Finite Group Databasebeta

← #10875920

Subgroups — #10875920

Order 768.

808 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#76341C211yesmaximal, characteristic, complemented
2#76202C211yesmaximal, characteristic, complemented
3C3×#1936C211yesmaximal, characteristic, complemented
4#70208C211yesmaximal, characteristic
5#70208C211yesmaximal, characteristic
6#70193C211yesmaximal, characteristic, complemented
7#70165C211yesmaximal, characteristic, complemented
8#69834C211yesmaximal, characteristic, complemented
9#69812C211yesmaximal, characteristic, complemented
10#69657C211yesmaximal, characteristic
11#69602C211yesmaximal, characteristic, complemented
12#68895C211yesmaximal, characteristic, complemented
13#68893C211yesmaximal, characteristic, complemented
14#68845C211yesmaximal, characteristic, complemented
15#68842C211yesmaximal, characteristic, complemented
16#27568—31nomaximal, Sylow
17C22×#213C2211yescharacteristic
18C2×#893C2211yescharacteristic
19C2×#893C2211yescharacteristic
20C2×D3×C4⋊C4—21no—
21C2×C4×D12C2211yescharacteristic
22C2×C3×#102C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C2×C3×#97C2211yescharacteristic
25C3×#390C2211yescharacteristic
26C3×#389C2211yescharacteristic
27#5221C2211yescharacteristic
28#5221C2211yescharacteristic
29C2×#832C2211yescharacteristic
30C2×#821C2211yescharacteristic
31C2×#821C2211yescharacteristic
32C2×Dic3⋊C8C2211yescharacteristic
33#4989—21no—
34#4984C2211yescharacteristic
35#4984C2211yescharacteristic
36#4977C2211yescharacteristic
37#4975C2211yescharacteristic
38C2×#809C2211yescharacteristic
39C2×#808C2211yescharacteristic
40#4953—21no—
41#4952—21no—
42#4951—21no—
43#4945—21no—
44C4×C4⋊Dic3C2211yescharacteristic
45C4×Dic3⋊C4C2211yescharacteristic
46#4668—21no—
47#4667—21no—
48C3×#340C2211yescharacteristic
49C3×#340C2211yescharacteristic
50#4551C2211yescharacteristic
51#4551C2211yescharacteristic
52#4550C2211yescharacteristic
53#4550C2211yescharacteristic
54#4550C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2513—31no—
61D4×C4⋊C4—31no—
62#2101—31no—
63#2099—31no—
64#2071—31no—
65#2065—31no—
66#1936D311yescharacteristic, complemented
67#1920—31no—
68#1917—31no—
69#1879—31no—
70#1879—31no—
71#1879—31no—
72#1864—31no—
73#1804—31no—
74#1762—31no—
75C22×C3×D4C2311yescharacteristic
76C22×D12C2311yescharacteristic
77C22×C4×D3—21no—
78C22×C4×D3—21no—
79C22×C4×D3—21no—
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C22⋊C4—21no—
87C2×C3×C22⋊C4—21no—
88C2×C3×C42C2311yescharacteristic
89C2×#217—21no—
90C2×#217—21no—
91C2×#217—21no—
92C2×#217—21no—
93C2×#213—22no—
94C2×#213—22no—
95C2×#213—22no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4D411yescharacteristic
107C2×Dic3⋊C4D411yescharacteristic
108C2×C4×Dic3—21no—
109C2×C4×Dic3C2311yescharacteristic
110C2×C4×Dic3C2311yescharacteristic
111C22×C3⋊C8C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113#893—22no—
114#893—22no—
115#893—22no—
116#893—41no—
117D3×C4⋊C4—44no—
118C4×D12—22no—
119C4×D12—41no—
120C3×#102—22no—
