Finite Group Databasebeta

← #10858833

Subgroups — #10858833

Order 768.

818 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82490C211yesmaximal, characteristic
2#75999C211yesmaximal, characteristic
3C3×#1863C211yesmaximal, characteristic, complemented
4#71626C211yesmaximal, characteristic, complemented
5#70632C211yesmaximal, characteristic
6#70610C211yesmaximal, characteristic, complemented
7#70270C211yesmaximal, characteristic
8#70268C211yesmaximal, characteristic
9#70262C211yesmaximal, characteristic, complemented
10#70231C211yesmaximal, characteristic, complemented
11#69623C211yesmaximal, characteristic, complemented
12D3×#340C211yesmaximal, characteristic
13#68799C211yesmaximal, characteristic, complemented
14#68666C211yesmaximal, characteristic
15#68643C211yesmaximal, characteristic, complemented
16#26414—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×#946C2211yescharacteristic
19C22×D3×C8C2211yescharacteristic
20C2×#897—21no—
21C2×D3×C4⋊C4C2211yescharacteristic
22C2×D3×C4⋊C4C2211yescharacteristic
23C2×#888C2211yescharacteristic
24C2×C3×#107C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C3×#389C2211yescharacteristic
28#5229C2211yescharacteristic
29C2×#835C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×(C2×D3)⋊C8C2211yescharacteristic
32C2×Dic3⋊C8C2211yescharacteristic
33#4989—21no—
34#4987C2211yescharacteristic
35#4984C2211yescharacteristic
36#4981—21no—
37#4976—21no—
38Dic3×C4⋊C4C2211yescharacteristic
39C2×#807C2211yescharacteristic
40C4×(C2×D3)⋊C4—21no—
41C4×(C2×D3)⋊C4C2211yescharacteristic
42#4935C2211yescharacteristic
43#4933—21no—
44#4928C2211yescharacteristic
45C2×C4⋊(C3⋊C8)C2211yescharacteristic
46#4668—21no—
47#4656—21no—
48C3×#340C2211yescharacteristic
49C3×#336C2211yescharacteristic
50#4551C2211yescharacteristic
51#4551C2211yescharacteristic
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4550C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2440—31no—
61#2360—31no—
62#2100—31no—
63#2087—31no—
64#2068—31no—
65#2024—31no—
66#1987—31no—
67#1921—31no—
68#1892—31no—
69#1872—31no—
70#1863D311yescharacteristic, complemented
71#1852—31no—
72#1850—31no—
73#1803—31no—
74C2×#340—31no—
75C22×C3×Q8C2311yescharacteristic
76C22×C4×D3—21no—
77C22×C4×D3—21no—
78C22×C4×D3C2311yescharacteristic
79C22×Dic6—21no—
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C3×C4×Q8—22no—
83C3×C4×Q8—41no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C42—21no—
90C2×C3×C42C2311yescharacteristic
91#946—22no—
92#946Q814yescomplemented
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4D411yescharacteristic
96C2×(C2×D3)⋊C4D411yescharacteristic
97C2×C4⋊Dic3—21no—
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3C2311yescharacteristic
111C22×C3⋊C8C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113C2×D3×C8—22no—
114C2×D3×C8—41no—
115C2×D3×C8—41no—
116#897—44no—
117D3×C4⋊C4—22no—
118D3×C4⋊C4—22no—
119D3×C4⋊C4—41no—
120D3×C4⋊C4—41no—
121#888—22no—
122#888—41no—
123C3×#107—22no—
124C3×#103—22no—
