Finite Group Databasebeta

← #10863701

Subgroups — #10863701

Order 768.

832 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#76306C211yesmaximal, characteristic
2#76028C211yesmaximal, characteristic, complemented
3C3×#2067C211yesmaximal, characteristic, complemented
4#70623C211yesmaximal, characteristic, complemented
5#70609C211yesmaximal, characteristic, complemented
6#70479C211yesmaximal, characteristic, complemented
7#70447C211yesmaximal, characteristic, complemented
8#70177C211yesmaximal, characteristic
9#70177C211yesmaximal, characteristic
10#70167C211yesmaximal, characteristic, complemented
11Dic3×#102C211yesmaximal, characteristic, complemented
12#69640C211yesmaximal, characteristic, complemented
13#69625C211yesmaximal, characteristic, complemented
14#68852C211yesmaximal, characteristic, complemented
15#68815C211yesmaximal, characteristic, complemented
16#26752—31nomaximal, Sylow
17C2×#940C2211yescharacteristic
18C2×D4×Dic3C2211yescharacteristic
19C2×#890—21no—
20C2×#67⋊C4—21no—
21C2×C4×D12C2211yescharacteristic
22C2×C3×#107C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C3×#390C2211yescharacteristic
27C3×#389C2211yescharacteristic
28C2×#832C2211yescharacteristic
29C2×#832C2211yescharacteristic
30C2×#821C2211yescharacteristic
31C2×#821C2211yescharacteristic
32C2×C8×Dic3C2211yescharacteristic
33#4988C2211yescharacteristic
34#4986—21no—
35#4984C2211yescharacteristic
36#4984C2211yescharacteristic
37Dic3×C4⋊C4C2211yescharacteristic
38Dic3×C4⋊C4C2211yescharacteristic
39#4972C2211yescharacteristic
40C2×#809C2211yescharacteristic
41C2×#809C2211yescharacteristic
42C2×#807C2211yescharacteristic
43#4958—21no—
44#4957—21no—
45#4956—21no—
46#4935C2211yescharacteristic
47C4×Dic3⋊C4C2211yescharacteristic
48#4661—21no—
49#4654—21no—
50C3×#340C2211yescharacteristic
51#4551C2211yescharacteristic
52#4551C2211yescharacteristic
53#4550C2211yescharacteristic
54#4550C2211yescharacteristic
55#4550C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2506—31no—
61#2366—31no—
62#2071—31no—
63#2067D311yescharacteristic, complemented
64#2041—31no—
65#2041—31no—
66#1934—31no—
67#1925—31no—
68#1918—31no—
69#1918—31no—
70#1916—31no—
71#1862—31no—
72#1862—31no—
73#1802—31no—
74C4×#102—31no—
75C22×C3×D4C2311yescharacteristic
76C22×#67—21no—
77C22×#67—21no—
78C23×Dic3—21no—
79C22×D12C2311yescharacteristic
80C22×C4×D3—21no—
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×C22⋊C4—21no—
88C2×C3×C22⋊C4—21no—
89C2×C3×C42C2311yescharacteristic
90C2×#217—21no—
91#940—22no—
92#940—22no—
93D4×Dic3—22no—
94D4×Dic3—41no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4D411yescharacteristic
107C2×Dic3⋊C4D411yescharacteristic
108C2×C4×Dic3—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3C2311yescharacteristic
111C2×C4×Dic3C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114#890—44no—
115#67⋊C4—44no—
116C4×D12—22no—
117C4×D12—41no—
118C3×#107—22no—
119C3×C4⋊C8—41no—
120C3×#102—22no—
121C3×#102—41no—
122C3×#95—21no—
123#832—22no—
124#832—22no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95D411yescharacteristic
131C3⋊#95D411yescharacteristic
