Finite Group Databasebeta

← #10796783

Subgroups — #10796783

Order 768.

1710 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5816C211yesmaximal, characteristic
2#85731C211yesmaximal, characteristic
3C2×C3×#419C211yesmaximal, characteristic, complemented
4C2×#5225C211yesmaximal, characteristic, complemented
5C2×#5132C211yesmaximal, characteristic
6#71982C211yesmaximal, characteristic, complemented
7#71574C211yesmaximal, characteristic, complemented
8#70345C214yesmaximal, complemented
9#70343C214yesmaximal, complemented
10#20955—31nomaximal, Sylow
11C23×C3×Q8C2211yescharacteristic
12C22×#235C2211yescharacteristic
13C22×C3×M16C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15#5816—24no—
16#5816C418yescomplemented
17C2×#947C2212yes—
18C2×#946—22no—
19C2×#944C2212yes—
20C2×#929—22no—
21C2×#926C2211yescharacteristic
22C2×#926C2211yescharacteristic
23C22×#208C2211yescharacteristic
24#5523—22no—
25#5522—22no—
26#5521C2212yes—
27D3×#117—22no—
28#5516C2212yes—
29#5504—22no—
30#5486—22no—
31C4×#235—22no—
32C3×#419—22no—
33C3×#419C2214yes—
34C2×C3×#103—22no—
35#5225—22no—
36#5225C2214yes—
37C2×#835—22no—
38C2×#824C2212yes—
39#5132—24no—
40C2×#823C2212yes—
41#4555C2214yes—
42#4532C2214yes—
43C2×#536—31no—
44#3428—31no—
45C2×#419—31no—
46C2×#419D311yescharacteristic, complemented
47C2×#413—31no—
48#1886—34no—
49#1885—34no—
50#1733—31no—
51#1733—31no—
52C22×C3×Q8—21no—
53C22×C3×Q8—22no—
54C22×C3×Q8—21no—
55C22×C3×Q8—21no—
56C22×C3×Q8—21no—
57C22×C3×Q8—21no—
58C22×C3×Q8—22no—
59C22×C3×Q8—22no—
60C22×C3×Q8C2×C414yescomplemented
61C22×C3×Q8C2312yes—
62C23×C3×C4—21no—
63C23×C3×C4C2311yescharacteristic
64C22×#67D412yes—
65C2×#235—21no—
66C2×#235—21no—
67C2×#235—21no—
68C2×#235—21no—
69C2×#235—22no—
70C2×#235—22no—
71C2×#235C2×C414yes—
72C2×#235D414yes—
73C22×D12C2311yescharacteristic
74C22×C4×D3D412yes—
75C22×Dic6C2311yescharacteristic
76C2×C3×M16—21no—
77C2×C3×M16—22no—
78C2×C3×M16—22no—
79C2×C3×M16C2312yes—
80C22×C3×C8—21no—
81C3×#117—21no—
82C3×#117—21no—
83C3×#117D412yes—
84C3×#117C2312yes—
85C2×C3×C4⋊C4D412yes—
86C2×C3×C22⋊C4—21no—
87C2×C3×C42—21no—
88C2×#217—21no—
89C2×#217—21no—
90#947—24no—
91#947C2×C418yescomplemented
92#946—216no—
93#944—24no—
94#944C2×C418yescomplemented
95#929—216no—
96C4×#67—28no—
97C2×(C2×D3)⋊C4—24no—
98#926—21no—
99#926—21no—
100#926—22no—
101#926C2×C414yescomplemented
102#926D412yes—
103#926C2312yes—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3D412yes—
106C2×Dic3⋊C4—24no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3C2312yes—
109C2×#208—21no—
110C2×#208—22no—
