Finite Group Databasebeta

← #10868366

Subgroups — #10868366

Order 768.

874 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#76048C211yesmaximal, characteristic, complemented
2#73653C211yesmaximal, characteristic, complemented
3C3×#1921C211yesmaximal, characteristic, complemented
4#70640C211yesmaximal, characteristic, complemented
5#70628C211yesmaximal, characteristic, complemented
6#69627C211yesmaximal, characteristic, complemented
7#69463C211yesmaximal, characteristic, complemented
8#68897C211yesmaximal, characteristic, complemented
9#68897C211yesmaximal, characteristic, complemented
10#68893C211yesmaximal, characteristic, complemented
11#68862C211yesmaximal, characteristic, complemented
12#68792C211yesmaximal, characteristic, complemented
13#68780C211yesmaximal, characteristic, complemented
14#68674C211yesmaximal, characteristic
15#68610C211yesmaximal, characteristic, complemented
16#27065—31nomaximal, Sylow
17C22×D24C2211yescharacteristic
18C22×D24C2211yescharacteristic
19C2×#893C2211yescharacteristic
20C2×#893C2211yescharacteristic
21C2×#874C2211yescharacteristic
22C2×C4×D12—21no—
23C2×C3×#107C2211yescharacteristic
24C3×#389C2211yescharacteristic
25C3×C4×C4⋊C4C2211yescharacteristic
26C2×#821C2211yescharacteristic
27C2×#821C2211yescharacteristic
28C2×Dic3⋊C8C2211yescharacteristic
29C2×C8×Dic3C2211yescharacteristic
30#4992—21no—
31#4990—21no—
32#4988C2211yescharacteristic
33#4988C2211yescharacteristic
34#4988C2211yescharacteristic
35#4988C2211yescharacteristic
36#4977C2211yescharacteristic
37Dic3×C4⋊C4C2211yescharacteristic
38C2×#809C2211yescharacteristic
39C2×#809C2211yescharacteristic
40C2×#809C2211yescharacteristic
41C2×#809C2211yescharacteristic
42C2×#809C2211yescharacteristic
43C2×#809C2211yescharacteristic
44C2×#808C2211yescharacteristic
45#4939—21no—
46#4939C2211yescharacteristic
47C4×C4⋊Dic3C2211yescharacteristic
48#4674—21no—
49#4670—21no—
50#4665—21no—
51#4665—21no—
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54C3×#340C2211yescharacteristic
55C3×#336C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2513—31no—
61#2366—31no—
62#2092—31no—
63#2054—31no—
64#2041—31no—
65#2011—31no—
66#1970—31no—
67#1926—31no—
68#1926—31no—
69#1921D311yescharacteristic, complemented
70#1916—31no—
71#1879—31no—
72#1879—31no—
73#1804—31no—
74C4×#102—31no—
75C22×D12—21no—
76C22×D12—21no—
77C22×D12C2311yescharacteristic
78C22×D12C2311yescharacteristic
79C22×D12C2311yescharacteristic
80C22×C4×D3—21no—
81C22×C4×D3—21no—
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×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×C4⋊Dic3D411yescharacteristic
104C2×C4⋊Dic3D411yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4×Dic3—21no—
107C2×C4×Dic3C2311yescharacteristic
108C2×C4×Dic3C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111C2×D24—22no—
112C2×D24—22no—
113C2×D24—22no—
114C2×D24—41no—
115C2×D24—41no—
116C2×D24—41no—
117#893—22no—
118#893—22no—
119#893—41no—
120#893—41no—
121#874—22no—
122#874—22no—
123C4×D12—44no—
124C3×#107—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—41no—
134Dic3⋊C8—41no—
135C8×Dic3—41no—
