Finite Group Databasebeta

← #10822318

Subgroups — #10822318

Order 768.

1986 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5816C211yesmaximal, characteristic, complemented
2C22×#942C211yesmaximal, characteristic, complemented
3C22×#904C211yesmaximal, characteristic
4C22×C3×#103C211yesmaximal, characteristic, complemented
5#82186C211yesmaximal, characteristic, complemented
6C2×#4966C211yesmaximal, characteristic, complemented
7#71965C211yesmaximal, characteristic, complemented
8#70283C212yesmaximal, complemented
9#70282C212yesmaximal, complemented
10#70278C212yesmaximal, complemented
11#70277C212yesmaximal, complemented
12#23427—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C22×#235C2211yescharacteristic
15C22×#235C2211yescharacteristic
16C23×C3×C8C2211yescharacteristic
17C22×C3×C4⋊C4C2211yescharacteristic
18#5816—24no—
19#5816—24no—
20C2×#947C2212yes—
21C2×#946—22no—
22C2×#944C2212yes—
23C22×#216C2212yes—
24C2×#942—24no—
25C2×#942—24no—
26C2×#942—24no—
27C22×#215C2212yes—
28C2×#929—22no—
29C2×#926C2211yescharacteristic
30C22×#208C2211yescharacteristic
31C22×Dic12C2211yescharacteristic
32C2×#904—24no—
33C2×#904—24no—
34C2×#904—24no—
35C22×D24C2211yescharacteristic
36C22×#204C2211yescharacteristic
37C22×#204C2211yescharacteristic
38C2×C3×#103—24no—
39C2×C3×#103C2212yes—
40C2×C3×#103C2212yes—
41#5213—22no—
42#4988C2212yes—
43#4987—22no—
44#4973C2212yes—
45C2×#810C2212yes—
46#4966—24no—
47C2×#809C2212yes—
48#4550C2212yes—
49#4550C2212yes—
50C22×#137—31no—
51C22×#135—31no—
52C2×#536—31no—
53C2×#417—31no—
54C22×#103D311yescharacteristic, complemented
55#2311—31no—
56#2016—32no—
57#2015—32no—
58#2013—32no—
59#2012—32no—
60#1729—31no—
61C22×C3×Q8—21no—
62C22×C3×Q8—22no—
63C22×C3×Q8—22no—
64C22×C3×Q8—22no—
65C22×C3×Q8—22no—
66C22×C3×Q8—22no—
67C22×C3×Q8D414yes—
68C22×C3×Q8C2312yes—
69C23×C3×C4—21no—
70C23×C3×C4—21no—
71C23×C3×C4C2311yescharacteristic
72C22×#67—21no—
73C22×#67—21no—
74C2×#235—22no—
75C2×#235—22no—
76C2×#235—22no—
77C2×#235—22no—
78C2×#235—22no—
79C2×#235—22no—
80C2×#235—22no—
81C2×#235—22no—
82C2×#235—22no—
83C2×#235—22no—
84C2×#235D414yes—
85C2×#235D414yes—
86C22×D12C2311yescharacteristic
87C22×D12C2311yescharacteristic
88C22×C4×D3—21no—
89C22×C4×D3—21no—
90C22×Dic6C2311yescharacteristic
91C22×Dic6C2311yescharacteristic
92C22×C3×C8—22no—
93C22×C3×C8—22no—
94C22×C3×C8D414yes—
95C22×C3×C8C2311yescharacteristic
96C22×C3×C8C2311yescharacteristic
97C2×C3×C4⋊C4—22no—
98C2×C3×C4⋊C4—22no—
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4C2312yes—
101C2×#217D412yes—
102#947—24no—
103#947—24no—
104#946—216no—
105#944—24no—
106#944—24no—
107C2×#216—24no—
108C2×#216—24no—
109C2×#216—24no—
110#942—232no—
111C2×#215—24no—
112C2×#215—24no—
113C2×#215—24no—
114#929—216no—
115C2×(C2×D3)⋊C4—24no—