121C3×#102—41no—
122C3×#97—41no—
123C3×#95—21no—
124#832—41no—
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—
134Dic3⋊C8—41no—
135#809—22no—
136#808—22no—
137C22×D8—31no—
138C2×C4×D4—31no—
139C2×C4×D4—31no—
140C2×C4×D4—31no—
141C22×C4⋊C4—61no—
142C2×#106—31no—
143C2×C4⋊C8—31no—
144C2×#102—31no—
145C2×#102—31no—
146C2×#102—31no—
147C2×#102—31no—
148C2×#102C2×D311yescharacteristic
149C2×#102C2×D311yescharacteristic
150C2×#97C2×D311yescharacteristic
151#397—61no—
152#397—61no—
153#390—31no—
154#390—31no—
155#390C2×D311yescharacteristic
156#389—31no—
157#389—31no—
158#389C2×D311yescharacteristic
159#388—61no—
160#388—61no—
161#385—61no—
162#385—61no—
163#384—31no—
164#382—31no—
165#380—61no—
166C4×C4⋊C4—31no—
167C4×C4⋊C4—31no—
168#340—31no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340C2×D311yescharacteristic
175#340C2×D311yescharacteristic
176#336—31no—
177#336—31no—
178#336—31no—
179#336—31no—
180C24×C3—21no—
181C23×D3—21no—
182C2×C3×D4—21no—
183C2×C3×D4—21no—
184C2×C3×D4—21no—
185C2×C3×D4—21no—
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2411yescharacteristic
194C22×Dic3—21no—
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#4112yes—
205C2×D12#4112yes—
206C2×D12#4112yes—
207C2×C4×D3—24no—
208C2×C4×D3—42no—
209C2×C4×D3—42no—
210C2×C4×D3—42no—
211C2×C4×D3—42no—
212C2×C4×D3—42no—
213C2×C4×D3—42no—
214C2×C4×D3—81no—
215C2×C4×D3—81no—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—42no—
228C3×C4⋊C4#4112yes—
229C3×C22⋊C4—42no—
230C3×C22⋊C4—81no—
231C3×C42—21no—
232C3×C42—21no—
233#217—42no—
234#217—81no—
235#217—81no—
236#217—81no—
237#213—44no—
238(C2×D3)⋊C4—24no—
239(C2×D3)⋊C4—42no—
240(C2×D3)⋊C4—42no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—42no—
248C4⋊Dic3—81no—
249Dic3⋊C4—22no—
250Dic3⋊C4—41no—
251Dic3⋊C4—42no—
252Dic3⋊C4—81no—
253C4×Dic3—21no—
254C4×Dic3—21no—
255C4×Dic3—21no—
256C4×Dic3—21no—
257C4×Dic3—81no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C22×D4—31no—
265C22×D4C22×D311yescharacteristic
266C23×C4—61no—
267C23×C4—61no—
268C23×C4—61no—
269C2×D8—62no—
270C2×D8—62no—
271C2×D8—62no—
272C22×C8—31no—
273C22×C8—31no—
274C22×C8C22×D311yescharacteristic
275C22×C8C22×D311yescharacteristic
276C4×D4—62no—
277C4×D4—62no—
278C4×D4—62no—
279C4×D4—62no—
280C4×D4—121no—
281C4×D4—121no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—31no—
286C2×C4⋊C4—31no—
287C2×C4⋊C4—31no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4—61no—
291C2×C4⋊C4—61no—
292C2×C4⋊C4—124no—
293C2×C4⋊C4C22×D311yescharacteristic
294C2×C4⋊C4C22×D311yescharacteristic
295C2×C4⋊C4C22×D311yescharacteristic
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—61no—
299C2×C22⋊C4—61no—
300C2×C22⋊C4—61no—
301C2×C22⋊C4—61no—
302C2×C22⋊C4—61no—
303C2×C22⋊C4—61no—
304C2×C22⋊C4—61no—
305C2×C42—31no—
306C2×C42—31no—
307C2×C42—61no—
308C2×C42C22×D311yescharacteristic
309#106—62no—
310C4⋊C8—121no—
311#102—22no—
312#102—41no—
313#102—62no—
314#102—62no—
315#102—62no—
316#102—121no—
317#97—41no—
318#95—21no—
