Finite Group Databasebeta

← #10868356

Subgroups — #10868356

Order 768.

848 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82343C211yesmaximal, characteristic
2#76522C211yesmaximal, characteristic, complemented
3C3×#1877C211yesmaximal, characteristic, complemented
4#72056C211yesmaximal, characteristic
5#72051C211yesmaximal, characteristic, complemented
6C4×#832C211yesmaximal, characteristic, complemented
7#70276C211yesmaximal, characteristic, complemented
8#70223C211yesmaximal, characteristic, complemented
9#70184C211yesmaximal, characteristic, complemented
10#70177C211yesmaximal, characteristic
11#69676C211yesmaximal, characteristic, complemented
12#68892C211yesmaximal, characteristic, complemented
13#68819C211yesmaximal, characteristic, complemented
14#68655C211yesmaximal, characteristic, complemented
15#68626C211yesmaximal, characteristic, complemented
16#27064—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C22×#215C2211yescharacteristic
20C2×#940C2211yescharacteristic
21C2×C4×#67—21no—
22C2×#893C2211yescharacteristic
23C2×C3×#103C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C3×#391C2211yescharacteristic
26#5229C2211yescharacteristic
27#5229C2211yescharacteristic
28C2×#835C2211yescharacteristic
29#5221C2211yescharacteristic
30C2×#832C2211yescharacteristic
31C2×#832C2211yescharacteristic
32#5213—21no—
33C2×#821C2211yescharacteristic
34C2×#821C2211yescharacteristic
35C2×#817C2211yescharacteristic
36C2×Dic3⋊C8C2211yescharacteristic
37#4984C2211yescharacteristic
38#4975C2211yescharacteristic
39Dic3×C4⋊C4C2211yescharacteristic
40C2×#809C2211yescharacteristic
41C2×#808C2211yescharacteristic
42#4957—21no—
43#4956—21no—
44#4939C2211yescharacteristic
45C4×C4⋊Dic3C2211yescharacteristic
46C4×Dic3⋊C4C2211yescharacteristic
47C2×C4×C3⋊C8C2211yescharacteristic
48#4659—21no—
49#4658—21no—
50#4655—21no—
51#4652—21no—
52C3×#340C2211yescharacteristic
53C3×#336C2211yescharacteristic
54#4551C2211yescharacteristic
55#4550C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2511—31no—
61#2366—31no—
62#2094—31no—
63#2055—31no—
64#2041—31no—
65#2014—31no—
66#1968—31no—
67#1927—31no—
68#1926—31no—
69#1920—31no—
70#1916—31no—
71#1877—31no—
72#1877D311yescharacteristic, complemented
73#1802—31no—
74C4×#102—31no—
75C22×C3×Q8C2311yescharacteristic
76C22×C3×D4C2311yescharacteristic
77C23×C3×C4—21no—
78C22×#67—21no—
79C22×#67—21no—
80C22×D12C2311yescharacteristic
81C22×C4×D3—21no—
82C2×C3×SD16—22no—
83C2×C3×SD16—41no—
84C2×C3×SD16—41no—
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C3×C4×D4—22no—
88C3×C4×D4—41no—
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C22⋊C4—21no—
92C2×C3×C42C2311yescharacteristic
93C2×#217—21no—
94C2×#217—21no—
95C2×#215—22no—
96C2×#215—22no—
97C2×#215—41no—
98#940—22no—
99#940—22no—
100C4×#67—44no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4D411yescharacteristic
111C2×Dic3⋊C4D411yescharacteristic
112C2×C4×Dic3—21no—
113C2×C4×Dic3—21no—
114C2×C4×Dic3C2311yescharacteristic
115C2×C4×Dic3C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118#893—22no—
119#893—41no—
120C3×#103—22no—
121C3×#102—41no—
122C3×#95—21no—
123C3×#95—21no—
124#835—41no—
125#832—22no—
126#832—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#821—22no—
