Finite Group Databasebeta

← #10767950

Subgroups — #10767950

Order 768.

950 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
1C3×#3491C211yesmaximal, characteristic
2#83153C212yesmaximal, complemented
3#82245C211yesmaximal, characteristic, complemented
4#71893C211yesmaximal, characteristic, complemented
5#71891C211yesmaximal, characteristic, complemented
6#71889C212yesmaximal, complemented
7#71887C212yesmaximal, complemented
8#71846C211yesmaximal, characteristic
9#71844C211yesmaximal, characteristic
10#71755C212yesmaximal, complemented
11#71572C211yesmaximal, characteristic, complemented
12#18177—31nomaximal, Sylow
13C3×#552—22no—
14C3×#552—22no—
15C3×#551—22no—
16C3×#551—22no—
17C3×#543C2211yescharacteristic
18C3×#539—21no—
19C3×#539C2212yes—
20C3×#534C2211yescharacteristic
21C3×#529—21no—
22C3×#529C2212yes—
23C2×C3×#123—21no—
24C3×#518—21no—
25C2×C3×#117C2211yescharacteristic
26C2×C4×Dic6C2211yescharacteristic, complemented
27C4×#217—21no—
28#5063—22no—
29#5054—22no—
30#5050C2212yescomplemented
31#5044C2212yescomplemented
32#5040C2212yes—
33#5034C2212yes—
34#5030—22no—
35#5018—22no—
36#4986C2211yescharacteristic, complemented
37#4982—21no—
38#4979C2211yescharacteristic, complemented
39#4949C2211yescharacteristic
40#4948—21no—
41#4947C2211yescharacteristic
42C4×C4⋊Dic3C2211yescharacteristic, complemented
43C4×Dic3⋊C4—21no—
44#4926C2212yescomplemented
45#4538C2212yescomplemented
46#4538C2212yescomplemented
47#4537C2212yes—
48#4537C2212yes—
49#4528C2212yescomplemented
50#4525C2211yescharacteristic
51#4525C2211yescharacteristic
52#3491D311yescharacteristic
53#3117—32no—
54#2396—31no—
55#1615—32no—
56#1614—31no—
57#1613—31no—
58#1603—32no—
59#1602—31no—
60#1601—31no—
61#1509—32no—
62#1464—31no—
63C22×C3×D4—21no—
64C23×C3×C4D411yescharacteristic
65C22×Dic6C2311yescharacteristic
66C3×#126—22no—
67C3×#126—22no—
68C3×#126—22no—
69C3×#126—22no—
70C3×#125—21no—
71C3×#125—22no—
72C3×#125—22no—
73C3×#125D412yes—
74C3×#124—21no—
75C3×#124D412yes—
76C3×#123—22no—
77C3×#123—22no—
78C3×#123—22no—
79C3×#123—22no—
80C3×#123—22no—
81C3×#123—22no—
82C3×#123—22no—
83C3×C22⋊Q8—22no—
84C3×C22⋊Q8—22no—
85C3×C22⋊Q8—22no—
86C3×C22⋊Q8—21no—
87C3×C22⋊Q8—22no—
88C3×C22⋊Q8C2312yes—
89C3×#121—21no—
90C3×#121C2312yes—
91C3×#120—22no—
92C3×C4×Q8—21no—
93C3×C4×Q8D412yes—
94C3×C4×D4—22no—
95C3×C4×D4—22no—
96C3×C4×D4—21no—
97C3×C4×D4—22no—
98C3×C4×D4—22no—
99C3×C4×D4D412yes—
100C3×#117—22no—
101C3×#117—22no—
102C3×#117—22no—
103C3×#117—22no—
104C3×#117—22no—
105C2×C3×C4⋊C4—21no—
106C2×C3×C4⋊C4—21no—
107C2×C3×C4⋊C4D411yescharacteristic
108C2×C3×C4⋊C4C2311yescharacteristic
109C2×C3×C22⋊C4—22no—
110C2×C3×C22⋊C4—21no—
111C2×C3×C22⋊C4—21no—
112C2×C3×C22⋊C4—21no—
113C2×C3×C22⋊C4D411yescharacteristic
114C2×C3×C22⋊C4C2311yescharacteristic
115C2×C3×C42D411yescharacteristic
116C2×C3×C42C2311yescharacteristic
