Finite Group Databasebeta

← #11662157

Subgroups — #11662157

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
1#87268C211yesmaximal, characteristic
2C22×#951C211yesmaximal, characteristic, complemented
3C22×#909C211yesmaximal, characteristic, complemented
4C2×#5516C211yesmaximal, characteristic, complemented
5C3×#2896C211yesmaximal, characteristic, complemented
6#83750C211yesmaximal, characteristic, complemented
7#82213C211yesmaximal, characteristic, complemented
8#78977C214yesmaximal, complemented
9#78945C214yesmaximal, complemented
10#78913C218yesmaximal, complemented
11#78865C214yesmaximal, complemented
12#78864C214yesmaximal, complemented
13#60424—31nomaximal, Sylow
14C22×C3×#41C2211yescharacteristic
15C22×#235C2211yescharacteristic
16C23×Dic6C2211yescharacteristic
17C22×C3×M16C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19#5833—22no—
20#5832C2214yes—
21#5830C2212yes—
22#5828—22no—
23#5825C2212yes—
24C2×#951—28no—
25C2×#951C2214yes—
26C2×#951C2214yescomplemented
27#5816C2212yes—
28C22×#216C2212yescomplemented
29#5805C2212yes—
30#5804—22no—
31C22×#214C2212yescomplemented
32#5791—22no—
33C2×#926C2211yescharacteristic
34C22×#208C2211yescharacteristic
35C2×#909—28no—
36C2×#909C2214yes—
37C2×#909C2214yescomplemented
38C22×Dic12C2212yescomplemented
39C22×#204C2212yescomplemented
40#5516—22no—
41#5516—22no—
42#5516C2214yescomplemented
43#5516C2214yescomplemented
44C2×#887C2214yescomplemented
45C2×#877—22no—
46C2×C4×Dic6—22no—
47C3×#419C2212yescomplemented
48C3×#418C2212yescomplemented
49C3×#417C2214yescomplemented
50C3×#416C2214yes—
51#5119C2214yescomplemented
52#5118C2214yescomplemented
53#5102C2214yescomplemented
54#5004C2212yescomplemented
55#5002C2214yes—
56#5001C2212yescomplemented
57#4966C2214yescomplemented
58#4798C22116yescomplemented
59#4795C22116yescomplemented
60#4767C22116yescomplemented
61#4762C22116yescomplemented
62C22×#137—31no—
63C22×#137—31no—
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#2896D311yescharacteristic, complemented
74C2×C3×#41—21no—
75C2×C3×#41—21no—
76C2×C3×#41—24no—
77C2×C3×#41—24no—
78C2×C3×#41C2312yes—
79C2×C3×#41C2312yes—
80C22×C3×Q8C2311yescharacteristic
81C22×C3×D4—21no—
82C22×C3×D4C2311yescharacteristic
83C23×C3×C4—21no—
84C23×C3×C4C2311yescharacteristic
85C22×#67—21no—
86C23×Dic3—21no—
87C2×#235—21no—
88C2×#235—21no—
89C2×#235—24no—
90C2×#235—24no—
91C2×#235C2312yes—
92C2×#235C2312yes—
93C22×D12C2311yescharacteristic
94C22×C4×D3—21no—
95C22×Dic6—21no—
96C22×Dic6—21no—
97C22×Dic6—21no—
98C22×Dic6—24no—
99C22×Dic6D418yescomplemented
100C22×Dic6C2311yescharacteristic
101C22×Dic6C2312yes—
102C22×Dic6C2312yes—
103C22×Dic6C2312yes—
104C2×C3×M16—24no—
105C2×C3×M16C2312yes—
106C2×C3×M16C2312yes—
107C22×C3×C8C2312yes—
108C3×#117—21no—
109C3×#117—21no—
110C3×#117C2312yes—
111C3×#117C2312yes—
112C2×C3×C4⋊C4C2312yes—
113C2×C3×C22⋊C4—21no—
114C2×C3×C42—21no—
115#953—24no—
116C2×#217D412yes—
117#951—216no—
118#951—216no—
119#947C2314yes—
120#946—24no—
121#944C2314yes—
122C2×#216—28no—
123C2×#216C2318yes—
124#940C2314yes—
125#939—28no—
126#938—24no—
127#937—24no—
128#936C2314yes—
129#935—24no—
130C2×#214—28no—
131C2×#214C2318yes—
132#930—24no—
133#929—24no—
134C4×#67—28no—
135#926D412yes—
136#926D412yes—
137#926C2312yes—
138#926C2312yes—
139C2×C4⋊Dic3D412yes—
140#924—24no—
141C2×Dic3⋊C4—24no—
142C2×C4×Dic3—24no—
143C2×C4×Dic3C2312yes—
144C2×#208—24no—
145C2×#208C2312yes—
146C2×#208C2312yes—
147C22×C3⋊C8C2312yes—
148#909—216no—
149#909—216no—
150C2×Dic12—28no—
151C2×Dic12C2318yes—
152C2×#204—28no—
153C2×#204C2318yes—
154#887—28no—
155#887C23116yes—
156#877—28no—
157#877—28no—
158C4×Dic6—28no—
159C4×Dic6—28no—
160C3×#103C2318yes—
161C3×#102C2318yes—
162Dic3⋊C8C23116yes—
163#810C2318yes—
164#809C2318yes—
165C22×#41—31no—
166C22×#41C2×D311yescharacteristic
167C23×Q8—31no—
168C2×#137—34no—
169C2×#137—34no—
