Finite Group Databasebeta

← #11662155

Subgroups — #11662155

Order 768.

2674 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#137C211yesmaximal, characteristic, complemented
2#87268C211yesmaximal, characteristic
3C2×#5812C211yesmaximal, characteristic, complemented
4C22×#942C211yesmaximal, characteristic, complemented
5#84072C211yesmaximal, characteristic, complemented
6#83970C214yesmaximal, complemented
7#83969C214yesmaximal, complemented
8#83966C218yesmaximal, complemented
9#83877C214yesmaximal, complemented
10#83849C214yesmaximal, complemented
11#83768C211yesmaximal, characteristic, complemented
12#83750C211yesmaximal, characteristic, complemented
13#60424—31nomaximal, Sylow
14C22×C3×#41C2211yescharacteristic
15C23×C3×Q8C2211yescharacteristic
16C22×#235C2211yescharacteristic
17C2×C3×#137—28no—
18C2×C3×#137C2214yes—
19C2×C3×#137C2214yescomplemented
20C22×C3×Q16C2212yescomplemented
21C22×C3×SD16C2212yescomplemented
22C22×C3×M16C2211yescharacteristic
23#5833—22no—
24#5832C2214yes—
25#5830C2212yes—
26#5828—22no—
27#5825C2212yes—
28#5816C2212yes—
29#5812—22no—
30#5812—22no—
31#5812C2214yescomplemented
32#5812C2214yescomplemented
33C2×Q8×Dic3—22no—
34C2×Q8×Dic3C2214yescomplemented
35C22×#216C2212yescomplemented
36C2×#942—28no—
37C2×#942C2214yes—
38C2×#942C2214yescomplemented
39C22×#215C2212yescomplemented
40#5805C2212yes—
41#5804—22no—
42#5791—22no—
43C2×#926—22no—
44C2×#926C2211yescharacteristic
45C2×#926C2211yescharacteristic
46C22×#208C2211yescharacteristic
47#5233C2214yes—
48#5232C2214yescomplemented
49#5225C2212yescomplemented
50#5215C2212yescomplemented
51#5185C22116yescomplemented
52#5182C22116yescomplemented
53#5166C22116yescomplemented
54#5163C22116yescomplemented
55#5133C2214yescomplemented
56#5131C2212yescomplemented
57#5120C2212yescomplemented
58#5119C2214yescomplemented
59#5118C2214yescomplemented
60#5113C2214yes—
61#5102C2214yescomplemented
62C22×#137—31no—
63C22×#137D311yescharacteristic, complemented
64#3449—31no—
65C2×#519—31no—
66#3044—34no—
67#3044—34no—
68#3043—34no—
69#3043—34no—
70#3037—38no—
71#2905—31no—
72#2896—31no—
73#2896—31no—
74C2×C3×#41—21no—
75C2×C3×#41—21no—
76C2×C3×#41—24no—
77C2×C3×#41—24no—
78C2×C3×#41C2312yes—
79C2×C3×#41C2312yes—
80C22×C3×Q8—21no—
81C22×C3×Q8—21no—
82C22×C3×Q8—21no—
83C22×C3×Q8—24no—
84C22×C3×Q8D418yescomplemented
85C22×C3×Q8C2311yescharacteristic
86C22×C3×Q8C2312yes—
87C22×C3×Q8C2312yes—
88C22×C3×Q8C2312yes—
89C22×C3×D4—21no—
90C22×C3×D4C2311yescharacteristic
