Finite Group Databasebeta

← #11662462

Subgroups — #11662462

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×#536C211yesmaximal, characteristic, complemented
2#87268C211yesmaximal, characteristic
3C2×#5819C211yesmaximal, characteristic, complemented
4C22×#942C211yesmaximal, characteristic, complemented
5C2×#5225C211yesmaximal, characteristic, complemented
6#83400C218yesmaximal, complemented
7#83399C214yesmaximal, complemented
8#83398C214yesmaximal, complemented
9#82732C218yesmaximal, complemented
10#82213C211yesmaximal, characteristic, complemented
11#82207C211yesmaximal, characteristic, complemented
12#60455—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C23×C3×Q8C2211yescharacteristic
15C22×#235C2211yescharacteristic
16C3×#536—22no—
17C3×#536—22no—
18C3×#536C2214yes—
19C3×#536C2214yescomplemented
20C2×C3×C4⋊Q8—21no—
21C2×C3×#124—21no—
22C2×C3×#123—22no—
23C2×C3×C22⋊Q8C2214yescomplemented
24C2×C3×#117C2211yescharacteristic
25#5833C2214yes—
26#5832—22no—
27#5830C2212yes—
28#5828—22no—
29#5825C2212yes—
30#5819—28no—
31C2×#948C2214yes—
32C2×#948C2214yescomplemented
33#5816—21no—
34C22×#216C2212yescomplemented
35C2×#942—28no—
36C2×#942C2214yes—
37C2×#942C2214yescomplemented
38C22×#215C2212yescomplemented
39#5805—21no—
40#5804—22no—
41#5791C2214yes—
42C2×#926C2211yescharacteristic
43C22×#208C2211yescharacteristic
44C22×#208C2211yescharacteristic
45C22×#208C2214yescomplemented
46#5225C2214yescomplemented
47#5225C2214yescomplemented
48C2×#835C2214yescomplemented
49#5052C22116yescomplemented
50#5051C22116yescomplemented
51#5049C22116yescomplemented
52#5048C22116yescomplemented
53#5004C2212yescomplemented
54#5002C2214yescomplemented
55#5001C2212yescomplemented
56#4998C2214yescomplemented
57#4995C2214yescomplemented
58#4966C2214yes—
59#4964C2214yescomplemented
60C22×#137—31no—
61C2×#568—31no—
62#3449—31no—
63C2×#536D311yescharacteristic, complemented
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×Q8—21no—
78C22×C3×Q8—21no—
79C22×C3×Q8—21no—
80C22×C3×Q8—24no—
81C22×C3×Q8—24no—
82C22×C3×Q8D411yescharacteristic
83C22×C3×Q8C2312yes—
84C22×C3×Q8C2312yes—
85C22×C3×Q8C2312yes—
86C22×C3×D4D412yes—
87C22×C3×D4D411yescharacteristic
88C23×C3×C4—21no—
89C23×C3×C4C2312yes—
90C23×C3×C4C2311yescharacteristic
91C22×#67—21no—
92C2×#235—21no—
93C2×#235—21no—
94C2×#235—24no—
95C2×#235—24no—
96C2×#235C2312yes—
97C2×#235C2312yes—
98C22×D12C2311yescharacteristic
99C22×C4×D3—21no—
100C22×Dic6C2311yescharacteristic
101C3×C4⋊Q8—24no—
102C3×C4⋊Q8—24no—
103C3×#124—24no—
104C3×#124—24no—
105C3×#123—28no—
106C3×#123—28no—
107C3×C22⋊Q8—28no—
