Finite Group Databasebeta

← #10867883

Subgroups — #10867883

Order 768.

784 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2361C211yesmaximal, characteristic
2#73510C211yesmaximal, characteristic, complemented
3#71680C211yesmaximal, characteristic
4#71679C211yesmaximal, characteristic
5C4×#832C211yesmaximal, characteristic, complemented
6#71606C211yesmaximal, characteristic, complemented
7#71555C211yesmaximal, characteristic, complemented
8#71554C211yesmaximal, characteristic, complemented
9#69653C211yesmaximal, characteristic, complemented
10#69641C211yesmaximal, characteristic, complemented
11#69619C211yesmaximal, characteristic, complemented
12#69601C211yesmaximal, characteristic, complemented
13#69468C211yesmaximal, characteristic, complemented
14#69461C211yesmaximal, characteristic, complemented
15#69451C211yesmaximal, characteristic, complemented
16#27033—31nomaximal, Sylow
17C2×C3×#121C2211yescharacteristic
18C2×C3×C4×D4—21no—
19C2×C3×C4×D4C2211yescharacteristic
20C22×C3×C4⋊C4C2211yescharacteristic
21C3×#390C2211yescharacteristic
22C3×#389C2211yescharacteristic
23C3×#388—21no—
24C3×#386—21no—
25C3×#385—21no—
26C3×C4×C4⋊C4C2211yescharacteristic
27C3×C4×C22⋊C4C2211yescharacteristic
28C2×#832C2211yescharacteristic
29C2×#832C2211yescharacteristic
30C2×#832C2211yescharacteristic
31C2×#832C2211yescharacteristic
32#5212C2211yescharacteristic
33#5212C2211yescharacteristic
34C2×#830C2211yescharacteristic
35C2×#830C2211yescharacteristic
36C2×#808C2211yescharacteristic
37C2×#807C2211yescharacteristic
38C2×#807C2211yescharacteristic
39C2×#807C2211yescharacteristic
40#4952—21no—
41#4945—21no—
42#4936C2211yescharacteristic
43C4×C4⋊Dic3C2211yescharacteristic
44#4928C2211yescharacteristic
45#4928C2211yescharacteristic
46C2×C4⋊(C3⋊C8)C2211yescharacteristic
47C2×C4×C3⋊C8C2211yescharacteristic
48C3⋊#401—21no—
49#4652—21no—
50#4529C2211yescharacteristic
51#4529C2211yescharacteristic
52#4529C2211yescharacteristic
53#4529C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4524C2211yescharacteristic
59#4524C2211yescharacteristic
60#2494—31no—
61#2361D311yescharacteristic
62#2073—31no—
63#2067—31no—
64#2034—31no—
65#2033—31no—
66#1949—31no—
67#1942—31no—
68#1923—31no—
69#1921—31no—
70#1919—31no—
71#1862—31no—
72#1852—31no—
73#1824—31no—
74C4×#102—31no—
75C22×C3×D4—21no—
76C22×C3×D4C2311yescharacteristic
77C23×C3×C4—21no—
78C23×C3×C4—21no—
79C23×C3×C4C2311yescharacteristic
80C3×#121—22no—
81C3×#121—22no—
82C3×C4×D4—22no—
83C3×C4×D4—41no—
84C3×C4×D4—44no—
85C2×C3×C4⋊C4—22no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—41no—
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4D411yescharacteristic
97C2×C3×C22⋊C4D411yescharacteristic
98C2×C3×C42—21no—
99C2×C3×C42C2311yescharacteristic
100C2×C3×C42C2311yescharacteristic
101C2×#217—21no—
102C2×#217—21no—
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×C4×Dic3—21no—
110C22×C3⋊C8C2311yescharacteristic
111C22×C3⋊C8C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114C3×#95—21no—
115C3×#95—21no—
116C3×#95D411yescharacteristic
117C3×#95D411yescharacteristic
118#832—22no—
119#832—22no—
120#832—22no—
121#832—41no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129#830—22no—
130#830—41no—
131#808—22no—
132#807—22no—
133#807—22no—
134#807—41no—
135C4⋊(C3⋊C8)—22no—
136C4×C3⋊C8—41no—
137C2×#121C2×D311yescharacteristic
138C2×C4×D4—21no—
139C2×C4×D4C2×D311yescharacteristic
140C22×C4⋊C4C2×D311yescharacteristic
141C2×#107—31no—
142C2×#107—31no—
143C2×#107—31no—
144C2×#106—31no—
145C2×C4⋊C8—31no—
146C2×#102—31no—
147C2×#102—31no—
148C2×#102—31no—
149C2×#102—31no—
150C2×C22⋊C8—31no—
151C2×C22⋊C8—31no—
152C2×C4×C8—31no—
153#401—61no—
154#397—61no—
155#390C2×D311yescharacteristic
156#389—31no—
157#389—31no—
158#389C2×D311yescharacteristic
159#388—21no—
160#388—61no—
161#387—61no—
162#386—21no—
163#385—21no—
164#382—31no—
165#380—31no—
166#380—31no—
167C4×C4⋊C4—31no—
168C4×C4⋊C4C2×D311yescharacteristic
169C4×C22⋊C4C2×D311yescharacteristic
