Finite Group Databasebeta

← #10822730

Subgroups — #10822730

Order 768.

1482 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5510C211yesmaximal, characteristic
2C2×C3×#428C211yesmaximal, characteristic, complemented
3C2×#5133C211yesmaximal, characteristic, complemented
4C2×#5129C211yesmaximal, characteristic, complemented
5#82270C211yesmaximal, characteristic, complemented
6C2×#5002C211yesmaximal, characteristic, complemented
7C2×#4529C211yesmaximal, characteristic, complemented
8#70635C212yesmaximal, complemented
9#70634C212yesmaximal, complemented
10#70582C212yesmaximal, complemented
11#70581C212yesmaximal, complemented
12#23485—31nomaximal, Sylow
13C22×#235C2211yescharacteristic
14C22×#235C2211yescharacteristic
15C22×C3×M16C2211yescharacteristic
16C2×C3×#117C2211yescharacteristic
17C22×C3×C4⋊C4C2211yescharacteristic
18C2×#930—22no—
19C22×C4⋊Dic3C2211yescharacteristic
20C23×C3⋊C8C2211yescharacteristic
21#5510—24no—
22#5510—24no—
23C2×#897—21no—
24C2×#897C2212yes—
25C2×#895—21no—
26C2×#895C2212yes—
27C3×#428—24no—
28C2×C3×#107C2212yes—
29C2×C3×#106C2212yes—
30#5133—24no—
31#5129—24no—
32C2×#821C2212yes—
33C2×(C2×D3)⋊C8C2212yes—
34C2×(C2×D3)⋊C8C2212yes—
35C2×#816C2212yes—
36#5002—24no—
37#4987C2212yes—
38#4987C2212yes—
39C2×#810C2211yescharacteristic
40C2×#810C2211yescharacteristic
41C2×#809C2211yescharacteristic
42C2×#809C2211yescharacteristic
43#4958—22no—
44#4939—21no—
45#4929—21no—
46#4529C2212yes—
47#4529C2212yes—
48C2×#537—31no—
49C2×#428D311yescharacteristic, complemented
50C2×#417—31no—
51C2×#416—31no—
52C2×#408—31no—
53#2311—31no—
54#2036—32no—
55#2035—32no—
56#2034—32no—
57#2033—32no—
58C2×#340—31no—
59C23×C3×C4—21no—
60C23×C3×C4C2311yescharacteristic
61C22×#67—21no—
62C22×#67D412yes—
63C23×Dic3—21no—
64C2×#235—22no—
65C2×#235—22no—
66C2×#235—22no—
67C2×#235—22no—
68C2×#235—22no—
69C2×#235—22no—
70C2×#235—22no—
71C2×#235—22no—
72C2×#235—22no—
73C2×#235—22no—
74C2×#235—22no—
75C2×#235D414yes—
76C22×D12—22no—
77C22×D12D411yescharacteristic
78C22×D12C2311yescharacteristic
79C22×C4×D3—21no—
80C22×C4×D3C2312yes—
81C22×Dic6D411yescharacteristic
82C22×Dic6C2311yescharacteristic
83C2×C3×M16—22no—
84C2×C3×M16—22no—
85C2×C3×M16D414yescomplemented
86C22×C3×C8C2312yes—
87C3×#117—22no—
88C3×#117—22no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—22no—
91C2×C3×C4⋊C4D414yes—
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2312yes—
95C2×C3×C22⋊C4—21no—
96C2×C3×C42—21no—
97#930—216no—
98C2×(C2×D3)⋊C4—22no—
99C2×(C2×D3)⋊C4—22no—
100C2×(C2×D3)⋊C4—22no—
101C2×(C2×D3)⋊C4—22no—
102C2×C4⋊Dic3—22no—
103C2×C4⋊Dic3—22no—
104C2×C4⋊Dic3—22no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3C2312yes—
107C22×C3⋊C8—22no—
108C22×C3⋊C8—22no—
