Finite Group Databasebeta

← #10822450

Subgroups — #10822450

Order 768.

1994 conjugacy classes of subgroups.

The subgroup lattice has 752 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#87268C211yesmaximal, characteristic
2C2×#5825C211yesmaximal, characteristic, complemented
3C2×#5526C211yesmaximal, characteristic, complemented
4C2×C3×#410C211yesmaximal, characteristic, complemented
5C2×#5006C211yesmaximal, characteristic, complemented
6#82230C211yesmaximal, characteristic, complemented
7#71593C211yesmaximal, characteristic, complemented
8#70031C218yesmaximal, complemented
9#23446—31nomaximal, Sylow
10C22×C3×#41C2211yescharacteristic
11C22×#235C2211yescharacteristic
12C23×Dic6C2211yescharacteristic
13C2×C3×#117C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15#5833—22no—
16#5832—22no—
17#5830—21no—
18#5828—22no—
19#5825—21no—
20#5825—24no—
21#5825—24no—
22#5816—21no—
23C2×#944—21no—
24#5805—21no—
25#5804C2214yes—
26C2×#936—21no—
27C2×#935C2214yes—
28#5791—22no—
29C2×#926C2211yescharacteristic
30C2×#926C2211yescharacteristic
31C2×#924—22no—
32#5526—24no—
33#5526—24no—
34C2×#897—22no—
35C2×#886C2214yes—
36C2×#875—21no—
37C2×#869—21no—
38C3×#410—24no—
39C2×C3×#99C2214yes—
40#5006—24no—
41#4973—22no—
42C2×#806C2214yes—
43#4943C2214yes—
44#4929—22no—
45#4527C2214yes—
46#3449—31no—
47C2×#536—31no—
48C2×#536—31no—
49C2×#410—31no—
50C2×#410D311yescharacteristic, complemented
51#2310—31no—
52#2022—38no—
53#1740—31no—
54C2×C3×#41—22no—
55C2×C3×#41—22no—
56C2×C3×#41—22no—
57C2×C3×#41—21no—
58C2×C3×#41—21no—
59C2×C3×#41—21no—
60C2×C3×#41—21no—
61C2×C3×#41—22no—
62C22×C3×Q8D411yescharacteristic
63C22×C3×D4D411yescharacteristic
64C22×C3×D4C2312yes—
65C23×C3×C4D412yes—
66C23×C3×C4C2311yescharacteristic
67C22×#67C2312yes—
68C23×Dic3C2312yes—
69C2×#235—22no—
70C2×#235—22no—
71C2×#235—22no—
72C2×#235—21no—
73C2×#235—21no—
74C2×#235—21no—
75C2×#235—21no—
76C2×#235—22no—
77C22×D12D411yescharacteristic
78C22×C4×D3D412yes—
79C22×Dic6—21no—
80C22×Dic6—21no—
81C22×Dic6—21no—
82C22×Dic6—21no—
83C22×Dic6—22no—
84C22×Dic6—21no—
85C22×Dic6—21no—
86C22×Dic6—22no—
87C22×Dic6—22no—
88C22×Dic6D411yescharacteristic
89C22×Dic6D414yescomplemented
90C3×#117—22no—
91C3×#117—22no—
92C3×#117—22no—
93C3×#117D414yescomplemented
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C22⋊C4C2312yes—
97C2×C3×C22⋊C4C2312yes—
98C2×C3×C42—21no—
99C2×C3×C42—21no—
100#953—24no—
101C2×#217—24no—
102C2×#217C2312yes—
