Finite Group Databasebeta

← #10796708

Subgroups — #10796708

Order 768.

1702 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#942C211yesmaximal, characteristic
2#85731C211yesmaximal, characteristic
3C2×C3×#419C211yesmaximal, characteristic, complemented
4#84062C211yesmaximal, characteristic, complemented
5C2×#5139C211yesmaximal, characteristic
6#71987C211yesmaximal, characteristic, complemented
7#71574C211yesmaximal, characteristic, complemented
8#70348C218yesmaximal, complemented
9#20944—31nomaximal, Sylow
10C23×C3×Q8C2211yescharacteristic
11C22×#235C2211yescharacteristic
12C22×C3×M16C2211yescharacteristic
13C2×C3×#117C2211yescharacteristic
14C22×#216—21no—
15C2×#942—24no—
16C2×#942—22no—
17C2×#942C418yescomplemented
18C2×#942C2214yes—
19C22×#215—21no—
20C2×#926C2211yescharacteristic
21C2×#926C2211yescharacteristic
22C22×#208C2211yescharacteristic
23#5523C2214yes—
24#5522—22no—
25#5521—21no—
26D3×#117—22no—
27#5516—21no—
28#5504—22no—
29#5486—22no—
30C4×#235—22no—
31C3×#419—22no—
32C3×#419C2214yescomplemented
33C2×C3×#103—22no—
34#5229—22no—
35#5229C2214yescomplemented
36#5211—22no—
37#5139—24no—
38C2×#825C2214yes—
39#4554C2214yescomplemented
40#4532C2214yescomplemented
41C22×#137—31no—
42#3428—31no—
43C2×#421—31no—
44C2×#419D311yescharacteristic, complemented
45#2310—31no—
46#1882—38no—
47#1742—31no—
48#1733—31no—
49C22×C3×Q8—21no—
50C22×C3×Q8—22no—
51C22×C3×Q8—21no—
52C22×C3×Q8—21no—
53C22×C3×Q8—21no—
54C22×C3×Q8—21no—
55C22×C3×Q8—22no—
56C22×C3×Q8—22no—
57C22×C3×Q8C2×C414yescomplemented
58C22×C3×Q8C2312yes—
59C23×C3×C4—21no—
60C23×C3×C4C2311yescharacteristic
61C22×#67—21no—
62C2×#235—21no—
63C2×#235—21no—
64C2×#235—22no—
65C2×#235—22no—
66C2×#235C2×C414yes—
67C2×#235D414yes—
68C2×#235D412yes—
69C2×#235C2312yes—
70C22×D12D411yescharacteristic
71C22×C4×D3—21no—
72C22×Dic6D411yescharacteristic
73C2×C3×M16—21no—
74C2×C3×M16—22no—
75C2×C3×M16—22no—
76C2×C3×M16C2312yes—
77C22×C3×C8—21no—
78C3×#117—21no—
79C3×#117—21no—
80C3×#117D412yes—
81C3×#117C2312yes—
82C2×C3×C4⋊C4D412yes—
83C2×C3×C22⋊C4—21no—
84C2×C3×C42—21no—
85C2×#217—21no—
86C2×#217D412yes—
87C2×#216—28no—
88C2×#216—24no—
89#942—216no—
90#942—28no—
91#942C2×C4116yescomplemented
92C2×#215—28no—
93C2×#215—24no—
94C4×#67—28no—
95#926—21no—
96#926—21no—
97#926—21no—
98#926—21no—
99#926—22no—
100#926—22no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3C2312yes—
103C2×C4×Dic3D412yes—
104C2×C4×Dic3C2312yes—
105C2×#208—22no—
106C2×#208—21no—
107C2×#208C2×C414yescomplemented
108C2×#208C2312yes—
109C22×C3⋊C8—21no—
110#893—24no—
111#892—24no—
112D3×C4⋊C4—24no—
113#887—24no—
114#67⋊C4—28no—
115D3×C22⋊C4—24no—
116#877—24no—
117C4×D12—24no—
118#872—24no—
119C42×D3—24no—
120C4×Dic6—24no—
121C3×#103—28no—
122C3⋊#95—28no—
123#825—28no—
124C22×#41—31no—
125C23×Q8C2×D311yescharacteristic
126C2×#137—38no—
127C2×#137—34no—
128C2×#137—62no—
129C2×#137—64no—
130C22×Q16—61no—
131C22×SD16—61no—
132C22×M16—31no—
133C22×M16C2×D311yescharacteristic
134#520—34no—
135#520—62no—
136#519—61no—
137#518—62no—
138#518—61no—
139C4×#41—62no—
140C4×#41—62no—
141C2×#117—31no—
