Finite Group Databasebeta

← #10686992

Subgroups — #10686992

Order 768.

1148 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5597C211yesmaximal, characteristic
2C2×C3×#327C211yesmaximal, characteristic, complemented
3#71677C211yesmaximal, characteristic, complemented
4#69932C211yesmaximal, characteristic, complemented
5C2×#4469C211yesmaximal, characteristic, complemented
6#69410C211yesmaximal, characteristic, complemented
7C2×#4451C211yesmaximal, characteristic, complemented
8#8404—31nomaximal, Sylow
9C22×#237—21no—
10C2×C3×#121C2211yescharacteristic
11C2×#939—21no—
12C2×#939—21no—
13C2×#930C2211yescharacteristic
14C22×C4⋊Dic3C2211yescharacteristic
15C22×Dic3⋊C4C2211yescharacteristic
16#5597—22no—
17#5597—22no—
18#5597—22no—
19#5597C414yescomplemented
20C2×#896—21no—
21C2×#884—21no—
22C2×#878—21no—
23C2×C3×C22⋊C8C2211yescharacteristic
24C2×C3×#95C2211yescharacteristic
25C2×#832—21no—
26C2×C3⋊#95—21no—
27C2×C3⋊#95—21no—
28C2×#830C2211yescharacteristic
29C2×#821—21no—
30C3×#327—22no—
31#4551—21no—
32#4529—21no—
33#4469—22no—
34#4451—22no—
35C2×#537—31no—
36#1874—31no—
37#1874—31no—
38C2×#327—31no—
39C2×#327D311yescharacteristic, complemented
40C2×#324—31no—
41#1441—31no—
42C22×C3×D4—21no—
43C22×C3×D4D411yescharacteristic
44C23×C3×C4—21no—
45C23×C3×C4C2311yescharacteristic
46C22×#67—21no—
47C22×#67—21no—
48C22×#67—21no—
49C23×Dic3—21no—
50C23×Dic3—21no—
51C23×Dic3D411yescharacteristic
52C23×Dic3D411yescharacteristic
53C2×#237—24no—
54C2×#237—24no—
55C2×#237—42no—
56C2×#237—42no—
57C2×#237—42no—
58C2×#237—42no—
59C22×D12D411yescharacteristic
60C22×C4×D3—21no—
61C22×Dic6—21no—
62C22×C3×C8—21no—
63C3×#121—21no—
64C3×#121—21no—
65C3×#121C2×C412yes—
66C3×#121D412yes—
67C2×C3×C4⋊C4D411yescharacteristic
68C2×C3×C22⋊C4—21no—
69C2×#217—21no—
70#939—24no—
71#939—24no—
72#939—24no—
73#939—24no—
74#930—21no—
75#930—21no—
76#930C2×C412yes—
77#930D412yes—
78C2×(C2×D3)⋊C4—21no—
79C2×(C2×D3)⋊C4—21no—
80C2×C4⋊Dic3—21no—
81C2×C4⋊Dic3—21no—
82C2×C4⋊Dic3—42no—
83C2×C4⋊Dic3D411yescharacteristic
84C2×Dic3⋊C4—21no—
85C2×Dic3⋊C4—21no—
86C2×Dic3⋊C4—21no—
87C2×Dic3⋊C4—21no—
88C2×Dic3⋊C4—21no—
89C2×Dic3⋊C4C2×C412yes—
90C2×Dic3⋊C4D412yes—
91C22×C3⋊C8—21no—
92#896—24no—
93#896—24no—
94#884—24no—
95#884—24no—
96#878—24no—
97#878—24no—
98C3×C22⋊C8—21no—
99C3×C22⋊C8—21no—
100C3×#95—21no—
101C3×#95—21no—
102#832—42no—
103C3⋊#95—42no—
104C3⋊#95—42no—
105#830—21no—
106#830—21no—
107#821—42no—
108C22×#41—61no—
109#537—34no—
110#537—62no—
111#537—62no—
112#537—62no—
113C2×C22⋊Q8—61no—
114C2×C22⋊Q8—61no—
115C2×#121—31no—
116C2×#121—61no—
117C2×#121—61no—
118C2×#121—61no—
119C2×#121C2×D311yescharacteristic
120C22×C4⋊C4—31no—
121C22×C4⋊C4—31no—
122C2×#102—61no—
