×C3×C4—28no—
58C23×C3×C4—44no—
59C23×C3×C4C2312yes—
60C23×C3×C4C2311yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C23×Dic3C2312yes—
64C23×Dic3C2312yes—
65C23×Dic3C2312yes—
66C23×Dic3C2312yes—
67C22×Dic6—22no—
68C22×Dic6—22no—
69C22×Dic6—22no—
70C22×Dic6—22no—
71C22×Dic6—22no—
72C22×Dic6—22no—
73C22×Dic6C2×C414yescomplemented
74C3×C4×D4—232no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4D414yes—
78C2×C3×C22⋊C4—24no—
79C2×C3×C22⋊C4—24no—
80C2×C3×C22⋊C4—24no—
81C2×C3×C42—22no—
82C2×C3×C42—22no—
83C2×C3×C42D414yes—
84C2×#217—28no—
85C2×#217—28no—
86C2×#217—28no—
87C2×#217—28no—
88C2×#217—44no—
89C2×#217C2×C4116yescomplemented
90#936—232no—
91C2×C4⋊Dic3—22no—
92C2×C4⋊Dic3—22no—
93C2×C4⋊Dic3—22no—
94C2×C4×Dic3—22no—
95C2×C4×Dic3—22no—
96C2×C4×Dic3—22no—
97C2×C4×Dic3—22no—
98C2×C4×Dic3—22no—
99C2×C4×Dic3C2×C414yescomplemented
100C3⋊#95—24no—
101C3⋊#95—24no—
102C3⋊#95—28no—
103C3⋊#95—24no—
104C3⋊#95—24no—
105C3⋊#95—44no—
106C23×Q8—31no—
107C23×D4C2×D311yescharacteristic
108C24×C4C2×D312yes—
109C2×#124—38no—
110C2×#124—64no—
111C2×#124—64no—
112C2×C4×D4—24no—
113C2×C4×D4—24no—
114C2×C4×D4—24no—
115C22×C4⋊C4—31no—
116C22×C4⋊C4C2×D311yescharacteristic
117C22×C22⋊C4—34no—
118C22×C22⋊C4—34no—
119C22×C22⋊C4C2×D312yes—
120C22×C42—31no—
121C22×C42—31no—
122C22×C42C2×D311yescharacteristic
123#391—64no—
124#391—64no—
125#390—64no—
126C2×#95—32no—
127C2×#95—32no—
128C2×#95—32no—
129C2×#95—34no—
130C2×#95—32no—
131C2×#95—34no—
132C24×C3—22no—
133C24×C3—22no—
134C24×C3—24no—
135C24×C3—24no—
136C24×C3—22no—
137C24×C3—22no—
138C24×C3—24no—
139C24×C3—24no—
140C24×C3C22×C414yes—
141C24×C3C2×D414yes—
142C24×C3C2×D414yes—
143C24×C3C2411yescharacteristic
144C2×C3×D4—28no—
145C2×C3×D4—28no—
146C2×C3×D4—28no—
147C2×C3×D4—28no—
148C2×C3×D4—28no—
149C2×C3×D4—28no—
150C2×C3×D4C22⋊C4116yes—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4—22no—
154C22×C3×C4—216no—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—24no—
161C22×C3×C4—24no—
162C22×C3×C4—22no—
163C22×C3×C4—416no—
164C22×C3×C4—416no—
165C22×C3×C4—48no—
166C22×C3×C4—48no—
167C22×C3×C4C22×C412yes—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D412yes—
170C22×C3×C4C2×D414yes—
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2×D412yes—
175C22×C3×C4#4112yes—
176C22×C3×C4#4112yes—
177C22×C3×C4#4114yes—
178C22×Dic3—22no—
179C22×Dic3—22no—
180C22×Dic3—22no—
181C22×Dic3—22no—
182C22×Dic3—22no—
183C22×Dic3—22no—
184C22×Dic3—22no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—22no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3—22no—
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3—24no—
200C22×Dic3—24no—
201C22×Dic3C22×C414yes—
202C22×Dic3C22×C414yes—
203C22×Dic3#4114yes—
204C2×Dic6—28no—
205C2×Dic6—28no—
206C2×Dic6—28no—
207C2×Dic6—44no—
208C2×Dic6—44no—
209C2×Dic6—44no—
210C2×Dic6—44no—
211C3×C4⋊C4—28no—
212C3×C22⋊C4—216no—
213C3×C42—28no—
214#217—232no—
215#217—416no—
