Finite Group Databasebeta

← #10796371

Subgroups — #10796371

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×#5819C211yesmaximal, characteristic, complemented
3C3×#2297C211yesmaximal, characteristic, complemented
4C2×#5005C211yesmaximal, characteristic, complemented
5C22×#805C211yesmaximal, characteristic, complemented
6#71578C211yesmaximal, characteristic, complemented
7C2×#4523C211yesmaximal, characteristic, complemented
8#69453C218yesmaximal, complemented
9#20903—31nomaximal, Sylow
10C22×C3×#41C2211yescharacteristic
11C22×#235C2211yescharacteristic
12C2×C3×#117C2211yescharacteristic
13C22×C3×C42C2211yescharacteristic
14#5833C414yescomplemented
15#5832—22no—
16#5830C412yescomplemented
17#5828—22no—
18#5825C412yescomplemented
19#5819—24no—
20#5819—24no—
21C2×#948—22no—
22C2×#948—22no—
23#5816—21no—
24#5805—21no—
25#5804—22no—
26#5791C2214yes—
27C2×#926C2211yescharacteristic
28C22×#208C2211yescharacteristic
29C22×#208C2211yescharacteristic
30C22×#208C2214yescomplemented
31C3×#391—21no—
32C3×#390—21no—
33C3×#386—22no—
34C3×C4×C22⋊C4C2214yescomplemented
35#5005—24no—
36C2×#805—24no—
37C2×#805C2214yes—
38C2×#805C2214yes—
39#4523C2214yescomplemented
40#4523C2214yescomplemented
41C2×#568—31no—
42#3449—31no—
43C2×#421—31no—
44C22×#104—31no—
45#2297D311yescharacteristic, complemented
46#1871—38no—
47#1733—31no—
48C2×#339—31no—
49C2×C3×#41—22no—
50C2×C3×#41—22no—
51C2×C3×#41—22no—
52C2×C3×#41—22no—
53C2×C3×#41C2×C412yes—
54C2×C3×#41C2×C412yes—
55C2×C3×#41D412yes—
56C2×C3×#41D412yes—
57C22×C3×Q8D411yescharacteristic
58C22×C3×D4D412yes—
59C22×C3×D4D411yescharacteristic
60C23×C3×C4—21no—
61C23×C3×C4C2312yes—
62C23×C3×C4C2311yescharacteristic
63C22×#67—21no—
64C2×#235—21no—
65C2×#235—21no—
66C2×#235—22no—
67C2×#235—22no—
68C2×#235—22no—
69C2×#235—22no—
70C2×#235C2×C412yes—
71C2×#235C2312yes—
72C22×D12C2×C411yescharacteristic
73C22×C4×D3—21no—
74C22×Dic6C2×C411yescharacteristic
75C3×#117—22no—
76C3×#117—22no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C22⋊C4—24no—
79C2×C3×C22⋊C4—21no—
80C2×C3×C42—22no—
81C2×C3×C42—22no—
82C2×C3×C42—21no—
83C2×C3×C42C2312yes—
84C2×C3×C42C2312yes—
85#953—24no—
86C2×#217—21no—
87#948—216no—
88#948—216no—
89#947—22no—
90#946—24no—
91#944—22no—
92#940—22no—
93#939—28no—
94#938—24no—
95#937—24no—
96#936—22no—
97#935—24no—
98#930C2×C418yescomplemented
99#929—24no—
100C4×#67—28no—
101#926—21no—
102#926—21no—
103#926C2×C412yes—
104#926C2312yes—
105C2×C4⋊Dic3C2×C412yes—
106#924C2×C418yescomplemented
107C2×C4×Dic3—21no—
108C2×#208—21no—
109C2×#208—21no—
110C2×#208—22no—
111C2×#208—22no—
