Finite Group Databasebeta

← #11662465

Subgroups — #11662465

Order 768.

2678 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×C3×#568C211yesmaximal, characteristic, complemented
2#87268C211yesmaximal, characteristic
3C2×#5825C211yesmaximal, characteristic, complemented
4C22×#909C211yesmaximal, characteristic, complemented
5#83808C218yesmaximal, complemented
6#83807C214yesmaximal, complemented
7#83798C214yesmaximal, complemented
8#83776C218yesmaximal, complemented
9C2×#5120C211yesmaximal, characteristic, complemented
10#83768C211yesmaximal, characteristic, complemented
11#83752C211yesmaximal, characteristic, complemented
12#60455—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C22×#235C2211yescharacteristic
15C23×Dic6C2211yescharacteristic
16C3×#568—28no—
17C2×C3×#131C2214yes—
18C2×C3×#131C2214yescomplemented
19C22×C3×M16C2211yescharacteristic
20C22×C3×M16C2211yescharacteristic
21C22×C3×M16C2214yescomplemented
22#5833C2214yes—
23#5832—22no—
24#5830C2212yes—
25#5828—22no—
26#5825—22no—
27#5825—22no—
28#5825C2214yes—
29#5825C2212yes—
30#5825C2214yescomplemented
31#5816—21no—
32C2×#944—21no—
33#5805—21no—
34#5804—22no—
35C2×#936—21no—
36C2×#935—22no—
37#5791C2214yes—
38C2×#926C2211yescharacteristic
39C2×#926C2211yescharacteristic
40C2×#924C2214yescomplemented
41C2×#909—28no—
42C2×#909C2214yes—
43C2×#909C2214yescomplemented
44C22×Dic12C2212yescomplemented
45C22×#204C2212yescomplemented
46#5137C22116yescomplemented
47#5134C22116yescomplemented
48#5133C2214yes—
49#5131C2212yescomplemented
50#5121C2214yescomplemented
51#5121C2214yescomplemented
52#5120C2212yescomplemented
53#5120C2214yescomplemented
54#5120C2214yescomplemented
55#5116C22116yescomplemented
56#5114C22116yescomplemented
57#5113C2214yescomplemented
58#5106C2214yescomplemented
59C2×#816C2214yescomplemented
60C22×#137—31no—
61C2×#568D311yescharacteristic, complemented
62#3449—31no—
63C2×#536—31no—
64#3063—38no—
65#3059—34no—
66#3057—34no—
67#3056—38no—
68#2911—31no—
69#2896—31no—
70C2×#419—31no—
71C2×C3×#41—24no—
72C2×C3×#41—24no—
73C2×C3×#41D412yes—
74C2×C3×#41D412yes—
75C2×C3×#41C2312yes—
76C2×C3×#41C2312yes—
77C22×C3×Q8D411yescharacteristic
78C22×C3×D4D412yes—
79C22×C3×D4D411yescharacteristic
80C23×C3×C4C2312yes—
81C23×C3×C4C2311yescharacteristic
82C22×#67—21no—
83C23×Dic3—21no—
84C2×#235—21no—
85C2×#235—21no—
86C2×#235—24no—
87C2×#235—24no—
88C2×#235C2312yes—
89C2×#235C2312yes—
90C22×D12C2311yescharacteristic
91C22×C4×D3—21no—
92C22×Dic6—24no—
93C22×Dic6—24no—
94C22×Dic6—21no—
95C22×Dic6—21no—
96C22×Dic6—21no—
97C22×Dic6C2312yes—
98C22×Dic6C2312yes—
99C22×Dic6C2311yescharacteristic
100C22×Dic6C2312yes—
101C3×#131—216no—
102C3×#131—216no—
103C2×C3×M16—24no—
104C2×C3×M16—216no—
105C2×C3×M16D418yescomplemented
106C2×C3×M16C2312yes—
107C2×C3×M16C2312yes—
108C2×C3×M16C2318yes—
109C2×C3×M16C2312yes—
110C2×C3×M16C2312yes—
111C22×C3×C8C2312yes—
112C22×C3×C8C2318yes—
113C22×C3×C8C2312yes—
114#953—24no—