216C4⋊Dic3—44no—
217C4×Dic3—28no—
218C4×Dic3—44no—
219C25C22×D312yes—
220C22×Q8—34no—
221C22×Q8—62no—
222C22×Q8—62no—
223C22×Q8—62no—
224C22×Q8—62no—
225C22×Q8—62no—
226C22×Q8—62no—
227C22×D4—22no—
228C22×D4—22no—
229C22×D4—22no—
230C22×D4—22no—
231C22×D4C4×D314yes—
232C22×D4D1214yes—
233C22×D4#6714yes—
234C23×C4—24no—
235C23×C4—24no—
236C23×C4—24no—
237C23×C4—28no—
238C23×C4—32no—
239C23×C4—32no—
240C23×C4—32no—
241C23×C4—32no—
242C23×C4—44no—
243C23×C4C22×D311yescharacteristic
244C23×C4C22×D312yes—
245C23×C4C22×D311yescharacteristic
246C23×C4C22×D311yescharacteristic
247#124—632no—
248C4×D4—232no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—62no—
252C2×C4⋊C4—62no—
253C2×C4⋊C4—62no—
254C2×C4⋊C4#6714yes—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—316no—
259C2×C22⋊C4—68no—
260C2×C22⋊C4—68no—
261C2×C22⋊C4—68no—
262C2×C22⋊C4—68no—
263C2×C22⋊C4—124no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—34no—
267C2×C42—62no—
268C2×C42—62no—
269C2×C42—62no—
270C2×C42—62no—
271C2×C42—62no—
272C2×C42#6714yes—
273#95—64no—
274#95—64no—
275#95—68no—
276#95—64no—
277#95—64no—
278#95—124no—
279C23×C3—24no—
280C23×C3—24no—
281C23×C3—28no—
282C23×C3—28no—
283C23×C3—24no—
284C23×C3—24no—
285C23×C3—24no—
286C23×C3—24no—
287C23×C3—28no—
288C23×C3—28no—
289C23×C3—28no—
290C23×C3—28no—
291C23×C3—42no—
292C23×C3—44no—
293C23×C3—42no—
294C23×C3—42no—
295C23×C3—42no—
296C23×C3—44no—
297C23×C3—44no—
298C23×C3—44no—
299C23×C3—44no—
300C23×C3—44no—
301C23×C3C2×C22⋊C418yes—
302C23×C3C23×C411yescharacteristic
303C23×C3C22×D411yescharacteristic
304C23×C3C22×D411yescharacteristic
305C23×C3C22×D411yescharacteristic
306C23×C3C22×D411yescharacteristic
307C23×C3C2×#4111yescharacteristic
308C23×C3C2×#4111yescharacteristic
309C23×C3C2×#4112yes—
310C23×C3C2×#4112yes—
311C23×C3H(5,𝔽2)12yes—
312C23×C32−1+412yes—
313C3×D4—232no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—24no—
317C2×C3×C4—24no—
318C2×C3×C4—22no—
319C2×C3×C4—24no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—28no—
323C2×C3×C4—432no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4—44no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4C2×C22⋊C414yes—
330C2×C3×C4C4×D414yes—
331C2×C3×C4C4×D414yes—
332C2×C3×C4#12014yes—
333C2×C3×C4#12114yes—
334C2×C3×C4#12114yes—
335C2×C3×C4#12314yes—
336C2×C3×C4#12414yescomplemented
337C2×Dic3—24no—
338C2×Dic3—24no—
339C2×Dic3—24no—
340C2×Dic3—24no—
341C2×Dic3—24no—
342C2×Dic3—24no—
343C2×Dic3—24no—
344C2×Dic3—24no—
345C2×Dic3—24no—
346C2×Dic3—42no—
347C2×Dic3—42no—
348C2×Dic3—42no—
349C2×Dic3—42no—
350C2×Dic3—42no—
351C2×Dic3—42no—
352C2×Dic3—42no—
353C2×Dic3—42no—
354C2×Dic3—42no—
355C2×Dic3—42no—
356C2×Dic3—42no—
357C2×Dic3—42no—
358C2×Dic3—42no—
359C2×Dic3—42no—
360C2×Dic3—42no—
361C2×Dic3—44no—
362C2×Dic3—44no—
363Dic6—416no—
364C24—22no—
365C24—22no—
366C24—24no—
367C24—24no—
368C24—24no—
369C24—24no—
370C24—22no—
371C24—22no—
372C24C2×C4×D314yes—
