Finite Group Databasebeta

← #10729196

Subgroups — #10729196

Order 768.

3030 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#929C212yesmaximal, complemented
2C22×#930C211yesmaximal, characteristic
3C22×C3×#95C211yesmaximal, characteristic, complemented
4C2×#5213C212yesmaximal, complemented
5#72127C211yesmaximal, characteristic, complemented
6#69428C214yesmaximal, complemented
7#69426C212yesmaximal, complemented
8#69420C212yesmaximal, complemented
9#14926—31nomaximal, Sylow
10C24×C3×C4C2211yescharacteristic
11C24×C3×C4C2211yescharacteristic
12C24×C3×C4C2211yescharacteristic
13C23×#67C2212yes—
14C23×D12C2211yescharacteristic
15C22×#217C2212yes—
16C2×#930—24no—
17C2×#930—28no—
18C2×#929—28no—
19C2×#929—28no—
20C2×#929—28no—
21C22×(C2×D3)⋊C4C2214yes—
22C22×(C2×D3)⋊C4C2212yes—
23C22×(C2×D3)⋊C4C2214yes—
24C22×C4⋊Dic3C2211yescharacteristic
25C22×Dic3⋊C4C2214yes—
26C2×C3×#95—24no—
27C2×C3×#95C2212yes—
28C2×C3×#95C2212yes—
29#5213—28no—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2214yes—
32C2×C3⋊#95C2212yes—
33C22×#123—32no—
34C22×#121—31no—
35C2×#386—32no—
36C22×#95D311yescharacteristic, complemented
37#1444—32no—
38#1439—34no—
39#1438—32no—
40C22⋊#95—31no—
41C25×C3C2311yescharacteristic
42C24×D3C2312yes—
43C23×C3×C4—22no—
44C23×C3×C4—22no—
45C23×C3×C4—24no—
46C23×C3×C4—24no—
47C23×C3×C4—24no—
48C23×C3×C4—22no—
49C23×C3×C4—24no—
50C23×C3×C4—24no—
51C23×C3×C4—24no—
52C23×C3×C4—42no—
53C23×C3×C4—42no—
54C23×C3×C4D414yes—
55C23×C3×C4D414yescomplemented
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2312yes—
58C23×C3×C4C2312yes—
59C23×C3×C4C2311yescharacteristic
60C22×#67—24no—
61C22×#67—24no—
62C22×#67—24no—
63C22×#67—24no—
64C22×#67—24no—
65C22×#67—24no—
66C22×#67D418yes—
67C23×Dic3C2312yes—
68C23×Dic3C2314yes—
69C22×D12—24no—
70C22×D12—22no—
71C22×D12—24no—
72C22×D12—42no—
73C2×#217—24no—
74C2×#217—24no—
75C2×#217D418yescomplemented
76#930—232no—
77#929—264no—
78C2×(C2×D3)⋊C4—28no—
79C2×(C2×D3)⋊C4—24no—
80C2×(C2×D3)⋊C4—28no—
81C2×(C2×D3)⋊C4—24no—
82C2×(C2×D3)⋊C4—28no—
83C2×(C2×D3)⋊C4—24no—
84C2×(C2×D3)⋊C4—28no—
85C2×(C2×D3)⋊C4—28no—
86C2×(C2×D3)⋊C4—44no—
87C2×C4⋊Dic3—22no—
88C2×C4⋊Dic3—24no—
89C2×Dic3⋊C4—28no—
90C2×Dic3⋊C4—28no—
91C2×Dic3⋊C4—28no—
92C3×#95—24no—
93C3×#95—24no—
94C3⋊#95—24no—
95C3⋊#95—28no—
96C3⋊#95—42no—
97C23×D4—32no—
98C23×D4—31no—
99C24×C4C2×D311yescharacteristic
100C24×C4C2×D311yescharacteristic
101C24×C4C2×D311yescharacteristic
102C2×#123—68no—
103C2×#123—68no—
104C2×#123—68no—
105C2×#121—68no—
106C2×#121—64no—
107C22×C4⋊C4—31no—
108C22×C4⋊C4—34no—
109C22×C22⋊C4—32no—
110C22×C22⋊C4—32no—
111C22×C22⋊C4—34no—
112C22×C22⋊C4—34no—
113#386—68no—
114C2×#95—24no—
115C2×#95—32no—
116C2×#95—34no—
117C2×#95—32no—
118C2×#95C2×D312yes—
119C2×#95C2×D312yes—
120C24×C3—22no—
121C24×C3—22no—
122C24×C3—22no—
123C24×C3—22no—
124C24×C3C2×D412yes—
125C24×C3C2×D414yes—
126C24×C3C2×D414yes—
127C24×C3C2×D412yes—
128C24×C3#4112yes—
129C24×C3C2411yescharacteristic
130C23×D3—22no—
131C23×D3—22no—
132C23×D3—22no—
133C23×D3—22no—
134C23×D3—22no—
135C23×D3—24no—
136C23×D3—24no—
