Finite Group Databasebeta

← #10822725

Subgroups — #10822725

Order 768.

1482 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5833C211yesmaximal, characteristic
2C2×C3×#417C211yesmaximal, characteristic, complemented
3C2×#5233C211yesmaximal, characteristic, complemented
4C2×#5129C211yesmaximal, characteristic, complemented
5C2×#5121C211yesmaximal, characteristic, complemented
6#82186C211yesmaximal, characteristic, complemented
7C2×#4529C211yesmaximal, characteristic, complemented
8#70244C212yesmaximal, complemented
9#70243C212yesmaximal, complemented
10#70176C212yesmaximal, complemented
11#70175C212yesmaximal, complemented
12#23485—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C22×#235C2211yescharacteristic
15C22×C3×M16C2211yescharacteristic
16C22×C3×C4⋊C4C2211yescharacteristic
17#5833—24no—
18#5833—24no—
19C2×#946C2212yes—
20C2×#939—22no—
21C2×#938C2212yes—
22C2×#930—21no—
23C2×#926C2211yescharacteristic
24C22×C4⋊Dic3C2211yescharacteristic
25C2×#924—21no—
26C23×C3⋊C8C2211yescharacteristic
27C3×#417—24no—
28C2×C3×#103C2212yes—
29C2×C3×#102C2212yes—
30#5233—24no—
31C2×#835C2211yescharacteristic
32C2×#835C2211yescharacteristic
33C2×#832C2211yescharacteristic
34C2×#832C2211yescharacteristic
35#5213—22no—
36#5129—24no—
37#5121—24no—
38C2×(C2×D3)⋊C8C2212yes—
39C2×(C2×D3)⋊C8C2212yes—
40C2×#818C2212yes—
41C2×#817C2212yes—
42#4988—21no—
43#4987C2212yes—
44#4987C2212yes—
45#4973—21no—
46#4529C2212yes—
47#4529C2212yes—
48C2×#537—31no—
49C2×#428—31no—
50C2×#417D311yescharacteristic, complemented
51C2×#416—31no—
52C2×#408—31no—
53#2311—31no—
54#2036—32no—
55#2035—32no—
56#2034—32no—
57#2033—32no—
58C2×#340—31no—
59C2×C3×#41—22no—
60C2×C3×#41—22no—
61C2×C3×#41—22no—
62C2×C3×#41—22no—
63C2×C3×#41—22no—
64C2×C3×#41D414yes—
65C22×C3×Q8C2311yescharacteristic
66C22×C3×D4—21no—
67C22×C3×D4C2311yescharacteristic
68C23×C3×C4—21no—
69C23×C3×C4—21no—
70C23×C3×C4C2311yescharacteristic
71C22×#67—22no—
72C22×#67D412yes—
73C23×Dic3—21no—
74C2×#235—22no—
75C2×#235—22no—
76C2×#235—22no—
77C2×#235—22no—
78C2×#235—22no—
79C2×#235—22no—
80C22×D12D411yescharacteristic
81C22×C4×D3C2312yes—
82C22×Dic6D411yescharacteristic
83C2×C3×M16—22no—
84C2×C3×M16—22no—
85C2×C3×M16D414yescomplemented
86C22×C3×C8C2312yes—
87C2×C3×C4⋊C4—22no—
88C2×C3×C4⋊C4—22no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4C2312yes—
91C2×#217—22no—
92C2×#217—21no—
93#946—24no—
94#946—24no—
95#939—216no—
96#938—24no—
97#938—24no—
98#930—28no—
99C2×(C2×D3)⋊C4—22no—
100C2×(C2×D3)⋊C4—22no—
101C2×(C2×D3)⋊C4—22no—
102#926—22no—
103#926—22no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3—22no—
106C2×C4⋊Dic3D414yes—
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2312yes—
110#924—28no—
