Finite Group Databasebeta

← #10858123

Subgroups — #10858123

Order 768.

804 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#428C211yesmaximal, characteristic
2#82243C211yesmaximal, characteristic
3#82172C211yesmaximal, characteristic
4C2×#4857C211yesmaximal, characteristic, complemented
5#71995C211yesmaximal, characteristic
6#71989C211yesmaximal, characteristic
7#71576C211yesmaximal, characteristic
8#71559C211yesmaximal, characteristic
9#70645C211yesmaximal, characteristic, complemented
10#70643C211yesmaximal, characteristic, complemented
11#70622C211yesmaximal, characteristic, complemented
12#70618C211yesmaximal, characteristic, complemented
13#70588C211yesmaximal, characteristic, complemented
14#70564C211yesmaximal, characteristic, complemented
15#70557C211yesmaximal, characteristic, complemented
16#26367—31nomaximal, Sylow
17C22×C3×M16C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C22×C3×C4⋊C4C2211yescharacteristic
20C2×#924C2211yescharacteristic
21C2×#890C2211yescharacteristic
22C2×#887C2211yescharacteristic
23C3×#428—24no—
24C2×C3×#107C2211yescharacteristic
25C2×C3×#107C2211yescharacteristic
26C2×C3×#106C2211yescharacteristic
27C2×C3×#106C2211yescharacteristic
28#5212—21no—
29#5212C2211yescharacteristic
30#5210—21no—
31C4×#217C2211yescharacteristic
32C2×#830C2211yescharacteristic
33C2×#816C2211yescharacteristic
34C2×#816C2211yescharacteristic
35C2×Dic3⋊C8C2211yescharacteristic
36C2×Dic3⋊C8C2211yescharacteristic
37#4986—21no—
38#4984C2211yescharacteristic
39#4982—21no—
40#4977C2211yescharacteristic
41Dic3×C4⋊C4C2211yescharacteristic
42#4973C2211yescharacteristic
43#4972C2211yescharacteristic
44#4972C2211yescharacteristic
45C2×#810C2211yescharacteristic
46C2×#810C2211yescharacteristic
47C2×#808C2211yescharacteristic
48C2×#807C2211yescharacteristic
49#4945—21no—
50#4944—21no—
51#4933—21no—
52C4×Dic3⋊C4—21no—
53#4857—24no—
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#479—31no—
63C2×#428D311yescharacteristic
64#2442—31no—
65#2362—31no—
66#2097—31no—
67#2087—31no—
68#2068—31no—
69#2059—31no—
70#1949—31no—
71#1919—31no—
72#1894—31no—
73#1872—31no—
74#1865—31no—
75#1854—31no—
76#1851—31no—
77C23×C3×C4—21no—
78C23×C3×C4C2311yescharacteristic
79C22×Dic6C2311yescharacteristic
80C2×C3×M16—22no—
81C2×C3×M16—22no—
82C2×C3×M16—22no—
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C3×#117—22no—
86C3×#117—41no—
87C2×C3×C4⋊C4—22no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—41no—
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C22⋊C4—21no—
95C2×C3×C42—21no—
96C2×#217—21no—
97C2×#217—21no—
98C2×#217D411yescharacteristic
99C2×#217D411yescharacteristic
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105#924—22no—
106#924Q814yes—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×C4×Dic3—21no—
113C2×C4×Dic3—21no—
114C2×C4×Dic3C2311yescharacteristic
115C2×C4×Dic3C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118#890—22no—
119#890—41no—
120#887—22no—
121#887—41no—
122C3×#107—22no—
123C3×#107—22no—
124C3×#106—22no—
125C3×#106D414yes—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95D411yescharacteristic
136C3⋊#95D411yescharacteristic
137#830—22no—
138#816—22no—
139#816—22no—
140Dic3⋊C8—22no—
141Dic3⋊C8—22no—
142#810—22no—
143#810—41no—
144#808—41no—
145#807—22no—
146C22×M16C2×D311yescharacteristic
147C2×#125—31no—
148C2×C22⋊Q8—31no—
149C2×C4×Q8—31no—
150C2×#117C2×D311yescharacteristic
151C22×C4⋊C4C2×D311yescharacteristic
152#479—64no—
153#428—24no—
154C2×#107—31no—
155C2×#107C2×D311yescharacteristic
156C2×#107C2×D311yescharacteristic
157C2×#106—31no—
158C2×#106C2×D311yescharacteristic
159C2×#106C2×D311yescharacteristic
160C2×C4⋊C8—31no—
161C2×C4⋊C8—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#103—31no—
166C2×C22⋊C8—31no—
167#400—61no—
168#398—61no—
169#391—31no—
170#389—31no—
171#389—31no—
172#389—31no—
173#388—61no—