137C23×D3—22no—
138C23×D3—24no—
139C23×D3—24no—
140C23×D3C2×D414yes—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—28no—
144C22×C3×C4—216no—
145C22×C3×C4—28no—
146C22×C3×C4—28no—
147C22×C3×C4—24no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—24no—
151C22×C3×C4—24no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—24no—
158C22×C3×C4—28no—
159C22×C3×C4—44no—
160C22×C3×C4—416no—
161C22×C3×C4—416no—
162C22×C3×C4—44no—
163C22×C3×C4—44no—
164C22×C3×C4—48no—
165C22×C3×C4—44no—
166C22×C3×C4—44no—
167C22×C3×C4C2×D414yes—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D412yes—
170C22×C3×C4#4114yes—
171C22×C3×C4#4114yes—
172C2×#67—216no—
173C2×#67—216no—
174C2×#67—216no—
175C2×#67—48no—
176C2×#67—48no—
177C2×#67—48no—
178C2×#67—48no—
179C22×Dic3—22no—
180C22×Dic3—22no—
181C22×Dic3—22no—
182C22×Dic3—24no—
183C22×Dic3—24no—
184C22×Dic3—22no—
185C22×Dic3—22no—
186C22×Dic3—24no—
187C22×Dic3—24no—
188C22×Dic3—22no—
189C22×Dic3—24no—
190C22×Dic3—24no—
191C22×Dic3C2×D418yes—
192C22×Dic3C2×D414yes—
193C2×D12—28no—
194C2×D12—416no—
195C2×D12—44no—
196C2×D12—44no—
197#217—216no—
198(C2×D3)⋊C4—232no—
199(C2×D3)⋊C4—216no—
200(C2×D3)⋊C4—416no—
201C4⋊Dic3—28no—
202Dic3⋊C4—232no—
203C25—32no—
204C25C22×D311yescharacteristic
205C22×D4—38no—
206C22×D4—64no—
207C22×D4—64no—
208C22×D4—64no—
209C22×D4—64no—
210C22×D4—64no—
211C22×D4—64no—
212C22×D4—64no—
213C22×D4—64no—
214C22×D4—62no—
215C22×D4—122no—
216C23×C4—22no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—34no—
226C23×C4—32no—
227C23×C4—42no—
228C23×C4—42no—
229C23×C4D1214yescomplemented
230C23×C4#6714yes—
231C23×C4C22×D312yes—
232C23×C4C22×D312yes—
233C23×C4C22×D311yescharacteristic
234C23×C4C22×D311yescharacteristic
235#123—664no—
236#121—632no—
237C2×C4⋊C4—64no—
238C2×C4⋊C4—68no—
239C2×C4⋊C4—68no—
240C2×C4⋊C4—68no—
241C2×C4⋊C4—62no—
242C2×C22⋊C4—38no—
243C2×C22⋊C4—64no—
244C2×C22⋊C4—68no—
245C2×C22⋊C4—64no—
246C2×C22⋊C4—68no—
247C2×C22⋊C4—64no—
248C2×C22⋊C4—64no—
249C2×C22⋊C4—68no—
250C2×C22⋊C4—64no—
251C2×C22⋊C4—68no—
252C2×C22⋊C4—68no—
253C2×C22⋊C4—124no—
254#95—24no—
255#95—24no—
256#95—68no—
257#95—64no—
258#95—122no—
259C23×C3—22no—
260C23×C3—22no—
261C23×C3—24no—
262C23×C3—24no—
263C23×C3—22no—
264C23×C3—22no—
265C23×C3—24no—
266C23×C3—24no—
267C23×C3—28no—
268C23×C3—28no—
269C23×C3—28no—
270C23×C3—28no—
271C23×C3#12018yes—
272C23×C3#12014yes—
273C23×C3#12118yes—
274C23×C3#12114yes—
275C23×C3#12714yes—
276C23×C3C22×D411yescharacteristic
277C23×C3C22×D412yes—
278C23×C3C22×D412yes—
279C23×C3C22×D411yescharacteristic
280C23×C3C2×#4111yescharacteristic
281C23×C3C2×#4112yes—
282C23×C3C2×#4112yes—
283C23×C3H(5,𝔽2)12yes—
284C23×C32−1+412yes—
285C22×D3—24no—
286C22×D3—24no—
287C22×D3—24no—
288C22×D3—28no—
289C22×D3—28no—
290C22×D3—42no—
291C22×D3—42no—
292C22×D3—42no—
293C22×D3—42no—
294C22×D3—42no—
295C22×D3—44no—
296C22×D3—42no—
297C22×D3—44no—
298C22×D3—42no—
299C22×D3—42no—
300C22×D3—44no—
301C22×D3—44no—
302C22×D3—44no—
303C22×D3—44no—
304C22×D3—44no—
305C22×D3—44no—
306C22×D3—44no—