111C2×#208—22no—
112C2×#208C2312yes—
113C22×C3⋊C8—21no—
114#893—24no—
115#892—24no—
116D3×C4⋊C4—24no—
117#887—24no—
118#67⋊C4—28no—
119D3×C22⋊C4—24no—
120#877—24no—
121C4×D12—24no—
122#872—24no—
123C42×D3—24no—
124C4×Dic6—24no—
125C3×#103—28no—
126#835—28no—
127#824—24no—
128#823—24no—
129C22×#41—31no—
130C23×Q8C2×D311yescharacteristic
131C22×M16—31no—
132C22×M16C2×D311yescharacteristic
133#536—38no—
134#536—64no—
135C2×C4⋊Q8—32no—
136C2×#124—32no—
137C2×#123—62no—
138C2×C22⋊Q8—62no—
139#520—62no—
140#520—62no—
141#519—32no—
142#518—32no—
143#518—62no—
144C4×#41—62no—
145C4×#41—62no—
146C2×#117—31no—
147C2×#117—31no—
148C2×#117—62no—
149C2×#117C2×D311yescharacteristic
150#419—22no—
151#419—34no—
152#419—62no—
153#419C2×D314yes—
154C2×#103—22no—
155C2×#103—62no—
156#413—64no—
157C2×#101—32no—
158C2×#100—32no—
159#339—34no—
160#339—34no—
161C24×C3C2×D411yescharacteristic
162C23×D3C2×D412yes—
163C2×C3×Q8—24no—
164C2×C3×Q8—22no—
165C2×C3×Q8—24no—
166C2×C3×Q8—24no—
167C2×C3×Q8—24no—
168C2×C3×Q8—24no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C2×C3×Q8—22no—
173C2×C3×Q8—41no—
174C2×C3×Q8—41no—
175C2×C3×Q8—42no—
176C2×C3×Q8—42no—
177C2×C3×Q8—42no—
178C2×C3×Q8—42no—
179C2×C3×Q8—42no—
180C2×C3×Q8C22×C414yes—
181C22×C3×C4—24no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—41no—
185C22×C3×C4—41no—
186C22×C3×C4—42no—
187C22×C3×C4C22×C412yes—
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D412yes—
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4#4112yes—
195C22×C3×C4#4112yes—
196C22×C3×C4C2411yescharacteristic
197C2×#67—22no—
198C2×#67—22no—
199C2×#67—24no—
200C2×#67C22⋊C418yes—
201C22×Dic3—22no—
202C22×Dic3—22no—
203C22×Dic3C2×D412yes—
204#235—24no—
205#235—24no—
206#235—24no—
207#235—24no—
208#235—24no—
209#235—24no—
210#235—24no—
211#235—24no—
212C2×D12—21no—
213C2×D12—21no—
214C2×D12—21no—
215C2×D12—21no—
216C2×D12C22×C412yes—
217C2×D12C2×D412yes—
218C2×C4×D3—22no—
219C2×C4×D3—22no—
220C2×C4×D3—24no—
221C2×C4×D3—28no—
222C2×C4×D3C22⋊C418yes—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6C22×C412yes—
228C2×Dic6C2×D412yes—
229C3×M16—22no—
230C3×M16—22no—
231C3×M16—42no—
232C2×C3×C8—22no—
233C2×C3×C8—42no—
234C3×C4⋊C4—22no—
235C3×C4⋊C4C2×D414yes—
236C3×C22⋊C4—22no—
237C3×C22⋊C4—22no—
238C3×C42—22no—
239C3×C42—22no—
240#217—24no—
241#217—22no—
242#217—22no—
243(C2×D3)⋊C4—28no—
244(C2×D3)⋊C4—216no—
245C4⋊Dic3—24no—
246C4⋊Dic3—22no—
247C4⋊Dic3C2×D414yes—
248Dic3⋊C4—28no—
249Dic3⋊C4—216no—
250C4×Dic3—28no—
251C4×Dic3—22no—
252C4×Dic3—22no—
253C4×Dic3—22no—
254C4×Dic3C22×C414yes—
