Finite Group Databasebeta

← #10732199

Subgroups — #10732199

Order 768.

2770 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×D4C211yesmaximal, characteristic
2C22×#924C212yesmaximal, complemented
3C2×#5212C212yesmaximal, complemented
4C2×#4929C211yesmaximal, characteristic, complemented
5#72131C212yesmaximal
6#71568C212yesmaximal, complemented
7#71547C214yesmaximal, complemented
8#71535C211yesmaximal, characteristic, complemented
9#15177—31nomaximal, Sylow
10C23×C3×D4C2211yescharacteristic
11C24×C3×C4C2212yes—
12C23×Dic6C2211yescharacteristic
13C2×C3×C4×D4—28no—
14C2×C3×C4×D4—24no—
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C22⋊C4C2212yes—
17C22×C3×C42C2211yescharacteristic
18C22×#217C2214yes—
19C22×#217C2214yes—
20C22×C4⋊Dic3C2211yescharacteristic
21C22×C4⋊Dic3C2212yes—
22C2×#924—28no—
23C2×#924—28no—
24C2×#924—28no—
25C22×Dic3⋊C4C2214yes—
26#5212—28no—
27C2×C3⋊#95C2214yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32#4929—24no—
33C22×C22⋊Q8—32no—
34C22×C4×D4D311yescharacteristic
35C2×#391—31no—
36C2×#380—32no—
37#1444—32no—
38#1439—34no—
39#1438—32no—
40C4⋊#95—31no—
41C25×C3C2312yes—
42C22×C3×D4—24no—
43C22×C3×D4—24no—
44C22×C3×D4—22no—
45C22×C3×D4—22no—
46C22×C3×D4Q814yescomplemented
47C23×C3×C4—24no—
48C23×C3×C4—24no—
49C23×C3×C4—24no—
50C23×C3×C4—24no—
51C23×C3×C4—24no—
52C23×C3×C4—24no—
53C23×C3×C4D418yes—
54C23×C3×C4C2312yes—
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2311yescharacteristic
58C23×Dic3C2312yes—
59C23×Dic3C2312yes—
60C23×Dic3C2314yes—
61C22×Dic6—22no—
62C22×Dic6—24no—
63C22×Dic6—24no—
64C22×Dic6—22no—
65C22×Dic6D414yescomplemented
66C3×C4×D4—232no—
67C2×C3×C4⋊C4—24no—
68C2×C3×C4⋊C4—22no—
69C2×C3×C22⋊C4—24no—
70C2×C3×C22⋊C4—24no—
71C2×C3×C22⋊C4—24no—
72C2×C3×C42—24no—
73C2×C3×C42D414yes—
74C2×#217—28no—
75C2×#217—28no—
76C2×#217—28no—
77C2×#217—28no—
78C2×#217—28no—
79C2×#217—44no—
80C2×C4⋊Dic3—24no—
81C2×C4⋊Dic3—24no—
82C2×C4⋊Dic3—22no—
83C2×C4⋊Dic3—24no—
84C2×C4⋊Dic3D418yescomplemented
85#924—264no—
86C2×Dic3⋊C4—28no—
87C2×Dic3⋊C4—28no—
88C2×Dic3⋊C4—28no—
89C3⋊#95—24no—
90C3⋊#95—24no—
91C3⋊#95—24no—
92C3⋊#95—28no—
93C3⋊#95—42no—
94C23×Q8—31no—
95C23×D4C2×D311yescharacteristic
96C24×C4C2×D312yes—
97C2×C22⋊Q8—68no—
98C2×C22⋊Q8—68no—
99C2×C22⋊Q8—68no—
100C2×C4×D4—28no—
101C2×C4×D4—24no—
102C22×C4⋊C4—32no—
103C22×C4⋊C4—34no—
104C22×C4⋊C4—31no—
105C22×C4⋊C4C2×D311yescharacteristic
106C22×C22⋊C4—34no—
107C22×C22⋊C4—34no—
108C22×C22⋊C4C2×D312yes—
109C22×C42C2×D311yescharacteristic
110#391—64no—
111#380—68no—
112C2×#95—32no—
113C2×#95—32no—
114C2×#95—32no—
115C2×#95—34no—
116C2×#95—32no—
117C24×C3—22no—
118C24×C3—22no—
119C24×C3—22no—
120C24×C3—22no—
121C24×C3—22no—
122C24×C3—22no—
123C24×C3—22no—
124C24×C3—22no—
125C24×C3—22no—
126C24×C3—22no—
127C24×C3—22no—
128C24×C3—22no—
129C24×C3C2×D414yes—
130C24×C3C2×Q814yes—
131C24×C3#4114yes—
132C24×C3C2411yescharacteristic
133C2×C3×D4—28no—
134C2×C3×D4—28no—
135C2×C3×D4—216no—
