Finite Group Databasebeta

← #10868336

Subgroups — #10868336

Order 768.

840 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#76049C211yesmaximal, characteristic, complemented
2#73653C211yesmaximal, characteristic
3C3×#1919C211yesmaximal, characteristic, complemented
4#70586C211yesmaximal, characteristic, complemented
5#70574C211yesmaximal, characteristic, complemented
6#69627C211yesmaximal, characteristic
7#69465C211yesmaximal, characteristic, complemented
8#68896C211yesmaximal, characteristic, complemented
9#68896C211yesmaximal, characteristic, complemented
10#68893C211yesmaximal, characteristic, complemented
11#68863C211yesmaximal, characteristic, complemented
12#68791C211yesmaximal, characteristic, complemented
13#68780C211yesmaximal, characteristic, complemented
14#68676C211yesmaximal, characteristic
15#68610C211yesmaximal, characteristic, complemented
16#27063—31nomaximal, Sylow
17C22×#204C2211yescharacteristic
18C22×#204C2211yescharacteristic
19C2×#893C2211yescharacteristic
20C2×#893C2211yescharacteristic
21C2×#874C2211yescharacteristic
22C2×C4×D12—21no—
23C2×C3×#106C2211yescharacteristic
24C3×#389C2211yescharacteristic
25C3×C4×C4⋊C4C2211yescharacteristic
26C2×#821C2211yescharacteristic
27C2×#821C2211yescharacteristic
28C2×Dic3⋊C8C2211yescharacteristic
29C2×C8×Dic3C2211yescharacteristic
30#4988C2211yescharacteristic
31#4988C2211yescharacteristic
32#4986—21no—
33#4980—21no—
34#4977C2211yescharacteristic
35Dic3×C4⋊C4C2211yescharacteristic
36#4973C2211yescharacteristic
37#4973C2211yescharacteristic
38C2×#810C2211yescharacteristic
39C2×#810C2211yescharacteristic
40C2×#809C2211yescharacteristic
41C2×#809C2211yescharacteristic
42C2×#809C2211yescharacteristic
43C2×#809C2211yescharacteristic
44C2×#807C2211yescharacteristic
45#4939—21no—
46C4×C4⋊Dic3C2211yescharacteristic
47#4929C2211yescharacteristic
48#4665—21no—
49#4665—21no—
50C3⋊#401—21no—
51#4651—21no—
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54C3×#340C2211yescharacteristic
55C3×#336C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2514—31no—
61#2366—31no—
62#2092—31no—
63#2055—31no—
64#2041—31no—
65#2014—31no—
66#1968—31no—
67#1928—31no—
68#1925—31no—
69#1919D311yescharacteristic, complemented
70#1916—31no—
71#1877—31no—
72#1877—31no—
73#1804—31no—
74C4×#102—31no—
75C22×D12—21no—
76C22×D12—21no—
77C22×D12C2311yescharacteristic
78C22×D12C2311yescharacteristic
79C22×C4×D3—21no—
80C22×C4×D3—21no—
81C22×Dic6C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C42—21no—
90C2×C3×C42C2311yescharacteristic
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×C4⋊Dic3D411yescharacteristic
98C2×C4⋊Dic3D411yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4×Dic3—21no—
101C2×C4×Dic3C2311yescharacteristic
102C2×C4×Dic3C2311yescharacteristic
103C22×C3⋊C8C2311yescharacteristic
104C22×C3⋊C8C2311yescharacteristic
105C2×#204—22no—
106C2×#204—22no—
107C2×#204—22no—
108C2×#204—41no—
109C2×#204—41no—
110C2×#204—41no—
111#893—22no—
112#893—22no—
113#893—41no—
114#893—41no—
115#874—22no—
116#874—22no—
117C4×D12—44no—
118C3×#106—22no—
119C3×#95—21no—