115C2×#217—24no—
116C2×#217—21no—
117C2×#217—21no—
118#947—22no—
119#946—24no—
120#944—24no—
121#944—24no—
122#944—22no—
123#940—22no—
124#939—28no—
125#938—24no—
126#937—24no—
127#936—24no—
128#936—24no—
129#936—22no—
130#935—28no—
131#935—28no—
132#935—24no—
133#930C2318yes—
134#929—24no—
135C4×#67—28no—
136#926—21no—
137#926—21no—
138#926—21no—
139#926—21no—
140#926C2312yes—
141#926C2312yes—
142#926C2312yes—
143#926C2312yes—
144C2×C4⋊Dic3C2312yes—
145C2×C4⋊Dic3C2312yes—
146#924—28no—
147#924C23116yes—
148#924C2318yes—
149C2×Dic3⋊C4—24no—
150C2×C4×Dic3—21no—
151C2×C4×Dic3—21no—
152#909—216no—
153#909—216no—
154C2×Dic12—28no—
155C2×Dic12C2318yes—
156C2×#204—28no—
157C2×#204C2318yes—
158#821C2318yes—
159#818C2318yes—
160#817C2318yes—
161#816C23116yes—
162#816C2318yes—
163C22×#41—31no—
164C22×#41C2×D311yescharacteristic
165C23×Q8—31no—
166C2×#137—34no—
167C2×#137—34no—
168C2×#137—68no—
169C22×Q16—32no—
170C22×SD16—32no—
171#568—28no—
172C2×#131C2×D314yes—
173C2×#131C2×D314yescomplemented
174C22×M16C2×D311yescharacteristic
175C22×M16C2×D311yescharacteristic
176C22×M16C2×D314yescomplemented
177#537—34no—
178#536—34no—
179#536—34no—
180#536—32no—
181#536—62no—
182#536—62no—
183#536—61no—
184#535—32no—
185#535—61no—
186#534—62no—
187#532—62no—
188C2×C4⋊Q8—61no—
189C2×#124—61no—
190#525—34no—
191#525—62no—
192C2×#123—62no—
193C2×C22⋊Q8—34no—
194C2×#117—31no—
195C2×#117—31no—
196#470—316no—
197#468—316no—
198#467—316no—
199#465—316no—
200#428—34no—
201#428—34no—
202#427—34no—
203#419—34no—
204#419—34no—
205#419—32no—
206#418—32no—
207#417—34no—
208#416—34no—
209C2×#103—34no—
210C24×C3C2×D412yes—
211C24×C3C2×D411yescharacteristic
212C23×D3—21no—
213C3×#41—28no—
214C3×#41—28no—
215C3×#41—24no—
216C3×#41—24no—
217C3×#41—24no—
218C3×#41—24no—
219C2×C3×Q8—24no—
220C2×C3×Q8C2×D412yes—
221C2×C3×Q8C2×D412yes—
222C2×C3×D4—28no—
223C2×C3×D4—24no—
224C2×C3×D4C2×D414yes—
225C2×C3×D4C2×D414yes—
226C2×C3×D4C2×D412yes—
227C2×C3×D4C2×D412yes—
228C22×C3×C4—24no—
229C22×C3×C4—24no—
230C22×C3×C4C2×D412yes—
231C22×C3×C4C2×D414yes—
232C22×C3×C4C2×D411yescharacteristic
233C22×C3×C4C2×D411yescharacteristic
234C22×C3×C4C2×D411yescharacteristic
235C22×C3×C4C2×D414yes—
236C22×C3×C4#4114yes—
237C22×C3×C4C2414yes—
238C22×C3×C4C2411yescharacteristic
239C22×C3×C4C2411yescharacteristic
240C22×C3×C4C2411yescharacteristic
241C2×#67—22no—
242C2×#67—22no—
243C2×#67—28no—
244C2×#67—44no—
245C22×Dic3—22no—
246C22×Dic3—22no—
247C22×Dic3—22no—
248C22×Dic3—22no—
249C22×Dic3—21no—
250C22×Dic3—22no—
251C22×Dic3—44no—
252#235—24no—
253#235—24no—
254#235—24no—
255#235—24no—
256#235—44no—
257#235—44no—
258C2×D12—22no—
259C2×D12—22no—
260C2×D12—21no—
261C2×D12C2412yes—
262C2×C4×D3—22no—
263C2×C4×D3—22no—
264C2×C4×D3—44no—
265C2×Dic6—24no—
266C2×Dic6—22no—
267C2×Dic6—24no—
268C2×Dic6—24no—
269C2×Dic6—24no—
270C2×Dic6—24no—
271C2×Dic6—24no—
272C2×Dic6—22no—
273C2×Dic6—22no—
274C2×Dic6—22no—
275C2×Dic6—22no—
