Finite Group Databasebeta

← #10875966

Subgroups — #10875966

Order 768.

836 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
1D4×Dic3⋊C4C211yesmaximal, characteristic, complemented
2#74333C211yesmaximal, characteristic, complemented
3C3×#1802C211yesmaximal, characteristic, complemented
4#71613C211yesmaximal, characteristic
5#70484C211yesmaximal, characteristic, complemented
6#70448C211yesmaximal, characteristic, complemented
7#70172C211yesmaximal, characteristic
8#70172C211yesmaximal, characteristic
9#70168C211yesmaximal, characteristic, complemented
10#70152C211yesmaximal, characteristic, complemented
11#69572C211yesmaximal, characteristic, complemented
12#68902C211yesmaximal, characteristic, complemented
13#68856C211yesmaximal, characteristic, complemented
14#68843C211yesmaximal, characteristic, complemented
15#68820C211yesmaximal, characteristic, complemented
16#27571—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C2×D4×Dic3C2211yescharacteristic
19C2×D4×Dic3C2211yescharacteristic
20C22×#213C2211yescharacteristic
21C22×Dic3⋊C4—21no—
22C2×#898—21no—
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C3×#384C2211yescharacteristic
27C2×#832C2211yescharacteristic
28#5210—21no—
29C2×#821C2211yescharacteristic
30C2×#821C2211yescharacteristic
31C2×#818C2211yescharacteristic
32C2×#815C2211yescharacteristic
33#4994—21no—
34#4992—21no—
35#4990—21no—
36#4988C2211yescharacteristic
37#4988C2211yescharacteristic
38#4984C2211yescharacteristic
39#4975C2211yescharacteristic
40Dic3×C4⋊C4C2211yescharacteristic
41#4972C2211yescharacteristic
42C2×#809C2211yescharacteristic
43#4946—21no—
44#4942—21no—
45#4939C2211yescharacteristic
46C4×Dic3⋊C4C2211yescharacteristic
47#4928C2211yescharacteristic
48C3×#340C2211yescharacteristic
49C3×#340C2211yescharacteristic
50#4551C2211yescharacteristic
51#4550C2211yescharacteristic
52#4550C2211yescharacteristic
53#4550C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2512—31no—
61D4×C4⋊C4—31no—
62#2100—31no—
63#2100—31no—
64#2071—31no—
65#2067—31no—
66#1936—31no—
67#1921—31no—
68#1917—31no—
69#1879—31no—
70#1879—31no—
71#1879—31no—
72#1862—31no—
73#1802D311yescharacteristic, complemented
74#1762—31no—
75C22×C3×D4C2311yescharacteristic
76C23×C3×C4—21no—
77C23×Dic3—21no—
78C23×Dic3—21no—
79C22×D12C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C3×C4×D4—22no—
83C3×C4×D4—41no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C22⋊C4—21no—
90C2×C3×C42C2311yescharacteristic
91C2×#217—21no—
92C2×#217—21no—
93D4×Dic3—22no—
94D4×Dic3—22no—
95D4×Dic3—22no—
96D4×Dic3—41no—
97C2×#213—22no—
98C2×#213—22no—
99C2×#213—22no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—21no—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—44no—
115C2×Dic3⋊C4D411yescharacteristic
116C2×Dic3⋊C4D411yescharacteristic
117C2×C4×Dic3—21no—
118C2×C4×Dic3C2311yescharacteristic
119C2×C4×Dic3C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122#898—44no—
123C3×C4⋊C8—41no—
124C3×#102—22no—
125C3×#102—22no—
126#832—22no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95D411yescharacteristic
132C3⋊#95D411yescharacteristic
133#821—22no—
134#821—41no—
135#818—22no—
136#815—41no—
137#809—41no—
138C22×D8—31no—
139C2×#126—61no—
140C2×C4×D4—31no—
141C2×C4×D4—31no—
142C2×C4×D4C2×D311yescharacteristic
143C22×C4⋊C4—61no—
144C2×#107—31no—
145C2×C4⋊C8C2×D311yescharacteristic
146C2×#102—31no—
147C2×#102—31no—
148C2×#102—31no—
149C2×#102—31no—
150C2×#102C2×D311yescharacteristic
151C2×#102C2×D311yescharacteristic
152C2×#97—31no—
153#397—61no—
154#396—61no—
155#390—31no—
156#390—31no—
157#390—31no—
158#389—31no—
159#389—31no—
160#389—31no—
161#388—61no—
162#385—61no—
163#385—61no—
164#384—31no—
165#384C2×D311yescharacteristic
166#380—61no—
167C4×C4⋊C4—31no—
168C4×C4⋊C4—31no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340C2×D311yescharacteristic
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#4112yes—
187C2×C3×D4#4112yes—