255#208—22no—
256#208—22no—
257#208—42no—
258C2×C3⋊C8—22no—
259C2×C3⋊C8—42no—
260C2×#41—34no—
261C2×#41—34no—
262C2×#41—62no—
263C2×#41—61no—
264C2×#41—61no—
265C2×#41—62no—
266C2×#41—61no—
267C2×#41—61no—
268C22×Q8—21no—
269C22×Q8—22no—
270C22×Q8—22no—
271C22×Q8—21no—
272C22×Q8—21no—
273C22×Q8—22no—
274C22×Q8—21no—
275C22×Q8—21no—
276C22×Q8—31no—
277C22×Q8C4×D314yescomplemented
278C22×Q8C22×D312yes—
279C22×D4—32no—
280C22×D4—31no—
281C23×C4—21no—
282C23×C4—32no—
283C23×C4C22×D311yescharacteristic
284C2×M16—21no—
285C2×M16—22no—
286C2×M16—22no—
287C2×M16—32no—
288C2×M16—61no—
289C2×M16—62no—
290C2×M16—62no—
291C2×M16C22×D312yes—
292C22×C8—21no—
293C22×C8—61no—
294C4⋊Q8—38no—
295C4⋊Q8—64no—
296#124—38no—
297#124—64no—
298#123—616no—
299C22⋊Q8—616no—
300C4×Q8—64no—
301C4×Q8—64no—
302C4×D4—64no—
303C4×D4—68no—
304C4×D4—68no—
305C4×D4—64no—
306#117—21no—
307#117—21no—
308#117—34no—
309#117—32no—
310#117—32no—
311#117—64no—
312#117—64no—
313#117—64no—
314#117—61no—
315#117—62no—
316#117—61no—
317#117#6712yes—
318#117C22×D312yes—
319C2×C4⋊C4—32no—
320C2×C4⋊C4—64no—
321C2×C4⋊C4—64no—
322C2×C4⋊C4—61no—
323C2×C4⋊C4#6712yes—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—64no—
326C2×C22⋊C4—64no—
327C2×C22⋊C4—61no—
328C2×C22⋊C4—61no—
329C2×C42—21no—
330C2×C42—32no—
331C2×C42—64no—
332C2×C42—61no—
333#103—28no—
334#103—68no—
335#101—64no—
336#100—64no—
337C23×C3—42no—
338C23×C3C2×C22⋊C412yes—
339C23×C3#12011yescharacteristic
340C23×C3#12011yescharacteristic
341C23×C3#12312yes—
342C23×C3C22×D411yescharacteristic
343C22×D3—22no—
344C22×D3—22no—
345C22×D3—44no—
346C22×D3C2×C22⋊C414yes—
347C3×Q8—24no—
348C3×Q8—24no—
349C3×Q8—42no—
350C3×Q8—42no—
351C3×Q8—44no—
352C3×Q8—44no—
353C2×C3×C4—24no—
354C2×C3×C4—24no—
355C2×C3×C4—42no—
356C2×C3×C4—42no—
357C2×C3×C4—44no—
358C2×C3×C4—44no—
359C2×C3×C4—82no—
360C2×C3×C4C2×C22⋊C412yes—
361C2×C3×C4C4×D412yes—
362C2×C3×C4C4×D412yes—
363C2×C3×C4C4×D412yes—
364C2×C3×C4C4×D412yes—
365C2×C3×C4#12011yescharacteristic
366C2×C3×C4#12011yescharacteristic
367C2×C3×C4#12112yes—
368C2×C3×C4#12112yes—
369C2×C3×C4#12112yes—
370C2×C3×C4#12112yes—
371C2×C3×C4#12312yes—
372C2×C3×C4C23×C411yescharacteristic
373C2×C3×C4C22×D411yescharacteristic
374C2×C3×C4C22×D411yescharacteristic
375C2×C3×C4C22×D411yescharacteristic
376C2×C3×C4C2×#4111yescharacteristic
377C2×C3×C4C2×#4111yescharacteristic
378#67—28no—
379#67—28no—
380C2×Dic3—22no—
381C2×Dic3—22no—
382C2×Dic3—24no—
383C2×Dic3—24no—
384C2×Dic3—44no—
385C2×Dic3—82no—
386C2×Dic3C2×C22⋊C414yes—
387D12—22no—
388D12—22no—
389D12—22no—
390D12—22no—
