Finite Group Databasebeta

← #10796376

Subgroups — #10796376

Order 768.

1782 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#87268C211yesmaximal, characteristic
2C2×#5745C211yesmaximal, characteristic, complemented
3C2×C3×#421C211yesmaximal, characteristic, complemented
4C22×#837C211yesmaximal, characteristic, complemented
5#82228C211yesmaximal, characteristic, complemented
6C2×#4555C211yesmaximal, characteristic, complemented
7#71578C211yesmaximal, characteristic, complemented
8#70381C218yesmaximal, complemented
9#20903—31nomaximal, Sylow
10C22×C3×#41C2211yescharacteristic
11C22×#235C2211yescharacteristic
12C22×C3×M16C2211yescharacteristic
13C2×C3×#117C2211yescharacteristic
14#5833C414yescomplemented
15#5832—22no—
16#5830—21no—
17#5828C2214yes—
18#5825—21no—
19#5816C412yescomplemented
20#5805C412yescomplemented
21#5804—22no—
22#5791—22no—
23C2×#926C2211yescharacteristic
24C22×C4×Dic3C2211yescharacteristic
25C22×#208C2211yescharacteristic
26#5745—24no—
27#5745—24no—
28C2×#907—22no—
29C2×D3×M16C2214yescomplemented
30C2×#901—22no—
31C3×#421—24no—
32C2×C3×#104C2214yes—
33C2×#837—24no—
34C2×#837C2214yes—
35#4988—21no—
36#4973—21no—
37#4958—22no—
38C4×(C2×D3)⋊C4C2214yescomplemented
39#4555C2214yescomplemented
40#4523C2214yescomplemented
41C2×#568—31no—
42#3449—31no—
43C2×#421D311yescharacteristic, complemented
44C22×#104—31no—
45#2297—31no—
46#1871—38no—
47#1733—31no—
48C2×#339—31no—
49C2×C3×#41—21no—
50C2×C3×#41—21no—
51C2×C3×#41—22no—
52C2×C3×#41—22no—
53C2×C3×#41—22no—
54C2×C3×#41—22no—
55C2×C3×#41C2×C412yes—
56C2×C3×#41C2312yes—
57C22×C3×Q8C2×C411yescharacteristic
58C22×C3×D4—21no—
59C22×C3×D4C2×C411yescharacteristic
60C23×C3×C4—21no—
61C23×C3×C4C2311yescharacteristic
62C22×#67D412yes—
63C23×Dic3—21no—
64C2×#235—22no—
65C2×#235—22no—
66C2×#235—22no—
67C2×#235—22no—
68C2×#235C2×C412yes—
69C2×#235C2×C412yes—
70C2×#235D412yes—
71C2×#235D412yes—
72C22×D12D411yescharacteristic
73C22×C4×D3C2312yes—
74C22×Dic6D411yescharacteristic
75C2×C3×M16—21no—
76C2×C3×M16—22no—
77C2×C3×M16D414yescomplemented
78C2×C3×M16C2312yes—
79C22×C3×C8—21no—
80C3×#117—22no—
81C3×#117—22no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C22⋊C4—21no—
84C2×C3×C42C2312yes—
85#953—24no—
86C2×#217—21no—
87#947—22no—
88#946C2×C418yescomplemented
89#944—22no—
90#940—22no—
91#939—28no—
92#938C2×C418yescomplemented
93#937—24no—
94#936—22no—
95#935—24no—
96#930—24no—
97#929—24no—
98C4×#67—28no—
99C2×(C2×D3)⋊C4—24no—
100#926—21no—
101#926—21no—
102#926C2×C412yes—
103#926C2312yes—
104C2×C4⋊Dic3C2×C412yes—
105#924—24no—
106C2×C4×Dic3—21no—
107C2×C4×Dic3—22no—
108C2×C4×Dic3—22no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3C2312yes—
111C2×#208—21no—
