Finite Group Databasebeta

← #10823162

Subgroups — #10823162

Order 768.

1458 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#950C211yesmaximal, characteristic
2C2×C3×#417C211yesmaximal, characteristic, complemented
3#84087C211yesmaximal, characteristic, complemented
4C2×#5133C211yesmaximal, characteristic, complemented
5C2×#5119C211yesmaximal, characteristic, complemented
6#82186C211yesmaximal, characteristic, complemented
7C2×#4529C211yesmaximal, characteristic, complemented
8#70285C212yesmaximal, complemented
9#70280C212yesmaximal, complemented
10#70217C212yesmaximal, complemented
11#70212C212yesmaximal, complemented
12#23543—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C22×#235C2211yescharacteristic
15C22×C3×M16C2211yescharacteristic
16C22×C3×C4⋊C4C2211yescharacteristic
17C2×#950—24no—
18C2×#950—24no—
19C2×#950—24no—
20C22×#216C2211yescharacteristic
21C22×#215C2211yescharacteristic
22C22×#214C2211yescharacteristic
23C22×#213C2211yescharacteristic
24C2×#926C2211yescharacteristic
25C22×C4⋊Dic3C2211yescharacteristic
26C23×C3⋊C8C2211yescharacteristic
27C3×#417—24no—
28C2×C3×#103C2212yes—
29C2×C3×#102C2212yes—
30#5229C2212yes—
31#5221C2212yes—
32#5220—22no—
33#5213—22no—
34#5212—21no—
35#5212—21no—
36#5133—24no—
37#5119—24no—
38C2×#821C2212yes—
39C2×#816C2212yes—
40C2×Dic3⋊C8C2212yes—
41C2×Dic3⋊C8C2212yes—
42#4988C2212yes—
43#4987—21no—
44#4987—21no—
45#4973C2212yes—
46#4529C2212yes—
47#4529C2212yes—
48C22×#135—31no—
49C2×#424—31no—
50C2×#417—31no—
51C2×#417D311yescharacteristic, complemented
52#2311—31no—
53#2311—31no—
54#2056—32no—
55#2055—32no—
56#2055—32no—
57#2054—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—21no—
72C23×Dic3—21no—
73C2×#235—22no—
74C2×#235—22no—
75C2×#235—22no—
76C2×#235—22no—
77C2×#235—22no—
78C2×#235D414yes—
79C22×D12C2311yescharacteristic
80C22×C4×D3—21no—
81C22×Dic6C2311yescharacteristic
82C2×C3×M16—22no—
83C2×C3×M16—22no—
84C2×C3×M16—22no—
85C22×C3×C8C2312yes—
86C2×C3×C4⋊C4—22no—
87C2×C3×C4⋊C4—22no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4C2312yes—
90C2×#217—22no—
91C2×#217—22no—
92C2×#217D412yes—
93#950—232no—
94C2×#216—22no—
95C2×#216—22no—
96C2×#216—22no—
97C2×#215—22no—
98C2×#215—22no—
99C2×#215—22no—
100C2×#214—22no—
101C2×#214—22no—
102C2×#214—22no—
103C2×#213—22no—
104C2×#213—22no—
105C2×#213—22no—
106C2×(C2×D3)⋊C4—22no—
107C2×(C2×D3)⋊C4—22no—
108#926—22no—
109#926—22no—
110C2×C4⋊Dic3—22no—
111C2×C4⋊Dic3—22no—
112C2×C4⋊Dic3—21no—
113C2×C4⋊Dic3D411yescharacteristic
114C2×C4⋊Dic3D411yescharacteristic
115C2×C4⋊Dic3C2312yes—
116C2×C4×Dic3C2312yes—
117C22×C3⋊C8—22no—
118C22×C3⋊C8—22no—
119C22×C3⋊C8D414yes—
120C22×C3⋊C8C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122C3×#103—24no—