142C2×#117—31no—
143C2×#117—62no—
144C2×#117C2×D311yescharacteristic
145#421—64no—
146C2×#104—34no—
147#419—22no—
148#419C2×D314yescomplemented
149C2×#103—22no—
150#391—34no—
151#391—62no—
152#381—62no—
153#343—34no—
154#339—34no—
155C24×C3C2×D411yescharacteristic
156C23×D3—21no—
157C2×C3×Q8—24no—
158C2×C3×Q8—22no—
159C2×C3×Q8—24no—
160C2×C3×Q8—24no—
161C2×C3×Q8—24no—
162C2×C3×Q8—24no—
163C2×C3×Q8—22no—
164C2×C3×Q8—22no—
165C2×C3×Q8—22no—
166C2×C3×Q8—22no—
167C2×C3×Q8—41no—
168C2×C3×Q8—41no—
169C2×C3×Q8—42no—
170C2×C3×Q8—42no—
171C2×C3×Q8—42no—
172C2×C3×Q8—42no—
173C2×C3×Q8—42no—
174C2×C3×Q8C22×C414yes—
175C22×C3×C4—24no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—41no—
179C22×C3×C4—41no—
180C22×C3×C4—42no—
181C22×C3×C4C22×C412yes—
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D412yes—
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4#4112yes—
189C22×C3×C4#4112yes—
190C22×C3×C4C2411yescharacteristic
191C2×#67—24no—
192C2×#67—22no—
193C2×#67—24no—
194C2×#67—22no—
195C22×Dic3—21no—
196C22×Dic3—22no—
197C22×Dic3C2×D414yes—
198#235—24no—
199#235—24no—
200#235—24no—
201#235—24no—
202#235—24no—
203#235—22no—
204#235—22no—
205#235C22⋊C418yes—
206#235C22×C414yes—
207#235C2×D414yes—
208C2×D12—21no—
209C2×D12—22no—
210C2×D12C22⋊C414yes—
211C2×D12C2×D412yes—
212C2×C4×D3—24no—
213C2×C4×D3—22no—
214C2×C4×D3—24no—
215C2×C4×D3—22no—
216C2×C4×D3—28no—
217C2×Dic6—21no—
218C2×Dic6—22no—
219C2×Dic6C22⋊C414yes—
220C2×Dic6C2×D412yes—
221C3×M16—22no—
222C3×M16—22no—
223C3×M16—42no—
224C2×C3×C8—22no—
225C2×C3×C8—42no—
226C3×C4⋊C4—22no—
227C3×C4⋊C4C2×D414yes—
228C3×C22⋊C4—22no—
229C3×C22⋊C4—22no—
230C3×C42—22no—
231C3×C42—22no—
232#217—22no—
233#217—22no—
234#217—42no—
235#216—216no—
236#215—216no—
237(C2×D3)⋊C4—28no—
238C4⋊Dic3—22no—
239C4⋊Dic3—22no—
240C4⋊Dic3—42no—
241Dic3⋊C4—28no—
242C4×Dic3—28no—
243C4×Dic3—22no—
244C4×Dic3—22no—
245C4×Dic3—22no—
246C4×Dic3—22no—
247#208—24no—
248#208—22no—
249#208C22×C414yes—
250C2×C3⋊C8—24no—
251C2×C3⋊C8—22no—
252C2×#41—34no—
253C2×#41—32no—
254C2×#41—34no—
255C2×#41—32no—
256C2×#41—62no—
257C2×#41—61no—
258C2×#41—62no—
259C2×#41—61no—
260C22×Q8—21no—
261C22×Q8—22no—
262C22×Q8—22no—
263C22×Q8—21no—
264C22×Q8—22no—
265C22×Q8—21no—
266C22×Q8—21no—
267C22×Q8—21no—
268C22×Q8—31no—
269C22×Q8C4×D314yescomplemented
270C22×Q8C22×D312yes—
271C22×D4—31no—
272C22×D4—61no—
273C23×C4—21no—
274C23×C4—61no—
275C23×C4C22×D311yescharacteristic
276#137—316no—
277#137—616no—
278#137—68no—
279C2×Q16—68no—
280C2×Q16—64no—
281C2×SD16—68no—
282C2×SD16—64no—
283C2×M16—21no—
284C2×M16—22no—
285C2×M16—22no—
286C2×M16—34no—
287C2×M16—32no—
288C2×M16—62no—
289C2×M16—61no—
290C2×M16C22×D312yes—
291C22×C8—21no—
292C22×C8—61no—
293C4×Q8—64no—
294C4×Q8—64no—
295C4×D4—64no—
296C4×D4—68no—
297C4×D4—64no—
298C4×D4—68no—
299#117—21no—
300#117—21no—
301#117—64no—
302#117—64no—
303#117—61no—
304#117—61no—
305#117—61no—
306#117—61no—
307#117—64no—
308#117—62no—
309#117—62no—
310#117#6712yes—
311#117C22×D312yes—
312C2×C4⋊C4—32no—
