Finite Group Databasebeta

← #10702078

Subgroups — #10702078

Order 768.

1252 conjugacy classes of subgroups.

The subgroup lattice has 796 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5748C211yesmaximal, characteristic
2C2×#5139C211yesmaximal, characteristic, complemented
3#83768C211yesmaximal, characteristic, complemented
4C2×#5132C211yesmaximal, characteristic, complemented
5#83762C211yesmaximal, characteristic, complemented
6C3×#1738C211yesmaximal, characteristic, complemented
7C2×#4556C211yesmaximal, characteristic, complemented
8#11037—31nomaximal, Sylow
9C22×#235C2211yescharacteristic
10C22×#235C2211yescharacteristic
11C22×C3×M16C2211yescharacteristic
12C22×C3×M16C2211yescharacteristic
13C22×C3×M16C2211yescharacteristic
14C2×#926C2211yescharacteristic
15C22×#208C2211yescharacteristic
16#5748—22no—
17#5748—22no—
18#5748—22no—
19#5748C414yescomplemented
20C2×#909—21no—
21C2×#909—21no—
22C2×#908—21no—
23C2×#908—21no—
24C2×#904—22no—
25#5139—22no—
26#5139—22no—
27C2×#825—22no—
28C2×#824—21no—
29#5133—22no—
30#5132—22no—
31#5132—22no—
32#5131—21no—
33C2×#823—21no—
34#5129—22no—
35#5126—22no—
36#5120—21no—
37#5113—22no—
38#5110—22no—
39C3×#341—22no—
40#4556—22no—
41C2×#575—31no—
42C2×#421—31no—
43#2896—31no—
44C2×#413—31no—
45#2879—31no—
46C2×#344—31no—
47#1738D311yescharacteristic, complemented
48C23×C3×C4C2311yescharacteristic
49C22×#67—21no—
50C22×#67—21no—
51C2×#235—21no—
52C2×#235—21no—
53C2×#235—21no—
54C2×#235—21no—
55C2×#235—21no—
56C2×#235—21no—
57C2×#235—21no—
58C2×#235—21no—
59C2×#235—21no—
60C2×#235—21no—
61C2×#235—21no—
62C2×#235—21no—
63C2×#235—21no—
64C2×#235—21no—
65C2×#235—21no—
66C2×#235—21no—
67C2×#235—21no—
68C2×#235—21no—
69C2×#235—21no—
70C2×#235—21no—
71C2×#235C2×C412yes—
72C2×#235C2×C412yes—
73C2×#235D412yes—
74C2×#235D412yes—
75C22×D12D411yescharacteristic
76C22×D12D411yescharacteristic
77C22×C4×D3—21no—
78C22×C4×D3—21no—
79C22×Dic6D411yescharacteristic
80C22×Dic6D411yescharacteristic
81C2×C3×M16—21no—
82C2×C3×M16—21no—
83C2×C3×M16—21no—
84C2×C3×M16—21no—
85C2×C3×M16—21no—
86C2×C3×M16—21no—
87C2×C3×M16—21no—
88C2×C3×M16—21no—
89C2×C3×M16—21no—
90C2×C3×M16—21no—
91C2×C3×M16—21no—
92C2×C3×M16—21no—
93C2×C3×M16—21no—
94C2×C3×M16—21no—
95C2×C3×M16C2×C412yes—
96C2×C3×M16D412yes—
97C2×C3×M16D412yes—
98C2×C3×M16D412yes—
99C22×C3×C8—21no—
100C22×C3×C8—21no—
101C22×C3×C8D412yes—
102C2×#217—21no—
103#926—21no—
104#926—21no—
105#926—21no—
106#926D412yes—
107C2×C4⋊Dic3—21no—
108C2×C4×Dic3—21no—
109C2×#208—21no—
110C2×#208—21no—
111C2×#208—21no—
112C2×#208—21no—
113C2×#208—21no—
114C2×#208D412yes—
115C22×C3⋊C8—21no—
116#909—24no—
117#909—24no—
118#909—24no—