109C22×C3⋊C8—22no—
110C22×C3⋊C8C2311yescharacteristic
111C22×C3⋊C8C2311yescharacteristic
112#897—28no—
113#897—24no—
114#897—24no—
115#895—28no—
116#895—24no—
117#895—24no—
118C3×#107—24no—
119C3×#106—24no—
120C3⋊#95—22no—
121C3⋊#95—22no—
122#821—24no—
123(C2×D3)⋊C8—24no—
124(C2×D3)⋊C8—24no—
125#816—24no—
126#810—22no—
127#810—22no—
128#809—22no—
129#809—22no—
130C22×#41—31no—
131C22×#41—31no—
132C22×M16C2×D311yescharacteristic
133C23×C8—31no—
134#537—64no—
135#537—64no—
136C2×C22⋊Q8—32no—
137C2×C22⋊Q8—61no—
138C2×#121—32no—
139C2×#121—61no—
140C2×#121—62no—
141C2×#117C2×D311yescharacteristic
142C22×C4⋊C4—31no—
143C22×C4⋊C4C2×D311yescharacteristic
144#428—24no—
145C2×#107C2×D312yes—
146C2×#106C2×D312yes—
147#417—64no—
148#416—64no—
149C2×#103—32no—
150C2×#103—31no—
151C2×#103—31no—
152C2×#102—32no—
153C2×#102—31no—
154C2×#102—31no—
155#408—64no—
156C2×C22⋊C8—32no—
157C2×C22⋊C8—32no—
158#391—61no—
159#390—61no—
160#386—62no—
161#380—32no—
162#380—32no—
163#340—32no—
164#340—32no—
165C24×C3C2×D411yescharacteristic
166C23×D3—21no—
167C23×D3C2×D412yes—
168C22×C3×C4—22no—
169C22×C3×C4—24no—
170C22×C3×C4—21no—
171C22×C3×C4—21no—
172C22×C3×C4—81no—
173C22×C3×C4—81no—
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D412yes—
177C22×C3×C4C2×D412yes—
178C22×C3×C4#4112yes—
179C22×C3×C4#4112yes—
180C22×C3×C4#4112yes—
181C22×C3×C4C2411yescharacteristic
182C2×#67—24no—
183C2×#67—24no—
184C2×#67—24no—
185C2×#67—24no—
186C2×#67—42no—
187C2×#67—42no—
188C22×Dic3—21no—
189C22×Dic3—22no—
190C22×Dic3—42no—
191C22×Dic3—81no—
192C22×Dic3—81no—
193C22×Dic3C2×D412yes—
194#235—28no—
195#235—28no—
196#235—44no—
197#235—44no—
198#235—44no—
199#235—44no—
200#235—44no—
201#235—44no—
202C2×D12—21no—
203C2×D12—21no—
204C2×D12—21no—
205C2×D12—21no—
206C2×D12—22no—
207C2×D12—22no—
208C2×D12—22no—
209C2×D12—21no—
210C2×D12—28no—
211C2×D12—82no—
212C2×D12—82no—
213C2×D12C2×D412yes—
214C2×C4×D3—24no—
215C2×C4×D3—22no—
216C2×C4×D3—22no—
217C2×C4×D3—22no—
218C2×C4×D3—22no—
219C2×C4×D3—22no—
220C2×C4×D3—22no—
221C2×C4×D3—42no—
222C2×C4×D3—42no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6—22no—
228C2×Dic6—22no—
229C2×Dic6—22no—
230C2×Dic6—21no—
231C2×Dic6C2×D412yes—
232C3×M16—28no—
233C2×C3×C8—22no—
234C2×C3×C8—22no—
235C2×C3×C8C2×D414yes—
236C3×C4⋊C4—24no—
237C3×C4⋊C4—22no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4C2×D414yes—
243C3×C22⋊C4—42no—
244C3×C42—42no—
245(C2×D3)⋊C4—28no—
246(C2×D3)⋊C4—28no—
247(C2×D3)⋊C4—82no—
248(C2×D3)⋊C4—82no—
249C4⋊Dic3—28no—
250C4⋊Dic3—22no—
251C4⋊Dic3—22no—
252C4⋊Dic3—42no—
253C2×C3⋊C8—21no—
254C2×C3⋊C8—21no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—44no—
260C2×#41—34no—
261C2×#41—62no—
262C2×#41—62no—
