Finite Group Databasebeta

← #10858050

Subgroups — #10858050

Order 768.

812 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82162C211yesmaximal, characteristic
2C2×#4964C211yesmaximal, characteristic
3C2×#4877C211yesmaximal, characteristic, complemented
4#73900C211yesmaximal, characteristic
5C3×#1852C211yesmaximal, characteristic
6#70644C211yesmaximal, characteristic, complemented
7#70618C211yesmaximal, characteristic, complemented
8#70615C211yesmaximal, characteristic, complemented
9#69951C211yesmaximal, characteristic
10#69931C211yesmaximal, characteristic
11#69516C211yesmaximal, characteristic
12#68912C211yesmaximal, characteristic, complemented
13#68906C211yesmaximal, characteristic, complemented
14#68807C211yesmaximal, characteristic, complemented
15#68788C211yesmaximal, characteristic, complemented
16#26362—31nomaximal, Sylow
17C22×C3×C4⋊C4C2211yescharacteristic
18C2×#926C2211yescharacteristic
19C2×#924C2211yescharacteristic
20C22×#208C2211yescharacteristic
21C2×#890C2211yescharacteristic
22C2×#887C2211yescharacteristic
23C2×C3×#107C2211yescharacteristic
24C2×C3×#107C2211yescharacteristic
25C2×C3×C22⋊C8C2211yescharacteristic
26C3×#380C2211yescharacteristic
27#5210—21no—
28C4×#217—21no—
29C4×#217C2211yescharacteristic
30C2×#816C2211yescharacteristic
31C2×#816C2211yescharacteristic
32C2×Dic3⋊C8C2211yescharacteristic
33C2×Dic3⋊C8C2211yescharacteristic
34#4984C2211yescharacteristic
35#4978—21no—
36#4976—21no—
37#4975C2211yescharacteristic
38Dic3×C4⋊C4C2211yescharacteristic
39#4973C2211yescharacteristic
40#4972C2211yescharacteristic
41#4972C2211yescharacteristic
42C2×#810C2211yescharacteristic
43C2×#810C2211yescharacteristic
44#4964—24no—
45C2×#808C2211yescharacteristic
46C2×#808C2211yescharacteristic
47C2×#807C2211yescharacteristic
48C2×#807C2211yescharacteristic
49#4949—21no—
50#4948—21no—
51#4877—24no—
52#4655—21no—
53#4647—21no—
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#479—31no—
63C2×#428—31no—
64#2441—31no—
65#2360—31no—
66#2095—31no—
67#2086—31no—
68#2066—31no—
69#2059—31no—
70#1947—31no—
71#1919—31no—
72#1894—31no—
73#1873—31no—
74#1863—31no—
75#1852D311yescharacteristic
76#1851—31no—
77C23×C3×C4—21no—
78C23×C3×C4C2311yescharacteristic
79C22×Dic6C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C2×C3×C4⋊C4—22no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—41no—
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C22⋊C4—21no—
90C2×#217—21no—
91C2×#217—21no—
92C2×#217D411yescharacteristic
93C2×#217D411yescharacteristic
94#926—22no—
95#926—41no—
96C2×C4⋊Dic3—21no—
97C2×C4⋊Dic3—21no—
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101#924—22no—
102#924Q814yes—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
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—
122#887—22no—
123#887—41no—
124C3×#107—22no—
125C3×#107—41no—
126C3×C22⋊C8—22no—
127C3×#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95D411yescharacteristic
135C3⋊#95D411yescharacteristic
136#816—22no—
137#816—41no—
138Dic3⋊C8—22no—
139Dic3⋊C8—22no—
140#810—22no—
141#810—22no—
142#808—22no—
143#808D414yes—
144#807—22no—
145#807—22no—
146C22×M16—31no—
147C2×#125—31no—
148C2×C22⋊Q8—31no—
149C2×C4×Q8—31no—
150C2×#117—31no—
151C22×C4⋊C4C2×D311yescharacteristic
152#479—64no—
153#428—64no—
154C2×#107—31no—
155C2×#107—31no—
156C2×#107C2×D311yescharacteristic
157C2×#107C2×D311yescharacteristic
158C2×#106—31no—
159C2×#106—31no—
160C2×C4⋊C8—31no—
161C2×C4⋊C8—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#103—31no—
166C2×C22⋊C8C2×D311yescharacteristic
167#399—61no—
168#396—61no—
169#391—31no—
170#389—31no—
171#389—31no—
172#389—31no—
173#389—61no—
174#389—61no—
175#387—61no—
176#387—61no—
177#385—61no—