111C2×Dic3⋊C4—22no—
112C2×C4×Dic3—21no—
113C22×C3⋊C8—22no—
114C22×C3⋊C8—22no—
115C22×C3⋊C8—22no—
116C22×C3⋊C8C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118C3×#103—24no—
119C3×#102—24no—
120#835—22no—
121#835—22no—
122#832—22no—
123#832—22no—
124C3⋊#95—22no—
125C3⋊#95—22no—
126(C2×D3)⋊C8—24no—
127(C2×D3)⋊C8—24no—
128#818—24no—
129#817—24no—
130C22×#41—31no—
131C22×#41C2×D311yescharacteristic
132C22×M16C2×D311yescharacteristic
133C23×C8—31no—
134#537—64no—
135#537—64no—
136C2×C22⋊Q8—32no—
137C2×C22⋊Q8—61no—
138C2×#121—32no—
139C2×#121—61no—
140C2×#121—62no—
141C2×#117—31no—
142C22×C4⋊C4—31no—
143C22×C4⋊C4C2×D311yescharacteristic
144#428—64no—
145C2×#107—32no—
146C2×#106—32no—
147#417—24no—
148#416—64no—
149C2×#103—31no—
150C2×#103—31no—
151C2×#103C2×D312yes—
152C2×#102—31no—
153C2×#102—31no—
154C2×#102C2×D312yes—
155#408—64no—
156C2×C22⋊C8—32no—
157C2×C22⋊C8—32no—
158#391—61no—
159#390—61no—
160#386—62no—
161#380—32no—
162#380—32no—
163#340—32no—
164#340—32no—
165C24×C3—21no—
166C24×C3C2×D411yescharacteristic
167C23×D3C2×D412yes—
168C3×#41—28no—
169C3×#41—28no—
170C3×#41—44no—
171C3×#41—44no—
172C2×C3×Q8—21no—
173C2×C3×Q8—21no—
174C2×C3×Q8—21no—
175C2×C3×Q8—21no—
176C2×C3×Q8—21no—
177C2×C3×Q8C2×D412yes—
178C2×C3×D4—24no—
179C2×C3×D4—21no—
180C2×C3×D4—21no—
181C2×C3×D4—21no—
182C2×C3×D4—21no—
183C2×C3×D4—21no—
184C2×C3×D4—42no—
185C2×C3×D4—42no—
186C2×C3×D4C2×D412yes—
187C22×C3×C4—21no—
188C22×C3×C4—24no—
189C22×C3×C4—22no—
190C22×C3×C4—42no—
191C22×C3×C4—42no—
192C22×C3×C4—42no—
193C22×C3×C4—81no—
194C22×C3×C4—81no—
195C22×C3×C4C2×D412yes—
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D412yes—
198C22×C3×C4C2×D412yes—
199C22×C3×C4#4112yes—
200C22×C3×C4#4112yes—
201C22×C3×C4#4112yes—
202C22×C3×C4C2411yescharacteristic
203C2×#67—24no—
204C2×#67—24no—
205C2×#67—24no—
206C2×#67—28no—
207C2×#67—82no—
208C2×#67—82no—
209C22×Dic3—22no—
210C22×Dic3—24no—
211C22×Dic3—21no—
212C22×Dic3—21no—
213C22×Dic3—81no—
214C22×Dic3—81no—
215C22×Dic3C2×D412yes—
216#235—44no—
217#235—44no—
218#235—44no—
219#235—44no—
220C2×D12—22no—
221C2×D12—22no—
222C2×D12—22no—
223C2×C4×D3—22no—
224C2×C4×D3—22no—
225C2×C4×D3—22no—
226C2×C4×D3—22no—
227C2×C4×D3—22no—
228C2×C4×D3—22no—
229C2×Dic6—22no—
230C2×Dic6—22no—
231C2×Dic6—22no—
232C3×M16—28no—
233C2×C3×C8—22no—
234C2×C3×C8—22no—
235C2×C3×C8C2×D414yes—
236C3×C4⋊C4—22no—
237C3×C4⋊C4—22no—
238C3×C4⋊C4—42no—
239#217—28no—
240#217—42no—
241(C2×D3)⋊C4—28no—
242(C2×D3)⋊C4—82no—
243(C2×D3)⋊C4—82no—
244C4⋊Dic3—24no—
245C4⋊Dic3—22no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3C2×D414yes—
251Dic3⋊C4—28no—
252C4×Dic3—42no—
253C2×C3⋊C8—21no—
254C2×C3⋊C8—21no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—44no—