132#821—22no—
133#821—22no—
134C8×Dic3—41no—
135#809—22no—
136#809—41no—
137#807—22no—
138C2×#127—31no—
139C2×#125—61no—
140C2×C4×D4—31no—
141C2×C4×D4—31no—
142C2×C4×D4—61no—
143C2×#107—31no—
144C2×#107C2×D311yescharacteristic
145C2×C4⋊C8C2×D311yescharacteristic
146C2×#102—31no—
147C2×#102—31no—
148C2×#102—31no—
149C2×#102—31no—
150C2×#102—31no—
151C2×#102—31no—
152C2×#102C2×D311yescharacteristic
153C2×#102C2×D311yescharacteristic
154C2×C4×C8—31no—
155#400—61no—
156#398—61no—
157#390—31no—
158#390—61no—
159#390C2×D311yescharacteristic
160#389—31no—
161#389—31no—
162#389—31no—
163#389C2×D311yescharacteristic
164#387—61no—
165#386—61no—
166#386—61no—
167#384—31no—
168C4×C4⋊C4—31no—
169C4×C4⋊C4—31no—
170C4×C4⋊C4—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340C2×D311yescharacteristic
177#336—31no—
178#336—31no—
179#336—31no—
180#336—31no—
181C24×C3—21no—
182C23×D3—21no—
183C2×C3×D4—21no—
184C2×C3×D4—21no—
185C2×C3×D4—21no—
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4#4112yes—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2411yescharacteristic
195C2×#67—24no—
196C2×#67—24no—
197C2×#67—42no—
198C2×#67—42no—
199C2×#67—81no—
200C2×#67—81no—
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3—42no—
205C22×Dic3—42no—
206C22×Dic3—81no—
207C22×Dic3C2×D411yescharacteristic
208C22×Dic3C2×D411yescharacteristic
209C22×Dic3C2×D411yescharacteristic
210C22×Dic3C2×D411yescharacteristic
211C2×D12—21no—
212C2×D12—21no—
213C2×D12—21no—
214C2×D12—21no—
215C2×D12—21no—
216C2×D12#4112yes—
217C2×C4×D3—42no—
218C2×C4×D3—42no—
219C2×C4×D3—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—21no—
231C3×C4⋊C4—42no—
232C3×C4⋊C4#4112yes—
233C3×C22⋊C4—42no—
234C3×C22⋊C4—81no—
235C3×C42—21no—
236C3×C42—21no—
237#217—42no—
238(C2×D3)⋊C4—42no—
239(C2×D3)⋊C4—42no—
240(C2×D3)⋊C4—81no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—42no—
247C4⋊Dic3—42no—
248C4⋊Dic3—42no—
249C4⋊Dic3—42no—
250C4⋊Dic3#4112yes—
251Dic3⋊C4—41no—
252Dic3⋊C4—42no—
253Dic3⋊C4D814yes—
254C4×Dic3—21no—
255C4×Dic3—21no—
256C4×Dic3—21no—
257C4×Dic3—21no—
258C4×Dic3—42no—
259C4×Dic3—42no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C22×D4—31no—
267C22×D4—61no—
268C22×D4—61no—
269C22×D4C22×D311yescharacteristic
270C23×C4—61no—
271C23×C4—61no—
272C22×C8—31no—
273C22×C8—31no—
274C22×C8C22×D311yescharacteristic
275C22×C8C22×D311yescharacteristic
276#127—62no—
277#127—62no—
278#125—124no—
279C4×D4—62no—
280C4×D4—62no—
281C4×D4—121no—
282C4×D4—121no—
283C4×D4—124no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—31no—
286C2×C4⋊C4—31no—
287C2×C4⋊C4—31no—
288C2×C4⋊C4—31no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—61no—
291C2×C4⋊C4—61no—
292C2×C4⋊C4—61no—
293C2×C4⋊C4—61no—
294C2×C4⋊C4—61no—
295C2×C4⋊C4C22×D311yescharacteristic
296C2×C4⋊C4C22×D311yescharacteristic
297C2×C4⋊C4C22×D311yescharacteristic
298C2×C22⋊C4—21no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—61no—
301C2×C22⋊C4—61no—
302C2×C22⋊C4—61no—
303C2×C22⋊C4—61no—
304C2×C42—31no—
305C2×C42—31no—
306C2×C42—61no—
307C2×C42—61no—
308C2×C42C22×D311yescharacteristic
309#107—22no—
310#107—62no—
311C4⋊C8—41no—