119#909—24no—
120#908—24no—
121#908—24no—
122#908—24no—
123#908—24no—
124C2×Dic12—22no—
125#904—28no—
126#904—28no—
127C2×D24—22no—
128C2×#204—22no—
129C2×#204—22no—
130#825—28no—
131#824—24no—
132#823—24no—
133#821—24no—
134(C2×D3)⋊C8—28no—
135#816—24no—
136C22×#41—31no—
137C22×#41—31no—
138#575—34no—
139#575—62no—
140#575—62no—
141#575—62no—
142C2×#137—61no—
143C2×#137—61no—
144C2×AGL(1,ℤ/8)—61no—
145C2×AGL(1,ℤ/8)—61no—
146C2×#135—62no—
147C22×M16—31no—
148C22×M16C2×D311yescharacteristic
149C22×M16C2×D311yescharacteristic
150C22×M16C2×D311yescharacteristic
151C2×#117—31no—
152#421—62no—
153#421—62no—
154C2×#104—62no—
155#419—61no—
156#418—61no—
157#417—62no—
158#416—62no—
159#413—62no—
160#413—62no—
161C2×#101—61no—
162C2×#100—61no—
163#408—62no—
164#408—62no—
165#407—62no—
166#344—62no—
167#341—22no—
168C24×C3#4111yescharacteristic
169C23×D3—21no—
170C23×D3—21no—
171C22×C3×C4C22×C411yescharacteristic
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4#4111yescharacteristic
183C22×C3×C4#4111yescharacteristic
184C22×C3×C4#4111yescharacteristic
185C2×#67—22no—
186C2×#67—22no—
187C2×#67—22no—
188C2×#67—22no—
189C2×#67—41no—
190C2×#67—41no—
191C2×#67—41no—
192C2×#67—41no—
193C2×#67—41no—
194C2×#67—41no—
195C2×#67—41no—
196C2×#67—41no—
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3—41no—
200#235—22no—
201#235—22no—
202#235—22no—
203#235—22no—
204#235—22no—
205#235—22no—
206#235—22no—
207#235—22no—
208#235—22no—
209#235—22no—
210#235—22no—
211#235—22no—
212#235—22no—
213#235—22no—
214#235—22no—
215#235—22no—
216#235—41no—
217#235—41no—
218#235—41no—
219#235—41no—
220#235—41no—
221#235—41no—
222#235—41no—
223#235—41no—
224#235—41no—
225#235—41no—
226#235—41no—
227#235—41no—
228#235—41no—
229#235—41no—
230#235—41no—
231#235—41no—
232C2×D12—21no—
233C2×D12—21no—
234C2×D12—21no—
235C2×D12—21no—
236C2×D12—21no—
237C2×D12—21no—
238C2×D12—21no—
239C2×D12—21no—
240C2×D12—21no—
241C2×D12—21no—
242C2×D12C22⋊C412yes—
243C2×D12C22⋊C412yes—
244C2×C4×D3—22no—
245C2×C4×D3—22no—
246C2×C4×D3—22no—
247C2×C4×D3—22no—
248C2×C4×D3—41no—
249C2×C4×D3—41no—
250C2×C4×D3—41no—
251C2×C4×D3—41no—
252C2×C4×D3—41no—
253C2×C4×D3—41no—
254C2×C4×D3—41no—
255C2×C4×D3—41no—
256C2×Dic6—21no—
257C2×Dic6—21no—
258C2×Dic6—21no—
259C2×Dic6—21no—
260C2×Dic6—21no—
261C2×Dic6—21no—
262C2×Dic6—21no—
263C2×Dic6—21no—
264C2×Dic6—21no—
265C2×Dic6—21no—
266C2×Dic6C22⋊C412yes—
267C2×Dic6C22⋊C412yes—
268C3×M16—22no—
269C3×M16—22no—
270C3×M16—22no—
271C3×M16—22no—
272C3×M16—22no—
273C3×M16—22no—
274C3×M16—22no—
275C3×M16—22no—
276C3×M16—41no—
277C3×M16—41no—
278C3×M16—41no—
279C3×M16—41no—
280C2×C3×C8—22no—
281C2×C3×C8—22no—
