Finite Group Databasebeta

← #10822116

Subgroups — #10822116

Order 768.

1954 conjugacy classes of subgroups.

The subgroup lattice has 750 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#536C211yesmaximal, characteristic, complemented
2#87268C211yesmaximal, characteristic
3C22×#942C211yesmaximal, characteristic, complemented
4C2×#5237C211yesmaximal, characteristic, complemented
5C2×#5225C211yesmaximal, characteristic, complemented
6C2×#5005C211yesmaximal, characteristic, complemented
7#71795C214yesmaximal, complemented
8#71731C214yesmaximal, complemented
9#71578C211yesmaximal, characteristic, complemented
10#23401—31nomaximal, Sylow
11C22×C3×#41C2211yescharacteristic
12C23×C3×Q8C2211yescharacteristic
13C22×#235C2211yescharacteristic
14C3×#536—24no—
15C3×#536—24no—
16C2×C3×C4⋊Q8C2212yescomplemented
17C2×C3×#124C2212yescomplemented
18C2×C3×#123—22no—
19C2×C3×C22⋊Q8—22no—
20C2×C3×#117C2211yescharacteristic
21#5833—22no—
22#5832—22no—
23#5830—21no—
24#5828—22no—
25#5825—21no—
26#5816C2212yes—
27C22×#216—21no—
28C2×#942—24no—
29C2×#942—22no—
30C2×#942—24no—
31C2×#942C2214yes—
32C22×#215—21no—
33#5805C2212yes—
34#5804—22no—
35#5791—22no—
36C2×#926C2211yescharacteristic
37C22×#208C2211yescharacteristic
38C22×#208C2211yescharacteristic
39#5237—24no—
40C2×#836C2212yescomplemented
41#5225—22no—
42#5225C2214yescomplemented
43C2×#835—22no—
44C2×#833C2212yescomplemented
45#5005—24no—
46C2×#805C2214yes—
47#4523C2214yescomplemented
48C22×#137—31no—
49#3449—31no—
50C2×#536D311yescharacteristic, complemented
51C2×#421—31no—
52C2×#419—31no—
53C2×#413—31no—
54#2005—34no—
55#2004—34no—
56#1733—31no—
57C2×C3×#41—22no—
58C2×C3×#41—21no—
59C2×C3×#41—21no—
60C2×C3×#41—21no—
61C2×C3×#41—21no—
62C2×C3×#41—22no—
63C2×C3×#41—22no—
64C2×C3×#41—22no—
65C22×C3×Q8—21no—
66C22×C3×Q8—22no—
67C22×C3×Q8—21no—
68C22×C3×Q8—21no—
69C22×C3×Q8—21no—
70C22×C3×Q8—21no—
71C22×C3×Q8—22no—
72C22×C3×Q8—22no—
73C22×C3×Q8D414yescomplemented
74C22×C3×Q8C2312yes—
75C22×C3×Q8C2311yescharacteristic
76C22×C3×D4D412yes—
77C22×C3×D4C2311yescharacteristic
78C23×C3×C4—21no—
79C23×C3×C4D412yes—
80C23×C3×C4C2311yescharacteristic
81C22×#67—21no—
82C2×#235—21no—
83C2×#235—21no—
84C2×#235—22no—
85C2×#235—22no—
86C2×#235—22no—
87C2×#235—22no—
88C2×#235D412yes—
89C2×#235C2312yes—
90C22×D12D411yescharacteristic
91C22×C4×D3—21no—
92C22×Dic6D411yescharacteristic
93C3×C4⋊Q8—24no—
94C3×C4⋊Q8—24no—
95C3×#124—24no—
96C3×#124—24no—
97C3×#123—216no—
98C3×C22⋊Q8—216no—
99C3×#117—22no—