260C2×#41—22no—
261C2×#41—22no—
262C2×#41—22no—
263C2×#41—22no—
264C2×#41—22no—
265C2×#41—62no—
266C2×#41—62no—
267C2×#41—62no—
268C2×#41—62no—
269C2×#41—62no—
270C2×#41—62no—
271C2×#41D1214yes—
272C22×Q8—31no—
273C22×Q8C22×D311yescharacteristic
274C22×D4—21no—
275C22×D4—32no—
276C22×D4—31no—
277C22×D4—62no—
278C22×D4C22×D311yescharacteristic
279C23×C4—21no—
280C23×C4—21no—
281C23×C4—32no—
282C23×C4—61no—
283C23×C4C22×D311yescharacteristic
284C2×M16—22no—
285C2×M16—22no—
286C2×M16#6714yescomplemented
287C22×C8—31no—
288C22×C8—31no—
289C22×C8—62no—
290C22×C8—62no—
291C22×C8—62no—
292C22×C8C22×D312yes—
293C22⋊Q8—64no—
294C22⋊Q8—68no—
295C22⋊Q8—64no—
296#121—64no—
297#121—68no—
298#121—616no—
299#121—64no—
300#117—62no—
301#117—62no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—34no—
306C2×C4⋊C4—32no—
307C2×C4⋊C4—31no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—62no—
310C2×C4⋊C4—62no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4C22×D312yes—
313C2×C22⋊C4—62no—
314C2×C22⋊C4—62no—
315C2×C22⋊C4—61no—
316C2×C22⋊C4—62no—
317C2×C22⋊C4—62no—
318C2×C42—61no—
319#107—64no—
320#106—64no—
321#103—24no—
322#103—62no—
323#103—62no—
324#102—24no—
325#102—62no—
326#102—62no—
327C22⋊C8—64no—
328C22⋊C8—64no—
329#95—62no—
330#95—62no—
331C23×C3—21no—
332C23×C3—21no—
333C23×C3—21no—
334C23×C3—21no—
335C23×C3—22no—
336C23×C3—41no—
337C23×C3—41no—
338C23×C3—41no—
339C23×C3—41no—
340C23×C3—41no—
341C23×C3—41no—
342C23×C3#12012yes—
343C23×C3#12112yes—
344C23×C3#12312yes—
345C23×C3C22×D411yescharacteristic
346C22×D3—22no—
347C22×D3—22no—
348C22×D3—22no—
349C22×D3—22no—
350C22×D3—22no—
351C22×D3—22no—
352C22×D3—22no—
353C3×Q8—22no—
354C3×Q8—22no—
355C3×Q8—41no—
356C3×Q8—41no—
357C3×D4—22no—
358C3×D4—22no—
359C3×D4—48no—
360C3×D4—41no—
361C3×D4—41no—
362C2×C3×C4—22no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—48no—
366C2×C3×C4—42no—
367C2×C3×C4—42no—
368C2×C3×C4—42no—
369C2×C3×C4—84no—
370C2×C3×C4#12012yes—
371C2×C3×C4#12114yes—
372C2×C3×C4#12112yes—
373C2×C3×C4#12114yes—
374C2×C3×C4#12312yes—
375C2×C3×C4#12314yes—
376C2×C3×C4#12414yes—
377C2×C3×C4C22×D411yescharacteristic
378C2×C3×C4C22×D411yescharacteristic
379C2×C3×C4C22×D411yescharacteristic
380C2×C3×C4C2×#4111yescharacteristic
381C2×C3×C4C2×#4111yescharacteristic
382C2×C3×C4C2×#4111yescharacteristic
383#67—48no—
384#67—88no—
385C2×Dic3—22no—
386C2×Dic3—24no—
387C2×Dic3—22no—
388C2×Dic3—22no—
389C2×Dic3—42no—
390C2×Dic3—42no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—84no—
398D12—44no—
399C4×D3—42no—
400C4×D3—42no—
401C4×D3—42no—
402C4×D3—42no—
403Dic6—44no—
404C3×C8—24no—
405C3⋊C8—41no—
406C3⋊C8—41no—
407C24—21no—
408C24—32no—
409C24D3×D411yescharacteristic
410#41—28no—
