Finite Group Databasebeta

← #10702675

Subgroups — #10702675

Order 768.

1190 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#390C211yesmaximal, characteristic
2#84044C211yesmaximal, characteristic
3C22×#833C211yesmaximal, characteristic
4C2×#4964C211yesmaximal, characteristic, complemented
5#71558C211yesmaximal, characteristic, complemented
6C2×#4523C211yesmaximal, characteristic, complemented
7C2×#4523C211yesmaximal, characteristic, complemented
8#11135—31nomaximal, Sylow
9C23×C3×D4C2211yescharacteristic
10C22×C3×C4⋊C4C2211yescharacteristic
11C22×C3×C22⋊C4—22no—
12C22×C3×C42C2211yescharacteristic
13C2×#926C2211yescharacteristic
14C2×#926C2211yescharacteristic
15C22×#208C2211yescharacteristic
16C22×#208C2211yescharacteristic
17C3×#390—22no—
18C3×#390—22no—
19#5247—22no—
20#5221—21no—
21#5221—21no—
22#5221—22no—
23C2×#833—22no—
24C2×#833—22no—
25C2×#833C414yescomplemented
26#4964—22no—
27#4964—22no—
28C2×#808—21no—
29C2×#807—21no—
30#4529—22no—
31#4523—22no—
32#4523—22no—
33C2×#428—31no—
34C22×#100—31no—
35#2309—31no—
36C2×#390D311yescharacteristic
37#1736—31no—
38C2×#339—31no—
39C2×#339—31no—
40C25×C3D412yes—
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×D4—21no—
53C22×C3×D4C2×C412yes—
54C22×C3×D4Q812yes—
55C23×C3×C4—21no—
56C23×C3×C4—21no—
57C23×C3×C4C2311yescharacteristic
58C2×C3×C4⋊C4—21no—
59C2×C3×C4⋊C4—21no—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4D412yes—
64C2×C3×C22⋊C4—28no—
65C2×C3×C22⋊C4—48no—
66C2×C3×C42—21no—
67C2×C3×C42—21no—
68C2×C3×C42—21no—
69C2×C3×C42—21no—
70C2×C3×C42—21no—
71C2×C3×C42D412yes—
72C2×#217—21no—
73C2×#217—21no—
74C2×#217—24no—
75C2×#217—24no—
76#926—21no—
77#926—21no—
78#926—21no—
79#926—21no—
80#926—21no—
81#926—21no—
82#926—21no—
83#926—21no—
84C2×C4⋊Dic3D411yescharacteristic
85C2×C4⋊Dic3D411yescharacteristic
86C2×C4⋊Dic3D412yes—
87C2×C4×Dic3—21no—
88C2×C4×Dic3—21no—
89C2×#208—21no—
90C2×#208—21no—
91C2×#208—21no—
92C2×#208—21no—
93C2×#208—21no—
94C2×#208—21no—
95C2×#208—21no—
96C2×#208—21no—
97C2×#208C2×C412yes—
98C2×#208C2×C412yes—
99C2×#208D412yes—
100C2×#208D412yes—
101C22×C3⋊C8—21no—
102C22×C3⋊C8—21no—
103#833—28no—
104#808—24no—
105#807—24no—
106C23×D4C2×D311yescharacteristic
107C22×M16—31no—
108C22×M16—31no—
109C2×#117—31no—
110C2×#117—31no—
111C22×C4⋊C4C2×D311yescharacteristic
112C22×C22⋊C4—22no—
113C22×C42C2×D311yescharacteristic
114#428—62no—
115#428—62no—
116C2×#107—61no—
117C2×#106—61no—
118C2×#100—34no—
119C2×#100—62no—
120C2×#100—62no—
121#390—22no—
122#390—22no—
123#390—62no—
124#390—61no—
125#390—61no—
126#379—62no—
127#340—62no—
128#339—62no—
129#339—62no—
130C24×C3—22no—
131C24×C3—22no—
132C24×C3—22no—
133C24×C3—22no—
134C24×C3—22no—
135C24×C3—22no—
136C24×C3—44no—
137C24×C3—44no—
138C24×C3C4⋊C414yes—
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—41no—
153C2×C3×D4—41no—
154C2×C3×D4—41no—
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×D4—41no—
168C22×C3×C4—24no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—42no—
172C22×C3×C4—41no—
173C22×C3×C4—41no—
174C22×C3×C4—44no—
175C22×C3×C4—41no—
176C22×C3×C4—41no—
177C22×C3×C4C22×C411yescharacteristic
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×Q811yescharacteristic
185C22×C3×C4#4111yescharacteristic