108C3×C22⋊Q8C23116yes—
109C3×#117—21no—
110C3×#117—21no—
111C3×#117C2312yes—
112C3×#117C2312yes—
113C2×C3×C4⋊C4—24no—
114C2×C3×C4⋊C4C2312yes—
115C2×C3×C22⋊C4—21no—
116C2×C3×C22⋊C4—24no—
117C2×C3×C42—21no—
118#953—24no—
119C2×#217—21no—
120#948—216no—
121#948—216no—
122#947—22no—
123#946—24no—
124#944—22no—
125C2×#216—28no—
126C2×#216C2318yes—
127#942—216no—
128#942—216no—
129C2×#215—28no—
130C2×#215C2318yes—
131#940—22no—
132#939—28no—
133#938—24no—
134#937—24no—
135#936—22no—
136#935—24no—
137#930C2318yes—
138#929—24no—
139C4×#67—28no—
140#926—21no—
141#926—21no—
142#926C2312yes—
143#926C2312yes—
144C2×C4⋊Dic3C2312yes—
145#924C2318yes—
146C2×C4×Dic3—21no—
147C2×#208—24no—
148C2×#208—216no—
149C2×#208D418yescomplemented
150C2×#208C2312yes—
151C2×#208C2312yes—
152C2×#208C2318yes—
153C2×#208C2312yes—
154C2×#208C2312yes—
155C22×C3⋊C8C2312yes—
156C22×C3⋊C8C2318yes—
157C22×C3⋊C8C2312yes—
158#835C23116yes—
159#810C2318yes—
160#809C2318yes—
161#808C2318yes—
162#807C2318yes—
163C22×#41—31no—
164C22×#41C2×D311yescharacteristic
165C23×Q8C2×D311yescharacteristic
166C2×#137—34no—
167C2×#137—34no—
168C2×#137—68no—
169C22×Q16—32no—
170C22×SD16—32no—
171#568—68no—
172C2×#131—34no—
173C2×#131—34no—
174C22×M16—31no—
175C22×M16—31no—
176C22×M16—34no—
177#537—34no—
178#536—22no—
179#536—22no—
180#536—32no—
181#536—61no—
182#536C2×D314yes—
183#536C2×D314yescomplemented
184#535—32no—
185#535—61no—
186#534—62no—
187#532—62no—
188C2×C4⋊Q8—21no—
189C2×#124—21no—
190#525—34no—
191#525—62no—
192C2×#123—22no—
193C2×C22⋊Q8C2×D314yescomplemented
194C2×#117—31no—
195C2×#117C2×D311yescharacteristic
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×Q8—22no—
221C2×C3×Q8—22no—
222C2×C3×Q8—22no—
223C2×C3×Q8—24no—
224C2×C3×Q8—24no—
225C2×C3×Q8—24no—
226C2×C3×Q8—24no—
227C2×C3×Q8—24no—
228C2×C3×Q8—24no—
229C2×C3×Q8—44no—
230C2×C3×Q8—44no—
231C2×C3×Q8—44no—
232C2×C3×Q8C2×D412yes—
233C2×C3×Q8C2×D412yes—
234C2×C3×Q8C2414yes—
235C2×C3×D4—28no—
236C2×C3×D4—24no—
237C2×C3×D4C2×D414yes—
238C2×C3×D4C2×D412yes—
239C2×C3×D4C2×D412yes—
240C2×C3×D4C2×D414yes—
241C22×C3×C4—24no—
242C22×C3×C4—24no—
243C22×C3×C4—22no—
244C22×C3×C4—22no—
245C22×C3×C4—22no—
246C22×C3×C4—22no—
247C22×C3×C4—44no—
248C22×C3×C4C2×D414yes—
249C22×C3×C4C2×D414yes—
250C22×C3×C4C2×D412yes—
251C22×C3×C4C2×D411yescharacteristic
252C22×C3×C4C2×D411yescharacteristic
253C22×C3×C4C2×D411yescharacteristic
254C22×C3×C4#4114yes—
255C22×C3×C4C2414yes—
256C22×C3×C4C2411yescharacteristic
257C22×C3×C4C2411yescharacteristic
258C22×C3×C4C2411yescharacteristic
259C2×#67—22no—
260C2×#67—22no—
261C2×#67—28no—