103C2×#217C2312yes—
104#947—22no—
105#946—24no—
106#944—28no—
107#944—22no—
108#940—22no—
109#939—28no—
110#938—24no—
111#937—24no—
112#936—28no—
113#936—22no—
114#935—28no—
115#935—24no—
116#935—28no—
117#930—24no—
118#929—24no—
119C4×#67—28no—
120C2×(C2×D3)⋊C4—24no—
121#926—21no—
122#926—21no—
123#926—22no—
124#926—21no—
125#926—21no—
126#926D414yescomplemented
127C2×C4⋊Dic3—21no—
128C2×C4⋊Dic3—24no—
129C2×C4⋊Dic3D412yes—
130#924—216no—
131#924—24no—
132C2×Dic3⋊C4—24no—
133C2×C4×Dic3—21no—
134C2×C4×Dic3D412yes—
135#897—216no—
136#886—28no—
137#886—28no—
138#875—28no—
139#869—28no—
140C3×#99—28no—
141C3⋊#95—28no—
142#806—28no—
143C22×#41—31no—
144C22×#41C2×D311yescharacteristic
145C23×Q8—31no—
146#537—62no—
147#536—64no—
148#536—64no—
149#536—64no—
150#536—61no—
151#536—61no—
152#536—64no—
153#535—61no—
154#535—61no—
155#534—34no—
156#532—62no—
157C2×C4⋊Q8—61no—
158C2×C4⋊Q8—61no—
159C2×#124—61no—
160C2×#124—61no—
161#525—62no—
162#525—62no—
163C2×#123—34no—
164C2×#123—34no—
165C2×C22⋊Q8—62no—
166C2×C22⋊Q8—62no—
167C2×#117—31no—
168C2×#117—31no—
169C2×#117C2×D311yescharacteristic
170C2×#117C2×D311yescharacteristic
171#410—24no—
172#410—64no—
173C2×#99—34no—
174C2×#99C2×D314yes—
175#391—62no—
176#391—62no—
177#381—34no—
178#342—34no—
179C24×C3C2×D412yes—
180C24×C3C2411yescharacteristic
181C23×D3C2×D412yes—
182C3×#41—42no—
183C3×#41—42no—
184C3×#41—42no—
185C3×#41—42no—
186C3×#41—42no—
187C3×#41—42no—
188C3×#41—42no—
189C3×#41—42no—
190C2×C3×Q8—22no—
191C2×C3×Q8—21no—
192C2×C3×Q8—21no—
193C2×C3×Q8—22no—
194C2×C3×D4—22no—
195C2×C3×D4—22no—
196C2×C3×D4—22no—
197C2×C3×D4—22no—
198C2×C3×D4—22no—
199C2×C3×D4—22no—
200C2×C3×D4—22no—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4—22no—
204C22×C3×C4—22no—
205C22×C3×C4—24no—
206C22×C3×C4—22no—
207C22×C3×C4—22no—
208C22×C3×C4—24no—
209C22×C3×C4—22no—
210C22×C3×C4C2×D412yes—
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2×D412yes—
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2×D412yes—
219C22×C3×C4C2×D412yes—
220C22×C3×C4#4112yes—
221C2×#67—22no—
222C2×#67—22no—
223C2×#67—22no—
224C2×#67—22no—
225C2×#67—22no—
226C2×#67—22no—
227C2×#67—28no—
228C22×Dic3—22no—
229C22×Dic3—22no—
230C22×Dic3—22no—
231C22×Dic3—22no—
232C22×Dic3—22no—
233C22×Dic3—22no—
234C22×Dic3—22no—
235C22×Dic3C2×D414yes—
236C22×Dic3C2×D412yes—
237C22×Dic3C2×D414yes—
238#235—42no—
239#235—42no—
240#235—42no—
241#235—42no—
242#235—42no—
243#235—42no—