136C2×C3×D4—216no—
137C2×C3×D4—28no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—21no—
148C22×C3×C4—22no—
149C22×C3×C4—21no—
150C22×C3×C4—216no—
151C22×C3×C4—216no—
152C22×C3×C4—216no—
153C22×C3×C4—48no—
154C22×C3×C4—48no—
155C22×C3×C4—48no—
156C22×C3×C4—48no—
157C22×C3×C4C2×D412yes—
158C22×C3×C4C2×D414yes—
159C22×C3×C4C2×D412yes—
160C22×C3×C4C2×D414yes—
161C22×C3×C4C2×D412yes—
162C22×C3×C4C2×D414yes—
163C22×C3×C4C2×D412yes—
164C22×C3×C4C2×Q812yes—
165C22×C3×C4#4112yes—
166C22×C3×C4#4112yes—
167C22×Dic3—22no—
168C22×Dic3—24no—
169C22×Dic3—24no—
170C22×Dic3—24no—
171C22×Dic3—24no—
172C22×Dic3—24no—
173C22×Dic3—24no—
174C22×Dic3—22no—
175C22×Dic3—24no—
176C22×Dic3—24no—
177C22×Dic3—22no—
178C22×Dic3—24no—
179C22×Dic3—24no—
180C22×Dic3—22no—
181C22×Dic3—22no—
182C22×Dic3C2×D418yes—
183C22×Dic3C2×D414yes—
184C2×Dic6—216no—
185C2×Dic6—28no—
186C2×Dic6—44no—
187C2×Dic6—48no—
188C2×Dic6—44no—
189C3×C4⋊C4—28no—
190C3×C22⋊C4—216no—
191C3×C42—28no—
192#217—232no—
193#217—416no—
194C4⋊Dic3—216no—
195C4⋊Dic3—44no—
196Dic3⋊C4—232no—
197C25C22×D312yes—
198C22×Q8—34no—
199C22×Q8—64no—
200C22×Q8—62no—
201C22×Q8—64no—
202C22×Q8—62no—
203C22×D4—24no—
204C22×D4—24no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4Dic614yescomplemented
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—32no—
215C23×C4—34no—
216C23×C4—32no—
217C23×C4#6718yes—
218C23×C4C22×D311yescharacteristic
219C23×C4C22×D312yes—
220C23×C4C22×D311yescharacteristic
221C23×C4C22×D311yescharacteristic
222C22⋊Q8—664no—
223C4×D4—232no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—38no—
227C2×C4⋊C4—68no—
228C2×C4⋊C4—68no—
229C2×C4⋊C4—64no—
230C2×C4⋊C4—68no—
231C2×C4⋊C4—64no—
232C2×C4⋊C4—64no—
233C2×C4⋊C4—62no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—68no—
238C2×C22⋊C4—68no—
239C2×C22⋊C4—68no—
240C2×C22⋊C4—68no—
241C2×C22⋊C4—68no—
242C2×C22⋊C4—124no—
243C2×C42—24no—
244C2×C42#6714yes—
245#95—68no—
246#95—64no—
247#95—64no—
248#95—64no—
249#95—122no—
250C23×C3—24no—
251C23×C3—24no—
252C23×C3—24no—
253C23×C3—24no—
254C23×C3—24no—
255C23×C3—24no—
256C23×C3—24no—
257C23×C3—24no—
258C23×C3—24no—
259C23×C3—24no—
260C23×C3—24no—
261C23×C3—24no—
262C23×C3—24no—
263C23×C3—24no—
264C23×C3—24no—
265C23×C3—24no—
266C23×C3—24no—
267C23×C3—24no—
268C23×C3—42no—
269C23×C3—42no—
270C23×C3—42no—
271C23×C3—42no—
272C23×C3—42no—
273C23×C3—42no—
274C23×C3—42no—
275C23×C3—42no—
276C23×C3—42no—
277C23×C3—42no—
278C23×C3—42no—
279C23×C3—42no—
280C23×C3—42no—
281C23×C3—42no—
282C23×C3—42no—
283C23×C3—42no—
284C23×C3C22⋊Q818yescomplemented
285C23×C3C22×D411yescharacteristic
286C23×C3C22×D412yes—
287C23×C3C22×D412yes—
288C23×C3C22×D411yescharacteristic
289C23×C3C22×Q811yescharacteristic
290C23×C3C2×#4112yes—
291C23×C3C2×#4111yescharacteristic
292C23×C3C2×#4111yescharacteristic
293C23×C3H(5,𝔽2)11yescharacteristic
294C23×C32−1+412yes—
295C23×C32−1+411yescharacteristic
296C3×D4—232no—
297C2×C3×C4—22no—