91C23×C3×C4—21no—
92C23×C3×C4—21no—
93C23×C3×C4C2311yescharacteristic
94C22×#67—21no—
95C2×#235—21no—
96C2×#235—21no—
97C2×#235—24no—
98C2×#235—24no—
99C2×#235C2312yes—
100C2×#235C2312yes—
101C22×D12C2311yescharacteristic
102C22×C4×D3—21no—
103C22×Dic6C2311yescharacteristic
104C3×#137—216no—
105C3×#137—216no—
106C2×C3×Q16—28no—
107C2×C3×Q16C2318yes—
108C2×C3×SD16—28no—
109C2×C3×SD16C2318yes—
110C2×C3×M16—24no—
111C2×C3×M16C2312yes—
112C2×C3×M16C2312yes—
113C22×C3×C8C2312yes—
114#953—24no—
115C2×#217—21no—
116C2×#217D412yes—
117#947C2314yes—
118#946—24no—
119Q8×Dic3—28no—
120Q8×Dic3—28no—
121Q8×Dic3—28no—
122Q8×Dic3C23116yes—
123#944C2314yes—
124C2×#216—28no—
125C2×#216C2318yes—
126#942—216no—
127#942—216no—
128C2×#215—28no—
129C2×#215C2318yes—
130#940C2314yes—
131#939—28no—
132#938—24no—
133#937—24no—
134#936C2314yes—
135#935—24no—
136#930—24no—
137#929—24no—
138C4×#67—28no—
139#926—21no—
140#926—21no—
141#926—28no—
142#926—28no—
143#926D412yes—
144#926D412yes—
145#926C2312yes—
146#926C2312yes—
147#926C2312yes—
148#926C2312yes—
149C2×C4⋊Dic3—24no—
150C2×C4⋊Dic3D412yes—
151C2×C4⋊Dic3C2312yes—
152#924—24no—
153C2×C4×Dic3—21no—
154C2×C4×Dic3—24no—
155C2×C4×Dic3C2312yes—
156C2×#208—24no—
157C2×#208C2312yes—
158C2×#208C2312yes—
159C22×C3⋊C8C2312yes—
160#835C2318yes—
161#832C2318yes—
162#821C2318yes—
163#816C2318yes—
164Dic3⋊C8C23116yes—
165C22×#41—31no—
166C22×#41C2×D311yescharacteristic
167C23×Q8C2×D311yescharacteristic
168C2×#137—28no—
169C2×#137—34no—
170C2×#137—34no—
171C2×#137—68no—
172C2×#137C2×D314yes—
173C2×#137C2×D314yescomplemented
174C22×Q16—32no—
175C22×Q16C2×D312yescomplemented
176C22×SD16—32no—
177C22×SD16C2×D312yescomplemented
178C22×M16—31no—
179C22×M16C2×D311yescharacteristic
180#537—62no—
181#536—32no—
182#536—32no—
183#535—32no—
184#535—32no—
185#534—62no—
186#532—34no—
187#525—62no—
188#525—62no—
189#519—34no—
190#519—34no—
191#519—62no—
192#519—62no—
193C2×C4×Q8—34no—
194C2×C4×Q8—62no—
195C2×#117—31no—
196C2×#117—31no—
197C2×#117—62no—
198#464—316no—
199#464—316no—
200#462—316no—
201#460—316no—
202#424—34no—
203#424—34no—
204#423—34no—
205#419—32no—
206#419—32no—
207#418—32no—
208#418—32no—
209#417—34no—
210#417—34no—
211#416—34no—
212#416—34no—
213C24×C3—21no—
214C24×C3C2×D411yescharacteristic
215C23×D3—21no—
216C3×#41—24no—
217C3×#41—24no—
218C3×#41—24no—
219C3×#41—24no—
220C3×#41—44no—
221C3×#41—44no—
222C2×C3×Q8—28no—
223C2×C3×Q8—28no—
224C2×C3×Q8—28no—
225C2×C3×Q8—22no—
226C2×C3×Q8—22no—
227C2×C3×Q8—22no—