276C2×Dic6—21no—
277C2×Dic6—44no—
278C2×Dic6—44no—
279C2×Dic6—44no—
280C2×Dic6C2414yes—
281C2×Dic6C2412yes—
282C3×M16—24no—
283C3×M16—24no—
284C3×M16—216no—
285C3×M16C2×D418yes—
286C3×M16C2×D418yes—
287C2×C3×C8—24no—
288C2×C3×C8—216no—
289C2×C3×C8C2×D418yes—
290C2×C3×C8C2414yes—
291C2×C3×C8C2414yes—
292#217—216no—
293#217—28no—
294#217—22no—
295#217—22no—
296#217—22no—
297#217—22no—
298(C2×D3)⋊C4—28no—
299C4⋊Dic3—22no—
300C4⋊Dic3—22no—
301C4⋊Dic3C2414yes—
302C4⋊Dic3C2414yes—
303Dic3⋊C4—216no—
304Dic3⋊C4—28no—
305C4×Dic3—22no—
306C4×Dic3—22no—
307C4×Dic3—22no—
308C4×Dic3—22no—
309Dic12—216no—
310#204—216no—
311C2×#41—24no—
312C2×#41—24no—
313C2×#41—32no—
314C2×#41—32no—
315C2×#41—61no—
316C2×#41—61no—
317C2×#41—64no—
318C2×#41—64no—
319C2×#41D1212yes—
320C2×#41D1212yes—
321C2×#41C22×D312yes—
322C2×#41C22×D312yes—
323C22×Q8—32no—
324C22×Q8—32no—
325C22×Q8—31no—
326C22×Q8—32no—
327C22×Q8—61no—
328C22×Q8—61no—
329C22×Q8—61no—
330C22×Q8—64no—
331C22×Q8—64no—
332C22×Q8D1211yescharacteristic
333C22×D4—31no—
334C22×D4—61no—
335C22×D4D1212yes—
336C22×D4D1211yescharacteristic
337C23×C4—61no—
338C23×C4—61no—
339C23×C4C22×D312yes—
340C23×C4C22×D311yescharacteristic
341#137—616no—
342#137—616no—
343C2×Q16—38no—
344C2×Q16—68no—
345C2×SD16—38no—
346C2×SD16—68no—
347#131—216no—
348#131—216no—
349C2×M16—24no—
350C2×M16—216no—
351C2×M16#6718yescomplemented
352C2×M16C22×D312yes—
353C2×M16C22×D312yes—
354C2×M16C22×D318yes—
355C2×M16C22×D312yes—
356C2×M16C22×D312yes—
357C22×C8C22×D312yes—
358C22×C8C22×D318yes—
359C22×C8C22×D312yes—
360C4⋊Q8—64no—
361C4⋊Q8—64no—
362C4⋊Q8—62no—
363#127—62no—
364#124—62no—
365#124—64no—
366#124—64no—
367#124—62no—
368#123—68no—
369#123—68no—
370#123—64no—
371#123—64no—
372C22⋊Q8—316no—
373C22⋊Q8—38no—
374C22⋊Q8—64no—
375C22⋊Q8—68no—
376#121—38no—
377#121—64no—
378#121—68no—
379#120—64no—
380#120—64no—
381C4×D4—68no—
382#117—32no—
383#117—32no—
384#117—32no—
385#117—32no—
386#117—61no—
387#117—61no—
388#117—61no—
389#117—61no—
390C2×C4⋊C4—32no—
391C2×C4⋊C4—32no—
392C2×C4⋊C4—64no—
393C2×C22⋊C4—61no—
394C2×C22⋊C4—61no—
395C2×C22⋊C4—64no—
396C2×C42—61no—
397C2×C42—61no—
398#107—38no—
399#106—38no—
400#103—316no—
401#103—38no—
402#102—38no—
403C23×C3—24no—
404C23×C3—42no—
405C23×C3—44no—
406C23×C3#12114yes—
407C23×C3C22×D414yes—
408C23×C3C22×D411yescharacteristic
409C23×C3C22×D411yescharacteristic
410C23×C3C22×D411yescharacteristic
411C22×D3—22no—
412C22×D3—42no—
413C22×D3—82no—
414C3×Q8—24no—
415C3×Q8—24no—
416C3×D4—28no—
417C3×D4—28no—
418C3×D4—24no—
419C3×D4—24no—
420C2×C3×C4—24no—
421C2×C3×C4—28no—
422C2×C3×C4—24no—
423C2×C3×C4—24no—
424C2×C3×C4#12114yes—
425C2×C3×C4#12114yes—
426C2×C3×C4#12114yes—
427C2×C3×C4C22×D414yes—
428C2×C3×C4C22×D411yescharacteristic
429C2×C3×C4C22×D411yescharacteristic
430C2×C3×C4C22×D411yescharacteristic
431C2×C3×C4C22×D412yes—
432C2×C3×C4C22×D412yes—
433C2×C3×C4C22×D412yes—
434C2×C3×C4C2×#4112yes—
435C2×C3×C4C2×#4112yes—
