Finite Group Databasebeta

← #10967472

Subgroups — #10967472

Order 768.

3114 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×Dic3C211yesmaximal, characteristic, complemented
2C2×#5600C211yesmaximal, characteristic, complemented
3C3×#2433C211yesmaximal, characteristic
4#84050C211yesmaximal, characteristic, complemented
5#83178C212yesmaximal, complemented
6Dic3×#127C212yesmaximal, complemented
7#82665C211yesmaximal, characteristic, complemented
8#82583C214yesmaximal, complemented
9Dic3×#120C214yesmaximal, complemented
10#76695C214yesmaximal, complemented
11#76666C214yesmaximal, complemented
12#76362C212yesmaximal, complemented
13#76292C212yesmaximal, complemented
14#73517C211yesmaximal, characteristic, complemented
15C4×#877C211yesmaximal, characteristic, complemented
16#34965—31nomaximal, Sylow
17C23×C3×D4C2211yescharacteristic
18C24×Dic3C2212yes—
19C2×C3×#127C2212yes—
20C2×C3×#121C2211yescharacteristic
21C2×C3×#120C2214yes—
22C22×#217C2212yes—
23C2×#936C2212yes—
24C2×#936C2211yescharacteristic, complemented
25C2×#935C2212yes—
26C2×D4×Dic3—28no—
27C2×D4×Dic3C2212yes—
28C2×D4×Dic3C2212yes—
29C2×D4×Dic3C2212yes—
30C2×D4×Dic3C2212yes—
31C2×D4×Dic3C2212yes—
32C2×D4×Dic3C2211yescharacteristic, complemented
33C2×D4×Dic3C2218yescomplemented
34C22×C4⋊Dic3C2211yescharacteristic
35C2×#924C2211yescharacteristic
36C22×C4×Dic3C2211yescharacteristic
37#5600—28no—
38C2×#890C2211yescharacteristic, complemented
39C2×#887C2211yescharacteristic, complemented
40C2×#879C2212yescomplemented
41C2×#877C2212yes—
42C2×#877C2212yescomplemented
43C3×#392C2214yes—
44C3×#390C2212yes—
45C3×C4×C22⋊C4C2211yescharacteristic
46#5247C2214yes—
47#5222C2214yes—
48#5222C2218yescomplemented
49#5221C2212yes—
50#5221C2218yescomplemented
51#5221C2212yescomplemented
52#5220C2214yes—
53#5220C2218yescomplemented
54C4×#217C2211yescharacteristic
55#4977C2212yescomplemented
56#4953C2214yescomplemented
57#4950C2214yes—
58#4950C2218yescomplemented
59#4948C2218yescomplemented
60#4944C2218yescomplemented
61Dic3×C22⋊C4C2212yes—
62Dic3×C22⋊C4C2218yescomplemented
63#4935C2212yescomplemented
64C4×Dic3⋊C4C2212yescomplemented
65C42×Dic3C2212yescomplemented
66#4929C2212yescomplemented
67#4661C2214yescomplemented
68#4649C2212yescomplemented
69#4648C2212yescomplemented
70C2×#529—31no—
71C22×C4×D4—31no—
72#2603—34no—
73#2566—32no—
74#2544—32no—
75#2504—31no—
76#2502—31no—
77#2435—31no—
78#2433D311yescharacteristic
79#2411—34no—
80#2392—34no—
81C4×#127—32no—
82C4×#124—32no—
83C4×#120—34no—
84C4×#117—31no—
85C25×C3C2312yes—
86C22×C3×D4—28no—
87C22×C3×D4—24no—
88C22×C3×D4C2312yes—
89C22×C3×D4C2312yes—
90C22×C3×D4C2311yescharacteristic
91C22×C3×D4C2318yes—
92C22×C3×D4C2312yes—
93C23×C3×C4C2311yescharacteristic
94C23×Dic3—28no—
95C23×Dic3—48no—
96C23×Dic3C2314yes—
97C23×Dic3C2312yes—
98C23×Dic3C2311yescharacteristic
99C23×Dic3C2311yescharacteristic
100C23×Dic3C2312yes—
101C23×Dic3C2314yes—
102C22×Dic6C2311yescharacteristic
103C3×#127—28no—
104C3×#121—24no—
105C3×#120—216no—
106C2×C3×C4⋊C4C2311yescharacteristic
107C2×C3×C22⋊C4C2318yes—
108C2×C3×C22⋊C4C2312yes—
109C2×C3×C22⋊C4C2312yes—
110C2×C3×C42C2312yes—
111C2×#217—28no—
112C2×#217C2312yes—
113C2×#217C2318yes—
114C2×#217C2314yes—
115C2×#217C2314yes—
