Finite Group Databasebeta

← #10863282

Subgroups — #10863282

Order 768.

826 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#76205C211yesmaximal, characteristic, complemented
2#73788C211yesmaximal, characteristic
3C3×#1920C211yesmaximal, characteristic, complemented
4#70582C211yesmaximal, characteristic, complemented
5#70569C211yesmaximal, characteristic, complemented
6#69615C211yesmaximal, characteristic, complemented
7C4×#810C211yesmaximal, characteristic, complemented
8#68910C211yesmaximal, characteristic, complemented
9#68887C211yesmaximal, characteristic
10#68880C211yesmaximal, characteristic
11#68872C211yesmaximal, characteristic, complemented
12#68863C211yesmaximal, characteristic, complemented
13#68800C211yesmaximal, characteristic, complemented
14#68686C211yesmaximal, characteristic, complemented
15#68657C211yesmaximal, characteristic, complemented
16#26724—31nomaximal, Sylow
17C22×#204C2211yescharacteristic
18C2×#897C2211yescharacteristic
19C2×#895C2211yescharacteristic
20C2×D3×C4⋊C4C2211yescharacteristic
21C2×#888C2211yescharacteristic
22C2×#875—21no—
23C2×C4×Dic6C2211yescharacteristic
24C2×C3×#106C2211yescharacteristic
25C3×#389C2211yescharacteristic
26C3×C4×C4⋊C4C2211yescharacteristic
27C2×#821C2211yescharacteristic
28C2×(C2×D3)⋊C8C2211yescharacteristic
29C2×(C2×D3)⋊C8C2211yescharacteristic
30C2×#816C2211yescharacteristic
31C2×Dic3⋊C8C2211yescharacteristic
32#4989—21no—
33#4988C2211yescharacteristic
34#4987C2211yescharacteristic
35Dic3×C4⋊C4C2211yescharacteristic
36#4973C2211yescharacteristic
37#4972—21no—
38C2×#810C2211yescharacteristic
39C2×#810C2211yescharacteristic
40C2×#809C2211yescharacteristic
41C2×#809C2211yescharacteristic
42#4940—21no—
43#4939C2211yescharacteristic
44C4×(C2×D3)⋊C4C2211yescharacteristic
45C2×C4×C3⋊C8C2211yescharacteristic
46#4673—21no—
47#4672—21no—
48#4666—21no—
49#4654—21no—
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52C3×#340C2211yescharacteristic
53C3×#336C2211yescharacteristic
54#4551C2211yescharacteristic
55#4550C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2491—31no—
61#2360—31no—
62#2055—31no—
63#2035—31no—
64#2024—31no—
65#2000—31no—
66#1988—31no—
67#1934—31no—
68#1926—31no—
69#1923—31no—
70#1920D311yescharacteristic, complemented
71#1880—31no—
72#1874—31no—
73#1824—31no—
74C4×#103—31no—
75C22×D12—21no—
76C22×D12C2311yescharacteristic
77C22×C4×D3—21no—
78C22×C4×D3C2311yescharacteristic
79C22×Dic6—21no—
80C22×Dic6C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C42—21no—
89C2×C3×C42C2311yescharacteristic
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4D411yescharacteristic
98C2×(C2×D3)⋊C4D411yescharacteristic
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×C4×Dic3—21no—
106C2×C4×Dic3—21no—
107C2×C4×Dic3C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110C2×#204—22no—
111C2×#204—41no—
112C2×#204—41no—
113#897—22no—
114#897—22no—
115#895—22no—
116#895—41no—
117D3×C4⋊C4—22no—
118D3×C4⋊C4—41no—
119#888—22no—
120#888—41no—
121#875—44no—
122C4×Dic6—22no—
123C4×Dic6—41no—
124C3×#106—22no—
125C3×#95—21no—
126C3×#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95D411yescharacteristic
132C3⋊#95D411yescharacteristic
133#821—22no—
134(C2×D3)⋊C8—41no—
135(C2×D3)⋊C8—41no—
136#816—41no—