125C3×#103—41no—
126C3×#95—21no—
127#835—41no—
128#835—41no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95D411yescharacteristic
136C3⋊#95D411yescharacteristic
137(C2×D3)⋊C8—22no—
138Dic3⋊C8—41no—
139#807—41no—
140C4⋊(C3⋊C8)—41no—
141C23×C8—31no—
142C2×C4⋊Q8—31no—
143C2×C22⋊Q8—31no—
144C2×C22⋊Q8—61no—
145C2×C4×Q8C2×D311yescharacteristic
146C22×C4⋊C4—31no—
147C22×C4⋊C4—31no—
148C2×#107—31no—
149C2×#107C2×D311yescharacteristic
150C2×C4⋊C8—31no—
151C2×C4⋊C8—31no—
152C2×#103—31no—
153C2×#103—31no—
154C2×#103C2×D311yescharacteristic
155C2×#103C2×D311yescharacteristic
156C2×C22⋊C8—31no—
157#398—61no—
158#395—61no—
159#391—31no—
160#389—31no—
161#389—31no—
162#389—61no—
163#389C2×D311yescharacteristic
164#387—61no—
165#385—61no—
166#385—61no—
167#384—31no—
168#380—31no—
169C4×C4⋊C4—31no—
170C4×C22⋊C4—31no—
171C4×C22⋊C4—61no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#340—31no—
178#340—31no—
179#340—31no—
180#340—31no—
181#340C2×D311yescharacteristic
182#336—31no—
183#336C2×D311yescharacteristic
184C23×D3C2×D411yescharacteristic
185C2×C3×Q8—21no—
186C2×C3×Q8—21no—
187C2×C3×Q8—21no—
188C2×C3×Q8—21no—
189C2×C3×Q8—21no—
190C2×C3×Q8C2×Q812yes—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2411yescharacteristic
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3C2×D411yescharacteristic
203C22×Dic3C2×D411yescharacteristic
204C22×Dic3C2×D411yescharacteristic
205C2×C4×D3—21no—
206C2×C4×D3—21no—
207C2×C4×D3—21no—
208C2×C4×D3—21no—
209C2×C4×D3—42no—
210C2×C4×D3—42no—
211C2×C4×D3—42no—
212C2×C4×D3—42no—
213C2×C4×D3—81no—
214C2×C4×D3—81no—
215C2×C4×D3C2×Q812yes—
216C2×C4×D3#4112yes—
217C2×Dic6—42no—
218C2×Dic6—42no—
219C2×Dic6—81no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—42no—
231C3×C4⋊C4—42no—
232C3×C4⋊C4#4112yes—
233C3×C4⋊C4#4112yes—
234C3×C42—21no—
235C3×C42—21no—
236C3×C42—42no—
237(C2×D3)⋊C4—24no—
238(C2×D3)⋊C4—22no—
239(C2×D3)⋊C4—42no—
240(C2×D3)⋊C4Q1614yes—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—42no—
249C4⋊Dic3—42no—
250C4⋊Dic3—81no—
251C4⋊Dic3C2×Q812yes—
252Dic3⋊C4—24no—
253Dic3⋊C4—42no—
254Dic3⋊C4—42no—
255Dic3⋊C4—42no—
256Dic3⋊C4—81no—
257C4×Dic3—21no—
258C4×Dic3—21no—
259C4×Dic3—81no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266D3×C8—44no—
267C22×Q8—61no—
268C22×Q8C22×D311yescharacteristic
269C23×C4—31no—
270C23×C4—61no—
271C23×C4—61no—
272C22×C8—31no—
273C22×C8—31no—
274C22×C8—62no—
275C22×C8—121no—
276C22×C8—121no—
277C22×C8C22×D311yescharacteristic
278C22×C8C22×D311yescharacteristic
279C4⋊Q8—62no—
280C4⋊Q8—121no—
281C22⋊Q8—34no—
282C22⋊Q8—62no—
283C22⋊Q8—124no—
284C4×Q8—22no—
285C4×Q8—41no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—31no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4—31no—
292C2×C4⋊C4—61no—
293C2×C4⋊C4—62no—
294C2×C4⋊C4—62no—
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—121no—
303C2×C4⋊C4—121no—
304C2×C4⋊C4C22×D311yescharacteristic
305C2×C4⋊C4C22×D311yescharacteristic
306C2×C4⋊C4C22×D311yescharacteristic
307C2×C22⋊C4—31no—
