Finite Group Databasebeta

← #10732609

Subgroups — #10732609

Order 768.

2038 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#936C211yesmaximal, characteristic
2C2×C3×#390C211yesmaximal, characteristic
3C2×#5221C211yesmaximal, characteristic
4C2×#4973C211yesmaximal, characteristic, complemented
5C2×#4972C211yesmaximal, characteristic, complemented
6C2×#4929C211yesmaximal, characteristic, complemented
7#71548C214yesmaximal, complemented
8#71547C214yesmaximal, complemented
9C4×C3⋊#95C211yesmaximal, characteristic, complemented
10#15226—31nomaximal, Sylow
11C23×C3×D4C2211yescharacteristic
12C23×Dic6C2211yescharacteristic
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C22⋊C4C2214yes—
15C22×C3×C42C2211yescharacteristic
16C22×#217C2212yes—
17C22×#217C2212yes—
18C22×#217C2214yes—
19C2×#936—24no—
20C2×#936—24no—
21C2×#936—24no—
22C22×C4⋊Dic3C2211yescharacteristic
23C22×Dic3⋊C4C2212yes—
24C22×C4×Dic3C2211yescharacteristic
25C22×C4×Dic3C2211yescharacteristic
26C3×#390—24no—
27#5221—24no—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2214yes—
31C2×C3⋊#95C2214yes—
32C2×C3⋊#95C2212yes—
33#4973—24no—
34#4972—24no—
35#4929—24no—
36C22×#124—31no—
37C2×#391—31no—
38C2×#391—31no—
39C2×#390—31no—
40C2×#390D311yescharacteristic
41C2×#389—31no—
42#1440—34no—
43#1439—34no—
44C4×#95—31no—
45C25×C3C2312yes—
46C22×C3×D4—22no—
47C22×C3×D4—22no—
48C22×C3×D4—22no—
49C22×C3×D4—22no—
50C22×C3×D4—22no—
51C22×C3×D4—22no—
52C22×C3×D4D414yes—
53C23×C3×C4C2312yes—
54C23×C3×C4C2312yes—
55C23×C3×C4C2311yescharacteristic
56C23×Dic3C2312yes—
57C23×Dic3C2311yescharacteristic
58C23×Dic3C2312yes—
59C23×Dic3C2311yescharacteristic
60C23×Dic3C2312yes—
61C22×Dic6—22no—
62C22×Dic6—22no—
63C22×Dic6—22no—
64C22×Dic6—22no—
65C22×Dic6—22no—
66C22×Dic6—22no—
67C22×Dic6D414yes—
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C22⋊C4—28no—
72C2×C3×C22⋊C4—28no—
73C2×C3×C22⋊C4—44no—
74C2×C3×C42—22no—
75C2×C3×C42—22no—
76C2×C3×C42—22no—
77C2×#217—24no—
78C2×#217—24no—
79C2×#217—24no—
80C2×#217—24no—
81C2×#217—24no—
82C2×#217—24no—
83C2×#217—28no—
84C2×#217—28no—
85C2×#217—44no—
86#936—232no—
87C2×C4⋊Dic3—22no—
88C2×C4⋊Dic3—22no—
89C2×C4⋊Dic3D414yes—
90C2×Dic3⋊C4—24no—
91C2×Dic3⋊C4—44no—
92C2×C4×Dic3—22no—
93C2×C4×Dic3—22no—
94C2×C4×Dic3—22no—
95C2×C4×Dic3—22no—
96C2×C4×Dic3—22no—
97C2×C4×Dic3D414yes—
98C3⋊#95—24no—
99C3⋊#95—28no—
100C3⋊#95—24no—
101C3⋊#95—28no—
102C3⋊#95—24no—
103C23×Q8—31no—
104C23×D4C2×D311yescharacteristic
105C2×#124—64no—
106C2×#124—64no—
107C2×#124—64no—
108C22×C4⋊C4—31no—
109C22×C4⋊C4—32no—
110C22×C4⋊C4C2×D311yescharacteristic
111C22×C22⋊C4—32no—
112C22×C22⋊C4—32no—
113C22×C22⋊C4—34no—
114C22×C22⋊C4C2×D314yes—
115C22×C42—31no—
116C22×C42—31no—
117C22×C42C2×D311yescharacteristic
118#391—64no—
119#391—64no—
120#390—24no—
121#390—64no—
122#389—64no—
123C2×#95—32no—
124C2×#95—32no—
125C2×#95—34no—
126C2×#95—34no—
127C2×#95—32no—
128C24×C3—22no—
129C24×C3—22no—
130C24×C3—22no—
131C24×C3—24no—
132C24×C3—24no—
133C24×C3—22no—
134C24×C3—22no—
135C24×C3—22no—
136C24×C3—24no—