137#821—41no—
138#817—41no—
139Dic3⋊C8—41no—
140#809—41no—
141#808—22no—
142C4×C3⋊C8—41no—
143C22×SD16—31no—
144C22×SD16C2×D311yescharacteristic
145C2×#127—31no—
146C2×C4×D4—31no—
147C2×C4×D4—61no—
148C2×C4×D4C2×D311yescharacteristic
149C2×#106—31no—
150C2×#106—31no—
151C2×C4⋊C8—31no—
152C2×#103—31no—
153C2×#103C2×D311yescharacteristic
154C2×#102—31no—
155C2×#102—31no—
156C2×#102—31no—
157C2×#102—31no—
158C2×#102—31no—
159C2×#102C2×D311yescharacteristic
160C2×C4×C8—31no—
161#400—61no—
162#400—61no—
163#391—31no—
164#391—31no—
165#391C2×D311yescharacteristic
166#390—31no—
167#390—31no—
168#390—61no—
169#389—31no—
170#387—61no—
171#387—61no—
172#386—61no—
173#386—61no—
174#384—31no—
175C4×C4⋊C4—31no—
176C4×C4⋊C4—31no—
177C4×C4⋊C4—31no—
178#340—31no—
179#340—31no—
180#340—31no—
181#340C2×D311yescharacteristic
182#336—31no—
183#336—31no—
184#336—31no—
185#336C2×D311yescharacteristic
186C24×C3—21no—
187C23×D3—21no—
188C2×C3×Q8—21no—
189C2×C3×Q8—21no—
190C2×C3×Q8—21no—
191C2×C3×Q8—21no—
192C2×C3×Q8—21no—
193C2×C3×Q8—21no—
194C2×C3×D4—21no—
195C2×C3×D4—21no—
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4#4112yes—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—42no—
204C22×C3×C4—42no—
205C22×C3×C4—81no—
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2411yescharacteristic
209C2×#67—24no—
210C2×#67—24no—
211C2×#67—42no—
212C2×#67—42no—
213C2×#67—81no—
214C2×#67—81no—
215C22×Dic3—21no—
216C22×Dic3—21no—
217C22×Dic3—21no—
218C22×Dic3C2×D411yescharacteristic
219C22×Dic3C2×D411yescharacteristic
220C22×Dic3C2×D411yescharacteristic
221C22×Dic3C2×D411yescharacteristic
222C2×D12—21no—
223C2×D12—21no—
224C2×D12—21no—
225C2×D12—21no—
226C2×D12—21no—
227C2×D12#4112yes—
228C2×C4×D3—42no—
229C2×C4×D3—42no—
230C2×C4×D3—81no—
231C3×SD16—44no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4#4112yes—
242C3×C22⋊C4—42no—
243C3×C42—21no—
244C3×C42—21no—
245#217—42no—
246#217—81no—
247#215—44no—
248(C2×D3)⋊C4—42no—
249(C2×D3)⋊C4—42no—
250(C2×D3)⋊C4—81no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—81no—
258Dic3⋊C4—41no—
259Dic3⋊C4—42no—
260Dic3⋊C4—81no—
261Dic3⋊C4SD1614yescomplemented
262C4×Dic3—21no—
263C4×Dic3—21no—
264C4×Dic3—21no—
265C4×Dic3—21no—
266C4×Dic3—42no—
267C4×Dic3—81no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8#4112yes—
274C22×Q8C22×D311yescharacteristic
275C22×D4—31no—
276C22×D4—61no—
277C22×D4—61no—
278C22×D4C22×D311yescharacteristic
279C23×C4—21no—
280C23×C4—61no—
281C2×SD16—22no—
282C2×SD16—41no—
283C2×SD16—41no—
284C2×SD16—62no—
285C2×SD16—62no—
286C2×SD16—121no—
287C22×C8—31no—
288C22×C8—31no—
289C22×C8C22×D311yescharacteristic
290C22×C8C22×D311yescharacteristic
291#127—62no—
292#127—62no—
293C4×D4—22no—
294C4×D4—41no—
295C4×D4—62no—
296C4×D4—121no—
297C4×D4—124no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4C22×D311yescharacteristic
307C2×C4⋊C4C22×D311yescharacteristic
308C2×C22⋊C4—21no—
309C2×C22⋊C4—61no—
310C2×C22⋊C4—61no—
311C2×C22⋊C4—61no—
312C2×C22⋊C4—61no—
313C2×C22⋊C4—61no—
314C2×C42—31no—
315C2×C42—31no—
316C2×C42—61no—
317C2×C42—61no—
318C2×C42C22×D311yescharacteristic
319#106—62no—
320#106—121no—
321C4⋊C8—121no—
322#103—22no—
323#103—121no—
324#102—41no—
325#102—62no—