188C2×C3×D4#4112yes—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—42no—
193C22×C3×C4—42no—
194C22×C3×C4—81no—
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2411yescharacteristic
198C22×Dic3—24no—
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—42no—
203C22×Dic3—42no—
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—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—42no—
230C3×C4⋊C4—81no—
231C3×C22⋊C4—42no—
232C3×C42—21no—
233C3×C42—21no—
234#217—24no—
235#217—42no—
236#213—44no—
237(C2×D3)⋊C4—42no—
238(C2×D3)⋊C4—42no—
239(C2×D3)⋊C4—42no—
240(C2×D3)⋊C4—42no—
241(C2×D3)⋊C4—81no—
242(C2×D3)⋊C4—81no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—42no—
249C4⋊Dic3#4112yes—
250Dic3⋊C4—22no—
251Dic3⋊C4—22no—
252Dic3⋊C4—42no—
253Dic3⋊C4—42no—
254Dic3⋊C4—42no—
255Dic3⋊C4—81no—
256C4×Dic3—21no—
257C4×Dic3—21no—
258C4×Dic3—21no—
259C4×Dic3—21no—
260C4×Dic3—42no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C22×D4—31no—
268C22×D4C22×D311yescharacteristic
269C23×C4—21no—
270C23×C4—61no—
271C23×C4—61no—
272C2×D8—62no—
273C2×D8—62no—
274C2×D8—62no—
275C22×C8—31no—
276C22×C8—31no—
277C22×C8C22×D311yescharacteristic
278C22×C8C22×D311yescharacteristic
279#126—124no—
280C4×D4—22no—
281C4×D4—41no—
282C4×D4—62no—
283C4×D4—62no—
284C4×D4—62no—
285C4×D4—121no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—31no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4—31no—
292C2×C4⋊C4—31no—
293C2×C4⋊C4—61no—
294C2×C4⋊C4—61no—
295C2×C4⋊C4—61no—
296C2×C4⋊C4—61no—
297C2×C4⋊C4—61no—
298C2×C4⋊C4—124no—
299C2×C4⋊C4C22×D311yescharacteristic
300C2×C4⋊C4C22×D311yescharacteristic
301C2×C4⋊C4C22×D311yescharacteristic
302C2×C22⋊C4—21no—
303C2×C22⋊C4—61no—
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×C22⋊C4—61no—
311C2×C42—31no—
312C2×C42—31no—
313C2×C42—61no—
314C2×C42C22×D311yescharacteristic
315#107—62no—
316C4⋊C8—41no—
317#102—22no—
318#102—22no—
319#102—62no—
320#102—62no—
321#102—121no—
322#102—121no—
323#97—121no—
324#95—31no—
325#95—31no—
326#95—61no—
327#95—61no—
328#95—61no—
329#95—61no—
330C23×C3—22no—
331C23×C3—22no—
332C23×C3—22no—
333C23×C3—41no—
334C23×C3—41no—
335C23×C3—41no—
336C23×C3—41no—
337C23×C3C22×D411yescharacteristic
338C22×D3—41no—
339C22×D3—41no—
340C22×D3—41no—
341C22×D3—41no—
342C22×D3—41no—
343C22×D3—41no—
344C22×D3—41no—
345C3×D4—22no—
346C3×D4—22no—
347C3×D4—22no—
348C3×D4#12614yescomplemented
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
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×C4—41no—
363C2×C3×C4—84no—
364C2×C3×C4AGL(1,ℤ/8)12yes—
365C2×C3×C4C2×#4111yescharacteristic
366C2×C3×C4C2×#4111yescharacteristic
367C2×C3×C4C2×#4111yescharacteristic
368C2×C3×C4C2×#4111yescharacteristic
369C2×C3×C4H(5,𝔽2)11yescharacteristic
370C2×C3×C42−1+411yescharacteristic
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—22no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—48no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—84no—
393C2×Dic3AGL(1,ℤ/8)12yes—
394D12—41no—
395D12—41no—
396D12—41no—
397D12—41no—
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—61no—
408C2×D4—61no—
409C2×D4—61no—
410C2×D4—61no—
411C2×D4—61no—
412C2×D4—61no—
413C2×D4#23512yes—
414C2×D4#23712yes—
415C2×D4#23912yes—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—31no—
420C22×C4—31no—
421C22×C4—31no—
422C22×C4—31no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—64no—
426C22×C4—61no—
427C22×C4—61no—
428C22×C4—61no—
429C22×C4—81no—
430C22×C4—122no—
431C22×C4—122no—
432C22×C4—122no—
433C22×C4—122no—
434C22×C4—241no—
435C22×C4D3×D411yescharacteristic
436C22×C4D3×D411yescharacteristic
437C22×C4C23×D311yescharacteristic
438D8—124no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—61no—
446C2×C8—61no—
447C2×C8—61no—
448C2×C8—61no—
449C2×C8—61no—
450C2×C8—61no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—32no—
458C4⋊C4—42no—
459C4⋊C4—62no—
460C4⋊C4—62no—
461C4⋊C4—61no—
462C4⋊C4—61no—
463C4⋊C4—61no—
464C4⋊C4—61no—
465C4⋊C4—61no—