137C24×C3—24no—
138C24×C3C2×D414yes—
139C24×C3C2411yescharacteristic
140C2×C3×D4—28no—
141C2×C3×D4—28no—
142C2×C3×D4—28no—
143C2×C3×D4—44no—
144C2×C3×D4—44no—
145C2×C3×D4—44no—
146C2×C3×D4—44no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—24no—
157C22×C3×C4—24no—
158C22×C3×C4C2×D412yes—
159C22×C3×C4C2×D412yes—
160C22×C3×C4C2×D412yes—
161C22×C3×C4C2×D412yes—
162C22×C3×C4#4112yes—
163C22×C3×C4#4112yes—
164C22×C3×C4#4114yes—
165C22×C3×C4#4112yes—
166C22×Dic3—22no—
167C22×Dic3—22no—
168C22×Dic3—22no—
169C22×Dic3—21no—
170C22×Dic3—21no—
171C22×Dic3—22no—
172C22×Dic3—21no—
173C22×Dic3—21no—
174C22×Dic3—22no—
175C22×Dic3—22no—
176C22×Dic3—22no—
177C22×Dic3—22no—
178C22×Dic3—22no—
179C22×Dic3—22no—
180C22×Dic3—22no—
181C22×Dic3—24no—
182C22×Dic3—22no—
183C22×Dic3—22no—
184C22×Dic3—22no—
185C22×Dic3—22no—
186C22×Dic3—24no—
187C22×Dic3—22no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3C2×D412yes—
191C22×Dic3C2×D414yes—
192C22×Dic3C2×D414yes—
193C22×Dic3C2×D412yes—
194C22×Dic3#4114yes—
195C2×Dic6—28no—
196C2×Dic6—28no—
197C2×Dic6—28no—
198C2×Dic6—44no—
199C2×Dic6—44no—
200C2×Dic6—44no—
201C2×Dic6—44no—
202C3×C4⋊C4—44no—
203C3×C22⋊C4—416no—
204C3×C42—44no—
205#217—216no—
206#217—216no—
207#217—416no—
208C4⋊Dic3—28no—
209Dic3⋊C4—48no—
210C4×Dic3—28no—
211C4×Dic3—44no—
212C25C22×D312yes—
213C22×Q8—34no—
214C22×Q8—62no—
215C22×Q8—62no—
216C22×Q8—62no—
217C22×Q8—62no—
218C22×Q8—62no—
219C22×Q8—62no—
220C22×D4—22no—
221C22×D4—22no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4#6714yes—
227C23×C4—32no—
228C23×C4—31no—
229C23×C4—32no—
230C23×C4—31no—
231C23×C4—32no—
232C23×C4C22×D312yes—
233C23×C4C22×D312yes—
234C23×C4C22×D311yescharacteristic
235#124—632no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—34no—
240C2×C4⋊C4—62no—
241C2×C4⋊C4—62no—
242C2×C4⋊C4—64no—
243C2×C4⋊C4—124no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—64no—
248C2×C22⋊C4—64no—
249C2×C22⋊C4—64no—
250C2×C22⋊C4—64no—
251C2×C22⋊C4—64no—
252C2×C22⋊C4—64no—
253C2×C22⋊C4—68no—
254C2×C22⋊C4—68no—
255C2×C22⋊C4—124no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—34no—
260C2×C42—62no—
261C2×C42—62no—
262C2×C42—62no—
263C2×C42—62no—
264C2×C42—62no—
265#95—64no—
266#95—64no—
267#95—68no—
268#95—68no—
269#95—64no—
270C23×C3—24no—
271C23×C3—24no—
272C23×C3—24no—
273C23×C3—28no—
274C23×C3—28no—
275C23×C3—42no—
276C23×C3—42no—
277C23×C3—42no—
278C23×C3—42no—
279C23×C3—44no—
280C23×C3—44no—
281C23×C3—42no—
282C23×C3—42no—
283C23×C3—42no—
284C23×C3—42no—
285C23×C3—44no—
286C23×C3—44no—
287C23×C3—44no—
288C23×C3—44no—
289C23×C3—44no—
290C23×C3—44no—
291C23×C3—44no—
292C23×C3—44no—
293C23×C3C22×D411yescharacteristic
294C23×C3C22×D411yescharacteristic
295C23×C3C22×D411yescharacteristic
296C23×C3C22×D411yescharacteristic
297C23×C3C2×#4111yescharacteristic
298C23×C3C2×#4111yescharacteristic
299C23×C3C2×#4111yescharacteristic
300C23×C3C2×#4112yes—
301C23×C3C2×#4112yes—
302C23×C3H(5,𝔽2)12yes—
303C23×C32−1+412yes—
304C3×D4—416no—
305C2×C3×C4—24no—
306C2×C3×C4—24no—
