Finite Group Databasebeta

← #10718671

Subgroups — #10718671

Order 768.

2358 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#936C211yesmaximal, characteristic
2C22×C4×Dic6C211yesmaximal, characteristic, complemented
3C2×C3×#390C211yesmaximal, characteristic
4C2×#5221C211yesmaximal, characteristic
5C2×#4973C211yesmaximal, characteristic, complemented
6#71563C211yesmaximal, characteristic, complemented
7#71547C214yesmaximal, complemented
8#71545C214yesmaximal, complemented
9C4×C3⋊#95C211yesmaximal, characteristic, complemented
10#14026—31nomaximal, Sylow
11C23×C3×D4C2211yescharacteristic
12C23×Dic6C2211yescharacteristic
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C22⋊C4C2212yes—
15C22×C3×C22⋊C4C2212yes—
16C22×C3×C42C2211yescharacteristic
17C22×#217C2214yes—
18C22×#217C2214yes—
19C2×#936—24no—
20C2×#936—24no—
21C2×#936C418yescomplemented
22C22×C4⋊Dic3C2211yescharacteristic
23C22×Dic3⋊C4C2212yes—
24C22×C4×Dic3C2211yescharacteristic
25C22×C4×Dic3C2212yes—
26C22×C4×Dic3C2211yescharacteristic
27C2×C4×Dic6—24no—
28C2×C4×Dic6—24no—
29C2×C4×Dic6—24no—
30C3×#390—24no—
31#5221—24no—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2214yes—
35C2×C3⋊#95C2214yes—
36#4973—24no—
37C22×#124—31no—
38C22×C4×Q8—31no—
39C2×#391—31no—
40C2×#390—31no—
41C2×#390D311yescharacteristic
42#1439—34no—
43#1437—34no—
44C4⋊#95—31no—
45C4×#95—31no—
46C25×C3C2312yes—
47C22×C3×D4—22no—
48C22×C3×D4—22no—
49C22×C3×D4—22no—
50C22×C3×D4—22no—
51C22×C3×D4—22no—
52C22×C3×D4—22no—
53C22×C3×D4C2×C414yes—
54C23×C3×C4C2312yes—
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2311yescharacteristic
58C23×Dic3C2312yes—
59C23×Dic3C2312yes—
60C23×Dic3C2312yes—
61C23×Dic3C2312yes—
62C22×Dic6—22no—
63C22×Dic6—22no—
64C22×Dic6—22no—
65C22×Dic6—22no—
66C22×Dic6C2×C414yes—
67C22×Dic6D414yes—
68C22×Dic6D414yes—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C22⋊C4—24no—
73C2×C3×C22⋊C4—24no—
74C2×C3×C22⋊C4—28no—
75C2×C3×C22⋊C4—44no—
76C2×C3×C42—22no—
77C2×C3×C42—22no—
78C2×C3×C42D414yes—
79C2×#217—28no—
80C2×#217—28no—
81C2×#217—28no—
82C2×#217—28no—
83C2×#217—44no—
84C2×#217C2×C4116yes—
85#936—232no—
86C2×C4⋊Dic3—22no—
87C2×C4⋊Dic3—22no—
88C2×C4⋊Dic3D414yes—
89C2×Dic3⋊C4—24no—
90C2×Dic3⋊C4—24no—
91C2×Dic3⋊C4—24no—
92C2×C4×Dic3—24no—
93C2×C4×Dic3—22no—
94C2×C4×Dic3—22no—
95C2×C4×Dic3—24no—
96C2×C4×Dic3—24no—
97C2×C4×Dic3—22no—
98C2×C4×Dic3—22no—
99C2×C4×Dic3—22no—
100C2×C4×Dic3C2×C414yes—
101C4×Dic6—232no—
102C3⋊#95—24no—
103C3⋊#95—28no—
104C3⋊#95—24no—
105C3⋊#95—44no—
106C23×Q8—31no—
107C23×D4C2×D311yescharacteristic
108C2×#124—38no—
109C2×#124—64no—
110C2×#124—64no—
111C2×C4×Q8—64no—
112C2×C4×Q8—64no—
113C2×C4×Q8—64no—
114C22×C4⋊C4—31no—
115C22×C4⋊C4—32no—
116C22×C4⋊C4C2×D311yescharacteristic
117C22×C22⋊C4—34no—
118C22×C22⋊C4—34no—
119C22×C22⋊C4C2×D312yes—
120C22×C22⋊C4C2×D312yes—
121C22×C42—31no—
122C22×C42—32no—
123C22×C42—31no—
124C22×C42C2×D311yescharacteristic
125#391—64no—
126#390—24no—
127#390—64no—
128C2×#95—32no—
129C2×#95—32no—
130C2×#95—34no—
131C2×#95—34no—
132C24×C3—22no—
133C24×C3—22no—
134C24×C3—22no—
135C24×C3—22no—
136C24×C3—24no—
137C24×C3—24no—
138C24×C3—22no—
139C24×C3—22no—
140C24×C3—24no—
141C24×C3—24no—