186C22×C3×C4#4111yescharacteristic
187C22×C3×C4#4111yescharacteristic
188C22×C3×C4#4111yescharacteristic
189C22×C3×C4#4111yescharacteristic
190C22×C3×C4#4111yescharacteristic
191C22×Dic3—21no—
192C22×Dic3—21no—
193C22×Dic3—22no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—41no—
197C3×C4⋊C4—41no—
198C3×C22⋊C4—416no—
199C3×C42—22no—
200C3×C42—22no—
201C3×C42—41no—
202C3×C42—41no—
203#217—41no—
204#217—41no—
205#217—41no—
206#217—41no—
207#217—84no—
208#217—84no—
209C4⋊Dic3—21no—
210C4⋊Dic3—21no—
211C4⋊Dic3—22no—
212C4⋊Dic3—21no—
213C4⋊Dic3—21no—
214C4⋊Dic3—22no—
215C4×Dic3—41no—
216C4×Dic3—41no—
217C4×Dic3—41no—
218C4×Dic3—41no—
219#208—22no—
220#208—22no—
221#208—22no—
222#208—22no—
223#208—22no—
224#208—41no—
225#208—41no—
226#208C22⋊C414yescomplemented
227C2×C3⋊C8—22no—
228C2×C3⋊C8—22no—
229C2×C3⋊C8—22no—
230C2×C3⋊C8—22no—
231C2×C3⋊C8—41no—
232C2×C3⋊C8—41no—
233C25#6712yes—
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×D4—21no—
245C22×D4—21no—
246C22×D4Dic612yes—
247C22×D4C4×D312yes—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4C22×D311yescharacteristic
251C2×M16—32no—
252C2×M16—32no—
253C2×M16—32no—
254C2×M16—32no—
255C2×M16—61no—
256C2×M16—61no—
257C2×M16—61no—
258C2×M16—61no—
259C2×M16—61no—
260C2×M16—61no—
261C2×M16—61no—
262C2×M16—61no—
263C22×C8—61no—
264C22×C8—61no—
265#117—61no—
266#117—61no—
267#117—61no—
268#117—61no—
269#117—61no—
270#117—61no—
271#117—61no—
272#117—61no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—32no—
279C2×C4⋊C4—31no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4#6712yes—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—48no—
284C2×C22⋊C4—61no—
285C2×C22⋊C4—61no—
286C2×C22⋊C4—64no—
287C2×C22⋊C4—64no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—61no—
294C2×C42—61no—
295C2×C42#6712yes—
296#107—64no—
297#106—64no—
298#100—68no—
299C23×C3—22no—
300C23×C3—22no—
301C23×C3—24no—
302C23×C3—24no—
303C23×C3—24no—
304C23×C3—48no—
305C23×C3—44no—
306C23×C3—44no—
307C23×C3—48no—
308C23×C3—42no—
309C23×C3—42no—
310C23×C3—42no—
311C23×C3—42no—
312C23×C3—44no—
313C23×C3—44no—
314C23×C3—84no—
315C23×C3—82no—
316C23×C3—82no—
317C23×C3—84no—
318C23×C3C2×C4⋊C411yescharacteristic
319C23×C3#12011yescharacteristic
320C23×C3#12011yescharacteristic
321C23×C3#12111yescharacteristic
322C23×C3#12111yescharacteristic
323C23×C3#12311yescharacteristic
324C23×C3#12311yescharacteristic
325C3×D4—42no—
326C3×D4—42no—
327C3×D4—42no—
328C3×D4—42no—
329C3×D4—42no—
330C3×D4—42no—
331C3×D4—42no—
332C3×D4—42no—
333C2×C3×C4—24no—
334C2×C3×C4—44no—
335C2×C3×C4—42no—
336C2×C3×C4—44no—
337C2×C3×C4—42no—
338C2×C3×C4—44no—
339C2×C3×C4—42no—
340C2×C3×C4—82no—
341C2×C3×C4—82no—
342C2×C3×C4C2×C22⋊C411yescharacteristic
343C2×C3×C4#11711yescharacteristic
344C2×C3×C4C4×D411yescharacteristic
345C2×C3×C4C4×D411yescharacteristic
346C2×C3×C4C4×D411yescharacteristic
347C2×C3×C4C4×D411yescharacteristic
348C2×C3×C4#12011yescharacteristic
349C2×C3×C4#12111yescharacteristic
350C2×C3×C4#12111yescharacteristic
351C2×C3×C4#12111yescharacteristic
352C2×C3×C4#12111yescharacteristic
353C2×C3×C4#12111yescharacteristic
354C2×C3×C4#12111yescharacteristic
355C2×C3×C4C22⋊Q811yescharacteristic
356C2×C3×C4C22⋊Q811yescharacteristic