136#809—22no—
137#809—22no—
138#809—22no—
139#809—41no—
140#809—41no—
141#809—41no—
142#808—41no—
143C22×D8—31no—
144C22×D8—31no—
145C2×#127—31no—
146C2×C4×D4—31no—
147C2×C4×D4—31no—
148C2×C4×D4—61no—
149C2×#107C2×D311yescharacteristic
150C2×#106—31no—
151C2×C4⋊C8—31no—
152C2×#102—31no—
153C2×#102—31no—
154C2×#102—31no—
155C2×#102—31no—
156C2×#102—31no—
157C2×#102—31no—
158C2×#102—31no—
159C2×#102—31no—
160C2×C4×C8—31no—
161#397—61no—
162#397—61no—
163#390—31no—
164#390—31no—
165#390—31no—
166#390—31no—
167#390—31no—
168#390—61no—
169#389C2×D311yescharacteristic
170#388—61no—
171#388—61no—
172#386—61no—
173#386—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—
188C23×D3—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2411yescharacteristic
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—21no—
213C2×D12—21no—
214C2×D12—21no—
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12—24no—
219C2×D12—42no—
220C2×D12—42no—
221C2×D12—81no—
222C2×D12—81no—
223C2×D12#4112yes—
224C2×D12#4112yes—
225C2×C4×D3—42no—
226C2×C4×D3—42no—
227C2×C4×D3—42no—
228C2×C4×D3—42no—
229C2×C4×D3—81no—
230C2×C4×D3—81no—
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—81no—
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—42no—
254(C2×D3)⋊C4—42no—
255(C2×D3)⋊C4—81no—
256(C2×D3)⋊C4—81no—
257(C2×D3)⋊C4—81no—
258(C2×D3)⋊C4—81no—
259(C2×D3)⋊C4—81no—
260(C2×D3)⋊C4—81no—
261C4⋊Dic3—22no—
262C4⋊Dic3—22no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4×Dic3—21no—
266C4×Dic3—21no—
267C4×Dic3—21no—
268C4×Dic3—21no—
269C4×Dic3—81no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276D24—44no—
277D24—44no—
278C22×D4—31no—
279C22×D4—31no—
280C22×D4—31no—
281C22×D4—61no—
282C22×D4—61no—
283C23×C4—61no—
284C23×C4—61no—
285C2×D8—62no—
286C2×D8—62no—
287C2×D8—62no—
288C2×D8—121no—
289C2×D8—121no—
290C2×D8—121no—
291C22×C8—31no—
292C22×C8—31no—
293C22×C8C22×D311yescharacteristic
294C22×C8C22×D311yescharacteristic
295#127—62no—
296#127—62no—
297C4×D4—62no—
298C4×D4—62no—
299C4×D4—124no—
300C4×D4—121no—
301C4×D4—121no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4C22×D311yescharacteristic
307C2×C4⋊C4C22×D311yescharacteristic
308C2×C4⋊C4C22×D311yescharacteristic
309C2×C4⋊C4C22×D311yescharacteristic
310C2×C22⋊C4—61no—
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×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C42—21no—
323C2×C42—31no—
324C2×C42—31no—
325C2×C42—61no—
326C2×C42C22×D311yescharacteristic
327#107—22no—
328#106—121no—
329C4⋊C8—121no—
330#102—62no—
331#102—62no—
332#102—62no—
333#102—62no—
334#102—121no—
335#102—121no—
336#102—121no—
337#102—121no—
338C4×C8—121no—
339#95—21no—
340#95—21no—
341#95—31no—
342#95—31no—
343#95—61no—
344#95—61no—
345#95—61no—
346C23×C3C22×D411yescharacteristic
347C22×D3—22no—
348C22×D3—22no—
349C22×D3—41no—
350C22×D3—41no—
351C22×D3—41no—
352C22×D3—41no—
353C22×D3—41no—
354C22×D3—41no—
355C22×D3—41no—
356C22×D3—41no—
357C22×D3—41no—
358C22×D3—41no—
359C22×D3—41no—
360C22×D3—41no—
361C22×D3—41no—
362C22×D3—41no—
363C22×D3—41no—
364C22×D3—41no—
365C22×D3—41no—
366C22×D3—41no—