142C24×C3C22×C414yes—
143C24×C3C2411yescharacteristic
144C2×C3×D4—28no—
145C2×C3×D4—28no—
146C2×C3×D4—28no—
147C2×C3×D4—44no—
148C2×C3×D4—44no—
149C2×C3×D4—44no—
150C2×C3×D4—44no—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4—22no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—24no—
161C22×C3×C4—22no—
162C22×C3×C4—24no—
163C22×C3×C4C22×C412yes—
164C22×C3×C4C2×D412yes—
165C22×C3×C4C2×D412yes—
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D412yes—
170C22×C3×C4#4112yes—
171C22×C3×C4#4114yes—
172C22×C3×C4#4112yes—
173C22×Dic3—22no—
174C22×Dic3—22no—
175C22×Dic3—22no—
176C22×Dic3—22no—
177C22×Dic3—22no—
178C22×Dic3—22no—
179C22×Dic3—22no—
180C22×Dic3—22no—
181C22×Dic3—22no—
182C22×Dic3—22no—
183C22×Dic3—22no—
184C22×Dic3—22no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—24no—
188C22×Dic3—22no—
189C22×Dic3—24no—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3C22×C414yes—
194C22×Dic3C22×C414yes—
195C22×Dic3C2×D414yes—
196C22×Dic3C2×D414yes—
197C22×Dic3C2×D414yes—
198C22×Dic3#4114yes—
199C2×Dic6—28no—
200C2×Dic6—28no—
201C2×Dic6—28no—
202C2×Dic6—28no—
203C2×Dic6—28no—
204C2×Dic6—28no—
205C2×Dic6C22⋊C4116yescomplemented
206C3×C4⋊C4—44no—
207C3×C22⋊C4—48no—
208C3×C22⋊C4—48no—
209C3×C42—28no—
210#217—232no—
211#217—416no—
212C4⋊Dic3—28no—
213Dic3⋊C4—216no—
214C4×Dic3—216no—
215C4×Dic3—28no—
216C4×Dic3—44no—
217C25C22×D312yes—
218C22×Q8—34no—
219C22×Q8—34no—
220C22×Q8—34no—
221C22×Q8—62no—
222C22×Q8—62no—
223C22×Q8—62no—
224C22×Q8—62no—
225C22×D4—22no—
226C22×D4—22no—
227C22×D4—22no—
228C22×D4—22no—
229C22×D4—22no—
230C22×D4—22no—
231C22×D4C4×D314yes—
232C23×C4—32no—
233C23×C4—32no—
234C23×C4—32no—
235C23×C4—32no—
236C23×C4C22×D312yes—
237C23×C4C22×D311yescharacteristic
238C23×C4C22×D311yescharacteristic
239C23×C4C22×D311yescharacteristic
240#124—632no—
241C4×Q8—632no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—34no—
246C2×C4⋊C4—64no—
247C2×C4⋊C4—64no—
248C2×C4⋊C4—62no—
249C2×C4⋊C4—64no—
250C2×C4⋊C4—62no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—316no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—68no—
257C2×C22⋊C4—68no—
258C2×C22⋊C4—68no—
259C2×C22⋊C4—68no—
260C2×C22⋊C4—124no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—34no—
264C2×C42—64no—
265C2×C42—64no—
266C2×C42—62no—
267C2×C42—62no—
268C2×C42—64no—
269C2×C42—62no—
270C2×C42—62no—
271C2×C42—62no—
272C2×C42#6714yes—
273#95—64no—
274#95—68no—
275#95—64no—
276#95—124no—
277C23×C3—24no—
278C23×C3—24no—
279C23×C3—24no—
280C23×C3—28no—
281C23×C3—28no—
282C23×C3—42no—
283C23×C3—42no—
284C23×C3—42no—
285C23×C3—42no—
286C23×C3—42no—
287C23×C3—44no—
288C23×C3—42no—
289C23×C3—42no—
290C23×C3—42no—
291C23×C3—44no—
292C23×C3—44no—
293C23×C3—44no—
294C23×C3—44no—
295C23×C3—44no—
296C23×C3—44no—
297C23×C3—44no—
298C23×C3—44no—
299C23×C3—44no—
300C23×C3C23×C411yescharacteristic
301C23×C3C22×D411yescharacteristic
302C23×C3C22×D411yescharacteristic
303C23×C3C22×D411yescharacteristic
304C23×C3C22×D411yescharacteristic
305C23×C3C2×#4111yescharacteristic
306C23×C3C2×#4111yescharacteristic
307C23×C3C2×#4112yes—
308C23×C3C2×#4112yes—
309C23×C3H(5,𝔽2)12yes—
310C23×C32−1+412yes—
311C3×D4—416no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—24no—
