Finite Group Databasebeta

← #10801954

Subgroups — #10801954

Order 768.

1010 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5101C211yesmaximal, characteristic, complemented
2#82633C211yesmaximal, characteristic
3C2×#5042C211yesmaximal, characteristic
4C2×#4757C211yesmaximal, characteristic, complemented
5#73871C211yesmaximal, characteristic, complemented
6C3×#1893C211yesmaximal, characteristic
7#71972C211yesmaximal, characteristic, complemented
8#70189C211yesmaximal, characteristic, complemented
9#70188C211yesmaximal, characteristic, complemented
10Dic3×#102C211yesmaximal, characteristic, complemented
11#69513C211yesmaximal, characteristic, complemented
12#68859C211yesmaximal, characteristic, complemented
13#68855C211yesmaximal, characteristic, complemented
14#68786C211yesmaximal, characteristic, complemented
15#68781C211yesmaximal, characteristic, complemented
16#21543—31nomaximal, Sylow
17C23×C3×C8C2211yescharacteristic
18C2×C3×#121C2211yescharacteristic
19C2×D4×Dic3C2211yescharacteristic
20C22×#214C2211yescharacteristic
21C2×#926C2211yescharacteristic
22C2×#924C2211yescharacteristic
23C22×#208C2211yescharacteristic
24C2×#887C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C3×#380C2211yescharacteristic
28#5221—21no—
29#5221C2211yescharacteristic
30#5220—21no—
31C2×#832C2211yescharacteristic
32#5212C2211yescharacteristic
33C2×#816C2211yescharacteristic
34C2×#816C2211yescharacteristic
35#5101—24no—
36C2×#815C2211yescharacteristic
37C2×#815C2211yescharacteristic
38C2×C8×Dic3C2211yescharacteristic
39C2×C8×Dic3C2211yescharacteristic
40#5042C418yescomplemented
41#4984C2211yescharacteristic
42Dic3×C4⋊C4C2211yescharacteristic
43Dic3×C4⋊C4C2211yescharacteristic
44#4973C2211yescharacteristic
45#4972C2211yescharacteristic
46C2×#810C2211yescharacteristic
47C2×#808C2211yescharacteristic
48C2×#807C2211yescharacteristic
49C2×#807C2211yescharacteristic
50#4951—21no—
51#4950—21no—
52#4945—21no—
53#4944—21no—
54#4757—24no—
55#4655—21no—
56#4647—21no—
57C3×#340C2211yescharacteristic
58C3×#340C2211yescharacteristic
59#4551C2211yescharacteristic
60#4551C2211yescharacteristic
61#4529C2211yescharacteristic
62#4529C2211yescharacteristic
63C2×#468—31no—
64C2×#440—31no—
65C2×#405—31no—
66#2324—31no—
67#2323—31no—
68#1968—31no—
69#1942—31no—
70#1936—31no—
71#1935—31no—
72#1893D311yescharacteristic
73#1892—31no—
74#1809—31no—
75#1776—31no—
76C4×#103—31no—
77C4×#102—31no—
78C22×C3×D4—21no—
79C22×C3×D4C2311yescharacteristic
80C23×C3×C4C2311yescharacteristic
81C23×Dic3—21no—
82C22×Dic6C2311yescharacteristic
83C22×C3×C8—22no—
84C22×C3×C8—22no—
85C22×C3×C8—22no—
86C22×C3×C8C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C3×#121—22no—
89C3×#121C2×C414yes—
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×#217—21no—
96C2×#217—21no—
97C2×#217—21no—