174#388—61no—
175#387—61no—
176#385—61no—
177#382—31no—
178#380—31no—
179#380—61no—
180C4×C4⋊C4—31no—
181C4×C4⋊C4—61no—
182C4×C22⋊C4—31no—
183#340—31no—
184#340—31no—
185#340—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#336—31no—
190#336—31no—
191C24×C3C2×D411yescharacteristic
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—42no—
201C22×C3×C4—42no—
202C22×C3×C4—81no—
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×Q812yes—
205C22×C3×C4#4112yes—
206C22×C3×C4C2411yescharacteristic
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3—21no—
211C22×Dic3C2×D411yescharacteristic
212C22×Dic3C2×D411yescharacteristic
213C22×Dic3C2×D411yescharacteristic
214C22×Dic3C2×D411yescharacteristic
215C2×Dic6—21no—
216C2×Dic6—21no—
217C2×Dic6—21no—
218C2×Dic6—21no—
219C2×Dic6—21no—
220C2×Dic6C2×Q812yes—
221C3×M16—28no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8C2×D412yes—
227C2×C3×C8#4112yes—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—42no—
235C3×C4⋊C4C2×D412yes—
236C3×C4⋊C4C2×D412yes—
237C3×C22⋊C4—42no—
238C3×C42—42no—
239#217—24no—
240#217—41no—
241#217—41no—
242#217—42no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—42no—
249C4⋊Dic3—81no—
250C4⋊Dic3C2×Q812yes—
251Dic3⋊C4—24no—
252Dic3⋊C4—42no—
253Dic3⋊C4—42no—
254Dic3⋊C4—42no—
255Dic3⋊C4—81no—
256C4×Dic3—21no—
257C4×Dic3—21no—
258C4×Dic3—21no—
259C4×Dic3—21no—
260C4×Dic3—42no—
261C4×Dic3—42no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C22×Q8—31no—
269C23×C4—21no—
270C23×C4C22×D311yescharacteristic
271C2×M16—22no—
272C2×M16—22no—
273C2×M16—22no—
274C22×C8—31no—
275C22×C8—31no—
276C22×C8C22×D311yescharacteristic
277C22×C8C22×D311yescharacteristic
278#125—62no—
279#125—121no—
280C22⋊Q8—34no—
281C22⋊Q8—62no—
282C4×Q8—62no—
283C4×Q8—121no—
284#117—22no—
285#117—41no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—31no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4—41no—
292C2×C4⋊C4—61no—
293C2×C4⋊C4—61no—
294C2×C4⋊C4—61no—
295C2×C4⋊C4—61no—
296C2×C4⋊C4—61no—
297C2×C4⋊C4—61no—
298C2×C4⋊C4—61no—
299C2×C4⋊C4C22×D311yescharacteristic
300C2×C4⋊C4C22×D311yescharacteristic
301C2×C4⋊C4C22×D311yescharacteristic
302C2×C4⋊C4C22×D311yescharacteristic
303C2×C22⋊C4—21no—
304C2×C22⋊C4—31no—
305C2×C22⋊C4—31no—
306C2×C22⋊C4—61no—
307C2×C22⋊C4—61no—
308C2×C42—21no—
309C2×C42—31no—
310C2×C42—31no—
311C2×C42—61no—
312C2×C42—61no—
313#107—22no—
314#107—22no—
315#107—62no—
316#106—22no—
317#106—121no—
318#106#6714yes—
319C4⋊C8—62no—
320C4⋊C8—62no—
321#103—62no—
322#103—62no—
323#103—62no—
324#103—121no—
325C22⋊C8—62no—
326#95—31no—
327#95—31no—
328#95—61no—
329#95—61no—
330#95—61no—
331#95—61no—
332#95—61no—
333#95—61no—
334#95—61no—
335#95—61no—
336#95—61no—
337C23×C3—21no—
338C23×C3—21no—
339C23×C3—21no—
340C23×C3—21no—
341C23×C3—21no—
342C23×C3—21no—
343C23×C3C22⋊Q812yes—
344C23×C3C22×D411yescharacteristic
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—84no—
364C2×C3×C4C22⋊Q812yes—
365C2×C3×C4C22×D411yescharacteristic
366C2×C3×C4C22×Q811yescharacteristic
367C2×C3×C4C2×#4111yescharacteristic
368C2×C3×C4C2×#4111yescharacteristic
369C2×C3×C4H(5,𝔽2)11yescharacteristic
370C2×C3×C42−1+411yescharacteristic
371C2×Dic3—22no—
372C2×Dic3—22no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395Dic6—22no—
396Dic6—22no—
397Dic6—41no—
398Dic6—41no—
399C3×C8—22no—
400C3×C8—22no—
401C3⋊C8—41no—
402C3⋊C8—41no—
403C24D3×D411yescharacteristic
404C2×Q8—32no—
405C2×Q8—61no—
406C2×Q8—61no—
407C2×Q8—61no—
408C2×Q8—61no—
409C2×Q8—61no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—31no—
419C22×C4—31no—
420C22×C4—31no—
421C22×C4—31no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—61no—
425C22×C4—61no—
426C22×C4—61no—
427C22×C4—61no—
428C22×C4—81no—
429C22×C4#23512yes—
430C22×C4D3×D411yescharacteristic
431C22×C4D3×Q812yes—
432C22×C4C23×D311yescharacteristic