263C2×#41—62no—
264C2×#41—62no—
265C2×#41—62no—
266C2×#41—62no—
267C2×#41—62no—
268C2×#41—62no—
269C2×#41—62no—
270C2×#41—62no—
271C2×#41—62no—
272C22×Q8—31no—
273C22×Q8—31no—
274C22×D4—32no—
275C22×D4—31no—
276C22×D4—31no—
277C22×D4—62no—
278C22×D4—61no—
279C23×C4—21no—
280C23×C4—32no—
281C23×C4—61no—
282C23×C4—61no—
283C23×C4C22×D311yescharacteristic
284C2×M16—22no—
285C2×M16—22no—
286C2×M16D1214yescomplemented
287C22×C8—31no—
288C22×C8—31no—
289C22×C8—62no—
290C22×C8—62no—
291C22×C8—62no—
292C22×C8C22×D312yes—
293C22⋊Q8—64no—
294C22⋊Q8—68no—
295C22⋊Q8—64no—
296#121—64no—
297#121—68no—
298#121—616no—
299#121—64no—
300#117—22no—
301#117—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—32no—
305C2×C4⋊C4—62no—
306C2×C4⋊C4—62no—
307C2×C4⋊C4—62no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4#6714yes—
310C2×C4⋊C4C22×D311yescharacteristic
311C2×C4⋊C4C22×D311yescharacteristic
312C2×C4⋊C4C22×D312yes—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—62no—
315C2×C22⋊C4—62no—
316C2×C22⋊C4—62no—
317C2×C22⋊C4—62no—
318C2×C42—21no—
319#107—24no—
320#106—24no—
321#103—62no—
322#103—62no—
323#103—64no—
324#102—62no—
325#102—62no—
326#102—64no—
327C22⋊C8—64no—
328C22⋊C8—64no—
329#95—62no—
330#95—62no—
331C23×C3—21no—
332C23×C3—21no—
333C23×C3—21no—
334C23×C3—21no—
335C23×C3#12012yes—
336C23×C3#12112yes—
337C23×C3#12312yes—
338C23×C3C22×D411yescharacteristic
339C22×D3—22no—
340C22×D3—22no—
341C22×D3—22no—
342C22×D3—22no—
343C22×D3—22no—
344C22×D3—22no—
345C22×D3—22no—
346C22×D3—22no—
347C22×D3—41no—
348C22×D3—41no—
349C22×D3—41no—
350C22×D3—41no—
351C22×D3—41no—
352C22×D3—41no—
353C2×C3×C4—24no—
354C2×C3×C4—42no—
355C2×C3×C4—42no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—84no—
363C2×C3×C4#12012yes—
364C2×C3×C4#12114yes—
365C2×C3×C4#12112yes—
366C2×C3×C4#12114yes—
367C2×C3×C4#12314yes—
368C2×C3×C4#12312yes—
369C2×C3×C4#12414yes—
370C2×C3×C4C22×D411yescharacteristic
371C2×C3×C4C22×D411yescharacteristic
372C2×C3×C4C22×D411yescharacteristic
373C2×C3×C4C2×#4111yescharacteristic
374C2×C3×C4C2×#4111yescharacteristic
375C2×C3×C4C2×#4111yescharacteristic
376#67—48no—
377#67—48no—
378C2×Dic3—22no—
379C2×Dic3—22no—
380C2×Dic3—22no—
381C2×Dic3—22no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—42no—
385C2×Dic3—42no—
386C2×Dic3—42no—
387C2×Dic3—84no—
388D12—22no—
389D12—22no—
390D12—41no—
391D12—41no—
392D12—44no—
393D12—88no—
394C4×D3—42no—
395C4×D3—42no—
396C4×D3—42no—
397C4×D3—42no—
398C4×D3—48no—
399Dic6—22no—
400Dic6—22no—
401Dic6—41no—
402Dic6—41no—
403Dic6—44no—
404C3×C8—24no—
405C3⋊C8—41no—
406C3⋊C8—41no—
407C24—32no—
408C24—61no—
409C24D3×D411yescharacteristic
410#41—68no—
411#41—68no—
412#41—124no—
413#41—124no—