228C2×C3×Q8—21no—
229C2×C3×Q8—24no—
230C2×C3×Q8—24no—
231C2×C3×Q8—24no—
232C2×C3×Q8—22no—
233C2×C3×Q8—22no—
234C2×C3×Q8C2×D418yes—
235C2×C3×Q8C2×D418yes—
236C2×C3×Q8C2×D418yes—
237C2×C3×Q8C2412yes—
238C2×C3×Q8C2414yes—
239C2×C3×D4—22no—
240C2×C3×D4—22no—
241C2×C3×D4—22no—
242C2×C3×D4—21no—
243C2×C3×D4—22no—
244C2×C3×D4—44no—
245C2×C3×D4C2412yes—
246C22×C3×C4—28no—
247C22×C3×C4—22no—
248C22×C3×C4—22no—
249C22×C3×C4—22no—
250C22×C3×C4—21no—
251C22×C3×C4—22no—
252C22×C3×C4—22no—
253C22×C3×C4—44no—
254C22×C3×C4C2×D411yescharacteristic
255C22×C3×C4C2×D411yescharacteristic
256C22×C3×C4C2×D411yescharacteristic
257C22×C3×C4C2×D414yes—
258C22×C3×C4#4114yes—
259C22×C3×C4C2411yescharacteristic
260C22×C3×C4C2411yescharacteristic
261C22×C3×C4C2411yescharacteristic
262C2×#67—22no—
263C2×#67—22no—
264C2×#67—28no—
265C2×#67—44no—
266C22×Dic3—22no—
267C22×Dic3—21no—
268C22×Dic3C2×D414yes—
269#235—24no—
270#235—24no—
271#235—24no—
272#235—24no—
273#235—44no—
274#235—44no—
275C2×D12—22no—
276C2×D12—22no—
277C2×D12—21no—
278C2×D12C2412yes—
279C2×C4×D3—22no—
280C2×C4×D3—22no—
281C2×C4×D3—44no—
282C2×Dic6—22no—
283C2×Dic6—22no—
284C2×Dic6—21no—
285C2×Dic6C2412yes—
286C3×Q16—216no—
287C3×SD16—216no—
288C3×M16—24no—
289C3×M16—24no—
290C2×C3×C8—24no—
291C2×C3×C8C2414yes—
292#217—22no—
293#217—22no—
294#217—28no—
295#217C2×D414yes—
296#217C2×D414yes—
297#216—216no—
298#215—216no—
299(C2×D3)⋊C4—28no—
300C4⋊Dic3—22no—
301C4⋊Dic3—216no—
302C4⋊Dic3C2×D414yes—
303C4⋊Dic3C2×D414yes—
304C4⋊Dic3C2414yes—
305Dic3⋊C4—28no—
306C4×Dic3—22no—
307C4×Dic3—22no—
308C4×Dic3—216no—
309C4×Dic3C2×D414yes—
310C4×Dic3C2414yes—
311#208—24no—
312#208—24no—
313C2×C3⋊C8—24no—
314C2×C3⋊C8C2414yes—
315C2×#41—21no—
316C2×#41—21no—
317C2×#41—24no—
318C2×#41—24no—
319C2×#41—32no—
320C2×#41—32no—
321C2×#41—61no—
322C2×#41—61no—
323C2×#41—64no—
324C2×#41—64no—
325C2×#41C22×D312yes—
326C2×#41C22×D312yes—
327C22×Q8—21no—
328C22×Q8—21no—
329C22×Q8—21no—
330C22×Q8—24no—
331C22×Q8—31no—
332C22×Q8#6718yescomplemented
333C22×Q8C22×D311yescharacteristic
334C22×Q8C22×D312yes—
335C22×Q8C22×D312yes—
336C22×Q8C22×D312yes—
337C22×D4—21no—
338C22×D4—31no—
339C22×D4—61no—
340C22×D4C22×D311yescharacteristic
341C23×C4—21no—
342C23×C4—21no—
343C23×C4—61no—
344C23×C4C22×D311yescharacteristic
345#137—216no—
346#137—216no—
347#137—616no—
348#137—616no—
349C2×Q16—28no—
350C2×Q16—38no—
351C2×Q16—68no—
352C2×Q16C22×D318yes—
353C2×SD16—28no—
354C2×SD16—38no—
355C2×SD16—68no—
356C2×SD16C22×D318yes—