137Dic3⋊C8—41no—
138#810—22no—
139#810—22no—
140#809—22no—
141#809—41no—
142C4×C3⋊C8—41no—
143C22×SD16—31no—
144C2×C4⋊Q8—31no—
145C2×#124—61no—
146C2×C22⋊Q8—31no—
147C2×#121—31no—
148C2×C4×Q8—31no—
149C22×C4⋊C4—31no—
150C2×#106C2×D311yescharacteristic
151C2×C4⋊C8—31no—
152C2×#103—31no—
153C2×#103—31no—
154C2×#103—31no—
155C2×#102—31no—
156C2×#102—31no—
157C2×#102—31no—
158C2×C22⋊C8—31no—
159C2×C22⋊C8—31no—
160C2×C4×C8—31no—
161#394—61no—
162#393—61no—
163#391—31no—
164#390—31no—
165#390—31no—
166#389—61no—
167#389C2×D311yescharacteristic
168#388—61no—
169#387—61no—
170#385—61no—
171#380—31no—
172C4×C4⋊C4—31no—
173C4×C4⋊C4C2×D311yescharacteristic
174C4×C22⋊C4—31no—
175C4×C22⋊C4—61no—
176#340—31no—
177#340—31no—
178#340—31no—
179#340—31no—
180#340—31no—
181#340C2×D311yescharacteristic
182#340C2×D311yescharacteristic
183#340C2×D311yescharacteristic
184#336—31no—
185#336C2×D311yescharacteristic
186C23×D3—21no—
187C23×D3C2×D411yescharacteristic
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2411yescharacteristic
195C22×Dic3—21no—
196C22×Dic3—21no—
197C22×Dic3—21no—
198C22×Dic3C2×D411yescharacteristic
199C22×Dic3C2×D411yescharacteristic
200C22×Dic3C2×D411yescharacteristic
201C2×D12—21no—
202C2×D12—21no—
203C2×D12—21no—
204C2×D12—21no—
205C2×D12—21no—
206C2×D12—21no—
207C2×D12—42no—
208C2×D12—42no—
209C2×D12—81no—
210C2×C4×D3—21no—
211C2×C4×D3—21no—
212C2×C4×D3—21no—
213C2×C4×D3—21no—
214C2×C4×D3—21no—
215C2×C4×D3—21no—
216C2×C4×D3—42no—
217C2×C4×D3—42no—
218C2×C4×D3—81no—
219C2×Dic6—21no—
220C2×Dic6—21no—
221C2×Dic6—21no—
222C2×Dic6—21no—
223C2×Dic6—21no—
224C2×Dic6—42no—
225C2×Dic6—42no—
226C2×Dic6—81no—
227C2×Dic6#4112yes—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—81no—
241C3×C4⋊C4#4112yes—
242C3×C4⋊C4#4112yes—
243C3×C42—21no—
244C3×C42—21no—
245C3×C42—42no—
246(C2×D3)⋊C4—24no—
247(C2×D3)⋊C4—41no—
248(C2×D3)⋊C4—42no—
249(C2×D3)⋊C4—42no—
250(C2×D3)⋊C4—42no—
251(C2×D3)⋊C4—42no—
252(C2×D3)⋊C4—42no—
253(C2×D3)⋊C4—81no—
254(C2×D3)⋊C4SD1614yescomplemented
255C4⋊Dic3—24no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3#4112yes—
260Dic3⋊C4—42no—
261Dic3⋊C4—42no—
262Dic3⋊C4—42no—
263C4×Dic3—21no—
264C4×Dic3—21no—
265C4×Dic3—42no—
266C4×Dic3—81no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273#204—44no—
274C22×Q8—31no—
275C22×Q8—61no—
276C22×D4—31no—
277C22×D4—61no—
278C23×C4—31no—
279C23×C4—61no—
280C2×SD16—62no—
281C2×SD16—121no—
282C2×SD16—121no—
283C22×C8—31no—
284C22×C8—31no—
285C22×C8C22×D311yescharacteristic
286C22×C8C22×D311yescharacteristic
287C4⋊Q8—62no—
288C4⋊Q8—121no—
289#124—124no—
290C22⋊Q8—62no—
291C22⋊Q8—62no—
292#121—62no—
293#121—121no—
294C4×Q8—62no—
295C4×Q8—121no—
296C2×C4⋊C4—21no—
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—121no—
305C2×C4⋊C4C22×D311yescharacteristic
306C2×C4⋊C4C22×D311yescharacteristic
307C2×C4⋊C4C22×D311yescharacteristic
308C2×C4⋊C4C22×D311yescharacteristic
309C2×C22⋊C4—31no—
310C2×C22⋊C4—31no—
311C2×C22⋊C4—61no—
312C2×C22⋊C4—61no—
313C2×C22⋊C4—61no—
314C2×C22⋊C4—61no—
315C2×C22⋊C4—61no—
316C2×C22⋊C4—61no—
317C2×C22⋊C4—61no—
318C2×C42—21no—
319C2×C42—31no—
320C2×C42—61no—
321C2×C42—61no—
322C2×C42C22×D311yescharacteristic
323#106—22no—
324C4⋊C8—121no—
325#103—62no—
326#103—62no—
327#103—121no—
328#102—62no—