123C2×#102—61no—
124C2×C22⋊C8—31no—
125C2×C22⋊C8C2×D311yescharacteristic
126C2×#95—61no—
127C2×#95—61no—
128C2×#95C2×D311yescharacteristic
129#340—61no—
130#340—61no—
131#327—22no—
132#327—62no—
133#324—62no—
134C24×C3—21no—
135C24×C3C2×D411yescharacteristic
136C23×D3—21no—
137C2×C3×D4—22no—
138C2×C3×D4—22no—
139C2×C3×D4—21no—
140C2×C3×D4—41no—
141C2×C3×D4—41no—
142C2×C3×D4—41no—
143C2×C3×D4—41no—
144C2×C3×D4—41no—
145C2×C3×D4—41no—
146C2×C3×D4C22⋊C412yes—
147C22×C3×C4—21no—
148C22×C3×C4—22no—
149C22×C3×C4—41no—
150C22×C3×C4—41no—
151C22×C3×C4—41no—
152C22×C3×C4—41no—
153C22×C3×C4—41no—
154C22×C3×C4—41no—
155C22×C3×C4—41no—
156C22×C3×C4—41no—
157C22×C3×C4C22×C411yescharacteristic
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C2×#67—24no—
164C2×#67—22no—
165C2×#67—22no—
166C2×#67—24no—
167C2×#67—42no—
168C2×#67—42no—
169C2×#67—41no—
170C2×#67—41no—
171C2×#67—41no—
172C2×#67—41no—
173C2×#67—81no—
174C2×#67—81no—
175C22×Dic3—21no—
176C22×Dic3—21no—
177C22×Dic3—21no—
178C22×Dic3—21no—
179C22×Dic3—24no—
180C22×Dic3—21no—
181C22×Dic3—21no—
182C22×Dic3—21no—
183C22×Dic3—22no—
184C22×Dic3—42no—
185C22×Dic3—42no—
186C22×Dic3—41no—
187C22×Dic3—41no—
188C22×Dic3—41no—
189C22×Dic3—41no—
190C22×Dic3—41no—
191C22×Dic3—41no—
192C22×Dic3—41no—
193C22×Dic3—41no—
194C22×Dic3C22⋊C412yes—
195C22×Dic3C22⋊C412yes—
196C22×Dic3D812yes—
197C22×Dic3SD1612yes—
198#237—48no—
199#237—48no—
200#237—48no—
201#237—48no—
202C2×D12—21no—
203C2×D12—41no—
204C2×D12—41no—
205C2×D12C22⋊C412yes—
206C2×C4×D3—24no—
207C2×C4×D3—42no—
208C2×C4×D3—42no—
209C2×Dic6—24no—
210C2×Dic6—42no—
211C2×Dic6—42no—
212C2×C3×C8—41no—
213C2×C3×C8—41no—
214C2×C3×C8—41no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4C22⋊C412yes—
217C3×C22⋊C4—22no—
218C3×C22⋊C4—22no—
219#217—24no—
220(C2×D3)⋊C4—24no—
221(C2×D3)⋊C4—22no—
222(C2×D3)⋊C4—22no—
223C4⋊Dic3—21no—
224C4⋊Dic3—22no—
225C4⋊Dic3—22no—
226C4⋊Dic3—41no—
227C4⋊Dic3—41no—
228C4⋊Dic3C22⋊C412yes—
229Dic3⋊C4—24no—
230Dic3⋊C4—22no—
231Dic3⋊C4—22no—
232Dic3⋊C4—22no—
233Dic3⋊C4—22no—
234C2×C3⋊C8—41no—
235C2×C3⋊C8—41no—
236C2×C3⋊C8—41no—
237C2×#41—64no—
238C2×#41—64no—
239C2×#41—122no—
240C2×#41—122no—
241C2×#41—122no—
242C2×#41—122no—
243C22×Q8—61no—
244C22×D4—21no—
245C22×D4—31no—
246C22×D4—61no—
247C22×D4—61no—
248C22×D4—61no—
249C22×D4D1211yescharacteristic
250C23×C4—21no—
251C23×C4—31no—
252C23×C4—31no—
253C23×C4—61no—
254C23×C4—61no—
255C23×C4—61no—
256C23×C4C22×D311yescharacteristic
257C22×C8—21no—
258C22×C8—61no—
259C22⋊Q8—64no—
260C22⋊Q8—64no—
261C22⋊Q8—64no—
262C22⋊Q8—64no—
263#121—21no—
264#121—21no—
265#121—32no—
266#121—32no—
267#121—64no—
268#121—64no—
269#121—64no—
270#121—64no—
271#121—64no—
272#121—64no—
273#121—61no—