357C2×M16—24no—
358C2×M16—32no—
359C2×M16—32no—
360C2×M16—64no—
361C2×M16C22×D312yes—
362C2×M16C22×D312yes—
363C22×C8—32no—
364C22×C8C22×D312yes—
365C4⋊Q8—34no—
366#127—34no—
367#124—34no—
368#124—34no—
369#123—64no—
370#123—64no—
371C22⋊Q8—64no—
372C22⋊Q8—64no—
373#121—64no—
374#121—64no—
375#121—68no—
376#120—64no—
377#120—64no—
378C4×Q8—316no—
379C4×Q8—68no—
380C4×Q8—68no—
381C4×Q8—68no—
382C4×D4—68no—
383#117—32no—
384#117—32no—
385#117—32no—
386#117—32no—
387#117—32no—
388#117—32no—
389#117—68no—
390#117—68no—
391#117—61no—
392#117—61no—
393C2×C4⋊C4—32no—
394C2×C4⋊C4—32no—
395C2×C4⋊C4—64no—
396C2×C22⋊C4—32no—
397C2×C22⋊C4—61no—
398C2×C42—32no—
399C2×C42—64no—
400C2×C42—61no—
401C4⋊C8—316no—
402#103—38no—
403#103—38no—
404#102—38no—
405#102—38no—
406C23×C3—22no—
407C23×C3—42no—
408C23×C3—42no—
409C23×C3—82no—
410C23×C3#12114yes—
411C23×C3C22×D411yescharacteristic
412C23×C3C22×D411yescharacteristic
413C23×C3C22×D411yescharacteristic
414C22×D3—22no—
415C22×D3—42no—
416C22×D3—82no—
417C3×Q8—28no—
418C3×Q8—28no—
419C3×Q8—28no—
420C3×Q8—24no—
421C3×Q8—22no—
422C3×Q8—22no—
423C3×Q8—42no—
424C3×Q8C22×D418yes—
425C3×D4—22no—
426C3×D4—22no—
427C3×D4—44no—
428C3×D4—44no—
429C3×D4—42no—
430C2×C3×C4—28no—
431C2×C3×C4—28no—
432C2×C3×C4—28no—
433C2×C3×C4—24no—
434C2×C3×C4—22no—
435C2×C3×C4—42no—
436C2×C3×C4—44no—
437C2×C3×C4—44no—
438C2×C3×C4#12114yes—
439C2×C3×C4#12114yes—
440C2×C3×C4#12114yes—
441C2×C3×C4C22×D411yescharacteristic
442C2×C3×C4C22×D411yescharacteristic
443C2×C3×C4C22×D411yescharacteristic
444C2×C3×C4C22×D412yes—
445C2×C3×C4C22×D412yes—
446C2×C3×C4C22×D412yes—
447C2×C3×C4C2×#4112yes—
448C2×C3×C4C2×#4112yes—
449C2×C3×C4C2×#4112yes—
450C2×C3×C4C2511yescharacteristic
451#67—44no—
452#67—44no—
453#67—88no—
454C2×Dic3—24no—
455C2×Dic3—22no—
456C2×Dic3—44no—
457C2×Dic3—44no—
458C2×Dic3—42no—
459C2×Dic3C22×D418yes—
460D12—22no—
461D12—22no—
462D12—42no—
463C4×D3—44no—
464C4×D3—44no—
465Dic6—22no—
466Dic6—22no—
467Dic6—42no—
468C3×C8—24no—
469C3⋊C8—24no—
470C24—21no—
471C24—61no—
472C24D3×D411yescharacteristic
473#41—24no—
474#41—24no—
475#41—24no—
476#41—24no—
477#41—44no—
478#41—44no—
479#41—64no—
480#41—64no—
481#41—64no—
482#41—64no—
483#41—124no—
484#41—124no—
485C2×Q8—28no—
486C2×Q8—28no—
487C2×Q8—28no—
488C2×Q8—22no—
489C2×Q8—22no—
490C2×Q8—22no—
491C2×Q8—21no—
492C2×Q8—24no—
493C2×Q8—24no—
494C2×Q8—24no—
495C2×Q8—22no—
496C2×Q8—22no—
497C2×Q8—32no—
498C2×Q8—61no—
499C2×Q8—62no—
500C2×Q8—62no—
501C2×Q8C2×#6718yes—