100C3×#117D414yescomplemented
101C2×C3×C4⋊C4—21no—
102C2×C3×C4⋊C4—24no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4—24no—
105C2×C3×C42C2312yes—
106#953—24no—
107C2×#217—21no—
108#947—22no—
109#946—24no—
110#944—22no—
111C2×#216—28no—
112C2×#216—24no—
113#942—216no—
114#942—28no—
115#942—28no—
116C2×#215—28no—
117C2×#215—24no—
118#940—22no—
119#939—28no—
120#938—24no—
121#937—24no—
122#936—22no—
123#935—24no—
124#930—24no—
125#929—24no—
126C4×#67—28no—
127#926—21no—
128#926—21no—
129#926D412yes—
130#926C2312yes—
131C2×C4⋊Dic3D412yes—
132#924—24no—
133C2×C4×Dic3—21no—
134C2×#208—21no—
135C2×#208—22no—
136C2×#208—21no—
137C2×#208—22no—
138C2×#208—22no—
139C2×#208D414yescomplemented
140C2×#208C2312yes—
141C2×#208C2312yes—
142C22×C3⋊C8—21no—
143C22×C3⋊C8—21no—
144#836—24no—
145#835—28no—
146#833—24no—
147#805—28no—
148C22×#41—31no—
149C22×#41C2×D311yescharacteristic
150C23×Q8C2×D311yescharacteristic
151C2×#137—34no—
152C2×#137—64no—
153C2×#137—62no—
154C2×#137—64no—
155C22×Q16—61no—
156C22×SD16—61no—
157C22×M16—31no—
158C22×M16—31no—
159#537—62no—
160#536—24no—
161#536—24no—
162#536—32no—
163#536—61no—
164#535—32no—
165#535—61no—
166#534—62no—
167#532—62no—
168C2×C4⋊Q8C2×D312yescomplemented
169C2×#124C2×D312yescomplemented
170#525—62no—
171#525—62no—
172C2×#123—22no—
173C2×C22⋊Q8—22no—
174C2×#117—31no—
175C2×#117C2×D311yescharacteristic
176#421—64no—
177C2×#104—34no—
178#419—34no—
179#419—62no—
180C2×#103—62no—
181#413—64no—
182C2×#101—32no—
183C2×#100—32no—
184#339—34no—
185C24×C3C2×D412yes—
186C24×C3C2×D411yescharacteristic
187C23×D3—21no—
188C3×#41—42no—
189C3×#41—42no—
190C3×#41—42no—
191C3×#41—42no—
192C3×#41—42no—
193C3×#41—42no—
194C3×#41—42no—
195C3×#41—42no—
196C2×C3×Q8—24no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—22no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8—21no—
204C2×C3×Q8—24no—
205C2×C3×Q8—24no—
206C2×C3×Q8—24no—
207C2×C3×Q8—24no—
208C2×C3×Q8—22no—
209C2×C3×Q8—22no—
210C2×C3×Q8—22no—
211C2×C3×Q8—22no—
212C2×C3×Q8—41no—
213C2×C3×Q8—41no—
214C2×C3×Q8—42no—
215C2×C3×Q8—42no—
216C2×C3×Q8—42no—
217C2×C3×Q8—42no—
218C2×C3×Q8—42no—
219C2×C3×Q8C2×D414yes—
220C2×C3×D4—22no—
221C2×C3×D4—22no—
222C2×C3×D4—24no—
223C2×C3×D4—24no—
224C2×C3×D4—21no—
225C2×C3×D4—21no—
226C2×C3×D4—21no—
227C2×C3×D4—21no—
228C2×C3×D4—21no—
229C2×C3×D4—21no—
230C22×C3×C4—24no—
231C22×C3×C4—22no—
232C22×C3×C4—22no—
233C22×C3×C4—22no—
234C22×C3×C4—22no—
235C22×C3×C4—24no—
236C22×C3×C4—24no—
237C22×C3×C4—41no—
238C22×C3×C4—41no—