123C3×#102—24no—
124C3⋊#95—22no—
125C3⋊#95—22no—
126C3⋊#95—22no—
127C3⋊#95—22no—
128#821—24no—
129#816—24no—
130Dic3⋊C8—24no—
131Dic3⋊C8—24no—
132C22×#41—31no—
133C22×#41C2×D311yescharacteristic
134C2×#135—64no—
135C2×#135—64no—
136C2×#135—64no—
137C22×Q16—31no—
138C22×SD16—31no—
139C22×SD16—31no—
140C22×D8—31no—
141C22×M16C2×D311yescharacteristic
142C23×C8—31no—
143C2×#117—31no—
144C22×C4⋊C4—31no—
145C22×C4⋊C4C2×D311yescharacteristic
146#424—64no—
147C2×C4⋊C8—32no—
148C2×C4⋊C8—32no—
149#417—24no—
150#417—64no—
151C2×#103—32no—
152C2×#103C2×D312yes—
153C2×#102—32no—
154C2×#102C2×D312yes—
155#391—32no—
156#391—32no—
157#390—32no—
158#390—32no—
159#386—62no—
160#386—62no—
161#380—61no—
162#380—61no—
163#380—61no—
164#380—61no—
165#340—32no—
166#340—32no—
167C24×C3—21no—
168C24×C3C2×D411yescharacteristic
169C23×D3—21no—
170C3×#41—28no—
171C3×#41—28no—
172C3×#41—44no—
173C3×#41—44no—
174C2×C3×Q8—21no—
175C2×C3×Q8—21no—
176C2×C3×Q8—21no—
177C2×C3×Q8—21no—
178C2×C3×Q8—21no—
179C2×C3×Q8C2×D412yes—
180C2×C3×D4—24no—
181C2×C3×D4—21no—
182C2×C3×D4—21no—
183C2×C3×D4—21no—
184C2×C3×D4—21no—
185C2×C3×D4—21no—
186C2×C3×D4—42no—
187C2×C3×D4—42no—
188C2×C3×D4C2×D412yes—
189C22×C3×C4—21no—
190C22×C3×C4—24no—
191C22×C3×C4—22no—
192C22×C3×C4—42no—
193C22×C3×C4—42no—
194C22×C3×C4—42no—
195C22×C3×C4—81no—
196C22×C3×C4—81no—
197C22×C3×C4C2×D412yes—
198C22×C3×C4C2×D412yes—
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D412yes—
201C22×C3×C4#4112yes—
202C22×C3×C4#4112yes—
203C22×C3×C4#4112yes—
204C22×C3×C4C2411yescharacteristic
205C2×#67—24no—
206C2×#67—42no—
207C2×#67—42no—
208C22×Dic3—21no—
209C22×Dic3—22no—
210C22×Dic3—42no—
211C22×Dic3—81no—
212C22×Dic3—81no—
213C22×Dic3C2×D412yes—
214C22×Dic3C2×D412yes—
215#235—28no—
216#235—28no—
217#235—44no—
218#235—44no—
219C2×D12—21no—
220C2×D12—21no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12C2×D412yes—
225C2×C4×D3—24no—
226C2×C4×D3—42no—
227C2×C4×D3—42no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6—21no—
232C2×Dic6—21no—
233C2×Dic6C2×D412yes—
234C3×M16—44no—
235C2×C3×C8—22no—
236C2×C3×C8—22no—
237C2×C3×C8—22no—
238C3×C4⋊C4—22no—
239C3×C4⋊C4—22no—
240C3×C4⋊C4—42no—
241#217—24no—
242#217—82no—
243#217—82no—
244#216—28no—
245#215—28no—
246#214—28no—
247#213—28no—
248(C2×D3)⋊C4—82no—
249(C2×D3)⋊C4—82no—
250C4⋊Dic3—22no—
251C4⋊Dic3—22no—
252C4⋊Dic3—22no—
253C4⋊Dic3—22no—
254C4⋊Dic3—42no—
255C4×Dic3—22no—
256C4×Dic3—22no—
257C2×C3⋊C8—28no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8C2×D412yes—
263C2×C3⋊C8C2×D412yes—
264C2×#41—22no—
265C2×#41—22no—
266C2×#41—22no—
267C2×#41—22no—
268C2×#41—22no—
269C2×#41—34no—
270C2×#41—62no—
271C2×#41—62no—