329#102—62no—
330#102—121no—
331C22⋊C8—121no—
332C22⋊C8—121no—
333C4×C8—121no—
334#95—21no—
335#95—21no—
336#95—31no—
337#95—31no—
338#95—61no—
339#95—61no—
340#95—61no—
341#95—61no—
342C23×C3C22×D411yescharacteristic
343C22×D3—21no—
344C22×D3—21no—
345C22×D3—21no—
346C22×D3—21no—
347C22×D3—21no—
348C22×D3—21no—
349C22×D3—41no—
350C22×D3—41no—
351C22×D3—41no—
352C22×D3—41no—
353C22×D3—41no—
354C22×D3—41no—
355C22×D3—41no—
356C22×D3C2×SD1612yes—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—22no—
360C2×C3×C4—21no—
361C2×C3×C4—22no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4C2×SD1612yes—
375C2×C3×C4C2×#4111yescharacteristic
376C2×C3×C4C2×#4111yescharacteristic
377C2×C3×C4C2×#4111yescharacteristic
378C2×C3×C4C2×#4111yescharacteristic
379C2×C3×C4H(5,𝔽2)11yescharacteristic
380C2×C3×C4H(5,𝔽2)11yescharacteristic
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—22no—
385C2×Dic3—21no—
386C2×Dic3—22no—
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3C2×SD1612yes—
399D12—41no—
400D12—41no—
401D12—41no—
402D12—41no—
403D12—84no—
404C4×D3—41no—
405C4×D3—41no—
406C4×D3—41no—
407C4×D3—41no—
408C4×D3—84no—
409Dic6—22no—
410Dic6—22no—
411Dic6—41no—
412Dic6—41no—
413Dic6—84no—
414C3×C8—41no—
415C3×C8—41no—
416C3⋊C8—41no—
417C3⋊C8—41no—
418C24—31no—
419C24—61no—
420C2×Q8—32no—
421C2×Q8—61no—
422C2×Q8—61no—
423C2×Q8—61no—
424C2×Q8—61no—
425C2×Q8—61no—
426C2×Q8—122no—
427C2×Q8—122no—
428C2×Q8—241no—
429C2×D4—61no—
430C2×D4—61no—
431C2×D4—61no—
432C2×D4—61no—
433C2×D4—61no—
434C2×D4—61no—
435C2×D4—122no—
436C2×D4—122no—
437C2×D4—241no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—31no—
443C22×C4—31no—
444C22×C4—31no—
445C22×C4—61no—
446C22×C4—61no—
447C22×C4—61no—
448C22×C4—61no—
449C22×C4—61no—
450C22×C4—61no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4—61no—
454C22×C4—122no—
455C22×C4—122no—
456C22×C4—241no—
457C22×C4D3×D411yescharacteristic
458C22×C4D3×D411yescharacteristic
459C22×C4C23×D311yescharacteristic
460SD16—124no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—61no—
468C2×C8—61no—
469C2×C8—61no—
470C2×C8—61no—
471C2×C8—61no—
472C2×C8—61no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—32no—
480C4⋊C4—64no—
481C4⋊C4—61no—
482C4⋊C4—61no—
483C4⋊C4—61no—
484C4⋊C4—81no—
485C4⋊C4—122no—
486C4⋊C4—122no—
487C4⋊C4—122no—
488C4⋊C4#23512yes—
489C4⋊C4#23912yes—
490C22⋊C4—34no—
491C22⋊C4—64no—
492C22⋊C4—121no—
493C22⋊C4—122no—
494C22⋊C4—122no—
495C22⋊C4—122no—
496C22⋊C4—122no—
497C22⋊C4—122no—
498C22⋊C4—241no—
499C42—21no—
500C42—21no—
501C42—42no—
502C42—61no—
503C42—61no—
504C42—122no—
505C42—241no—
506C22×C3#52511yescharacteristic
507C22×C3#54611yescharacteristic
508C22×C3#54611yescharacteristic
509C22×C3C22×SD1611yescharacteristic
510C22×C3#57511yescharacteristic
511C22×C3#57611yescharacteristic
512C22×C3#57811yescharacteristic
513C2×D3—22no—
514C2×D3—22no—
515C2×D3—22no—
516C2×D3—41no—
517C2×D3—41no—
518C2×D3—41no—
519C2×D3—41no—
520C2×D3—81no—
521C2×D3—81no—
522C2×D3—81no—
523C2×D3—81no—
524C2×D3—81no—
525C2×D3—81no—
526C2×D3—81no—
527C3×C4—22no—
528C3×C4—42no—
529C3×C4—42no—
530C3×C4—41no—
531C3×C4—81no—
532C3×C4—81no—
533C3×C4#52911yescharacteristic
534C3×C4#53811yescharacteristic
535C3×C4#55511yescharacteristic
536C3×C4#55911yescharacteristic
