Finite Group Databasebeta

← #10858097

Subgroups — #10858097

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
1#82748C211yesmaximal, characteristic
2#82172C211yesmaximal, characteristic
3C2×#4964C211yesmaximal, characteristic
4C2×#4799C211yesmaximal, characteristic, complemented
5C3×#1872C211yesmaximal, characteristic
6#71697C211yesmaximal, characteristic
7#70261C211yesmaximal, characteristic, complemented
8#70251C211yesmaximal, characteristic, complemented
9#70235C211yesmaximal, characteristic, complemented
10#69951C211yesmaximal, characteristic
11#69929C211yesmaximal, characteristic
12#68829C211yesmaximal, characteristic, complemented
13#68821C211yesmaximal, characteristic, complemented
14#68809C211yesmaximal, characteristic, complemented
15#68797C211yesmaximal, characteristic, complemented
16#26365—31nomaximal, Sylow
17C2×C3×C22⋊Q8C2211yescharacteristic
18C22×C3×C4⋊C4C2211yescharacteristic
19C2×Q8×Dic3C2211yescharacteristic
20C2×#926C2211yescharacteristic
21C22×#208C2211yescharacteristic
22C2×#890C2211yescharacteristic
23C2×C3×#103C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×C22⋊C8C2211yescharacteristic
26#5229C2211yescharacteristic
27C2×#835C2211yescharacteristic
28C2×#835C2211yescharacteristic
29#5212—21no—
30#5212C2211yescharacteristic
31#5210—21no—
32C4×#217C2211yescharacteristic
33C2×#818C2211yescharacteristic
34C2×#817C2211yescharacteristic
35C2×Dic3⋊C8C2211yescharacteristic
36C2×Dic3⋊C8C2211yescharacteristic
37#4986—21no—
38#4984C2211yescharacteristic
39#4984C2211yescharacteristic
40#4982—21no—
41#4980—21no—
42#4977C2211yescharacteristic
43Dic3×C4⋊C4—21no—
44Dic3×C4⋊C4C2211yescharacteristic
45#4972C2211yescharacteristic
46#4964—24no—
47C2×#808C2211yescharacteristic
48C2×#808C2211yescharacteristic
49C2×#807C2211yescharacteristic
50C2×#807C2211yescharacteristic
51#4951—21no—
52#4951—21no—
53#4799—24no—
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#479—31no—
63C2×#428—31no—
64#2442—31no—
65#2362—31no—
66#2097—31no—
67#2088—31no—
68#2066—31no—
69#2059—31no—
70#1949—31no—
71#1921—31no—
72#1894—31no—
73#1872D311yescharacteristic
74#1863—31no—
75#1854—31no—
76#1851—31no—
77C22×C3×Q8C2311yescharacteristic
78C23×C3×C4—21no—
79C23×C3×C4C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C3×C22⋊Q8—22no—
83C3×C22⋊Q8Q814yes—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—22no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—41no—
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C22⋊C4—21no—
92C2×#217—21no—
93C2×#217—21no—
94C2×#217D411yescharacteristic
95C2×#217D411yescharacteristic
96Q8×Dic3—22no—
97Q8×Dic3—41no—
98#926—22no—
99#926—41no—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×C4×Dic3—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3—21no—
113C2×C4×Dic3C2311yescharacteristic
114C2×C4×Dic3C2311yescharacteristic
115C2×#208—22no—
116C2×#208—22no—
117C2×#208—22no—
118C22×C3⋊C8C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120#890—22no—
121#890—41no—
122C3×#103—22no—
123C3×#103—41no—
124C3×C22⋊C8—22no—
125#835—22no—
126#835—22no—
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⋊#95—21no—
136C3⋊#95D411yescharacteristic
137C3⋊#95D411yescharacteristic
138#818—41no—
139#817—22no—
140Dic3⋊C8—22no—
141Dic3⋊C8—22no—
142#808—22no—
143#808—22no—
144#807—22no—
145#807D414yes—
146C22×M16—31no—
147C2×#125—31no—
148C2×C22⋊Q8C2×D311yescharacteristic
149C2×C4×Q8—31no—
150C2×#117—31no—