272C2×#41—62no—
273C2×#41—62no—
274C2×#41—62no—
275C2×#41#6714yes—
276C22×Q8—31no—
277C22×Q8C22×D311yescharacteristic
278C22×D4—21no—
279C22×D4—31no—
280C22×D4—61no—
281C22×D4C22×D311yescharacteristic
282C23×C4—21no—
283C23×C4—21no—
284C23×C4—61no—
285C23×C4—61no—
286C23×C4C22×D311yescharacteristic
287#135—632no—
288C2×Q16—62no—
289C2×Q16—62no—
290C2×Q16—62no—
291C2×SD16—62no—
292C2×SD16—62no—
293C2×SD16—62no—
294C2×SD16—62no—
295C2×SD16—62no—
296C2×SD16—62no—
297C2×D8—62no—
298C2×D8—62no—
299C2×D8—62no—
300C2×M16—22no—
301C2×M16—22no—
302C2×M16—22no—
303C22×C8—34no—
304C22×C8—31no—
305C22×C8—31no—
306C22×C8—62no—
307C22×C8—62no—
308C22×C8C22×D312yes—
309#117—62no—
310#117—62no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—31no—
315C2×C4⋊C4—31no—
316C2×C4⋊C4—32no—
317C2×C4⋊C4—62no—
318C2×C4⋊C4—62no—
319C2×C4⋊C4—61no—
320C2×C4⋊C4C22×D312yes—
321C2×C22⋊C4—32no—
322C2×C22⋊C4—62no—
323C2×C22⋊C4—62no—
324C2×C22⋊C4—62no—
325C2×C22⋊C4—62no—
326C2×C42—32no—
327C4⋊C8—64no—
328C4⋊C8—64no—
329#103—24no—
330#103—64no—
331#102—24no—
332#102—64no—
333#95—62no—
334#95—62no—
335#95—62no—
336#95—62no—
337C23×C3—21no—
338C23×C3—21no—
339C23×C3—21no—
340C23×C3—21no—
341C23×C3—22no—
342C23×C3—41no—
343C23×C3—41no—
344C23×C3—41no—
345C23×C3—41no—
346C23×C3—41no—
347C23×C3—41no—
348C23×C3#12112yes—
349C23×C3#12112yes—
350C23×C3#12112yes—
351C23×C3C22×D411yescharacteristic
352C22×D3—22no—
353C22×D3—41no—
354C22×D3—41no—
355C22×D3—41no—
356C22×D3—41no—
357C22×D3—41no—
358C22×D3—41no—
359C3×Q8—22no—
360C3×Q8—22no—
361C3×Q8—41no—
362C3×Q8—41no—
363C3×D4—22no—
364C3×D4—22no—
365C3×D4—48no—
366C3×D4—41no—
367C3×D4—41no—
368C2×C3×C4—22no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—48no—
372C2×C3×C4—42no—
373C2×C3×C4—42no—
374C2×C3×C4—42no—
375C2×C3×C4—84no—
376C2×C3×C4#12014yes—
377C2×C3×C4#12112yes—
378C2×C3×C4#12112yes—
379C2×C3×C4#12112yes—
380C2×C3×C4#12314yes—
381C2×C3×C4#12314yes—
382C2×C3×C4#12414yes—
383C2×C3×C4C22×D411yescharacteristic
384C2×C3×C4C22×D411yescharacteristic
385C2×C3×C4C22×D411yescharacteristic
386C2×C3×C4C2×#4111yescharacteristic
387C2×C3×C4C2×#4111yescharacteristic
388C2×C3×C4C2×#4111yescharacteristic
389#67—48no—
390C2×Dic3—22no—
391C2×Dic3—22no—
392C2×Dic3—22no—
393C2×Dic3—22no—
394C2×Dic3—22no—
395C2×Dic3—22no—
396C2×Dic3—22no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—42no—
400C2×Dic3—42no—
401C2×Dic3—42no—
402C2×Dic3—84no—
403D12—22no—
404D12—22no—
405D12—41no—
406D12—41no—
407C4×D3—48no—
408Dic6—22no—
409Dic6—22no—
410Dic6—41no—
411Dic6—41no—
412C3×C8—42no—
413C3⋊C8—22no—
414C3⋊C8—22no—
415C24—21no—
416C24—61no—
417C24D3×D411yescharacteristic
418#41—28no—
419#41—28no—
420#41—44no—
421#41—44no—
