Finite Group Databasebeta

← #10796776

Subgroups — #10796776

Order 768.

1694 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5825C211yesmaximal, characteristic
2C2×#5516C211yesmaximal, characteristic, complemented
3C3×#2896C211yesmaximal, characteristic
4#84072C211yesmaximal, characteristic
5C22×#824C211yesmaximal, characteristic, complemented
6#71982C211yesmaximal, characteristic, complemented
7#71574C211yesmaximal, characteristic, complemented
8#70345C214yesmaximal, complemented
9#70325C214yesmaximal, complemented
10#20954—31nomaximal, Sylow
11C22×C3×#41C2211yescharacteristic
12C23×Dic6C2211yescharacteristic
13C22×C3×M16C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15#5825—24no—
16#5825C418yescomplemented
17C2×#944C2212yes—
18C2×#936C2212yes—
19C2×#935—22no—
20C2×#926C2211yescharacteristic
21C2×#926C2211yescharacteristic
22C2×#924—22no—
23C22×#208C2211yescharacteristic
24#5516—22no—
25#5516—22no—
26#5516—22no—
27#5516C2214yes—
28C2×#887—22no—
29C2×#877—22no—
30C2×C4×Dic6—22no—
31C3×#419C2212yes—
32C3×#418C2212yes—
33C3×#417—22no—
34C3×#416—22no—
35#5233—22no—
36#5232—22no—
37#5225C2212yes—
38#5215C2212yes—
39C2×#824—24no—
40C2×#824C2214yes—
41#4555C2214yes—
42#4532C2214yes—
43C2×#536—31no—
44C2×#519—31no—
45#2896—31no—
46#2896D311yescharacteristic
47C22×#101—31no—
48#1888—34no—
49#1886—34no—
50#1733—31no—
51#1733—31no—
52C2×C3×#41—22no—
53C2×C3×#41—21no—
54C2×C3×#41—21no—
55C2×C3×#41—21no—
56C2×C3×#41—21no—
57C2×C3×#41—22no—
58C2×C3×#41—22no—
59C2×C3×#41C2×C414yes—
60C22×C3×Q8C2311yescharacteristic
61C22×C3×D4—21no—
62C22×C3×D4C2311yescharacteristic
63C23×C3×C4—21no—
64C23×C3×C4C2311yescharacteristic
65C23×Dic3D412yes—
66C22×Dic6—21no—
67C22×Dic6—21no—
68C22×Dic6—21no—
69C22×Dic6—21no—
70C22×Dic6—22no—
71C22×Dic6—22no—
72C22×Dic6C2×C414yes—
73C22×Dic6D412yes—
74C22×Dic6D414yes—
75C22×Dic6C2312yes—
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—
90C2×#217—24no—
91#944—24no—
92#944C2×C418yescomplemented
93#936—24no—
94#936C2×C418yescomplemented
95#935—216no—
96#926—21no—
97#926—21no—
98#926—22no—
99#926C2×C414yes—
100#926D412yes—
101#926C2312yes—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3D412yes—
104#924—216no—
105C2×Dic3⋊C4—24no—
106C2×Dic3⋊C4—24no—
107C2×C4×Dic3—24no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3C2312yes—
110C2×#208—21no—
111C2×#208—22no—
112C2×#208—22no—
113C2×#208C2312yes—
114C22×C3⋊C8—21no—
115#887—28no—
116#887—28no—
117#877—28no—
118#877—28no—
119C4×Dic6—28no—
120C4×Dic6—28no—
121C3×#103—24no—
122C3×#102—24no—
123#835—24no—
124#832—24no—
125#824—28no—
126C22×#41C2×D311yescharacteristic
127C23×Q8—31no—
128C22×M16—31no—
129C22×M16C2×D311yescharacteristic
130#536—38no—
131#536—64no—
132C2×C4⋊Q8—32no—
133C2×#124—32no—
134C2×#123—62no—
135C2×C22⋊Q8—62no—
136#519—34no—
137#519—62no—
138#519—62no—
139#519—62no—
140C2×C4×Q8—62no—
141C2×C4×Q8—62no—
142C2×#117—31no—
143C2×#117—31no—
144C2×#117—62no—
145C2×#117C2×D311yescharacteristic
146#419—32no—
147#419C2×D312yes—
148#418—32no—
149#418C2×D312yes—
150#417—22no—
151#417—62no—
152#416—22no—
153#416—62no—
154C2×#101—34no—