282C2×C3×C8—22no—
283C2×C3×C8—22no—
284C2×C3×C8—41no—
285C2×C3×C8—41no—
286C2×C3×C8—41no—
287C2×C3×C8—41no—
288C2×C3×C8C22⋊C414yes—
289#217—22no—
290#217—41no—
291C4⋊Dic3—22no—
292C4⋊Dic3—41no—
293C4×Dic3—22no—
294C4×Dic3—41no—
295#208—22no—
296#208—22no—
297#208—41no—
298#208—41no—
299C2×C3⋊C8—22no—
300C2×C3⋊C8—41no—
301C2×C3⋊C8—41no—
302Dic12—28no—
303D24—28no—
304#204—28no—
305#204—28no—
306C2×#41—32no—
307C2×#41—32no—
308C2×#41—32no—
309C2×#41—32no—
310C2×#41—61no—
311C2×#41—61no—
312C2×#41—61no—
313C2×#41—61no—
314C2×#41—61no—
315C2×#41—61no—
316C2×#41—61no—
317C2×#41—61no—
318C2×#41—61no—
319C2×#41—61no—
320C2×#41—61no—
321C2×#41—61no—
322C2×#41—61no—
323C2×#41—61no—
324C2×#41—61no—
325C2×#41—61no—
326C2×#41—61no—
327C2×#41—61no—
328C2×#41—61no—
329C2×#41—61no—
330C22×Q8—31no—
331C22×Q8—31no—
332C22×D4—31no—
333C22×D4—31no—
334C22×D4—61no—
335C22×D4—61no—
336C23×C4—61no—
337C23×C4—61no—
338C23×C4C22×D311yescharacteristic
339#137—64no—
340#137—64no—
341#137—64no—
342#137—64no—
343AGL(1,ℤ/8)—64no—
344AGL(1,ℤ/8)—64no—
345AGL(1,ℤ/8)—64no—
346AGL(1,ℤ/8)—64no—
347#135—68no—
348#135—68no—
349C2×Q16—62no—
350C2×SD16—62no—
351C2×SD16—62no—
352C2×D8—62no—
353C2×M16—21no—
354C2×M16—21no—
355C2×M16—21no—
356C2×M16—21no—
357C2×M16—21no—
358C2×M16—21no—
359C2×M16—21no—
360C2×M16—21no—
361C2×M16—21no—
362C2×M16—21no—
363C2×M16—21no—
364C2×M16—21no—
365C2×M16—21no—
366C2×M16—21no—
367C2×M16—32no—
368C2×M16—61no—
369C2×M16—61no—
370C2×M16—61no—
371C2×M16—61no—
372C2×M16—61no—
373C2×M16C4×D312yes—
374C2×M16D1212yes—
375C2×M16D1212yes—
376C2×M16D1212yes—
377C22×C8—21no—
378C22×C8—21no—
379C22×C8—61no—
380C22×C8#6712yes—
381#117—32no—
382#117—61no—
383#117—61no—
384#117—61no—
385C2×C4⋊C4—61no—
386C2×C22⋊C4—61no—
387C2×C42—61no—
388#104—68no—
389#103—64no—
390#102—64no—
391#101—64no—
392#100—64no—
393C22⋊C8—68no—
394C23×C3—81no—
395C23×C3C4×D411yescharacteristic
396C23×C3#12111yescharacteristic
397C23×C3#12111yescharacteristic
398C23×C3#12111yescharacteristic
399C23×C3#12311yescharacteristic
400C23×C3#12311yescharacteristic
401C23×C3#12411yescharacteristic
402C22×D3—22no—
403C22×D3—22no—
404C22×D3—41no—
405C22×D3—41no—
406C22×D3—41no—
407C22×D3—41no—
408C22×D3—82no—
409C22×D3—82no—
410C2×C3×C4C2×C22⋊C411yescharacteristic
411C2×C3×C4C2×C22⋊C411yescharacteristic
412C2×C3×C4C2×C22⋊C411yescharacteristic
413C2×C3×C4C4×D411yescharacteristic
414C2×C3×C4C4×D411yescharacteristic
415C2×C3×C4C4×D411yescharacteristic
416C2×C3×C4#12011yescharacteristic
417C2×C3×C4#12011yescharacteristic
418C2×C3×C4#12011yescharacteristic
419C2×C3×C4#12011yescharacteristic
420C2×C3×C4#12011yescharacteristic
421C2×C3×C4#12011yescharacteristic