98C2×#217—21no—
99C2×#217—21no—
100C2×#217D411yescharacteristic
101C2×#217D411yescharacteristic
102D4×Dic3—22no—
103D4×Dic3—22no—
104C2×#214—22no—
105C2×#214—22no—
106C2×#214C2×C414yes—
107#926—22no—
108#926—22no—
109C2×C4⋊Dic3—21no—
110C2×C4⋊Dic3D411yescharacteristic
111C2×C4⋊Dic3D411yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115#924—22no—
116#924C2×C414yescomplemented
117C2×Dic3⋊C4—21no—
118C2×Dic3⋊C4—21no—
119C2×Dic3⋊C4—21no—
120C2×C4×Dic3—21no—
121C2×C4×Dic3—21no—
122C2×C4×Dic3—21no—
123C2×C4×Dic3C2311yescharacteristic
124C2×C4×Dic3C2311yescharacteristic
125C2×#208—22no—
126C2×#208—22no—
127C2×#208C2×C414yescomplemented
128C22×C3⋊C8C2311yescharacteristic
129C22×C3⋊C8C2311yescharacteristic
130#887—22no—
131#887—22no—
132C3×#102—22no—
133C3×#102—22no—
134C3×#95—21no—
135#832C2×C414yes—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142C3⋊#95—21no—
143#816—22no—
144#816—22no—
145#815—22no—
146#815D414yescomplemented
147C8×Dic3—22no—
148C8×Dic3—22no—
149#810C2×C414yescomplemented
150#808—22no—
151#807—22no—
152#807C2×C414yescomplemented
153C22×SD16—31no—
154C22×M16—31no—
155C23×C8C2×D311yescharacteristic
156C2×C22⋊Q8—31no—
157C2×#121C2×D311yescharacteristic
158C2×C4×Q8—31no—
159C2×C4×D4—31no—
160C2×#117—31no—
161#468—38no—
162#440—64no—
163C2×#107—31no—
164C2×#107—31no—
165C2×#106—31no—
166C2×#103—31no—
167C2×#103—31no—
168C2×#103—31no—
169C2×#102—31no—
170C2×#102C2×D311yescharacteristic
171C2×#102C2×D311yescharacteristic
172#405—64no—
173C2×#97—31no—
174C2×#97—31no—
175C2×C4×C8—31no—
176C2×C4×C8—31no—
177#391—31no—
178#390—31no—
179#390—61no—
180#389—31no—
181#389—31no—
182#389—61no—
183#388—61no—
184#388—61no—
185#388—61no—
186#387—61no—
187#386—61no—
188#385—61no—
189#380—31no—
190#380C2×D311yescharacteristic
191C4×C4⋊C4—31no—
192C4×C4⋊C4—31no—
193#340—31no—
194#340—31no—
195#340—31no—
196#340—31no—
197#340C2×D311yescharacteristic
198#340C2×D311yescharacteristic
199C24×C3—21no—
200C24×C3C2×D411yescharacteristic
201C2×C3×D4—24no—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—81no—
208C2×C3×D4—81no—
209C2×C3×D4C22×C412yes—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4—21no—
217C22×C3×C4C22×C412yes—
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4#4112yes—
220C22×C3×C4C2411yescharacteristic
221C22×Dic3—21no—
222C22×Dic3—21no—
223C22×Dic3—21no—
224C22×Dic3—21no—
225C22×Dic3—42no—
226C22×Dic3—81no—
227C22×Dic3—81no—
228C22×Dic3C2×D411yescharacteristic
229C22×Dic3C2×D411yescharacteristic
230C22×Dic3C2×D411yescharacteristic
231C22×Dic3C2×D411yescharacteristic
232C2×Dic6—21no—
233C2×Dic6—21no—
234C2×Dic6—21no—
235C2×Dic6—21no—
236C2×Dic6—21no—
237C2×Dic6C22×C412yes—
238C2×C3×C8—28no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8C2×D412yes—
244C2×C3×C8#4112yes—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4C22×C412yes—