373C24C2×D1214yes—
374C24C2×#6714yes—
375C24C23×D311yescharacteristic
376C2×Q8—68no—
377C2×Q8—68no—
378C2×Q8—68no—
379C2×Q8—124no—
380C2×Q8—124no—
381C2×Q8—124no—
382C2×Q8—124no—
383C2×D4—28no—
384C2×D4—28no—
385C2×D4—28no—
386C2×D4—28no—
387C2×D4—28no—
388C2×D4—28no—
389C2×D4(C2×D3)⋊C4116yes—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—216no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—34no—
403C22×C4—34no—
404C22×C4—34no—
405C22×C4—416no—
406C22×C4—416no—
407C22×C4—48no—
408C22×C4—48no—
409C22×C4—62no—
410C22×C4—62no—
411C22×C4—62no—
412C22×C4—62no—
413C22×C4—62no—
414C22×C4—62no—
415C22×C4—62no—
416C22×C4—62no—
417C22×C4—62no—
418C22×C4—62no—
419C22×C4—62no—
420C22×C4—62no—
421C22×C4—62no—
422C22×C4—62no—
423C22×C4—62no—
424C22×C4—62no—
425C22×C4—62no—
426C22×C4—62no—
427C22×C4—62no—
428C22×C4—62no—
429C22×C4—64no—
430C22×C4—64no—
431C22×C4—62no—
432C22×C4C2×C4×D312yes—
433C22×C4C2×D1212yes—
434C22×C4#23512yes—
435C22×C4#23512yes—
436C22×C4#23714yes—
437C22×C4C2×#6712yes—
438C22×C4C2×#6714yes—
439C22×C4C2×#6712yes—
440C22×C4C2×#6712yes—
441C22×C4C2×#6712yes—
442C22×C4C2×#6712yes—
443C4⋊C4—28no—
444C4⋊C4—124no—
445C22⋊C4—216no—
446C22⋊C4—632no—
447C22⋊C4—1216no—
448C42—28no—
449C42—68no—
450C42—124no—
451C22×C3—216no—
452C22×C3—28no—
453C22×C3—44no—
454C22×C3—48no—
455C22×C3—48no—
456C22×C3—44no—
457C22×C3—44no—
458C22×C3—48no—
459C22×C3—44no—
460C22×C3—44no—
461C22×C3—44no—
462C22×C3C22×C22⋊C411yescharacteristic
463C22×C3C2×C4×D411yescharacteristic
464C22×C3C2×C4×D411yescharacteristic
465C22×C3C4×#4112yes—
466C22×C3#52012yes—
467C22×C3C2×#12011yescharacteristic
468C22×C3C2×#12111yescharacteristic
469C22×C3C2×#12111yescharacteristic
470C22×C3C2×#12311yescharacteristic
471C22×C3C2×#12411yescharacteristic
472C22×C3#52912yes—
473C22×C3#53211yescharacteristic
474C22×C3#53511yescharacteristic
475C22×C3#53611yescharacteristic
476C22×C3#54212yes—
477C22×C3#54612yes—
478C22×C3#54612yes—
479C22×C3#54612yes—
480C22×C3#54712yes—
481C22×C3#54712yes—
482C22×C3#54812yes—
483C22×C3#55612yes—
484C22×C3#55912yes—
485C3×C4—24no—
486C3×C4—42no—
487C3×C4—44no—
488C3×C4#38618yes—
489Dic3—44no—
490Dic3—44no—
491Dic3—44no—
492Dic3—82no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—28no—
499C23—28no—
500C23—28no—
501C23—28no—
502C23—28no—
503C23—28no—
504C23—24no—
505C23—42no—
506C23—44no—
507C23—42no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—42no—
513C23—44no—
514C23—42no—
515C23C2×(C2×D3)⋊C418yes—
516C23C22×C4×D311yescharacteristic
517C23C22×D1211yescharacteristic
518C23C2×#23511yescharacteristic
519C23C2×#23511yescharacteristic
520C23C2×#23712yes—
521C23#100412yes—
522C23D3×#4112yes—
523C23#101012yes—
524C23C22×#6711yescharacteristic
525C23C22×#6711yescharacteristic
526C23C22×#6711yescharacteristic
527Q8—1216no—
528D4—232no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—28no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—432no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—64no—