112C2×#208—24no—
113C2×#208—28no—
114C2×#208—28no—
115C2×#208D414yescomplemented
116C2×#208D414yescomplemented
117C2×#208C2312yes—
118C2×#208C2312yes—
119C22×C3⋊C8—21no—
120C22×C3⋊C8—21no—
121C22×C3⋊C8—24no—
122C3×#95—24no—
123#805—28no—
124#805—28no—
125C22×#41—31no—
126C22×#41C2×D311yescharacteristic
127#568—64no—
128#568—64no—
129C2×#131—62no—
130C2×#131—62no—
131C22×M16—31no—
132C22×M16—31no—
133C22×M16—34no—
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×C42C2×D311yescharacteristic
146#421—64no—
147C2×#104—34no—
148C2×#104—34no—
149C2×#104—64no—
150#391—21no—
151#390—21no—
152#386—22no—
153C4×C22⋊C4C2×D314yescomplemented
154#339—34no—
155#339—34no—
156C24×C3C2×D412yes—
157C24×C3C2×D411yescharacteristic
158C23×D3—21no—
159C3×#41—24no—
160C3×#41—24no—
161C3×#41—24no—
162C3×#41—24no—
163C3×#41—24no—
164C3×#41—24no—
165C3×#41—24no—
166C3×#41—24no—
167C2×C3×Q8—22no—
168C2×C3×Q8—22no—
169C2×C3×Q8C22⋊C412yes—
170C2×C3×Q8C22⋊C412yes—
171C2×C3×D4—24no—
172C2×C3×D4—24no—
173C2×C3×D4—22no—
174C2×C3×D4—22no—
175C2×C3×D4C22⋊C414yes—
176C2×C3×D4C22⋊C412yes—
177C2×C3×D4C22⋊C412yes—
178C2×C3×D4C22⋊C414yes—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4—44no—
186C22×C3×C4—42no—
187C22×C3×C4C22×C414yes—
188C22×C3×C4C22×C411yescharacteristic
189C22×C3×C4C22×C411yescharacteristic
190C22×C3×C4C2×D414yes—
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D412yes—
194C22×C3×C4C2×D412yes—
195C22×C3×C4C2×D412yes—
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4#4112yes—
198C22×C3×C4#4112yes—
199C22×C3×C4C2411yescharacteristic
200C2×#67—22no—
201C2×#67—22no—
202C2×#67—28no—
203C2×#67—42no—
204C2×#67—42no—
205C22×Dic3—21no—
206C22×Dic3—22no—
207#235—24no—
208#235—24no—
209#235—22no—
210#235—22no—
211#235—22no—
212#235—22no—
213#235—42no—
214#235—42no—
215#235—42no—
216#235—42no—
217C2×D12—22no—
218C2×D12—22no—
219C2×D12—21no—
220C2×D12C22×C412yes—
221C2×C4×D3—22no—
222C2×C4×D3—22no—
223C2×C4×D3—42no—
224C2×C4×D3—42no—
225C2×Dic6—22no—
226C2×Dic6—22no—
227C2×Dic6—21no—
228C2×Dic6C22×C412yes—
229C3×C4⋊C4—42no—
230C3×C22⋊C4—48no—
231C3×C22⋊C4—42no—
232C3×C42—22no—
233C3×C42—22no—
234C3×C42—22no—
235C3×C42—22no—
236C3×C42—42no—
237#217—28no—
238#217—22no—
239#217—22no—
240(C2×D3)⋊C4—28no—
241C4⋊Dic3—22no—
242C4⋊Dic3C22×C414yes—
243Dic3⋊C4—28no—
244C4×Dic3—22no—
245C4×Dic3—22no—
246#208—216no—
247#208—24no—
248#208—24no—
249#208—22no—
250#208—22no—
251#208C2×D414yes—
252#208C2×D414yes—
253C2×C3⋊C8—216no—
254C2×C3⋊C8—24no—
255C2×C3⋊C8—24no—
256C2×C3⋊C8—22no—
257C2×C3⋊C8—22no—
258C2×#41—22no—
259C2×#41—22no—
260C2×#41—22no—