313C2×C4⋊C4—64no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4#6712yes—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—32no—
318C2×C22⋊C4—64no—
319C2×C22⋊C4—61no—
320C2×C42—21no—
321C2×C42—32no—
322C2×C42—32no—
323C2×C42—64no—
324#104—68no—
325#103—28no—
326#95—68no—
327C23×C3—42no—
328C23×C3C4×D412yes—
329C23×C3#12011yescharacteristic
330C23×C3#12011yescharacteristic
331C23×C3#12112yes—
332C23×C3C22×D411yescharacteristic
333C22×D3—22no—
334C22×D3—24no—
335C22×D3—82no—
336C3×Q8—24no—
337C3×Q8—24no—
338C3×Q8—42no—
339C3×Q8—42no—
340C3×Q8—44no—
341C3×Q8—44no—
342C2×C3×C4—24no—
343C2×C3×C4—24no—
344C2×C3×C4—42no—
345C2×C3×C4—42no—
346C2×C3×C4—44no—
347C2×C3×C4—44no—
348C2×C3×C4—82no—
349C2×C3×C4C2×C22⋊C412yes—
350C2×C3×C4C2×C22⋊C412yes—
351C2×C3×C4C4×D412yes—
352C2×C3×C4C4×D412yes—
353C2×C3×C4C4×D412yes—
354C2×C3×C4#12011yescharacteristic
355C2×C3×C4#12011yescharacteristic
356C2×C3×C4#12112yes—
357C2×C3×C4#12112yes—
358C2×C3×C4#12112yes—
359C2×C3×C4#12312yes—
360C2×C3×C4#12312yes—
361C2×C3×C4C23×C411yescharacteristic
362C2×C3×C4C22×D411yescharacteristic
363C2×C3×C4C22×D411yescharacteristic
364C2×C3×C4C22×D411yescharacteristic
365C2×C3×C4C2×#4111yescharacteristic
366C2×C3×C4C2×#4111yescharacteristic
367#67—24no—
368#67—24no—
369#67—28no—
370C2×Dic3—22no—
371C2×Dic3—24no—
372C2×Dic3—24no—
373C2×Dic3—44no—
374C2×Dic3—44no—
375C2×Dic3—42no—
376D12—22no—
377D12—24no—
378D12C2×C22⋊C414yes—
379C4×D3—24no—
380C4×D3—24no—
381C4×D3—28no—
382C4×D3—88no—
383Dic6—22no—
384Dic6—24no—
385Dic6C2×C22⋊C414yes—
386C3×C8—42no—
387C3⋊C8—24no—
388C24—61no—
389C24D3×D411yescharacteristic
390#41—38no—
391#41—34no—
392#41—34no—
393#41—64no—
394#41—64no—
395#41—64no—
396#41—64no—
397#41—64no—
398#41—62no—
399#41—62no—
400C2×Q8—24no—
401C2×Q8—22no—
402C2×Q8—24no—
403C2×Q8—24no—
404C2×Q8—24no—
405C2×Q8—24no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—34no—
411C2×Q8—32no—
412C2×Q8—42no—
413C2×Q8—42no—
414C2×Q8—42no—
415C2×Q8—41no—
416C2×Q8—41no—
417C2×Q8—42no—
418C2×Q8—42no—
419C2×Q8—62no—
420C2×Q8—61no—
421C2×Q8C2×C4×D314yes—
422C2×D4—34no—
423C2×D4—32no—
424C2×D4—64no—
425C2×D4—64no—
426C2×D4—62no—
427C2×D4—62no—
428C2×D4—62no—
429C2×D4—61no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—34no—
434C22×C4—42no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—68no—
438C22×C4—61no—
439C22×C4—64no—
440C22×C4—62no—
441C22×C4—64no—
442C22×C4—62no—
443C22×C4—62no—
444C22×C4C2×C4×D312yes—
445C22×C4#23512yes—
446C22×C4D3×D411yescharacteristic
447C22×C4D3×D411yescharacteristic
448C22×C4D3×D412yes—
449C22×C4D3×D411yescharacteristic
450C22×C4#23712yes—
451C22×C4C2×#6711yescharacteristic
452C22×C4C2×#6711yescharacteristic
453C22×C4C23×D311yescharacteristic
454Q16—616no—
455SD16—616no—
456M16—22no—
457M16—22no—
458M16—34no—
459M16—42no—
460M16—64no—
461M16—62no—
462C2×C8—22no—
463C2×C8—42no—
464C2×C8—64no—
465C2×C8—62no—
466C4⋊C4—22no—
467C4⋊C4—68no—
468C4⋊C4—62no—
469C4⋊C4—62no—
470C4⋊C4—122no—
471C4⋊C4C2×#6714yes—
472C22⋊C4—22no—
473C22⋊C4—22no—
474C22⋊C4—68no—
475C22⋊C4—62no—
476C22⋊C4—62no—
477C22⋊C4—122no—
478C42—22no—
479C42—22no—
480C42—68no—
481C42—62no—
482C42—62no—