155C2×#101—64no—
156#339—34no—
157#339—34no—
158C24×C3—21no—
159C24×C3C2×D411yescharacteristic
160C3×#41—24no—
161C3×#41—24no—
162C3×#41—24no—
163C3×#41—24no—
164C3×#41—42no—
165C3×#41—42no—
166C3×#41—42no—
167C3×#41—42no—
168C2×C3×Q8—21no—
169C2×C3×Q8—21no—
170C2×C3×Q8—21no—
171C2×C3×Q8—21no—
172C2×C3×Q8—21no—
173C2×C3×Q8C22×C412yes—
174C2×C3×D4—24no—
175C2×C3×D4—21no—
176C2×C3×D4—21no—
177C2×C3×D4—21no—
178C2×C3×D4—21no—
179C2×C3×D4—21no—
180C2×C3×D4—41no—
181C2×C3×D4—41no—
182C2×C3×D4—42no—
183C2×C3×D4C22×C412yes—
184C22×C3×C4—22no—
185C22×C3×C4—21no—
186C22×C3×C4—24no—
187C22×C3×C4—41no—
188C22×C3×C4—41no—
189C22×C3×C4—42no—
190C22×C3×C4C22×C412yes—
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D412yes—
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4#4112yes—
198C22×C3×C4#4112yes—
199C22×C3×C4C2411yescharacteristic
200C22×Dic3—22no—
201C22×Dic3—22no—
202C22×Dic3—24no—
203C22×Dic3—22no—
204C22×Dic3—22no—
205C22×Dic3C22⋊C418yes—
206C22×Dic3C2×D414yes—
207C2×Dic6—24no—
208C2×Dic6—24no—
209C2×Dic6—24no—
210C2×Dic6—24no—
211C2×Dic6—22no—
212C2×Dic6—22no—
213C2×Dic6—24no—
214C2×Dic6—24no—
215C2×Dic6—24no—
216C2×Dic6—24no—
217C2×Dic6—24no—
218C2×Dic6—22no—
219C2×Dic6—22no—
220C2×Dic6—22no—
221C2×Dic6—22no—
222C2×Dic6C22⋊C418yes—
223C2×Dic6C22×C414yes—
224C2×Dic6C2×D414yes—
225C3×M16—22no—
226C3×M16—22no—
227C3×M16—42no—
228C2×C3×C8—22no—
229C2×C3×C8—42no—
230C3×C4⋊C4—22no—
231C3×C4⋊C4C2×D414yes—
232C3×C22⋊C4—22no—
233C3×C22⋊C4—22no—
234C3×C42—22no—
235C3×C42—22no—
236#217—24no—
237#217—22no—
238#217—22no—
239#217—216no—
240C4⋊Dic3—24no—
241C4⋊Dic3—22no—
242C4⋊Dic3C2×D414yes—
243Dic3⋊C4—216no—
244Dic3⋊C4—216no—
245C4×Dic3—216no—
246C4×Dic3—22no—
247C4×Dic3—22no—
248C4×Dic3—22no—
249C4×Dic3C22×C414yes—
250#208—22no—
251#208—22no—
252#208—42no—
253C2×C3⋊C8—22no—
254C2×C3⋊C8—42no—
255C2×#41—22no—
256C2×#41—22no—
257C2×#41—21no—
258C2×#41—21no—
259C2×#41—22no—
260C2×#41—21no—
261C2×#41—21no—
262C2×#41C4×D314yes—
263C22×Q8—34no—
264C22×Q8—32no—
265C22×Q8—34no—
266C22×Q8—32no—
267C22×Q8—62no—
268C22×Q8—61no—
269C22×Q8—61no—
270C22×Q8—62no—
271C22×Q8—61no—
272C22×Q8—61no—
273C22×Q8C22×D311yescharacteristic
274C22×D4—21no—
275C22×D4C22×D311yescharacteristic
276C23×C4—21no—
277C23×C4—32no—
278C23×C4C22×D311yescharacteristic
279C2×M16—21no—
280C2×M16—22no—
281C2×M16—22no—
282C2×M16—32no—
283C2×M16—61no—
284C2×M16—62no—
285C2×M16—62no—
286C2×M16C22×D312yes—
287C22×C8—21no—
288C22×C8—61no—
289C4⋊Q8—38no—
290C4⋊Q8—64no—
291#124—38no—
292#124—64no—
293#123—616no—
294C22⋊Q8—616no—
295C4×Q8—68no—
296C4×Q8—68no—
297C4×Q8—68no—
298C4×Q8—68no—
299#117—21no—
300#117—21no—
301#117—34no—
302#117—32no—
303#117—32no—
304#117—68no—
305#117—68no—
306#117—61no—
307#117—62no—
308#117—61no—
309#117#6712yes—
310#117C22×D312yes—
311C2×C4⋊C4—32no—
312C2×C4⋊C4—64no—
313C2×C4⋊C4—64no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4#6712yes—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—64no—
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C42—21no—
321C2×C42—32no—
322C2×C42—64no—
323C2×C42—61no—
324#103—24no—