274#121—61no—
275#121C4×D312yes—
276#121#6712yes—
277C2×C4⋊C4—31no—
278C2×C4⋊C4—32no—
279C2×C4⋊C4—32no—
280C2×C4⋊C4—61no—
281C2×C4⋊C4—61no—
282C2×C4⋊C4—61no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—122no—
288C2×C4⋊C4D1211yescharacteristic
289C2×C22⋊C4—21no—
290C2×C22⋊C4—61no—
291C2×C22⋊C4—61no—
292C2×C22⋊C4—61no—
293#102—122no—
294#102—122no—
295C22⋊C8—21no—
296C22⋊C8—21no—
297C22⋊C8—61no—
298C22⋊C8—61no—
299#95—21no—
300#95—21no—
301#95—122no—
302#95—122no—
303C23×C3—21no—
304C23×C3—21no—
305C23×C3—21no—
306C23×C3—21no—
307C23×C3—22no—
308C23×C3—41no—
309C23×C3—41no—
310C23×C3—41no—
311C23×C3—41no—
312C23×C3—41no—
313C23×C3—41no—
314C23×C3C2×C22⋊C411yescharacteristic
315C23×C3#12011yescharacteristic
316C23×C3#12011yescharacteristic
317C23×C3C2×D811yescharacteristic
318C23×C3C2×SD1611yescharacteristic
319C23×C3AGL(1,ℤ/8)11yescharacteristic
320C23×C3AGL(1,ℤ/8)11yescharacteristic
321C22×D3—22no—
322C22×D3—41no—
323C22×D3—41no—
324C22×D3—41no—
325C22×D3—41no—
326C22×D3—41no—
327C22×D3—41no—
328C3×D4—42no—
329C3×D4—42no—
330C3×D4—42no—
331C3×D4—42no—
332C3×D4—42no—
333C3×D4—81no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—22no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—42no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4—81no—
349C2×C3×C4—81no—
350C2×C3×C4—81no—
351C2×C3×C4—81no—
352C2×C3×C4C2×C22⋊C411yescharacteristic
353C2×C3×C4C2×C22⋊C411yescharacteristic
354C2×C3×C4#12011yescharacteristic
355C2×C3×C4#12011yescharacteristic
356#67—48no—
357#67—42no—
358#67—42no—
359#67—42no—
360#67—42no—
361#67—84no—
362C2×Dic3—22no—
363C2×Dic3—22no—
364C2×Dic3—22no—
365C2×Dic3—22no—
366C2×Dic3—22no—
367C2×Dic3—22no—
368C2×Dic3—22no—
369C2×Dic3—22no—
370C2×Dic3—48no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—42no—
376C2×Dic3—42no—
377C2×Dic3—42no—
378C2×Dic3—42no—
379C2×Dic3—81no—
380C2×Dic3—81no—
381C2×Dic3—81no—
382C2×Dic3—81no—
383C2×Dic3—81no—
384C2×Dic3#10214yescomplemented
385D12—42no—
386D12—81no—
387C4×D3—48no—
388Dic6—48no—
389C3×C8—81no—
390C3⋊C8—81no—
391C24—21no—
392C24—61no—
393C24D3×D411yescharacteristic
394#41—128no—
395#41—128no—
396#41—128no—
397#41—128no—
398C2×Q8—64no—
399C2×Q8—122no—
400C2×Q8—122no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—21no—
404C2×D4—32no—
405C2×D4—41no—
406C2×D4—41no—
407C2×D4—41no—
408C2×D4—41no—
409C2×D4—41no—
410C2×D4—41no—
411C2×D4—64no—
412C2×D4—62no—
413C2×D4—62no—
414C2×D4—64no—
415C2×D4—61no—
416C2×D4—121no—
417C2×D4—121no—
418C2×D4—122no—
419C2×D4—122no—
420C2×D4—121no—
421C2×D4—121no—
422C2×D4—121no—
423C2×D4—121no—
424C2×D4—241no—
425C2×D4—241no—
426C2×D4(C2×D3)⋊C412yes—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—32no—
430C22×C4—32no—