502C2×Q8C2×#6718yes—
503C2×Q8C2×#6718yes—
504C2×Q8C23×D312yes—
505C2×Q8C23×D314yes—
506C2×D4—22no—
507C2×D4—22no—
508C2×D4—21no—
509C2×D4—22no—
510C2×D4—22no—
511C2×D4—32no—
512C2×D4—44no—
513C2×D4—68no—
514C2×D4—62no—
515C2×D4—62no—
516C2×D4—61no—
517C2×D4—62no—
518C2×D4—62no—
519C2×D4—124no—
520C2×D4C23×D312yes—
521C22×C4—28no—
522C22×C4—22no—
523C22×C4—22no—
524C22×C4—22no—
525C22×C4—21no—
526C22×C4—22no—
527C22×C4—22no—
528C22×C4—34no—
529C22×C4—44no—
530C22×C4—62no—
531C22×C4—61no—
532C22×C4—62no—
533C22×C4—62no—
534C22×C4—124no—
535C22×C4D3×D411yescharacteristic
536C22×C4D3×D411yescharacteristic
537C22×C4D3×D411yescharacteristic
538C22×C4#23714yes—
539C22×C4C2×#6714yes—
540C22×C4C23×D311yescharacteristic
541C22×C4C23×D311yescharacteristic
542C22×C4C23×D311yescharacteristic
543Q16—216no—
544Q16—616no—
545SD16—216no—
546SD16—616no—
547M16—24no—
548M16—24no—
549M16—64no—
550M16—64no—
551C2×C8—24no—
552C2×C8—34no—
553C2×C8—64no—
554C2×C8C23×D314yes—
555C4⋊C4—34no—
556C4⋊C4—34no—
557C4⋊C4—34no—
558C4⋊C4—616no—
559C4⋊C4—68no—
560C4⋊C4—62no—
561C22⋊C4—34no—
562C22⋊C4—34no—
563C22⋊C4—68no—
564C22⋊C4—68no—
565C22⋊C4—62no—
566C22⋊C4—62no—
567C42—34no—
568C42—34no—
569C42—616no—
570C42—62no—
571C42—62no—
572C22×C3—22no—
573C22×C3—22no—
574C22×C3—22no—
575C22×C3—44no—
576C22×C3—44no—
577C22×C3—42no—
578C22×C3—84no—
579C22×C3C2×#12112yes—
580C22×C3C2×#12112yes—
581C22×C3C2×#12112yes—
582C22×C3C23×D411yescharacteristic
583C2×D3—44no—
584C2×D3—42no—
585C2×D3—84no—
586C3×C4—28no—
587C3×C4—42no—
588C3×C4C2×#12112yes—
589C3×C4C2×#12112yes—
590C3×C4C2×#12112yes—
591C3×C4C23×D411yescharacteristic
592C3×C4C22×#4111yescharacteristic
593Dic3—28no—
594Dic3—44no—
595Dic3—42no—
596C23—22no—
597C23—42no—
598C23—42no—
599C23—62no—
600C23—82no—
601C23—122no—
602C23—242no—
603C23#93914yes—
604C23C2×D3×D411yescharacteristic
605C23C2×D3×D411yescharacteristic
606C23C2×D3×D411yescharacteristic
607Q8—24no—
608Q8—28no—
609Q8—28no—
610Q8—28no—
611Q8—22no—
612Q8—22no—
613Q8—42no—
614Q8—62no—
615Q8—62no—
616Q8—122no—
617Q8C22×#6718yes—
618D4—22no—
619D4—22no—
620D4—44no—
621D4—44no—
622D4—42no—
623D4—62no—
624D4—62no—
625D4—124no—
626D4—124no—
627D4—122no—
628D4—248no—
629C2×C4—24no—
630C2×C4—28no—
631C2×C4—28no—
632C2×C4—28no—
633C2×C4—22no—
634C2×C4—38no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—64no—
639C2×C4—62no—
640C2×C4—124no—
641C2×C4—124no—
642C2×C4—122no—
643C2×C4—124no—
644C2×C4—124no—
645C2×C4#93914yes—
646C2×C4#93914yes—