116C2×(C2×D3)⋊C4—24no—
117#926—22no—
118#926—22no—
119C2×C4⋊Dic3D412yes—
120C2×Dic3⋊C4—24no—
121C2×C4×Dic3C2312yes—
122C2×#208—22no—
123C2×#208—22no—
124C2×#208D414yes—
125C22×C3⋊C8C2312yes—
126C2×Dic12—22no—
127C2×Dic12—22no—
128C2×Dic12—22no—
129#904—232no—
130C2×D24—22no—
131C2×D24—22no—
132C2×D24—22no—
133C2×#204—22no—
134C2×#204—22no—
135C2×#204—22no—
136C2×#204—22no—
137C2×#204—22no—
138C2×#204—22no—
139C3×#103—24no—
140C3×#103—24no—
141C3⋊#95—24no—
142#810—24no—
143#809—24no—
144C22×#41—31no—
145C22×#41—31no—
146C23×Q8C2×D311yescharacteristic
147C2×#137—64no—
148C2×#137—64no—
149C2×#137—64no—
150C2×#135—64no—
151C2×#135—64no—
152C2×#135—64no—
153C22×Q16—32no—
154C22×Q16—31no—
155C22×SD16—32no—
156C22×SD16—31no—
157C22×SD16—31no—
158C22×D8—31no—
159C22×M16—31no—
160C23×C8C2×D311yescharacteristic
161#536—64no—
162#536—64no—
163C2×C4⋊Q8—32no—
164C2×#124—32no—
165C2×#123—62no—
166C2×C22⋊Q8—62no—
167C2×#117—31no—
168C22×C4⋊C4C2×D311yescharacteristic
169#417—64no—
170C2×#103—24no—
171C2×#103—32no—
172C2×#103C2×D312yes—
173C2×#103C2×D312yes—
174C2×#102—32no—
175#391—32no—
176#390—32no—
177#386—62no—
178#380—62no—
179#336—32no—
180#336—32no—
181C24×C3C2×D411yescharacteristic
182C23×D3—21no—
183C23×D3—21no—
184C2×C3×Q8—24no—
185C2×C3×Q8—22no—
186C2×C3×Q8—28no—
187C2×C3×Q8—28no—
188C2×C3×Q8—22no—
189C2×C3×Q8—22no—
190C2×C3×Q8—22no—
191C2×C3×Q8—22no—
192C2×C3×Q8—44no—
193C2×C3×Q8—42no—
194C2×C3×Q8—44no—
195C2×C3×Q8—42no—
196C2×C3×Q8C2×D414yes—
197C22×C3×C4—24no—
198C22×C3×C4—22no—
199C22×C3×C4—22no—
200C22×C3×C4—42no—
201C22×C3×C4—42no—
202C22×C3×C4—42no—
203C22×C3×C4—82no—
204C22×C3×C4C2×D412yes—
205C22×C3×C4C2×D412yes—
206C22×C3×C4C2×D412yes—
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D412yes—
209C22×C3×C4C2×D412yes—
210C22×C3×C4#4112yes—
211C22×C3×C4C2411yescharacteristic
212C2×#67—24no—
213C2×#67—24no—
214C2×#67—42no—
215C2×#67—42no—
216C2×#67—42no—
217C2×#67—42no—
218C22×Dic3—21no—
219C22×Dic3—21no—
220C22×Dic3C2×D414yes—
221#235—28no—
222#235—28no—
223#235—28no—
224#235—28no—
225#235—44no—
226#235—44no—
227#235—44no—
228#235—44no—
229C2×D12—21no—
230C2×D12—21no—
231C2×D12—21no—
232C2×D12—21no—
233C2×D12—21no—
234C2×D12—21no—
235C2×D12—21no—
236C2×D12—21no—
237C2×D12—21no—
238C2×D12—21no—
239C2×D12C2×D412yes—
240C2×D12C2×D412yes—
241C2×C4×D3—24no—
242C2×C4×D3—24no—
243C2×C4×D3—42no—
244C2×C4×D3—42no—
245C2×C4×D3—42no—
246C2×C4×D3—42no—
247C2×Dic6—21no—
248C2×Dic6—21no—
249C2×Dic6—21no—
250C2×Dic6—21no—
251C2×Dic6—21no—
252C2×Dic6—21no—
253C2×Dic6—21no—
254C2×Dic6—21no—
255C2×Dic6—21no—
256C2×Dic6—21no—
257C2×Dic6C2×D412yes—
258C2×Dic6C2×D412yes—
259C2×C3×C8—28no—
260C2×C3×C8—21no—
261C2×C3×C8—21no—
262C2×C3×C8—21no—
263C2×C3×C8—21no—
264C2×C3×C8C2×D412yes—
265C2×C3×C8C2×D412yes—