436C2×C3×C4C2×#4112yes—
437C2×C3×C4C2511yescharacteristic
438#67—44no—
439#67—44no—
440#67—88no—
441C2×Dic3—24no—
442C2×Dic3—24no—
443C2×Dic3—24no—
444C2×Dic3—22no—
445C2×Dic3—44no—
446C2×Dic3—44no—
447C2×Dic3—44no—
448C2×Dic3—44no—
449C2×Dic3—42no—
450C2×Dic3—82no—
451D12—22no—
452D12—22no—
453D12—42no—
454C4×D3—44no—
455C4×D3—44no—
456Dic6—24no—
457Dic6—22no—
458Dic6—24no—
459Dic6—22no—
460Dic6—44no—
461Dic6—44no—
462Dic6—44no—
463Dic6—42no—
464C3×C8—24no—
465C3×C8C22×D418yes—
466C24—61no—
467C24C2×D1212yes—
468C24C2×D1211yescharacteristic
469#41—28no—
470#41—28no—
471#41—24no—
472#41—24no—
473#41—24no—
474#41—24no—
475#41—64no—
476#41—64no—
477#41—64no—
478#41—64no—
479#41—124no—
480#41—124no—
481C2×Q8—24no—
482C2×Q8—34no—
483C2×Q8—32no—
484C2×Q8—62no—
485C2×Q8—62no—
486C2×Q8—62no—
487C2×Q8—61no—
488C2×Q8—64no—
489C2×Q8—64no—
490C2×Q8—64no—
491C2×Q8—62no—
492C2×Q8—64no—
493C2×Q8—62no—
494C2×Q8—64no—
495C2×Q8—64no—
496C2×Q8—124no—
497C2×Q8—124no—
498C2×Q8—124no—
499C2×Q8C2×D1212yes—
500C2×Q8C2×D1212yes—
501C2×D4—28no—
502C2×D4—24no—
503C2×D4—32no—
504C2×D4—62no—
505C2×D4—62no—
506C2×D4—68no—
507C2×D4—61no—
508C2×D4—62no—
509C2×D4—62no—
510C2×D4—124no—
511C2×D4C2×D1214yes—
512C2×D4C2×D1214yes—
513C2×D4C2×D1212yes—
514C2×D4C2×D1212yes—
515C22×C4—24no—
516C22×C4—24no—
517C22×C4—62no—
518C22×C4—62no—
519C22×C4—61no—
520C22×C4—62no—
521C22×C4—62no—
522C22×C4—62no—
523C22×C4—62no—
524C22×C4—62no—
525C22×C4—124no—
526C22×C4—124no—
527C22×C4C2×D1212yes—
528C22×C4C2×D1214yes—
529C22×C4C2×D1211yescharacteristic
530C22×C4C2×D1211yescharacteristic
531C22×C4C2×D1211yescharacteristic
532C22×C4#23514yes—
533C22×C4C2×#6714yes—
534C22×C4C23×D314yes—
535C22×C4C23×D311yescharacteristic
536C22×C4C23×D311yescharacteristic
537C22×C4C23×D311yescharacteristic
538Q16—616no—
539SD16—616no—
540M16—24no—
541M16—24no—
542M16—216no—
543M16C2×#6718yes—
544M16C2×#6718yes—
545C2×C8—24no—
546C2×C8—216no—
547C2×C8C2×#6718yes—
548C2×C8C23×D314yes—
549C2×C8C23×D314yes—
550C4⋊C4—34no—
551C4⋊C4—34no—
552C4⋊C4—62no—
553C4⋊C4—62no—
554C4⋊C4—616no—
555C4⋊C4—68no—
556C22⋊C4—62no—
557C22⋊C4—62no—
558C22⋊C4—62no—
559C22⋊C4—62no—
560C22⋊C4—616no—
561C22⋊C4—68no—
562C22⋊C4—68no—
563C42—62no—
564C42—62no—
565C42—62no—
566C42—62no—
567C22×C3—22no—
568C22×C3—22no—
569C22×C3—22no—
570C22×C3—28no—
571C22×C3—24no—
572C22×C3—44no—
573C22×C3—84no—
574C22×C3C2×#12112yes—
575C22×C3C2×#12112yes—
576C22×C3C2×#12112yes—
577C22×C3C23×D411yescharacteristic
578C2×D3—44no—
579C2×D3—42no—
580C2×D3—84no—
581C3×C4—24no—
582C3×C4C2×#12112yes—
583C3×C4C2×#12112yes—
584C3×C4C2×#12112yes—
585C3×C4C23×D411yescharacteristic
586C3×C4C22×#4111yescharacteristic
587Dic3—44no—
588Dic3—44no—
589Dic3—44no—
590Dic3—42no—
591C23—24no—
592C23—42no—
593C23—44no—
594C23—62no—
595C23—122no—
596C23—242no—
597C23#93014yes—
598C23C22×D1214yes—
599C23C22×D1211yescharacteristic
600C23C22×D1211yescharacteristic
601C23C22×D1211yescharacteristic
602Q8—24no—