151C22×C4⋊C4C2×D311yescharacteristic
152#479—64no—
153#428—64no—
154C2×#107—31no—
155C2×#107—31no—
156C2×#107—31no—
157C2×#106—31no—
158C2×#106—31no—
159C2×#106—31no—
160C2×C4⋊C8—31no—
161C2×C4⋊C8—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103C2×D311yescharacteristic
165C2×#103C2×D311yescharacteristic
166C2×C22⋊C8C2×D311yescharacteristic
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#340C2×D311yescharacteristic
188#340C2×D311yescharacteristic
189#336—31no—
190#336—31no—
191C24×C3C2×D411yescharacteristic
192C2×C3×Q8—21no—
193C2×C3×Q8—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8C2×Q812yes—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—42no—
207C22×C3×C4—42no—
208C22×C3×C4—81no—
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×Q812yes—
211C22×C3×C4#4112yes—
212C22×C3×C4C2411yescharacteristic
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3—21no—
217C22×Dic3C2×D411yescharacteristic
218C22×Dic3C2×D411yescharacteristic
219C22×Dic3C2×D411yescharacteristic
220C22×Dic3C2×D411yescharacteristic
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8—21no—
227C3×C4⋊C4—24no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—42no—
233C3×C4⋊C4C2×D412yes—
234C3×C4⋊C4C2×Q812yes—
235C3×C22⋊C4—24no—
236#217—41no—
237#217—41no—
238#217—42no—
239#217—42no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—42no—
248C4⋊Dic3—42no—
249C4⋊Dic3—42no—
250C4⋊Dic3—81no—
251C4⋊Dic3C2×D412yes—
252Dic3⋊C4—42no—
253Dic3⋊C4—81no—
254C4×Dic3—21no—
255C4×Dic3—21no—
256C4×Dic3—21no—
257C4×Dic3—21no—
258C4×Dic3—42no—
259C4×Dic3—42no—
260C4×Dic3—42no—
261#208—28no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8C2×D412yes—
267C2×C3⋊C8#4112yes—
268C22×Q8C22×D311yescharacteristic
269C23×C4—21no—
270C23×C4C22×D311yescharacteristic
271C2×M16—62no—
272C2×M16—62no—
273C2×M16—62no—
274C22×C8—31no—
275C22×C8—31no—
276C22×C8C22×D311yescharacteristic
277C22×C8C22×D311yescharacteristic
278#125—62no—
279#125—121no—
280C22⋊Q8—22no—
281C22⋊Q8Dic614yes—
282C4×Q8—62no—
283C4×Q8—121no—
284#117—62no—
285#117—121no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4—31no—
292C2×C4⋊C4—31no—
293C2×C4⋊C4—41no—
294C2×C4⋊C4—61no—
295C2×C4⋊C4—61no—
296C2×C4⋊C4—61no—
297C2×C4⋊C4—61no—
298C2×C4⋊C4—61no—
299C2×C4⋊C4—61no—
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—31no—
309C2×C42—31no—
310C2×C42—61no—
311C2×C42—61no—
312C2×C42—61no—
313#107—34no—
314#107—62no—
315#107—121no—
316#106—62no—
317#106—62no—
318#106—62no—
319C4⋊C8—62no—
320C4⋊C8—62no—
321#103—22no—
322#103—41no—
323#103—62no—
324#103—62no—
325C22⋊C8—22no—
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
345C3×Q8—22no—
346C3×Q8—22no—
347C3×Q8—41no—
348C3×Q8—41no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—84no—
368C2×C3×C4C22⋊Q812yes—
369C2×C3×C4C22×D411yescharacteristic
370C2×C3×C4C22×Q811yescharacteristic
371C2×C3×C4C2×#4111yescharacteristic
372C2×C3×C4C2×#4111yescharacteristic
373C2×C3×C4H(5,𝔽2)11yescharacteristic
374C2×C3×C42−1+411yescharacteristic
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—22no—
378C2×Dic3—21no—
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—41no—
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—
399C3×C8—41no—
400C3×C8—41no—
401C3⋊C8—22no—
402C3⋊C8—22no—
403C24D3×D411yescharacteristic
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8C2×Dic612yes—
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×C4C2×Dic612yes—