325#103—64no—
326#102—24no—
327#102—64no—
328#101—68no—
329C23×C3—22no—
330C23×C3—42no—
331C23×C3—44no—
332C23×C3—41no—
333C23×C3—41no—
334C23×C3C2×C22⋊C412yes—
335C23×C3#12011yescharacteristic
336C23×C3#12011yescharacteristic
337C23×C3#12312yes—
338C23×C3C22×D411yescharacteristic
339C3×Q8—22no—
340C3×Q8—22no—
341C3×Q8—41no—
342C3×Q8—41no—
343C3×D4—22no—
344C3×D4—22no—
345C3×D4—44no—
346C3×D4—44no—
347C3×D4—41no—
348C3×D4—41no—
349C2×C3×C4—22no—
350C2×C3×C4—24no—
351C2×C3×C4—44no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—44no—
355C2×C3×C4—44no—
356C2×C3×C4C2×C22⋊C412yes—
357C2×C3×C4C4×D412yes—
358C2×C3×C4C4×D412yes—
359C2×C3×C4C4×D412yes—
360C2×C3×C4C4×D412yes—
361C2×C3×C4#12011yescharacteristic
362C2×C3×C4#12011yescharacteristic
363C2×C3×C4#12112yes—
364C2×C3×C4#12112yes—
365C2×C3×C4#12112yes—
366C2×C3×C4#12112yes—
367C2×C3×C4#12312yes—
368C2×C3×C4C23×C411yescharacteristic
369C2×C3×C4C22×D411yescharacteristic
370C2×C3×C4C22×D411yescharacteristic
371C2×C3×C4C22×D411yescharacteristic
372C2×C3×C4C2×#4111yescharacteristic
373C2×C3×C4C2×#4111yescharacteristic
374C2×Dic3—24no—
375C2×Dic3—24no—
376C2×Dic3—28no—
377C2×Dic3—28no—
378C2×Dic3—24no—
379C2×Dic3—24no—
380C2×Dic3—44no—
381C2×Dic3—44no—
382C2×Dic3C2×C22⋊C418yes—
383Dic6—24no—
384Dic6—24no—
385Dic6—24no—
386Dic6—24no—
387Dic6—28no—
388Dic6—28no—
389C3×C8—42no—
390C3⋊C8—42no—
391C24—21no—
392C24D3×D411yescharacteristic
393#41—24no—
394#41—24no—
395#41—24no—
396#41—24no—
397#41—42no—
398#41—42no—
399#41—42no—
400#41—42no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—38no—
407C2×Q8—34no—
408C2×Q8—34no—
409C2×Q8—64no—
410C2×Q8—64no—
411C2×Q8—64no—
412C2×Q8—64no—
413C2×Q8—64no—
414C2×Q8—64no—
415C2×Q8—64no—
416C2×Q8—62no—
417C2×Q8—62no—
418C2×Q8—64no—
419C2×Q8—64no—
420C2×Q8—62no—
421C2×Q8—62no—
422C2×Q8—62no—
423C2×Q8—62no—
424C2×Q8C2×C4×D312yes—
425C2×D4—24no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—42no—
432C2×D4—41no—
433C2×D4—41no—
434C2×D4C2×C4×D312yes—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—24no—
438C22×C4—38no—
439C22×C4—34no—
440C22×C4—42no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—64no—
444C22×C4—62no—
445C22×C4—62no—
446C22×C4—62no—
447C22×C4—62no—
448C22×C4C2×C4×D312yes—
449C22×C4#23512yes—
450C22×C4D3×D411yescharacteristic
451C22×C4D3×D412yes—
452C22×C4D3×D411yescharacteristic
453C22×C4D3×D411yescharacteristic
454C22×C4#23712yes—
455C22×C4C2×#6711yescharacteristic
456C22×C4C2×#6711yescharacteristic
457C22×C4C23×D311yescharacteristic
458M16—22no—
459M16—22no—
460M16—42no—
461M16—62no—
462M16—62no—
463M16—122no—
464C2×C8—22no—
465C2×C8—42no—
466C2×C8—62no—
467C2×C8—122no—
468C4⋊C4—22no—
469C4⋊C4—34no—
470C4⋊C4—616no—
471C4⋊C4—616no—
472C4⋊C4—64no—
473C4⋊C4—62no—
474C4⋊C4C2×#6714yes—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—616no—
478C22⋊C4—64no—
479C22⋊C4—62no—
480C22⋊C4—62no—
481C42—22no—
482C42—22no—
483C42—34no—
484C42—616no—
485C42—62no—
486C42—62no—
487C42—62no—
488C22×C3—22no—
489C22×C3—22no—
490C22×C3—22no—
491C22×C3—42no—
492C22×C3—42no—
493C22×C3—44no—
494C22×C3—42no—
495C22×C3—82no—