326#102—62no—
327#102—62no—
328#102—121no—
329#102—121no—
330C4×C8—121no—
331#95—21no—
332#95—21no—
333#95—31no—
334#95—31no—
335#95—61no—
336#95—61no—
337#95—61no—
338#95—61no—
339#95—61no—
340#95—61no—
341#95—61no—
342C23×C3—22no—
343C23×C3—41no—
344C23×C3—41no—
345C23×C3—41no—
346C23×C3—41no—
347C23×C3—41no—
348C23×C3—41no—
349C23×C3C22×D411yescharacteristic
350C22×D3—22no—
351C22×D3—41no—
352C22×D3—41no—
353C22×D3—41no—
354C22×D3—41no—
355C22×D3—41no—
356C22×D3—41no—
357C3×Q8—41no—
358C3×Q8—41no—
359C3×Q8—41no—
360C3×Q8—41no—
361C3×D4—22no—
362C3×D4—22no—
363C3×D4—41no—
364C3×D4—41no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—22no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—84no—
380C2×C3×C4C2×SD1612yes—
381C2×C3×C4C2×#4111yescharacteristic
382C2×C3×C4C2×#4111yescharacteristic
383C2×C3×C4C2×#4111yescharacteristic
384C2×C3×C4C2×#4111yescharacteristic
385C2×C3×C4H(5,𝔽2)11yescharacteristic
386C2×C3×C4H(5,𝔽2)11yescharacteristic
387#67—48no—
388#67—84no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—21no—
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×Dic3C2×SD1612yes—
409C2×Dic3C2×SD1612yes—
410D12—22no—
411D12—22no—
412D12—41no—
413D12—41no—
414C4×D3—84no—
415C3×C8—41no—
416C3×C8—41no—
417C3⋊C8—22no—
418C3⋊C8—41no—
419C24—21no—
420C24—61no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—32no—
433C2×D4—64no—
434C2×D4—64no—
435C2×D4—61no—
436C2×D4—61no—
437C2×D4—61no—
438C2×D4—61no—
439C2×D4—61no—
440C2×D4—122no—
441C2×D4—122no—
442C2×D4—241no—
443C2×D4—241no—
444C2×D4#23512yes—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—31no—
449C22×C4—31no—
450C22×C4—31no—
451C22×C4—31no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—61no—
455C22×C4—61no—
456C22×C4—61no—
457C22×C4—81no—
458C22×C4—122no—
459C22×C4—122no—
460C22×C4—241no—
461C22×C4D3×D411yescharacteristic
462C22×C4D3×D411yescharacteristic
463C22×C4C23×D311yescharacteristic
464SD16—44no—
465SD16—124no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—32no—
473C2×C8—61no—
474C2×C8—61no—
475C2×C8—61no—
476C2×C8—61no—
477C2×C8—61no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—34no—
482C4⋊C4—61no—
483C4⋊C4—61no—
484C4⋊C4—61no—
485C4⋊C4—61no—
486C4⋊C4—61no—
487C4⋊C4—61no—
488C4⋊C4—121no—
489C4⋊C4—122no—
490C4⋊C4—241no—
491C4⋊C4—241no—
492C4⋊C4#23712yes—
493C22⋊C4—42no—
494C22⋊C4—122no—
495C22⋊C4—122no—
496C22⋊C4—122no—
497C22⋊C4—241no—
498C22⋊C4—241no—
499C42—21no—
500C42—21no—
501C42—61no—
502C42—61no—
503C42—61no—
504C42—61no—
505C42—122no—
506C42—241no—
507C22×C3—42no—
508C22×C3—42no—
509C22×C3—42no—
510C22×C3—81no—
511C22×C3—81no—
512C22×C3—81no—
513C22×C3—81no—
514C22×C3#52511yescharacteristic
515C22×C3#54611yescharacteristic
516C22×C3#55011yescharacteristic
517C22×C3C22×SD1611yescharacteristic
518C22×C3#57511yescharacteristic
519C22×C3#57611yescharacteristic
520C22×C3#57711yescharacteristic
521C2×D3—42no—
522C2×D3—42no—
523C2×D3—42no—
524C2×D3—81no—
525C2×D3—81no—
526C2×D3—81no—
527C2×D3—81no—
528C3×C4—22no—
529C3×C4—42no—
530C3×C4—41no—
531C3×C4—81no—
532C3×C4—81no—
533C3×C4#52911yescharacteristic
534C3×C4#53811yescharacteristic
535C3×C4#55111yescharacteristic
536C3×C4#55311yescharacteristic
537Dic3—22no—
538Dic3—22no—
539Dic3—41no—
540Dic3—41no—