307C22×D3—44no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—28no—
311C2×C3×C4—28no—
312C2×C3×C4—22no—
313C2×C3×C4—24no—
314C2×C3×C4—22no—
315C2×C3×C4—24no—
316C2×C3×C4—22no—
317C2×C3×C4—28no—
318C2×C3×C4—416no—
319C2×C3×C4—416no—
320C2×C3×C4—42no—
321C2×C3×C4—44no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—44no—
326C2×C3×C4—44no—
327C2×C3×C4—416no—
328C2×C3×C4#12314yescomplemented
329#67—432no—
330C2×Dic3—24no—
331C2×Dic3—28no—
332C2×Dic3—24no—
333C2×Dic3—24no—
334C2×Dic3—28no—
335C2×Dic3—28no—
336C2×Dic3—42no—
337C2×Dic3—42no—
338C2×Dic3—42no—
339C2×Dic3—42no—
340C2×Dic3—44no—
341C2×Dic3—44no—
342C2×Dic3—44no—
343C2×Dic3—44no—
344D12—416no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—34no—
350C24—62no—
351C24—64no—
352C24—62no—
353C24—64no—
354C24—62no—
355C24—62no—
356C24—62no—
357C24—64no—
358C24—64no—
359C24—62no—
360C24C2×D1212yes—
361C24#23512yes—
362C24D3×D414yes—
363C24D3×D414yes—
364C24C2×#6712yes—
365C24C23×D311yescharacteristic
366C2×D4—68no—
367C2×D4—616no—
368C2×D4—616no—
369C2×D4—616no—
370C2×D4—1216no—
371C2×D4—124no—
372C2×D4—128no—
373C2×D4—128no—
374C2×D4—128no—
375C2×D4—128no—
376C2×D4—124no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—28no—
380C22×C4—216no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—28no—
395C22×C4—38no—
396C22×C4—34no—
397C22×C4—44no—
398C22×C4—416no—
399C22×C4—416no—
400C22×C4—44no—
401C22×C4—48no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—64no—
406C22×C4—62no—
407C22×C4—64no—
408C22×C4—64no—
409C22×C4—62no—
410C22×C4—62no—
411C22×C4—64no—
412C22×C4—62no—
413C22×C4—62no—
414C22×C4—64no—
415C22×C4—64no—
416C22×C4—62no—
417C22×C4C2×D1214yes—
418C22×C4#23514yes—
419C22×C4#23514yes—
420C22×C4C2×#6712yes—
421C22×C4C2×#6712yes—
422C4⋊C4—68no—
423C4⋊C4—632no—
424C22⋊C4—616no—
425C22⋊C4—632no—
426C22⋊C4—616no—
427C22⋊C4—1216no—
428C22×C3—24no—
429C22×C3—24no—
430C22×C3—24no—
431C22×C3—24no—
432C22×C3—216no—
433C22×C3—216no—
434C22×C3—24no—
435C22×C3—24no—
436C22×C3#39218yes—
437C22×C3C2×#12012yes—
438C22×C3C2×#12011yescharacteristic
439C22×C3C2×#12112yes—
440C22×C3C2×#12111yescharacteristic
441C22×C3C2×#12311yescharacteristic
442C22×C3#52511yescharacteristic
443C22×C3#52912yes—
444C22×C3C2×#12711yescharacteristic
445C22×C3#53212yes—
446C22×C3#53411yescharacteristic
447C22×C3#53611yescharacteristic
448C22×C3#53712yes—
449C22×C3#53912yes—
450C22×C3#54614yes—
451C22×C3#54614yes—
452C22×C3#54714yes—
453C22×C3#54714yes—
454C2×D3—44no—
455C2×D3—44no—
456C2×D3—44no—
457C2×D3—48no—
458C2×D3—48no—
459C2×D3—82no—
460C2×D3—82no—
461C2×D3—84no—
462C2×D3—84no—
463C2×D3—82no—
464C2×D3—82no—
465C3×C4—24no—
466C3×C4—44no—
467C3×C4—42no—
468C3×C4—44no—
469Dic3—44no—
470Dic3—48no—
471C23—24no—
472C23—24no—
473C23—28no—
474C23—28no—
475C23—28no—
476C23—28no—
477C23—24no—
478C23—24no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—64no—
484C23—64no—
485C23—68no—
486C23—68no—
487C23—64no—
488C23—124no—
489C23—124no—
490C23—122no—
491C23—122no—
492C23—124no—
493C23—124no—
494C23—124no—
495C23—124no—
496C23—124no—
497C23—124no—
498C23—124no—
499C23—124no—