262C2×#67—44no—
263C22×Dic3—21no—
264C22×Dic3—22no—
265#235—24no—
266#235—24no—
267#235—24no—
268#235—24no—
269#235—44no—
270#235—44no—
271C2×D12—22no—
272C2×D12—22no—
273C2×D12—21no—
274C2×D12C2412yes—
275C2×C4×D3—22no—
276C2×C4×D3—22no—
277C2×C4×D3—44no—
278C2×Dic6—22no—
279C2×Dic6—22no—
280C2×Dic6—21no—
281C2×Dic6C2412yes—
282C3×C4⋊C4—216no—
283C3×C4⋊C4—22no—
284C3×C4⋊C4C2414yes—
285C3×C22⋊C4—22no—
286C3×C22⋊C4—22no—
287C3×C22⋊C4—216no—
288C3×C42—22no—
289C3×C42—22no—
290#217—28no—
291#217—22no—
292#217—22no—
293#216—216no—
294#215—216no—
295(C2×D3)⋊C4—28no—
296C4⋊Dic3—22no—
297C4⋊Dic3C2414yes—
298Dic3⋊C4—28no—
299C4×Dic3—22no—
300C4×Dic3—22no—
301#208—24no—
302#208—24no—
303#208—216no—
304#208C2×D418yes—
305#208C2×D418yes—
306C2×C3⋊C8—24no—
307C2×C3⋊C8—216no—
308C2×C3⋊C8C2×D418yes—
309C2×C3⋊C8C2414yes—
310C2×C3⋊C8C2414yes—
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×#41#6712yes—
320C2×#41#6712yes—
321C2×#41C22×D312yes—
322C2×#41C22×D312yes—
323C22×Q8—21no—
324C22×Q8—21no—
325C22×Q8—21no—
326C22×Q8—24no—
327C22×Q8—24no—
328C22×Q8—31no—
329C22×Q8#6711yescharacteristic
330C22×Q8C22×D312yes—
331C22×Q8C22×D312yes—
332C22×Q8C22×D312yes—
333C22×D4—31no—
334C22×D4—61no—
335C22×D4#6712yes—
336C22×D4#6711yescharacteristic
337C23×C4—21no—
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—616no—
348#131—616no—
349C2×M16—38no—
350C2×M16—32no—
351C2×M16—32no—
352C2×M16—32no—
353C2×M16—32no—
354C2×M16—38no—
355C2×M16—616no—
356C2×M16—64no—
357C22×C8—32no—
358C22×C8—32no—
359C22×C8—38no—
360C4⋊Q8—24no—
361C4⋊Q8—24no—
362C4⋊Q8—62no—
363#127—62no—
364#124—24no—
365#124—24no—
366#124—62no—
367#124—62no—
368#123—28no—
369#123—28no—
370#123—64no—
371#123—64no—
372C22⋊Q8—28no—
373C22⋊Q8—38no—
374C22⋊Q8—64no—
375C22⋊Q8C22×D3116yes—
376#121—38no—
377#121—64no—
378#121—68no—
379#120—64no—
380#120—64no—
381C4×D4—68no—
382#117—21no—
383#117—21no—
384#117—32no—
385#117—32no—
386#117—61no—
387#117—61no—
388#117C22×D312yes—
389#117C22×D312yes—
390C2×C4⋊C4—24no—
391C2×C4⋊C4—32no—
392C2×C4⋊C4C22×D312yes—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—24no—
395C2×C22⋊C4—61no—
396C2×C42—21no—
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×Q8—24no—
417C3×Q8—24no—
418C3×Q8—44no—
419C3×Q8—44no—
420C3×Q8—44no—
421C3×D4—28no—
422C3×D4—28no—
423C3×D4—24no—
424C3×D4—24no—
425C2×C3×C4—24no—
426C2×C3×C4—28no—
427C2×C3×C4—24no—
428C2×C3×C4—24no—
429C2×C3×C4—24no—
430C2×C3×C4—24no—
431C2×C3×C4—44no—
432C2×C3×C4—44no—
433C2×C3×C4—44no—
434C2×C3×C4—44no—
435C2×C3×C4#12114yes—
436C2×C3×C4#12114yes—
437C2×C3×C4#12114yes—
438C2×C3×C4C22×D412yes—
