Finite Group Databasebeta

← #10690327

Subgroups — #10690327

Order 768.

1050 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5217C211yesmaximal, characteristic
2#73333C211yesmaximal, characteristic, complemented
3C3×#1438C211yesmaximal, characteristic, complemented
4#72128C211yesmaximal, characteristic, complemented
5C2×#4451C211yesmaximal, characteristic, complemented
6#67613C211yesmaximal, characteristic, complemented
7#67612C211yesmaximal, characteristic, complemented
8#9041—31nomaximal, Sylow
9C23×C3×D4C2211yescharacteristic
10C22×C3×C22⋊C4—21no—
11C22×C3×C22⋊C4—21no—
12C22×#213—21no—
13C2×#930C2211yescharacteristic
14C2×C4×#67—21no—
15C2×C3×#95C2211yescharacteristic
16C2×C3×#95C2211yescharacteristic
17#5217—22no—
18#5217C414yescomplemented
19C2×#832—21no—
20C2×#832—21no—
21C2×#832—21no—
22#5213—21no—
23#5212C2211yescharacteristic
24C2×#830C2211yescharacteristic
25C2×#830C2211yescharacteristic
26#4670—21no—
27#4668—21no—
28#4665—21no—
29#4653—21no—
30#4451—22no—
31C2×#447—31no—
32#2325—31no—
33#1786—31no—
34C2×#327—31no—
35#1438D311yescharacteristic, complemented
36#1294—31no—
37#1280—31no—
38C25×C3—21no—
39C22×C3×D4—21no—
40C22×C3×D4—21no—
41C22×C3×D4—21no—
42C22×C3×D4—21no—
43C22×C3×D4—21no—
44C22×C3×D4—42no—
45C22×C3×D4—42no—
46C22×C3×D4C2×C412yescomplemented
47C23×C3×C4—21no—
48C23×C3×C4D411yescharacteristic
49C23×C3×C4D411yescharacteristic
50C23×C3×C4C2311yescharacteristic
51C22×#67—21no—
52C22×D12D411yescharacteristic
53C22×C4×D3—21no—
54C2×C3×C22⋊C4—42no—
55C2×C3×C22⋊C4—42no—
56C2×C3×C22⋊C4—42no—
57C2×C3×C22⋊C4—42no—
58C2×C3×C22⋊C4—42no—
59C2×C3×C22⋊C4—42no—
60C2×#217—21no—
61C2×#213—24no—
62C2×#213—42no—
63C2×#213—42no—
64#930—21no—
65#930—21no—
66#930C2×C412yes—
67#930D412yes—
68C4×#67—42no—
69C4×#67—42no—
70C2×(C2×D3)⋊C4—21no—
71C2×(C2×D3)⋊C4—21no—
72C2×(C2×D3)⋊C4—21no—
73C2×(C2×D3)⋊C4—21no—
74C2×C4⋊Dic3D411yescharacteristic
75C2×C4⋊Dic3D411yescharacteristic
76C2×C4⋊Dic3D411yescharacteristic
77C2×Dic3⋊C4—21no—
78C2×C4×Dic3—21no—
79C22×C3⋊C8—21no—
80C22×C3⋊C8—21no—
81C3×#95—21no—
82C3×#95—21no—
83C3×#95—21no—
84C3×#95—21no—
85#832—24no—
86#832—42no—
87#832—42no—
88C3⋊#95—21no—
89C3⋊#95—21no—
90C3⋊#95—21no—
91#830—21no—
92#830—21no—
93#830—21no—
94#830C2×C412yescomplemented
95C23×D4C2×D311yescharacteristic
96C22×D8—61no—
97C2×#121—31no—
98C2×C4×D4—61no—
99C22×C22⋊C4—21no—
100C22×C22⋊C4—21no—
101#447—34no—
102#447—62no—
103C2×#102—61no—
104C2×#102—61no—
105C2×#102—61no—
106C2×C22⋊C8—31no—
107C2×C22⋊C8—31no—
108#388—61no—
109#387—61no—
110#386—61no—
111#386—61no—
112#385—61no—
113#380—31no—
114C2×#95C2×D311yescharacteristic
115C2×#95C2×D311yescharacteristic