315C2×C3×C4—24no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—44no—
325C2×C3×C4—44no—
326C2×C3×C4C2×C22⋊C414yes—
327C2×C3×C4C4×D414yes—
328C2×C3×C4C4×D414yes—
329C2×C3×C4#12014yes—
330C2×C3×C4#12114yes—
331C2×C3×C4#12114yes—
332C2×C3×C4#12314yes—
333C2×C3×C4#12414yescomplemented
334C2×Dic3—24no—
335C2×Dic3—24no—
336C2×Dic3—24no—
337C2×Dic3—24no—
338C2×Dic3—24no—
339C2×Dic3—24no—
340C2×Dic3—24no—
341C2×Dic3—24no—
342C2×Dic3—24no—
343C2×Dic3—24no—
344C2×Dic3—24no—
345C2×Dic3—24no—
346C2×Dic3—28no—
347C2×Dic3—24no—
348C2×Dic3—42no—
349C2×Dic3—42no—
350C2×Dic3—42no—
351C2×Dic3—42no—
352C2×Dic3—42no—
353C2×Dic3—42no—
354C2×Dic3—42no—
355C2×Dic3—42no—
356C2×Dic3—42no—
357C2×Dic3—42no—
358C2×Dic3—44no—
359C2×Dic3C2×C22⋊C418yes—
360Dic6—232no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—24no—
366C24—22no—
367C24—24no—
368C24—24no—
369C24—24no—
370C24—22no—
371C24C2×C4×D314yes—
372C24C23×D311yescharacteristic
373C2×Q8—316no—
374C2×Q8—68no—
375C2×Q8—68no—
376C2×Q8—68no—
377C2×Q8—68no—
378C2×Q8—68no—
379C2×Q8—68no—
380C2×D4—28no—
381C2×D4—28no—
382C2×D4—28no—
383C2×D4—44no—
384C2×D4—44no—
385C2×D4—44no—
386C2×D4—44no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—34no—
400C22×C4—34no—
401C22×C4—34no—
402C22×C4—34no—
403C22×C4—34no—
404C22×C4—34no—
405C22×C4—62no—
406C22×C4—62no—
407C22×C4—62no—
408C22×C4—62no—
409C22×C4—62no—
410C22×C4—62no—
411C22×C4—62no—
412C22×C4—62no—
413C22×C4—62no—
414C22×C4—62no—
415C22×C4—62no—
416C22×C4—62no—
417C22×C4—62no—
418C22×C4—64no—
419C22×C4—62no—
420C22×C4—62no—
421C22×C4—64no—
422C22×C4—62no—
423C22×C4—62no—
424C22×C4—62no—
425C22×C4C2×C4×D312yes—
426C22×C4#23512yes—
427C22×C4D3×D412yes—
428C22×C4D3×D412yes—
429C22×C4D3×D412yes—
430C22×C4#23714yes—
431C22×C4#23712yes—
432C22×C4C2×#6712yes—
433C22×C4C2×#6712yes—
434C22×C4C2×#6712yes—
435C4⋊C4—44no—
436C4⋊C4—616no—
437C4⋊C4—68no—
438C22⋊C4—48no—
439C22⋊C4—48no—
440C22⋊C4—632no—
441C22⋊C4—1216no—
442C42—28no—
443C42—616no—
444C42—68no—
445C42—124no—
446C22×C3—44no—
447C22×C3—48no—
448C22×C3—48no—
449C22×C3—44no—
450C22×C3—44no—
451C22×C3—84no—
452C22×C3—82no—
453C22×C3—82no—
454C22×C3—84no—
455C22×C3—82no—
456C22×C3—82no—
457C22×C3C22×C22⋊C411yescharacteristic
458C22×C3C2×C4×D411yescharacteristic
459C22×C3C2×C4×D411yescharacteristic
460C22×C3C4×#4112yes—
461C22×C3#52012yes—
462C22×C3C2×#12011yescharacteristic
463C22×C3C2×#12111yescharacteristic
464C22×C3C2×#12111yescharacteristic
465C22×C3C2×#12311yescharacteristic
466C22×C3C2×#12411yescharacteristic
467C22×C3#52912yes—
468C22×C3#53211yescharacteristic
469C22×C3#53511yescharacteristic
470C22×C3#53611yescharacteristic
471C22×C3#54212yes—
472C22×C3#54612yes—
473C22×C3#54812yes—
474C22×C3#54812yes—
475C22×C3#54812yes—
476C22×C3#55012yes—
477C22×C3#55012yes—
478C22×C3#55612yes—
479C22×C3#55912yes—
480C3×C4—24no—
481C3×C4—42no—
482C3×C4—82no—
483C3×C4#38618yes—
484Dic3—28no—
485Dic3—44no—
486Dic3—44no—
487Dic3—44no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—28no—
492C23—28no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—44no—
504C23—44no—
505C23—42no—
506C23—42no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23C22×C4×D311yescharacteristic
