Finite Group Databasebeta

← #10796790

Subgroups — #10796790

Order 768.

1902 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5805C211yesmaximal, characteristic
2#85731C211yesmaximal, characteristic, complemented
3C2×C3×#418C211yesmaximal, characteristic, complemented
4C2×#5215C211yesmaximal, characteristic, complemented
5C2×#5132C211yesmaximal, characteristic, complemented
6#71982C211yesmaximal, characteristic, complemented
7#71574C211yesmaximal, characteristic, complemented
8#70325C214yesmaximal, complemented
9#70323C214yesmaximal, complemented
10#20956—31nomaximal, Sylow
11C23×C3×D4C2211yescharacteristic
12C22×#235C2211yescharacteristic
13C22×C3×M16C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C2×#953—22no—
16#5805—24no—
17#5805C418yescomplemented
18C2×#940C2212yes—
19C2×#938—22no—
20C2×#936C2212yes—
21C2×#926C2211yescharacteristic
22C2×#926C2211yescharacteristic
23C22×#208C2211yescharacteristic
24#5523—22no—
25#5522—22no—
26#5521C2212yescomplemented
27D3×#117—22no—
28#5516C2212yescomplemented
29#5504—22no—
30#5486—22no—
31C4×#235—22no—
32C3×#418—22no—
33C3×#418C2214yes—
34C2×C3×#102—22no—
35#5215—22no—
36#5215C2214yes—
37C2×#832—22no—
38C2×#824C2212yescomplemented
39#5132—24no—
40C2×#823C2212yescomplemented
41#4555C2214yescomplemented
42#4532C2214yescomplemented
43C2×#535—31no—
44#3428—31no—
45C2×#418—31no—
46C2×#418D311yescharacteristic, complemented
47C2×#413—31no—
48#1888—34no—
49#1887—34no—
50#1733—31no—
51#1733—31no—
52C25×C3—21no—
53C22×C3×D4—21no—
54C22×C3×D4—21no—
55C22×C3×D4—21no—
56C22×C3×D4—21no—
57C22×C3×D4—21no—
58C22×C3×D4—22no—
59C22×C3×D4—22no—
60C22×C3×D4—22no—
61C22×C3×D4C2×C414yescomplemented
62C22×C3×D4C2312yes—
63C23×C3×C4C2311yescharacteristic
64C22×#67—24no—
65C22×#67D412yes—
66C2×#235—21no—
67C2×#235—21no—
68C2×#235—21no—
69C2×#235—21no—
70C2×#235—22no—
71C2×#235—22no—
72C2×#235C2×C414yes—
73C2×#235D414yes—
74C22×D12C2311yescharacteristic
75C22×C4×D3D412yes—
76C22×Dic6C2311yescharacteristic
77C2×C3×M16—21no—
78C2×C3×M16—22no—
79C2×C3×M16—22no—
80C2×C3×M16C2312yes—
81C22×C3×C8—21no—
82C3×#117—21no—
83C3×#117—21no—
84C3×#117D412yescomplemented
85C3×#117C2312yes—
86C2×C3×C4⋊C4D412yescomplemented
87C2×C3×C22⋊C4—21no—
88C2×C3×C42—21no—
89#953—216no—
90C2×#217—21no—
91C2×#217—21no—
92C2×#217—24no—
93#940—24no—
94#940C2×C418yescomplemented
95#938—216no—
96#936—24no—
97#936C2×C418yescomplemented
98C4×#67—28no—
99#926—21no—
100#926—21no—
101#926—22no—
102#926C2×C414yescomplemented
103#926D412yescomplemented
104#926C2312yes—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3D412yescomplemented