541Dic3—81no—
542Dic3—81no—
543Dic3—81no—
544C23—22no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—62no—
552C23—121no—
553C23—121no—
554C23—121no—
555C23—121no—
556C23—121no—
557C23—121no—
558C23C2×D3×D411yescharacteristic
559Q8—41no—
560Q8—41no—
561Q8—41no—
562Q8—41no—
563D4—22no—
564D4—22no—
565D4—41no—
566D4—41no—
567D4—62no—
568D4—62no—
569D4—128no—
570D4—121no—
571D4—121no—
572D4—244no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—32no—
580C2×C4—32no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—61no—
590C2×C4—61no—
591C2×C4—61no—
592C2×C4—61no—
593C2×C4—61no—
594C2×C4—61no—
595C2×C4—61no—
596C2×C4—61no—
597C2×C4—61no—
598C2×C4—61no—
599C2×C4—84no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4—121no—
603C2×C4—121no—
604C2×C4—121no—
605C2×C4—121no—
606C2×C4—121no—
607C2×C4—121no—
608C2×C4—121no—
609C2×C4—244no—
610C2×C4D3×SD1612yes—
611C2×C4C2×#23511yescharacteristic
612C2×C4C2×#23711yescharacteristic
613C2×C4#100411yescharacteristic
614C2×C4#100411yescharacteristic
615C2×C4D3×#4111yescharacteristic
616C2×C4D3×#4111yescharacteristic
617C8—41no—
618C8—41no—
619C8—62no—
620C8—121no—
621C2×C3—82no—
622C2×C3#246711yescharacteristic
623C2×C3#310111yescharacteristic
624C2×C3#311811yescharacteristic
625C2×C3#329511yescharacteristic
626C2×C3#331711yescharacteristic
627C2×C3#339211yescharacteristic
628C2×C3#340311yescharacteristic
629D3—82no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—81no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22—122no—
638C22—122no—
639C22—122no—
640C22—241no—
641C22—241no—
642C22—241no—
643C22—241no—
644C22#554711yescharacteristic
645C22#559111yescharacteristic
646C22#559911yescharacteristic
647C22C2×D3×SD1611yescharacteristic
648C22#576811yescharacteristic
649C22#577411yescharacteristic
650C22#577811yescharacteristic
651C4—22no—
652C4—42no—
653C4—41no—
654C4—62no—
655C4—62no—
656C4—81no—
657C4—81no—
658C4—121no—
659C4—121no—
660C4—241no—
661C4—241no—
662C4—241no—
663C4#549411yescharacteristic
664C4#553811yescharacteristic
665C4#561411yescharacteristic
666C4#565911yescharacteristic
667C3#2706411yescharacteristic, complemented, Sylow
668C2—82no—
669C2—242no—
670C2#7472111yescharacteristic
671C2#7965111yescharacteristic
672C2#7978511yescharacteristic
673C2#8015811yescharacteristic
674C2#8025611yescharacteristic
675C2#8042011yescharacteristic
676C2#8053011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1877C2
socleC23×C3C22×D4
prime radicalC2×#940C22
hypercenter#1877D3
hypocenterC3#27064
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1877

Complements

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

NH ≅ G/Ntype
2. #76522C2semidirect
3. C3×#1877C2semidirect
5. #72051C2semidirect
6. C4×#832C2semidirect
7. #70276C2semidirect
8. #70223C2semidirect
9. #70184C2semidirect
11. #69676C2semidirect
12. #68892C2semidirect
13. #68819C2semidirect
14. #68655C2semidirect
15. #68626C2semidirect
72. #1877D3semidirect
261. Dic3⋊C4SD16semidirect
667. C3#27064semidirect

Maximal subgroups

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

subgroupindex
#823432
#765222
C3×#18772
#720562
#720512
C4×#8322
#702762
#702232
#701842
#701772
#696762
#688922
#688192
#686552
#686262
#270643