116C2×#217C2312yes—
117C2×#217C2312yes—
118C2×#217C2312yes—
119C2×#217C2312yes—
120C2×#217C2318yes—
121C2×#217C2318yes—
122#936—28no—
123#936—24no—
124#935—28no—
125D4×Dic3—28no—
126D4×Dic3—28no—
127D4×Dic3—28no—
128D4×Dic3—28no—
129D4×Dic3—28no—
130D4×Dic3—24no—
131D4×Dic3—416no—
132C2×C4⋊Dic3—24no—
133C2×C4⋊Dic3C2312yes—
134C2×C4⋊Dic3C2312yes—
135C2×C4⋊Dic3C2311yescharacteristic
136C2×C4⋊Dic3C2312yes—
137C2×C4⋊Dic3C2318yes—
138#924—24no—
139C2×Dic3⋊C4C2314yes—
140C2×Dic3⋊C4C2312yes—
141C2×Dic3⋊C4C2312yes—
142C2×Dic3⋊C4C2312yes—
143C2×C4×Dic3D418yescomplemented
144C2×C4×Dic3C2314yes—
145C2×C4×Dic3C2314yes—
146C2×C4×Dic3C2312yes—
147C2×C4×Dic3C2312yes—
148C2×C4×Dic3C2311yescharacteristic
149C2×C4×Dic3C2311yescharacteristic
150C2×C4×Dic3C2311yescharacteristic
151C2×C4×Dic3C2311yescharacteristic
152#890—24no—
153#887—24no—
154#879—28no—
155#877—28no—
156#877—28no—
157C3×#95C2312yes—
158C3⋊#95C2312yes—
159C3⋊#95C2314yes—
160C3⋊#95C2318yes—
161C3⋊#95C2318yes—
162C23×D4C2×D311yescharacteristic
163C24×C4—32no—
164C2×#127C2×D312yes—
165#529—68no—
166C2×#126—32no—
167C2×#125—31no—
168C2×#124—31no—
169C2×#124—32no—
170C2×#123—32no—
171C2×C22⋊Q8—31no—
172C2×#121C2×D311yescharacteristic
173C2×#120C2×D314yes—
174C2×C4×Q8—31no—
175C2×C4×D4—31no—
176C2×C4×D4—32no—
177C2×C4×D4—32no—
178C2×C4×D4—32no—
179C2×C4×D4—32no—
180C2×C4×D4—32no—
181C2×C4×D4—38no—
182C2×C4×D4—68no—
183C2×#117—32no—
184C2×#117—32no—
185C22×C4⋊C4—31no—
186C22×C22⋊C4—32no—
187C22×C42—31no—
188#398—34no—
189#397—34no—
190#396—38no—
191#394—34no—
192#394—38no—
193#392C2×D314yes—
194#391—32no—
195#390—32no—
196#390—38no—
197#390—32no—
198#390C2×D312yes—
199#388—38no—
200#386—38no—
201#386—34no—
202#385—38no—
203#385—34no—
204#384—32no—
205#382—32no—
206#379—34no—
207C4×C4⋊C4—32no—
208C4×C4⋊C4—32no—
209C4×C22⋊C4—31no—
210C4×C22⋊C4—32no—
211C4×C22⋊C4—38no—
212C4×C22⋊C4C2×D311yescharacteristic
213#376—32no—
214C43—32no—
215C24×C3—24no—
216C24×C3—28no—
217C24×C3—28no—
218C24×C3—44no—
219C24×C3C2412yes—
220C24×C3C2414yes—
221C24×C3C2411yescharacteristic
222C2×C3×D4—232no—
223C2×C3×D4—28no—
224C2×C3×D4—24no—
225C2×C3×D4—28no—
226C2×C3×D4—216no—
227C2×C3×D4—24no—
228C2×C3×D4—24no—
229C2×C3×D4—22no—
230C2×C3×D4—24no—
231C2×C3×D4—216no—
232C2×C3×D4—24no—
233C2×C3×D4—48no—
234C2×C3×D4#4118yes—
235C22×C3×C4—24no—
236C22×C3×C4C2×D414yes—
237C22×C3×C4C2412yes—
238C22×C3×C4C2411yescharacteristic
239C22×C3×C4C2414yes—
240C22×C3×C4C2411yescharacteristic
241C22×Dic3—24no—
242C22×Dic3—24no—
243C22×Dic3—24no—
244C22×Dic3—24no—
245C22×Dic3—28no—
246C22×Dic3—28no—
247C22×Dic3—28no—
248C22×Dic3—216no—
249C22×Dic3—48no—
250C22×Dic3—44no—
251C22×Dic3—432no—
252C22×Dic3C2×D414yes—
253C22×Dic3C2×D414yes—
254C22×Dic3C2414yes—
255C22×Dic3C2412yes—
256C22×Dic3C2414yes—
257C22×Dic3C2412yes—
258C22×Dic3C2411yescharacteristic
259C22×Dic3C2411yescharacteristic
260C22×Dic3C2411yescharacteristic
261C22×Dic3C2411yescharacteristic
262C2×Dic6—22no—
263C2×Dic6—24no—
264C3×C4⋊C4—22no—
265C3×C22⋊C4—24no—
266C3×C22⋊C4—216no—
267C3×C22⋊C4#4118yes—
268C3×C42—24no—
269#217—24no—
270#217—24no—
271#217—216no—
272#217—24no—
273#217—24no—
274#217—24no—
275#217—28no—
276#217—28no—
277#217—48no—
278#217—48no—
279C4⋊Dic3—22no—
280C4⋊Dic3—24no—
281C4⋊Dic3—24no—