357C2×C3×C4C22⋊Q811yescharacteristic
358C2×C3×C4C22⋊Q811yescharacteristic
359C2×C3×C4C22⋊Q811yescharacteristic
360C2×C3×C4C22⋊Q811yescharacteristic
361C2×C3×C4#12311yescharacteristic
362C2×C3×C4#12311yescharacteristic
363C2×C3×C4#12311yescharacteristic
364C2×C3×C4#12411yescharacteristic
365C2×C3×C4#12411yescharacteristic
366C2×C3×C4#12411yescharacteristic
367C2×C3×C4#12611yescharacteristic
368C2×C3×C4#12611yescharacteristic
369C2×C3×C4#12711yescharacteristic
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—42no—
373C2×Dic3—42no—
374C2×Dic3—42no—
375C2×Dic3—42no—
376C2×Dic3—42no—
377C3⋊C8—24no—
378C3⋊C8—42no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24—22no—
383C24—22no—
384C24—22no—
385C24—44no—
386C24—44no—
387C24Dic3⋊C414yes—
388C24C2×#6711yescharacteristic
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—41no—
402C2×D4—41no—
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×D4—41no—
416C2×D4—41no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—42no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—44no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—62no—
427C22×C4—61no—
428C22×C4—61no—
429C22×C4C2×Dic611yescharacteristic
430C22×C4C2×C4×D311yescharacteristic
431C22×C4#23511yescharacteristic
432C22×C4#23511yescharacteristic
433C22×C4D3×D411yescharacteristic
434C22×C4D3×D411yescharacteristic
435C22×C4D3×D411yescharacteristic
436C22×C4D3×D411yescharacteristic
437C22×C4#23711yescharacteristic
438C22×C4#23711yescharacteristic
439C22×C4#23711yescharacteristic
440C22×C4#23711yescharacteristic
441C22×C4C2×#6711yescharacteristic
442C22×C4C2×#6711yescharacteristic
443M16—34no—
444M16—62no—
445M16—62no—
446M16—62no—
447M16—62no—
448M16—62no—
449M16—121no—
450M16—121no—
451C2×C8—62no—
452C2×C8—62no—
453C2×C8—62no—
454C2×C8—62no—
455C2×C8—121no—
456C2×C8—121no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—62no—
462C4⋊C4—62no—
463C4⋊C4—61no—
464C4⋊C4—61no—
465C4⋊C4—61no—
466C4⋊C4—61no—
467C22⋊C4—416no—
468C22⋊C4—121no—
469C22⋊C4—121no—
470C22⋊C4—121no—
471C22⋊C4—121no—
472C22⋊C4—244no—
473C22⋊C4—244no—
474C42—22no—
475C42—22no—
476C42—41no—
477C42—41no—
478C42—121no—
479C42—121no—
480C42—121no—
481C42—121no—
482C22×C3—22no—
483C22×C3—22no—
484C22×C3—21no—
485C22×C3—21no—
486C22×C3—21no—
487C22×C3—21no—
488C22×C3—22no—
489C22×C3—22no—
490C22×C3—48no—
491C22×C3—48no—
492C22×C3—44no—
493C22×C3—84no—
494C22×C3—84no—
495C22×C3—84no—
496C22×C3—82no—
497C22×C3—82no—
498C22×C3—84no—
499C22×C3#38011yescharacteristic
500C22×C3#38511yescharacteristic
501C22×C3#38511yescharacteristic
502C22×C3#39211yescharacteristic
503C22×C3#39411yescharacteristic
504C22×C3#39411yescharacteristic
505C22×C3#39711yescharacteristic
506C22×C3#45312yes—
507C22×C3#45512yes—
508C3×C4—44no—
509C3×C4—82no—
510C3×C4#38611yescharacteristic
511C3×C4#38811yescharacteristic
512C3×C4#38811yescharacteristic
513C3×C4#39011yescharacteristic
514C3×C4#39311yescharacteristic
515C3×C4#39611yescharacteristic
516C3×C4#39611yescharacteristic
517C3×C4#39911yescharacteristic
518Dic3—81no—
519Dic3—81no—
520Dic3—82no—
521C23—22no—
522C23—22no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—48no—
527C23—44no—
528C23—48no—
529C23—44no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—44no—