112C2×#208—22no—
113C2×#208D414yescomplemented
114C2×#208C2312yes—
115C22×C3⋊C8—21no—
116#907—216no—
117D3×M16—28no—
118D3×M16—28no—
119#901—216no—
120C2×ΓL(1,𝔽25)—24no—
121C2×D3×C8—24no—
122C3×#104—28no—
123#837—28no—
124C3⋊#95—24no—
125C22×#41—31no—
126C22×#41C2×D311yescharacteristic
127#568—64no—
128#568—64no—
129C2×#131—62no—
130C2×#131—62no—
131C22×M16—31no—
132C22×M16—34no—
133C22×M16C2×D311yescharacteristic
134#537—34no—
135#536—32no—
136#536—61no—
137#535—32no—
138#535—61no—
139#534—62no—
140#532—62no—
141#525—34no—
142#525—62no—
143C2×#117—31no—
144C2×#117C2×D311yescharacteristic
145C22×C42—31no—
146#421—24no—
147C2×#104—34no—
148C2×#104—64no—
149C2×#104C2×D314yes—
150#391—61no—
151#390—61no—
152#386—62no—
153C4×C22⋊C4—34no—
154#339—34no—
155#339—34no—
156C24×C3—21no—
157C24×C3C2×D411yescharacteristic
158C23×D3C2×D412yes—
159C3×#41—24no—
160C3×#41—24no—
161C3×#41—22no—
162C3×#41—22no—
163C3×#41—22no—
164C3×#41—22no—
165C3×#41—42no—
166C3×#41—42no—
167C3×#41—42no—
168C3×#41—42no—
169C2×C3×Q8—21no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C2×C3×Q8C22×C412yes—
173C2×C3×D4—22no—
174C2×C3×D4—22no—
175C2×C3×D4—21no—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4—42no—
179C2×C3×D4—42no—
180C2×C3×D4C22×C412yes—
181C22×C3×C4—21no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4—42no—
186C22×C3×C4—42no—
187C22×C3×C4C22×C411yescharacteristic
188C22×C3×C4C22×C411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D412yes—
193C22×C3×C4C2×D412yes—
194C22×C3×C4#4112yes—
195C22×C3×C4#4112yes—
196C22×C3×C4C2411yescharacteristic
197C2×#67—24no—
198C2×#67—24no—
199C2×#67—28no—
200C2×#67C22⋊C414yes—
201C2×#67C22⋊C414yes—
202C22×Dic3—22no—
203C22×Dic3—22no—
204C22×Dic3—44no—
205C22×Dic3—42no—
206C22×Dic3C2×D412yes—
207#235—24no—
208#235—24no—
209#235—24no—
210#235—24no—
211#235—24no—
212#235—24no—
213#235—24no—
214#235—24no—
215C2×D12—22no—
216C2×D12—22no—
217C2×D12C22⋊C412yes—
218C2×D12C22⋊C412yes—
219C2×C4×D3—22no—
220C2×C4×D3—22no—
221C2×C4×D3—22no—
222C2×C4×D3—22no—
223C2×C4×D3C22×C414yes—
224C2×C4×D3C2×D414yes—
225C2×Dic6—22no—
226C2×Dic6—22no—
227C2×Dic6C22⋊C412yes—
228C2×Dic6C22⋊C412yes—
229C3×M16—24no—
230C3×M16—22no—
231C3×M16C2×D414yes—
232C2×C3×C8—24no—
233C2×C3×C8—22no—
234C3×C4⋊C4—42no—
235C3×C22⋊C4—42no—
236C3×C42—22no—
237C3×C42—22no—
238#217—22no—
239#217—22no—
240#217—28no—
241(C2×D3)⋊C4—28no—
242(C2×D3)⋊C4—48no—
243C4⋊Dic3—22no—
244C4⋊Dic3C22×C414yes—
245Dic3⋊C4—28no—
246C4×Dic3—22no—
247C4×Dic3—22no—
248C4×Dic3—22no—
249C4×Dic3—22no—
250C4×Dic3—42no—
251#208—24no—
252#208—22no—
253#208C2×D414yes—
254C2×C3⋊C8—24no—
255C2×C3⋊C8—22no—
256ΓL(1,𝔽25)—216no—
257D3×C8—216no—
258C2×#41—21no—
259C2×#41—21no—
