Finite Group Databasebeta

← #10801120

Subgroups — #10801120

Order 768.

1422 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#1004C211yesmaximal, characteristic
2#82328C211yesmaximal, characteristic, complemented
3C2×#4746C211yesmaximal, characteristic
4C4×#909C212yesmaximal, complemented
5C3×C4×#100C211yesmaximal, characteristic, complemented
6#70086C211yesmaximal, characteristic
7#70069C211yesmaximal, characteristic, complemented
8#70068C211yesmaximal, characteristic, complemented
9#69866C212yesmaximal, complemented
10#69004C212yesmaximal, complemented
11#69002C212yesmaximal, complemented
12#21474—31nomaximal, Sylow
13C2×#1004C2211yescharacteristic
14C2×C3×C4×D4C2211yescharacteristic
15#5798C2211yescharacteristic
16C2×#936C2211yescharacteristic
17C2×C4×#67—22no—
18C2×#909C2212yes—
19#5546—22no—
20#5545—22no—
21C4×D3×D4—22no—
22C4×#237—22no—
23#5504C2212yes—
24#5486—22no—
25C4×#235C2212yes—
26C2×C4×Dic6C2211yescharacteristic, complemented
27C2×C3×#100C2211yescharacteristic
28C2×C3×#100C2211yescharacteristic
29C3×C4×M16C2212yescomplemented
30#5221C2211yescharacteristic
31#5210—22no—
32C2×#825C2212yes—
33C2×#825C2212yes—
34#5133C2212yes—
35#5121C2212yescomplemented
36#5120C2212yescomplemented
37#4975C2211yescharacteristic, complemented
38#4973C2211yescharacteristic, complemented
39#4944—22no—
40#4932C2211yescharacteristic, complemented
41#4746C418yescomplemented
42#4716—22no—
43#4710—22no—
44C4×Dic12—22no—
45C4×#204—22no—
46C3×#343C2212yescomplemented
47#4555C2212yescomplemented
48#4555C2212yescomplemented
49C4×H(5,𝔽2)—31no—
50C2×#455—31no—
51C4×#137—32no—
52#2326—31no—
53#1885—31no—
54#1882—32no—
55#1859—32no—
56#1856—31no—
57#1792—31no—
58#1759—32no—
59C4×#100D311yescharacteristic, complemented
60C22×C3×D4C2311yescharacteristic
61C23×C3×C4D412yes—
62C22×#67—22no—
63#1004—22no—
64#1004—24no—
65#1004C2×C414yes—
66C2×#237—22no—
67C2×D3×D4—22no—
68C2×#235C2312yes—
69C22×C4×D3—22no—
70C22×Dic6C2311yescharacteristic
71C2×C3×M16C2312yes—
72C2×C3×M16C2312yes—
73C3×C4×D4—22no—
74C3×C4×D4—22no—
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C22⋊C4D412yes—
77C2×C3×C42C2311yescharacteristic
78C2×#217—22no—
79C2×#217—22no—
80C2×#217—22no—
81#936—22no—
82#936C2×C414yescomplemented
83D4×Dic3—22no—
84D4×Dic3—22no—
85C4×#67—24no—
86C4×#67—24no—
87C4×#67—22no—
88C4×#67—22no—
89C4×#67—24no—
90C2×(C2×D3)⋊C4—22no—
91#926C2312yes—
92C2×C4⋊Dic3C2311yescharacteristic
93C2×Dic3⋊C4—22no—
94C2×Dic3⋊C4—21no—
95C2×Dic3⋊C4—22no—
96C2×C4×Dic3—22no—
97C2×C4×Dic3—21no—
98C2×C4×Dic3C2311yescharacteristic
99C2×C4×Dic3C2311yescharacteristic
100#909—24no—
101#909C2×C418yescomplemented
102C2×Dic12—22no—
103C2×#204—22no—
104#893—22no—
105#892—22no—
106D3×C4⋊C4—22no—
107#887—22no—
108#67⋊C4—24no—
109#67⋊C4—24no—
110D3×C22⋊C4—24no—
111#877—24no—
112C4×D12D412yes—
113C4×D12D412yes—
114#872—22no—
115C42×D3—22no—
116C4×Dic6—22no—
117C4×Dic6—22no—
118C4×Dic6D412yes—
119C4×Dic6D412yes—
120C3×#100—22no—
121C3×#100C2×C414yescomplemented
122C3×#97—22no—
123C3×C4×C8—22no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—22no—
127#825—24no—
128#825C2×C418yescomplemented
129#821—22no—
130#818—22no—
131#817—22no—
132#816—22no—
133#816—22no—
134#816—22no—
135C2×H(5,𝔽2)—31no—
136C2×#137—32no—
137C2×#124—31no—
138#520—32no—
139#520—62no—
140#518—31no—
141#518—62no—
142#518—62no—
143C4×#41—32no—
144C4×#41—62no—
145C2×C4×Q8—31no—
146C2×C4×D4—62no—
147C2×C4×D4—62no—
148C2×C4×D4C2×D311yescharacteristic
149#455—38no—
150#441—62no—
151#440—62no—
152C4×Q16—62no—
153C4×SD16—62no—
154#428—32no—
155C2×#104—32no—
156C2×#104—32no—
157#419—32no—
158#417—32no—
159C2×#100C2×D311yescharacteristic
160C2×#100C2×D311yescharacteristic
161C4×M16C2×D312yescomplemented
162#391—31no—
163#390—31no—
164#388—62no—
165#385—62no—
166#384—31no—
167#376—31no—
168#343C2×D312yescomplemented