391C4×D3—28no—
392C4×D3—28no—
393C4×D3—88no—
394Dic6—22no—
395Dic6—22no—
396Dic6—22no—
397Dic6—22no—
398C3×C8—42no—
399C3⋊C8—42no—
400C24—32no—
401C24D3×D411yescharacteristic
402#41—64no—
403#41—64no—
404#41—64no—
405#41—64no—
406#41—64no—
407#41—64no—
408#41—64no—
409#41—64no—
410C2×Q8—24no—
411C2×Q8—22no—
412C2×Q8—24no—
413C2×Q8—24no—
414C2×Q8—24no—
415C2×Q8—24no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—32no—
421C2×Q8—32no—
422C2×Q8—42no—
423C2×Q8—42no—
424C2×Q8—42no—
425C2×Q8—41no—
426C2×Q8—41no—
427C2×Q8—42no—
428C2×Q8—42no—
429C2×Q8—61no—
430C2×Q8—61no—
431C2×Q8—61no—
432C2×Q8—61no—
433C2×Q8C2×C4×D314yes—
434C2×D4—38no—
435C2×D4—32no—
436C2×D4—32no—
437C2×D4—64no—
438C2×D4—62no—
439C2×D4—62no—
440C2×D4—61no—
441C2×D4—61no—
442C2×D4—61no—
443C2×D4—61no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—32no—
448C22×C4—38no—
449C22×C4—42no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—68no—
453C22×C4—64no—
454C22×C4—62no—
455C22×C4—62no—
456C22×C4—62no—
457C22×C4—62no—
458C22×C4C2×C4×D312yes—
459C22×C4#23512yes—
460C22×C4D3×D411yescharacteristic
461C22×C4D3×D412yes—
462C22×C4D3×D411yescharacteristic
463C22×C4D3×D411yescharacteristic
464C22×C4#23712yes—
465C22×C4C2×#6711yescharacteristic
466C22×C4C2×#6711yescharacteristic
467C22×C4C23×D311yescharacteristic
468M16—22no—
469M16—22no—
470M16—42no—
471M16—62no—
472M16—62no—
473M16—122no—
474C2×C8—22no—
475C2×C8—42no—
476C2×C8—62no—
477C2×C8—122no—
478C4⋊C4—22no—
479C4⋊C4—34no—
480C4⋊C4—616no—
481C4⋊C4—68no—
482C4⋊C4—64no—
483C4⋊C4—62no—
484C4⋊C4C2×#6714yes—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—616no—
488C22⋊C4—68no—
489C22⋊C4—64no—
490C22⋊C4—62no—
491C22⋊C4—62no—
492C42—22no—
493C42—22no—
494C42—34no—
495C42—68no—
496C42—62no—
497C42—62no—
498C42—62no—
499C22×C3—22no—
500C22×C3—22no—
501C22×C3—22no—
502C22×C3—42no—
503C22×C3—42no—
504C22×C3#38612yes—
505C22×C3#38612yes—
506C22×C3C22×C22⋊C411yescharacteristic
507C22×C3C2×#12011yescharacteristic
508C22×C3C2×#12311yescharacteristic
509C2×D3—24no—
510C2×D3—44no—
511C2×D3—44no—
512C2×D3—82no—
513C3×C4—44no—
514C3×C4—44no—
515C3×C4#38612yes—
516C3×C4#38612yes—
517C3×C4C2×C4×D411yescharacteristic
518C3×C4C2×C4×D411yescharacteristic
519C3×C4C2×#12111yescharacteristic
520C3×C4C2×#12111yescharacteristic
521Dic3—24no—
522Dic3—44no—
523Dic3—44no—
524C23—34no—
525C23—42no—
526C23—62no—
527C23—62no—
528C23—124no—
529C23D3×C22⋊C412yes—
530C23#88312yes—
531C23#93711yescharacteristic
532C23#93711yescharacteristic
533C23C2×D3×D411yescharacteristic
534Q8—24no—
535Q8—24no—
536Q8—44no—
537Q8—44no—