308C2×C22⋊C4—31no—
309C2×C22⋊C4—61no—
310C2×C22⋊C4—61no—
311C2×C42—21no—
312C2×C42—31no—
313C2×C42—61no—
314C2×C42C22×D311yescharacteristic
315#107—22no—
316#107—121no—
317C4⋊C8—121no—
318C4⋊C8—121no—
319#103—22no—
320#103—41no—
321#103—121no—
322#103—121no—
323C22⋊C8—62no—
324#95—21no—
325#95—31no—
326#95—31no—
327#95—61no—
328#95—61no—
329#95—61no—
330#95—61no—
331#95—61no—
332#95—61no—
333C23×C3C22×D411yescharacteristic
334C22×D3—21no—
335C22×D3—21no—
336C22×D3—21no—
337C22×D3—21no—
338C22×D3C22⋊Q812yes—
339C22×D3C2×Q1612yes—
340C22×D3AGL(1,ℤ/8)12yes—
341C3×Q8—22no—
342C3×Q8—22no—
343C3×Q8—41no—
344C3×Q8—41no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
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×C4C2×Q1612yes—
363C2×C3×C4C22×Q811yescharacteristic
364C2×C3×C4C2×#4111yescharacteristic
365C2×C3×C4C2×#4111yescharacteristic
366C2×C3×C4C2×#4111yescharacteristic
367C2×C3×C4H(5,𝔽2)11yescharacteristic
368C2×C3×C4H(5,𝔽2)11yescharacteristic
369C2×Dic3—21no—
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—22no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3C22⋊Q812yes—
389C2×Dic3C2×Q1612yes—
390C4×D3—22no—
391C4×D3—22no—
392C4×D3—22no—
393C4×D3—22no—
394C4×D3—84no—
395C4×D3—84no—
396Dic6—84no—
397C3×C8—41no—
398C3×C8—41no—
399C3⋊C8—41no—
400C3⋊C8—41no—
401C24—31no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—122no—
408C2×Q8—122no—
409C2×Q8—241no—
410C2×Q8D3×Q812yes—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—31no—
416C22×C4—31no—
417C22×C4—32no—
418C22×C4—32no—
419C22×C4—31no—
420C22×C4—61no—
421C22×C4—61no—
422C22×C4—61no—
423C22×C4—61no—
424C22×C4—61no—
425C22×C4—61no—
426C22×C4—61no—
427C22×C4—61no—
428C22×C4—122no—
429C22×C4—122no—
430C22×C4—122no—
431C22×C4—122no—
432C22×C4—241no—
433C22×C4—241no—
434C22×C4D3×D411yescharacteristic
435C22×C4D3×D411yescharacteristic
436C22×C4C23×D311yescharacteristic
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—61no—
444C2×C8—61no—
445C2×C8—61no—
446C2×C8—61no—
447C2×C8—61no—
448C2×C8—61no—
449C2×C8—124no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—32no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—64no—
458C4⋊C4—61no—
459C4⋊C4—61no—
460C4⋊C4—61no—
461C4⋊C4—61no—
462C4⋊C4—61no—
463C4⋊C4—61no—
464C4⋊C4—61no—
465C4⋊C4—122no—
466C4⋊C4—122no—
467C4⋊C4—122no—
468C4⋊C4—122no—
469C4⋊C4—122no—
470C4⋊C4—241no—
471C4⋊C4—241no—
472C4⋊C4#23512yes—
473C4⋊C4#23712yes—
474C22⋊C4—34no—
475C22⋊C4—62no—
476C22⋊C4—64no—
477C22⋊C4—122no—
478C42—21no—
479C42—21no—
480C42—42no—
481C42—61no—
482C42—61no—
483C42—241no—
484C22×C3C2×C22⋊Q811yescharacteristic
485C22×C3#54611yescharacteristic
486C22×C3#54611yescharacteristic
487C22×C3C22×Q1611yescharacteristic
488C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
489C22×C3#57711yescharacteristic
490C22×C3#57711yescharacteristic
491C2×D3—22no—
492C2×D3—22no—
493C2×D3—22no—
494C2×D3—22no—
495C2×D3—22no—
496C2×D3—22no—
497C2×D3#47714yes—
498C3×C4—22no—
499C3×C4—42no—
500C3×C4—42no—
501C3×C4—41no—
502C3×C4—42no—