298C2×C3×C4—24no—
299C2×C3×C4—22no—
300C2×C3×C4—24no—
301C2×C3×C4—24no—
302C2×C3×C4—22no—
303C2×C3×C4—24no—
304C2×C3×C4—24no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—432no—
314C2×C3×C4#12018yes—
315C2×C3×C4#12014yes—
316C2×C3×C4#12114yes—
317C2×C3×C4#12118yes—
318C2×C3×C4C22⋊Q814yescomplemented
319C2×C3×C4#12714yes—
320C2×Dic3—28no—
321C2×Dic3—28no—
322C2×Dic3—24no—
323C2×Dic3—28no—
324C2×Dic3—28no—
325C2×Dic3—42no—
326C2×Dic3—44no—
327C2×Dic3—44no—
328C2×Dic3—42no—
329C2×Dic3—44no—
330C2×Dic3—44no—
331C2×Dic3—44no—
332C2×Dic3—42no—
333C2×Dic3—42no—
334C2×Dic3—42no—
335C2×Dic3—44no—
336C2×Dic3—44no—
337Dic6—416no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24C2×Dic614yes—
351C24#23514yes—
352C24C2×#6714yes—
353C24C23×D311yescharacteristic
354C2×Q8—616no—
355C2×Q8—68no—
356C2×Q8—128no—
357C2×Q8—124no—
358C2×Q8—124no—
359C2×D4—28no—
360C2×D4—216no—
361C2×D4—28no—
362C2×D4—216no—
363C2×D4—28no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—216no—
377C22×C4—216no—
378C22×C4—216no—
379C22×C4—38no—
380C22×C4—34no—
381C22×C4—48no—
382C22×C4—48no—
383C22×C4—48no—
384C22×C4—48no—
385C22×C4—64no—
386C22×C4—64no—
387C22×C4—64no—
388C22×C4—64no—
389C22×C4—62no—
390C22×C4—64no—
391C22×C4—64no—
392C22×C4—62no—
393C22×C4—64no—
394C22×C4—64no—
395C22×C4—64no—
396C22×C4—64no—
397C22×C4—62no—
398C22×C4—62no—
399C22×C4—62no—
400C22×C4C2×Dic612yes—
401C22×C4#23512yes—
402C22×C4D3×D412yes—
403C22×C4D3×D414yes—
404C22×C4#23712yes—
405C22×C4C2×#6712yes—
406C22×C4C2×#6714yes—
407C22×C4C2×#6712yes—
408C22×C4C2×#6714yes—
409C22×C4C2×#6712yes—
410C4⋊C4—28no—
411C4⋊C4—632no—
412C4⋊C4—616no—
413C4⋊C4—124no—
414C22⋊C4—216no—
415C22⋊C4—632no—
416C22⋊C4—1216no—
417C42—28no—
418C22×C3—28no—
419C22×C3—28no—
420C22×C3—28no—
421C22×C3—44no—
422C22×C3—44no—
423C22×C3—44no—
424C22×C3—44no—
425C22×C3—44no—
426C22×C3—44no—
427C22×C3—44no—
428C22×C3—44no—
429C22×C3—44no—
430C22×C3—44no—
431C22×C3—44no—
432C22×C3—44no—
433C22×C3C2×#12012yes—
434C22×C3C2×#12011yescharacteristic
435C22×C3C2×#12111yescharacteristic
436C22×C3C2×#12112yes—
437C22×C3C2×C22⋊Q812yes—
438C22×C3C2×C22⋊Q811yescharacteristic
439C22×C3#52512yes—
440C22×C3#52511yescharacteristic
441C22×C3C2×#12711yescharacteristic
442C22×C3#53612yes—
443C22×C3#53611yescharacteristic
444C22×C3#53712yes—
445C22×C3#53711yescharacteristic
446C22×C3#54612yes—
447C22×C3#54712yes—
448C22×C3#54812yes—
449C22×C3#54812yes—
450C22×C3D4×Q811yescharacteristic
451C22×C3D4×Q812yes—
452C22×C3#55011yescharacteristic
453C22×C3#55212yes—
454C22×C3#55411yescharacteristic
455C22×C3#55611yescharacteristic
456C3×C4—24no—
457C3×C4—42no—
458C3×C4—44no—
459C3×C4#39218yes—
460Dic3—44no—
461Dic3—48no—
462Dic3—82no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23#92418yescomplemented
498C23C22×Dic611yescharacteristic
499C23C2×#23512yes—
500C23C2×#23511yescharacteristic
501C23C2×D3×D411yescharacteristic
502C23C2×D3×D412yes—
503C23C2×#23711yescharacteristic