239C22×C3×C4—42no—
240C22×C3×C4C2×D411yescharacteristic
241C22×C3×C4C2×D411yescharacteristic
242C22×C3×C4C2×D412yes—
243C22×C3×C4C2×D411yescharacteristic
244C22×C3×C4C2×D411yescharacteristic
245C22×C3×C4C2×D412yes—
246C22×C3×C4C2×D411yescharacteristic
247C22×C3×C4C2×D412yes—
248C22×C3×C4C2×D412yes—
249C22×C3×C4#4112yes—
250C22×C3×C4C2411yescharacteristic
251C2×#67—22no—
252C2×#67—22no—
253C2×#67—28no—
254C2×#67—42no—
255C2×#67—42no—
256C22×Dic3—21no—
257C22×Dic3—22no—
258#235—24no—
259#235—24no—
260#235—22no—
261#235—22no—
262#235—22no—
263#235—22no—
264#235—42no—
265#235—42no—
266#235—42no—
267#235—42no—
268C2×D12—22no—
269C2×D12—22no—
270C2×D12—21no—
271C2×D12C2×D412yes—
272C2×C4×D3—22no—
273C2×C4×D3—22no—
274C2×C4×D3—42no—
275C2×C4×D3—42no—
276C2×Dic6—22no—
277C2×Dic6—22no—
278C2×Dic6—21no—
279C2×Dic6C2×D412yes—
280C3×C4⋊C4—216no—
281C3×C4⋊C4—24no—
282C3×C22⋊C4—216no—
283C3×C22⋊C4—24no—
284C3×C42—22no—
285C3×C42C2×D414yes—
286#217—28no—
287#217—22no—
288#217—22no—
289#216—216no—
290#215—216no—
291(C2×D3)⋊C4—28no—
292C4⋊Dic3—22no—
293C4⋊Dic3C2×D414yes—
294Dic3⋊C4—28no—
295C4×Dic3—22no—
296C4×Dic3—22no—
297#208—24no—
298#208—22no—
299#208—22no—
300#208—22no—
301#208—42no—
302#208C2×D414yes—
303C2×C3⋊C8—24no—
304C2×C3⋊C8—22no—
305C2×C3⋊C8—22no—
306C2×C3⋊C8—42no—
307C2×#41—21no—
308C2×#41—21no—
309C2×#41—21no—
310C2×#41—21no—
311C2×#41—22no—
312C2×#41—22no—
313C2×#41—22no—
314C2×#41—22no—
315C2×#41—32no—
316C2×#41—32no—
317C2×#41—61no—
318C2×#41—61no—
319C2×#41—62no—
320C2×#41—62no—
321C2×#41—62no—
322C2×#41—62no—
323C22×Q8—21no—
324C22×Q8—22no—
325C22×Q8—21no—
326C22×Q8—21no—
327C22×Q8—22no—
328C22×Q8—21no—
329C22×Q8—21no—
330C22×Q8—22no—
331C22×Q8—31no—
332C22×Q8#6714yescomplemented
333C22×Q8C22×D312yes—
334C22×Q8C22×D311yescharacteristic
335C22×D4—31no—
336C22×D4—61no—
337C22×D4#6712yes—
338C22×D4C22×D311yescharacteristic
339C23×C4—21no—
340C23×C4—61no—
341C23×C4#6712yes—
342C23×C4C22×D311yescharacteristic
343#137—68no—
344#137—616no—
345#137—68no—
346C2×Q16—64no—
347C2×Q16—68no—
348C2×SD16—64no—
349C2×SD16—68no—
350C2×M16—34no—
351C2×M16—32no—
352C2×M16—32no—
353C2×M16—61no—
354C2×M16—62no—
355C2×M16—61no—
356C2×M16—62no—
357C2×M16—62no—
358C22×C8—61no—
359C22×C8—61no—
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—216no—
369#123—64no—
370#123—64no—
371C22⋊Q8—216no—
372C22⋊Q8—64no—
373C22⋊Q8—64no—
374#121—64no—
375#121—64no—
376#121—68no—
377#120—64no—
378#120—64no—
379C4×D4—68no—
380#117—22no—
381#117—32no—
382#117—32no—
383#117—61no—