414#41—124no—
415#41—124no—
416#41—124no—
417#41—124no—
418C2×Q8—32no—
419C2×Q8—62no—
420C2×Q8—62no—
421C2×Q8—62no—
422C2×Q8—61no—
423C2×Q8—61no—
424C2×Q8—61no—
425C2×Q8—61no—
426C2×Q8—61no—
427C2×D4—32no—
428C2×D4—68no—
429C2×D4—64no—
430C2×D4—64no—
431C2×D4—64no—
432C2×D4—64no—
433C2×D4—62no—
434C2×D4—62no—
435C2×D4—62no—
436C2×D4—61no—
437C2×D4—61no—
438C2×D4—61no—
439C2×D4—61no—
440C2×D4—61no—
441C2×D4—122no—
442C2×D4—122no—
443C2×D4—242no—
444C2×D4—242no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—32no—
450C22×C4—61no—
451C22×C4—64no—
452C22×C4—62no—
453C22×C4—62no—
454C22×C4—62no—
455C22×C4—62no—
456C22×C4—62no—
457C22×C4—62no—
458C22×C4—62no—
459C22×C4—81no—
460C22×C4—81no—
461C22×C4—122no—
462C22×C4—122no—
463C22×C4—122no—
464C22×C4—241no—
465C22×C4—241no—
466C22×C4C2×D1212yes—
467C22×C4#23512yes—
468C22×C4D3×D411yescharacteristic
469C22×C4D3×D412yes—
470C22×C4#23912yes—
471C22×C4#23912yes—
472C22×C4C2×#6712yes—
473C22×C4C23×D311yescharacteristic
474M16—28no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—61no—
478C2×C8—61no—
479C2×C8—61no—
480C2×C8—61no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8—124no—
484C2×C8C2×D1214yes—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—68no—
492C4⋊C4—62no—
493C4⋊C4—62no—
494C4⋊C4—122no—
495C4⋊C4C2×#6714yes—
496C22⋊C4—42no—
497C22⋊C4—68no—
498C22⋊C4—68no—
499C22⋊C4—242no—
500C22⋊C4—242no—
501C42—42no—
502C22×C3—22no—
503C22×C3—22no—
504C22×C3—22no—
505C22×C3—22no—
506C22×C3—22no—
507C22×C3—22no—
508C22×C3#39414yes—
509C22×C3C2×#12011yescharacteristic
510C22×C3C2×#12111yescharacteristic
511C22×C3C2×#12311yescharacteristic
512C22×C3#57611yescharacteristic
513C22×C3#57711yescharacteristic
514C22×C3#57711yescharacteristic
515C22×C3#57811yescharacteristic
516C2×D3—42no—
517C2×D3—42no—
518C2×D3—42no—
519C2×D3—42no—
520C2×D3—42no—
521C2×D3—42no—
522C2×D3—42no—
523C2×D3—42no—
524C2×D3—42no—
525C2×D3—42no—
526C2×D3—81no—
527C2×D3—81no—
528C2×D3—81no—
529C2×D3—81no—
530C3×C4—44no—
531C3×C4—81no—
532C3×C4—81no—
533C3×C4#39414yes—
534C3×C4C2×#12111yescharacteristic
535C3×C4C2×#12111yescharacteristic
536C3×C4C2×#12311yescharacteristic
537C3×C4C2×#12411yescharacteristic
538Dic3—42no—
539Dic3—42no—
540Dic3—82no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—62no—
546C23—62no—
547C23—62no—
548C23—62no—
549C23—62no—
550C23—62no—
551C23—62no—
552C23—62no—
553C23—121no—
554C23—121no—
555C23—121no—
556C23—121no—
557C23—121no—
558C23—121no—
559C23#89412yes—
560C23#89512yes—
561C23#93712yes—
562C23C2×D3×D411yescharacteristic
563Q8—62no—
564Q8—62no—
565Q8—124no—
566Q8—121no—
567Q8—121no—
568D4—62no—
569D4—62no—
570D4—128no—
571D4—128no—
572D4—124no—
573D4—121no—
574D4—121no—
575D4—248no—