422#41—68no—
423#41—68no—
424#41—124no—
425#41—124no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—32no—
432C2×Q8—61no—
433C2×Q8—61no—
434C2×Q8—61no—
435C2×Q8—61no—
436C2×Q8—61no—
437C2×Q8C2×#6712yes—
438C2×D4—24no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—32no—
445C2×D4—42no—
446C2×D4—42no—
447C2×D4—64no—
448C2×D4—61no—
449C2×D4—61no—
450C2×D4—61no—
451C2×D4—61no—
452C2×D4—61no—
453C2×D4—122no—
454C2×D4—122no—
455C2×D4C2×#6712yes—
456C22×C4—21no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—32no—
460C22×C4—32no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—61no—
465C22×C4—64no—
466C22×C4—62no—
467C22×C4—81no—
468C22×C4—81no—
469C22×C4—122no—
470C22×C4—122no—
471C22×C4—122no—
472C22×C4—241no—
473C22×C4—241no—
474C22×C4#23512yes—
475C22×C4#23512yes—
476C22×C4D3×D412yes—
477C22×C4D3×D411yescharacteristic
478C22×C4D3×D412yes—
479C22×C4#23712yes—
480C22×C4C2×#6712yes—
481C22×C4C23×D311yescharacteristic
482Q16—68no—
483SD16—68no—
484SD16—68no—
485D8—68no—
486M16—44no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—32no—
491C2×C8—32no—
492C2×C8—68no—
493C2×C8—61no—
494C2×C8—61no—
495C2×C8—61no—
496C2×C8—61no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—42no—
500C4⋊C4—62no—
501C4⋊C4—62no—
502C4⋊C4—62no—
503C4⋊C4—62no—
504C4⋊C4—122no—
505C22⋊C4—64no—
506C22⋊C4—242no—
507C22⋊C4—242no—
508C22⋊C4—242no—
509C22⋊C4—242no—
510C42—62no—
511C42—62no—
512C22×C3—22no—
513C22×C3—22no—
514C22×C3—22no—
515C22×C3—22no—
516C22×C3—22no—
517C22×C3—22no—
518C22×C3—42no—
519C22×C3—42no—
520C22×C3—42no—
521C22×C3—81no—
522C22×C3—81no—
523C22×C3—81no—
524C22×C3—81no—
525C22×C3#39414yes—
526C22×C3C2×#12111yescharacteristic
527C22×C3C2×#12111yescharacteristic
528C22×C3C2×#12111yescharacteristic
529C22×C3#57611yescharacteristic
530C22×C3#57711yescharacteristic
531C22×C3#57711yescharacteristic
532C22×C3#57811yescharacteristic
533C2×D3—42no—
534C2×D3—42no—
535C2×D3—42no—
536C2×D3—81no—
537C2×D3—81no—
538C2×D3—81no—
539C2×D3—81no—
540C3×C4—42no—
541C3×C4—82no—
542C3×C4#39414yes—
543C3×C4C2×#12011yescharacteristic
544C3×C4C2×#12311yescharacteristic
545C3×C4C2×#12311yescharacteristic
546C3×C4C2×#12411yescharacteristic
547Dic3—42no—
548Dic3—42no—
549Dic3—42no—
550Dic3—82no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—22no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—62no—
563C23—121no—
564C23—121no—
565C23—121no—
566C23—121no—
567C23—121no—
568C23—121no—
569C23#88412yes—
570C23#88412yes—
571C23#93912yes—
572C23C2×D3×D411yescharacteristic
573Q8—22no—
574Q8—22no—
575Q8—41no—
576Q8—41no—
577Q8—62no—
578Q8—62no—
579Q8—121no—
580Q8—121no—
581D4—22no—
582D4—22no—
583D4—48no—
584D4—41no—
585D4—41no—