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#340—31no—
178#336—31no—
179#336—31no—
180C24×C3—21no—
181C24×C3C2×D411yescharacteristic
182C2×C3×D4—24no—
183C2×C3×D4—21no—
184C2×C3×D4—21no—
185C2×C3×D4—21no—
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4—42no—
189C2×C3×D4—42no—
190C2×C3×D4#4112yes—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—42no—
201C22×C3×C4—42no—
202C22×C3×C4—42no—
203C22×C3×C4—42no—
204C22×C3×C4—81no—
205C22×C3×C4—81no—
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2411yescharacteristic
212C22×Dic3—21no—
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—42no—
221C3×C4⋊C4—42no—
222C3×C4⋊C4#4112yes—
223C3×C4⋊C4#4112yes—
224C3×C22⋊C4—24no—
225C3×C22⋊C4—22no—
226C3×C22⋊C4—22no—
227C3×C22⋊C4—42no—
228C3×C22⋊C4—42no—
229C3×C22⋊C4—81no—
230C3×C42—21no—
231C3×C42—21no—
232C3×C42—21no—
233C3×C42—21no—
234C3×C42—42no—
235#217—42no—
236#217—81no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245Dic3⋊C4—42no—
246Dic3⋊C4—81no—
247C4×Dic3—81no—
248C2×C3⋊C8—21no—
249C2×C3⋊C8—21no—
250C2×C3⋊C8—21no—
251C2×C3⋊C8—21no—
252C2×C3⋊C8—21no—
253C2×C3⋊C8—21no—
254C2×C3⋊C8—21no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8#4112yes—
260C22×D4—21no—
261C22×D4C22×D311yescharacteristic
262C23×C4—21no—
263C23×C4—21no—
264C23×C4C22×D311yescharacteristic
265C22×C8—31no—
266C22×C8—31no—
267C22×C8—31no—
268C22×C8—31no—
269#121—22no—
270#121—22no—
271C4×D4—22no—
272C4×D4—41no—
273C4×D4—44no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—31no—
278C2×C4⋊C4—31no—
279C2×C4⋊C4—31no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—41no—
282C2×C4⋊C4—61no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4C22×D311yescharacteristic
285C2×C4⋊C4C22×D311yescharacteristic
286C2×C4⋊C4C22×D311yescharacteristic
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—61no—
292C2×C22⋊C4—61no—
293C2×C22⋊C4#6711yescharacteristic
294C2×C22⋊C4#6711yescharacteristic
295C2×C42—21no—
296C2×C42—61no—
297C2×C42C22×D311yescharacteristic
298C2×C42C22×D311yescharacteristic
299#107—62no—
300#107—62no—
301#107—121no—
302#106—62no—
303C4⋊C8—62no—
304#102—62no—
305#102—62no—
306#102—62no—
307#102—121no—
308C22⋊C8—62no—
309C22⋊C8—121no—
310C4×C8—121no—
311#95—21no—
312#95—21no—
313#95—61no—
314#95—61no—
315#95—61no—
316#95—61no—
317#95—61no—
318#95—61no—
319#95—61no—
320#95#6711yescharacteristic
321#95#6711yescharacteristic
322C23×C3—21no—
323C23×C3—21no—
324C23×C3—21no—
325C23×C3—21no—
326C23×C3—21no—
327C23×C3—21no—
328C23×C3—22no—
329C23×C3—41no—
330C23×C3—41no—
331C23×C3—41no—
332C23×C3—41no—
333C23×C3—41no—
334C23×C3—41no—
335C23×C3#13512yes—
336C23×C3C22×D411yescharacteristic
337C3×D4—22no—
338C3×D4—22no—
339C3×D4—48no—
340C3×D4—41no—
341C3×D4—41no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—22no—
346C2×C3×C4—21no—
347C2×C3×C4—22no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
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—84no—
370C2×C3×C4—84no—
371C2×C3×C4AGL(1,ℤ/8)12yes—
372C2×C3×C4C2×#4111yescharacteristic
373C2×C3×C4C2×#4111yescharacteristic
374C2×C3×C4C2×#4111yescharacteristic
375C2×C3×C4C2×#4111yescharacteristic
376C2×C3×C4H(5,𝔽2)11yescharacteristic
377C2×C3×C42−1+411yescharacteristic
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×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C3⋊C8—22no—
391C3⋊C8—41no—
392C3⋊C8—41no—
393C3⋊C8—41no—
394C24—21no—
395C24C2×#6711yescharacteristic
396C2×D4—24no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4#23512yes—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—61no—
419C22×C4—61no—
420C22×C4—61no—
421C22×C4—61no—
422C22×C4—81no—
423C22×C4—81no—
424C22×C4C2×#6711yescharacteristic
425C22×C4C2×#6711yescharacteristic
426C22×C4C2×#6711yescharacteristic
427C22×C4C2×#6711yescharacteristic
428C22×C4C2×#6711yescharacteristic
429C22×C4C23×D311yescharacteristic
430C2×C8—32no—
431C2×C8—61no—
432C2×C8—61no—
433C2×C8—61no—
434C2×C8—61no—
435C2×C8—61no—
436C2×C8—61no—
437C2×C8—61no—
438C2×C8—61no—
439C2×C8—61no—
440C2×C8—61no—