170C2×#137—34no—
171C2×#137—34no—
172C2×#137—68no—
173C2×#137—68no—
174C22×Q16—32no—
175C22×Q16—32no—
176C22×SD16—32no—
177C22×SD16—32no—
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—62no—
197C2×#117C2×D311yescharacteristic
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#419C2×D312yescomplemented
207#418—32no—
208#418C2×D312yescomplemented
209#417—34no—
210#417C2×D314yescomplemented
211#416—34no—
212#416C2×D314yes—
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—21no—
223C2×C3×Q8—22no—
224C2×C3×Q8—22no—
225C2×C3×Q8C2412yes—
226C2×C3×D4—22no—
227C2×C3×D4—22no—
228C2×C3×D4—22no—
229C2×C3×D4—21no—
230C2×C3×D4—22no—
231C2×C3×D4—44no—
232C2×C3×D4C2412yes—
233C22×C3×C4—22no—
234C22×C3×C4—21no—
235C22×C3×C4—22no—
236C22×C3×C4—22no—
237C22×C3×C4—44no—
238C22×C3×C4C2×D411yescharacteristic
239C22×C3×C4C2×D411yescharacteristic
240C22×C3×C4C2×D411yescharacteristic
241C22×C3×C4C2×D414yes—
242C22×C3×C4#4114yes—
243C22×C3×C4C2411yescharacteristic
244C22×C3×C4C2411yescharacteristic
245C22×C3×C4C2411yescharacteristic
246C2×#67—22no—
247C2×#67—22no—
248C2×#67—28no—
249C2×#67—44no—
250C22×Dic3—28no—
251C22×Dic3—22no—
252C22×Dic3—22no—
253C22×Dic3—22no—
254C22×Dic3—21no—
255C22×Dic3C2×D414yes—
256#235—24no—
257#235—24no—
258#235—24no—
259#235—24no—
260#235—44no—
261#235—44no—
262C2×D12—22no—
263C2×D12—22no—
264C2×D12—21no—
265C2×D12C2412yes—
266C2×C4×D3—22no—
267C2×C4×D3—22no—
268C2×C4×D3—44no—
269C2×Dic6—28no—
270C2×Dic6—28no—
271C2×Dic6—28no—
272C2×Dic6—22no—
273C2×Dic6—22no—
274C2×Dic6—22no—
275C2×Dic6—24no—
276C2×Dic6—24no—
277C2×Dic6—24no—
278C2×Dic6—22no—
279C2×Dic6—22no—
280C2×Dic6—21no—
281C2×Dic6C2×D418yes—
282C2×Dic6C2×D418yes—
283C2×Dic6C2×D418yes—
284C2×Dic6C2412yes—
285C2×Dic6C2414yes—
286C3×M16—24no—
287C3×M16—24no—
288C2×C3×C8—24no—
289C2×C3×C8C2414yes—
290C3×C4⋊C4—22no—
291C3×C4⋊C4C2414yes—
292C3×C22⋊C4—22no—
293C3×C22⋊C4—22no—
294C3×C42—22no—
295C3×C42—22no—
296#217—28no—
297#217C2×D414yes—
298#217C2×D414yes—
299#216—216no—
300#214—216no—
301(C2×D3)⋊C4—28no—
302C4⋊Dic3C2×D414yes—
303C4⋊Dic3C2×D414yes—
304Dic3⋊C4—216no—
305Dic3⋊C4—28no—
306C4×Dic3—216no—
307C4×Dic3C2×D414yes—
308C4×Dic3C2414yes—
309#208—24no—
310#208—24no—
311C2×C3⋊C8—24no—
312C2×C3⋊C8C2414yes—
313Dic12—216no—
314#204—216no—
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—38no—
328C22×Q8—32no—
329C22×Q8—32no—
330C22×Q8—32no—
331C22×Q8—31no—
332C22×Q8—64no—
333C22×Q8—61no—
334C22×Q8—61no—
335C22×Q8—61no—
336C22×Q8C22×D311yescharacteristic
337C22×D4—21no—
338C22×D4—31no—
339C22×D4—61no—
340C22×D4C22×D311yescharacteristic
341C23×C4—21no—
342C23×C4—61no—
343C23×C4—61no—
344C23×C4C22×D311yescharacteristic
345#137—616no—
346#137—616no—
347#137—616no—
348#137—616no—
349C2×Q16—38no—
350C2×Q16—38no—
351C2×Q16—68no—
352C2×Q16—68no—
353C2×SD16—38no—
354C2×SD16—38no—
355C2×SD16—68no—
356C2×SD16—68no—
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—21no—
384#117—21no—
385#117—32no—
386#117—32no—
387#117—32no—
388#117—32no—
389#117—68no—
390#117—68no—
391#117C22×D312yes—
392#117C22×D312yes—
393C2×C4⋊C4—32no—
394C2×C4⋊C4—64no—
395C2×C4⋊C4C22×D312yes—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—32no—
398C2×C42—21no—
399C2×C42—32no—
400C2×C42—64no—
401C4⋊C8—316no—
402#103—38no—
403#103C22×D318yes—
404#102—38no—
405#102C22×D318yes—
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—22no—
418C3×Q8—22no—
419C3×Q8—42no—
420C3×D4—22no—
421C3×D4—22no—
422C3×D4—44no—
423C3×D4—44no—
424C3×D4—42no—
425C2×C3×C4—24no—
426C2×C3×C4—22no—
427C2×C3×C4—44no—
428C2×C3×C4—42no—
429C2×C3×C4—44no—
430C2×C3×C4—44no—
431C2×C3×C4#12114yes—
432C2×C3×C4#12114yes—
433C2×C3×C4#12114yes—
434C2×C3×C4C22×D411yescharacteristic
435C2×C3×C4C22×D411yescharacteristic
436C2×C3×C4C22×D411yescharacteristic
437C2×C3×C4C22×D412yes—
438C2×C3×C4C22×D412yes—
439C2×C3×C4C22×D412yes—
440C2×C3×C4C2×#4112yes—
