Finite Group Databasebeta

← #10702221

Subgroups — #10702221

Order 768.

1174 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#418C211yesmaximal, characteristic
2C2×#5221C211yesmaximal, characteristic
3C2×#5215C211yesmaximal, characteristic
4#82234C211yesmaximal, characteristic, complemented
5C2×#4554C211yesmaximal, characteristic, complemented
6C2×#4555C211yesmaximal, characteristic, complemented
7C2×#4532C211yesmaximal, characteristic, complemented
8#11063—31nomaximal, Sylow
9C23×C3×D4C2211yescharacteristic
10C22×C3×M16C2211yescharacteristic
11C2×C3×#117C2211yescharacteristic
12C22×#217—22no—
13C2×#926C2211yescharacteristic
14C22×C4⋊Dic3C2211yescharacteristic
15C22×C4×Dic3C2211yescharacteristic
16C22×#208C2211yescharacteristic
17C3×#418—22no—
18C3×#418—22no—
19C2×C3×#102—21no—
20C2×C3×#102—21no—
21#5221—22no—
22#5221—22no—
23#5215—22no—
24#5215C414yescomplemented
25C2×#832—22no—
26#4977—21no—
27#4975—21no—
28#4946—22no—
29#4928—22no—
30#4555—22no—
31#4554—22no—
32#4532—22no—
33C2×#418—31no—
34C2×#418D311yescharacteristic
35#2317—31no—
36C2×#390—31no—
37C2×#343—31no—
38C2×#339—31no—
39C2×#339—31no—
40C25×C3—21no—
41C22×C3×D4—21no—
42C22×C3×D4—21no—
43C22×C3×D4—21no—
44C22×C3×D4—21no—
45C22×C3×D4—21no—
46C22×C3×D4—21no—
47C22×C3×D4—21no—
48C22×C3×D4—21no—
49C22×C3×D4—21no—
50C22×C3×D4—21no—
51C22×C3×D4—21no—
52C22×C3×D4C2×C412yes—
53C22×C3×D4D412yes—
54C22×C3×D4Q812yes—
55C23×C3×C4C2311yescharacteristic
56C23×Dic3—21no—
57C23×Dic3D412yes—
58C2×C3×M16—21no—
59C2×C3×M16—21no—
60C2×C3×M16—21no—
61C2×C3×M16—21no—
62C2×C3×M16—21no—
63C2×C3×M16D412yes—
64C22×C3×C8—21no—
65C3×#117—21no—
66C3×#117—21no—
67C3×#117—21no—
68C3×#117—21no—
69C2×C3×C4⋊C4D411yescharacteristic
70C2×C3×C4⋊C4D411yescharacteristic
71C2×C3×C22⋊C4—21no—
72C2×C3×C42—21no—
73C2×#217—21no—
74C2×#217—28no—
75C2×#217—48no—
76#926—21no—
77#926—21no—
78#926C2×C412yes—
79#926D412yes—
80C2×C4⋊Dic3—21no—
81C2×C4⋊Dic3—21no—
82C2×C4⋊Dic3—21no—
83C2×C4⋊Dic3—21no—
84C2×C4⋊Dic3—21no—
85C2×C4⋊Dic3—21no—
86C2×C4⋊Dic3D412yes—
87C2×Dic3⋊C4—24no—
88C2×C4×Dic3—21no—
89C2×C4×Dic3—21no—
90C2×C4×Dic3—21no—
91C2×C4×Dic3—21no—
92C2×C4×Dic3—21no—
93C2×C4×Dic3—21no—
94C2×C4×Dic3D412yes—
95C2×#208—21no—
96C2×#208—21no—
97C2×#208—21no—
98C2×#208—21no—
99C2×#208—21no—
100C2×#208C2×C412yes—
101C22×C3⋊C8—21no—
102C3×#102—24no—
103C3×#102—24no—
104#832—28no—
105C3⋊#95—24no—
106C23×D4C2×D311yescharacteristic
107C22×M16—31no—
108C22×M16C2×D311yescharacteristic
109C2×#117—31no—
110C2×#117C2×D311yescharacteristic
111C22×C4⋊C4—31no—
112C22×C22⋊C4—62no—
113C22×C42—31no—
114#418—22no—
115#418—22no—
116#418—34no—
117#418—62no—
118C2×#102—21no—
119C2×#102—21no—
120C2×#102—62no—
121#390—62no—
122#390—62no—
123#389—62no—
124#385—62no—
125#384—61no—
126#382—61no—
127#343—62no—