512C23C2×#23511yescharacteristic
513C23C2×D3×D411yescharacteristic
514C23C2×D3×D411yescharacteristic
515C23C2×D3×D411yescharacteristic
516C23C2×#23711yescharacteristic
517C23C2×#23712yes—
518C23#100412yes—
519C23D3×#4112yes—
520C23#101012yes—
521C23C22×#6711yescharacteristic
522Q8—632no—
523D4—416no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—38no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—64no—
540C2×C4—64no—
541C2×C4—64no—
542C2×C4—64no—
543C2×C4—64no—
544C2×C4—64no—
545C2×C4—64no—
546C2×C4—64no—
547C2×C4—64no—
548C2×C4—64no—
549C2×C4—64no—
550C2×C4—64no—
551C2×C4—64no—
552C2×C4—68no—
553C2×C4—122no—
554C2×C4—122no—
555C2×C4—122no—
556C2×C4—122no—
557C2×C4—122no—
558C2×C4—122no—
559C2×C4—122no—
560C2×C4—122no—
561C2×C4—122no—
562C2×C4—122no—
563C2×C4—124no—
564C2×C4D3×C22⋊C414yes—
565C2×C4#67⋊C414yes—
566C2×C4#88314yes—
567C2×C4#88414yes—
568C2×C4C4×#6714yes—
569C2×C4#93614yescomplemented
570C2×C4#93714yes—
571C2×C4#93814yes—
572C2×C3—84no—
573C2×C3C2×#38611yescharacteristic
574C2×C3C4×#12411yescharacteristic
575C2×C3#232611yescharacteristic
576C2×C3#234112yes—
577C2×C3#236712yes—
578C2×C3#242612yes—
579C2×C3#243111yescharacteristic
580C2×C3#243611yescharacteristic
581C2×C3#244812yes—
582C2×C3#254511yescharacteristic
583C2×C3#254811yescharacteristic
584C22—48no—
585C22—48no—
586C22—44no—
587C22—44no—
588C22—44no—
589C22—82no—
590C22—84no—
591C22—84no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22C2×D3×C22⋊C411yescharacteristic
596C22C2×#67⋊C411yescharacteristic
597C22C2×#88311yescharacteristic
598C22C2×#88411yescharacteristic
599C22C4×#23712yes—
600C22#554612yes—
601C22#555712yes—
602C22#556112yes—
603C22#559812yes—
604C22#559812yes—
605C22#559912yes—
606C22#561212yes—
607C22#567112yes—
608C22#567712yes—
609C22#571312yes—
610C22#571812yes—
611C22C2×C4×#6711yescharacteristic
612C22C2×#93611yescharacteristic
613C22C2×#93711yescharacteristic
614C22C2×#93811yescharacteristic
615C22#580511yescharacteristic
616C22#582511yescharacteristic
617C22#583211yescharacteristic
618C4—24no—
619C4—42no—
620C4—68no—
621C4—82no—
622C4—124no—
623C4—124no—
624C4—124no—
625C4#495618yes—
626C3#1402611yescharacteristic, complemented, Sylow
627C2—84no—
628C2#7630112yes—
629C2#7630712yes—
630C2#7632012yes—
631C2#7644312yes—
632C2C2×#495611yescharacteristic
633C2C4×#93611yescharacteristic
634C2#8232811yescharacteristic
635C2#8265211yescharacteristic
636C2#8265411yescharacteristic
637C2#8297111yescharacteristic
638C2#8317911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC2×C3×#390C2
socle, cosocleC24×C3C24
prime radicalC23×C3×D4C22
hypercenterC2×#390D3
hypocenterC3#14026
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
2. C22×C4×Dic6C2semidirect
5. C2×#4973C2semidirect
6. #71563C2semidirect
7. #71547C2semidirect
8. #71545C2semidirect
9. C4×C3⋊#95C2semidirect
21. C2×#936C4semidirect
205. C2×Dic6C22⋊C4semidirect
333. C2×C3×C4#124semidirect
569. C2×C4#936semidirect
626. C3#14026semidirect

Maximal subgroups

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

subgroupindex
C22×#9362
C22×C4×Dic62
C2×C3×#3902
C2×#52212
C2×#49732
#715632
#715472
#715452
C4×C3⋊#952
#140263