538Q8—42no—
539Q8—42no—
540Q8—62no—
541Q8—62no—
542Q8—62no—
543Q8—62no—
544D4—68no—
545D4—68no—
546D4—62no—
547D4—62no—
548D4—62no—
549D4—62no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—34no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—62no—
558C2×C4—62no—
559C2×C4—68no—
560C2×C4—68no—
561C2×C4—64no—
562C2×C4—64no—
563C2×C4—82no—
564C2×C4—124no—
565C2×C4—248no—
566C2×C4—242no—
567C2×C4D3×C22⋊C412yes—
568C2×C4#67⋊C412yes—
569C2×C4#67⋊C412yes—
570C2×C4#88312yes—
571C2×C4#88412yes—
572C2×C4#88412yes—
573C2×C4C4×#6712yes—
574C2×C4C4×#6712yes—
575C2×C4#93711yescharacteristic
576C2×C4#93711yescharacteristic
577C2×C4#93812yes—
578C2×C4#93812yes—
579C2×C4C22×C4×D311yescharacteristic
580C2×C4C2×#23511yescharacteristic
581C2×C4C2×D3×D411yescharacteristic
582C2×C4C2×D3×D411yescharacteristic
583C2×C4C2×#23711yescharacteristic
584C2×C4C22×#6711yescharacteristic
585C8—42no—
586C8—122no—
587C2×C3—21no—
588C2×C3—21no—
589C2×C3—22no—
590C2×C3—22no—
591C2×C3C2×#38611yescharacteristic
592C2×C3#302212yes—
593D3—44no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—42no—
598C22—42no—
599C22—64no—
600C22—124no—
601C22—124no—
602C22—242no—
603C22#495612yes—
604C22#495612yes—
605C22C2×D3×C22⋊C411yescharacteristic
606C22C2×#88311yescharacteristic
607C22C2×#93711yescharacteristic
608C4—44no—
609C4—44no—
610C4—64no—
611C4—124no—
612C4—124no—
613C4#495612yes—
614C4#495612yes—
615C4C2×#67⋊C411yescharacteristic
616C4C2×#88411yescharacteristic
617C4C2×C4×#6711yescharacteristic
618C4C2×#93811yescharacteristic
619C3#2095511yescharacteristic, complemented, Sylow
620C2—21no—
621C2—21no—
622C2—22no—
623C2—22no—
624C2—124no—
625C2C2×#495611yescharacteristic
626C2#8399312yes—
subgroupG/H
centerC22C2×#937
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC2×C3×#419C2
socleC22×C3C2×#120
prime radicalC22×#235C22
cosocleC22×C3×C4C24
hypercenterC2×#419D3
hypocenterC3#20955
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#419

Complements

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

NH ≅ G/Ntype
3. C2×C3×#419C2semidirect
4. C2×#5225C2semidirect
6. #71982C2semidirect
7. #71574C2semidirect
8. #70345C2semidirect
9. #70343C2semidirect
16. #5816C4semidirect
46. C2×#419D3semidirect
60. C22×C3×Q8C2×C4semidirect
91. #947C2×C4semidirect
94. #944C2×C4semidirect
101. #926C2×C4semidirect
277. C22×Q8C4×D3semidirect
619. C3#20955semidirect

Maximal subgroups

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

subgroupindex
C2×#58162
#857312
C2×C3×#4192
C2×#52252
C2×#51322
#719822
#715742
#703452
#703432
#209553