116#327—62no—
117C24×C3—22no—
118C24×C3—22no—
119C24×C3—22no—
120C24×C3—41no—
121C24×C3—41no—
122C24×C3—41no—
123C24×C3—41no—
124C24×C3—41no—
125C24×C3—41no—
126C24×C3—41no—
127C24×C3—41no—
128C24×C3—41no—
129C24×C3—41no—
130C24×C3—41no—
131C24×C3—41no—
132C24×C3C2×D411yescharacteristic
133C23×D3—21no—
134C2×C3×D4—22no—
135C2×C3×D4—22no—
136C2×C3×D4—22no—
137C2×C3×D4—22no—
138C2×C3×D4—22no—
139C2×C3×D4—22no—
140C2×C3×D4—41no—
141C2×C3×D4—41no—
142C2×C3×D4—41no—
143C2×C3×D4—41no—
144C2×C3×D4—44no—
145C2×C3×D4—44no—
146C2×C3×D4—44no—
147C2×C3×D4—44no—
148C2×C3×D4—41no—
149C2×C3×D4—41no—
150C22×C3×C4—21no—
151C22×C3×C4—21no—
152C22×C3×C4—22no—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4—41no—
157C22×C3×C4—41no—
158C22×C3×C4—41no—
159C22×C3×C4—41no—
160C22×C3×C4—41no—
161C22×C3×C4—41no—
162C22×C3×C4—41no—
163C22×C3×C4—41no—
164C22×C3×C4—41no—
165C22×C3×C4—41no—
166C22×C3×C4—41no—
167C22×C3×C4—41no—
168C22×C3×C4D812yescomplemented
169C22×C3×C4C22×C411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4#4111yescharacteristic
174C22×C3×C4#4111yescharacteristic
175C2×#67—22no—
176C2×#67—22no—
177C2×#67—41no—
178C2×#67—41no—
179C2×#67—41no—
180C2×#67—41no—
181C22×Dic3—21no—
182C22×Dic3—21no—
183C22×Dic3—21no—
184C2×D12—21no—
185C2×D12—41no—
186C2×D12—41no—
187C2×D12C22⋊C412yescomplemented
188C2×C4×D3—42no—
189C2×C4×D3—81no—
190C2×C4×D3—81no—
191C3×C22⋊C4—48no—
192C3×C22⋊C4—48no—
193#217—41no—
194#217—41no—
195#213—48no—
196(C2×D3)⋊C4—22no—
197(C2×D3)⋊C4—22no—
198(C2×D3)⋊C4—42no—
199(C2×D3)⋊C4—81no—
200(C2×D3)⋊C4—81no—
201C4⋊Dic3—21no—
202C4⋊Dic3—41no—
203C4⋊Dic3—41no—
204C4⋊Dic3C22⋊C412yescomplemented
205Dic3⋊C4—42no—
206C4×Dic3—42no—
207C2×C3⋊C8—22no—
208C2×C3⋊C8—41no—
209C2×C3⋊C8—41no—
210C2×C3⋊C8—41no—
211C2×C3⋊C8—41no—
212C2×C3⋊C8—41no—
213C25—21no—
214C22×D4—21no—
215C22×D4—21no—
216C22×D4—21no—
217C22×D4—21no—
218C22×D4—21no—
219C22×D4—31no—
220C22×D4—42no—
221C22×D4—42no—
222C22×D4—61no—
223C22×D4C4×D312yescomplemented
224C23×C4—21no—
225C23×C4—61no—
226C23×C4#6711yescharacteristic
227C23×C4#6711yescharacteristic
228C23×C4C22×D311yescharacteristic
229C2×D8—64no—
230C2×D8—122no—
231C2×D8—122no—
232C22×C8—61no—
233C22×C8—61no—
234#121—32no—
235#121—32no—
236#121—61no—
237#121—61no—
238C4×D4—122no—
239C4×D4—122no—
240C2×C4⋊C4—31no—
241C2×C4⋊C4—31no—
242C2×C4⋊C4—31no—
243C2×C4⋊C4—61no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—61no—
251C2×C22⋊C4—61no—
252C2×C22⋊C4—61no—
253C2×C22⋊C4—61no—
254C2×C22⋊C4—61no—
255C2×C42—61no—
256#102—64no—
257#102—122no—
258#102—122no—
259C22⋊C8—32no—
260C22⋊C8—61no—
261C22⋊C8—61no—
262C22⋊C8—61no—
263#95—21no—
264#95—21no—
265#95—21no—