422C2×C3×C4#12011yescharacteristic
423C2×C3×C4#12011yescharacteristic
424C2×C3×C4#12111yescharacteristic
425C2×C3×C4#12111yescharacteristic
426C2×C3×C4#12111yescharacteristic
427C2×C3×C4#12111yescharacteristic
428C2×C3×C4#12111yescharacteristic
429C2×C3×C4#12111yescharacteristic
430C2×C3×C4#12111yescharacteristic
431C2×C3×C4#12111yescharacteristic
432C2×C3×C4#12111yescharacteristic
433C2×C3×C4#12311yescharacteristic
434C2×C3×C4#12311yescharacteristic
435C2×C3×C4#12411yescharacteristic
436C2×C3×C4#12711yescharacteristic
437C2×C3×C4#12711yescharacteristic
438#67—42no—
439#67—42no—
440#67—42no—
441#67—42no—
442#67—42no—
443#67—42no—
444#67—42no—
445#67—42no—
446C2×Dic3—22no—
447C2×Dic3—22no—
448C2×Dic3—41no—
449C2×Dic3—41no—
450C2×Dic3—41no—
451C2×Dic3—41no—
452C2×Dic3—42no—
453C2×Dic3—82no—
454D12—22no—
455D12—22no—
456D12—22no—
457D12—22no—
458D12—41no—
459D12—41no—
460D12—41no—
461D12—41no—
462C4×D3—42no—
463C4×D3—42no—
464C4×D3—42no—
465C4×D3—42no—
466C4×D3—42no—
467C4×D3—42no—
468C4×D3—42no—
469C4×D3—42no—
470Dic6—22no—
471Dic6—22no—
472Dic6—22no—
473Dic6—22no—
474Dic6—41no—
475Dic6—41no—
476Dic6—41no—
477Dic6—41no—
478C3×C8—24no—
479C3×C8—42no—
480C3×C8—42no—
481C3⋊C8—42no—
482C24—61no—
483C24—61no—
484C24#23911yescharacteristic
485#41—62no—
486#41—62no—
487#41—62no—
488#41—62no—
489#41—62no—
490#41—62no—
491#41—62no—
492#41—62no—
493#41—62no—
494#41—62no—
495#41—62no—
496#41—62no—
497#41—62no—
498#41—62no—
499#41—62no—
500#41—62no—
501#41—121no—
502#41—121no—
503#41—121no—
504#41—121no—
505#41—121no—
506#41—121no—
507#41—121no—
508#41—121no—
509#41—121no—
510#41—121no—
511#41—121no—
512#41—121no—
513#41—121no—
514#41—121no—
515#41—121no—
516#41—121no—
517C2×Q8—32no—
518C2×Q8—32no—
519C2×Q8—61no—
520C2×Q8—61no—
521C2×Q8—61no—
522C2×Q8—61no—
523C2×Q8—61no—
524C2×Q8—61no—
525C2×Q8—61no—
526C2×Q8—61no—
527C2×Q8—61no—
528C2×Q8—61no—
529C2×D4—32no—
530C2×D4—32no—
531C2×D4—62no—
532C2×D4—62no—
533C2×D4—62no—
534C2×D4—62no—
535C2×D4—61no—
536C2×D4—61no—
537C2×D4—61no—
538C2×D4—61no—
539C2×D4—61no—
540C2×D4—61no—
541C2×D4—61no—
542C2×D4—61no—
543C2×D4—61no—
544C2×D4—61no—
545C2×D4—121no—
546C2×D4—121no—
547C2×D4—121no—
548C2×D4—121no—
549C2×D4—121no—
550C2×D4—121no—
551C2×D4—121no—
552C2×D4—121no—
553C22×C4—61no—
554C22×C4—61no—
555C22×C4—62no—
556C22×C4—62no—
557C22×C4—62no—
558C22×C4—62no—
559C22×C4—121no—
560C22×C4—121no—
561C22×C4—121no—
562C22×C4—121no—
563C22×C4—121no—
564C22×C4—121no—
565C22×C4—121no—
566C22×C4—121no—
567C22×C4—121no—
568C22×C4C2×C4×D311yescharacteristic
569C22×C4C2×D1211yescharacteristic
570C22×C4C2×D1211yescharacteristic
571C22×C4C2×D1211yescharacteristic