260C2×#41—22no—
261C2×#41—22no—
262C2×#41—22no—
263C2×#41—22no—
264C2×#41—32no—
265C2×#41—32no—
266C2×#41—32no—
267C2×#41—32no—
268C2×#41—62no—
269C2×#41—62no—
270C2×#41—62no—
271C2×#41—62no—
272C2×#41C4×D312yes—
273C2×#41C22×D312yes—
274C22×Q8—31no—
275C22×Q8C4×D311yescharacteristic
276C22×D4—21no—
277C22×D4—32no—
278C22×D4—31no—
279C22×D4C4×D311yescharacteristic
280C23×C4—21no—
281C23×C4—32no—
282C23×C4—61no—
283C23×C4C22×D311yescharacteristic
284#131—616no—
285#131—616no—
286C2×M16—21no—
287C2×M16—22no—
288C2×M16—34no—
289C2×M16—32no—
290C2×M16—64no—
291C2×M16—68no—
292C2×M16—61no—
293C2×M16—62no—
294C2×M16—68no—
295C2×M16#6714yescomplemented
296C2×M16C22×D312yes—
297C22×C8—21no—
298C22×C8—64no—
299C22×C8—61no—
300C4⋊Q8—62no—
301#127—62no—
302#124—62no—
303#124—62no—
304#123—64no—
305#123—64no—
306C22⋊Q8—38no—
307C22⋊Q8—64no—
308#121—38no—
309#121—64no—
310#121—68no—
311#120—64no—
312#120—64no—
313C4×D4—68no—
314#117—22no—
315#117—22no—
316#117—32no—
317#117—32no—
318#117—61no—
319#117—61no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—32no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—61no—
324C2×C22⋊C4—64no—
325C2×C42—32no—
326C2×C42—61no—
327C2×C42—62no—
328C2×C42—62no—
329C2×C42—61no—
330C2×C42C22×D312yes—
331#104—28no—
332#104—68no—
333#95—64no—
334C23×C3—22no—
335C23×C3—42no—
336C23×C3—42no—
337C23×C3—82no—
338C23×C3C2×C22⋊C411yescharacteristic
339C23×C3C2×C22⋊C411yescharacteristic
340C23×C3#12112yes—
341C23×C3#12112yes—
342C23×C3C22×D411yescharacteristic
343C22×D3—22no—
344C22×D3—22no—
345C22×D3—44no—
346C22×D3C2×C22⋊C414yes—
347C3×Q8—22no—
348C3×Q8—22no—
349C3×Q8—42no—
350C3×D4—22no—
351C3×D4—22no—
352C3×D4—42no—
353C3×D4—42no—
354C3×D4—44no—
355C3×D4—42no—
356C2×C3×C4—22no—
357C2×C3×C4—42no—
358C2×C3×C4—42no—
359C2×C3×C4—42no—
360C2×C3×C4—44no—
361C2×C3×C4—44no—
362C2×C3×C4—42no—
363C2×C3×C4C2×C22⋊C411yescharacteristic
364C2×C3×C4C2×C22⋊C411yescharacteristic
365C2×C3×C4C4×D412yes—
366C2×C3×C4C4×D412yes—
367C2×C3×C4C4×D412yes—
368C2×C3×C4C4×D412yes—
369C2×C3×C4#12012yes—
370C2×C3×C4#12012yes—
371C2×C3×C4#12112yes—
372C2×C3×C4#12112yes—
373C2×C3×C4#12312yes—
374C2×C3×C4#12312yes—
375C2×C3×C4C23×C411yescharacteristic
376C2×C3×C4C22×D411yescharacteristic
377C2×C3×C4C22×D411yescharacteristic
378C2×C3×C4C22×D411yescharacteristic
379C2×C3×C4C2×#4111yescharacteristic
380C2×C3×C4C2×#4111yescharacteristic
381#67—28no—
382#67—28no—
383#67—88no—
384C2×Dic3—22no—
385C2×Dic3—22no—
386C2×Dic3—24no—
387C2×Dic3—24no—
388C2×Dic3—42no—
389C2×Dic3—48no—
390C2×Dic3—42no—
391C2×Dic3—82no—
392C2×Dic3C2×C22⋊C414yes—
393D12—24no—
394D12—24no—
395C4×D3—24no—
396C4×D3—24no—
397C4×D3—24no—
398C4×D3—24no—
399Dic6—24no—
400Dic6—24no—