266#95—21no—
267#95—61no—
268#95—61no—
269#95—61no—
270C23×C3—21no—
271C23×C3—21no—
272C23×C3—21no—
273C23×C3—21no—
274C23×C3—24no—
275C23×C3—42no—
276C23×C3—42no—
277C23×C3—42no—
278C23×C3—42no—
279C23×C3—42no—
280C23×C3—42no—
281C23×C3—42no—
282C23×C3—42no—
283C23×C3—42no—
284C23×C3—42no—
285C23×C3—42no—
286C23×C3—42no—
287C23×C3—42no—
288C23×C3—42no—
289C23×C3—42no—
290C23×C3—42no—
291C23×C3—42no—
292C23×C3—42no—
293C23×C3—81no—
294C23×C3—81no—
295C23×C3—81no—
296C23×C3—81no—
297C23×C3—81no—
298C23×C3—81no—
299C23×C3—81no—
300C23×C3—81no—
301C23×C3—81no—
302C23×C3—81no—
303C23×C3—81no—
304C23×C3—81no—
305C23×C3—81no—
306C23×C3—81no—
307C23×C3—81no—
308C23×C3—81no—
309C23×C3C4×D411yescharacteristic
310C23×C3#12011yescharacteristic
311C23×C3#12111yescharacteristic
312C23×C3C2×D811yescharacteristic
313C23×C3#13511yescharacteristic
314C23×C3AGL(1,ℤ/8)11yescharacteristic
315C23×C3AGL(1,ℤ/8)11yescharacteristic
316C22×D3—22no—
317C22×D3—41no—
318C22×D3—41no—
319C22×D3—41no—
320C22×D3—41no—
321C22×D3—41no—
322C22×D3—41no—
323C3×D4—24no—
324C3×D4—24no—
325C3×D4—42no—
326C3×D4—42no—
327C3×D4—42no—
328C3×D4—42no—
329C3×D4—42no—
330C3×D4—42no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—22no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—42no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4—42no—
346C2×C3×C4—42no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—81no—
351C2×C3×C4—81no—
352C2×C3×C4—81no—
353C2×C3×C4—81no—
354C2×C3×C4—81no—
355C2×C3×C4—81no—
356C2×C3×C4C2×C22⋊C411yescharacteristic
357C2×C3×C4C4×D411yescharacteristic
358C2×C3×C4#12111yescharacteristic
359C2×C3×C4#12311yescharacteristic
360#67—42no—
361#67—42no—
362#67—42no—
363#67—42no—
364C2×Dic3—22no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3—41no—
368C2×Dic3—41no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373D12—42no—
374D12—81no—
375C4×D3—84no—
376C3⋊C8—42no—
377C3⋊C8—81no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—41no—
382C24—41no—
383C24—41no—
384C24—41no—
385C24—41no—
386C24—41no—
387C24—41no—
388C24—41no—
389C24—41no—
390C24—41no—
391C24—41no—
392C24—41no—
393C24—61no—
394C24C2×#6711yescharacteristic
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—32no—
402C2×D4—41no—
403C2×D4—41no—
404C2×D4—41no—
405C2×D4—41no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—41no—
411C2×D4—41no—
412C2×D4—62no—
413C2×D4—62no—
414C2×D4—61no—
415C2×D4—121no—
416C2×D4—121no—
417C2×D4—121no—
418C2×D4—121no—
419C2×D4—121no—
420C2×D4—121no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—61no—
440C22×C4—61no—
441C22×C4—61no—
442C22×C4—122no—
443C22×C4—241no—
444C22×C4—241no—