572C22×C4#23511yescharacteristic
573C22×C4#23511yescharacteristic
574C22×C4D3×D411yescharacteristic
575C22×C4D3×D411yescharacteristic
576C22×C4D3×D411yescharacteristic
577C22×C4D3×D411yescharacteristic
578C22×C4D3×D411yescharacteristic
579C22×C4D3×D411yescharacteristic
580C22×C4#23911yescharacteristic
581C22×C4C2×#6711yescharacteristic
582Q16—68no—
583SD16—68no—
584SD16—68no—
585D8—68no—
586M16—22no—
587M16—22no—
588M16—22no—
589M16—22no—
590M16—22no—
591M16—22no—
592M16—22no—
593M16—22no—
594M16—41no—
595M16—41no—
596M16—41no—
597M16—41no—
598M16—62no—
599M16—62no—
600M16—121no—
601M16—121no—
602C2×C8—22no—
603C2×C8—22no—
604C2×C8—22no—
605C2×C8—22no—
606C2×C8—41no—
607C2×C8—41no—
608C2×C8—41no—
609C2×C8—41no—
610C2×C8—62no—
611C2×C8—121no—
612C2×C8—121no—
613C2×C8(C2×D3)⋊C414yes—
614C4⋊C4—62no—
615C4⋊C4—121no—
616C22⋊C4—62no—
617C22⋊C4—121no—
618C42—62no—
619C42—121no—
620C22×C3—41no—
621C22×C3—41no—
622C22×C3—41no—
623C22×C3—41no—
624C22×C3—41no—
625C22×C3—41no—
626C22×C3—41no—
627C22×C3#38611yescharacteristic
628C22×C3#38611yescharacteristic
629C22×C3#39011yescharacteristic
630C22×C3#39211yescharacteristic
631C22×C3#39411yescharacteristic
632C22×C3#39411yescharacteristic
633C22×C3#39411yescharacteristic
634C2×D3—44no—
635C2×D3—42no—
636C2×D3—42no—
637C2×D3—82no—
638C2×D3—82no—
639C2×D3—82no—
640C2×D3—82no—
641C2×D3—82no—
642C3×C4#37911yescharacteristic
643C3×C4#38611yescharacteristic
644C3×C4#38611yescharacteristic
645C3×C4#39011yescharacteristic
646C3×C4#39211yescharacteristic
647C3×C4#39211yescharacteristic
648C3×C4#39211yescharacteristic
649C3×C4#39411yescharacteristic
650Dic3—42no—
651Dic3—42no—
652Dic3—82no—
653C23—62no—
654C23—62no—
655C23—81no—
656C23—121no—
657C23—121no—
658C23—121no—
659C23—121no—
660C23—242no—
661C23—242no—
662C23#89311yescharacteristic
663C23#89411yescharacteristic
664C23#89411yescharacteristic
665C23#89511yescharacteristic
666C23#89511yescharacteristic
667C23#89511yescharacteristic
668C23#94711yescharacteristic
669Q8—62no—
670Q8—62no—
671Q8—62no—
672Q8—62no—
673Q8—121no—
674Q8—121no—
675Q8—121no—
676Q8—121no—
677D4—62no—
678D4—62no—
679D4—62no—
680D4—62no—
681D4—122no—
682D4—122no—
683D4—122no—
684D4—122no—
685D4—122no—
686D4—122no—
687D4—122no—
688D4—122no—
689D4—121no—
690D4—121no—
691D4—121no—
692D4—121no—
693C2×C4—62no—
694C2×C4—62no—
695C2×C4—121no—
696C2×C4—121no—
697C2×C4—121no—
698C2×C4—121no—
699C2×C4—122no—
700C2×C4—122no—
701C2×C4—122no—
702C2×C4—122no—
703C2×C4—122no—
704C2×C4—122no—
705C2×C4—122no—
706C2×C4—122no—
707C2×C4—122no—
708C2×C4—242no—
709C2×C4C4×D1211yescharacteristic
710C2×C4C4×D1211yescharacteristic
711C2×C4#87411yescharacteristic