244#235—42no—
245#235—42no—
246C2×D12—22no—
247C2×D12—21no—
248C2×D12—21no—
249C2×D12—22no—
250C2×C4×D3—24no—
251C2×C4×D3—22no—
252C2×C4×D3—22no—
253C2×C4×D3—24no—
254C2×Dic6—22no—
255C2×Dic6—24no—
256C2×Dic6—24no—
257C2×Dic6—24no—
258C2×Dic6—21no—
259C2×Dic6—24no—
260C2×Dic6—21no—
261C2×Dic6—24no—
262C2×Dic6—24no—
263C2×Dic6—22no—
264C2×Dic6—41no—
265C2×Dic6—41no—
266C2×Dic6—41no—
267C2×Dic6—41no—
268C2×Dic6—42no—
269C2×Dic6—42no—
270C2×Dic6—42no—
271C2×Dic6—42no—
272C2×Dic6—42no—
273C2×Dic6—42no—
274C3×C4⋊C4—24no—
275C3×C4⋊C4—42no—
276C3×C22⋊C4—22no—
277C3×C22⋊C4—22no—
278C3×C22⋊C4—22no—
279C3×C22⋊C4C2×D414yes—
280C3×C42—24no—
281C3×C42—42no—
282#217—216no—
283#217—28no—
284#217—22no—
285#217—22no—
286#217—22no—
287#217C2×D414yes—
288(C2×D3)⋊C4—216no—
289(C2×D3)⋊C4—28no—
290C4⋊Dic3—216no—
291C4⋊Dic3—24no—
292C4⋊Dic3—22no—
293C4⋊Dic3—22no—
294Dic3⋊C4—216no—
295Dic3⋊C4—28no—
296C4×Dic3—24no—
297C4×Dic3—22no—
298C4×Dic3—22no—
299C2×#41—22no—
300C2×#41—21no—
301C2×#41—21no—
302C2×#41—21no—
303C2×#41—22no—
304C2×#41—22no—
305C2×#41—21no—
306C2×#41—22no—
307C2×#41—61no—
308C2×#41—61no—
309C2×#41—62no—
310C2×#41—61no—
311C2×#41—62no—
312C2×#41—61no—
313C2×#41—62no—
314C2×#41—62no—
315C22×Q8—31no—
316C22×Q8—34no—
317C22×Q8—61no—
318C22×Q8—61no—
319C22×Q8—62no—
320C22×Q8—61no—
321C22×Q8—61no—
322C22×Q8—62no—
323C22×Q8—61no—
324C22×Q8—61no—
325C22×Q8—62no—
326C22×Q8D1211yescharacteristic
327C22×D4—32no—
328C22×D4—31no—
329C22×D4D1211yescharacteristic
330C22×D4C22×D312yes—
331C23×C4—32no—
332C23×C4—32no—
333C23×C4D1212yes—
334C23×C4C22×D311yescharacteristic
335C4⋊Q8—68no—
336C4⋊Q8—68no—
337C4⋊Q8—62no—
338#127—62no—
339#124—68no—
340#124—68no—
341#124—62no—
342#124—62no—
343#123—68no—
344#123—68no—
345#123—68no—
346#123—64no—
347#123—64no—
348#123—68no—
349C22⋊Q8—616no—
350C22⋊Q8—616no—
351C22⋊Q8—64no—
352C22⋊Q8—64no—
353#121—64no—
354#121—68no—
355#121—64no—
356#120—64no—
357#120—64no—
358C4×D4—68no—
359#117—22no—
360#117—22no—
361#117—22no—
362#117—34no—
363#117—61no—
364#117—61no—
365#117—61no—
366#117—61no—
367#117—62no—
368#117#6714yescomplemented
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—32no—
372C2×C4⋊C4—64no—
373C2×C4⋊C4—61no—
374C2×C4⋊C4—64no—
375C2×C22⋊C4—32no—
376C2×C22⋊C4—32no—
377C2×C22⋊C4—64no—
378C2×C22⋊C4—64no—
379C2×C22⋊C4C22×D312yes—
380C2×C22⋊C4C22×D312yes—
381C2×C42—21no—
382C2×C42—21no—
383C2×C42—32no—
384C2×C42—61no—
385#99—28no—
386#99—68no—
387#95—68no—