251C3×C22⋊C4—24no—
252C3×C22⋊C4—81no—
253#217—24no—
254#217—22no—
255#217—22no—
256#217—42no—
257#217—81no—
258#217—81no—
259#217—81no—
260#214—28no—
261C4⋊Dic3—21no—
262C4⋊Dic3—22no—
263C4⋊Dic3—22no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—81no—
269C4⋊Dic3C22×C412yes—
270Dic3⋊C4—24no—
271Dic3⋊C4—42no—
272Dic3⋊C4—81no—
273C4×Dic3—21no—
274C4×Dic3—21no—
275C4×Dic3—42no—
276C4×Dic3—81no—
277C4×Dic3—81no—
278C4×Dic3C2×D412yes—
279C4×Dic3#4112yes—
280#208—28no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8C22×C412yes—
284C2×C3⋊C8C22×C412yes—
285C2×C3⋊C8C2×D412yes—
286C2×C3⋊C8#4112yes—
287C22×Q8—31no—
288C22×D4—21no—
289C22×D4C22×D311yescharacteristic
290C23×C4—61no—
291C23×C4C22×D311yescharacteristic
292C2×SD16—34no—
293C2×SD16—62no—
294C2×SD16—62no—
295C2×M16—34no—
296C2×M16—62no—
297C2×M16—62no—
298C22×C8—22no—
299C22×C8—22no—
300C22×C8—22no—
301C22×C8—31no—
302C22×C8—31no—
303C22×C8C22×D311yescharacteristic
304C22×C8C22×D311yescharacteristic
305C22⋊Q8—34no—
306C22⋊Q8—62no—
307#121—22no—
308#121C4×D314yes—
309C4×Q8—62no—
310C4×Q8—62no—
311C4×D4—62no—
312C4×D4—62no—
313#117—62no—
314#117—62no—
315C2×C4⋊C4—31no—
316C2×C4⋊C4—31no—
317C2×C4⋊C4—31no—
318C2×C4⋊C4—31no—
319C2×C4⋊C4—31no—
320C2×C4⋊C4—61no—
321C2×C4⋊C4—61no—
322C2×C4⋊C4—61no—
323C2×C4⋊C4—61no—
324C2×C4⋊C4C22×D311yescharacteristic
325C2×C4⋊C4C22×D311yescharacteristic
326C2×C4⋊C4C22×D311yescharacteristic
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—31no—
330C2×C22⋊C4—31no—
331C2×C22⋊C4—61no—
332C2×C22⋊C4—61no—
333C2×C22⋊C4—61no—
334C2×C22⋊C4—61no—
335C2×C22⋊C4—61no—
336C2×C42—31no—
337C2×C42—31no—
338C2×C42—61no—
339C2×C42—61no—
340C2×C42—61no—
341#107—34no—
342#107—62no—
343#106—62no—
344#103—34no—
345#103—62no—
346#103—62no—
347#102—22no—
348#102—22no—
349#102—34no—
350#97—34no—
351#97—62no—
352C4×C8—62no—
353C4×C8—62no—
354#95—21no—
355#95—61no—
356#95—61no—
357#95—61no—
358#95—61no—
359#95—61no—
360#95—61no—
361#95—61no—
362C23×C3—21no—
363C23×C3—21no—
364C23×C3—21no—
365C23×C3—21no—
366C23×C3—21no—
367C23×C3—21no—
368C23×C3—22no—
369C23×C3—41no—
370C23×C3—41no—
371C23×C3—41no—
372C23×C3—41no—
373C23×C3—41no—
374C23×C3—41no—
375C23×C3C4×D412yes—
376C23×C3C22×D411yescharacteristic
377C3×D4—22no—
378C3×D4—22no—
379C3×D4—41no—
380C3×D4—41no—
381C3×D4—84no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—22no—
385C2×C3×C4—22no—
386C2×C3×C4—22no—
387C2×C3×C4—22no—
388C2×C3×C4—22no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4C4×D412yes—
398C2×C3×C4C23×C411yescharacteristic
399C2×C3×C4C22×D411yescharacteristic
400C2×C3×C4C2×#4111yescharacteristic
401C2×C3×C4C2×#4111yescharacteristic
402C2×C3×C4C2×#4111yescharacteristic
403C2×C3×C4C2×#4111yescharacteristic
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—22no—