545C2×C4—64no—
546C2×C4—64no—
547C2×C4—64no—
548C2×C4—64no—
549C2×C4—64no—
550C2×C4—64no—
551C2×C4—64no—
552C2×C4—64no—
553C2×C4—122no—
554C2×C4—122no—
555C2×C4—122no—
556C2×C4—122no—
557C2×C4—122no—
558C2×C4—122no—
559C2×C4—122no—
560C2×C4—122no—
561C2×C4—122no—
562C2×C4—122no—
563C2×C4—122no—
564C2×C4—122no—
565C2×C4—122no—
566C2×C4—122no—
567C2×C4—124no—
568C2×C4—124no—
569C2×C4—122no—
570C2×C4C2×(C2×D3)⋊C414yes—
571C2×C4C4×#6714yes—
572C2×C4C4×#6714yes—
573C2×C4#92914yes—
574C2×C4#93014yes—
575C2×C4#93014yes—
576C2×C4#93614yescomplemented
577C2×C4#95314yes—
578C2×C3—48no—
579C2×C3C2×#38611yescharacteristic
580C2×C3C4×#12411yescharacteristic
581C2×C3#232611yescharacteristic
582C2×C3#233912yes—
583C2×C3#236712yes—
584C2×C3#242712yes—
585C2×C3#243211yescharacteristic
586C2×C3#243511yescharacteristic
587C2×C3#244712yes—
588C2×C3#254411yescharacteristic
589C2×C3#254711yescharacteristic
590C22—28no—
591C22—216no—
592C22—44no—
593C22—44no—
594C22—44no—
595C22—44no—
596C22—48no—
597C22—48no—
598C22—44no—
599C22—44no—
600C22—48no—
601C22C4×#23712yes—
602C22#554212yes—
603C22#554612yes—
604C22#555412yes—
605C22#555512yes—
606C22#555712yes—
607C22#555912yes—
608C22#556112yes—
609C22C22×(C2×D3)⋊C411yescharacteristic
610C22C2×C4×#6711yescharacteristic
611C22C2×C4×#6711yescharacteristic
612C22C2×#92911yescharacteristic
613C22C2×#93011yescharacteristic
614C22C2×#93011yescharacteristic
615C22C2×#93611yescharacteristic
616C22#580511yescharacteristic
617C22#580612yes—
618C22#580612yes—
619C22#582412yes—
620C22#582511yescharacteristic
621C22#583112yes—
622C22#583211yescharacteristic
623C22C2×#95311yescharacteristic
624C4—24no—
625C4—44no—
626C4—42no—
627C4—124no—
628C4—124no—
629C4—124no—
630C4—242no—
631C4#521318yes—
632C3#1402711yescharacteristic, complemented, Sylow
633C2—48no—
634C2C4×#93611yescharacteristic
635C2#8232811yescharacteristic
636C2#8233012yes—
637C2#8234412yes—
638C2#8235912yes—
639C2#8238111yescharacteristic
640C2#8238211yescharacteristic
641C2#8240812yes—
642C2C2×#521311yescharacteristic
643C2#8405011yescharacteristic
644C2#8409311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC22×C3×C4×D4C2
socle, cosocleC24×C3C24
prime radicalC23×C3×D4C22
hypercenterC22×C4×D4D3
hypocenterC3#14027
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×C4×D4

Complements

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

NH ≅ G/Ntype
4. C2×#4973C2semidirect
5. C2×#4929C2semidirect
6. #72127C2semidirect
7. #71563C2semidirect
8. #71549C2semidirect
9. C4×C3⋊#95C2semidirect
24. C2×#936C4semidirect
73. C22×Dic6C2×C4semidirect
89. C2×#217C2×C4semidirect
99. C2×C4×Dic3C2×C4semidirect
336. C2×C3×C4#124semidirect
576. C2×C4#936semidirect
632. C3#14027semidirect

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×#9362
C2×#52212
C2×#49732
C2×#49292
#721272
#715632
#715492
C4×C3⋊#952
#140273