117C2×#217—21no—
118C2×#217—21no—
119C2×#216—22no—
120C2×#214—22no—
121C2×C4⋊Dic3—21no—
122C2×C4⋊Dic3C2311yescharacteristic
123C2×Dic3⋊C4—21no—
124C2×Dic3⋊C4—21no—
125C2×Dic3⋊C4—21no—
126C2×C4×Dic3—21no—
127C2×C4×Dic3—21no—
128C4×Dic6—22no—
129C4×Dic6—22no—
130#835—22no—
131#832—22no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95C2311yescharacteristic
137C3⋊#95C2311yescharacteristic
138#830C2312yes—
139#830C2312yes—
140#810—22no—
141#810—22no—
142#808—22no—
143#807—22no—
144C4⋊(C3⋊C8)—22no—
145#803—22no—
146#552—22no—
147#552—22no—
148#551—22no—
149#551—22no—
150#543C2×D311yescharacteristic
151#539—21no—
152#539C2×D312yes—
153#534C2×D311yescharacteristic
154#529—21no—
155#529C2×D312yes—
156C2×#123—21no—
157#518—21no—
158C2×C4×Q8—31no—
159C2×#117C2×D311yescharacteristic
160#484—32no—
161#481—32no—
162#479—62no—
163#464—62no—
164#451—32no—
165#448—32no—
166#441—62no—
167#440—62no—
168#432—32no—
169#400—31no—
170#399—31no—
171#398—61no—
172#396—61no—
173#395—31no—
174#393—31no—
175C4×C4⋊C4—31no—
176C4×C4⋊C4—61no—
177C4×C22⋊C4—61no—
178#328—32no—
179#328—32no—
180#327—32no—
181#327—32no—
182#324—32no—
183#323—31no—
184#323—31no—
185C24×C3—21no—
186C24×C3C2×D411yescharacteristic
187C2×C3×Q8—21no—
188C2×C3×Q8C2×D412yes—
189C2×C3×D4—22no—
190C2×C3×D4—22no—
191C2×C3×D4—21no—
192C2×C3×D4—42no—
193C2×C3×D4—42no—
194C2×C3×D4C2×D412yes—
195C22×C3×C4—21no—
196C22×C3×C4—22no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—22no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—22no—
203C22×C3×C4—41no—
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2411yescharacteristic
210C22×Dic3—21no—
211C22×Dic3—21no—
212C22×Dic3—21no—
213C22×Dic3—21no—
214C2×Dic6—22no—
215C2×Dic6—42no—
216C3×C4⋊C4—22no—
217C3×C4⋊C4—22no—
218C3×C4⋊C4—22no—
219C3×C4⋊C4—22no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—22no—
223C3×C4⋊C4—22no—
224C3×C4⋊C4—22no—
225C3×C4⋊C4—22no—
226C3×C4⋊C4—22no—
227C3×C4⋊C4—22no—
228C3×C4⋊C4C2×D412yes—
229C3×C4⋊C4C2×D412yes—
230C3×C22⋊C4—22no—
231C3×C22⋊C4—22no—
232C3×C22⋊C4—22no—
233C3×C22⋊C4—22no—
234C3×C22⋊C4—22no—
235C3×C22⋊C4—22no—
236C3×C22⋊C4—22no—
237C3×C22⋊C4—22no—
238C3×C22⋊C4—22no—
239C3×C22⋊C4—22no—
240C3×C22⋊C4—22no—
241C3×C22⋊C4—22no—
242C3×C42—22no—
243C3×C42—21no—
244C3×C42—21no—
245C3×C42—22no—
246C3×C42C2×D412yes—
247C3×C42C2×D412yes—
248#217—42no—
249#217—42no—
250#216—44no—
251#214—44no—
252C4⋊Dic3—22no—
253C4⋊Dic3—42no—
254Dic3⋊C4—42no—
255Dic3⋊C4—42no—
256Dic3⋊C4—42no—
257C4×Dic3—42no—
258C4×Dic3—41no—
259C4×Dic3—41no—
260C2×C3⋊C8—22no—
261C2×C3⋊C8—22no—
262C22×Q8—31no—
263C22×D4—21no—
264C23×C4#6711yescharacteristic
265C2×Q16—62no—
266C2×SD16—62no—
267#126—22no—
268#126—22no—
269#126—22no—
270#126—22no—
271#125—21no—
272#125—22no—
273#125—22no—
274#125#6712yes—
275#124—21no—
276#124#6712yes—
277#123—22no—
278#123—22no—
279#123—22no—
280#123—22no—