441C2×C3×C4C2×#4112yes—
442C2×C3×C4C2×#4112yes—
443C2×C3×C4C2511yescharacteristic
444#67—44no—
445#67—44no—
446#67—88no—
447C2×Dic3—28no—
448C2×Dic3—28no—
449C2×Dic3—28no—
450C2×Dic3—24no—
451C2×Dic3—22no—
452C2×Dic3—44no—
453C2×Dic3—42no—
454C2×Dic3C22×D418yes—
455D12—22no—
456D12—22no—
457D12—42no—
458C4×D3—44no—
459C4×D3—44no—
460Dic6—28no—
461Dic6—28no—
462Dic6—28no—
463Dic6—24no—
464Dic6—22no—
465Dic6—22no—
466Dic6—42no—
467Dic6C22×D418yes—
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—21no—
486C2×Q8—22no—
487C2×Q8—22no—
488C2×Q8—38no—
489C2×Q8—38no—
490C2×Q8—38no—
491C2×Q8—34no—
492C2×Q8—32no—
493C2×Q8—68no—
494C2×Q8—68no—
495C2×Q8—68no—
496C2×Q8—64no—
497C2×Q8—64no—
498C2×Q8—64no—
499C2×Q8—62no—
500C2×Q8—62no—
501C2×Q8—62no—
502C2×Q8—61no—
503C2×Q8—62no—
504C2×Q8—62no—
505C2×Q8C23×D312yes—
506C2×D4—22no—
507C2×D4—22no—
508C2×D4—22no—
509C2×D4—22no—
510C2×D4—21no—
511C2×D4—32no—
512C2×D4—44no—
513C2×D4—68no—
514C2×D4—62no—
515C2×D4—62no—
516C2×D4—62no—
517C2×D4—62no—
518C2×D4—61no—
519C2×D4—124no—
520C2×D4C23×D312yes—
521C22×C4—22no—
522C22×C4—21no—
523C22×C4—22no—
524C22×C4—22no—
525C22×C4—34no—
526C22×C4—44no—
527C22×C4—68no—
528C22×C4—62no—
529C22×C4—62no—
530C22×C4—62no—
531C22×C4—61no—
532C22×C4—62no—
533C22×C4—62no—
534C22×C4—124no—
535C22×C4#23514yes—
536C22×C4D3×D411yescharacteristic
537C22×C4D3×D411yescharacteristic
538C22×C4D3×D411yescharacteristic
539C22×C4D3×D414yes—
540C22×C4C23×D311yescharacteristic
541C22×C4C23×D311yescharacteristic
542C22×C4C23×D311yescharacteristic
543Q16—616no—
544Q16—616no—
545SD16—616no—
546SD16—616no—
547M16—24no—
548M16—24no—
549M16—64no—
550M16—64no—
551C2×C8—24no—
552C2×C8—34no—
553C2×C8—64no—
554C2×C8C23×D314yes—
555C4⋊C4—22no—
556C4⋊C4—34no—
557C4⋊C4—34no—
558C4⋊C4—616no—
559C4⋊C4—68no—
560C4⋊C4C23×D314yes—
561C22⋊C4—22no—
562C22⋊C4—22no—
563C22⋊C4—34no—
564C22⋊C4—34no—
565C22⋊C4—68no—
566C22⋊C4—68no—
567C42—22no—
568C42—22no—
569C42—34no—
570C42—34no—
571C42—616no—
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—44no—
587C3×C4—42no—
588C3×C4C2×#12112yes—
589C3×C4C2×#12112yes—
590C3×C4C2×#12112yes—
591C3×C4C23×D411yescharacteristic
592C3×C4C22×#4111yescharacteristic
593Dic3—28no—
594Dic3—28no—
595Dic3—42no—
596C23—22no—
597C23—42no—
598C23—42no—
599C23—62no—
600C23—82no—
601C23—122no—
602C23—242no—
603C23#88414yes—
604C23C2×D3×D411yescharacteristic
605C23C2×D3×D411yescharacteristic
606C23C2×D3×D411yescharacteristic
607Q8—22no—
608Q8—22no—
609Q8—38no—
610Q8—42no—
611Q8—64no—
612Q8—68no—
613Q8—68no—
614Q8—68no—
615Q8—62no—
616Q8—62no—
617Q8—122no—
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—22no—
631C2×C4—38no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—64no—
637C2×C4—68no—
638C2×C4—68no—
639C2×C4—68no—
640C2×C4—62no—
641C2×C4—124no—
642C2×C4—122no—
643C2×C4—124no—
644C2×C4—124no—
645C2×C4#88414yes—
646C2×C4#88414yes—
647C2×C4#88414yes—
648C2×C4C2×#23512yes—
649C2×C4C2×#23512yes—
650C2×C4C2×#23512yes—
651C2×C4C2×D3×D412yes—
652C2×C4C2×D3×D412yes—
653C2×C4C2×D3×D412yes—
654C2×C4C2×D3×D411yescharacteristic
655C2×C4C2×D3×D411yescharacteristic
656C2×C4C2×D3×D411yescharacteristic
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×#88412yes—
678C22C2×#88412yes—
679C22C2×#88412yes—
680C22C22×D3×D411yescharacteristic
681C4—44no—
682C4—42no—
683C4—68no—
684C4—68no—
685C4—122no—
686C4C2×#88412yes—
687C4C2×#88412yes—
688C4C2×#88412yes—
689C4C22×#23511yescharacteristic
690C4C22×D3×D411yescharacteristic
691C3#6042411yescharacteristic, complemented, Sylow
692C2—22no—
693C2—22no—
694C2—22no—
695C2—82no—
696C2—242no—
697C2C22×#88411yescharacteristic
698C2#8720112yes—
subgroupG/H
centerC22C22×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC3×#2896C2
socleC22×C3C23×D4
prime radical#87268C2
hypercenter#2896D3
hypocenterC3#60424
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2896