586D4—62no—
587D4—62no—
588D4—128no—
589D4—121no—
590D4—121no—
591C2×C4—22no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—48no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—62no—
599C2×C4—62no—
600C2×C4—62no—
601C2×C4—62no—
602C2×C4—62no—
603C2×C4—62no—
604C2×C4—62no—
605C2×C4—84no—
606C2×C4—121no—
607C2×C4—121no—
608C2×C4—128no—
609C2×C4—122no—
610C2×C4—122no—
611C2×C4—122no—
612C2×C4—244no—
613C2×C4#88314yes—
614C2×C4#88412yes—
615C2×C4#88412yes—
616C2×C4#88514yes—
617C2×C4#92914yes—
618C2×C4#93714yes—
619C2×C4#93912yes—
620C2×C4C2×#23511yescharacteristic
621C2×C4C2×#23511yescharacteristic
622C2×C4C2×D3×D411yescharacteristic
623C2×C4C2×D3×D411yescharacteristic
624C2×C4C2×#23711yescharacteristic
625C2×C4C22×#6711yescharacteristic
626C8—42no—
627C8—62no—
628C8—62no—
629C2×C3—24no—
630C2×C3—82no—
631C2×C3C2×#39411yescharacteristic
632C2×C3#304511yescharacteristic
633C2×C3#304511yescharacteristic
634C2×C3#304611yescharacteristic
635C2×C3#304611yescharacteristic
636C2×C3#306011yescharacteristic
637C2×C3#306111yescharacteristic
638D3—82no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—81no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22—122no—
653C22—122no—
654C22—122no—
655C22—241no—
656C22—241no—
657C22—241no—
658C22—241no—
659C22#495914yes—
660C22C2×#88411yescharacteristic
661C22C2×#88411yescharacteristic
662C22#577411yescharacteristic
663C22#577511yescharacteristic
664C22#577811yescharacteristic
665C22#577911yescharacteristic
666C22C2×#93911yescharacteristic
667C4—42no—
668C4—82no—
669C4—122no—
670C4—122no—
671C4—122no—
672C4—242no—
673C4#495914yes—
674C4C2×#88311yescharacteristic
675C4C2×#88511yescharacteristic
676C4C2×#92911yescharacteristic
677C4C2×#93711yescharacteristic
678C3#2354311yescharacteristic, complemented, Sylow
679C2—24no—
680C2—82no—
681C2—242no—
682C2#7889011yescharacteristic
683C2#7889111yescharacteristic
684C2#7891111yescharacteristic
685C2#7891211yescharacteristic
686C2C2×#495911yescharacteristic
687C2#8393611yescharacteristic
688C2#8397111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#417C2
socleC23×C3C22×D4
prime radicalC22×#950C2
hypercenterC2×#417D3
hypocenterC3#23543
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. #84087C2semidirect
4. C2×#5133C2semidirect
5. C2×#5119C2semidirect
6. #82186C2semidirect
7. C2×#4529C2semidirect
8. #70285C2semidirect
9. #70280C2semidirect
10. #70217C2semidirect
11. #70212C2semidirect
51. C2×#417D3semidirect
678. C3#23543semidirect

Maximal subgroups

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

subgroupindex
C22×#9502
C2×C3×#4172
#840872
C2×#51332
C2×#51192
#821862
C2×#45292
#702852
#702802
#702172
#702122
#235433