384#117—61no—
385#117#6714yescomplemented
386C2×C4⋊C4—21no—
387C2×C4⋊C4—24no—
388C2×C4⋊C4—32no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—24no—
391C2×C22⋊C4—61no—
392C2×C42—61no—
393C2×C42C22×D312yes—
394#104—68no—
395#103—68no—
396#101—64no—
397#100—64no—
398C23×C3—22no—
399C23×C3—22no—
400C23×C3—22no—
401C23×C3—42no—
402C23×C3—44no—
403C23×C3#12011yescharacteristic
404C23×C3#12011yescharacteristic
405C23×C3#12012yes—
406C23×C3#12112yes—
407C23×C3C22×D411yescharacteristic
408C22×D3—22no—
409C22×D3—42no—
410C22×D3—82no—
411C3×Q8—24no—
412C3×Q8—24no—
413C3×Q8—41no—
414C3×Q8—42no—
415C3×Q8—41no—
416C3×Q8—41no—
417C3×Q8—42no—
418C3×Q8—41no—
419C3×Q8—44no—
420C3×Q8—44no—
421C3×D4—44no—
422C3×D4—44no—
423C3×D4—41no—
424C3×D4—41no—
425C3×D4—41no—
426C3×D4—41no—
427C2×C3×C4—22no—
428C2×C3×C4—24no—
429C2×C3×C4—24no—
430C2×C3×C4—22no—
431C2×C3×C4—22no—
432C2×C3×C4—44no—
433C2×C3×C4—42no—
434C2×C3×C4—42no—
435C2×C3×C4—44no—
436C2×C3×C4—44no—
437C2×C3×C4—44no—
438C2×C3×C4—44no—
439C2×C3×C4#12012yes—
440C2×C3×C4#12012yes—
441C2×C3×C4#12011yescharacteristic
442C2×C3×C4#12011yescharacteristic
443C2×C3×C4#12012yes—
444C2×C3×C4#12112yes—
445C2×C3×C4#12112yes—
446C2×C3×C4#12112yes—
447C2×C3×C4#12112yes—
448C2×C3×C4#12112yes—
449C2×C3×C4#12712yes—
450C2×C3×C4#12712yes—
451C2×C3×C4C22×D411yescharacteristic
452C2×C3×C4C22×D411yescharacteristic
453C2×C3×C4C22×D411yescharacteristic
454C2×C3×C4C22×D411yescharacteristic
455C2×C3×C4C22×D411yescharacteristic
456C2×C3×C4C2×#4111yescharacteristic
457#67—42no—
458#67—42no—
459#67—44no—
460#67—88no—
461C2×Dic3—24no—
462C2×Dic3—22no—
463C2×Dic3—42no—
464C2×Dic3—82no—
465D12—22no—
466D12—22no—
467D12—42no—
468C4×D3—42no—
469C4×D3—42no—
470C4×D3—44no—
471Dic6—22no—
472Dic6—22no—
473Dic6—42no—
474C3⋊C8—24no—
475C3⋊C8—42no—
476C24—61no—
477C24C2×#6712yes—
478C24C2×#6711yescharacteristic
479#41—42no—
480#41—42no—
481#41—42no—
482#41—42no—
483#41—42no—
484#41—42no—
485#41—42no—
486#41—42no—
487#41—64no—
488#41—64no—
489#41—62no—
490#41—62no—
491#41—62no—
492#41—62no—
493#41—122no—
494#41—122no—
495#41—122no—
496#41—122no—
497C2×Q8—24no—
498C2×Q8—21no—
499C2×Q8—21no—
500C2×Q8—21no—
501C2×Q8—21no—
502C2×Q8—22no—
503C2×Q8—21no—
504C2×Q8—21no—
505C2×Q8—24no—
506C2×Q8—24no—
507C2×Q8—24no—
508C2×Q8—24no—
509C2×Q8—22no—
510C2×Q8—22no—
511C2×Q8—22no—
512C2×Q8—22no—
513C2×Q8—32no—
514C2×Q8—41no—
515C2×Q8—41no—
516C2×Q8—42no—
517C2×Q8—42no—
518C2×Q8—42no—
519C2×Q8—42no—
520C2×Q8—42no—
521C2×Q8—61no—
522C2×Q8—62no—