266C3×C4⋊C4—22no—
267C3×C4⋊C4—22no—
268C3×C4⋊C4—42no—
269#217—24no—
270#216—216no—
271#215—216no—
272(C2×D3)⋊C4—216no—
273(C2×D3)⋊C4—84no—
274C4⋊Dic3—24no—
275Dic3⋊C4—216no—
276C4×Dic3—22no—
277C4×Dic3—22no—
278#208—28no—
279C2×C3⋊C8—22no—
280C2×C3⋊C8—22no—
281C2×C3⋊C8C2×D414yes—
282Dic12—28no—
283D24—28no—
284#204—28no—
285#204—28no—
286C2×#41—34no—
287C2×#41—34no—
288C2×#41—62no—
289C2×#41—62no—
290C2×#41—62no—
291C2×#41—62no—
292C2×#41—62no—
293C2×#41—62no—
294C2×#41—62no—
295C2×#41—62no—
296C2×#41—62no—
297C2×#41—62no—
298C22×Q8—21no—
299C22×Q8—22no—
300C22×Q8—22no—
301C22×Q8—22no—
302C22×Q8—22no—
303C22×Q8—22no—
304C22×Q8—31no—
305C22×Q8—31no—
306C22×Q8D1214yes—
307C22×Q8C22×D312yes—
308C22×D4—31no—
309C22×D4—31no—
310C22×D4—61no—
311C22×D4—61no—
312C23×C4—21no—
313C23×C4—21no—
314C23×C4—61no—
315C23×C4—61no—
316C23×C4C22×D311yescharacteristic
317#137—632no—
318#135—632no—
319C2×Q16—62no—
320C2×Q16—62no—
321C2×Q16—64no—
322C2×Q16—64no—
323C2×Q16—64no—
324C2×Q16—62no—
325C2×SD16—62no—
326C2×SD16—62no—
327C2×SD16—62no—
328C2×SD16—62no—
329C2×SD16—64no—
330C2×SD16—64no—
331C2×SD16—64no—
332C2×SD16—62no—
333C2×SD16—62no—
334C2×D8—62no—
335C2×D8—62no—
336C2×D8—62no—
337C2×M16—34no—
338C2×M16—62no—
339C2×M16—62no—
340C22×C8—22no—
341C22×C8—22no—
342C22×C8—32no—
343C22×C8#6714yes—
344C22×C8C22×D311yescharacteristic
345C22×C8C22×D311yescharacteristic
346C4⋊Q8—64no—
347C4⋊Q8—64no—
348#124—64no—
349#124—64no—
350#123—616no—
351C22⋊Q8—616no—
352#117—62no—
353#117—62no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—32no—
358C2×C4⋊C4—64no—
359C2×C4⋊C4C22×D312yes—
360C2×C22⋊C4—32no—
361C2×C22⋊C4—64no—
362C2×C22⋊C4—64no—
363C2×C42—32no—
364#103—24no—
365#103—24no—
366#103—64no—
367#102—64no—
368#95—64no—
369C23×C3—22no—
370C23×C3—22no—
371C23×C3#12012yes—
372C23×C3#12112yes—
373C23×C3#12712yes—
374C23×C3C22×D411yescharacteristic
375C22×D3—22no—
376C22×D3—22no—
377C22×D3—41no—
378C22×D3—41no—
379C22×D3—41no—
380C22×D3—41no—
381C22×D3—42no—
382C22×D3—42no—
383C22×D3—42no—
384C22×D3—42no—
385C3×Q8—24no—
386C3×Q8—24no—
387C3×Q8—42no—
388C3×Q8—42no—
389C3×Q8—48no—
390C2×C3×C4—24no—
391C2×C3×C4—42no—
392C2×C3×C4—42no—
393C2×C3×C4—48no—
394C2×C3×C4—42no—
395C2×C3×C4—44no—
396C2×C3×C4—84no—
397C2×C3×C4#12014yes—
398C2×C3×C4#12012yes—
399C2×C3×C4#12014yes—
400C2×C3×C4#12114yes—
401C2×C3×C4#12112yes—
402C2×C3×C4#12114yes—
403C2×C3×C4#12712yes—
404C2×C3×C4C22×D411yescharacteristic
405C2×C3×C4C22×D411yescharacteristic
406C2×C3×C4C22×D411yescharacteristic
407C2×C3×C4C22×D411yescharacteristic
408C2×C3×C4C22×D411yescharacteristic
409C2×C3×C4C2×#4111yescharacteristic
410#67—48no—
411#67—48no—
412C2×Dic3—24no—
413C2×Dic3—22no—
414C2×Dic3—22no—
415C2×Dic3—24no—
416C2×Dic3—24no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421D12—22no—