712C2×C4#87511yescharacteristic
713C2×C4D3×C22⋊C411yescharacteristic
714C2×C4D3×C22⋊C411yescharacteristic
715C2×C4#88211yescharacteristic
716C2×C4#88211yescharacteristic
717C2×C4#88211yescharacteristic
718C2×C4#88211yescharacteristic
719C2×C4#88211yescharacteristic
720C2×C4#88211yescharacteristic
721C2×C4#88411yescharacteristic
722C2×C4#88411yescharacteristic
723C2×C4#88411yescharacteristic
724C2×C4#88411yescharacteristic
725C2×C4#89311yescharacteristic
726C2×C4#89411yescharacteristic
727C2×C4#89411yescharacteristic
728C2×C4#89511yescharacteristic
729C2×C4#89511yescharacteristic
730C2×C4#89511yescharacteristic
731C2×C4C2×(C2×D3)⋊C411yescharacteristic
732C2×C4#93011yescharacteristic
733C2×C4#93011yescharacteristic
734C2×C4#93711yescharacteristic
735C2×C4#93711yescharacteristic
736C2×C4#94011yescharacteristic
737C8—24no—
738C8—42no—
739C8—42no—
740C8—122no—
741C2×C3—21no—
742C2×C3—21no—
743C2×C3—21no—
744C2×C3—21no—
745C2×C3—21no—
746C2×C3—21no—
747C2×C3—21no—
748C2×C3#143811yescharacteristic
749D3—82no—
750D3—82no—
751C22—41no—
752C22—41no—
753C22—41no—
754C22—41no—
755C22—41no—
756C22—41no—
757C22—41no—
758C22—124no—
759C22—122no—
760C22—122no—
761C22—242no—
762C22—242no—
763C22—242no—
764C22—242no—
765C22—242no—
766C22#466511yescharacteristic
767C22#466511yescharacteristic
768C22#467111yescharacteristic
769C22#467211yescharacteristic
770C22#498811yescharacteristic
771C22#499111yescharacteristic
772C22#499111yescharacteristic
773C4—122no—
774C4—122no—
775C4—242no—
776C4#466511yescharacteristic
777C4#466511yescharacteristic
778C4#467111yescharacteristic
779C4#467211yescharacteristic
780C4#493911yescharacteristic
781C4#495511yescharacteristic
782C4#496011yescharacteristic
783C4#496011yescharacteristic
784C3#1103711yescharacteristic, complemented, Sylow
785C2—21no—
786C2—21no—
787C2—21no—
788C2—21no—
789C2—21no—
790C2—21no—
791C2—21no—
792C2—242no—
793C2—242no—
794C2#6942611yescharacteristic
subgroupG/H
centerC4#4960
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1738C2
socleC2×C3#1438
prime radicalC2×#5748C2
cosocleC23×C3×C4C23
hypercenter#1738D3
hypocenterC3#11037
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C23×C4 ◁ #1738

Complements

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

NH ≅ G/Ntype
2. C2×#5139C2semidirect
3. #83768C2semidirect
4. C2×#5132C2semidirect
5. #83762C2semidirect
6. C3×#1738C2semidirect
7. C2×#4556C2semidirect
19. #5748C4semidirect
47. #1738D3semidirect
784. C3#11037semidirect

Maximal subgroups

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

subgroupindex
C2×#57482
C2×#51392
#837682
C2×#51322
#837622
C3×#17382
C2×#45562
#110373