523C2×Q8—62no—
524C2×Q8C2×#6714yes—
525C2×D4—22no—
526C2×D4—22no—
527C2×D4—24no—
528C2×D4—24no—
529C2×D4—21no—
530C2×D4—21no—
531C2×D4—21no—
532C2×D4—21no—
533C2×D4—21no—
534C2×D4—21no—
535C2×D4—32no—
536C2×D4—62no—
537C2×D4—62no—
538C2×D4—68no—
539C2×D4—62no—
540C2×D4—61no—
541C2×D4—62no—
542C2×D4—122no—
543C2×D4—122no—
544C22×C4—24no—
545C22×C4—22no—
546C22×C4—22no—
547C22×C4—22no—
548C22×C4—22no—
549C22×C4—24no—
550C22×C4—24no—
551C22×C4—41no—
552C22×C4—41no—
553C22×C4—42no—
554C22×C4—61no—
555C22×C4—62no—
556C22×C4—62no—
557C22×C4—62no—
558C22×C4—122no—
559C22×C4—122no—
560C22×C4D3×D411yescharacteristic
561C22×C4D3×D411yescharacteristic
562C22×C4D3×D412yes—
563C22×C4D3×D411yescharacteristic
564C22×C4D3×D411yescharacteristic
565C22×C4#23712yes—
566C22×C4C2×#6712yes—
567C22×C4C2×#6711yescharacteristic
568C22×C4C2×#6712yes—
569C22×C4C2×#6712yes—
570C22×C4C23×D311yescharacteristic
571Q16—616no—
572SD16—616no—
573M16—34no—
574M16—64no—
575M16—62no—
576M16—62no—
577M16—62no—
578M16—122no—
579C2×C8—64no—
580C2×C8—62no—
581C2×C8—62no—
582C2×C8—122no—
583C4⋊C4—216no—
584C4⋊C4—24no—
585C4⋊C4—34no—
586C4⋊C4—68no—
587C4⋊C4—62no—
588C22⋊C4—216no—
589C22⋊C4—24no—
590C22⋊C4—68no—
591C22⋊C4—68no—
592C22⋊C4—62no—
593C22⋊C4—62no—
594C42—22no—
595C42—62no—
596C42—62no—
597C42C2×#6714yes—
598C22×C3—22no—
599C22×C3—22no—
600C22×C3—22no—
601C22×C3—42no—
602C22×C3—42no—
603C22×C3—44no—
604C22×C3—42no—
605C22×C3—44no—
606C22×C3—82no—
607C22×C3#39212yes—
608C22×C3#39212yes—
609C22×C3C2×#12011yescharacteristic
610C22×C3C2×#12011yescharacteristic
611C22×C3C2×#12111yescharacteristic
612C2×D3—42no—
613C2×D3—44no—
614C2×D3—84no—
615C3×C4—44no—
616C3×C4—42no—
617C3×C4—44no—
618C3×C4#39212yes—
619C3×C4#39212yes—
620C3×C4C2×#12011yescharacteristic
621C3×C4C2×#12111yescharacteristic
622C3×C4C2×#12111yescharacteristic
623C3×C4C2×#12711yescharacteristic
624Dic3—42no—
625Dic3—44no—
626C23—22no—
627C23—22no—
628C23—22no—
629C23—42no—
630C23—44no—
631C23—62no—
632C23—122no—
633C23—242no—
634C23#93711yescharacteristic
635C23#93711yescharacteristic
636C23#93912yes—
637C23#95312yes—
638C23C22×#6711yescharacteristic
639Q8—24no—
640Q8—24no—
641Q8—44no—
642Q8—44no—
643Q8—41no—
644Q8—42no—
645Q8—41no—
646Q8—41no—
647Q8—42no—
648Q8—41no—
649Q8—62no—
650Q8—62no—
651Q8—122no—
652D4—44no—
653D4—44no—
654D4—41no—
655D4—41no—
656D4—41no—
657D4—41no—
658D4—62no—
659D4—62no—
660D4—124no—
661D4—122no—
662D4—122no—
663D4—122no—
664D4—248no—
665C2×C4—22no—
666C2×C4—22no—
667C2×C4—24no—