483C42—62no—
484C42—62no—
485C22×C3—22no—
486C22×C3—22no—
487C22×C3—22no—
488C22×C3—42no—
489C22×C3—42no—
490C22×C3#38612yes—
491C22×C3#38612yes—
492C22×C3C2×C4×D411yescharacteristic
493C22×C3C2×#12011yescharacteristic
494C22×C3C2×#12111yescharacteristic
495C2×D3—24no—
496C2×D3—44no—
497C2×D3—84no—
498C3×C4—44no—
499C3×C4—44no—
500C3×C4#38612yes—
501C3×C4#38612yes—
502C3×C4C22×C22⋊C411yescharacteristic
503C3×C4C2×C4×D411yescharacteristic
504C3×C4C2×#12111yescharacteristic
505C3×C4C2×#12311yescharacteristic
506Dic3—24no—
507Dic3—44no—
508Dic3—82no—
509C23—42no—
510C23—62no—
511C23—64no—
512C23—242no—
513C23#67⋊C412yes—
514C23#88412yes—
515C23#93711yescharacteristic
516C23#93711yescharacteristic
517C23C2×D3×D411yescharacteristic
518Q8—24no—
519Q8—24no—
520Q8—34no—
521Q8—44no—
522Q8—44no—
523Q8—42no—
524Q8—42no—
525Q8—64no—
526Q8—62no—
527D4—34no—
528D4—68no—
529D4—64no—
530D4—64no—
531D4—64no—
532D4—62no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—64no—
540C2×C4—62no—
541C2×C4—68no—
542C2×C4—64no—
543C2×C4—64no—
544C2×C4—64no—
545C2×C4—82no—
546C2×C4—124no—
547C2×C4—124no—
548C2×C4—122no—
549C2×C4—248no—
550C2×C4D3×C22⋊C412yes—
551C2×C4D3×C22⋊C412yes—
552C2×C4#67⋊C412yes—
553C2×C4#88312yes—
554C2×C4#88312yes—
555C2×C4#88412yes—
556C2×C4C4×#6712yes—
557C2×C4C4×#6712yes—
558C2×C4#93711yescharacteristic
559C2×C4#93711yescharacteristic
560C2×C4#93812yes—
561C2×C4#93812yes—
562C2×C4C22×C4×D311yescharacteristic
563C2×C4C2×#23511yescharacteristic
564C2×C4C2×D3×D411yescharacteristic
565C2×C4C2×D3×D411yescharacteristic
566C2×C4C2×#23711yescharacteristic
567C2×C4C22×#6711yescharacteristic
568C8—42no—
569C8—64no—
570C2×C3—21no—
571C2×C3—22no—
572C2×C3—21no—
573C2×C3—22no—
574C2×C3C2×#38611yescharacteristic
575C2×C3#302212yes—
576D3—44no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—42no—
581C22—42no—
582C22—64no—
583C22—124no—
584C22—244no—
585C22#495612yes—
586C22#495612yes—
587C22C2×#67⋊C411yescharacteristic
588C22C2×#88411yescharacteristic
589C22C2×#93711yescharacteristic
590C4—44no—
591C4—44no—
592C4—64no—
593C4—124no—
594C4—242no—
595C4#495612yes—
596C4#495612yes—
597C4C2×D3×C22⋊C411yescharacteristic
598C4C2×#88311yescharacteristic
599C4C2×C4×#6711yescharacteristic
600C4C2×#93811yescharacteristic
601C3#2094411yescharacteristic, complemented, Sylow
602C2—21no—
603C2—22no—
604C2—22no—
605C2—21no—
606C2—124no—
607C2C2×#495611yescharacteristic
608C2#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#20944
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. #84062C2semidirect
6. #71987C2semidirect
7. #71574C2semidirect
8. #70348C2semidirect
17. C2×#942C4semidirect
32. C3×#419C22semidirect
35. #5229C22semidirect
39. #4554C22semidirect
40. #4532C22semidirect
44. C2×#419D3semidirect
57. C22×C3×Q8C2×C4semidirect
91. #942C2×C4semidirect
107. C2×#208C2×C4semidirect
148. #419C2×D3semidirect
269. C22×Q8C4×D3semidirect
601. C3#20944semidirect

Maximal subgroups

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

subgroupindex
C22×#9422
#857312
C2×C3×#4192
#840622
C2×#51392
#719872
#715742
#703482
#209443