Finite Group Databasebeta

← #10858057

Subgroups — #10858057

Order 768.

860 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2363C211yesmaximal, characteristic
2C2×#5009C211yesmaximal, characteristic, complemented
3C2×#4995C211yesmaximal, characteristic, complemented
4#73495C211yesmaximal, characteristic, complemented
5#71682C211yesmaximal, characteristic
6#71677C211yesmaximal, characteristic
7#71606C211yesmaximal, characteristic, complemented
8#71576C211yesmaximal, characteristic, complemented
9#69654C211yesmaximal, characteristic, complemented
10#69652C211yesmaximal, characteristic, complemented
11#69624C211yesmaximal, characteristic, complemented
12#69606C211yesmaximal, characteristic, complemented
13#69468C211yesmaximal, characteristic, complemented
14#69458C211yesmaximal, characteristic, complemented
15#69450C211yesmaximal, characteristic, complemented
16#26363—31nomaximal, Sylow
17C2×C3×#121C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C2×C3×#117C2211yescharacteristic
20C22×C4⋊Dic3C2211yescharacteristic
21C22×#208C2211yescharacteristic
22C2×#870C2211yescharacteristic
23C3×#390—21no—
24C3×#390C2211yescharacteristic
25C3×#389C2211yescharacteristic
26C3×#388—21no—
27C3×#386—21no—
28C3×C4×C4⋊C4C2211yescharacteristic
29C3×C4×C22⋊C4—21no—
30C3×C4×C22⋊C4C2211yescharacteristic
31C2×#832C2211yescharacteristic
32C2×#832C2211yescharacteristic
33C2×#832C2211yescharacteristic
34C2×#832C2211yescharacteristic
35#5212C2211yescharacteristic
36C2×#830C2211yescharacteristic
37#5009—24no—
38#4995—24no—
39C2×#808C2211yescharacteristic
40C2×#808C2211yescharacteristic
41C2×#808C2211yescharacteristic
42C2×#807C2211yescharacteristic
43C2×#807C2211yescharacteristic
44C2×#807C2211yescharacteristic
45#4946—21no—
46#4942—21no—
47#4936C2211yescharacteristic
48#4928C2211yescharacteristic
49#4928C2211yescharacteristic
50C2×C4⋊(C3⋊C8)C2211yescharacteristic
51C2×C4⋊(C3⋊C8)C2211yescharacteristic
52#4660—21no—
53#4657—21no—
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4524C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#478—31no—
63C2×#428—31no—
64#2442—31no—
65#2363D311yescharacteristic
66#2092—31no—
67#2088—31no—
68#2065—31no—
69#2059—31no—
70#1949—31no—
71#1918—31no—
72#1895—31no—
73#1874—31no—
74#1862—31no—
75#1854—31no—
76#1851—31no—
77C22×C3×D4—21no—
78C22×C3×D4C2311yescharacteristic
79C23×C3×C4—21no—
80C23×C3×C4C2311yescharacteristic
81C23×Dic3—21no—
82C3×#121—22no—
83C3×#121Q814yes—
84C3×C4×D4—22no—
85C3×C4×D4—41no—
86C3×#117—22no—
87C3×#117—41no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4D411yescharacteristic
98C2×C3×C22⋊C4D411yescharacteristic
99C2×C3×C42—21no—
100C2×C3×C42—21no—
101C2×C3×C42C2311yescharacteristic
102C2×C3×C42C2311yescharacteristic
103C2×#217—21no—
104C2×C4⋊Dic3—22no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3—41no—
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×#208—22no—
116C2×#208—22no—
117C2×#208—22no—
118C22×C3⋊C8C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122#870—22no—
123#870—41no—
124C3×#95—21no—
125C3×#95—21no—
126C3×#95D411yescharacteristic
127C3×#95D411yescharacteristic
128#832—22no—
129#832—22no—
130#832—22no—
131#832—41no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137#830—22no—
138#808—22no—
139#808—22no—
140#808—22no—
141#807—22no—
142#807—41no—
143#807D414yescomplemented
144C4⋊(C3⋊C8)—22no—
145C4⋊(C3⋊C8)—22no—
146C22×M16—31no—
147C2×#125—31no—
148C2×#121C2×D311yescharacteristic
149C2×C4×D4C2×D311yescharacteristic
150C2×#117C2×D311yescharacteristic
151C22×C4⋊C4—31no—
152#478—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×#102—31no—
163C2×#102—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×C22⋊C8—31no—
167#400—61no—
168#398—61no—
169#390—21no—
170#390C2×D311yescharacteristic
171#389—31no—
172#389—31no—
173#389C2×D311yescharacteristic
174#388—21no—
175#386—21no—
176#385—61no—
177#382—31no—
178#380—31no—
179#380—61no—
180C4×C4⋊C4C2×D311yescharacteristic
181C4×C22⋊C4—21no—
182C4×C22⋊C4C2×D311yescharacteristic
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×C3—21no—