409C2×Dic3—22no—
410C2×Dic3—21no—
411C2×Dic3—21no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—21no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3—41no—
426C2×Dic3—84no—
427C2×Dic3#12112yes—
428C2×Dic3#12112yes—
429Dic6—22no—
430Dic6—22no—
431Dic6—41no—
432Dic6—41no—
433C3×C8—22no—
434C3×C8—22no—
435C3⋊C8—22no—
436C3⋊C8C4×D414yes—
437C24—21no—
438C24D3×D411yescharacteristic
439C2×Q8—32no—
440C2×Q8—61no—
441C2×Q8—61no—
442C2×Q8—61no—
443C2×Q8—61no—
444C2×Q8—61no—
445C2×D4—24no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—81no—
452C2×D4—81no—
453C2×D4C2×C4×D312yes—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—31no—
462C22×C4—31no—
463C22×C4—31no—
464C22×C4—31no—
465C22×C4—61no—
466C22×C4—61no—
467C22×C4—61no—
468C22×C4—61no—
469C22×C4—122no—
470C22×C4—241no—
471C22×C4—241no—
472C22×C4C2×C4×D312yes—
473C22×C4D3×D411yescharacteristic
474C22×C4#23712yes—
475C22×C4C23×D311yescharacteristic
476SD16—68no—
477M16—68no—
478C2×C8—28no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—32no—
484C2×C8—32no—
485C2×C8—32no—
486C2×C8—32no—
487C2×C8—61no—
488C2×C8—61no—
489C2×C8D3×D412yes—
490C2×C8#23712yes—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—32no—
497C4⋊C4—62no—
498C4⋊C4—62no—
499C4⋊C4—64no—
500C4⋊C4—61no—
501C4⋊C4—61no—
502C4⋊C4—61no—
503C4⋊C4—61no—
504C4⋊C4—61no—
505C4⋊C4—122no—
506C4⋊C4—241no—
507C4⋊C4—241no—
508C4⋊C4C2×C4×D312yes—
509C22⋊C4—24no—
510C22⋊C4—62no—
511C22⋊C4—62no—
512C22⋊C4—64no—
513C22⋊C4—81no—
514C22⋊C4—122no—
515C22⋊C4—241no—
516C22⋊C4—241no—
517C22⋊C4—241no—
518C42—32no—
519C42—32no—
520C42—61no—
521C42—61no—
522C42—122no—
523C42—241no—
524C42—241no—
525C22×C3—22no—
526C22×C3—22no—
527C22×C3—22no—
528C22×C3—41no—
529C22×C3—41no—
530C22×C3—41no—
531C22×C3—41no—
532C22×C3—42no—
533C22×C3—42no—
534C22×C3—42no—
535C22×C3—81no—
536C22×C3—81no—
537C22×C3—81no—
538C22×C3—81no—
539C22×C3C2×C4×D411yescharacteristic
540C22×C3C2×#12111yescharacteristic
541C22×C3#53211yescharacteristic
542C22×C3#57611yescharacteristic
543C22×C3#57711yescharacteristic
544C22×C3#57711yescharacteristic
545C22×C3#57811yescharacteristic
546C3×C4—22no—
547C3×C4—42no—
548C3×C4—81no—
549C3×C4—81no—
550C3×C4C2×C4×D411yescharacteristic
551C3×C4C4×#4111yescharacteristic
552C3×C4#52511yescharacteristic
553C3×C4#52911yescharacteristic
554Dic3—22no—
555Dic3—22no—
556Dic3—42no—
557Dic3—41no—
558Dic3—41no—
559Dic3—42no—
560Dic3—81no—
561Dic3—81no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—22no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23#67⋊C412yes—
576C23C2×D3×D411yescharacteristic
577Q8—62no—
578Q8—62no—
579Q8—121no—
580Q8—121no—
581D4—22no—
582D4—22no—
583D4—41no—
584D4—41no—
585D4—84no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—32no—
594C2×C4—32no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—61no—