439C2×C3×C4C22×D412yes—
440C2×C3×C4C22×D412yes—
441C2×C3×C4C22×D414yes—
442C2×C3×C4C22×D411yescharacteristic
443C2×C3×C4C22×D411yescharacteristic
444C2×C3×C4C22×D411yescharacteristic
445C2×C3×C4C2×#4112yes—
446C2×C3×C4C2×#4112yes—
447C2×C3×C4C2×#4112yes—
448C2×C3×C4C2511yescharacteristic
449#67—44no—
450#67—44no—
451#67—88no—
452C2×Dic3—24no—
453C2×Dic3—22no—
454C2×Dic3—42no—
455C2×Dic3—82no—
456D12—22no—
457D12—22no—
458D12—42no—
459C4×D3—44no—
460C4×D3—44no—
461Dic6—22no—
462Dic6—22no—
463Dic6—42no—
464C3⋊C8—24no—
465C3⋊C8C22×D418yes—
466C24—61no—
467C24C2×#6712yes—
468C24C2×#6711yescharacteristic
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—22no—
483C2×Q8—22no—
484C2×Q8—22no—
485C2×Q8—24no—
486C2×Q8—24no—
487C2×Q8—24no—
488C2×Q8—24no—
489C2×Q8—24no—
490C2×Q8—24no—
491C2×Q8—32no—
492C2×Q8—44no—
493C2×Q8—44no—
494C2×Q8—44no—
495C2×Q8—61no—
496C2×Q8—62no—
497C2×Q8—62no—
498C2×Q8C2×#6712yes—
499C2×Q8C2×#6712yes—
500C2×Q8C23×D314yes—
501C2×D4—28no—
502C2×D4—24no—
503C2×D4—32no—
504C2×D4—62no—
505C2×D4—62no—
506C2×D4—68no—
507C2×D4—62no—
508C2×D4—62no—
509C2×D4—61no—
510C2×D4—124no—
511C2×D4C2×#6714yes—
512C2×D4C2×#6714yes—
513C2×D4C2×#6712yes—
514C2×D4C2×#6712yes—
515C22×C4—24no—
516C22×C4—24no—
517C22×C4—22no—
518C22×C4—22no—
519C22×C4—22no—
520C22×C4—22no—
521C22×C4—44no—
522C22×C4—61no—
523C22×C4—62no—
524C22×C4—62no—
525C22×C4—62no—
526C22×C4—124no—
527C22×C4D3×D414yes—
528C22×C4#23714yes—
529C22×C4C2×#6712yes—
530C22×C4C2×#6714yes—
531C22×C4C2×#6711yescharacteristic
532C22×C4C2×#6711yescharacteristic
533C22×C4C2×#6711yescharacteristic
534C22×C4C23×D314yes—
535C22×C4C23×D311yescharacteristic
536C22×C4C23×D311yescharacteristic
537C22×C4C23×D311yescharacteristic
538Q16—616no—
539SD16—616no—
540M16—38no—
541M16—38no—
542M16—616no—
543M16—64no—
544M16—64no—
545C2×C8—38no—
546C2×C8—34no—
547C2×C8—34no—
548C2×C8—616no—
549C2×C8—64no—
550C4⋊C4—216no—
551C4⋊C4—22no—
552C4⋊C4—34no—
553C4⋊C4—62no—
554C4⋊C4—68no—
555C4⋊C4C23×D314yes—
556C22⋊C4—22no—
557C22⋊C4—22no—
558C22⋊C4—216no—
559C22⋊C4—62no—
560C22⋊C4—62no—
561C22⋊C4—68no—
562C22⋊C4—68no—
563C42—22no—
564C42—22no—
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×C4—44no—
583C3×C4—44no—
584C3×C4C2×#12112yes—
585C3×C4C2×#12112yes—
586C3×C4C2×#12112yes—
587C3×C4C23×D411yescharacteristic
588C3×C4C22×#4111yescharacteristic
589Dic3—44no—
590Dic3—42no—
591C23—24no—
592C23—42no—
593C23—44no—
594C23—62no—
595C23—122no—
596C23—242no—
597C23#93914yes—
598C23C22×#6714yes—
599C23C22×#6711yescharacteristic
600C23C22×#6711yescharacteristic
601C23C22×#6711yescharacteristic