120C3×#95—21no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95D411yescharacteristic
131C3⋊#95D411yescharacteristic
132#821—22no—
133#821—41no—
134Dic3⋊C8—41no—
135C8×Dic3—41no—
136#810—22no—
137#810—41no—
138#809—22no—
139#809—22no—
140#809—41no—
141#809—41no—
142#807—41no—
143C22×SD16—31no—
144C22×SD16—31no—
145C2×#127—31no—
146C2×C4×D4—31no—
147C2×C4×D4—31no—
148C2×C4×D4—61no—
149C2×#107—31no—
150C2×#106C2×D311yescharacteristic
151C2×C4⋊C8—31no—
152C2×#103—31no—
153C2×#103—31no—
154C2×#102—31no—
155C2×#102—31no—
156C2×#102—31no—
157C2×#102—31no—
158C2×#102—31no—
159C2×#102—31no—
160C2×C4×C8—31no—
161#401—61no—
162#400—61no—
163#391—31no—
164#391—31no—
165#391—31no—
166#390—31no—
167#390—31no—
168#390—61no—
169#389C2×D311yescharacteristic
170#387—61no—
171#386—61no—
172#386—61no—
173#383—61no—
174#382—31no—
175C4×C4⋊C4—31no—
176C4×C4⋊C4—31no—
177C4×C4⋊C4C2×D311yescharacteristic
178#340—31no—
179#340C2×D311yescharacteristic
180#340C2×D311yescharacteristic
181#340C2×D311yescharacteristic
182#336—31no—
183#336—31no—
184#336—31no—
185#336C2×D311yescharacteristic
186C23×D3—21no—
187C23×D3—21no—
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×Dic3C2×D411yescharacteristic
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—21no—
208C2×D12—21no—
209C2×D12—24no—
210C2×D12—21no—
211C2×D12—21no—
212C2×D12—24no—
213C2×D12—42no—
214C2×D12—42no—
215C2×D12—81no—
216C2×D12—81no—
217C2×D12#4112yes—
218C2×D12#4112yes—
219C2×C4×D3—42no—
220C2×C4×D3—42no—
221C2×C4×D3—42no—
222C2×C4×D3—42no—
223C2×C4×D3—81no—
224C2×C4×D3—81no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8#4112yes—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—42no—
245C3×C4⋊C4#4112yes—
246C3×C42—21no—
247C3×C42—21no—
248C3×C42—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)⋊C4—81no—
255C4⋊Dic3—22no—
256C4⋊Dic3—22no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4×Dic3—21no—
260C4×Dic3—21no—
261C4×Dic3—21no—
262C4×Dic3—21no—
263C4×Dic3—42no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270#204—44no—
271#204—44no—
272C22×Q8—31no—
273C22×D4—31no—
274C22×D4—31no—
275C22×D4—61no—
276C22×D4—61no—
277C23×C4—61no—
278C23×C4—61no—
279C2×SD16—62no—
280C2×SD16—62no—
281C2×SD16—62no—
282C2×SD16—121no—
283C2×SD16—121no—
284C2×SD16—121no—
285C22×C8—31no—
286C22×C8—31no—
287C22×C8C22×D311yescharacteristic
288C22×C8C22×D311yescharacteristic
289#127—62no—
290#127—62no—
291C4×D4—62no—
292C4×D4—62no—
293C4×D4—124no—
294C4×D4—121no—
295C4×D4—121no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4C22×D311yescharacteristic
301C2×C4⋊C4C22×D311yescharacteristic
302C2×C4⋊C4C22×D311yescharacteristic
303C2×C4⋊C4C22×D311yescharacteristic
304C2×C22⋊C4—61no—
305C2×C22⋊C4—61no—
306C2×C22⋊C4—61no—
307C2×C22⋊C4—61no—
308C2×C22⋊C4—61no—
309C2×C22⋊C4—61no—
310C2×C42—21no—
311C2×C42—31no—
312C2×C42—31no—
313C2×C42—61no—
314C2×C42C22×D311yescharacteristic
315#107—121no—
316#106—22no—
317C4⋊C8—121no—
318#103—62no—
319#103—121no—
320#102—62no—
321#102—62no—
322#102—62no—
323#102—121no—
324#102—121no—
325#102—121no—
326C4×C8—121no—
327#95—21no—
328#95—21no—
329#95—31no—
330#95—31no—
331#95—61no—