401C3×C8—24no—
402C3⋊C8—24no—
403C24—21no—
404C24—32no—
405C24D3×D411yescharacteristic
406#41—24no—
407#41—24no—
408#41—22no—
409#41—22no—
410#41—22no—
411#41—22no—
412#41—42no—
413#41—42no—
414#41—42no—
415#41—42no—
416#41—64no—
417#41—64no—
418#41—64no—
419#41—64no—
420#41—64no—
421#41—64no—
422#41—64no—
423#41—64no—
424C2×Q8—21no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—32no—
428C2×Q8—32no—
429C2×Q8—62no—
430C2×Q8—62no—
431C2×Q8C2×C4×D312yes—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—21no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—34no—
438C2×D4—34no—
439C2×D4—32no—
440C2×D4—32no—
441C2×D4—42no—
442C2×D4—42no—
443C2×D4—64no—
444C2×D4—64no—
445C2×D4—68no—
446C2×D4—62no—
447C2×D4—62no—
448C2×D4C2×C4×D312yes—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—32no—
454C22×C4—34no—
455C22×C4—34no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—62no—
459C22×C4—62no—
460C22×C4—62no—
461C22×C4—62no—
462C22×C4—62no—
463C22×C4—62no—
464C22×C4—124no—
465C22×C4—122no—
466C22×C4C2×C4×D311yescharacteristic
467C22×C4C2×C4×D311yescharacteristic
468C22×C4#23512yes—
469C22×C4D3×D411yescharacteristic
470C22×C4D3×D411yescharacteristic
471C22×C4D3×D411yescharacteristic
472C22×C4D3×D412yes—
473C22×C4#23712yes—
474C22×C4C2×#6712yes—
475C22×C4C23×D311yescharacteristic
476M16—24no—
477M16—22no—
478M16—34no—
479M16—616no—
480M16—64no—
481M16—62no—
482M16C2×#6714yes—
483C2×C8—24no—
484C2×C8—22no—
485C2×C8—616no—
486C2×C8—64no—
487C2×C8—62no—
488C4⋊C4—34no—
489C4⋊C4—42no—
490C4⋊C4—68no—
491C4⋊C4—62no—
492C22⋊C4—42no—
493C22⋊C4—68no—
494C22⋊C4—68no—
495C22⋊C4—62no—
496C22⋊C4—62no—
497C22⋊C4—128no—
498C42—22no—
499C42—22no—
500C42—62no—
501C42—62no—
502C42—62no—
503C42—62no—
504C42—122no—
505C22×C3—22no—
506C22×C3—22no—
507C22×C3—22no—
508C22×C3—42no—
509C22×C3—42no—
510C22×C3—44no—
511C22×C3—42no—
512C22×C3—84no—
513C22×C3#38612yes—
514C22×C3#38612yes—
515C22×C3C22×C22⋊C411yescharacteristic
516C22×C3C2×#12111yescharacteristic
517C22×C3C2×#12111yescharacteristic
518C2×D3—28no—
519C2×D3—24no—
520C2×D3—82no—
521C2×D3—82no—
522C3×C4—42no—
523C3×C4—82no—
524C3×C4#38612yes—
525C3×C4#38612yes—
526C3×C4C2×C4×D411yescharacteristic
527C3×C4C2×C4×D411yescharacteristic
528C3×C4C2×#12011yescharacteristic
529C3×C4C2×#12311yescharacteristic
530Dic3—24no—
531Dic3—44no—
532Dic3—44no—
533C23—22no—
534C23—34no—
535C23—42no—
536C23—42no—
537C23—62no—
538C23—62no—
539C23—82no—
540C23—124no—
541C23D3×C22⋊C411yescharacteristic
542C23D3×C22⋊C411yescharacteristic
543C23#88412yes—
544C23#93812yes—
545C23C2×D3×D411yescharacteristic
546Q8—22no—
547Q8—22no—
548Q8—42no—
549Q8—64no—
550Q8—64no—
551D4—22no—
552D4—22no—
553D4—44no—
554D4—42no—