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×#102—28no—
126#832—28no—
127#824—24no—
128#823—24no—
129C22×#41—31no—
130C23×D4C2×D311yescharacteristic
131C22×M16—31no—
132C22×M16C2×D311yescharacteristic
133#535—38no—
134#535—64no—
135C2×#127—32no—
136C2×#124—32no—
137C2×#121—62no—
138C2×#120—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#418—22no—
151#418—34no—
152#418—62no—
153#418C2×D314yes—
154C2×#102—22no—
155C2×#102—62no—
156#413—64no—
157C2×#101—32no—
158C2×#100—32no—
159#339—34no—
160#339—34no—
161C24×C3—22no—
162C24×C3—24no—
163C24×C3—41no—
164C24×C3—41no—
165C24×C3—42no—
166C24×C3—84no—
167C24×C3C2×D411yescharacteristic
168C23×D3C2×D412yes—
169C2×C3×D4—24no—
170C2×C3×D4—24no—
171C2×C3×D4—24no—
172C2×C3×D4—24no—
173C2×C3×D4—24no—
174C2×C3×D4—22no—
175C2×C3×D4—22no—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4—41no—
180C2×C3×D4—41no—
181C2×C3×D4—42no—
182C2×C3×D4—42no—
183C2×C3×D4—42no—
184C2×C3×D4—42no—
185C2×C3×D4—42no—
186C2×C3×D4C22×C414yes—
187C22×C3×C4—22no—
188C22×C3×C4C22×C412yes—
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D412yes—
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4#4112yes—
196C22×C3×C4#4112yes—
197C22×C3×C4C2411yescharacteristic
198C2×#67—22no—
199C2×#67—22no—
200C2×#67—24no—
201C2×#67—216no—
202C2×#67—88no—
203C2×#67C22⋊C418yes—
204C22×Dic3—22no—
205C22×Dic3—22no—
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—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12—21no—
219C2×D12C22×C412yes—
220C2×D12C2×D412yes—
221C2×C4×D3—22no—
222C2×C4×D3—22no—
223C2×C4×D3—24no—
224C2×C4×D3—28no—
225C2×C4×D3C22⋊C418yes—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6C22×C412yes—
231C2×Dic6C2×D412yes—
232C3×M16—22no—
233C3×M16—22no—
234C3×M16—42no—
235C2×C3×C8—22no—
236C2×C3×C8—42no—
237C3×C4⋊C4—22no—
238C3×C4⋊C4C2×D414yes—
239C3×C22⋊C4—22no—
240C3×C22⋊C4—22no—
241C3×C42—22no—
242C3×C42—22no—
243#217—24no—
244#217—22no—
245#217—22no—
246#217—216no—
247(C2×D3)⋊C4—28no—
248C4⋊Dic3—24no—
249C4⋊Dic3—22no—
250C4⋊Dic3C2×D414yes—
251Dic3⋊C4—28no—
252C4×Dic3—28no—
253C4×Dic3—22no—
254C4×Dic3—22no—
255C4×Dic3—22no—
256C4×Dic3C22×C414yes—
257#208—22no—
258#208—22no—
259#208—42no—
260C2×C3⋊C8—22no—
261C2×C3⋊C8—42no—
262C25—21no—
263C2×#41—34no—
264C2×#41—34no—
265C2×#41—62no—
266C2×#41—61no—
267C2×#41—61no—
268C2×#41—62no—
269C2×#41—61no—
270C2×#41—61no—
271C22×Q8—31no—
272C22×D4—22no—
273C22×D4—21no—
274C22×D4—21no—
275C22×D4—21no—
276C22×D4—22no—
277C22×D4—22no—
278C22×D4—21no—
279C22×D4—21no—
280C22×D4—32no—
281C22×D4—31no—
282C22×D4—64no—
283C22×D4C4×D314yescomplemented
284C22×D4C22×D312yes—