576C2×C4—24no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—62no—
586C2×C4—62no—
587C2×C4—62no—
588C2×C4—62no—
589C2×C4—84no—
590C2×C4—121no—
591C2×C4—121no—
592C2×C4—128no—
593C2×C4—122no—
594C2×C4—122no—
595C2×C4—122no—
596C2×C4—122no—
597C2×C4—122no—
598C2×C4—122no—
599C2×C4—122no—
600C2×C4—244no—
601C2×C4#89412yes—
602C2×C4#89414yes—
603C2×C4#89512yes—
604C2×C4#89514yes—
605C2×C4#93014yes—
606C2×C4#93712yes—
607C2×C4#94714yes—
608C2×C4C22×D1211yescharacteristic
609C2×C4C2×#23511yescharacteristic
610C2×C4C2×D3×D411yescharacteristic
611C2×C4C2×#23911yescharacteristic
612C2×C4C2×#23911yescharacteristic
613C2×C4C22×#6711yescharacteristic
614C8—24no—
615C8—121no—
616C8—121no—
617C2×C3—24no—
618C2×C3C2×#39411yescharacteristic
619C2×C3#301211yescharacteristic
620C2×C3#301311yescharacteristic
621C2×C3#306211yescharacteristic
622C2×C3#306311yescharacteristic
623C2×C3#309811yescharacteristic
624C2×C3#309911yescharacteristic
625D3—82no—
626D3—82no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—122no—
634C22—122no—
635C22—122no—
636C22—122no—
637C22—122no—
638C22—122no—
639C22—122no—
640C22—122no—
641C22—122no—
642C22—122no—
643C22—241no—
644C22—241no—
645C22—241no—
646C22—241no—
647C22#499114yes—
648C22C2×#89411yescharacteristic
649C22C2×#89511yescharacteristic
650C22#577411yescharacteristic
651C22#577511yescharacteristic
652C22#577811yescharacteristic
653C22#577911yescharacteristic
654C22C2×#93711yescharacteristic
655C4—44no—
656C4—81no—
657C4—81no—
658C4—122no—
659C4—122no—
660C4—242no—
661C4#499114yes—
662C4C2×#89411yescharacteristic
663C4C2×#89511yescharacteristic
664C4C2×#93011yescharacteristic
665C4C2×#94711yescharacteristic
666C3#2348511yescharacteristic, complemented, Sylow
667C2—24no—
668C2—242no—
669C2—242no—
670C2#7921311yescharacteristic
671C2#7921411yescharacteristic
672C2#7922111yescharacteristic
673C2#7922211yescharacteristic
674C2C2×#499111yescharacteristic
675C2#8394911yescharacteristic
676C2#8398411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#428C2
socleC23×C3C22×D4
prime radicalC2×#5510C2
hypercenterC2×#428D3
hypocenterC3#23485
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
2. C2×C3×#428C2semidirect
3. C2×#5133C2semidirect
4. C2×#5129C2semidirect
5. #82270C2semidirect
6. C2×#5002C2semidirect
7. C2×#4529C2semidirect
8. #70635C2semidirect
9. #70634C2semidirect
10. #70582C2semidirect
11. #70581C2semidirect
49. C2×#428D3semidirect
85. C2×C3×M16D4semidirect
286. C2×M16D12semidirect
666. C3#23485semidirect

Maximal subgroups

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

subgroupindex
C2×#55102
C2×C3×#4282
C2×#51332
C2×#51292
#822702
C2×#50022
C2×#45292
#706352
#706342
#705822
#705812
#234853