307C2×C3×C4—24no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—44no—
316C2×C3×C4—44no—
317C2×C3×C4#12014yes—
318C2×C3×C4#12114yes—
319C2×C3×C4#12114yes—
320C2×C3×C4#12114yes—
321C2×C3×C4#12314yes—
322C2×C3×C4#12314yes—
323C2×C3×C4#12414yes—
324C2×Dic3—22no—
325C2×Dic3—22no—
326C2×Dic3—24no—
327C2×Dic3—24no—
328C2×Dic3—24no—
329C2×Dic3—24no—
330C2×Dic3—24no—
331C2×Dic3—22no—
332C2×Dic3—22no—
333C2×Dic3—22no—
334C2×Dic3—22no—
335C2×Dic3—24no—
336C2×Dic3—28no—
337C2×Dic3—24no—
338C2×Dic3—42no—
339C2×Dic3—42no—
340C2×Dic3—42no—
341C2×Dic3—42no—
342C2×Dic3—42no—
343C2×Dic3—42no—
344C2×Dic3—42no—
345C2×Dic3—42no—
346C2×Dic3—42no—
347C2×Dic3—42no—
348C2×Dic3—42no—
349C2×Dic3—42no—
350C2×Dic3—42no—
351C2×Dic3—42no—
352C2×Dic3—44no—
353Dic6—416no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—24no—
360C24—22no—
361C24—24no—
362C24—24no—
363C24—24no—
364C24C2×#6714yes—
365C24C23×D311yescharacteristic
366C2×Q8—68no—
367C2×Q8—68no—
368C2×Q8—68no—
369C2×Q8—124no—
370C2×Q8—124no—
371C2×Q8—124no—
372C2×Q8—124no—
373C2×D4—28no—
374C2×D4—28no—
375C2×D4—28no—
376C2×D4—44no—
377C2×D4—44no—
378C2×D4—44no—
379C2×D4—44no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—32no—
392C22×C4—34no—
393C22×C4—34no—
394C22×C4—32no—
395C22×C4—34no—
396C22×C4—62no—
397C22×C4—62no—
398C22×C4—62no—
399C22×C4—62no—
400C22×C4—62no—
401C22×C4—62no—
402C22×C4—62no—
403C22×C4—61no—
404C22×C4—61no—
405C22×C4—62no—
406C22×C4—62no—
407C22×C4—62no—
408C22×C4—62no—
409C22×C4—62no—
410C22×C4—62no—
411C22×C4—64no—
412C22×C4—62no—
413C22×C4—62no—
414C22×C4—64no—
415C22×C4—62no—
416C22×C4—62no—
417C22×C4—61no—
418C22×C4—61no—
419C22×C4—62no—
420C22×C4#23512yes—
421C22×C4#23512yes—
422C22×C4D3×D412yes—
423C22×C4D3×D412yes—
424C22×C4D3×D412yes—
425C22×C4#23714yes—
426C22×C4#23712yes—
427C22×C4C2×#6712yes—
428C4⋊C4—44no—
429C4⋊C4—68no—
430C4⋊C4—128no—
431C22⋊C4—416no—
432C22⋊C4—616no—
433C22⋊C4—616no—
434C22⋊C4—1216no—
435C42—44no—
436C42—68no—
437C42—124no—
438C22×C3—44no—
439C22×C3—48no—
440C22×C3—44no—
441C22×C3—44no—
442C22×C3—48no—
443C22×C3—82no—
444C22×C3—82no—
445C22×C3—84no—
446C22×C3—84no—
447C22×C3—82no—
448C22×C3—82no—
449C22×C3C2×#12011yescharacteristic
450C22×C3C2×#12111yescharacteristic
451C22×C3C2×#12111yescharacteristic
452C22×C3C2×#12111yescharacteristic
453C22×C3C2×#12311yescharacteristic
454C22×C3C2×#12311yescharacteristic
455C22×C3C2×#12411yescharacteristic
456C22×C3#52912yes—
457C22×C3#52912yes—
458C22×C3#53211yescharacteristic
459C22×C3#53211yescharacteristic
460C22×C3#53511yescharacteristic
461C22×C3#53611yescharacteristic
462C22×C3#54212yes—
463C22×C3#54212yes—
464C22×C3#54612yes—
465C22×C3#54612yes—
466C22×C3#54712yes—
467C22×C3#54812yes—
468C22×C3#54812yes—
469C22×C3#55012yes—
470C22×C3#55612yes—
471C22×C3#55912yes—
472C3×C4—44no—
473C3×C4—82no—
474C3×C4#39418yes—
475Dic3—44no—
476Dic3—44no—
477Dic3—42no—
478Dic3—42no—
479Dic3—44no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—28no—
484C23—28no—
485C23—42no—
486C23—42no—
487C23—44no—
488C23—42no—
489C23—42no—