441C2×C8—61no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C4⋊C4—61no—
451C4⋊C4—61no—
452C4⋊C4—61no—
453C4⋊C4—61no—
454C4⋊C4—61no—
455C4⋊C4—61no—
456C4⋊C4—122no—
457C4⋊C4—241no—
458C4⋊C4#23512yes—
459C4⋊C4#23712yes—
460C22⋊C4—24no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—42no—
464C22⋊C4—42no—
465C22⋊C4—81no—
466C22⋊C4—122no—
467C22⋊C4—241no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—42no—
473C42—241no—
474C22×C3—22no—
475C22×C3—22no—
476C22×C3—22no—
477C22×C3—41no—
478C22×C3—41no—
479C22×C3—41no—
480C22×C3—41no—
481C22×C3—42no—
482C22×C3—42no—
483C22×C3—42no—
484C22×C3—81no—
485C22×C3—81no—
486C22×C3—81no—
487C22×C3—81no—
488C22×C3#52511yescharacteristic
489C22×C3#54611yescharacteristic
490C22×C3#54711yescharacteristic
491C22×C3C2×#13511yescharacteristic
492C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
493C22×C3#57611yescharacteristic
494C22×C3#57811yescharacteristic
495C3×C4—22no—
496C3×C4—41no—
497C3×C4—41no—
498C3×C4—42no—
499C3×C4—41no—
500C3×C4—41no—
501C3×C4—42no—
502C3×C4—81no—
503C3×C4#52911yescharacteristic
504C3×C4#53911yescharacteristic
505C3×C4#55311yescharacteristic
506C3×C4#55611yescharacteristic
507Dic3—81no—
508Dic3—81no—
509Dic3—81no—
510Dic3—81no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—22no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23#95012yes—
525C23C22×#6711yescharacteristic
526D4—22no—
527D4—22no—
528D4—48no—
529D4—41no—
530D4—41no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
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—84no—
559C2×C4—84no—
560C2×C4—121no—
561C2×C4—121no—
562C2×C4—121no—
563C2×C4—121no—
564C2×C4—121no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4—121no—
569C2×C4—121no—
570C2×C4—121no—
571C2×C4—121no—
572C2×C4#93212yes—
573C2×C4C2×#23511yescharacteristic
574C2×C4C2×#23511yescharacteristic
575C2×C4C2×#23711yescharacteristic
576C2×C4#100411yescharacteristic
577C2×C4D3×#4111yescharacteristic
578C2×C4#101011yescharacteristic
579C8—62no—
580C8—121no—
581C8—121no—
582C8—121no—
583C2×C3—42no—
584C2×C3—82no—
585C2×C3#246611yescharacteristic
586C2×C3#310511yescharacteristic
587C2×C3#312411yescharacteristic
588C2×C3#330011yescharacteristic
589C2×C3#331811yescharacteristic
590C2×C3#333911yescharacteristic
591C2×C3#335311yescharacteristic
592C22—22no—
593C22—22no—
594C22—22no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—81no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22#579111yescharacteristic
607C22C2×#93211yescharacteristic
608C22#580611yescharacteristic
609C22C2×#95011yescharacteristic
610C22#582411yescharacteristic
611C22#583511yescharacteristic
612C22#583811yescharacteristic
613C4—22no—
614C4—42no—
615C4—41no—
616C4—41no—
617C4—42no—
618C4—41no—
619C4—41no—
620C4—81no—
621C4—241no—
622C4—241no—
623C4—241no—
624C4—241no—
625C4#553811yescharacteristic
626C4#554211yescharacteristic
627C4#555811yescharacteristic
628C4#556411yescharacteristic
629C3#2703311yescharacteristic, complemented, Sylow
630C2—42no—
631C2—82no—
632C2#8230911yescharacteristic
633C2#8236911yescharacteristic
634C2#8256911yescharacteristic
635C2#8348511yescharacteristic
636C2#8352011yescharacteristic
637C2#8353411yescharacteristic
638C2#8360711yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2361C2
socleC23×C3C22×D4
prime radicalC2×C3×#121C22
hypercenter#2361D3
hypocenterC3#27033
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2361

Complements

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

NH ≅ G/Ntype
2. #73510C2semidirect
5. C4×#832C2semidirect
6. #71606C2semidirect
7. #71555C2semidirect
8. #71554C2semidirect
9. #69653C2semidirect
10. #69641C2semidirect
11. #69619C2semidirect
12. #69601C2semidirect
13. #69468C2semidirect
14. #69461C2semidirect
15. #69451C2semidirect
629. C3#27033semidirect

Maximal subgroups

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

subgroupindex
C3×#23612
#735102
#716802
#716792
C4×#8322
#716062
#715552
#715542
#696532
#696412
#696192
#696012
#694682
#694612
#694512
#270333