178#384—31no—
179#380C2×D311yescharacteristic
180C4×C4⋊C4—31no—
181C4×C22⋊C4—31no—
182C4×C22⋊C4—61no—
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
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—
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—21no—
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×C22⋊C4—81no—
237#217—24no—
238#217—22no—
239#217—41no—
240#217—42no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—42no—
246C4⋊Dic3—81no—
247C4⋊Dic3C2×D412yes—
248C4⋊Dic3C2×Q812yes—
249Dic3⋊C4—24no—
250Dic3⋊C4—42no—
251Dic3⋊C4—42no—
252Dic3⋊C4—42no—
253Dic3⋊C4—42no—
254Dic3⋊C4—81no—
255Dic3⋊C4—81no—
256C4×Dic3—21no—
257C4×Dic3—21no—
258C4×Dic3—21no—
259C4×Dic3—21no—
260C4×Dic3—42no—
261C4×Dic3—42no—
262C4×Dic3—81no—
263#208—28no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8C2×D412yes—
269C2×C3⋊C8#4112yes—
270C22×Q8—31no—
271C23×C4—21no—
272C23×C4C22×D311yescharacteristic
273C2×M16—62no—
274C2×M16—62no—
275C2×M16—62no—
276C22×C8—31no—
277C22×C8—31no—
278C22×C8C22×D311yescharacteristic
279C22×C8C22×D311yescharacteristic
280#125—62no—
281#125—121no—
282C22⋊Q8—34no—
283C22⋊Q8—62no—
284C4×Q8—62no—
285C4×Q8—121no—
286#117—62no—
287#117—121no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
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⋊C4—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4C22×D311yescharacteristic
304C2×C4⋊C4C22×D311yescharacteristic
305C2×C4⋊C4C22×D311yescharacteristic
306C2×C4⋊C4C22×D311yescharacteristic
307C2×C22⋊C4—21no—
308C2×C22⋊C4—31no—
309C2×C22⋊C4—31no—
310C2×C22⋊C4—61no—
311C2×C22⋊C4—61no—
312C2×C42—31no—
313C2×C42—31no—
314C2×C42—61no—
315C2×C42—61no—
316C2×C42—61no—
317#107—22no—
318#107—41no—
319#107—62no—
320#107—62no—
321#106—34no—
322#106—62no—
323C4⋊C8—62no—
324C4⋊C8—62no—
325#103—62no—
326#103—62no—
327#103—62no—
328#103—121no—
329C22⋊C8—22no—
330#95—21no—
331#95—31no—
332#95—31no—
333#95—61no—
334#95—61no—
335#95—61no—
336#95—61no—
337#95—61no—
338#95—61no—
339C23×C3—21no—
340C23×C3—21no—
341C23×C3—21no—
342C23×C3—21no—
343C23×C3—21no—
344C23×C3—21no—
345C23×C3C22⋊Q812yes—
346C23×C3C22×D411yescharacteristic
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
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—41no—
364C2×C3×C4—41no—
365C2×C3×C4—84no—
366C2×C3×C4C22⋊Q812yes—
367C2×C3×C4C22×D411yescharacteristic
368C2×C3×C4C22×Q811yescharacteristic
369C2×C3×C4C2×#4111yescharacteristic
370C2×C3×C4C2×#4111yescharacteristic
371C2×C3×C4H(5,𝔽2)11yescharacteristic
372C2×C3×C42−1+411yescharacteristic
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—22no—
376C2×Dic3—22no—
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—22no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—41no—
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#13712yes—
397Dic6—22no—
398Dic6—22no—
399Dic6—41no—
400Dic6—41no—
401C3×C8—41no—
402C3×C8—41no—
403C3⋊C8—22no—
404C3⋊C8—22no—
405C24D3×D411yescharacteristic
406C2×Q8—32no—
407C2×Q8—61no—
408C2×Q8—61no—
409C2×Q8—61no—
410C2×Q8—61no—
411C2×Q8—61no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—31no—
421C22×C4—31no—
422C22×C4—31no—
423C22×C4—31no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—61no—
427C22×C4—61no—
428C22×C4—61no—
429C22×C4—61no—
430C22×C4—81no—
431C22×C4#23512yes—
432C22×C4D3×D411yescharacteristic
433C22×C4D3×Q812yes—
434C22×C4C23×D311yescharacteristic
435M16—68no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—32no—
443C2×C8—32no—
444C2×C8—61no—
445C2×C8—61no—
446C2×C8—61no—
447C2×C8—61no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—32no—
456C4⋊C4—32no—
457C4⋊C4—42no—
458C4⋊C4—64no—
459C4⋊C4—61no—
460C4⋊C4—61no—
461C4⋊C4—61no—
462C4⋊C4—61no—
463C4⋊C4—122no—