282C4⋊Dic3—24no—
283C4⋊Dic3—48no—
284Dic3⋊C4—24no—
285Dic3⋊C4—28no—
286Dic3⋊C4—24no—
287Dic3⋊C4—24no—
288C4×Dic3—28no—
289C4×Dic3—24no—
290C4×Dic3—24no—
291C4×Dic3—28no—
292C4×Dic3C2×D414yes—
293C4×Dic3C2×D414yes—
294C4×Dic3C2×D414yes—
295C4×Dic3C2×D414yes—
296C25C22×D312yes—
297C22×Q8—31no—
298C22×D4—28no—
299C22×D4—24no—
300C22×D4C22×D312yes—
301C22×D4C22×D312yes—
302C22×D4C22×D311yescharacteristic
303C22×D4C22×D318yes—
304C22×D4C22×D312yes—
305C23×C4—31no—
306C23×C4—31no—
307C23×C4—32no—
308C23×C4—34no—
309C23×C4—32no—
310C23×C4—34no—
311C23×C4—68no—
312C23×C4—128no—
313C23×C4C22×D311yescharacteristic
314#127—28no—
315#126—68no—
316#125—64no—
317#124—64no—
318#124—68no—
319#123—68no—
320C22⋊Q8—64no—
321#121—24no—
322#120—216no—
323C4×Q8—64no—
324C4×D4—68no—
325C4×D4—68no—
326C4×D4—68no—
327C4×D4—68no—
328C4×D4—64no—
329C4×D4—68no—
330C4×D4—1216no—
331#117—68no—
332#117—68no—
333C2×C4⋊C4—32no—
334C2×C4⋊C4—32no—
335C2×C4⋊C4—32no—
336C2×C4⋊C4—32no—
337C2×C4⋊C4—34no—
338C2×C4⋊C4—31no—
339C2×C4⋊C4—32no—
340C2×C4⋊C4—32no—
341C2×C4⋊C4—38no—
342C2×C4⋊C4—64no—
343C2×C4⋊C4C22×D311yescharacteristic
344C2×C22⋊C4—34no—
345C2×C22⋊C4—34no—
346C2×C22⋊C4—32no—
347C2×C22⋊C4—32no—
348C2×C22⋊C4—32no—
349C2×C22⋊C4—32no—
350C2×C22⋊C4—32no—
351C2×C22⋊C4—38no—
352C2×C22⋊C4—38no—
353C2×C22⋊C4—38no—
354C2×C22⋊C4—68no—
355C2×C22⋊C4C22×D312yes—
356C2×C22⋊C4C22×D318yes—
357C2×C22⋊C4C22×D312yes—
358C2×C42—38no—
359C2×C42—31no—
360C2×C42—31no—
361C2×C42—31no—
362C2×C42—31no—
363C2×C42—32no—
364C2×C42—34no—
365C2×C42—34no—
366C2×C42—32no—
367C2×C42C22×D312yes—
368#95—34no—
369#95—32no—
370#95—38no—
371#95—38no—
372#95C22×D312yes—
373C23×C3—22no—
374C23×C3—22no—
375C23×C3—216no—
376C23×C3—24no—
377C23×C3—28no—
378C23×C3—24no—
379C23×C3—28no—
380C23×C3—22no—
381C23×C3—24no—
382C23×C3—24no—
383C23×C3—24no—
384C23×C3—432no—
385C23×C3—48no—
386C23×C3—48no—
387C23×C3C22×D412yes—
388C23×C3C2×#4118yes—
389C23×C3C2×#4114yes—
390C23×C3C2511yescharacteristic
391C3×D4—216no—
392C3×D4—216no—
393C3×D4—432no—
394C3×D4—48no—
395C3×D4—44no—
396C2×C3×C4—24no—
397C2×C3×C4—22no—
398C2×C3×C4—24no—
399C2×C3×C4—28no—
400C2×C3×C4—22no—
401C2×C3×C4—24no—
402C2×C3×C4C22×D412yes—
403C2×C3×C4C22×D412yes—
404C2×C3×C4C2×#4118yes—
405C2×C3×C4C2×#4114yes—
406C2×Dic3—28no—
407C2×Dic3—22no—
408C2×Dic3—22no—
409C2×Dic3—22no—
410C2×Dic3—28no—
411C2×Dic3—22no—
412C2×Dic3—22no—
413C2×Dic3—24no—
414C2×Dic3—24no—
415C2×Dic3—24no—
416C2×Dic3—24no—
417C2×Dic3—24no—
418C2×Dic3—24no—
419C2×Dic3—28no—
420C2×Dic3—416no—
421C2×Dic3—48no—
422C2×Dic3—48no—
423C2×Dic3—416no—
424C2×Dic3C22×D412yes—
425C2×Dic3C22×D412yes—
426C2×Dic3C22×D412yes—
427C2×Dic3C22×D412yes—
428C2×Dic3H(5,𝔽2)14yes—
429Dic6—44no—
430C24—24no—
431C24—28no—
432C24—28no—
433C24—44no—
434C24C23×D312yes—
435C24C23×D314yes—
436C24C23×D311yescharacteristic
437C2×Q8—62no—
438C2×Q8—64no—
439C2×D4—232no—
440C2×D4—28no—
441C2×D4—24no—
442C2×D4—28no—
443C2×D4—216no—
444C2×D4—24no—
445C2×D4—24no—
446C2×D4—22no—
447C2×D4—24no—
448C2×D4—216no—
449C2×D4—24no—
450C2×D4—48no—
451C2×D4#23718yes—
452C22×C4—24no—
453C22×C4—31no—
454C22×C4—34no—
455C22×C4—34no—
456C22×C4—31no—
457C22×C4—31no—
458C22×C4—31no—
459C22×C4—34no—
460C22×C4—32no—
461C22×C4—34no—
462C22×C4—32no—