388C23×C3—22no—
389C23×C3—22no—
390C23×C3—22no—
391C23×C3—42no—
392C23×C3—44no—
393C23×C3C22×D411yescharacteristic
394C23×C3C22×D411yescharacteristic
395C23×C3C22×D411yescharacteristic
396C23×C3C22×D411yescharacteristic
397C23×C3C22×D411yescharacteristic
398C23×C3C22×D411yescharacteristic
399C23×C3C2×#4111yescharacteristic
400C22×D3—22no—
401C22×D3—22no—
402C22×D3—22no—
403C22×D3—44no—
404C3×Q8—42no—
405C3×Q8—42no—
406C3×D4—42no—
407C3×D4—42no—
408C3×D4—42no—
409C3×D4—42no—
410C3×D4—42no—
411C3×D4—42no—
412C2×C3×C4—22no—
413C2×C3×C4—22no—
414C2×C3×C4—24no—
415C2×C3×C4—22no—
416C2×C3×C4—44no—
417C2×C3×C4—44no—
418C2×C3×C4—44no—
419C2×C3×C4—44no—
420C2×C3×C4—42no—
421C2×C3×C4#12012yes—
422C2×C3×C4#12011yescharacteristic
423C2×C3×C4#12011yescharacteristic
424C2×C3×C4#12012yes—
425C2×C3×C4#12011yescharacteristic
426C2×C3×C4#12011yescharacteristic
427C2×C3×C4#12012yes—
428C2×C3×C4#12012yes—
429C2×C3×C4#12112yes—
430C2×C3×C4#12112yes—
431C2×C3×C4#12112yes—
432C2×C3×C4#12112yes—
433C2×C3×C4#12112yes—
434C2×C3×C4#12112yes—
435C2×C3×C4#12712yes—
436C2×C3×C4#12712yes—
437#67—42no—
438#67—42no—
439#67—42no—
440#67—42no—
441#67—88no—
442C2×Dic3—24no—
443C2×Dic3—24no—
444C2×Dic3—24no—
445C2×Dic3—22no—
446C2×Dic3—22no—
447C2×Dic3—24no—
448C2×Dic3—24no—
449C2×Dic3—22no—
450C2×Dic3—44no—
451C2×Dic3—42no—
452C2×Dic3—42no—
453C2×Dic3—42no—
454C2×Dic3—42no—
455C2×Dic3—44no—
456D12—42no—
457D12—42no—
458C4×D3—44no—
459C4×D3—44no—
460Dic6—44no—
461Dic6—44no—
462Dic6—44no—
463Dic6—44no—
464Dic6—42no—
465Dic6—42no—
466C24—32no—
467C24C2×D1212yes—
468C24C23×D311yescharacteristic
469#41—42no—
470#41—42no—
471#41—42no—
472#41—42no—
473#41—42no—
474#41—42no—
475#41—42no—
476#41—42no—
477#41—122no—
478#41—122no—
479#41—122no—
480#41—122no—
481#41—122no—
482#41—122no—
483#41—122no—
484#41—122no—
485C2×Q8—21no—
486C2×Q8—21no—
487C2×Q8—22no—
488C2×Q8—22no—
489C2×Q8—64no—
490C2×Q8—61no—
491C2×Q8—64no—
492C2×Q8—61no—
493C2×Q8—64no—
494C2×Q8—64no—
495C2×Q8—62no—
496C2×Q8—64no—
497C2×Q8—64no—
498C2×Q8—62no—
499C2×Q8—121no—
500C2×Q8—121no—
501C2×Q8—121no—
502C2×Q8—121no—
503C2×Q8—122no—
504C2×Q8—122no—
505C2×Q8—122no—
506C2×Q8—122no—
507C2×Q8—122no—
508C2×Q8—122no—
509C2×D4—22no—
510C2×D4—22no—
511C2×D4—22no—
512C2×D4—22no—
513C2×D4—22no—
514C2×D4—22no—
515C2×D4—21no—
516C2×D4—22no—
517C2×D4—21no—
518C2×D4—22no—
519C2×D4—62no—
520C2×D4—62no—
521C2×D4—62no—
522C2×D4—62no—
523C2×D4—62no—
524C2×D4—62no—
525C2×D4—68no—