Complements

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

NH ≅ G/Ntype
2. C22×#951C2semidirect
3. C22×#909C2semidirect
4. C2×#5516C2semidirect
5. C3×#2896C2semidirect
6. #83750C2semidirect
7. #82213C2semidirect
8. #78977C2semidirect
9. #78945C2semidirect
10. #78913C2semidirect
11. #78865C2semidirect
12. #78864C2semidirect
26. C2×#951C22semidirect
28. C22×#216C22semidirect
31. C22×#214C22semidirect
37. C2×#909C22semidirect
38. C22×Dic12C22semidirect
39. C22×#204C22semidirect
42. #5516C22semidirect
43. #5516C22semidirect
44. C2×#887C22semidirect
47. C3×#419C22semidirect
48. C3×#418C22semidirect
49. C3×#417C22semidirect
51. #5119C22semidirect
52. #5118C22semidirect
53. #5102C22semidirect
54. #5004C22semidirect
56. #5001C22semidirect
57. #4966C22semidirect
58. #4798C22semidirect
59. #4795C22semidirect
60. #4767C22semidirect
61. #4762C22semidirect
73. #2896D3semidirect
99. C22×Dic6D4semidirect
206. #419C2×D3semidirect
208. #418C2×D3semidirect
210. #417C2×D3semidirect
691. C3#60424semidirect

Maximal subgroups

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

subgroupindex
#872682
C22×#9512
C22×#9092
C2×#55162
C3×#28962
#837502
#822132
#789772
#789452
#789132
#788652
#788642
#604243