668C2×C4—24no—
669C2×C4—22no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—64no—
678C2×C4—62no—
679C2×C4—122no—
680C2×C4—124no—
681C2×C4—122no—
682C2×C4—122no—
683C2×C4—242no—
684C2×C4#93711yescharacteristic
685C2×C4#93711yescharacteristic
686C2×C4#93712yes—
687C2×C4#93712yes—
688C2×C4#93812yes—
689C2×C4#93812yes—
690C2×C4#93912yes—
691C2×C4#93912yes—
692C2×C4#93912yes—
693C2×C4#94012yes—
694C2×C4#94012yes—
695C2×C4#95312yes—
696C2×C4C2×D3×D411yescharacteristic
697C2×C4C2×D3×D411yescharacteristic
698C2×C4C2×D3×D411yescharacteristic
699C2×C4C2×#23711yescharacteristic
700C2×C4C22×#6711yescharacteristic
701C2×C4C22×#6711yescharacteristic
702C8—64no—
703C8—122no—
704C2×C3—21no—
705C2×C3—22no—
706C2×C3—21no—
707C2×C3—22no—
708C2×C3—82no—
709C2×C3C2×#39211yescharacteristic
710C2×C3#302212yes—
711D3—82no—
712C22—22no—
713C22—22no—
714C22—22no—
715C22—42no—
716C22—42no—
717C22—44no—
718C22—42no—
719C22—44no—
720C22—82no—
721C22—122no—
722C22—124no—
723C22—244no—
724C22#522312yes—
725C22#522312yes—
726C22C2×#93711yescharacteristic
727C22C2×#93911yescharacteristic
728C22C2×#95311yescharacteristic
729C4—44no—
730C4—42no—
731C4—44no—
732C4—122no—
733C4—124no—
734C4#522312yes—
735C4#522312yes—
736C4C2×#93711yescharacteristic
737C4C2×#93811yescharacteristic
738C4C2×#93911yescharacteristic
739C4C2×#94011yescharacteristic
740C3#2340111yescharacteristic, complemented, Sylow
741C2—21no—
742C2—21no—
743C2—22no—
744C2—22no—
745C2—82no—
746C2—242no—
747C2#8341612yes—
748C2C2×#522311yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#536C2
socleC22×C3C2×#120
prime radical#87268C2
hypercenterC2×#536D3
hypocenterC3#23401
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#536

Complements

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

NH ≅ G/Ntype
1. C2×C3×#536C2semidirect
3. C22×#942C2semidirect
4. C2×#5237C2semidirect
5. C2×#5225C2semidirect
6. C2×#5005C2semidirect
7. #71795C2semidirect
8. #71731C2semidirect
9. #71578C2semidirect
16. C2×C3×C4⋊Q8C22semidirect
17. C2×C3×#124C22semidirect
40. C2×#836C22semidirect
42. #5225C22semidirect
44. C2×#833C22semidirect
47. #4523C22semidirect
50. C2×#536D3semidirect
73. C22×C3×Q8D4semidirect
100. C3×#117D4semidirect
139. C2×#208D4semidirect
168. C2×C4⋊Q8C2×D3semidirect
169. C2×#124C2×D3semidirect
332. C22×Q8#67semidirect
385. #117#67semidirect
740. C3#23401semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5362
#872682
C22×#9422
C2×#52372
C2×#52252
C2×#50052
#717952
#717312
#715782
#234013