466C4⋊C4—81no—
467C4⋊C4—122no—
468C4⋊C4—122no—
469C4⋊C4—122no—
470C4⋊C4—122no—
471C4⋊C4—241no—
472C22⋊C4—42no—
473C22⋊C4—64no—
474C22⋊C4—122no—
475C22⋊C4—122no—
476C22⋊C4—122no—
477C22⋊C4—122no—
478C22⋊C4—122no—
479C22⋊C4—241no—
480C22⋊C4—241no—
481C42—21no—
482C42—21no—
483C42—61no—
484C42—61no—
485C42—61no—
486C42—61no—
487C42—122no—
488C22×C3—24no—
489C22×C3—42no—
490C22×C3—42no—
491C22×C3—42no—
492C22×C3—42no—
493C22×C3—42no—
494C22×C3—42no—
495C22×C3#52511yescharacteristic
496C22×C3#54611yescharacteristic
497C22×C3#54811yescharacteristic
498C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
499C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
500C22×C3#57611yescharacteristic
501C22×C3#57811yescharacteristic
502C2×D3—81no—
503C2×D3—81no—
504C2×D3—81no—
505C2×D3—81no—
506C2×D3—81no—
507C2×D3—81no—
508C2×D3—81no—
509C3×C4—22no—
510C3×C4—41no—
511C3×C4—81no—
512C3×C4—81no—
513C3×C4—81no—
514C3×C4C2×#12611yescharacteristic
515C3×C4#53911yescharacteristic
516C3×C4#55211yescharacteristic
517C3×C4#55311yescharacteristic
518Dic3—22no—
519Dic3—41no—
520Dic3—42no—
521Dic3—41no—
522Dic3—41no—
523Dic3—81no—
524Dic3—81no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—121no—
533C23—121no—
534C23—121no—
535C23—121no—
536C23—121no—
537C23—121no—
538C23—121no—
539C23C2×D3×D411yescharacteristic
540D4—22no—
541D4—22no—
542D4—22no—
543D4—121no—
544D4—121no—
545D4—121no—
546D4—121no—
547D4#89814yescomplemented
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—32no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—61no—
564C2×C4—61no—
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—84no—
576C2×C4—128no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—244no—
586C2×C4#91412yes—
587C2×C4C2×#23511yescharacteristic
588C2×C4C2×#23711yescharacteristic
589C2×C4#100411yescharacteristic
590C2×C4C2×#23911yescharacteristic
591C2×C4D3×#4111yescharacteristic
592C2×C4#101011yescharacteristic
593C8—41no—
594C8—41no—
595C8—121no—
596C8—121no—
597C2×C3—44no—
598C2×C3#248111yescharacteristic
599C2×C3#310911yescharacteristic
600C2×C3#312411yescharacteristic
601C2×C3#331811yescharacteristic
602C2×C3#333011yescharacteristic
603C2×C3#336611yescharacteristic
604C2×C3#336711yescharacteristic
605D3—161no—
606C22—24no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—241no—
614C22—241no—
615C22—241no—
616C22—241no—
617C22—241no—
618C22—241no—
619C22—241no—
620C22#549011yescharacteristic
621C22#559811yescharacteristic
622C22#562711yescharacteristic
623C22#575711yescharacteristic
624C22C2×#91411yescharacteristic
625C22D3×AGL(1,ℤ/8)11yescharacteristic
626C22#577911yescharacteristic
627C4—22no—
628C4—41no—
629C4—62no—
630C4—81no—
631C4—81no—
632C4—81no—
633C4—121no—
634C4—121no—
635C4—122no—
636C4—121no—
637C4—241no—
638C4—241no—
639C4C2×#89811yescharacteristic
640C4#553111yescharacteristic
641C4#560111yescharacteristic
642C4#564511yescharacteristic
643C3#2757111yescharacteristic, complemented, Sylow
644C2—44no—
645C2—481no—
646C2#7391011yescharacteristic
647C2#7876411yescharacteristic
648C2#7890611yescharacteristic
649C2#8032111yescharacteristic
650C2#8041611yescharacteristic
651C2#8095311yescharacteristic
652C2#8104611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1802C2
socleC23×C3C22×D4
prime radicalC22×#213C22
hypercenter#1802D3
hypocenterC3#27571
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
1. D4×Dic3⋊C4C2semidirect
2. #74333C2semidirect
3. C3×#1802C2semidirect
5. #70484C2semidirect
6. #70448C2semidirect
9. #70168C2semidirect
10. #70152C2semidirect
11. #69572C2semidirect
12. #68902C2semidirect
13. #68856C2semidirect
14. #68843C2semidirect
15. #68820C2semidirect
73. #1802D3semidirect
348. C3×D4#126semidirect
547. D4#898semidirect
643. C3#27571semidirect

Maximal subgroups

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

subgroupindex
D4×Dic3⋊C42
#743332
C3×#18022
#716132
#704842
#704482
#701722
#701682
#701522
#695722
#689022
#688562
#688432
#688202
#275713