463C22×C4—64no—
464C22×C4—64no—
465C22×C4—68no—
466C22×C4—68no—
467C22×C4—68no—
468C22×C4—616no—
469C22×C4—64no—
470C22×C4—64no—
471C22×C4—128no—
472C22×C4—124no—
473C22×C4—1232no—
474C22×C4D3×D414yes—
475C22×C4C23×D312yes—
476C22×C4C23×D311yescharacteristic
477C22×C4C23×D314yes—
478C22×C4C23×D311yescharacteristic
479C4⋊C4—22no—
480C4⋊C4—64no—
481C4⋊C4—64no—
482C4⋊C4—64no—
483C4⋊C4—64no—
484C4⋊C4—64no—
485C4⋊C4—64no—
486C4⋊C4—62no—
487C4⋊C4—68no—
488C4⋊C4—128no—
489C22⋊C4—24no—
490C22⋊C4—216no—
491C22⋊C4—68no—
492C22⋊C4—68no—
493C22⋊C4—64no—
494C22⋊C4—64no—
495C22⋊C4—64no—
496C22⋊C4—64no—
497C22⋊C4—64no—
498C22⋊C4—616no—
499C22⋊C4—128no—
500C22⋊C4—128no—
501C22⋊C4#23718yes—
502C42—24no—
503C42—34no—
504C42—34no—
505C42—34no—
506C42—34no—
507C42—64no—
508C42—68no—
509C42—68no—
510C42—64no—
511C22×C3—22no—
512C22×C3—22no—
513C22×C3—28no—
514C22×C3—28no—
515C22×C3—216no—
516C22×C3—432no—
517C22×C3—44no—
518C22×C3—44no—
519C22×C3—48no—
520C22×C3—42no—
521C22×C3—42no—
522C22×C3—42no—
523C22×C3—48no—
524C22×C3#53214yes—
525C22×C3C23×D411yescharacteristic
526C22×C3C22×#4112yes—
527C22×C3C22×#4112yes—
528C22×C3C2×H(5,𝔽2)12yes—
529C3×C4—22no—
530C3×C4—28no—
531C3×C4—42no—
532C3×C4#52514yes—
533Dic3—22no—
534Dic3—22no—
535Dic3—22no—
536Dic3—42no—
537Dic3—42no—
538Dic3—44no—
539Dic3—44no—
540Dic3#53514yes—
541C23—22no—
542C23—22no—
543C23—216no—
544C23—24no—
545C23—28no—
546C23—24no—
547C23—28no—
548C23—22no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—432no—
553C23—48no—
554C23—48no—
555C23C2×D3×D412yes—
556C23C2×#23718yes—
557C23C2×#23714yes—
558C23C24×D311yescharacteristic
559Q8—124no—
560D4—216no—
561D4—216no—
562D4—432no—
563D4—48no—
564D4—44no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—28no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—32no—
572C2×C4—34no—
573C2×C4—32no—
574C2×C4—32no—
575C2×C4—32no—
576C2×C4—68no—
577C2×C4—62no—
578C2×C4—62no—
579C2×C4—62no—
580C2×C4—68no—
581C2×C4—62no—
582C2×C4—62no—
583C2×C4—64no—
584C2×C4—64no—
585C2×C4—64no—
586C2×C4—64no—
587C2×C4—64no—
588C2×C4—64no—
589C2×C4—68no—
590C2×C4—1216no—
591C2×C4—128no—
592C2×C4—1216no—
593C2×C4—128no—
594C2×C4C2×D3×D412yes—
595C2×C4C2×D3×D412yes—
596C2×C4C2×#23718yes—
597C2×C4C2×#23714yes—
598C2×C3—24no—
599C2×C3—48no—
600C2×C3—82no—
601C2×C3C2×#52511yescharacteristic
602C2×C3C2×#53211yescharacteristic
603C2×C3C2×#53511yescharacteristic
604C2×C3#348114yes—
605C22—22no—
606C22—28no—
607C22—22no—
608C22—216no—
609C22—28no—
610C22—432no—
611C22—44no—
612C22—48no—
613C22—44no—
614C22—42no—
615C22—48no—
616C22—42no—
617C22—42no—
618C22#571614yes—
619C22C22×D3×D411yescharacteristic
620C22C22×#23712yes—
621C22C22×#23712yes—
622C22D3×H(5,𝔽2)12yes—
623C4—28no—
624C4—22no—
625C4—34no—
626C4—42no—
627C4—62no—
628C4—62no—
629C4—62no—
630C4—124no—
631C4—122no—
632C4—124no—
633C4—122no—
634C4#558614yes—
635C3#3496511yescharacteristic, complemented, Sylow
636C2—24no—
637C2—48no—
638C2—82no—
639C2C2×#558611yescharacteristic
640C2C2×#571611yescharacteristic
641C2D3×#53511yescharacteristic
642C2#8660614yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
Fitting, prime radicalC3×#2433C2
hypercenter#2433D3
hypocenterC3#34965
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2433