312#102—22no—
313#102—41no—
314#102—62no—
315#102—62no—
316#102—62no—
317#102—62no—
318#102—62no—
319#102—121no—
320C4×C8—121no—
321#95—21no—
322#95—31no—
323#95—31no—
324#95—61no—
325#95—61no—
326#95—61no—
327#95—61no—
328#95—61no—
329C23×C3—22no—
330C23×C3—41no—
331C23×C3—41no—
332C23×C3—41no—
333C23×C3—41no—
334C23×C3—41no—
335C23×C3—41no—
336C23×C3C22×D411yescharacteristic
337C22×D3—22no—
338C22×D3—41no—
339C22×D3—41no—
340C22×D3—41no—
341C22×D3—41no—
342C22×D3—41no—
343C22×D3—41no—
344C3×D4—22no—
345C3×D4—22no—
346C3×D4—41no—
347C3×D4—41no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—22no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
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×D812yes—
363C2×C3×C4C2×#4111yescharacteristic
364C2×C3×C4C2×#4111yescharacteristic
365C2×C3×C4C2×#4111yescharacteristic
366C2×C3×C4C2×#4111yescharacteristic
367C2×C3×C4H(5,𝔽2)11yescharacteristic
368C2×C3×C42−1+411yescharacteristic
369#67—48no—
370#67—84no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—22no—
378C2×Dic3—21no—
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—84no—
391C2×Dic3C2×D812yes—
392C2×Dic3C2×D812yes—
393D12—22no—
394D12—22no—
395D12—41no—
396D12—41no—
397C4×D3—84no—
398C3×C8—41no—
399C3×C8—41no—
400C3⋊C8—41no—
401C3⋊C8—41no—
402C24—21no—
403C24—61no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—32no—
410C2×D4—64no—
411C2×D4—64no—
412C2×D4—61no—
413C2×D4—61no—
414C2×D4—61no—
415C2×D4—61no—
416C2×D4—61no—
417C2×D4—122no—
418C2×D4—122no—
419C2×D4—241no—
420C2×D4—241no—
421C2×D4#23912yes—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—31no—
426C22×C4—31no—
427C22×C4—31no—
428C22×C4—31no—
429C22×C4—61no—
430C22×C4—61no—
431C22×C4—61no—
432C22×C4—122no—
433C22×C4—122no—
434C22×C4—122no—
435C22×C4—122no—
436C22×C4—241no—
437C22×C4—241no—
438C22×C4D3×D411yescharacteristic
439C22×C4D3×D411yescharacteristic
440C22×C4C23×D311yescharacteristic
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—61no—
448C2×C8—61no—
449C2×C8—61no—
450C2×C8—61no—
451C2×C8—61no—
452C2×C8—61no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—34no—
459C4⋊C4—32no—
460C4⋊C4—42no—
461C4⋊C4—61no—
462C4⋊C4—61no—
463C4⋊C4—61no—
464C4⋊C4—61no—
465C4⋊C4—61no—
466C4⋊C4—121no—
467C4⋊C4—122no—
468C4⋊C4—122no—
469C4⋊C4—122no—
470C4⋊C4—122no—
471C4⋊C4—122no—
472C4⋊C4#23712yes—
473C22⋊C4—42no—
474C22⋊C4—81no—
475C22⋊C4—122no—
476C22⋊C4—122no—
477C22⋊C4—122no—
478C22⋊C4—241no—
479C42—21no—
480C42—21no—
481C42—61no—
482C42—61no—
483C42—61no—
484C42—61no—
485C42—122no—
486C42—122no—
487C22×C3—42no—
488C22×C3—42no—
489C22×C3—42no—
490C22×C3—81no—
491C22×C3—81no—
492C22×C3—81no—
493C22×C3—81no—
494C22×C3#52511yescharacteristic
495C22×C3#54611yescharacteristic
496C22×C3#54811yescharacteristic
497C22×C3C22×D811yescharacteristic
498C22×C3#57511yescharacteristic
499C22×C3#57711yescharacteristic
500C22×C3#57711yescharacteristic
501C2×D3—42no—
502C2×D3—42no—
503C2×D3—42no—
504C2×D3—81no—
505C2×D3—81no—
506C2×D3—81no—
507C2×D3—81no—
508C3×C4—22no—
509C3×C4—42no—
510C3×C4—41no—
511C3×C4—81no—
512C3×C4—81no—
513C3×C4#52911yescharacteristic