319#95—31no—
320#95—31no—
321#95—61no—
322#95—61no—
323#95—61no—
324#95—61no—
325#95—61no—
326C23×C3—41no—
327C23×C3—41no—
328C23×C3—41no—
329C23×C3—41no—
330C23×C3—41no—
331C23×C3—41no—
332C23×C3—41no—
333C23×C3C22×D411yescharacteristic
334C22×D3—22no—
335C22×D3—22no—
336C22×D3—22no—
337C22×D3—41no—
338C22×D3—41no—
339C22×D3—41no—
340C22×D3—41no—
341C3×D4—41no—
342C3×D4—41no—
343C3×D4—41no—
344C3×D4—41no—
345C2×C3×C4—22no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
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×C4C2×#4111yescharacteristic
361C2×C3×C4C2×#4111yescharacteristic
362C2×C3×C4C2×#4111yescharacteristic
363C2×C3×C4C2×#4111yescharacteristic
364C2×C3×C4H(5,𝔽2)11yescharacteristic
365C2×C3×C42−1+411yescharacteristic
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3AGL(1,ℤ/8)12yes—
387D12—22no—
388D12—22no—
389D12—22no—
390D12#12614yes—
391C4×D3—48no—
392C4×D3—84no—
393C4×D3—84no—
394C3×C8—41no—
395C3×C8—41no—
396C3⋊C8—41no—
397C3⋊C8—41no—
398C24—21no—
399C24—61no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—32no—
407C2×D4—32no—
408C2×D4—32no—
409C2×D4—61no—
410C2×D4—61no—
411C2×D4—61no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—31no—
416C22×C4—31no—
417C22×C4—31no—
418C22×C4—31no—
419C22×C4—64no—
420C22×C4—61no—
421C22×C4—61no—
422C22×C4—61no—
423C22×C4—122no—
424C22×C4—122no—
425C22×C4—122no—
426C22×C4—122no—
427C22×C4—122no—
428C22×C4—122no—
429C22×C4—241no—
430C22×C4—241no—
431C22×C4D3×D411yescharacteristic
432C22×C4D3×D411yescharacteristic
433C22×C4C23×D311yescharacteristic
434D8—124no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—61no—
442C2×C8—61no—
443C2×C8—61no—
444C2×C8—61no—
445C2×C8—61no—
446C2×C8—61no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—42no—
453C4⋊C4—62no—
454C4⋊C4—61no—
455C4⋊C4—61no—
456C4⋊C4—61no—
457C4⋊C4—61no—
458C4⋊C4—61no—
459C4⋊C4—61no—
460C4⋊C4—121no—
461C4⋊C4—122no—
462C4⋊C4—122no—
463C4⋊C4—241no—
464C4⋊C4—241no—
465C4⋊C4#23512yes—
466C22⋊C4—42no—
467C22⋊C4—64no—
468C22⋊C4—81no—
469C22⋊C4—122no—
470C22⋊C4—122no—
471C22⋊C4—122no—
472C22⋊C4—241no—
473C22⋊C4—241no—
474C22⋊C4—241no—
475C42—21no—
476C42—21no—
477C42—61no—
478C42—61no—
479C42—61no—
480C42—61no—
481C42—241no—
482C22×C3—81no—
483C22×C3—81no—
484C22×C3—81no—
485C22×C3—81no—
486C22×C3—81no—
487C22×C3—81no—
488C22×C3—81no—
489C22×C3#52511yescharacteristic
490C22×C3#54611yescharacteristic
491C22×C3#54811yescharacteristic
492C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
493C22×C3#57511yescharacteristic
494C22×C3#57611yescharacteristic
495C22×C3#57711yescharacteristic
496C2×D3—24no—
497C2×D3—42no—
498C2×D3—42no—
499C2×D3—42no—
500C2×D3—42no—
501C2×D3—42no—
502C2×D3—42no—
503C3×C4—42no—
504C3×C4—41no—
505C3×C4—41no—
506C3×C4—81no—
507C3×C4—81no—
508C3×C4C2×#12611yescharacteristic
509C3×C4#53911yescharacteristic
510C3×C4#55211yescharacteristic
511C3×C4#55311yescharacteristic
512Dic3—22no—
513Dic3—41no—
514Dic3—41no—
515Dic3—41no—
516Dic3—81no—
517Dic3—81no—