332#95—61no—
333#95—61no—
334#95—61no—
335#95—61no—
336#95—61no—
337#95—61no—
338#95—61no—
339#95—61no—
340C23×C3C22×D411yescharacteristic
341C22×D3—22no—
342C22×D3—22no—
343C22×D3—41no—
344C22×D3—41no—
345C22×D3—41no—
346C22×D3—41no—
347C22×D3—41no—
348C22×D3—41no—
349C22×D3—41no—
350C22×D3—41no—
351C22×D3—41no—
352C22×D3—41no—
353C22×D3—41no—
354C22×D3—41no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—22no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
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#13512yes—
373C2×C3×C4C2×#4111yescharacteristic
374C2×C3×C4C2×#4111yescharacteristic
375C2×C3×C4C2×#4111yescharacteristic
376C2×C3×C4C2×#4111yescharacteristic
377C2×C3×C4H(5,𝔽2)11yescharacteristic
378C2×C3×C4H(5,𝔽2)11yescharacteristic
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3AGL(1,ℤ/8)12yes—
397D12—22no—
398D12—22no—
399D12—22no—
400D12—22no—
401D12—48no—
402D12—41no—
403D12—41no—
404D12—41no—
405D12—41no—
406D12—84no—
407C4×D3—84no—
408C4×D3—84no—
409Dic6—41no—
410Dic6—41no—
411Dic6—41no—
412Dic6—41no—
413C3×C8—22no—
414C3×C8—41no—
415C3⋊C8—41no—
416C3⋊C8—41no—
417C24—61no—
418C24—61no—
419C2×Q8—61no—
420C2×Q8—61no—
421C2×Q8—61no—
422C2×Q8—61no—
423C2×Q8—61no—
424C2×Q8—61no—
425C2×D4—32no—
426C2×D4—32no—
427C2×D4—64no—
428C2×D4—64no—
429C2×D4—61no—
430C2×D4—61no—
431C2×D4—61no—
432C2×D4—61no—
433C2×D4—61no—
434C2×D4—61no—
435C2×D4—61no—
436C2×D4—61no—
437C2×D4—61no—
438C2×D4—61no—
439C2×D4—122no—
440C2×D4—122no—
441C2×D4—241no—
442C2×D4—241no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—31no—
448C22×C4—31no—
449C22×C4—31no—
450C22×C4—31no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4—122no—
454C22×C4—122no—
455C22×C4—122no—
456C22×C4—122no—
457C22×C4—241no—
458C22×C4—241no—
459C22×C4D3×D411yescharacteristic
460C22×C4D3×D411yescharacteristic
461C22×C4C23×D311yescharacteristic
462SD16—124no—
463SD16—124no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—61no—
470C2×C8—61no—
471C2×C8—61no—
472C2×C8—61no—
473C2×C8—61no—
474C2×C8—61no—
475C2×C8#23512yes—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—42no—
484C4⋊C4—62no—
485C4⋊C4—62no—
486C4⋊C4—61no—
487C4⋊C4—61no—
488C4⋊C4#23912yes—
489C22⋊C4—122no—
490C22⋊C4—122no—
491C22⋊C4—122no—
492C22⋊C4—122no—
493C22⋊C4—241no—
494C22⋊C4—241no—
495C42—21no—
496C42—21no—
497C42—42no—
498C42—61no—
499C42—61no—
500C42—61no—
501C42—61no—
502C42—122no—
503C22×C3#52511yescharacteristic
504C22×C3#54611yescharacteristic
505C22×C3#55011yescharacteristic
506C22×C3C2×#13511yescharacteristic
507C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
508C22×C3#57711yescharacteristic
509C22×C3#57711yescharacteristic
510C2×D3—42no—
511C2×D3—42no—
512C2×D3—42no—
513C2×D3—42no—
514C2×D3—42no—
515C2×D3—42no—
516C2×D3—81no—
517C2×D3—81no—
518C2×D3—81no—
519C2×D3—81no—
520C2×D3—81no—
521C2×D3—81no—
522C2×D3—81no—
523C2×D3—81no—
524C3×C4—22no—
525C3×C4—41no—
526C3×C4—42no—
527C3×C4—81no—
528C3×C4—81no—
529C3×C4—81no—
530C3×C4#52911yescharacteristic
531C3×C4#53811yescharacteristic
532C3×C4#55111yescharacteristic
533C3×C4#55311yescharacteristic
534Dic3—22no—
535Dic3—41no—