433M16—28no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—61no—
439C2×C8—61no—
440C2×C8—61no—
441C2×C8—61no—
442C2×C8—61no—
443C2×C8—61no—
444C2×C8#23512yes—
445C2×C8C2×#6712yes—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—32no—
453C4⋊C4—42no—
454C4⋊C4—64no—
455C4⋊C4—61no—
456C4⋊C4—61no—
457C4⋊C4—61no—
458C4⋊C4—61no—
459C4⋊C4—61no—
460C4⋊C4—122no—
461C4⋊C4—122no—
462C4⋊C4—122no—
463C4⋊C4—122no—
464C4⋊C4—241no—
465C4⋊C4—241no—
466C4⋊C4C2×#6712yes—
467C4⋊C4C2×#6712yes—
468C22⋊C4—42no—
469C22⋊C4—64no—
470C22⋊C4—121no—
471C22⋊C4—121no—
472C22⋊C4—122no—
473C42—42no—
474C42—61no—
475C42—61no—
476C42—61no—
477C42—61no—
478C42—122no—
479C42—122no—
480C22×C3—22no—
481C22×C3—22no—
482C22×C3—22no—
483C22×C3—41no—
484C22×C3—41no—
485C22×C3—41no—
486C22×C3—41no—
487C22×C3C2×C22⋊Q811yescharacteristic
488C22×C3D4211yescharacteristic
489C22×C3#54711yescharacteristic
490C22×C3#57511yescharacteristic
491C22×C3#57511yescharacteristic
492C22×C3#57711yescharacteristic
493C22×C3#57811yescharacteristic
494C3×C4—22no—
495C3×C4—42no—
496C3×C4—42no—
497C3×C4—81no—
498C3×C4—81no—
499C3×C4#52511yescharacteristic
500C3×C4#53911yescharacteristic
501C3×C4D4×Q811yescharacteristic
502C3×C4#55411yescharacteristic
503Dic3—42no—
504Dic3—41no—
505Dic3—41no—
506Dic3—42no—
507Dic3—41no—
508Dic3—41no—
509Dic3—81no—
510Dic3—81no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23#89612yes—
518C23C2×D3×D411yescharacteristic
519Q8—62no—
520Q8—62no—
521Q8—121no—
522Q8—121no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—61no—
542C2×C4—61no—
543C2×C4—62no—
544C2×C4—62no—
545C2×C4—61no—
546C2×C4—61no—
547C2×C4—61no—
548C2×C4—61no—
549C2×C4—61no—
550C2×C4—61no—
551C2×C4—61no—
552C2×C4—61no—
553C2×C4—61no—
554C2×C4—61no—
555C2×C4—84no—
556C2×C4—121no—
557C2×C4—121no—
558C2×C4—121no—
559C2×C4—121no—
560C2×C4—121no—
561C2×C4—121no—
562C2×C4—121no—
563C2×C4—121no—
564C2×C4—121no—
565C2×C4—121no—
566C2×C4#89612yes—
567C2×C4C2×#23511yescharacteristic
568C2×C4C2×#23511yescharacteristic
569C2×C4#100411yescharacteristic
570C2×C4C2×D3×Q811yescharacteristic
571C2×C4#100711yescharacteristic
572C2×C4C22×#6711yescharacteristic
573C8—22no—
574C8—22no—
575C8—121no—
576C8—121no—
577C2×C3—42no—
578C2×C3#244911yescharacteristic
579C2×C3#308011yescharacteristic
580C2×C3#308211yescharacteristic
581C2×C3#326911yescharacteristic
582C2×C3#327511yescharacteristic
583C2×C3#334911yescharacteristic
584C2×C3#335111yescharacteristic
585C22—22no—
586C22—22no—
587C22—22no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22#550711yescharacteristic
593C22C2×#89611yescharacteristic
594C22#577311yescharacteristic
595C22#577511yescharacteristic
596C22#577711yescharacteristic
597C22#577911yescharacteristic
598C22D4×#6711yescharacteristic
599C4—22no—
600C4—42no—
601C4—42no—
602C4—81no—
603C4—81no—
604C4—121no—
605C4—121no—
606C4—121no—
607C4—122no—
608C4—121no—
609C4—122no—
610C4—241no—
611C4—241no—
612C4#551111yescharacteristic
613C4#551511yescharacteristic
614C4#579111yescharacteristic
615C4Q8×#6711yescharacteristic
616C3#2636711yescharacteristic, complemented, Sylow
617C2—42no—
618C2#7922911yescharacteristic
619C2#7923211yescharacteristic
620C2#7924111yescharacteristic
621C2#7924611yescharacteristic
622C2#8217311yescharacteristic
623C2#8396211yescharacteristic
624C2#8398811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#428C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenterC2×#428D3
hypocenterC3#26367
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
4. C2×#4857C2semidirect
9. #70645C2semidirect
10. #70643C2semidirect
11. #70622C2semidirect
12. #70618C2semidirect
13. #70588C2semidirect
14. #70564C2semidirect
15. #70557C2semidirect
616. C3#26367semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4282
#822432
#821722
C2×#48572
#719952
#719892
#715762
#715592
#706452
#706432
#706222
#706182
#705882
#705642
#705572
#263673