602Q8—24no—
603Q8—24no—
604Q8—24no—
605Q8—24no—
606Q8—44no—
607Q8—44no—
608Q8—44no—
609Q8—62no—
610Q8—62no—
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—24no—
627C2×C4—24no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—64no—
633C2×C4—62no—
634C2×C4—122no—
635C2×C4—124no—
636C2×C4—124no—
637C2×C4—242no—
638C2×C4#93914yes—
639C2×C4#93914yes—
640C2×C4#93914yes—
641C2×C4C2×D3×D412yes—
642C2×C4C2×D3×D412yes—
643C2×C4C2×D3×D412yes—
644C2×C4C2×#23712yes—
645C2×C4C2×#23712yes—
646C2×C4C2×#23712yes—
647C2×C4C22×#6714yes—
648C2×C4C22×#6711yescharacteristic
649C2×C4C22×#6711yescharacteristic
650C2×C4C22×#6711yescharacteristic
651C2×C4C24×D311yescharacteristic
652C8—38no—
653C8—64no—
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—24no—
665C22—28no—
666C22—44no—
667C22—84no—
668C22—122no—
669C22—124no—
670C22—244no—
671C22C2×#93912yes—
672C22C2×#93912yes—
673C22C2×#93912yes—
674C22C23×#6711yescharacteristic
675C4—24no—
676C4—44no—
677C4—44no—
678C4—122no—
679C4—124no—
680C4C2×#93912yes—
681C4C2×#93912yes—
682C4C2×#93912yes—
683C4C22×D3×D411yescharacteristic
684C4C22×#23711yescharacteristic
685C3#6045511yescharacteristic, complemented, Sylow
686C2—22no—
687C2—22no—
688C2—22no—
689C2—44no—
690C2—242no—
691C2C22×#93911yescharacteristic
692C2#8728712yes—
subgroupG/H
centerC22C23×#67
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#536C2
socleC22×C3C23×D4
prime radical#87268C2
hypercenterC2×#536D3
hypocenterC3#60455
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×#41 ◁ C2×#536

Complements

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

NH ≅ G/Ntype
1. C2×C3×#536C2semidirect
3. C2×#5819C2semidirect
4. C22×#942C2semidirect
5. C2×#5225C2semidirect
6. #83400C2semidirect
7. #83399C2semidirect
8. #83398C2semidirect
9. #82732C2semidirect
10. #82213C2semidirect
11. #82207C2semidirect
19. C3×#536C22semidirect
23. C2×C3×C22⋊Q8C22semidirect
32. C2×#948C22semidirect
34. C22×#216C22semidirect
37. C2×#942C22semidirect
38. C22×#215C22semidirect
45. C22×#208C22semidirect
46. #5225C22semidirect
47. #5225C22semidirect
48. C2×#835C22semidirect
49. #5052C22semidirect
50. #5051C22semidirect
51. #5049C22semidirect
52. #5048C22semidirect
53. #5004C22semidirect
54. #5002C22semidirect
55. #5001C22semidirect
56. #4998C22semidirect
57. #4995C22semidirect
59. #4964C22semidirect
63. C2×#536D3semidirect
149. C2×#208D4semidirect
183. #536C2×D3semidirect
193. C2×C22⋊Q8C2×D3semidirect
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×#5362
#872682
C2×#58192
C22×#9422
C2×#52252
#834002
#833992
#833982
#827322
#822132
#822072
#604553