536Dic3—41no—
537Dic3—41no—
538Dic3—81no—
539Dic3—81no—
540C23—62no—
541C23—62no—
542C23—121no—
543C23—121no—
544C23—121no—
545C23—121no—
546C23—121no—
547C23—121no—
548C23—121no—
549C23—121no—
550C23—121no—
551C23—121no—
552C23—121no—
553C23—121no—
554C23C2×D3×D411yescharacteristic
555Q8—121no—
556Q8—121no—
557Q8—121no—
558Q8—121no—
559D4—62no—
560D4—62no—
561D4—62no—
562D4—62no—
563D4—128no—
564D4—121no—
565D4—121no—
566D4—121no—
567D4—121no—
568D4—244no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
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—41no—
587C2×C4—61no—
588C2×C4—61no—
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—121no—
599C2×C4—121no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4—121no—
603C2×C4—121no—
604C2×C4—244no—
605C2×C4—244no—
606C2×C4#91612yes—
607C2×C4C2×#23511yescharacteristic
608C2×C4C2×#23911yescharacteristic
609C2×C4D3×#4111yescharacteristic
610C2×C4D3×#4111yescharacteristic
611C2×C4#100911yescharacteristic
612C2×C4#100911yescharacteristic
613C8—22no—
614C8—41no—
615C8—121no—
616C8—121no—
617C2×C3#246711yescharacteristic
618C2×C3#310511yescharacteristic
619C2×C3#312311yescharacteristic
620C2×C3#330411yescharacteristic
621C2×C3#331011yescharacteristic
622C2×C3#339411yescharacteristic
623C2×C3#339711yescharacteristic
624D3—82no—
625D3—82no—
626C22—122no—
627C22—122no—
628C22—122no—
629C22—122no—
630C22—122no—
631C22—122no—
632C22—241no—
633C22—241no—
634C22—241no—
635C22—241no—
636C22—241no—
637C22—241no—
638C22—241no—
639C22—241no—
640C22#555311yescharacteristic
641C22#563011yescharacteristic
642C22#565611yescharacteristic
643C22C2×#91611yescharacteristic
644C22#577011yescharacteristic
645C22D3×AGL(1,ℤ/8)11yescharacteristic
646C22#577811yescharacteristic
647C4—22no—
648C4—42no—
649C4—41no—
650C4—62no—
651C4—81no—
652C4—81no—
653C4—81no—
654C4—121no—
655C4—121no—
656C4—121no—
657C4—241no—
658C4—241no—
659C4#552711yescharacteristic
660C4#558011yescharacteristic
661C4#569511yescharacteristic
662C4#570411yescharacteristic
663C3#2706311yescharacteristic, complemented, Sylow
664C2—242no—
665C2—242no—
666C2#7500211yescharacteristic
667C2#7963811yescharacteristic
668C2#7976311yescharacteristic
669C2#8102311yescharacteristic
670C2#8118911yescharacteristic
671C2#8130911yescharacteristic
672C2#8149811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1919C2
socleC23×C3C22×D4
prime radicalC2×#874C22
hypercenter#1919D3
hypocenterC3#27063
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1919

Complements

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

NH ≅ G/Ntype
1. #76049C2semidirect
3. C3×#1919C2semidirect
4. #70586C2semidirect
5. #70574C2semidirect
7. #69465C2semidirect
8. #68896C2semidirect
9. #68896C2semidirect
10. #68893C2semidirect
11. #68863C2semidirect
12. #68791C2semidirect
13. #68780C2semidirect
15. #68610C2semidirect
69. #1919D3semidirect
663. C3#27063semidirect

Maximal subgroups

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

subgroupindex
#760492
#736532
C3×#19192
#705862
#705742
#696272
#694652
#688962
#688932
#688632
#687912
#687802
#686762
#686102
#270633