Complements

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

NH ≅ G/Ntype
1. C22×D4×Dic3C2semidirect
2. C2×#5600C2semidirect
4. #84050C2semidirect
5. #83178C2semidirect
6. Dic3×#127C2semidirect
7. #82665C2semidirect
8. #82583C2semidirect
9. Dic3×#120C2semidirect
10. #76695C2semidirect
11. #76666C2semidirect
12. #76362C2semidirect
13. #76292C2semidirect
14. #73517C2semidirect
15. C4×#877C2semidirect
24. C2×#936C22semidirect
32. C2×D4×Dic3C22semidirect
33. C2×D4×Dic3C22semidirect
38. C2×#890C22semidirect
39. C2×#887C22semidirect
40. C2×#879C22semidirect
42. C2×#877C22semidirect
48. #5222C22semidirect
50. #5221C22semidirect
51. #5221C22semidirect
53. #5220C22semidirect
55. #4977C22semidirect
56. #4953C22semidirect
58. #4950C22semidirect
59. #4948C22semidirect
60. #4944C22semidirect
62. Dic3×C22⋊C4C22semidirect
63. #4935C22semidirect
64. C4×Dic3⋊C4C22semidirect
65. C42×Dic3C22semidirect
66. #4929C22semidirect
67. #4661C22semidirect
68. #4649C22semidirect
69. #4648C22semidirect
143. C2×C4×Dic3D4semidirect
635. C3#34965semidirect

Maximal subgroups

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

subgroupindex
C22×D4×Dic32
C2×#56002
C3×#24332
#840502
#831782
Dic3×#1272
#826652
#825832
Dic3×#1202
#766952
#766662
#763622
#762922
#735172
C4×#8772
#349653