192C24×C3C2×D411yescharacteristic
193C2×C3×D4—24no—
194C2×C3×D4—21no—
195C2×C3×D4—21no—
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4—81no—
200C2×C3×D4—81no—
201C2×C3×D4C2×Q812yes—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—42no—
210C22×C3×C4—42no—
211C22×C3×C4—81no—
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×Q812yes—
218C22×C3×C4#4112yes—
219C22×C3×C4C2411yescharacteristic
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3—21no—
223C22×Dic3—21no—
224C22×Dic3—42no—
225C22×Dic3—42no—
226C22×Dic3—81no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—81no—
232C3×C4⋊C4C2×D412yes—
233C3×C4⋊C4C2×Q812yes—
234C3×C22⋊C4—24no—
235C3×C22⋊C4—41no—
236C3×C22⋊C4—41no—
237C3×C22⋊C4—42no—
238C3×C22⋊C4—42no—
239C3×C22⋊C4—42no—
240C3×C22⋊C4—81no—
241C3×C42—21no—
242C3×C42—21no—
243C3×C42—21no—
244C3×C42—21no—
245C3×C42—42no—
246C3×C42—81no—
247#217—42no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—42no—
256C4⋊Dic3C2×D412yes—
257Dic3⋊C4—42no—
258Dic3⋊C4—42no—
259Dic3⋊C4—81no—
260Dic3⋊C4—81no—
261#208—28no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8C2×D412yes—
273C2×C3⋊C8#4112yes—
274C22×D4—21no—
275C22×D4C22×D311yescharacteristic
276C23×C4—21no—
277C23×C4—61no—
278C23×C4C22×D311yescharacteristic
279C2×M16—62no—
280C2×M16—62no—
281C2×M16—62no—
282C22×C8—31no—
283C22×C8—31no—
284C22×C8—31no—
285C22×C8—31no—
286#125—62no—
287#125—121no—
288#121—22no—
289#121Dic614yes—
290C4×D4—22no—
291C4×D4—41no—
292#117—22no—
293#117—41no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—31no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—62no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—121no—
306C2×C4⋊C4C22×D311yescharacteristic
307C2×C4⋊C4C22×D311yescharacteristic
308C2×C4⋊C4C22×D311yescharacteristic
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—61no—
315C2×C22⋊C4#6711yescharacteristic
316C2×C22⋊C4#6711yescharacteristic
317C2×C42—21no—
318C2×C42—21no—
319C2×C42C22×D311yescharacteristic
320C2×C42C22×D311yescharacteristic
321#107—34no—
322#107—62no—
323#107—121no—
324#106—62no—
325#106—62no—
326#106—62no—
327C4⋊C8—62no—
328C4⋊C8—62no—
329#102—62no—
330#102—62no—
331#102—62no—
332#102—121no—
333C22⋊C8—62no—
334#95—21no—
335#95—21no—
336#95—61no—
337#95—61no—
338#95—61no—
339#95—61no—
340#95—61no—
341#95#6711yescharacteristic
342#95#6711yescharacteristic
343C23×C3—21no—
344C23×C3—21no—
345C23×C3—21no—
346C23×C3—21no—
347C23×C3—21no—
348C23×C3—21no—
349C23×C3—22no—
350C23×C3—41no—
351C23×C3—41no—
352C23×C3—41no—
353C23×C3—41no—
354C23×C3—41no—
355C23×C3—41no—
356C23×C3C22⋊Q812yes—
357C23×C3C22×D411yescharacteristic
358C3×D4—22no—
359C3×D4—22no—
360C3×D4—41no—
361C3×D4—41no—
362C3×D4—84no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—22no—
366C2×C3×C4—22no—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—22no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—84no—
391C2×C3×C4C22⋊Q812yes—
392C2×C3×C4C22×D411yescharacteristic
393C2×C3×C4C22×Q811yescharacteristic
394C2×C3×C4C2×#4111yescharacteristic
395C2×C3×C4C2×#4111yescharacteristic
396C2×C3×C4H(5,𝔽2)11yescharacteristic
397C2×C3×C42−1+411yescharacteristic
398C2×Dic3—22no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—84no—
411C3⋊C8—22no—
412C3⋊C8—22no—
413C3⋊C8—41no—
414C3⋊C8—41no—
415C24—21no—
416C24C2×#6711yescharacteristic
417C2×D4—24no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—81no—
424C2×D4—81no—
425C2×D4C2×Dic612yes—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—61no—
436C22×C4—61no—
437C22×C4—61no—
438C22×C4—61no—
439C22×C4—81no—
440C22×C4—122no—
441C22×C4—122no—
442C22×C4—241no—
443C22×C4C2×Dic612yes—
444C22×C4#23512yes—
445C22×C4C2×#6711yescharacteristic
446C22×C4C2×#6711yescharacteristic
447C22×C4C2×#6711yescharacteristic
448C22×C4C2×#6711yescharacteristic
449C22×C4C2×#6711yescharacteristic
450C22×C4C23×D311yescharacteristic
451M16—68no—
452C2×C8—32no—
453C2×C8—32no—
454C2×C8—61no—
455C2×C8—61no—
456C2×C8—61no—
457C2×C8—61no—
458C2×C8—61no—
459C2×C8—61no—
460C2×C8—61no—