647C2×C4#93914yes—
648C2×C4C2×D3×D411yescharacteristic
649C2×C4C2×D3×D411yescharacteristic
650C2×C4C2×D3×D411yescharacteristic
651C2×C4C2×#23712yes—
652C2×C4C2×#23712yes—
653C2×C4C2×#23712yes—
654C2×C4C22×#6712yes—
655C2×C4C22×#6712yes—
656C2×C4C22×#6712yes—
657C2×C4C24×D311yescharacteristic
658C8—24no—
659C8—64no—
660C2×C3—22no—
661C2×C3—22no—
662C2×C3—22no—
663C2×C3—82no—
664C2×C3C22×#12111yescharacteristic
665C2×C3#358612yes—
666D3—82no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—44no—
671C22—44no—
672C22—42no—
673C22—84no—
674C22—124no—
675C22—122no—
676C22—244no—
677C22C2×#93912yes—
678C22C2×#93912yes—
679C22C2×#93912yes—
680C22C22×D3×D411yescharacteristic
681C4—28no—
682C4—42no—
683C4—68no—
684C4—124no—
685C4—122no—
686C4C2×#93912yes—
687C4C2×#93912yes—
688C4C2×#93912yes—
689C4C22×#23711yescharacteristic
690C4C23×#6711yescharacteristic
691C3#6042411yescharacteristic, complemented, Sylow
692C2—22no—
693C2—22no—
694C2—22no—
695C2—82no—
696C2—242no—
697C2#8719712yes—
698C2C22×#93911yescharacteristic
subgroupG/H
centerC22C22×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC22×C3×#137C2
socleC22×C3C23×D4
prime radical#87268C2
hypercenterC22×#137D3
hypocenterC3#60424
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#137

Complements

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

NH ≅ G/Ntype
1. C22×C3×#137C2semidirect
3. C2×#5812C2semidirect
4. C22×#942C2semidirect
5. #84072C2semidirect
6. #83970C2semidirect
7. #83969C2semidirect
8. #83966C2semidirect
9. #83877C2semidirect
10. #83849C2semidirect
11. #83768C2semidirect
12. #83750C2semidirect
19. C2×C3×#137C22semidirect
20. C22×C3×Q16C22semidirect
21. C22×C3×SD16C22semidirect
31. #5812C22semidirect
32. #5812C22semidirect
34. C2×Q8×Dic3C22semidirect
35. C22×#216C22semidirect
38. C2×#942C22semidirect
39. C22×#215C22semidirect
48. #5232C22semidirect
49. #5225C22semidirect
50. #5215C22semidirect
51. #5185C22semidirect
52. #5182C22semidirect
53. #5166C22semidirect
54. #5163C22semidirect
55. #5133C22semidirect
56. #5131C22semidirect
57. #5120C22semidirect
58. #5119C22semidirect
59. #5118C22semidirect
61. #5102C22semidirect
63. C22×#137D3semidirect
84. C22×C3×Q8D4semidirect
173. C2×#137C2×D3semidirect
175. C22×Q16C2×D3semidirect
177. C22×SD16C2×D3semidirect
332. C22×Q8#67semidirect
691. C3#60424semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#1372
#872682
C2×#58122
C22×#9422
#840722
#839702
#839692
#839662
#838772
#838492
#837682
#837502
#604243