431C22×C4—32no—
432C22×C4—32no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—64no—
442C22×C4—61no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4—61no—
446C22×C4—64no—
447C22×C4—61no—
448C22×C4—61no—
449C22×C4—61no—
450C22×C4—62no—
451C22×C4—122no—
452C22×C4—122no—
453C22×C4—122no—
454C22×C4—122no—
455C22×C4—121no—
456C22×C4—121no—
457C22×C4—121no—
458C22×C4—121no—
459C22×C4—121no—
460C22×C4—121no—
461C22×C4—121no—
462C22×C4—121no—
463C22×C4C2×C4×D311yescharacteristic
464C22×C4C2×D1211yescharacteristic
465C22×C4D3×D411yescharacteristic
466C22×C4D3×D411yescharacteristic
467C22×C4D3×D411yescharacteristic
468C22×C4C2×#6711yescharacteristic
469C2×C8—41no—
470C2×C8—41no—
471C2×C8—41no—
472C2×C8—121no—
473C2×C8—121no—
474C2×C8—121no—
475C4⋊C4—21no—
476C4⋊C4—32no—
477C4⋊C4—64no—
478C4⋊C4—62no—
479C4⋊C4—62no—
480C4⋊C4—62no—
481C4⋊C4—62no—
482C4⋊C4—62no—
483C4⋊C4—62no—
484C4⋊C4—61no—
485C4⋊C4—121no—
486C4⋊C4—121no—
487C4⋊C4(C2×D3)⋊C412yes—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—64no—
491C22⋊C4—64no—
492C22⋊C4—62no—
493C22⋊C4—62no—
494C22×C3—21no—
495C22×C3—21no—
496C22×C3—21no—
497C22×C3—21no—
498C22×C3—21no—
499C22×C3—21no—
500C22×C3—21no—
501C22×C3—21no—
502C22×C3—21no—
503C22×C3—21no—
504C22×C3—21no—
505C22×C3—21no—
506C22×C3—42no—
507C22×C3—42no—
508C22×C3—42no—
509C22×C3—81no—
510C22×C3—81no—
511C22×C3—81no—
512C22×C3—81no—
513C22×C3#37911yescharacteristic
514C22×C3C2×#10211yescharacteristic
515C22×C3#41811yescharacteristic
516C22×C3#44711yescharacteristic
517C22×C3#44711yescharacteristic
518C22×C3#45011yescharacteristic
519C22×C3#45011yescharacteristic
520C22×C3#45411yescharacteristic
521C22×C3#45611yescharacteristic
522C22×C3B(4,𝔽2)11yescharacteristic
523C22×C3#45811yescharacteristic
524C2×D3—42no—
525C2×D3—42no—
526C2×D3—42no—
527C2×D3—81no—
528C2×D3—81no—
529C2×D3—81no—
530C2×D3—81no—
531C3×C4—21no—
532C3×C4—21no—
533C3×C4—21no—
534C3×C4—21no—
535C3×C4—42no—
536C3×C4—82no—
537Dic3—24no—
538Dic3—42no—
539Dic3—42no—
540Dic3—82no—
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—121no—
554C23—121no—
555C23—121no—
556C23—121no—
557C23—121no—
558C23—121no—
559C23D3×C22⋊C411yescharacteristic
560C23#88211yescharacteristic
561C23D3×D811yescharacteristic
562C23#91111yescharacteristic
563C23D3×SD1611yescharacteristic
564C23#91411yescharacteristic
565C23#93711yescharacteristic
566Q8—128no—
567D4—42no—
568D4—42no—
569D4—42no—
570D4—42no—
571D4—42no—
572D4—81no—
573D4—122no—
574D4—128no—
575D4—122no—
576D4—122no—
577D4—122no—
578D4—122no—
579D4—241no—
580D4—244no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—34no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—62no—
597C2×C4—62no—
598C2×C4—62no—
599C2×C4—62no—