496C22×C3—82no—
497C22×C3#38612yes—
498C22×C3#38612yes—
499C22×C3C22×C22⋊C411yescharacteristic
500C22×C3C2×#12011yescharacteristic
501C22×C3C2×#12311yescharacteristic
502C3×C4—44no—
503C3×C4—42no—
504C3×C4#38612yes—
505C3×C4#38612yes—
506C3×C4C2×C4×D411yescharacteristic
507C3×C4C2×C4×D411yescharacteristic
508C3×C4C2×#12111yescharacteristic
509C3×C4C2×#12111yescharacteristic
510Dic3—28no—
511Dic3—44no—
512Dic3—44no—
513C23—22no—
514C23—42no—
515C23—44no—
516C23—41no—
517C23—41no—
518C23D3×C22⋊C412yes—
519C23#88312yes—
520C23#93711yescharacteristic
521C23#93711yescharacteristic
522C23C2×D3×D411yescharacteristic
523Q8—22no—
524Q8—22no—
525Q8—41no—
526Q8—41no—
527Q8—68no—
528Q8—68no—
529Q8—64no—
530Q8—64no—
531Q8—64no—
532Q8—64no—
533D4—22no—
534D4—22no—
535D4—44no—
536D4—44no—
537D4—41no—
538D4—41no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—38no—
542C2×C4—44no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—68no—
548C2×C4—68no—
549C2×C4—64no—
550C2×C4—64no—
551C2×C4—64no—
552C2×C4—64no—
553C2×C4—124no—
554C2×C4—124no—
555C2×C4D3×C22⋊C412yes—
556C2×C4#67⋊C412yes—
557C2×C4#67⋊C412yes—
558C2×C4#88312yes—
559C2×C4#88412yes—
560C2×C4#88412yes—
561C2×C4C4×#6712yes—
562C2×C4C4×#6712yes—
563C2×C4#93711yescharacteristic
564C2×C4#93711yescharacteristic
565C2×C4#93812yes—
566C2×C4#93812yes—
567C2×C4C22×C4×D311yescharacteristic
568C2×C4C2×#23511yescharacteristic
569C2×C4C2×D3×D411yescharacteristic
570C2×C4C2×D3×D411yescharacteristic
571C2×C4C2×#23711yescharacteristic
572C2×C4C22×#6711yescharacteristic
573C8—42no—
574C8—122no—
575C2×C3—21no—
576C2×C3—21no—
577C2×C3—22no—
578C2×C3—22no—
579C2×C3—82no—
580C2×C3C2×#38611yescharacteristic
581C2×C3#302012yes—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—42no—
586C22—42no—
587C22—44no—
588C22—42no—
589C22—82no—
590C22—82no—
591C22#495612yes—
592C22#495612yes—
593C22C2×D3×C22⋊C411yescharacteristic
594C22C2×#88311yescharacteristic
595C22C2×#93711yescharacteristic
596C4—44no—
597C4—42no—
598C4—68no—
599C4—124no—
600C4—124no—
601C4#495612yes—
602C4#495612yes—
603C4C2×#67⋊C411yescharacteristic
604C4C2×#88411yescharacteristic
605C4C2×C4×#6711yescharacteristic
606C4C2×#93811yescharacteristic
607C3#2095411yescharacteristic, complemented, Sylow
608C2—21no—
609C2—21no—
610C2—22no—
611C2—22no—
612C2—82no—
613C2C2×#495611yescharacteristic
614C2#8400612yes—
subgroupG/H
centerC22C2×#937
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#2896C2
socleC22×C3C2×#120
prime radicalC22×C3×#41C22
cosocleC22×C3×C4C24
hypercenter#2896D3
hypocenterC3#20954
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2896

Complements

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

NH ≅ G/Ntype
2. C2×#5516C2semidirect
5. C22×#824C2semidirect
6. #71982C2semidirect
7. #71574C2semidirect
8. #70345C2semidirect
9. #70325C2semidirect
16. #5825C4semidirect
92. #944C2×C4semidirect
94. #936C2×C4semidirect
607. C3#20954semidirect

Maximal subgroups

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

subgroupindex
C2×#58252
C2×#55162
C3×#28962
#840722
C22×#8242
#719822
#715742
#703452
#703252
#209543