518Dic3—81no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—62no—
527C23—62no—
528C23—62no—
529C23—121no—
530C23—121no—
531C23—121no—
532C23—121no—
533C23C2×D3×D411yescharacteristic
534D4—34no—
535D4—41no—
536D4—41no—
537D4—41no—
538D4—41no—
539D4—62no—
540D4—62no—
541D4—62no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—32no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—61no—
559C2×C4—61no—
560C2×C4—61no—
561C2×C4—61no—
562C2×C4—61no—
563C2×C4—61no—
564C2×C4—61no—
565C2×C4—61no—
566C2×C4—61no—
567C2×C4—61no—
568C2×C4—61no—
569C2×C4—128no—
570C2×C4—121no—
571C2×C4—121no—
572C2×C4—121no—
573C2×C4—121no—
574C2×C4—121no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—244no—
580C2×C4—244no—
581C2×C4C2×#23511yescharacteristic
582C2×C4#100411yescharacteristic
583C2×C4D3×#4111yescharacteristic
584C2×C4D3×#4111yescharacteristic
585C2×C4D3×#4111yescharacteristic
586C2×C4#101011yescharacteristic
587C8—41no—
588C8—41no—
589C8—121no—
590C8—121no—
591C2×C3—161no—
592C2×C3#248111yescharacteristic
593C2×C3#311411yescharacteristic
594C2×C3#311911yescharacteristic
595C2×C3#330711yescharacteristic
596C2×C3#332511yescharacteristic
597C2×C3#337011yescharacteristic
598C2×C3#337311yescharacteristic
599D3—44no—
600C22—64no—
601C22—81no—
602C22—81no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22—81no—
608C22—122no—
609C22—122no—
610C22—122no—
611C22—122no—
612C22—122no—
613C22—122no—
614C22#554711yescharacteristic
615C22#561011yescharacteristic
616C22#567711yescharacteristic
617C22D3×AGL(1,ℤ/8)11yescharacteristic
618C22#577311yescharacteristic
619C22#577411yescharacteristic
620C22#577811yescharacteristic
621C4—41no—
622C4—41no—
623C4—42no—
624C4—62no—
625C4—81no—
626C4—81no—
627C4—121no—
628C4—121no—
629C4—121no—
630C4—241no—
631C4—241no—
632C4—241no—
633C4#552911yescharacteristic
634C4#555811yescharacteristic
635C4#566011yescharacteristic
636C4D3×#12611yescharacteristic
637C3#2756811yescharacteristic, complemented, Sylow
638C2—124no—
639C2—161no—
640C2#7554211yescharacteristic
641C2#7981211yescharacteristic
642C2#7987111yescharacteristic
643C2#8081911yescharacteristic
644C2#8088411yescharacteristic
645C2#8167411yescharacteristic
646C2#8171011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1936C2
socleC23×C3C22×D4
prime radicalC22×#213C22
hypercenter#1936D3
hypocenterC3#27568
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1936

Complements

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

NH ≅ G/Ntype
1. #76341C2semidirect
2. #76202C2semidirect
3. C3×#1936C2semidirect
6. #70193C2semidirect
7. #70165C2semidirect
8. #69834C2semidirect
9. #69812C2semidirect
11. #69602C2semidirect
12. #68895C2semidirect
13. #68893C2semidirect
14. #68845C2semidirect
15. #68842C2semidirect
66. #1936D3semidirect
637. C3#27568semidirect

Maximal subgroups

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

subgroupindex
#763412
#762022
C3×#19362
#702082
#701932
#701652
#698342
#698122
#696572
#696022
#688952
#688932
#688452
#688422
#275683