504C23#100911yescharacteristic
505C23#101012yes—
506C23#101011yescharacteristic
507C23C22×#6712yes—
508C23C22×#6711yescharacteristic
509Q8—1216no—
510D4—232no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—432no—
528C2×C4—68no—
529C2×C4—68no—
530C2×C4—64no—
531C2×C4—68no—
532C2×C4—68no—
533C2×C4—124no—
534C2×C4—124no—
535C2×C4—122no—
536C2×C4—124no—
537C2×C4—124no—
538C2×C4—124no—
539C2×C4—122no—
540C2×C4—122no—
541C2×C4—122no—
542C2×C4—124no—
543C2×C4—124no—
544C2×C4—122no—
545C2×C4#92414yescomplemented
546C2×C4#93718yes—
547C2×C4#93814yes—
548C2×C4#93918yes—
549C2×C4#94014yes—
550C2×C4#95314yes—
551C2×C3—48no—
552C2×C3C2×#39211yescharacteristic
553C2×C3#240712yes—
554C2×C3#240912yes—
555C2×C3#241512yes—
556C2×C3#241712yes—
557C2×C3#253611yescharacteristic
558C2×C3#253811yescharacteristic
559C2×C3#254011yescharacteristic
560C2×C3#254211yescharacteristic
561C2×C3#261611yescharacteristic
562C2×C3#284811yescharacteristic
563C22—28no—
564C22—28no—
565C22—28no—
566C22—44no—
567C22—44no—
568C22—44no—
569C22—44no—
570C22—44no—
571C22—44no—
572C22—44no—
573C22—44no—
574C22—44no—
575C22—44no—
576C22—44no—
577C22—44no—
578C22D4×Dic611yescharacteristic
579C22D4×Dic612yes—
580C22#554112yes—
581C22#554211yescharacteristic
582C22#554311yescharacteristic
583C22#555012yes—
584C22#555112yes—
585C22#555212yes—
586C22#555212yes—
587C22#555311yescharacteristic
588C22C2×#92412yes—
589C22C2×#92411yescharacteristic
590C22#579112yes—
591C22#579111yescharacteristic
592C22C2×#93712yes—
593C22C2×#93811yescharacteristic
594C22C2×#93912yes—
595C22C2×#94011yescharacteristic
596C22#582512yes—
597C22#582511yescharacteristic
598C22#583312yes—
599C22#583311yescharacteristic
600C22C2×#95311yescharacteristic
601C4—24no—
602C4—44no—
603C4—42no—
604C4—124no—
605C4—128no—
606C4—242no—
607C4#522318yes—
608C3#1517711yescharacteristic, complemented, Sylow
609C2—48no—
610C2#8234912yes—
611C2#8235112yes—
612C2#8235312yes—
613C2#8237511yescharacteristic
614C2#8237611yescharacteristic
615C2#8237711yescharacteristic
616C2#8237911yescharacteristic
617C2#8240212yes—
618C2#8403111yescharacteristic
619C2C2×#522311yescharacteristic
620C2#8409411yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC22×C3×C4×D4C2
prime radicalC23×C3×D4C22
hypercenterC22×C4×D4D3
hypocenterC3#15177
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×C4×D4

Complements

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

NH ≅ G/Ntype
2. C22×#924C2semidirect
3. C2×#5212C2semidirect
4. C2×#4929C2semidirect
6. #71568C2semidirect
7. #71547C2semidirect
8. #71535C2semidirect
46. C22×C3×D4Q8semidirect
65. C22×Dic6D4semidirect
84. C2×C4⋊Dic3D4semidirect
207. C22×D4Dic6semidirect
284. C23×C3C22⋊Q8semidirect
318. C2×C3×C4C22⋊Q8semidirect
497. C23#924semidirect
545. C2×C4#924semidirect
608. C3#15177semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4×D42
C22×#9242
C2×#52122
C2×#49292
#721312
#715682
#715472
#715352
#151773