169#339—32no—
170#339—32no—
171C24×C3C2×D412yes—
172C23×D3—22no—
173C2×C3×D4—22no—
174C2×C3×D4—22no—
175C2×C3×D4C22×C412yes—
176C2×C3×D4#4112yes—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—82no—
180C22×C3×C4C2×D412yes—
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2411yescharacteristic
184C2×#67—24no—
185C2×#67—28no—
186C2×#67—28no—
187C2×#67—22no—
188C22×Dic3—22no—
189C22×Dic3—21no—
190C22×Dic3—21no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—21no—
194C22×Dic3—21no—
195#237—28no—
196#237—28no—
197D3×D4—28no—
198D3×D4—28no—
199#235—22no—
200#235—22no—
201#235C22×C414yes—
202#235#4114yes—
203C2×D12C2×D412yes—
204C2×C4×D3—22no—
205C2×C4×D3—22no—
206C2×C4×D3—24no—
207C2×C4×D3—22no—
208C2×C4×D3—84no—
209C2×Dic6—22no—
210C2×Dic6—22no—
211C2×Dic6—21no—
212C2×Dic6C22×C412yescomplemented
213C2×Dic6C2×D412yes—
214C3×M16—22no—
215C3×M16—22no—
216C3×M16—22no—
217C3×M16C22×C414yescomplemented
218C2×C3×C8—22no—
219C2×C3×C8—22no—
220C3×C4⋊C4—22no—
221C3×C22⋊C4—24no—
222C3×C42C2×D412yes—
223C3×C42C2×D412yes—
224#217—24no—
225#217—28no—
226#217—22no—
227#217—82no—
228(C2×D3)⋊C4—24no—
229(C2×D3)⋊C4—22no—
230(C2×D3)⋊C4—22no—
231C4⋊Dic3—22no—
232C4⋊Dic3C2×D412yes—
233C4⋊Dic3C2×D412yes—
234Dic3⋊C4—24no—
235Dic3⋊C4—22no—
236Dic3⋊C4—22no—
237Dic3⋊C4—41no—
238Dic3⋊C4—41no—
239Dic3⋊C4—82no—
240C4×Dic3—24no—
241C4×Dic3—22no—
242C4×Dic3—22no—
243C4×Dic3—22no—
244C4×Dic3—21no—
245C4×Dic3—22no—
246C4×Dic3—42no—
247C4×Dic3C22×C412yescomplemented
248Dic12—28no—
249#204—28no—
250H(5,𝔽2)—34no—
251H(5,𝔽2)—64no—
252H(5,𝔽2)—62no—
253C2×#41—32no—
254C2×#41—62no—
255C22×Q8—31no—
256C22×D4—62no—
257C22×D4—62no—
258C22×D4C22×D311yescharacteristic
259C23×C4—62no—
260C23×C4D1212yes—
261#137—38no—
262#137—64no—
263C2×Q16—62no—
264C2×SD16—62no—
265C2×M16C22×D312yes—
266C2×M16C22×D312yes—
267#124—34no—
268#124—62no—
269C4×Q8—32no—
270C4×Q8—32no—
271C4×Q8—62no—
272C4×Q8—62no—
273C4×Q8—62no—
274C4×D4—22no—
275C4×D4—22no—
276C4×D4—32no—
277C4×D4—32no—
278C4×D4—64no—
279C4×D4—64no—
280C4×D4—62no—
281C4×D4—62no—
282C4×D4—62no—
283C4×D4—64no—
284C4×D4—64no—
285C4×D4—62no—
286C4×D4—64no—
287C4×D4—62no—
288#117—32no—
289#117—64no—
290#117—62no—
291#117—62no—
292C2×C4⋊C4—31no—
293C2×C4⋊C4—62no—
294C2×C4⋊C4—62no—
295C2×C4⋊C4—61no—
296C2×C4⋊C4—62no—
297C2×C4⋊C4C22×D311yescharacteristic
298C2×C22⋊C4—64no—
299C2×C22⋊C4—62no—
300C2×C22⋊C4—62no—
301C2×C22⋊C4—62no—
302C2×C22⋊C4—62no—
303C2×C22⋊C4D1212yes—
304C2×C42—31no—
305C2×C42—31no—
306C2×C42—62no—
307C2×C42—62no—
308C2×C42—61no—
309C2×C42C22×D311yescharacteristic
310#107—62no—
311#106—62no—
312#104—38no—
313#104—64no—
314#103—62no—
315#103—62no—
316#103—62no—
317#102—62no—
318#100—22no—
319#100C4×D314yescomplemented
320#97—22no—
321C4×C8—22no—
322#95—61no—
323#95—61no—
324#95—62no—
325C23×C3—22no—
326C23×C3—22no—
327C23×C3—42no—
328C23×C3—42no—
329C23×C3C4×D414yes—
330C23×C3C22×D411yescharacteristic
331C22×D3—22no—
332C22×D3—28no—
333C22×D3—82no—
334C22×D3—82no—
335C3×D4—24no—
336C3×D4—24no—
337C2×C3×C4—22no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—44no—
341C2×C3×C4—42no—
342C2×C3×C4—82no—
343C2×C3×C4#12011yescharacteristic
344C2×C3×C4#12012yes—
345C2×C3×C4#12011yescharacteristic
346C2×C3×C4C23×C411yescharacteristic
347C2×C3×C4C22×D412yes—
348C2×C3×C4C2×#4111yescharacteristic
349C2×C3×C4C2×#4112yes—
350#67—216no—
351#67—24no—
352#67—24no—
353C2×Dic3—28no—
354C2×Dic3—22no—
355C2×Dic3—22no—
356C2×Dic3—22no—
357C2×Dic3—22no—
358C2×Dic3—22no—
359C2×Dic3—41no—
360C2×Dic3—41no—
361C2×Dic3—41no—
362C2×Dic3—41no—
363C2×Dic3—42no—
364C2×Dic3—41no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3—41no—
368C2×Dic3—82no—
369C2×Dic3—82no—
370D12—22no—
371D12C4×D414yes—
372C4×D3—24no—
373C4×D3—24no—
374C4×D3—44no—
375C4×D3—82no—
376C4×D3—84no—
377Dic6—22no—
378Dic6—24no—
379Dic6—42no—