261C2×#41—22no—
262C2×#41—32no—
263C2×#41—32no—
264C2×#41—61no—
265C2×#41—61no—
266C2×#41—62no—
267C2×#41—62no—
268C2×#41—62no—
269C2×#41—62no—
270C2×#41C4×D312yes—
271C2×#41C4×D312yes—
272C2×#41D1212yes—
273C2×#41D1212yes—
274C22×Q8—31no—
275C22×Q8#6711yescharacteristic
276C22×D4—31no—
277C22×D4—61no—
278C22×D4#6712yes—
279C22×D4#6711yescharacteristic
280C23×C4—21no—
281C23×C4—61no—
282C23×C4C22×D312yes—
283C23×C4C22×D311yescharacteristic
284#131—616no—
285#131—616no—
286C2×M16—34no—
287C2×M16—34no—
288C2×M16—32no—
289C2×M16—32no—
290C2×M16—68no—
291C2×M16—68no—
292C2×M16—64no—
293C2×M16—61no—
294C2×M16—61no—
295C2×M16—62no—
296C2×M16—62no—
297C22×C8—64no—
298C22×C8—61no—
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—24no—
324C2×C22⋊C4—61no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—21no—
328C2×C42—61no—
329C2×C42C22×D312yes—
330C2×C42C22×D312yes—
331#104—68no—
332#104—68no—
333#95—24no—
334C23×C3—22no—
335C23×C3—22no—
336C23×C3—42no—
337C23×C3—44no—
338C23×C3C2×C22⋊C414yes—
339C23×C3C2×C22⋊C411yescharacteristic
340C23×C3C2×C22⋊C411yescharacteristic
341C23×C3#12112yes—
342C23×C3#12112yes—
343C23×C3C22×D411yescharacteristic
344C22×D3—22no—
345C22×D3—42no—
346C22×D3—82no—
347C3×Q8—24no—
348C3×Q8—24no—
349C3×D4—28no—
350C3×D4—28no—
351C3×D4—24no—
352C3×D4—24no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—24no—
356C2×C3×C4—24no—
357C2×C3×C4—24no—
358C2×C3×C4—24no—
359C2×C3×C4—24no—
360C2×C3×C4—42no—
361C2×C3×C4—48no—
362C2×C3×C4—42no—
363C2×C3×C4—44no—
364C2×C3×C4—42no—
365C2×C3×C4C2×C22⋊C414yes—
366C2×C3×C4C2×C22⋊C411yescharacteristic
367C2×C3×C4C2×C22⋊C411yescharacteristic
368C2×C3×C4C4×D412yes—
369C2×C3×C4C4×D412yes—
370C2×C3×C4C4×D412yes—
371C2×C3×C4C4×D412yes—
372C2×C3×C4#12012yes—
373C2×C3×C4#12012yes—
374C2×C3×C4#12112yes—
375C2×C3×C4#12112yes—
376C2×C3×C4#12312yes—
377C2×C3×C4#12312yes—
378C2×C3×C4C23×C411yescharacteristic
379C2×C3×C4C22×D411yescharacteristic
380C2×C3×C4C22×D411yescharacteristic
381C2×C3×C4C22×D411yescharacteristic
382C2×C3×C4C2×#4111yescharacteristic
383C2×C3×C4C2×#4111yescharacteristic
384#67—42no—
385#67—42no—
386#67—44no—
387#67—88no—
388C2×Dic3—24no—
389C2×Dic3—22no—
390C2×Dic3—42no—
391C2×Dic3—82no—
392D12—22no—
393D12—22no—
394D12—42no—
395C4×D3—42no—
396C4×D3—42no—
397C4×D3—44no—
398Dic6—22no—
399Dic6—22no—
400Dic6—42no—
401C3⋊C8—24no—
402C3⋊C8—24no—
403C24—61no—
404C24C2×#6712yes—
405C24C2×#6711yescharacteristic
406#41—24no—
407#41—24no—
408#41—24no—
409#41—24no—
410#41—24no—
411#41—24no—
412#41—24no—
413#41—24no—
414#41—64no—
415#41—64no—
416#41—62no—
417#41—62no—