461C2×C8—61no—
462C2×C8—61no—
463C2×C8—61no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—32no—
469C4⋊C4—61no—
470C4⋊C4—61no—
471C4⋊C4—61no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C4⋊C4—61no—
475C4⋊C4—61no—
476C4⋊C4—81no—
477C4⋊C4—122no—
478C4⋊C4—122no—
479C4⋊C4—122no—
480C4⋊C4—241no—
481C4⋊C4—241no—
482C4⋊C4C2×Dic612yes—
483C4⋊C4D3×D412yes—
484C22⋊C4—24no—
485C22⋊C4—41no—
486C22⋊C4—41no—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C22⋊C4—42no—
490C22⋊C4—81no—
491C22⋊C4—122no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—42no—
497C42—81no—
498C22×C3—22no—
499C22×C3—22no—
500C22×C3—22no—
501C22×C3—41no—
502C22×C3—41no—
503C22×C3—41no—
504C22×C3—41no—
505C22×C3—42no—
506C22×C3—42no—
507C22×C3—42no—
508C22×C3—81no—
509C22×C3—81no—
510C22×C3—81no—
511C22×C3—81no—
512C22×C3C2×C22⋊Q811yescharacteristic
513C22×C3D4211yescharacteristic
514C22×C3#54711yescharacteristic
515C22×C3#57511yescharacteristic
516C22×C3#57511yescharacteristic
517C22×C3#57611yescharacteristic
518C22×C3#57711yescharacteristic
519C3×C4—22no—
520C3×C4—42no—
521C3×C4—41no—
522C3×C4—41no—
523C3×C4—41no—
524C3×C4—41no—
525C3×C4—42no—
526C3×C4—81no—
527C3×C4#52511yescharacteristic
528C3×C4#53911yescharacteristic
529C3×C4D4×Q811yescharacteristic
530C3×C4#55411yescharacteristic
531Dic3—42no—
532Dic3—81no—
533Dic3—81no—
534Dic3—81no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—22no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23#92412yes—
549C23C22×#6711yescharacteristic
550D4—22no—
551D4—22no—
552D4—41no—
553D4—41no—
554D4—84no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—62no—
583C2×C4—84no—
584C2×C4—121no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4—121no—
590C2×C4—121no—
591C2×C4—121no—
592C2×C4—121no—
593C2×C4—121no—
594C2×C4—121no—
595C2×C4—244no—
596C2×C4#92412yes—
597C2×C4C22×Dic611yescharacteristic
598C2×C4C2×#23511yescharacteristic
599C2×C4C2×D3×D411yescharacteristic
600C2×C4#100411yescharacteristic
601C2×C4D3×#4111yescharacteristic
602C2×C4#101011yescharacteristic
603C8—62no—
604C8—62no—
605C8—121no—
606C8—121no—
607C2×C3—42no—
608C2×C3—82no—
609C2×C3#244911yescharacteristic
610C2×C3#308011yescharacteristic
611C2×C3#308111yescharacteristic
612C2×C3#327111yescharacteristic
613C2×C3#327311yescharacteristic
614C2×C3#334811yescharacteristic
615C2×C3#335511yescharacteristic
616C22—22no—
617C22—22no—
618C22—22no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22C2×#92411yescharacteristic
631C22D4×#6711yescharacteristic
632C22#582211yescharacteristic
633C22#582211yescharacteristic
634C22#582411yescharacteristic
635C22#583511yescharacteristic
636C22#583611yescharacteristic
637C4—22no—
638C4—41no—
639C4—42no—
640C4—42no—
641C4—41no—
642C4—41no—
643C4—41no—
644C4—81no—
645C4—122no—
646C4—241no—
647C4—241no—
648C4—241no—
649C4D4×Dic611yescharacteristic
650C4#553911yescharacteristic
651C4#554711yescharacteristic
652C4#556411yescharacteristic
653C3#2636311yescharacteristic, complemented, Sylow
654C2—42no—
655C2—82no—
656C2#8236011yescharacteristic
657C2#8250911yescharacteristic
658C2#8253911yescharacteristic
659C2#8350411yescharacteristic
660C2#8358311yescharacteristic
661C2#8361311yescharacteristic
662C2#8364911yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2363C2
socleC23×C3C22×D4
prime radicalC2×C3×#121C22
hypercenter#2363D3
hypocenterC3#26363
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2363

Complements

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

NH ≅ G/Ntype
2. C2×#5009C2semidirect
3. C2×#4995C2semidirect
4. #73495C2semidirect
7. #71606C2semidirect
8. #71576C2semidirect
9. #69654C2semidirect
10. #69652C2semidirect
11. #69624C2semidirect
12. #69606C2semidirect
13. #69468C2semidirect
14. #69458C2semidirect
15. #69450C2semidirect
143. #807D4semidirect
653. C3#26363semidirect

Maximal subgroups

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

subgroupindex
C3×#23632
C2×#50092
C2×#49952
#734952
#716822
#716772
#716062
#715762
#696542
#696522
#696242
#696062
#694682
#694582
#694502
#263633