445C22×C4#21312yescomplemented
446C22×C4C2×C4×D311yescharacteristic
447C22×C4#23511yescharacteristic
448C22×C4D3×D411yescharacteristic
449C22×C4D3×D411yescharacteristic
450C22×C4D3×D411yescharacteristic
451C22×C4#23711yescharacteristic
452D8—128no—
453C2×C8—62no—
454C2×C8—121no—
455C2×C8—121no—
456C2×C8—121no—
457C2×C8—121no—
458C2×C8—121no—
459C4⋊C4—32no—
460C4⋊C4—61no—
461C4⋊C4—121no—
462C4⋊C4—121no—
463C4⋊C4—122no—
464C22⋊C4—48no—
465C22⋊C4—48no—
466C22⋊C4—62no—
467C22⋊C4—62no—
468C22⋊C4—122no—
469C22⋊C4—121no—
470C22⋊C4—121no—
471C22⋊C4—241no—
472C22⋊C4—241no—
473C42—122no—
474C22×C3—21no—
475C22×C3—21no—
476C22×C3—21no—
477C22×C3—21no—
478C22×C3—21no—
479C22×C3—21no—
480C22×C3—21no—
481C22×C3—21no—
482C22×C3—21no—
483C22×C3—21no—
484C22×C3—21no—
485C22×C3—21no—
486C22×C3—44no—
487C22×C3—44no—
488C22×C3—44no—
489C22×C3—82no—
490C22×C3—82no—
491C22×C3—82no—
492C22×C3—82no—
493C22×C3—82no—
494C22×C3—82no—
495C22×C3—82no—
496C22×C3—82no—
497C22×C3—82no—
498C22×C3—82no—
499C22×C3—82no—
500C22×C3—82no—
501C22×C3#38611yescharacteristic
502C22×C3C4×D811yescharacteristic
503C22×C3#44211yescharacteristic
504C22×C3#44711yescharacteristic
505C22×C3#44911yescharacteristic
506C22×C3#45311yescharacteristic
507C22×C3#45411yescharacteristic
508C22×C3B(4,𝔽2)11yescharacteristic
509C22×C3B(4,𝔽2)11yescharacteristic
510C22×C3#45911yescharacteristic
511C22×C3#46311yescharacteristic
512C2×D3—42no—
513C2×D3—42no—
514C2×D3—42no—
515C2×D3—81no—
516C2×D3—81no—
517C2×D3—81no—
518C2×D3—81no—
519C3×C4—21no—
520C3×C4—21no—
521C3×C4—21no—
522C3×C4—21no—
523C3×C4—42no—
524C3×C4—82no—
525C3×C4—81no—
526Dic3—42no—
527Dic3—81no—
528Dic3—81no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—24no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—62no—
553C23—81no—
554C23—81no—
555C23—81no—
556C23—81no—
557C23—81no—
558C23—81no—
559C23—81no—
560C23—81no—
561C23—81no—
562C23—81no—
563C23—81no—
564C23—81no—
565C23—81no—
566C23—81no—
567C23—81no—
568C23—81no—
569C23—121no—
570C23—121no—
571C23—121no—
572C23—121no—
573C23—121no—
574C23—121no—
575C23C4×#6711yescharacteristic
576C23C2×#21311yescharacteristic
577C23#93211yescharacteristic
578C23#93711yescharacteristic
579C23#93811yescharacteristic
580C23#94911yescharacteristic
581C23#95011yescharacteristic
582D4—24no—
583D4—24no—
584D4—42no—
585D4—42no—
586D4—42no—
587D4—42no—
588D4—42no—
589D4—42no—
590D4—122no—
591D4—122no—
592D4—122no—
593D4—122no—
594D4—122no—
595D4—241no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—62no—
616C2×C4—81no—
617C2×C4—81no—
618C2×C4—81no—
619C2×C4—81no—
620C2×C4—81no—
621C2×C4—81no—
622C2×C4—121no—
623C2×C4—121no—