411#41—28no—
412#41—44no—
413#41—44no—
414#41—124no—
415#41—124no—
416#41—124no—
417#41—124no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—62no—
424C2×Q8—62no—
425C2×Q8—62no—
426C2×Q8C2×D1212yes—
427C2×D4—24no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—42no—
434C2×D4—42no—
435C2×D4—64no—
436C2×D4—64no—
437C2×D4—64no—
438C2×D4—68no—
439C2×D4—62no—
440C2×D4—62no—
441C2×D4—62no—
442C2×D4—242no—
443C2×D4—242no—
444C2×D4C2×D1212yes—
445C22×C4—21no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—32no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—62no—
453C22×C4—64no—
454C22×C4—62no—
455C22×C4—62no—
456C22×C4—62no—
457C22×C4—62no—
458C22×C4—62no—
459C22×C4—62no—
460C22×C4—61no—
461C22×C4—61no—
462C22×C4—81no—
463C22×C4—81no—
464C22×C4—241no—
465C22×C4—241no—
466C22×C4C2×D1212yes—
467C22×C4#23512yes—
468C22×C4D3×D411yescharacteristic
469C22×C4D3×D412yes—
470C22×C4#23712yes—
471C22×C4#23712yes—
472C22×C4C2×#6712yes—
473C22×C4C23×D311yescharacteristic
474M16—28no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—61no—
478C2×C8—61no—
479C2×C8—61no—
480C2×C8—61no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8—124no—
484C2×C8C2×#6714yes—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—34no—
488C4⋊C4—42no—
489C4⋊C4—68no—
490C4⋊C4—64no—
491C4⋊C4—62no—
492C4⋊C4—61no—
493C4⋊C4—61no—
494C4⋊C4—61no—
495C4⋊C4—61no—
496C22⋊C4—68no—
497C22⋊C4—68no—
498C22⋊C4—122no—
499C22⋊C4—242no—
500C22⋊C4—242no—
501C42—122no—
502C22×C3—22no—
503C22×C3—22no—
504C22×C3—22no—
505C22×C3—22no—
506C22×C3—22no—
507C22×C3—22no—
508C22×C3—42no—
509C22×C3—42no—
510C22×C3—42no—
511C22×C3—81no—
512C22×C3—81no—
513C22×C3—81no—
514C22×C3—81no—
515C22×C3#39414yes—
516C22×C3C2×#12011yescharacteristic
517C22×C3C2×#12111yescharacteristic
518C22×C3C2×#12311yescharacteristic
519C22×C3#57611yescharacteristic
520C22×C3#57711yescharacteristic
521C22×C3#57711yescharacteristic
522C22×C3#57811yescharacteristic
523C2×D3—42no—
524C2×D3—42no—
525C2×D3—42no—
526C2×D3—42no—
527C2×D3—42no—
528C2×D3—42no—
529C2×D3—42no—
530C3×C4—42no—
531C3×C4—82no—
532C3×C4#39414yes—
533C3×C4C2×#12111yescharacteristic
534C3×C4C2×#12111yescharacteristic
535C3×C4C2×#12311yescharacteristic
536C3×C4C2×#12411yescharacteristic
537Dic3—42no—
538Dic3—44no—
539Dic3—81no—
540Dic3—81no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—22no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—62no—
553C23—62no—
554C23—62no—
555C23—62no—
556C23—62no—
557C23—62no—
558C23—62no—
559C23#88212yes—
560C23#88312yes—
561C23#93812yes—
562C23C2×D3×D411yescharacteristic
563Q8—22no—
564Q8—22no—
565Q8—41no—
566Q8—41no—
567Q8—124no—
568D4—22no—
569D4—22no—
570D4—48no—
571D4—41no—
572D4—41no—
573D4—128no—
574D4—124no—