418#41—62no—
419#41—62no—
420#41—122no—
421#41—122no—
422#41—122no—
423#41—122no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—32no—
427C2×Q8—61no—
428C2×Q8—62no—
429C2×Q8—62no—
430C2×Q8(C2×D3)⋊C412yes—
431C2×Q8(C2×D3)⋊C412yes—
432C2×D4—24no—
433C2×D4—24no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—32no—
437C2×D4—62no—
438C2×D4—62no—
439C2×D4—68no—
440C2×D4—62no—
441C2×D4—62no—
442C2×D4—61no—
443C2×D4—122no—
444C2×D4—122no—
445C2×D4(C2×D3)⋊C414yes—
446C2×D4(C2×D3)⋊C414yes—
447C2×D4(C2×D3)⋊C412yes—
448C2×D4(C2×D3)⋊C412yes—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—44no—
456C22×C4—42no—
457C22×C4—61no—
458C22×C4—62no—
459C22×C4—62no—
460C22×C4—62no—
461C22×C4—122no—
462C22×C4—122no—
463C22×C4C2×C4×D314yes—
464C22×C4C2×C4×D311yescharacteristic
465C22×C4C2×C4×D311yescharacteristic
466C22×C4C2×D1214yes—
467C22×C4C2×D1211yescharacteristic
468C22×C4C2×D1211yescharacteristic
469C22×C4D3×D412yes—
470C22×C4D3×D412yes—
471C22×C4#23712yes—
472C22×C4#23712yes—
473C22×C4C2×#6712yes—
474C22×C4C2×#6711yescharacteristic
475C22×C4C23×D311yescharacteristic
476M16—34no—
477M16—34no—
478M16—616no—
479M16—64no—
480M16—64no—
481M16—62no—
482M16—62no—
483C2×C8—616no—
484C2×C8—64no—
485C2×C8—64no—
486C2×C8—62no—
487C2×C8—62no—
488C4⋊C4—34no—
489C4⋊C4—42no—
490C4⋊C4—62no—
491C4⋊C4—68no—
492C22⋊C4—48no—
493C22⋊C4—42no—
494C22⋊C4—62no—
495C22⋊C4—68no—
496C22⋊C4—62no—
497C22⋊C4—68no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—42no—
503C42—62no—
504C42—62no—
505C22×C3—22no—
506C22×C3—22no—
507C22×C3—22no—
508C22×C3—28no—
509C22×C3—24no—
510C22×C3—42no—
511C22×C3—42no—
512C22×C3—82no—
513C22×C3—82no—
514C22×C3#38612yes—
515C22×C3#38612yes—
516C22×C3C22×C22⋊C411yescharacteristic
517C22×C3C2×#12111yescharacteristic
518C22×C3C2×#12111yescharacteristic
519C2×D3—44no—
520C2×D3—42no—
521C2×D3—84no—
522C3×C4—24no—
523C3×C4—44no—
524C3×C4—82no—
525C3×C4#38612yes—
526C3×C4#38612yes—
527C3×C4C2×C4×D411yescharacteristic
528C3×C4C2×C4×D411yescharacteristic
529C3×C4C2×#12011yescharacteristic
530C3×C4C2×#12311yescharacteristic
531Dic3—44no—
532Dic3—42no—
533C23—22no—
534C23—22no—
535C23—42no—
536C23—44no—
537C23—62no—
538C23—122no—
539C23—242no—
540C23C2×(C2×D3)⋊C414yes—
541C23C2×(C2×D3)⋊C411yescharacteristic
542C23C2×(C2×D3)⋊C411yescharacteristic
543C23#93912yes—
544C23#93912yes—
545C23C22×#6711yescharacteristic
546Q8—24no—
547Q8—24no—
548Q8—62no—
549Q8—62no—
550Q8—122no—
551D4—28no—
552D4—28no—
553D4—24no—
554D4—24no—
555D4—62no—
556D4—62no—
557D4—124no—
558D4—122no—
559D4—122no—
560D4—122no—