430C22×C4D3×D411yescharacteristic
431C22×C4#23712yes—
432C22×C4C23×D311yescharacteristic
433M16—68no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—32no—
441C2×C8—32no—
442C2×C8—61no—
443C2×C8—61no—
444C2×C8—61no—
445C2×C8—61no—
446C4⋊C4—24no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—32no—
452C4⋊C4—42no—
453C4⋊C4—61no—
454C4⋊C4—61no—
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×Dic612yes—
467C4⋊C4D3×D412yes—
468C22⋊C4—24no—
469C22⋊C4—121no—
470C22⋊C4—121no—
471C22⋊C4—122no—
472C22⋊C4—122no—
473C42—61no—
474C42—61no—
475C42—61no—
476C42—61no—
477C42—122no—
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—42no—
498C3×C4—81no—
499C3×C4#52511yescharacteristic
500C3×C4#53911yescharacteristic
501C3×C4D4×Q811yescharacteristic
502C3×C4#55411yescharacteristic
503Dic3—41no—
504Dic3—41no—
505Dic3—41no—
506Dic3—42no—
507Dic3—41no—
508Dic3—81no—
509Dic3—81no—
510Dic3—81no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23#87812yes—
518C23C2×D3×D411yescharacteristic
519Q8—22no—
520Q8—22no—
521Q8—41no—
522Q8—41no—
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—22no—
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—61no—
544C2×C4—61no—
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—62no—
554C2×C4—84no—
555C2×C4—121no—
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#87812yes—
567C2×C4C22×Dic611yescharacteristic
568C2×C4C2×D3×D411yescharacteristic
569C2×C4C2×#23711yescharacteristic
570C2×C4#100411yescharacteristic
571C2×C4D3×#4111yescharacteristic
572C2×C4#101011yescharacteristic
573C8—41no—
574C8—41no—
575C8—62no—
576C8—62no—
577C2×C3—42no—
578C2×C3#244911yescharacteristic
579C2×C3#308011yescharacteristic
580C2×C3#308211yescharacteristic
581C2×C3#326711yescharacteristic
582C2×C3#327711yescharacteristic
583C2×C3#335011yescharacteristic
584C2×C3#335211yescharacteristic
585C22—22no—
586C22—22no—
587C22—22no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22C2×#87811yescharacteristic
593C22#559011yescharacteristic
594C22#565511yescharacteristic
595C22#576211yescharacteristic
596C22#576611yescharacteristic
597C22#577311yescharacteristic
598C22#577711yescharacteristic
599C4—22no—
600C4—42no—
601C4—42no—
602C4—42no—
603C4—81no—
604C4—121no—
605C4—122no—
606C4—121no—
607C4—121no—
608C4—121no—
609C4—241no—
610C4—241no—
611C4—241no—
612C4D4×Dic611yescharacteristic
613C4#553911yescharacteristic
614C4#560211yescharacteristic
615C4#560611yescharacteristic
616C3#2636511yescharacteristic, complemented, Sylow
617C2—42no—
618C2#7470911yescharacteristic
619C2#7877911yescharacteristic
620C2#7887111yescharacteristic
621C2#8015711yescharacteristic
622C2#8021011yescharacteristic
623C2#8143911yescharacteristic
624C2#8148011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1872C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#1872D3
hypocenterC3#26365
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1872

Complements

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

NH ≅ G/Ntype
4. C2×#4799C2semidirect
7. #70261C2semidirect
8. #70251C2semidirect
9. #70235C2semidirect
12. #68829C2semidirect
13. #68821C2semidirect
14. #68809C2semidirect
15. #68797C2semidirect
616. C3#26365semidirect

Maximal subgroups

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

subgroupindex
#827482
#821722
C2×#49642
C2×#47992
C3×#18722
#716972
#702612
#702512
#702352
#699512
#699292
#688292
#688212
#688092
#687972
#263653