575D4—248no—
576C2×C4—22no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—48no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—64no—
584C2×C4—62no—
585C2×C4—62no—
586C2×C4—62no—
587C2×C4—84no—
588C2×C4—122no—
589C2×C4—122no—
590C2×C4—122no—
591C2×C4—122no—
592C2×C4—122no—
593C2×C4—122no—
594C2×C4—121no—
595C2×C4—121no—
596C2×C4—121no—
597C2×C4—121no—
598C2×C4—121no—
599C2×C4—121no—
600C2×C4—244no—
601C2×C4#88212yes—
602C2×C4#88312yes—
603C2×C4#88514yes—
604C2×C4#88614yes—
605C2×C4#93014yes—
606C2×C4#93812yes—
607C2×C4#93914yes—
608C2×C4C22×D1211yescharacteristic
609C2×C4C2×#23511yescharacteristic
610C2×C4C2×D3×D411yescharacteristic
611C2×C4C2×#23711yescharacteristic
612C2×C4C2×#23711yescharacteristic
613C2×C4C22×#6711yescharacteristic
614C8—24no—
615C8—121no—
616C8—121no—
617C2×C3—24no—
618C2×C3—82no—
619C2×C3C2×#39411yescharacteristic
620C2×C3#301211yescharacteristic
621C2×C3#301311yescharacteristic
622C2×C3#306211yescharacteristic
623C2×C3#306311yescharacteristic
624C2×C3#309811yescharacteristic
625C2×C3#309911yescharacteristic
626D3—82no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—122no—
641C22—122no—
642C22—122no—
643C22—122no—
644C22—122no—
645C22—122no—
646C22—122no—
647C22#496114yes—
648C22C2×#88211yescharacteristic
649C22C2×#88311yescharacteristic
650C22#577411yescharacteristic
651C22#577511yescharacteristic
652C22#577811yescharacteristic
653C22#577911yescharacteristic
654C22C2×#93811yescharacteristic
655C4—42no—
656C4—82no—
657C4—124no—
658C4—122no—
659C4—241no—
660C4—241no—
661C4#496114yes—
662C4C2×#88511yescharacteristic
663C4C2×#88611yescharacteristic
664C4C2×#93011yescharacteristic
665C4C2×#93911yescharacteristic
666C3#2348511yescharacteristic, complemented, Sylow
667C2—24no—
668C2—82no—
669C2—242no—
670C2#7887411yescharacteristic
671C2#7887511yescharacteristic
672C2#7888611yescharacteristic
673C2#7888711yescharacteristic
674C2C2×#496111yescharacteristic
675C2#8395111yescharacteristic
676C2#8398611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#417C2
socleC23×C3C22×D4
prime radicalC2×#5833C2
hypercenterC2×#417D3
hypocenterC3#23485
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
2. C2×C3×#417C2semidirect
3. C2×#5233C2semidirect
4. C2×#5129C2semidirect
5. C2×#5121C2semidirect
6. #82186C2semidirect
7. C2×#4529C2semidirect
8. #70244C2semidirect
9. #70243C2semidirect
10. #70176C2semidirect
11. #70175C2semidirect
50. C2×#417D3semidirect
85. C2×C3×M16D4semidirect
286. C2×M16#67semidirect
666. C3#23485semidirect

Maximal subgroups

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

subgroupindex
C2×#58332
C2×C3×#4172
C2×#52332
C2×#51292
C2×#51212
#821862
C2×#45292
#702442
#702432
#701762
#701752
#234853