604C2×C4—61no—
605C2×C4—61no—
606C2×C4—61no—
607C2×C4—62no—
608C2×C4—62no—
609C2×C4—61no—
610C2×C4—61no—
611C2×C4—61no—
612C2×C4—61no—
613C2×C4—61no—
614C2×C4—61no—
615C2×C4—121no—
616C2×C4—121no—
617C2×C4—121no—
618C2×C4—121no—
619C2×C4—121no—
620C2×C4—121no—
621C2×C4—121no—
622C2×C4—121no—
623C2×C4—121no—
624C2×C4—121no—
625C2×C4—244no—
626C2×C4#67⋊C412yes—
627C2×C4C22×C4×D311yescharacteristic
628C2×C4C2×D3×D411yescharacteristic
629C2×C4C2×#23711yescharacteristic
630C2×C4C2×#23711yescharacteristic
631C2×C4D3×#4111yescharacteristic
632C2×C4D3×#4111yescharacteristic
633C8—22no—
634C8—22no—
635C8—34no—
636C8—62no—
637C2×C3—42no—
638C2×C3—82no—
639C2×C3C4×#12111yescharacteristic
640C2×C3#294611yescharacteristic
641C2×C3#294711yescharacteristic
642C2×C3#306211yescharacteristic
643C2×C3#306311yescharacteristic
644C2×C3#317411yescharacteristic
645C2×C3#317511yescharacteristic
646C22—22no—
647C22—22no—
648C22—22no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—81no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22C2×#67⋊C411yescharacteristic
661C22#559611yescharacteristic
662C22D3×#12111yescharacteristic
663C22#575711yescharacteristic
664C22#576211yescharacteristic
665C22#577011yescharacteristic
666C22#577111yescharacteristic
667C4—22no—
668C4—42no—
669C4—62no—
670C4—62no—
671C4—81no—
672C4—81no—
673C4—121no—
674C4—121no—
675C4—122no—
676C4—122no—
677C4—241no—
678C4—241no—
679C4C4×#23711yescharacteristic
680C4C4×D3×D411yescharacteristic
681C4#558611yescharacteristic
682C4#564911yescharacteristic
683C3#2154311yescharacteristic, complemented, Sylow
684C2—42no—
685C2—82no—
686C2#7466011yescharacteristic
687C2#7872611yescharacteristic
688C2#7882111yescharacteristic
689C2#8063511yescharacteristic
690C2#8066211yescharacteristic
691C2#8069811yescharacteristic
692C2#8071811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1893C2
socleC23×C3C22×D4
prime radicalC2×C3×#121C22
cosocleC22×C3×C4C24
hypercenter#1893D3
hypocenterC3#21543
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1893

Complements

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

NH ≅ G/Ntype
1. C2×#5101C2semidirect
4. C2×#4757C2semidirect
5. #73871C2semidirect
7. #71972C2semidirect
8. #70189C2semidirect
9. #70188C2semidirect
10. Dic3×#102C2semidirect
11. #69513C2semidirect
12. #68859C2semidirect
13. #68855C2semidirect
14. #68786C2semidirect
15. #68781C2semidirect
40. #5042C4semidirect
116. #924C2×C4semidirect
127. C2×#208C2×C4semidirect
146. #815D4semidirect
149. #810C2×C4semidirect
152. #807C2×C4semidirect
683. C3#21543semidirect

Maximal subgroups

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

subgroupindex
C2×#51012
#826332
C2×#50422
C2×#47572
#738712
C3×#18932
#719722
#701892
#701882
Dic3×#1022
#695132
#688592
#688552
#687862
#687812
#215433