561D4—248no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—42no—
570C2×C4—48no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—64no—
575C2×C4—62no—
576C2×C4—122no—
577C2×C4—124no—
578C2×C4—122no—
579C2×C4—122no—
580C2×C4—242no—
581C2×C4#67⋊C412yes—
582C2×C4#67⋊C412yes—
583C2×C4#67⋊C412yes—
584C2×C4#67⋊C412yes—
585C2×C4#88212yes—
586C2×C4#88212yes—
587C2×C4#88612yes—
588C2×C4#88612yes—
589C2×C4C2×(C2×D3)⋊C414yes—
590C2×C4C2×(C2×D3)⋊C411yescharacteristic
591C2×C4C2×(C2×D3)⋊C411yescharacteristic
592C2×C4#93912yes—
593C2×C4#93912yes—
594C2×C4C22×C4×D311yescharacteristic
595C2×C4C22×D1211yescharacteristic
596C2×C4C2×D3×D411yescharacteristic
597C2×C4C2×D3×D411yescharacteristic
598C2×C4C2×#23711yescharacteristic
599C2×C4C2×#23711yescharacteristic
600C8—64no—
601C8—64no—
602C2×C3—21no—
603C2×C3—21no—
604C2×C3—22no—
605C2×C3—22no—
606C2×C3—44no—
607C2×C3C2×#38611yescharacteristic
608C2×C3#290112yes—
609D3—82no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—28no—
614C22—24no—
615C22—42no—
616C22—42no—
617C22—82no—
618C22—82no—
619C22—124no—
620C22—122no—
621C22—244no—
622C22#495812yes—
623C22#495812yes—
624C22C22×(C2×D3)⋊C411yescharacteristic
625C22C2×#93911yescharacteristic
626C22C2×#93911yescharacteristic
627C4—24no—
628C4—44no—
629C4—82no—
630C4—124no—
631C4—122no—
632C4#495812yes—
633C4#495812yes—
634C4C2×#67⋊C411yescharacteristic
635C4C2×#67⋊C411yescharacteristic
636C4C2×#88211yescharacteristic
637C4C2×#88611yescharacteristic
638C3#2090311yescharacteristic, complemented, Sylow
639C2—21no—
640C2—21no—
641C2—22no—
642C2—22no—
643C2—44no—
644C2—242no—
645C2C2×#495811yescharacteristic
646C2#8253612yes—
subgroupG/H
centerC2×C4C22×D12
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#2297C2
socleC22×C3C22×C22⋊C4
prime radical#87268C2
cosocleC22×C3×C4C24
hypercenter#2297D3
hypocenterC3#20903
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C22×#41 ◁ #2297

Complements

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

NH ≅ G/Ntype
2. C2×#5819C2semidirect
3. C3×#2297C2semidirect
4. C2×#5005C2semidirect
5. C22×#805C2semidirect
6. #71578C2semidirect
7. C2×#4523C2semidirect
8. #69453C2semidirect
14. #5833C4semidirect
16. #5830C4semidirect
18. #5825C4semidirect
30. C22×#208C22semidirect
34. C3×C4×C22⋊C4C22semidirect
39. #4523C22semidirect
40. #4523C22semidirect
45. #2297D3semidirect
98. #930C2×C4semidirect
106. #924C2×C4semidirect
115. C2×#208D4semidirect
116. C2×#208D4semidirect
153. C4×C22⋊C4C2×D3semidirect
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×#58192
C3×#22972
C2×#50052
C22×#8052
#715782
C2×#45232
#694532
#209033