526C2×D4—61no—
527C2×D4—61no—
528C2×D4—62no—
529C2×D4—62no—
530C22×C4—22no—
531C22×C4—22no—
532C22×C4—22no—
533C22×C4—24no—
534C22×C4—24no—
535C22×C4—22no—
536C22×C4—32no—
537C22×C4—34no—
538C22×C4—34no—
539C22×C4—62no—
540C22×C4—62no—
541C22×C4—62no—
542C22×C4—62no—
543C22×C4—62no—
544C22×C4—62no—
545C22×C4—62no—
546C22×C4—62no—
547C22×C4—62no—
548C22×C4—64no—
549C22×C4—64no—
550C22×C4C2×D1212yes—
551C22×C4C2×D1211yescharacteristic
552C22×C4C2×D1211yescharacteristic
553C22×C4#23512yes—
554C22×C4D3×D411yescharacteristic
555C22×C4D3×D411yescharacteristic
556C22×C4D3×D412yes—
557C22×C4D3×D411yescharacteristic
558C22×C4D3×D411yescharacteristic
559C22×C4D3×D412yes—
560C22×C4C2×#6712yes—
561C4⋊C4—24no—
562C4⋊C4—42no—
563C4⋊C4—616no—
564C4⋊C4—616no—
565C4⋊C4—68no—
566C4⋊C4—62no—
567C4⋊C4—62no—
568C4⋊C4—64no—
569C22⋊C4—22no—
570C22⋊C4—22no—
571C22⋊C4—22no—
572C22⋊C4—34no—
573C22⋊C4—616no—
574C22⋊C4—68no—
575C22⋊C4—616no—
576C22⋊C4—68no—
577C22⋊C4—62no—
578C22⋊C4—62no—
579C22⋊C4—62no—
580C22⋊C4C2×#6714yes—
581C42—24no—
582C42—42no—
583C42—62no—
584C42—62no—
585C42—64no—
586C22×C3—22no—
587C22×C3—22no—
588C22×C3—22no—
589C22×C3—41no—
590C22×C3—41no—
591C22×C3—41no—
592C22×C3—41no—
593C22×C3—44no—
594C22×C3—42no—
595C22×C3—44no—
596C22×C3—82no—
597C22×C3C2×#12011yescharacteristic
598C22×C3C2×#12011yescharacteristic
599C22×C3C2×#12011yescharacteristic
600C22×C3C2×#12111yescharacteristic
601C22×C3C2×#12111yescharacteristic
602C22×C3C2×#12111yescharacteristic
603C22×C3C2×#12711yescharacteristic
604C2×D3—44no—
605C2×D3—42no—
606C2×D3—44no—
607C2×D3—82no—
608C3×C4—44no—
609C3×C4—42no—
610C3×C4—82no—
611C3×C4#39212yes—
612C3×C4#39212yes—
613C3×C4#39212yes—
614C3×C4#39212yes—
615Dic3—44no—
616Dic3—44no—
617Dic3—42no—
618Dic3—44no—
619C23—22no—
620C23—22no—
621C23—22no—
622C23—42no—
623C23—44no—
624C23—62no—
625C23—62no—
626C23—62no—
627C23—124no—
628C23C22×D1211yescharacteristic
629C23C2×#23511yescharacteristic
630C23C2×D3×D411yescharacteristic
631C23C2×D3×D411yescharacteristic
632C23C2×D3×D411yescharacteristic
633C23C2×D3×D411yescharacteristic
634C23C22×#6711yescharacteristic
635Q8—42no—
636Q8—42no—
637Q8—124no—
638Q8—124no—
639Q8—124no—
640Q8—124no—
641Q8—122no—
642Q8—122no—
643D4—42no—
644D4—42no—
645D4—42no—
646D4—42no—
647D4—42no—
648D4—42no—
649D4—122no—
650D4—122no—
651D4—122no—
652D4—122no—
653D4—122no—
654D4—122no—
655D4—248no—
656C2×C4—22no—
657C2×C4—22no—
658C2×C4—22no—