367C22×D3—41no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—22no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4#13512yes—
386C2×C3×C4C2×#4111yescharacteristic
387C2×C3×C4C2×#4111yescharacteristic
388C2×C3×C4C2×#4111yescharacteristic
389C2×C3×C4C2×#4111yescharacteristic
390C2×C3×C4H(5,𝔽2)11yescharacteristic
391C2×C3×C4H(5,𝔽2)11yescharacteristic
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3AGL(1,ℤ/8)12yes—
407D12—22no—
408D12—22no—
409D12—22no—
410D12—22no—
411D12—48no—
412D12—41no—
413D12—41no—
414D12—41no—
415D12—41no—
416D12—41no—
417D12—41no—
418D12—41no—
419D12—41no—
420D12—84no—
421C4×D3—84no—
422C4×D3—84no—
423C3×C8—22no—
424C3×C8—41no—
425C3⋊C8—41no—
426C3⋊C8—41no—
427C24—61no—
428C24—61no—
429C24—61no—
430C2×D4—32no—
431C2×D4—32no—
432C2×D4—64no—
433C2×D4—64no—
434C2×D4—61no—
435C2×D4—61no—
436C2×D4—61no—
437C2×D4—61no—
438C2×D4—61no—
439C2×D4—61no—
440C2×D4—61no—
441C2×D4—61no—
442C2×D4—61no—
443C2×D4—61no—
444C2×D4—61no—
445C2×D4—61no—
446C2×D4—61no—
447C2×D4—61no—
448C2×D4—61no—
449C2×D4—61no—
450C2×D4—122no—
451C2×D4—122no—
452C2×D4—241no—
453C2×D4—241no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—31no—
459C22×C4—31no—
460C22×C4—31no—
461C22×C4—31no—
462C22×C4—61no—
463C22×C4—122no—
464C22×C4—122no—
465C22×C4—122no—
466C22×C4—122no—
467C22×C4—241no—
468C22×C4—241no—
469C22×C4D3×D411yescharacteristic
470C22×C4D3×D411yescharacteristic
471C22×C4C23×D311yescharacteristic
472D8—124no—
473D8—124no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—61no—
480C2×C8—61no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8—61no—
485C2×C8#23512yes—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—62no—
494C4⋊C4—62no—
495C4⋊C4—61no—
496C4⋊C4—61no—
497C4⋊C4—81no—
498C4⋊C4#23912yes—
499C22⋊C4—122no—
500C22⋊C4—122no—
501C22⋊C4—122no—
502C22⋊C4—122no—
503C22⋊C4—122no—
504C22⋊C4—122no—
505C22⋊C4—241no—
506C22⋊C4—241no—
507C22⋊C4—241no—
508C22⋊C4—241no—
509C22⋊C4—241no—
510C22⋊C4—241no—
511C42—21no—
512C42—21no—
513C42—42no—
514C42—61no—
515C42—61no—
516C42—61no—
517C42—61no—
518C42—241no—
519C22×C3#52511yescharacteristic
520C22×C3#54611yescharacteristic
521C22×C3#55011yescharacteristic
522C22×C3C2×#13511yescharacteristic
523C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
524C22×C3#57611yescharacteristic
525C22×C3#57611yescharacteristic
526C2×D3—42no—
527C2×D3—42no—
528C2×D3—42no—
529C2×D3—42no—
530C2×D3—42no—
531C2×D3—42no—
532C2×D3—81no—
533C2×D3—81no—
534C2×D3—81no—
535C2×D3—81no—
536C2×D3—81no—
537C2×D3—81no—
538C2×D3—81no—
539C2×D3—81no—
540C2×D3—81no—
541C2×D3—81no—
542C2×D3—81no—
543C2×D3—81no—
544C2×D3—81no—
545C2×D3—81no—
546C2×D3—81no—
547C3×C4—22no—
548C3×C4—41no—
549C3×C4—42no—
550C3×C4—81no—
551C3×C4—81no—
552C3×C4—81no—
553C3×C4#52911yescharacteristic
554C3×C4#53811yescharacteristic
555C3×C4#55111yescharacteristic
556C3×C4#55311yescharacteristic
557Dic3—22no—
558Dic3—41no—
559Dic3—41no—
560Dic3—41no—
561Dic3—81no—
562C23—62no—
563C23—62no—
564C23—121no—
565C23—121no—
566C23—121no—
567C23—121no—
568C23—121no—
569C23—121no—
570C23—121no—
571C23—121no—
572C23—121no—
573C23—121no—
574C23—121no—
575C23—121no—
576C23—121no—