500C23—122no—
501C23—122no—
502C23—122no—
503C23—122no—
504C23—122no—
505C23—122no—
506C23#88218yes—
507C23#88418yes—
508C23#93014yes—
509C23#93714yes—
510C23#94014yes—
511C23C22×D1211yescharacteristic
512C23C2×#23512yes—
513C23C2×#23512yes—
514C23C2×#23511yescharacteristic
515C23C2×D3×D412yes—
516C23C2×D3×D412yes—
517C23#100412yes—
518C23#100712yes—
519C23C22×#6711yescharacteristic
520D4—1216no—
521D4—1232no—
522C2×C4—28no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—28no—
526C2×C4—28no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—416no—
533C2×C4—416no—
534C2×C4—416no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—68no—
543C2×C4—64no—
544C2×C4—64no—
545C2×C4—68no—
546C2×C4—68no—
547C2×C4—64no—
548C2×C4—124no—
549C2×C4—122no—
550C2×C4—122no—
551C2×C4—124no—
552C2×C4—124no—
553C2×C4—124no—
554C2×C4—122no—
555C2×C4—122no—
556C2×C4#92914yescomplemented
557C2×C3—28no—
558C2×C3C2×#39211yescharacteristic
559C2×C3#240511yescharacteristic
560C2×C3#240611yescharacteristic
561C2×C3#242514yes—
562C2×C3#243014yes—
563C2×C3#250111yescharacteristic
564C2×C3#253911yescharacteristic
565C2×C3#261811yescharacteristic
566C2×C3#285411yescharacteristic
567D3—84no—
568C22—24no—
569C22—24no—
570C22—24no—
571C22—216no—
572C22—216no—
573C22—24no—
574C22—24no—
575C22—24no—
576C22—124no—
577C22—128no—
578C22—128no—
579C22—124no—
580C22—124no—
581C22—242no—
582C22—242no—
583C22—244no—
584C22—244no—
585C22—242no—
586C22—242no—
587C22#496018yes—
588C22#547712yes—
589C22#547912yes—
590C22C2×#88212yes—
591C22C2×#88412yes—
592C22#549014yes—
593C22#549014yes—
594C22#550714yes—
595C22#550714yes—
596C22#551012yes—
597C22#551512yes—
598C22C2×#92911yescharacteristic
599C22C2×#93011yescharacteristic
600C22#579111yescharacteristic
601C22C2×#93711yescharacteristic
602C22C2×#94011yescharacteristic
603C22#580411yescharacteristic
604C22#581611yescharacteristic
605C4—24no—
606C4—44no—
607C4—44no—
608C4—42no—
609C4—128no—
610C4—124no—
611C3#1492611yescharacteristic, complemented, Sylow
612C2—28no—
613C2—244no—
614C2#7334914yes—
615C2#7337614yes—
616C2#8208011yescharacteristic
617C2#8211511yescharacteristic
618C2C2×#496011yescharacteristic
619C2#8212611yescharacteristic
620C2#8217711yescharacteristic
621C2#8219211yescharacteristic
622C2#8219911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC22×C3×#95C2
prime radicalC22×#930C2
hypercenterC22×#95D3
hypocenterC3#14926
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×#95

Complements

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

NH ≅ G/Ntype
1. C22×#929C2semidirect
3. C22×C3×#95C2semidirect
4. C2×#5213C2semidirect
5. #72127C2semidirect
6. #69428C2semidirect
7. #69426C2semidirect
8. #69420C2semidirect
36. C22×#95D3semidirect
55. C23×C3×C4D4semidirect
75. C2×#217D4semidirect
229. C23×C4D12semidirect
328. C2×C3×C4#123semidirect
556. C2×C4#929semidirect
611. C3#14926semidirect

Maximal subgroups

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

subgroupindex
C22×#9292
C22×#9302
C22×C3×#952
C2×#52132
#721272
#694282
#694262
#694202
#149263