380Dic6C4×D414yes—
381C3×C8—24no—
382C3×C8—42no—
383C24—62no—
384C24C2×D1212yes—
385#41—34no—
386#41—34no—
387#41—68no—
388#41—68no—
389#41—62no—
390#41—62no—
391C2×Q8—32no—
392C2×Q8—32no—
393C2×Q8—61no—
394C2×Q8—62no—
395C2×Q8—62no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—32no—
399C2×D4—68no—
400C2×D4—68no—
401C2×D4—64no—
402C2×D4—68no—
403C2×D4—68no—
404C2×D4—62no—
405C2×D4C2×C4×D312yes—
406C2×D4#23512yes—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—64no—
410C22×C4—62no—
411C22×C4—62no—
412C22×C4—61no—
413C22×C4—61no—
414C22×C4—62no—
415C22×C4—62no—
416C22×C4—62no—
417C22×C4—62no—
418C22×C4—61no—
419C22×C4—61no—
420C22×C4—82no—
421C22×C4—244no—
422C22×C4C2×D1212yes—
423C22×C4C2×D1211yescharacteristic
424C22×C4C2×D1211yescharacteristic
425C22×C4C23×D311yescharacteristic
426Q16—68no—
427SD16—68no—
428M16—22no—
429M16—22no—
430M16—22no—
431M16C2×C4×D314yescomplemented
432C2×C8—22no—
433C2×C8—22no—
434C4⋊C4—22no—
435C4⋊C4—32no—
436C4⋊C4—32no—
437C4⋊C4—64no—
438C4⋊C4—62no—
439C4⋊C4—62no—
440C4⋊C4—62no—
441C4⋊C4—121no—
442C4⋊C4—121no—
443C4⋊C4—242no—
444C22⋊C4—24no—
445C22⋊C4—64no—
446C22⋊C4—64no—
447C22⋊C4—68no—
448C22⋊C4—62no—
449C22⋊C4—62no—
450C22⋊C4—62no—
451C22⋊C4—242no—
452C42—32no—
453C42—64no—
454C42—62no—
455C42—62no—
456C42—61no—
457C42—62no—
458C42—62no—
459C42—122no—
460C42D3×D412yes—
461C42D3×D412yes—
462C22×C3—21no—
463C22×C3—21no—
464C22×C3—24no—
465C22×C3—44no—
466C22×C3—42no—
467C22×C3—42no—
468C22×C3—82no—
469C22×C3C2×C4×D411yescharacteristic
470C22×C3C2×#12011yescharacteristic
471C22×C3#52511yescharacteristic
472C2×D3—24no—
473C2×D3—42no—
474C2×D3—42no—
475C2×D3—88no—
476C3×C4—21no—
477C3×C4—21no—
478C3×C4—22no—
479C3×C4—42no—
480C3×C4C2×C4×D412yes—
481C3×C4#52512yes—
482Dic3—24no—
483Dic3—42no—
484Dic3—42no—
485Dic3—42no—
486Dic3—42no—
487Dic3—81no—
488Dic3—81no—
489C23—22no—
490C23—22no—
491C23—42no—
492C23—42no—
493C23—68no—
494C23—62no—
495C23—242no—
496C23—242no—
497C23C4×D1214yes—
498C23C22×D1211yescharacteristic
499Q8—34no—
500Q8—62no—
501Q8—64no—
502Q8—122no—
503D4—24no—
504D4—24no—
505D4—34no—
506D4—616no—
507D4—64no—
508D4—64no—
509D4—62no—
510C2×C4—22no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—68no—
516C2×C4—62no—
517C2×C4—64no—
518C2×C4—64no—
519C2×C4—62no—
520C2×C4—62no—
521C2×C4—62no—
522C2×C4—62no—
523C2×C4—82no—
524C2×C4—124no—
525C2×C4—121no—
526C2×C4—121no—
527C2×C4—121no—
528C2×C4—121no—
529C2×C4—121no—
530C2×C4—122no—
531C2×C4—121no—
532C2×C4—121no—
533C2×C4—121no—
534C2×C4—242no—
535C2×C4—244no—
536C2×C4—242no—
537C2×C4—242no—
538C2×C4#88211yescharacteristic
539C2×C4#88212yes—
540C2×C4#88211yescharacteristic
541C2×C4C22×C4×D311yescharacteristic
542C2×C4C2×#23511yescharacteristic
543C2×C4C2×D3×D412yes—
544C2×C4D3×#4112yes—
545C8—24no—
546C8—42no—
547C2×C3—21no—
548C2×C3—21no—
549C2×C3—44no—
550C2×C3C4×#12011yescharacteristic
551C2×C3#301911yescharacteristic
552C2×C3#302211yescharacteristic
553D3—44no—
554C22—21no—
555C22—21no—
556C22—24no—
557C22—44no—
558C22—42no—
559C22—42no—
560C22—64no—
561C22—82no—
562C22—122no—
563C22—122no—
564C22—248no—
565C22C2×C4×D1211yescharacteristic
566C22C2×#88211yescharacteristic
567C22#552411yescharacteristic
568C4—21no—
569C4—21no—
570C4—22no—
571C4—42no—
572C4—64no—
573C4—122no—
574C4—122no—
575C4—122no—
576C4—122no—
577C4—241no—
578C4—241no—
579C4C4×D3×D412yes—
580C4#554712yes—
581C3#2147411yescharacteristic, complemented, Sylow
582C2—21no—
583C2—21no—
584C2—44no—
585C2—124no—
586C2C4×#88211yescharacteristic
587C2#7870011yescharacteristic
588C2#7871811yescharacteristic
subgroupG/H
centerC22C2×#882
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×C4×#100C2
socleC22×C3C2×#120
prime radicalC2×#1004C22
cosocleC22×C3×C4C24
hypercenterC4×#100D3
hypocenterC3#21474
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#100