128#339—62no—
129#339—62no—
130C24×C3—22no—
131C24×C3—22no—
132C24×C3—22no—
133C24×C3—44no—
134C24×C3—44no—
135C24×C3—41no—
136C24×C3—41no—
137C24×C3—41no—
138C24×C3—41no—
139C24×C3C2×D411yescharacteristic
140C2×C3×D4—22no—
141C2×C3×D4—22no—
142C2×C3×D4—22no—
143C2×C3×D4—22no—
144C2×C3×D4—22no—
145C2×C3×D4—22no—
146C2×C3×D4—22no—
147C2×C3×D4—22no—
148C2×C3×D4—22no—
149C2×C3×D4—22no—
150C2×C3×D4—22no—
151C2×C3×D4—22no—
152C2×C3×D4—22no—
153C2×C3×D4—22no—
154C2×C3×D4—22no—
155C2×C3×D4—41no—
156C2×C3×D4—41no—
157C2×C3×D4—41no—
158C2×C3×D4—41no—
159C2×C3×D4—41no—
160C2×C3×D4—41no—
161C2×C3×D4—41no—
162C2×C3×D4—41no—
163C2×C3×D4—41no—
164C2×C3×D4—41no—
165C2×C3×D4—41no—
166C2×C3×D4—41no—
167C2×C3×D4C4⋊C414yescomplemented
168C22×C3×C4—21no—
169C22×C3×C4—21no—
170C22×C3×C4C22×C411yescharacteristic
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2×Q811yescharacteristic
178C22×C3×C4#4111yescharacteristic
179C22×C3×C4#4111yescharacteristic
180C22×C3×C4#4111yescharacteristic
181C22×C3×C4#4111yescharacteristic
182C22×C3×C4#4111yescharacteristic
183C22×C3×C4#4111yescharacteristic
184C22×Dic3—24no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—22no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—42no—
191C22×Dic3—41no—
192C22×Dic3—41no—
193C22×Dic3—44no—
194C3×M16—22no—
195C3×M16—22no—
196C3×M16—41no—
197C3×M16—41no—
198C2×C3×C8—22no—
199C2×C3×C8—41no—
200C2×C3×C8—41no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—21no—
205C3×C22⋊C4—41no—
206C3×C22⋊C4—41no—
207C3×C42—41no—
208C3×C42—41no—
209#217—22no—
210#217—22no—
211#217—416no—
212C4⋊Dic3—22no—
213C4⋊Dic3—41no—
214C4⋊Dic3—41no—
215C4⋊Dic3—41no—
216C4⋊Dic3—41no—
217C4⋊Dic3C22⋊C414yescomplemented
218Dic3⋊C4—84no—
219C4×Dic3—22no—
220C4×Dic3—22no—
221C4×Dic3—22no—
222C4×Dic3—22no—
223C4×Dic3—41no—
224C4×Dic3—41no—
225#208—22no—
226#208—22no—
227#208—41no—
228#208—41no—
229C2×C3⋊C8—22no—
230C2×C3⋊C8—41no—
231C2×C3⋊C8—41no—
232C25—21no—
233C22×D4—21no—
234C22×D4—21no—
235C22×D4—21no—
236C22×D4—21no—
237C22×D4—21no—
238C22×D4—21no—
239C22×D4—21no—
240C22×D4—21no—
241C22×D4—21no—
242C22×D4—21no—
243C22×D4—21no—
244C22×D4Dic612yes—
245C22×D4C4×D312yes—
246C22×D4#6712yes—
247C23×C4—32no—
248C23×C4—61no—
249C23×C4C22×D311yescharacteristic
250C2×M16—21no—
251C2×M16—21no—
252C2×M16—21no—
253C2×M16—21no—
254C2×M16—21no—
255C2×M16—32no—
256C2×M16—61no—
257C2×M16—61no—
258C2×M16—61no—
259C2×M16—61no—
260C2×M16—61no—
261C2×M16#6712yes—
262C22×C8—21no—
263C22×C8—61no—
264#117—21no—
265#117—21no—
266#117—21no—
267#117—21no—
268#117—32no—
269#117—32no—
270#117—61no—
271#117—61no—
272C2×C4⋊C4—32no—
273C2×C4⋊C4—61no—
274C2×C4⋊C4—61no—
275C2×C4⋊C4—61no—
276C2×C4⋊C4—61no—
277C2×C4⋊C4—61no—
278C2×C4⋊C4—61no—
279C2×C4⋊C4—64no—
280C2×C4⋊C4#6711yescharacteristic