514C3×C4#53911yescharacteristic
515C3×C4#55211yescharacteristic
516C3×C4#55311yescharacteristic
517Dic3—22no—
518Dic3—22no—
519Dic3—41no—
520Dic3—42no—
521Dic3—41no—
522Dic3—81no—
523Dic3—81no—
524C23—22no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—62no—
532C23—121no—
533C23—121no—
534C23—121no—
535C23—121no—
536C23—121no—
537C23—121no—
538C23C2×D3×D411yescharacteristic
539D4—22no—
540D4—22no—
541D4—41no—
542D4—41no—
543D4—62no—
544D4—62no—
545D4—128no—
546D4—121no—
547D4—121no—
548D4—244no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—32no—
556C2×C4—32no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—61no—
566C2×C4—61no—
567C2×C4—61no—
568C2×C4—61no—
569C2×C4—61no—
570C2×C4—61no—
571C2×C4—61no—
572C2×C4—61no—
573C2×C4—61no—
574C2×C4—62no—
575C2×C4—61no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—244no—
585C2×C4—244no—
586C2×C4D3×D812yes—
587C2×C4C2×#23711yescharacteristic
588C2×C4#100411yescharacteristic
589C2×C4C2×#23911yescharacteristic
590C2×C4D3×#4111yescharacteristic
591C2×C4D3×#4111yescharacteristic
592C2×C4#101011yescharacteristic
593C8—41no—
594C8—41no—
595C8—121no—
596C8—121no—
597C2×C3—82no—
598C2×C3#246011yescharacteristic
599C2×C3#310011yescharacteristic
600C2×C3#312311yescharacteristic
601C2×C3#329411yescharacteristic
602C2×C3#331711yescharacteristic
603C2×C3#335811yescharacteristic
604C2×C3#337111yescharacteristic
605D3—82no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—122no—
614C22—122no—
615C22—122no—
616C22—241no—
617C22—241no—
618C22—241no—
619C22—241no—
620C22#559811yescharacteristic
621C22#561011yescharacteristic
622C22#562711yescharacteristic
623C22C2×D3×D811yescharacteristic
624C22#577011yescharacteristic
625C22#577511yescharacteristic
626C22#577711yescharacteristic
627C4—22no—
628C4—41no—
629C4—42no—
630C4—62no—
631C4—62no—
632C4—81no—
633C4—81no—
634C4—122no—
635C4—121no—
636C4—121no—
637C4—241no—
638C4—241no—
639C4#561411yescharacteristic
640C4#562411yescharacteristic
641C4#564511yescharacteristic
642C4#570811yescharacteristic
643C3#2675211yescharacteristic, complemented, Sylow
644C2—82no—
645C2—242no—
646C2#7729411yescharacteristic
647C2#8031811yescharacteristic
648C2#8058411yescharacteristic
649C2#8073111yescharacteristic
650C2#8082411yescharacteristic
651C2#8094811yescharacteristic
652C2#8114711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2067C2
socleC23×C3C22×D4
prime radicalC2×#940C22
hypercenter#2067D3
hypocenterC3#26752
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2067

Complements

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

NH ≅ G/Ntype
2. #76028C2semidirect
3. C3×#2067C2semidirect
4. #70623C2semidirect
5. #70609C2semidirect
6. #70479C2semidirect
7. #70447C2semidirect
10. #70167C2semidirect
11. Dic3×#102C2semidirect
12. #69640C2semidirect
13. #69625C2semidirect
14. #68852C2semidirect
15. #68815C2semidirect
63. #2067D3semidirect
643. C3#26752semidirect

Maximal subgroups

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

subgroupindex
#763062
#760282
C3×#20672
#706232
#706092
#704792
#704472
#701772
#701672
Dic3×#1022
#696402
#696252
#688522
#688152
#267523