281#123—22no—
282#123—22no—
283#123—22no—
284C22⋊Q8—22no—
285C22⋊Q8—22no—
286C22⋊Q8—22no—
287C22⋊Q8—21no—
288C22⋊Q8—22no—
289C22⋊Q8C22×D312yes—
290#121—21no—
291#121C22×D312yes—
292#120—22no—
293C4×Q8—21no—
294C4×Q8—62no—
295C4×Q8—62no—
296C4×Q8#6712yes—
297C4×D4—22no—
298C4×D4—22no—
299C4×D4—21no—
300C4×D4—22no—
301C4×D4—22no—
302C4×D4#6712yes—
303#117—22no—
304#117—22no—
305#117—22no—
306#117—22no—
307#117—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4#6711yescharacteristic
316C2×C4⋊C4C22×D311yescharacteristic
317C2×C22⋊C4—22no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—61no—
322C2×C22⋊C4—61no—
323C2×C22⋊C4#6711yescharacteristic
324C2×C22⋊C4C22×D311yescharacteristic
325C2×C42—61no—
326C2×C42—61no—
327C2×C42#6711yescharacteristic
328C2×C42C22×D311yescharacteristic
329#107—62no—
330#106—62no—
331C4⋊C8—62no—
332#103—62no—
333#103—62no—
334#103—62no—
335#102—62no—
336C22⋊C8—32no—
337C22⋊C8—32no—
338#97—62no—
339#95—31no—
340#95—31no—
341#95—61no—
342#95—61no—
343#95—61no—
344#95—61no—
345C23×C3—21no—
346C23×C3—21no—
347C23×C3—22no—
348C23×C3—41no—
349C23×C3—41no—
350C23×C3—42no—
351C23×C3—81no—
352C23×C3—81no—
353C23×C3#13712yes—
354C23×C3C22×D411yescharacteristic
355C3×Q8—42no—
356C3×Q8—81no—
357C3×D4—42no—
358C3×D4—44no—
359C3×D4—42no—
360C3×D4—42no—
361C3×D4—81no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—22no—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—42no—
375C2×C3×C4—42no—
376C2×C3×C4—42no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—82no—
380C2×C3×C4—81no—
381C2×C3×C4—82no—
382C2×C3×C4—81no—
383C2×C3×C4#12011yescharacteristic
384C2×C3×C4#12011yescharacteristic
385C2×C3×C4#12012yes—
386C2×C3×C4C22×D412yes—
387C2×Dic3—22no—
388C2×Dic3—22no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399Dic6—24no—
400Dic6—42no—
401C3⋊C8—42no—
402C3⋊C8—42no—
403C24—21no—
404C24C2×#6711yescharacteristic
405C2×Q8—21no—
406C2×Q8—62no—
407C2×Q8—122no—
408C2×Q8C2×#6712yes—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—21no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4C2×#6712yes—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—41no—
424C22×C4—61no—
425C22×C4—61no—
426C22×C4—61no—
427C22×C4—61no—
428C22×C4C2×#6711yescharacteristic
429C22×C4C2×#6711yescharacteristic
430C22×C4C2×#6711yescharacteristic
431C22×C4C2×#6711yescharacteristic
432C22×C4C2×#6711yescharacteristic
433C22×C4C23×D311yescharacteristic
434Q16—124no—
435SD16—124no—
436C2×C8—62no—
437C2×C8—62no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—62no—
451C4⋊C4—122no—
452C4⋊C4—122no—
453C4⋊C4—122no—
454C4⋊C4—122no—
455C4⋊C4C2×#6712yes—
456C4⋊C4C2×#6712yes—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—122no—
470C22⋊C4—122no—
471C42—22no—
472C42—21no—
473C42—21no—
474C42—22no—
475C42—122no—
476C42—121no—
477C42—121no—
478C42C2×#6712yes—
479C42C2×#6712yes—