281C2×C4⋊C4#6711yescharacteristic
282C2×C22⋊C4—21no—
283C2×C22⋊C4—61no—
284C2×C22⋊C4—68no—
285C2×C22⋊C4—128no—
286C2×C42—21no—
287C2×C42—32no—
288C2×C42—61no—
289C2×C42—61no—
290C2×C42—61no—
291C2×C42—61no—
292C2×C42—61no—
293C2×C42—61no—
294#102—24no—
295#102—24no—
296#102—68no—
297#95—64no—
298C23×C3—22no—
299C23×C3—22no—
300C23×C3—24no—
301C23×C3—44no—
302C23×C3—48no—
303C23×C3—44no—
304C23×C3—48no—
305C23×C3—42no—
306C23×C3—42no—
307C23×C3—42no—
308C23×C3—42no—
309C23×C3—42no—
310C23×C3—42no—
311C23×C3—84no—
312C23×C3—82no—
313C23×C3—82no—
314C23×C3—84no—
315C23×C3—82no—
316C23×C3—82no—
317C23×C3C4×D411yescharacteristic
318C23×C3#12011yescharacteristic
319C23×C3#12011yescharacteristic
320C23×C3#12111yescharacteristic
321C23×C3#12111yescharacteristic
322C23×C3C22⋊Q811yescharacteristic
323C23×C3#12411yescharacteristic
324C3×D4—24no—
325C3×D4—24no—
326C3×D4—42no—
327C3×D4—42no—
328C3×D4—42no—
329C3×D4—42no—
330C3×D4—42no—
331C3×D4—42no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4C2×C22⋊C411yescharacteristic
337C2×C3×C4C2×C4⋊C411yescharacteristic
338C2×C3×C4#11711yescharacteristic
339C2×C3×C4C4×D411yescharacteristic
340C2×C3×C4C4×D411yescharacteristic
341C2×C3×C4C4×D411yescharacteristic
342C2×C3×C4#12011yescharacteristic
343C2×C3×C4#12111yescharacteristic
344C2×C3×C4#12111yescharacteristic
345C2×C3×C4#12111yescharacteristic
346C2×C3×C4#12111yescharacteristic
347C2×C3×C4#12111yescharacteristic
348C2×C3×C4#12111yescharacteristic
349C2×C3×C4C22⋊Q811yescharacteristic
350C2×C3×C4C22⋊Q811yescharacteristic
351C2×C3×C4C22⋊Q811yescharacteristic
352C2×C3×C4C22⋊Q811yescharacteristic
353C2×C3×C4C22⋊Q811yescharacteristic
354C2×C3×C4#12311yescharacteristic
355C2×C3×C4#12311yescharacteristic
356C2×C3×C4#12311yescharacteristic
357C2×C3×C4#12311yescharacteristic
358C2×C3×C4#12311yescharacteristic
359C2×C3×C4#12411yescharacteristic
360C2×C3×C4#12411yescharacteristic
361C2×C3×C4#12611yescharacteristic
362C2×C3×C4#12611yescharacteristic
363C2×C3×C4#12711yescharacteristic
364C2×Dic3—24no—
365C2×Dic3—24no—
366C2×Dic3—44no—
367C2×Dic3—42no—
368C2×Dic3—44no—
369C2×Dic3—42no—
370C2×Dic3—42no—
371C2×Dic3—42no—
372C2×Dic3—44no—
373C2×Dic3—42no—
374C2×Dic3—44no—
375C2×Dic3—82no—
376C3×C8—42no—
377C3⋊C8—42no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—44no—
382C24—44no—
383C24—41no—
384C24—41no—
385C24—41no—
386C24—41no—
387C24D3×D411yescharacteristic
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—41no—
404C2×D4—41no—
405C2×D4—41no—
406C2×D4—41no—
407C2×D4—41no—
408C2×D4—41no—
409C2×D4—41no—
410C2×D4—41no—
411C2×D4—41no—
412C2×D4—41no—
413C2×D4—41no—
414C2×D4—41no—
415C2×D4Dic3⋊C414yescomplemented
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—64no—
419C22×C4—62no—
420C22×C4—62no—
421C22×C4—62no—
422C22×C4—62no—