422D12—22no—
423D12—22no—
424D12—22no—
425D12—41no—
426D12—41no—
427D12—41no—
428D12—41no—
429C4×D3—48no—
430C4×D3—48no—
431Dic6—22no—
432Dic6—22no—
433Dic6—22no—
434Dic6—22no—
435Dic6—41no—
436Dic6—41no—
437Dic6—41no—
438Dic6—41no—
439C3×C8—22no—
440C3×C8—22no—
441C3⋊C8—24no—
442C24—61no—
443C24—61no—
444C24D3×D411yescharacteristic
445#41—68no—
446#41—68no—
447#41—68no—
448#41—68no—
449#41—124no—
450#41—124no—
451#41—124no—
452#41—124no—
453C2×Q8—24no—
454C2×Q8—22no—
455C2×Q8—28no—
456C2×Q8—28no—
457C2×Q8—22no—
458C2×Q8—22no—
459C2×Q8—22no—
460C2×Q8—22no—
461C2×Q8—32no—
462C2×Q8—32no—
463C2×Q8—44no—
464C2×Q8—42no—
465C2×Q8—44no—
466C2×Q8—42no—
467C2×Q8—61no—
468C2×Q8—61no—
469C2×Q8—61no—
470C2×Q8—61no—
471C2×Q8—61no—
472C2×Q8—61no—
473C2×Q8—61no—
474C2×Q8—61no—
475C2×Q8—61no—
476C2×Q8—61no—
477C2×Q8C2×D1214yes—
478C2×D4—32no—
479C2×D4—32no—
480C2×D4—64no—
481C2×D4—64no—
482C2×D4—61no—
483C2×D4—61no—
484C2×D4—61no—
485C2×D4—61no—
486C2×D4—61no—
487C2×D4—61no—
488C2×D4—61no—
489C2×D4—61no—
490C2×D4—61no—
491C2×D4—61no—
492C2×D4—122no—
493C2×D4—122no—
494C2×D4—122no—
495C2×D4—122no—
496C22×C4—24no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—34no—
500C22×C4—42no—
501C22×C4—42no—
502C22×C4—42no—
503C22×C4—61no—
504C22×C4—61no—
505C22×C4—64no—
506C22×C4—64no—
507C22×C4—82no—
508C22×C4—122no—
509C22×C4—122no—
510C22×C4—122no—
511C22×C4—122no—
512C22×C4C2×D1212yes—
513C22×C4#23512yes—
514C22×C4D3×D411yescharacteristic
515C22×C4D3×D412yes—
516C22×C4D3×D412yes—
517C22×C4D3×D412yes—
518C22×C4C2×#6712yes—
519C22×C4C23×D311yescharacteristic
520Q16—68no—
521Q16—616no—
522SD16—68no—
523SD16—68no—
524SD16—616no—
525D8—68no—
526M16—68no—
527C2×C8—28no—
528C2×C8—21no—
529C2×C8—21no—
530C2×C8—21no—
531C2×C8—21no—
532C2×C8—34no—
533C2×C8—62no—
534C2×C8—62no—
535C2×C8C2×#6712yes—
536C2×C8C2×#6712yes—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—42no—
540C4⋊C4—616no—
541C4⋊C4—64no—
542C22⋊C4—616no—
543C22⋊C4—64no—
544C22⋊C4—244no—
545C42—62no—
546C42—62no—
547C22×C3—22no—
548C22×C3—22no—
549C22×C3—24no—
550C22×C3—24no—
551C22×C3#39214yes—
552C22×C3C2×#12011yescharacteristic
553C22×C3C2×#12111yescharacteristic
554C22×C3C2×#12711yescharacteristic
555C22×C3#57712yes—
556C22×C3#57812yes—
557C2×D3—42no—
558C2×D3—42no—
559C2×D3—44no—
560C2×D3—44no—
561C2×D3—81no—
562C2×D3—81no—
563C2×D3—81no—
564C2×D3—81no—
565C2×D3—82no—
566C2×D3—82no—
567C3×C4—44no—
568C3×C4—82no—
569C3×C4#39214yes—
570C3×C4C2×#12011yescharacteristic
571C3×C4C2×#12011yescharacteristic
572C3×C4C2×#12111yescharacteristic
573C3×C4C2×#12111yescharacteristic
574Dic3—42no—
575Dic3—42no—
576Dic3—44no—
577C23—22no—
578C23—22no—
579C23—62no—
580C23—62no—
581C23—121no—
582C23—121no—
583C23—122no—
584C23—122no—
585C23—122no—
586C23—122no—