285C23×C4—32no—
286C23×C4C22×D311yescharacteristic
287C2×M16—21no—
288C2×M16—22no—
289C2×M16—22no—
290C2×M16—32no—
291C2×M16—61no—
292C2×M16—62no—
293C2×M16—62no—
294C2×M16C22×D312yes—
295C22×C8—21no—
296C22×C8—61no—
297#127—38no—
298#127—64no—
299#124—38no—
300#124—64no—
301#121—616no—
302#120—616no—
303C4×Q8—64no—
304C4×Q8—64no—
305C4×D4—64no—
306C4×D4—68no—
307C4×D4—68no—
308C4×D4—64no—
309#117—21no—
310#117—21no—
311#117—34no—
312#117—32no—
313#117—32no—
314#117—64no—
315#117—64no—
316#117—64no—
317#117—61no—
318#117—62no—
319#117—61no—
320#117#6712yescomplemented
321#117C22×D312yes—
322C2×C4⋊C4—32no—
323C2×C4⋊C4—64no—
324C2×C4⋊C4—61no—
325C2×C4⋊C4#6712yescomplemented
326C2×C22⋊C4—21no—
327C2×C22⋊C4—64no—
328C2×C22⋊C4—64no—
329C2×C22⋊C4—61no—
330C2×C22⋊C4—61no—
331C2×C42—21no—
332C2×C42—32no—
333C2×C42—64no—
334C2×C42—61no—
335#102—28no—
336#102—68no—
337#101—64no—
338#100—64no—
339C23×C3—24no—
340C23×C3—42no—
341C23×C3—48no—
342C23×C3—42no—
343C23×C3—42no—
344C23×C3—44no—
345C23×C3—44no—
346C23×C3—84no—
347C23×C3—84no—
348C23×C3—88no—
349C23×C3—88no—
350C23×C3C2×C22⋊C412yes—
351C23×C3#12011yescharacteristic
352C23×C3#12011yescharacteristic
353C23×C3#12312yes—
354C23×C3C22×D411yescharacteristic
355C22×D3—22no—
356C22×D3—22no—
357C22×D3—44no—
358C22×D3C2×C22⋊C414yes—
359C3×D4—24no—
360C3×D4—24no—
361C3×D4—42no—
362C3×D4—42no—
363C3×D4—44no—
364C3×D4—44no—
365C2×C3×C4—24no—
366C2×C3×C4—82no—
367C2×C3×C4C2×C22⋊C412yes—
368C2×C3×C4C4×D412yes—
369C2×C3×C4C4×D412yes—
370C2×C3×C4C4×D412yes—
371C2×C3×C4C4×D412yes—
372C2×C3×C4#12011yescharacteristic
373C2×C3×C4#12011yescharacteristic
374C2×C3×C4#12112yes—
375C2×C3×C4#12112yes—
376C2×C3×C4#12112yes—
377C2×C3×C4#12112yes—
378C2×C3×C4#12312yes—
379C2×C3×C4C23×C411yescharacteristic
380C2×C3×C4C22×D411yescharacteristic
381C2×C3×C4C22×D411yescharacteristic
382C2×C3×C4C22×D411yescharacteristic
383C2×C3×C4C2×#4111yescharacteristic
384C2×C3×C4C2×#4111yescharacteristic
385#67—28no—
386#67—28no—
387#67—816no—
388C2×Dic3—22no—
389C2×Dic3—22no—
390C2×Dic3—24no—
391C2×Dic3—24no—
392C2×Dic3—44no—
393C2×Dic3—82no—
394C2×Dic3C2×C22⋊C414yes—
395D12—22no—
396D12—22no—
397D12—22no—
398D12—22no—
399C4×D3—28no—
400C4×D3—28no—
401C4×D3—88no—
402Dic6—22no—
403Dic6—22no—
404Dic6—22no—
405Dic6—22no—
406C3×C8—42no—
407C3⋊C8—42no—
408C24—22no—
409C24—24no—
410C24—32no—
411C24—42no—
412C24—41no—
413C24—41no—
414C24—84no—
415C24D3×D411yescharacteristic
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—32no—
425C2×Q8—32no—
426C2×Q8—61no—
427C2×Q8—61no—
428C2×Q8—61no—
429C2×Q8—61no—
430C2×D4—24no—
431C2×D4—24no—
432C2×D4—24no—