423C22×C4—62no—
424C22×C4—122no—
425C22×C4—121no—
426C22×C4—121no—
427C22×C4—124no—
428C22×C4C2×Dic611yescharacteristic
429C22×C4C2×C4×D311yescharacteristic
430C22×C4#23511yescharacteristic
431C22×C4#23511yescharacteristic
432C22×C4D3×D411yescharacteristic
433C22×C4D3×D411yescharacteristic
434C22×C4D3×D411yescharacteristic
435C22×C4#23711yescharacteristic
436C22×C4#23711yescharacteristic
437C22×C4#23711yescharacteristic
438C22×C4#23711yescharacteristic
439C22×C4C2×#6711yescharacteristic
440C22×C4C2×#6711yescharacteristic
441C22×C4C2×#6711yescharacteristic
442M16—22no—
443M16—22no—
444M16—41no—
445M16—41no—
446M16—62no—
447M16—62no—
448M16—121no—
449M16—121no—
450C2×C8—22no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—62no—
454C2×C8—121no—
455C2×C8—121no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—34no—
461C4⋊C4—62no—
462C4⋊C4—121no—
463C4⋊C4—121no—
464C4⋊C4—121no—
465C4⋊C4—121no—
466C4⋊C4—244no—
467C22⋊C4—41no—
468C22⋊C4—41no—
469C22⋊C4—62no—
470C22⋊C4—62no—
471C22⋊C4—1216no—
472C42—41no—
473C42—41no—
474C42—62no—
475C42—62no—
476C42—62no—
477C42—62no—
478C42—121no—
479C42—121no—
480C22×C3—22no—
481C22×C3—22no—
482C22×C3—21no—
483C22×C3—21no—
484C22×C3—21no—
485C22×C3—21no—
486C22×C3—22no—
487C22×C3—22no—
488C22×C3—48no—
489C22×C3—44no—
490C22×C3—84no—
491C22×C3—84no—
492C22×C3—84no—
493C22×C3—84no—
494C22×C3—82no—
495C22×C3—82no—
496C22×C3—84no—
497C22×C3#38511yescharacteristic
498C22×C3#38611yescharacteristic
499C22×C3#38811yescharacteristic
500C22×C3#39211yescharacteristic
501C22×C3#39311yescharacteristic
502C22×C3#39411yescharacteristic
503C22×C3#39611yescharacteristic
504C22×C3#45312yes—
505C22×C3#45412yes—
506C3×C4—81no—
507C3×C4—81no—
508C3×C4#38011yescharacteristic
509C3×C4#38511yescharacteristic
510C3×C4#38811yescharacteristic
511C3×C4#39011yescharacteristic
512C3×C4#39411yescharacteristic
513C3×C4#39611yescharacteristic
514C3×C4#39711yescharacteristic
515C3×C4#39911yescharacteristic
516Dic3—44no—
517Dic3—44no—
518Dic3—82no—
519C23—22no—
520C23—22no—
521C23—24no—
522C23—44no—
523C23—44no—
524C23—48no—
525C23—48no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—82no—
533C23—82no—
534C23—84no—
535C23—84no—
536C23—82no—
537C23—82no—
538C23#87811yescharacteristic
539C23#67⋊C411yescharacteristic
540C23#88411yescharacteristic
541C23#88511yescharacteristic
542C23#93711yescharacteristic
543C23#93711yescharacteristic
544C23#93811yescharacteristic
545D4—24no—
546D4—24no—
547D4—42no—
548D4—42no—
549D4—42no—
550D4—42no—
551D4—42no—
552D4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—64no—
558C2×C4—64no—
559C2×C4—122no—
560C2×C4—124no—
561C2×C4—124no—
562C2×C4—122no—
563C2×C4—122no—
564C2×C4—122no—
565C2×C4—124no—
566C2×C4—124no—
567C2×C4—122no—
568C2×C4—242no—
569C2×C4#87711yescharacteristic