600C2×C4—62no—
601C2×C4—62no—
602C2×C4—62no—
603C2×C4—62no—
604C2×C4—81no—
605C2×C4—81no—
606C2×C4—81no—
607C2×C4—81no—
608C2×C4—128no—
609C2×C4—128no—
610C2×C4—121no—
611C2×C4—121no—
612C2×C4—121no—
613C2×C4—121no—
614C2×C4—122no—
615C2×C4—122no—
616C2×C4—122no—
617C2×C4—122no—
618C2×C4—241no—
619C2×C4—241no—
620C2×C4—241no—
621C2×C4—241no—
622C2×C4—241no—
623C2×C4D3×C22⋊C411yescharacteristic
624C2×C4#88211yescharacteristic
625C2×C4C2×(C2×D3)⋊C411yescharacteristic
626C2×C4#93711yescharacteristic
627C8—81no—
628C8—241no—
629C2×C3—21no—
630C2×C3—21no—
631C2×C3—21no—
632C2×C3—21no—
633C2×C3—82no—
634C2×C3#159811yescharacteristic
635C2×C3#160411yescharacteristic
636C2×C3#161011yescharacteristic
637C2×C3#161611yescharacteristic
638C2×C3#178611yescharacteristic
639C2×C3#179011yescharacteristic
640C2×C3#179311yescharacteristic
641D3—82no—
642C22—21no—
643C22—21no—
644C22—21no—
645C22—21no—
646C22—21no—
647C22—21no—
648C22—21no—
649C22—21no—
650C22—21no—
651C22—21no—
652C22—21no—
653C22—21no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22—122no—
662C22—122no—
663C22—122no—
664C22—241no—
665C22—241no—
666C22—241no—
667C22—241no—
668C22#474111yescharacteristic
669C22#474211yescharacteristic
670C22D3×#10211yescharacteristic
671C22#477011yescharacteristic
672C22#477311yescharacteristic
673C22#477611yescharacteristic
674C22#482611yescharacteristic
675C22#482711yescharacteristic
676C22#495511yescharacteristic
677C22#515611yescharacteristic
678C22#516911yescharacteristic
679C4—21no—
680C4—21no—
681C4—21no—
682C4—21no—
683C4—42no—
684C4—64no—
685C4—82no—
686C4—122no—
687C4—122no—
688C4—242no—
689C3#840411yescharacteristic, complemented, Sylow
690C2—21no—
691C2—21no—
692C2—21no—
693C2—21no—
694C2—82no—
695C2—242no—
696C2#6829311yescharacteristic
697C2#6829511yescharacteristic
698C2#6829711yescharacteristic
699C2#6829911yescharacteristic
700C2#7003311yescharacteristic
701C2#7019811yescharacteristic
702C2#7038711yescharacteristic
subgroupG/H
centerC23#882
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#327C2
socleC23×C3#120
prime radicalC2×#5597C2
cosocleC23×C3×C4C23
hypercenterC2×#327D3
hypocenterC3#8404
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#327

Complements

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

NH ≅ G/Ntype
2. C2×C3×#327C2semidirect
3. #71677C2semidirect
4. #69932C2semidirect
5. C2×#4469C2semidirect
6. #69410C2semidirect
7. C2×#4451C2semidirect
19. #5597C4semidirect
39. C2×#327D3semidirect
384. C2×Dic3#102semidirect
689. C3#8404semidirect

Maximal subgroups

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

subgroupindex
C2×#55972
C2×C3×#3272
#716772
#699322
C2×#44692
#694102
C2×#44512
#84043