490C23—44no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23C2×#23511yescharacteristic
504C23C2×#23511yescharacteristic
505C23C2×D3×D411yescharacteristic
506C23C2×D3×D411yescharacteristic
507C23C2×D3×D411yescharacteristic
508C23C2×#23712yes—
509C23C2×#23711yescharacteristic
510C23#100412yes—
511C23D3×#4112yes—
512C23#101012yes—
513C23C22×#6711yescharacteristic
514Q8—1216no—
515D4—416no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—62no—
529C2×C4—62no—
530C2×C4—64no—
531C2×C4—62no—
532C2×C4—62no—
533C2×C4—64no—
534C2×C4—64no—
535C2×C4—64no—
536C2×C4—62no—
537C2×C4—62no—
538C2×C4—64no—
539C2×C4—64no—
540C2×C4—64no—
541C2×C4—68no—
542C2×C4—122no—
543C2×C4—122no—
544C2×C4—122no—
545C2×C4—122no—
546C2×C4—122no—
547C2×C4—122no—
548C2×C4—122no—
549C2×C4—122no—
550C2×C4—122no—
551C2×C4—122no—
552C2×C4—122no—
553C2×C4—124no—
554C2×C4—122no—
555C2×C4—122no—
556C2×C4—122no—
557C2×C4#88314yes—
558C2×C4#88414yes—
559C2×C4#88414yes—
560C2×C4#88514yes—
561C2×C4#92914yes—
562C2×C4#93714yes—
563C2×C4#93914yes—
564C2×C3—84no—
565C2×C3C2×#39411yescharacteristic
566C2×C3#242612yes—
567C2×C3#242712yes—
568C2×C3#243111yescharacteristic
569C2×C3#243211yescharacteristic
570C2×C3#244712yes—
571C2×C3#244812yes—
572C2×C3#254411yescharacteristic
573C2×C3#254511yescharacteristic
574C2×C3#264011yescharacteristic
575C2×C3#286111yescharacteristic
576C22—44no—
577C22—48no—
578C22—44no—
579C22—48no—
580C22—44no—
581C22—84no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22—84no—
587C22C2×#88311yescharacteristic
588C22C2×#88411yescharacteristic
589C22C2×#88411yescharacteristic
590C22C2×#88511yescharacteristic
591C22#554212yes—
592C22#555712yes—
593C22#555912yes—
594C22#556112yes—
595C22#559812yes—
596C22#560012yes—
597C22#560312yes—
598C22#561212yes—
599C22#566811yescharacteristic
600C22#567511yescharacteristic
601C22#567712yes—
602C22#571511yescharacteristic
603C22#571611yescharacteristic
604C22#571812yes—
605C22C2×#92911yescharacteristic
606C22C2×#93711yescharacteristic
607C22C2×#93911yescharacteristic
608C22#580612yes—
609C22#582412yes—
610C4—44no—
611C4—82no—
612C4—122no—
613C4—124no—
614C4—124no—
615C4—124no—
616C4—122no—
617C4#495918yes—
618C3#1522611yescharacteristic, complemented, Sylow
619C2—84no—
620C2#7632112yes—
621C2#7635011yescharacteristic
622C2#7635111yescharacteristic
623C2#7637311yescharacteristic
624C2#7642611yescharacteristic
625C2#7644312yes—
626C2C2×#495911yescharacteristic
627C2#8235912yes—
628C2#8264712yes—
629C2#8295911yescharacteristic
630C2#8318711yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC2×C3×#390C2
prime radicalC23×C3×D4C22
hypercenterC2×#390D3
hypocenterC3#15226
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
4. C2×#4973C2semidirect
5. C2×#4972C2semidirect
6. C2×#4929C2semidirect
7. #71548C2semidirect
8. #71547C2semidirect
9. C4×C3⋊#95C2semidirect
618. C3#15226semidirect

Maximal subgroups

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

subgroupindex
C22×#9362
C2×C3×#3902
C2×#52212
C2×#49732
C2×#49722
C2×#49292
#715482
#715472
C4×C3⋊#952
#152263