570C2×C4#87811yescharacteristic
571C2×C4#87811yescharacteristic
572C2×C4#87811yescharacteristic
573C2×C4#87911yescharacteristic
574C2×C4#87911yescharacteristic
575C2×C4D3×C22⋊C411yescharacteristic
576C2×C4#67⋊C411yescharacteristic
577C2×C4#88311yescharacteristic
578C2×C4#88311yescharacteristic
579C2×C4#88411yescharacteristic
580C2×C4#88511yescharacteristic
581C2×C4C2×Dic3⋊C411yescharacteristic
582C2×C4#92411yescharacteristic
583C2×C4#92411yescharacteristic
584C2×C4C4×#6711yescharacteristic
585C2×C4C4×#6711yescharacteristic
586C2×C4#92911yescharacteristic
587C2×C4#93511yescharacteristic
588C2×C4#93511yescharacteristic
589C2×C4#93611yescharacteristic
590C2×C4#93811yescharacteristic
591C2×C4#93911yescharacteristic
592C2×C4#93911yescharacteristic
593C2×C4#93911yescharacteristic
594C2×C4#93911yescharacteristic
595C2×C4#94011yescharacteristic
596C2×C4#95311yescharacteristic
597C8—42no—
598C8—122no—
599C2×C3—21no—
600C2×C3—21no—
601C2×C3—21no—
602C2×C3—21no—
603C2×C3—84no—
604C2×C3#143911yescharacteristic
605C2×C3#188312yes—
606C2×C3#200211yescharacteristic
607C2×C3#200311yescharacteristic
608C2×C3#202712yes—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—48no—
618C22—44no—
619C22—84no—
620C22—84no—
621C22—84no—
622C22—82no—
623C22—82no—
624C22—84no—
625C22—84no—
626C22#494411yescharacteristic
627C22#494611yescharacteristic
628C22#494711yescharacteristic
629C22#494811yescharacteristic
630C22#495611yescharacteristic
631C22#495911yescharacteristic
632C22#519812yes—
633C22#520112yes—
634C22#522311yescharacteristic
635C4—81no—
636C4—81no—
637C4—124no—
638C4—124no—
639C4—242no—
640C4#494211yescharacteristic
641C4#494511yescharacteristic
642C4#494811yescharacteristic
643C4#494911yescharacteristic
644C4#495211yescharacteristic
645C4#495711yescharacteristic
646C4#521011yescharacteristic
647C4#522211yescharacteristic
648C3#1106311yescharacteristic, complemented, Sylow
649C2—21no—
650C2—21no—
651C2—21no—
652C2—21no—
653C2—84no—
654C2#7031312yes—
655C2#7032812yes—
656C2#7154711yescharacteristic
657C2#7209411yescharacteristic
658C2#7209611yescharacteristic
subgroupG/H
centerC23#937
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#418C2
socleC23×C3#120
prime radicalC23×C3×D4C22
cosocleC23×C3×C4C23
hypercenterC2×#418D3
hypocenterC3#11063
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#418

Complements

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

NH ≅ G/Ntype
4. #82234C2semidirect
5. C2×#4554C2semidirect
6. C2×#4555C2semidirect
7. C2×#4532C2semidirect
24. #5215C4semidirect
167. C2×C3×D4C4⋊C4semidirect
217. C4⋊Dic3C22⋊C4semidirect
415. C2×D4Dic3⋊C4semidirect
648. C3#11063semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4182
C2×#52212
C2×#52152
#822342
C2×#45542
C2×#45552
C2×#45322
#110633