480C22×C3—21no—
481C22×C3—21no—
482C22×C3—22no—
483C22×C3—42no—
484C22×C3—41no—
485C22×C3—41no—
486C22×C3—42no—
487C22×C3—42no—
488C22×C3—42no—
489C22×C3—81no—
490C22×C3—162no—
491C22×C3C2×#12011yescharacteristic
492C22×C3C2×#13711yescharacteristic
493C22×C3#57511yescharacteristic
494C3×C4—21no—
495C3×C4—21no—
496C3×C4—22no—
497C3×C4—22no—
498C3×C4—41no—
499C3×C4—42no—
500C3×C4—41no—
501C3×C4—42no—
502C3×C4—41no—
503C3×C4—42no—
504Dic3—42no—
505Dic3—42no—
506Dic3—81no—
507Dic3—81no—
508C23—21no—
509C23—21no—
510C23—22no—
511C23—41no—
512C23—41no—
513C23—42no—
514C23—81no—
515C23—81no—
516C23#95112yes—
517C23C22×#6711yescharacteristic
518Q8—42no—
519Q8—64no—
520Q8—81no—
521Q8—122no—
522D4—42no—
523D4—44no—
524D4—42no—
525D4—42no—
526D4—81no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—62no—
545C2×C4—62no—
546C2×C4—82no—
547C2×C4—81no—
548C2×C4—82no—
549C2×C4—81no—
550C2×C4—121no—
551C2×C4—121no—
552C2×C4—121no—
553C2×C4—121no—
554C2×C4—121no—
555C2×C4—121no—
556C2×C4—121no—
557C2×C4—121no—
558C2×C4—121no—
559C2×C4—121no—
560C2×C4#95311yescharacteristic
561C2×C4#95311yescharacteristic
562C2×C4#95312yes—
563C2×C4C22×#6712yes—
564C8—122no—
565C8—122no—
566C2×C3—21no—
567C2×C3—21no—
568C2×C3—42no—
569C2×C3—82no—
570C2×C3#300711yescharacteristic
571C2×C3#301811yescharacteristic
572C2×C3#302511yescharacteristic
573C22—21no—
574C22—21no—
575C22—22no—
576C22—42no—
577C22—41no—
578C22—41no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—81no—
583C22—162no—
584C22#582211yescharacteristic
585C22C2×#95111yescharacteristic
586C22C2×#95311yescharacteristic
587C4—21no—
588C4—21no—
589C4—22no—
590C4—22no—
591C4—41no—
592C4—42no—
593C4—41no—
594C4—42no—
595C4—41no—
596C4—42no—
597C4—122no—
598C4—122no—
599C4—241no—
600C4—241no—
601C3#1817711yescharacteristic, complemented, Sylow
602C2—21no—
603C2—21no—
604C2—42no—
605C2—82no—
606C2#8408411yescharacteristic
607C2#8412111yescharacteristic
608C2#8412511yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3491C2
socleC22×C3C2×#120
prime radicalC3×#534C22
hypercenter#3491D3
hypocenterC3#18177
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3491

Complements

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

NH ≅ G/Ntype
2. #83153C2semidirect
3. #82245C2semidirect
4. #71893C2semidirect
5. #71891C2semidirect
6. #71889C2semidirect
7. #71887C2semidirect
10. #71755C2semidirect
11. #71572C2semidirect
26. C2×C4×Dic6C22semidirect
30. #5050C22semidirect
31. #5044C22semidirect
36. #4986C22semidirect
38. #4979C22semidirect
42. C4×C4⋊Dic3C22semidirect
44. #4926C22semidirect
45. #4538C22semidirect
46. #4538C22semidirect
49. #4528C22semidirect
601. C3#18177semidirect

Maximal subgroups

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

subgroupindex
C3×#34912
#831532
#822452
#718932
#718912
#718892
#718872
#718462
#718442
#717552
#715722
#181773