659C2×C4—24no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—42no—
665C2×C4—64no—
666C2×C4—64no—
667C2×C4—64no—
668C2×C4—62no—
669C2×C4—62no—
670C2×C4—64no—
671C2×C4—64no—
672C2×C4—62no—
673C2×C4—124no—
674C2×C4—122no—
675C2×C4—122no—
676C2×C4—122no—
677C2×C4—122no—
678C2×C4—124no—
679C2×C4—124no—
680C2×C4—124no—
681C2×C4#88212yes—
682C2×C4#88212yes—
683C2×C4#88211yescharacteristic
684C2×C4#88211yescharacteristic
685C2×C4#88211yescharacteristic
686C2×C4#88211yescharacteristic
687C2×C4#88412yes—
688C2×C4#88412yes—
689C2×C4#88412yes—
690C2×C4#88412yes—
691C2×C4#93012yes—
692C2×C4#93012yes—
693C2×C4#93712yes—
694C2×C4#93712yes—
695C2×C4#94012yes—
696C2×C4#94012yes—
697C2×C3—21no—
698C2×C3—21no—
699C2×C3—21no—
700C2×C3—21no—
701C2×C3—21no—
702C2×C3—21no—
703C2×C3—82no—
704C2×C3C2×#39211yescharacteristic
705C2×C3#302812yes—
706D3—82no—
707C22—22no—
708C22—22no—
709C22—22no—
710C22—41no—
711C22—41no—
712C22—41no—
713C22—41no—
714C22—44no—
715C22—44no—
716C22—42no—
717C22—82no—
718C22—124no—
719C22—124no—
720C22—122no—
721C22—242no—
722C22C2×#88211yescharacteristic
723C22C2×#88211yescharacteristic
724C22C2×#88411yescharacteristic
725C22C2×#88411yescharacteristic
726C22C2×#93011yescharacteristic
727C22C2×#93711yescharacteristic
728C22C2×#94011yescharacteristic
729C4—44no—
730C4—42no—
731C4—82no—
732C4—124no—
733C4—124no—
734C4—122no—
735C4—124no—
736C4#496012yes—
737C4#496012yes—
738C4#496012yes—
739C4#496012yes—
740C3#2344611yescharacteristic, complemented, Sylow
741C2—21no—
742C2—21no—
743C2—21no—
744C2—21no—
745C2—21no—
746C2—21no—
747C2—82no—
748C2—242no—
749C2#7869212yes—
750C2C2×#496011yescharacteristic
subgroupG/H
centerC22C2×#882
commutator, cosocleC24×C3C24
FrattiniC24C23×D3
FittingC2×C3×#410C2
socleC22×C3C2×#120
prime radical#87268C2
hypercenterC2×#410D3
hypocenterC3#23446
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#410

Complements

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

NH ≅ G/Ntype
2. C2×#5825C2semidirect
3. C2×#5526C2semidirect
4. C2×C3×#410C2semidirect
5. C2×#5006C2semidirect
6. #82230C2semidirect
7. #71593C2semidirect
8. #70031C2semidirect
50. C2×#410D3semidirect
89. C22×Dic6D4semidirect
93. C3×#117D4semidirect
126. #926D4semidirect
368. #117#67semidirect
740. C3#23446semidirect

Maximal subgroups

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

subgroupindex
#872682
C2×#58252
C2×#55262
C2×C3×#4102
C2×#50062
#822302
#715932
#700312
#234463