603Q8—24no—
604Q8—64no—
605Q8—62no—
606Q8—64no—
607Q8—62no—
608Q8—124no—
609Q8—124no—
610Q8—124no—
611Q8—122no—
612D4—28no—
613D4—28no—
614D4—24no—
615D4—24no—
616D4—62no—
617D4—62no—
618D4—124no—
619D4—124no—
620D4—122no—
621D4—248no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—28no—
626C2×C4—64no—
627C2×C4—64no—
628C2×C4—62no—
629C2×C4—64no—
630C2×C4—124no—
631C2×C4—122no—
632C2×C4—124no—
633C2×C4—124no—
634C2×C4—124no—
635C2×C4—124no—
636C2×C4—124no—
637C2×C4—242no—
638C2×C4#93014yes—
639C2×C4#93014yes—
640C2×C4#93014yes—
641C2×C4C22×D1214yes—
642C2×C4C22×D1211yescharacteristic
643C2×C4C22×D1211yescharacteristic
644C2×C4C22×D1211yescharacteristic
645C2×C4C2×#23512yes—
646C2×C4C2×#23512yes—
647C2×C4C2×#23512yes—
648C2×C4C22×#6712yes—
649C2×C4C22×#6712yes—
650C2×C4C22×#6712yes—
651C2×C4C24×D311yescharacteristic
652C8—24no—
653C8C22×#6718yes—
654C2×C3—22no—
655C2×C3—22no—
656C2×C3—22no—
657C2×C3—44no—
658C2×C3C22×#12111yescharacteristic
659C2×C3#358612yes—
660D3—82no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—28no—
665C22—24no—
666C22—44no—
667C22—84no—
668C22—124no—
669C22—122no—
670C22—244no—
671C22C2×#93012yes—
672C22C2×#93012yes—
673C22C2×#93012yes—
674C22C23×D1211yescharacteristic
675C4—24no—
676C4—124no—
677C4—124no—
678C4—122no—
679C4—124no—
680C4C2×#93012yes—
681C4C2×#93012yes—
682C4C2×#93012yes—
683C4C22×#23511yescharacteristic
684C4C23×#6711yescharacteristic
685C3#6045511yescharacteristic, complemented, Sylow
686C2—22no—
687C2—22no—
688C2—22no—
689C2—44no—
690C2—242no—
691C2#8716812yes—
692C2C22×#93011yescharacteristic
subgroupG/H
centerC22C23×D12
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#568C2
socleC22×C3C23×D4
prime radical#87268C2
hypercenterC2×#568D3
hypocenterC3#60455
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×#41 ◁ C2×#568

Complements

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

NH ≅ G/Ntype
1. C2×C3×#568C2semidirect
3. C2×#5825C2semidirect
4. C22×#909C2semidirect
5. #83808C2semidirect
6. #83807C2semidirect
7. #83798C2semidirect
8. #83776C2semidirect
9. C2×#5120C2semidirect
10. #83768C2semidirect
11. #83752C2semidirect
18. C2×C3×#131C22semidirect
21. C22×C3×M16C22semidirect
30. #5825C22semidirect
40. C2×#924C22semidirect
43. C2×#909C22semidirect
44. C22×Dic12C22semidirect
45. C22×#204C22semidirect
46. #5137C22semidirect
47. #5134C22semidirect
49. #5131C22semidirect
50. #5121C22semidirect
51. #5121C22semidirect
52. #5120C22semidirect
53. #5120C22semidirect
54. #5120C22semidirect
55. #5116C22semidirect
56. #5114C22semidirect
57. #5113C22semidirect
58. #5106C22semidirect
59. C2×#816C22semidirect
61. C2×#568D3semidirect
105. C2×C3×M16D4semidirect
173. C2×#131C2×D3semidirect
176. C22×M16C2×D3semidirect
351. C2×M16#67semidirect
685. C3#60455semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5682
#872682
C2×#58252
C22×#9092
#838082
#838072
#837982
#837762
C2×#51202
#837682
#837522
#604553