535C23—44no—
536C23—84no—
537C23—84no—
538C23—82no—
539C23—82no—
540C23C2×Dic3⋊C411yescharacteristic
541C23#92911yescharacteristic
542C23#93511yescharacteristic
543C23#93711yescharacteristic
544C23#93911yescharacteristic
545C23#93911yescharacteristic
546C23#95311yescharacteristic
547D4—42no—
548D4—42no—
549D4—42no—
550D4—42no—
551D4—42no—
552D4—42no—
553D4—42no—
554D4—42no—
555C2×C4—24no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—122no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4—122no—
568C2×C4—122no—
569C2×C4—122no—
570C2×C4—122no—
571C2×C4#87711yescharacteristic
572C2×C4#87811yescharacteristic
573C2×C4#87811yescharacteristic
574C2×C4#87811yescharacteristic
575C2×C4#87811yescharacteristic
576C2×C4#87911yescharacteristic
577C2×C4#87911yescharacteristic
578C2×C4D3×C22⋊C411yescharacteristic
579C2×C4#67⋊C411yescharacteristic
580C2×C4#67⋊C411yescharacteristic
581C2×C4#88311yescharacteristic
582C2×C4#88311yescharacteristic
583C2×C4#88411yescharacteristic
584C2×C4#88411yescharacteristic
585C2×C4#88511yescharacteristic
586C2×C4#88511yescharacteristic
587C2×C4#92411yescharacteristic
588C2×C4#92411yescharacteristic
589C2×C4C4×#6711yescharacteristic
590C2×C4C4×#6711yescharacteristic
591C2×C4#93511yescharacteristic
592C2×C4#93611yescharacteristic
593C2×C4#93711yescharacteristic
594C2×C4#93811yescharacteristic
595C2×C4#93811yescharacteristic
596C2×C4#93911yescharacteristic
597C2×C4#93911yescharacteristic
598C2×C4#94011yescharacteristic
599C8—64no—
600C8—122no—
601C2×C3—21no—
602C2×C3—21no—
603C2×C3—21no—
604C2×C3—21no—
605C2×C3—84no—
606C2×C3#143911yescharacteristic
607C2×C3#185612yes—
608C2×C3#200211yescharacteristic
609C2×C3#200411yescharacteristic
610C2×C3#203712yes—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—21no—
619C22—48no—
620C22—48no—
621C22—44no—
622C22—84no—
623C22—84no—
624C22—84no—
625C22—84no—
626C22—82no—
627C22—82no—
628C22#494211yescharacteristic
629C22#494611yescharacteristic
630C22#495211yescharacteristic
631C22#495911yescharacteristic
632C22#506112yes—
633C22#507712yes—
634C22#521011yescharacteristic
635C22#522211yescharacteristic
636C22#522311yescharacteristic
637C4—44no—
638C4—82no—
639C4—242no—
640C4—241no—
641C4—241no—
642C4#494411yescharacteristic
643C4#494511yescharacteristic
644C4#494711yescharacteristic
645C4#494811yescharacteristic
646C4#494811yescharacteristic
647C4#494911yescharacteristic
648C4#495611yescharacteristic
649C4#495711yescharacteristic
650C3#1113511yescharacteristic, complemented, Sylow
651C2—21no—
652C2—21no—
653C2—21no—
654C2—21no—
655C2—84no—
656C2#6964212yes—
657C2#6966512yes—
658C2#7154711yescharacteristic
659C2#7173111yescharacteristic
660C2#7176911yescharacteristic
subgroupG/H
centerC23#937
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#390C2
socleC23×C3#120
prime radicalC23×C3×D4C22
cosocleC23×C3×C4C23
hypercenterC2×#390D3
hypocenterC3#11135
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
4. C2×#4964C2semidirect
5. #71558C2semidirect
6. C2×#4523C2semidirect
7. C2×#4523C2semidirect
25. C2×#833C4semidirect
226. #208C22⋊C4semidirect
650. C3#11135semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3902
#840442
C22×#8332
C2×#49642
#715582
C2×#45232
#111353