464C4⋊C4—122no—
465C4⋊C4—122no—
466C4⋊C4—122no—
467C4⋊C4—122no—
468C4⋊C4—241no—
469C4⋊C4—241no—
470C4⋊C4—241no—
471C4⋊C4D3×D412yes—
472C22⋊C4—62no—
473C22⋊C4—64no—
474C22⋊C4—81no—
475C22⋊C4—121no—
476C22⋊C4—122no—
477C42—61no—
478C42—61no—
479C42—61no—
480C42—61no—
481C42—122no—
482C42—122no—
483C42—241no—
484C22×C3—22no—
485C22×C3—22no—
486C22×C3—22no—
487C22×C3—41no—
488C22×C3—41no—
489C22×C3—41no—
490C22×C3—41no—
491C22×C3C2×C22⋊Q811yescharacteristic
492C22×C3D4211yescharacteristic
493C22×C3#54711yescharacteristic
494C22×C3C2×#13711yescharacteristic
495C22×C3#57511yescharacteristic
496C22×C3#57611yescharacteristic
497C22×C3#57811yescharacteristic
498C3×C4—22no—
499C3×C4—42no—
500C3×C4—81no—
501C3×C4—81no—
502C3×C4—81no—
503C3×C4#52511yescharacteristic
504C3×C4#53911yescharacteristic
505C3×C4D4×Q811yescharacteristic
506C3×C4#55411yescharacteristic
507Dic3—22no—
508Dic3—42no—
509Dic3—41no—
510Dic3—42no—
511Dic3—41no—
512Dic3—41no—
513Dic3—42no—
514Dic3—81no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23#89612yes—
522C23C2×D3×D411yescharacteristic
523Q8—62no—
524Q8—62no—
525Q8—121no—
526Q8—121no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—32no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—61no—
547C2×C4—61no—
548C2×C4—61no—
549C2×C4—61no—
550C2×C4—62no—
551C2×C4—62no—
552C2×C4—61no—
553C2×C4—61no—
554C2×C4—61no—
555C2×C4—61no—
556C2×C4—61no—
557C2×C4—61no—
558C2×C4—61no—
559C2×C4—62no—
560C2×C4—84no—
561C2×C4—121no—
562C2×C4—121no—
563C2×C4—121no—
564C2×C4—121no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4—121no—
569C2×C4—121no—
570C2×C4#89612yes—
571C2×C4C2×#23511yescharacteristic
572C2×C4C2×D3×D411yescharacteristic
573C2×C4#100411yescharacteristic
574C2×C4C2×D3×Q811yescharacteristic
575C2×C4D3×#4111yescharacteristic
576C2×C4#101011yescharacteristic
577C8—41no—
578C8—41no—
579C8—62no—
580C8—62no—
581C2×C3—42no—
582C2×C3#244911yescharacteristic
583C2×C3#307811yescharacteristic
584C2×C3#308311yescharacteristic
585C2×C3#326811yescharacteristic
586C2×C3#327611yescharacteristic
587C2×C3#334611yescharacteristic
588C2×C3#335211yescharacteristic
589C22—22no—
590C22—22no—
591C22—22no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22C2×#89611yescharacteristic
597C22#559011yescharacteristic
598C22#565511yescharacteristic
599C22#575711yescharacteristic
600C22#576611yescharacteristic
601C22#577311yescharacteristic
602C22D3×#13711yescharacteristic
603C4—22no—
604C4—42no—
605C4—62no—
606C4—81no—
607C4—81no—
608C4—81no—
609C4—122no—
610C4—121no—
611C4—121no—
612C4—122no—
613C4—121no—
614C4—122no—
615C4—241no—
616C4#554711yescharacteristic
617C4#556411yescharacteristic
618C4#563211yescharacteristic
619C4#563611yescharacteristic
620C3#2636211yescharacteristic, complemented, Sylow
621C2—42no—
622C2#7442111yescharacteristic
623C2#7916211yescharacteristic
624C2#7924511yescharacteristic
625C2#8001311yescharacteristic
626C2#8020611yescharacteristic
627C2#8129511yescharacteristic
628C2#8148011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1852C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#1852D3
hypocenterC3#26362
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1852

Complements

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

NH ≅ G/Ntype
3. C2×#4877C2semidirect
6. #70644C2semidirect
7. #70618C2semidirect
8. #70615C2semidirect
12. #68912C2semidirect
13. #68906C2semidirect
14. #68807C2semidirect
15. #68788C2semidirect
620. C3#26362semidirect

Maximal subgroups

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

subgroupindex
#821622
C2×#49642
C2×#48772
#739002
C3×#18522
#706442
#706182
#706152
#699512
#699312
#695162
#689122
#689062
#688072
#687882
#263623