587C23—121no—
588C23—121no—
589C23#88212yes—
590C23#88412yes—
591C23#94012yes—
592C23C2×D3×D411yescharacteristic
593Q8—24no—
594Q8—24no—
595Q8—48no—
596Q8—42no—
597Q8—42no—
598Q8—62no—
599Q8—62no—
600Q8—62no—
601Q8—62no—
602Q8—121no—
603Q8—121no—
604Q8—121no—
605Q8—121no—
606D4—62no—
607D4—62no—
608D4—62no—
609D4—62no—
610D4—128no—
611D4—128no—
612D4—121no—
613D4—121no—
614D4—121no—
615D4—121no—
616C2×C4—24no—
617C2×C4—48no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—64no—
623C2×C4—62no—
624C2×C4—62no—
625C2×C4—64no—
626C2×C4—64no—
627C2×C4—84no—
628C2×C4—121no—
629C2×C4—121no—
630C2×C4—121no—
631C2×C4—121no—
632C2×C4—128no—
633C2×C4—128no—
634C2×C4#88212yes—
635C2×C4#88214yes—
636C2×C4#88412yes—
637C2×C4#88414yes—
638C2×C4#93014yes—
639C2×C4#93714yes—
640C2×C4#94012yes—
641C2×C4C22×D1211yescharacteristic
642C2×C4C2×#23511yescharacteristic
643C2×C4C2×D3×D411yescharacteristic
644C2×C4C2×D3×D411yescharacteristic
645C2×C4C2×D3×D411yescharacteristic
646C2×C4C22×#6711yescharacteristic
647C8—22no—
648C8—22no—
649C8—64no—
650C2×C3—24no—
651C2×C3C2×#39211yescharacteristic
652C2×C3#301312yes—
653C2×C3#304612yes—
654C2×C3#315411yescharacteristic
655C2×C3#315611yescharacteristic
656D3—82no—
657D3—82no—
658C22—22no—
659C22—22no—
660C22—24no—
661C22—24no—
662C22—124no—
663C22—122no—
664C22—122no—
665C22—124no—
666C22—241no—
667C22—241no—
668C22—241no—
669C22—241no—
670C22—242no—
671C22—242no—
672C22#496014yes—
673C22C2×#88211yescharacteristic
674C22C2×#88411yescharacteristic
675C22#576212yes—
676C22#576612yes—
677C22C2×#94011yescharacteristic
678C4—44no—
679C4—82no—
680C4—124no—
681C4—122no—
682C4—122no—
683C4#496014yes—
684C4C2×#88211yescharacteristic
685C4C2×#88411yescharacteristic
686C4C2×#93011yescharacteristic
687C4C2×#93711yescharacteristic
688C3#2342711yescharacteristic, complemented, Sylow
689C2—24no—
690C2—242no—
691C2—242no—
692C2#7879512yes—
693C2#7881512yes—
694C2C2×#496011yescharacteristic
695C2#8385411yescharacteristic
696C2#8388111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC22×C3×#103C2
socleC23×C3C22×D4
prime radicalC22×#904C2
hypercenterC22×#103D3
hypocenterC3#23427
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#103

Complements

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

NH ≅ G/Ntype
1. C2×#5816C2semidirect
2. C22×#942C2semidirect
4. C22×C3×#103C2semidirect
5. #82186C2semidirect
6. C2×#4966C2semidirect
7. #71965C2semidirect
8. #70283C2semidirect
9. #70282C2semidirect
10. #70278C2semidirect
11. #70277C2semidirect
54. C22×#103D3semidirect
688. C3#23427semidirect

Maximal subgroups

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

subgroupindex
C2×#58162
C22×#9422
C22×#9042
C22×C3×#1032
#821862
C2×#49662
#719652
#702832
#702822
#702782
#702772
#234273