433C2×D4—24no—
434C2×D4—24no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—38no—
441C2×D4—32no—
442C2×D4—32no—
443C2×D4—42no—
444C2×D4—42no—
445C2×D4—42no—
446C2×D4—41no—
447C2×D4—41no—
448C2×D4—42no—
449C2×D4—42no—
450C2×D4—64no—
451C2×D4—62no—
452C2×D4—62no—
453C2×D4—616no—
454C2×D4—61no—
455C2×D4—61no—
456C2×D4—61no—
457C2×D4—61no—
458C2×D4—248no—
459C2×D4C2×C4×D314yes—
460C22×C4—22no—
461C22×C4—32no—
462C22×C4—38no—
463C22×C4—68no—
464C22×C4—64no—
465C22×C4—62no—
466C22×C4—62no—
467C22×C4—62no—
468C22×C4—62no—
469C22×C4C2×C4×D312yes—
470C22×C4#23512yes—
471C22×C4D3×D411yescharacteristic
472C22×C4D3×D412yes—
473C22×C4D3×D411yescharacteristic
474C22×C4D3×D411yescharacteristic
475C22×C4#23712yes—
476C22×C4C2×#6711yescharacteristic
477C22×C4C2×#6711yescharacteristic
478C22×C4C23×D311yescharacteristic
479M16—22no—
480M16—22no—
481M16—42no—
482M16—62no—
483M16—62no—
484M16—122no—
485C2×C8—22no—
486C2×C8—42no—
487C2×C8—62no—
488C2×C8—122no—
489C4⋊C4—22no—
490C4⋊C4—34no—
491C4⋊C4—68no—
492C4⋊C4—64no—
493C4⋊C4—62no—
494C4⋊C4C2×#6714yes—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—68no—
498C22⋊C4—616no—
499C22⋊C4—64no—
500C22⋊C4—62no—
501C22⋊C4—62no—
502C42—22no—
503C42—22no—
504C42—34no—
505C42—68no—
506C42—62no—
507C42—62no—
508C42—62no—
509C22×C3—22no—
510C22×C3—22no—
511C22×C3—22no—
512C22×C3—42no—
513C22×C3—42no—
514C22×C3—48no—
515C22×C3—44no—
516C22×C3—84no—
517C22×C3—88no—
518C22×C3—88no—
519C22×C3—84no—
520C22×C3#38612yes—
521C22×C3#38612yes—
522C22×C3C22×C22⋊C411yescharacteristic
523C22×C3C2×#12011yescharacteristic
524C22×C3C2×#12311yescharacteristic
525C2×D3—24no—
526C2×D3—44no—
527C2×D3—44no—
528C2×D3—82no—
529C3×C4—44no—
530C3×C4#38612yes—
531C3×C4#38612yes—
532C3×C4C2×C4×D411yescharacteristic
533C3×C4C2×C4×D411yescharacteristic
534C3×C4C2×#12111yescharacteristic
535C3×C4C2×#12111yescharacteristic
536Dic3—24no—
537Dic3—44no—
538Dic3—44no—
539C23—24no—
540C23—34no—
541C23—42no—
542C23—48no—
543C23—44no—
544C23—44no—
545C23—42no—
546C23—42no—
547C23—62no—
548C23—62no—
549C23—84no—
550C23—84no—
551C23—88no—
552C23—88no—
553C23—124no—
554C23D3×C22⋊C412yes—
555C23#88312yes—
556C23#93711yescharacteristic
557C23#93711yescharacteristic
558C23C2×D3×D411yescharacteristic
559Q8—62no—
560Q8—62no—
561Q8—62no—
562Q8—62no—
563D4—24no—
564D4—24no—
565D4—44no—
566D4—44no—
567D4—42no—
568D4—42no—
569D4—68no—
570D4—68no—
571D4—62no—
572D4—62no—
573D4—62no—
574D4—62no—
575D4—2416no—
576C2×C4—24no—
577C2×C4—34no—
578C2×C4—62no—
579C2×C4—62no—
580C2×C4—68no—
581C2×C4—68no—
582C2×C4—64no—
583C2×C4—64no—
584C2×C4—82no—
585C2×C4—124no—
586C2×C4—248no—