Complements

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

NH ≅ G/Ntype
2. #82328C2semidirect
4. C4×#909C2semidirect
5. C3×C4×#100C2semidirect
7. #70069C2semidirect
8. #70068C2semidirect
9. #69866C2semidirect
10. #69004C2semidirect
11. #69002C2semidirect
26. C2×C4×Dic6C22semidirect
29. C3×C4×M16C22semidirect
35. #5121C22semidirect
36. #5120C22semidirect
37. #4975C22semidirect
38. #4973C22semidirect
40. #4932C22semidirect
41. #4746C4semidirect
46. C3×#343C22semidirect
47. #4555C22semidirect
48. #4555C22semidirect
59. C4×#100D3semidirect
82. #936C2×C4semidirect
101. #909C2×C4semidirect
121. C3×#100C2×C4semidirect
128. #825C2×C4semidirect
161. C4×M16C2×D3semidirect
168. #343C2×D3semidirect
212. C2×Dic6C22×C4semidirect
217. C3×M16C22×C4semidirect
247. C4×Dic3C22×C4semidirect
319. #100C4×D3semidirect
431. M16C2×C4×D3semidirect
581. C3#21474semidirect

Maximal subgroups

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

subgroupindex
C4×#10042
#823282
C2×#47462
C4×#9092
C3×C4×#1002
#700862
#700692
#700682
#698662
#690042
#690022
#214743