537Dic3—22no—
538Dic3—42no—
539Dic3—41no—
540Dic3—42no—
541Dic3—41no—
542Dic3—81no—
543C23—32no—
544C23—61no—
545C23—61no—
546C23—61no—
547C23—61no—
548C23—61no—
549C23—61no—
550C23—121no—
551C23—121no—
552C23—121no—
553C23—121no—
554C23—121no—
555C23—121no—
556C23—121no—
557C23C2×D3×D411yescharacteristic
558Q8—62no—
559Q8—62no—
560Q8—121no—
561Q8—121no—
562Q8—244no—
563D4—121no—
564D4—121no—
565D4—121no—
566D4—121no—
567D4—244no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—32no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—61no—
587C2×C4—61no—
588C2×C4—62no—
589C2×C4—61no—
590C2×C4—61no—
591C2×C4—61no—
592C2×C4—62no—
593C2×C4—61no—
594C2×C4—61no—
595C2×C4—61no—
596C2×C4—121no—
597C2×C4—121no—
598C2×C4—121no—
599C2×C4—121no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4—121no—
603C2×C4—121no—
604C2×C4—121no—
605C2×C4—121no—
606C2×C4—121no—
607C2×C4—244no—
608C2×C4D3×SD1612yes—
609C2×C4C2×#23511yescharacteristic
610C2×C4C2×#23911yescharacteristic
611C2×C4D3×#4111yescharacteristic
612C2×C4D3×#4111yescharacteristic
613C2×C4#100911yescharacteristic
614C2×C4#100911yescharacteristic
615C8—41no—
616C8—41no—
617C8—121no—
618C8—121no—
619C2×C3#245911yescharacteristic
620C2×C3#310311yescharacteristic
621C2×C3#312411yescharacteristic
622C2×C3#329511yescharacteristic
623C2×C3#329811yescharacteristic
624C2×C3#331911yescharacteristic
625C2×C3#332711yescharacteristic
626D3—42no—
627D3—161no—
628C22—62no—
629C22—62no—
630C22—62no—
631C22—121no—
632C22—121no—
633C22—121no—
634C22—121no—
635C22—241no—
636C22—241no—
637C22—241no—
638C22—241no—
639C22—241no—
640C22—241no—
641C22—241no—
642C22#554711yescharacteristic
643C22#563011yescharacteristic
644C22#565411yescharacteristic
645C22C2×D3×SD1611yescharacteristic
646C22#576811yescharacteristic
647C22#577411yescharacteristic
648C22#577911yescharacteristic
649C4—22no—
650C4—42no—
651C4—42no—
652C4—41no—
653C4—62no—
654C4—81no—
655C4—81no—
656C4—121no—
657C4—122no—
658C4—121no—
659C4—122no—
660C4—241no—
661C4#553211yescharacteristic
662C4#558511yescharacteristic
663C4#569111yescharacteristic
664C4#570411yescharacteristic
665C3#2672411yescharacteristic, complemented, Sylow
666C2—122no—
667C2—481no—
668C2#7532611yescharacteristic
669C2#7965311yescharacteristic
670C2#7978811yescharacteristic
671C2#8107511yescharacteristic
672C2#8125211yescharacteristic
673C2#8140211yescharacteristic
674C2#8145011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1920C2
socleC23×C3C22×D4
prime radicalC2×#895C22
hypercenter#1920D3
hypocenterC3#26724
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1920

Complements

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

NH ≅ G/Ntype
1. #76205C2semidirect
3. C3×#1920C2semidirect
4. #70582C2semidirect
5. #70569C2semidirect
6. #69615C2semidirect
7. C4×#810C2semidirect
8. #68910C2semidirect
11. #68872C2semidirect
12. #68863C2semidirect
13. #68800C2semidirect
14. #68686C2semidirect
15. #68657C2semidirect
70. #1920D3semidirect
254. (C2×D3)⋊C4SD16semidirect
665. C3#26724semidirect

Maximal subgroups

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

subgroupindex
#762052
#737882
C3×#19202
#705822
#705692
#696152
C4×#8102
#689102
#688872
#688802
#688722
#688632
#688002
#686862
#686572
#267243