555D4—42no—
556D4—42no—
557D4—68no—
558D4—68no—
559D4—64no—
560D4—64no—
561D4—248no—
562C2×C4—22no—
563C2×C4—34no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—62no—
571C2×C4—62no—
572C2×C4—64no—
573C2×C4—64no—
574C2×C4—64no—
575C2×C4—64no—
576C2×C4—64no—
577C2×C4—64no—
578C2×C4—122no—
579C2×C4—128no—
580C2×C4—122no—
581C2×C4—242no—
582C2×C4D3×C22⋊C411yescharacteristic
583C2×C4D3×C22⋊C411yescharacteristic
584C2×C4#67⋊C412yes—
585C2×C4#67⋊C412yes—
586C2×C4#88312yes—
587C2×C4#88312yes—
588C2×C4#88412yes—
589C2×C4C4×#6712yes—
590C2×C4C4×#6712yes—
591C2×C4#93712yes—
592C2×C4#93712yes—
593C2×C4#93812yes—
594C2×C4C22×C4×D311yescharacteristic
595C2×C4C2×#23511yescharacteristic
596C2×C4C2×D3×D411yescharacteristic
597C2×C4C2×D3×D411yescharacteristic
598C2×C4C2×#23711yescharacteristic
599C2×C4C22×#6711yescharacteristic
600C8—24no—
601C8—64no—
602C2×C3—21no—
603C2×C3—21no—
604C2×C3—22no—
605C2×C3—22no—
606C2×C3—82no—
607C2×C3C2×#38611yescharacteristic
608C2×C3#290112yes—
609D3—44no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—42no—
614C22—42no—
615C22—44no—
616C22—42no—
617C22—68no—
618C22—64no—
619C22—84no—
620C22—242no—
621C22—242no—
622C22#495612yes—
623C22#495612yes—
624C22C2×D3×C22⋊C411yescharacteristic
625C22C2×#88411yescharacteristic
626C22C2×#93811yescharacteristic
627C4—42no—
628C4—64no—
629C4—82no—
630C4—124no—
631C4—124no—
632C4#495612yes—
633C4#495612yes—
634C4C2×#67⋊C411yescharacteristic
635C4C2×#88311yescharacteristic
636C4C2×C4×#6711yescharacteristic
637C4C2×#93711yescharacteristic
638C3#2090311yescharacteristic, complemented, Sylow
639C2—21no—
640C2—21no—
641C2—22no—
642C2—22no—
643C2—82no—
644C2—124no—
645C2#7899212yes—
646C2C2×#495611yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC2×C3×#421C2
socleC22×C3C22×C22⋊C4
prime radical#87268C2
cosocleC22×C3×C4C24
hypercenterC2×#421D3
hypocenterC3#20903
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#421

Complements

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

NH ≅ G/Ntype
2. C2×#5745C2semidirect
3. C2×C3×#421C2semidirect
4. C22×#837C2semidirect
5. #82228C2semidirect
6. C2×#4555C2semidirect
7. #71578C2semidirect
8. #70381C2semidirect
14. #5833C4semidirect
19. #5816C4semidirect
20. #5805C4semidirect
29. C2×D3×M16C22semidirect
38. C4×(C2×D3)⋊C4C22semidirect
39. #4555C22semidirect
40. #4523C22semidirect
43. C2×#421D3semidirect
77. C2×C3×M16D4semidirect
88. #946C2×C4semidirect
92. #938C2×C4semidirect
113. C2×#208D4semidirect
295. C2×M16#67semidirect
638. C3#20903semidirect

Maximal subgroups

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

subgroupindex
#872682
C2×#57452
C2×C3×#4212
C22×#8372
#822282
C2×#45552
#715782
#703812
#209033