577C23—121no—
578C23—121no—
579C23—121no—
580C23—121no—
581C23—121no—
582C23—121no—
583C23C2×D3×D411yescharacteristic
584D4—62no—
585D4—62no—
586D4—62no—
587D4—62no—
588D4—128no—
589D4—121no—
590D4—121no—
591D4—121no—
592D4—121no—
593D4—121no—
594D4—121no—
595D4—121no—
596D4—121no—
597D4—244no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—32no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—61no—
617C2×C4—61no—
618C2×C4—61no—
619C2×C4—61no—
620C2×C4—61no—
621C2×C4—61no—
622C2×C4—61no—
623C2×C4—61no—
624C2×C4—61no—
625C2×C4—61no—
626C2×C4—61no—
627C2×C4—121no—
628C2×C4—121no—
629C2×C4—121no—
630C2×C4—244no—
631C2×C4—244no—
632C2×C4#91912yes—
633C2×C4C2×#23511yescharacteristic
634C2×C4C2×#23911yescharacteristic
635C2×C4D3×#4111yescharacteristic
636C2×C4D3×#4111yescharacteristic
637C2×C4#100911yescharacteristic
638C2×C4#100911yescharacteristic
639C8—22no—
640C8—41no—
641C8—121no—
642C8—121no—
643C2×C3#246711yescharacteristic
644C2×C3#310511yescharacteristic
645C2×C3#312211yescharacteristic
646C2×C3#330511yescharacteristic
647C2×C3#331811yescharacteristic
648C2×C3#339611yescharacteristic
649C2×C3#339911yescharacteristic
650D3—82no—
651D3—82no—
652D3—161no—
653C22—122no—
654C22—122no—
655C22—122no—
656C22—122no—
657C22—122no—
658C22—122no—
659C22—241no—
660C22—241no—
661C22—241no—
662C22—241no—
663C22—241no—
664C22—241no—
665C22—241no—
666C22—241no—
667C22—241no—
668C22—241no—
669C22—241no—
670C22—241no—
671C22—241no—
672C22—241no—
673C22—241no—
674C22#555311yescharacteristic
675C22#563011yescharacteristic
676C22#565611yescharacteristic
677C22C2×#91911yescharacteristic
678C22#576911yescharacteristic
679C22D3×AGL(1,ℤ/8)11yescharacteristic
680C22#577411yescharacteristic
681C4—22no—
682C4—42no—
683C4—41no—
684C4—62no—
685C4—81no—
686C4—81no—
687C4—81no—
688C4—121no—
689C4—121no—
690C4—121no—
691C4—241no—
692C4#552711yescharacteristic
693C4#558011yescharacteristic
694C4#569511yescharacteristic
695C4#570411yescharacteristic
696C3#2706511yescharacteristic, complemented, Sylow
697C2—242no—
698C2—242no—
699C2—481no—
700C2#7500211yescharacteristic
701C2#7971211yescharacteristic
702C2#7985711yescharacteristic
703C2#8112011yescharacteristic
704C2#8119111yescharacteristic
705C2#8135411yescharacteristic
706C2#8145311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1921C2
socleC23×C3C22×D4
prime radical#68674C2
hypercenter#1921D3
hypocenterC3#27065
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1921

Complements

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

NH ≅ G/Ntype
1. #76048C2semidirect
2. #73653C2semidirect
3. C3×#1921C2semidirect
4. #70640C2semidirect
5. #70628C2semidirect
6. #69627C2semidirect
7. #69463C2semidirect
8. #68897C2semidirect
9. #68897C2semidirect
10. #68893C2semidirect
11. #68862C2semidirect
12. #68792C2semidirect
13. #68780C2semidirect
15. #68610C2semidirect
69. #1921D3semidirect
696. C3#27065semidirect

Maximal subgroups

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

subgroupindex
#760482
#736532
C3×#19212
#706402
#706282
#696272
#694632
#688972
#688932
#688622
#687922
#687802
#686742
#686102
#270653