624C2×C4—121no—
625C2×C4—121no—
626C2×C4—121no—
627C2×C4—121no—
628C2×C4—121no—
629C2×C4—121no—
630C2×C4—244no—
631C2×C4D3×C22⋊C411yescharacteristic
632C2×C4#67⋊C411yescharacteristic
633C2×C4#88311yescharacteristic
634C2×C4#88411yescharacteristic
635C8—122no—
636C8—241no—
637C2×C3—21no—
638C2×C3—21no—
639C2×C3—21no—
640C2×C3—21no—
641C2×C3—84no—
642C2×C3#159511yescharacteristic
643C2×C3#159711yescharacteristic
644C2×C3#159811yescharacteristic
645C2×C3#160011yescharacteristic
646C2×C3#187911yescharacteristic
647C2×C3#188311yescharacteristic
648C2×C3#188911yescharacteristic
649D3—82no—
650C22—21no—
651C22—21no—
652C22—21no—
653C22—21no—
654C22—21no—
655C22—21no—
656C22—21no—
657C22—21no—
658C22—21no—
659C22—21no—
660C22—21no—
661C22—21no—
662C22—44no—
663C22—44no—
664C22—44no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22—82no—
677C22—122no—
678C22—122no—
679C22—122no—
680C22—241no—
681C22—241no—
682C22—241no—
683C22—241no—
684C22#495611yescharacteristic
685C22C4×#21311yescharacteristic
686C22#501411yescharacteristic
687C22#503211yescharacteristic
688C22#503211yescharacteristic
689C22#503611yescharacteristic
690C22#503711yescharacteristic
691C22#505711yescharacteristic
692C22#506211yescharacteristic
693C22#507211yescharacteristic
694C22#507711yescharacteristic
695C4—21no—
696C4—21no—
697C4—21no—
698C4—21no—
699C4—42no—
700C4—82no—
701C4—81no—
702C4—122no—
703C4—241no—
704C4—241no—
705C3#904111yescharacteristic, complemented, Sylow
706C2—21no—
707C2—21no—
708C2—21no—
709C2—21no—
710C2—84no—
711C2—242no—
712C2#6949911yescharacteristic
713C2#6965711yescharacteristic
714C2#6966311yescharacteristic
715C2#6969611yescharacteristic
716C2#6969711yescharacteristic
717C2#6970211yescharacteristic
718C2#6970311yescharacteristic
subgroupG/H
centerC23#937
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1438C2
socleC23×C3#120
prime radicalC2×#5217C2
cosocleC23×C3×C4C23
hypercenter#1438D3
hypocenterC3#9041
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1438

Complements

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

NH ≅ G/Ntype
2. #73333C2semidirect
3. C3×#1438C2semidirect
4. #72128C2semidirect
5. C2×#4451C2semidirect
6. #67613C2semidirect
7. #67612C2semidirect
18. #5217C4semidirect
35. #1438D3semidirect
46. C22×C3×D4C2×C4semidirect
94. #830C2×C4semidirect
168. C22×C3×C4D8semidirect
187. C2×D12C22⋊C4semidirect
204. C4⋊Dic3C22⋊C4semidirect
223. C22×D4C4×D3semidirect
445. C22×C4#213semidirect
705. C3#9041semidirect

Maximal subgroups

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

subgroupindex
C2×#52172
#733332
C3×#14382
#721282
C2×#44512
#676132
#676122
#90413