503C3×C4—81no—
504C3×C4#52911yescharacteristic
505C3×C4#53811yescharacteristic
506C3×C4#55411yescharacteristic
507C3×C4#55411yescharacteristic
508Dic3—22no—
509Dic3—22no—
510Dic3—42no—
511Dic3—41no—
512Dic3—81no—
513Dic3—81no—
514Dic3—81no—
515C23—32no—
516C23—32no—
517C23—32no—
518C23—61no—
519C23—61no—
520C23—61no—
521C23—61no—
522C23C2×D3×D411yescharacteristic
523Q8—22no—
524Q8—22no—
525Q8—41no—
526Q8—41no—
527Q8—244no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—32no—
537C2×C4—32no—
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—41no—
547C2×C4—61no—
548C2×C4—61no—
549C2×C4—61no—
550C2×C4—61no—
551C2×C4—62no—
552C2×C4—62no—
553C2×C4—62no—
554C2×C4—62no—
555C2×C4—61no—
556C2×C4—61no—
557C2×C4—61no—
558C2×C4—62no—
559C2×C4—121no—
560C2×C4—121no—
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—244no—
571C2×C4—244no—
572C2×C4D3×Q1612yes—
573C2×C4C2×#23511yescharacteristic
574C2×C4C2×#23711yescharacteristic
575C2×C4#100411yescharacteristic
576C2×C4C2×D3×Q811yescharacteristic
577C2×C4D3×#4111yescharacteristic
578C2×C4#100911yescharacteristic
579C8—41no—
580C8—41no—
581C8—121no—
582C8—121no—
583C2×C3#245011yescharacteristic
584C2×C3C2×#47711yescharacteristic
585C2×C3#308411yescharacteristic
586C2×C3#329911yescharacteristic
587C2×C3#329911yescharacteristic
588C2×C3#330611yescharacteristic
589C2×C3#331011yescharacteristic
590D3—24no—
591C22—34no—
592C22—62no—
593C22—62no—
594C22—62no—
595C22—62no—
596C22—62no—
597C22—62no—
598C22#555011yescharacteristic
599C22#559111yescharacteristic
600C22D3×C22⋊Q811yescharacteristic
601C22C2×D3×Q1611yescharacteristic
602C22#577011yescharacteristic
603C22D3×AGL(1,ℤ/8)11yescharacteristic
604C22#577811yescharacteristic
605C4—22no—
606C4—42no—
607C4—42no—
608C4—41no—
609C4—42no—
610C4—62no—
611C4—62no—
612C4—81no—
613C4—122no—
614C4—121no—
615C4—241no—
616C4—241no—
617C4—241no—
618C4#551111yescharacteristic
619C4#555711yescharacteristic
620C4#563511yescharacteristic
621C4#565711yescharacteristic
622C3#2641411yescharacteristic, complemented, Sylow
623C2—64no—
624C2#7461011yescharacteristic
625C2#7973511yescharacteristic
626C2#7980011yescharacteristic
627C2#8010611yescharacteristic
628C2#8019311yescharacteristic
629C2D3×#47711yescharacteristic
630C2#8123511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1863C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#1863D3
hypocenterC3#26414
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1863

Complements

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

NH ≅ G/Ntype
3. C3×#1863C2semidirect
4. #71626C2semidirect
6. #70610C2semidirect
9. #70262C2semidirect
10. #70231C2semidirect
11. #69623C2semidirect
13. #68799C2semidirect
15. #68643C2semidirect
70. #1863D3semidirect
92. #946Q8semidirect
622. C3#26414semidirect

Maximal subgroups

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

subgroupindex
#824902
#759992
C3×#18632
#716262
#706322
#706102
#702702
#702682
#702622
#702312
#696232
D3×#3402
#687992
#686662
#686432
#264143