587C2×C4—242no—
588C2×C4D3×C22⋊C412yes—
589C2×C4#67⋊C412yes—
590C2×C4#67⋊C412yes—
591C2×C4#88312yes—
592C2×C4#88412yes—
593C2×C4#88412yes—
594C2×C4C4×#6712yes—
595C2×C4C4×#6712yes—
596C2×C4#93711yescharacteristic
597C2×C4#93711yescharacteristic
598C2×C4#93812yes—
599C2×C4#93812yes—
600C2×C4C22×C4×D311yescharacteristic
601C2×C4C2×#23511yescharacteristic
602C2×C4C2×D3×D411yescharacteristic
603C2×C4C2×D3×D411yescharacteristic
604C2×C4C2×#23711yescharacteristic
605C2×C4C22×#6711yescharacteristic
606C8—42no—
607C8—122no—
608C2×C3—21no—
609C2×C3—21no—
610C2×C3—22no—
611C2×C3—22no—
612C2×C3—84no—
613C2×C3C2×#38611yescharacteristic
614C2×C3#302112yes—
615D3—44no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—42no—
620C22—42no—
621C22—48no—
622C22—44no—
623C22—64no—
624C22—84no—
625C22—88no—
626C22—88no—
627C22—84no—
628C22—124no—
629C22—124no—
630C22—242no—
631C22#495612yes—
632C22#495612yes—
633C22C2×D3×C22⋊C411yescharacteristic
634C22C2×#88311yescharacteristic
635C22C2×#93711yescharacteristic
636C4—44no—
637C4—64no—
638C4—124no—
639C4—124no—
640C4#495612yes—
641C4#495612yes—
642C4C2×#67⋊C411yescharacteristic
643C4C2×#88411yescharacteristic
644C4C2×C4×#6711yescharacteristic
645C4C2×#93811yescharacteristic
646C3#2095611yescharacteristic, complemented, Sylow
647C2—21no—
648C2—21no—
649C2—22no—
650C2—22no—
651C2—84no—
652C2—124no—
653C2C2×#495611yescharacteristic
654C2#8395712yes—
subgroupG/H
centerC22C2×#937
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC2×C3×#418C2
socleC22×C3C2×#120
prime radicalC2×#5805C2
cosocleC22×C3×C4C24
hypercenterC2×#418D3
hypocenterC3#20956
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#418

Complements

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

NH ≅ G/Ntype
2. #85731C2semidirect
3. C2×C3×#418C2semidirect
4. C2×#5215C2semidirect
5. C2×#5132C2semidirect
6. #71982C2semidirect
7. #71574C2semidirect
8. #70325C2semidirect
9. #70323C2semidirect
17. #5805C4semidirect
26. #5521C22semidirect
28. #5516C22semidirect
38. C2×#824C22semidirect
40. C2×#823C22semidirect
41. #4555C22semidirect
42. #4532C22semidirect
46. C2×#418D3semidirect
61. C22×C3×D4C2×C4semidirect
84. C3×#117D4semidirect
86. C2×C3×C4⋊C4D4semidirect
94. #940C2×C4semidirect
97. #936C2×C4semidirect
102. #926C2×C4semidirect
103. #926D4semidirect
106. C2×C4⋊Dic3D4semidirect
283. C22×D4C4×D3semidirect
320. #117#67semidirect
325. C2×C4⋊C4#67semidirect
646. C3#20956semidirect

Maximal subgroups

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

subgroupindex
C2×#58052
#857312
C2×C3×#4182
C2×#52152
C2×#51322
#719822
#715742
#703252
#703232
#209563