Finite Group Databasebeta

← #10863709

Subgroups — #10863709

Order 768.

804 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
1C3×#2508C211yesmaximal, characteristic, complemented
2#73653C211yesmaximal, characteristic
3#69630C211yesmaximal, characteristic, complemented
4#69627C211yesmaximal, characteristic
5#69627C211yesmaximal, characteristic
6#69626C211yesmaximal, characteristic, complemented
7#69626C211yesmaximal, characteristic, complemented
8#69615C211yesmaximal, characteristic, complemented
9#69603C211yesmaximal, characteristic, complemented
10#69601C211yesmaximal, characteristic, complemented
11#69557C211yesmaximal, characteristic, complemented
12#69556C211yesmaximal, characteristic, complemented
13#69482C211yesmaximal, characteristic, complemented
14C4×#809C211yesmaximal, characteristic, complemented
15#69462C211yesmaximal, characteristic, complemented
16#26753—31nomaximal, Sylow
17C2×#893—21no—
18C2×#874C2211yescharacteristic
19C2×C4×D12C2211yescharacteristic
20C2×C4×D12C2211yescharacteristic
21C3×#400—21no—
22C3×#398—21no—
23C3×#389C2211yescharacteristic
24C3×#389C2211yescharacteristic
25C3×#389C2211yescharacteristic
26C3×#389C2211yescharacteristic
27C3×#387—21no—
28C3×#382—21no—
29C3×#382C2211yescharacteristic
30C3×C4×C4⋊C4C2211yescharacteristic
31C3×C4×C4⋊C4C2211yescharacteristic
32#4988—21no—
33C2×#809C2211yescharacteristic
34C2×#809C2211yescharacteristic
35C2×#809C2211yescharacteristic
36C2×#809C2211yescharacteristic
37C2×#809C2211yescharacteristic
38C2×#809C2211yescharacteristic
39C2×#809C2211yescharacteristic
40C2×#809C2211yescharacteristic
41C2×#808C2211yescharacteristic
42C2×#807C2211yescharacteristic
43#4939C2211yescharacteristic
44#4939C2211yescharacteristic
45C4×C4⋊Dic3C2211yescharacteristic
46C2×C4⋊(C3⋊C8)C2211yescharacteristic
47C2×C4×C3⋊C8C2211yescharacteristic
48#4665—21no—
49#4665—21no—
50#4529C2211yescharacteristic
51#4529C2211yescharacteristic
52#4529C2211yescharacteristic
53#4529C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4524C2211yescharacteristic
57#4524C2211yescharacteristic
58#4524C2211yescharacteristic
59#4524C2211yescharacteristic
60#2508D311yescharacteristic, complemented
61#2366—31no—
62#2073—31no—
63#2067—31no—
64#2041—31no—
65#2041—31no—
66#1934—31no—
67#1925—31no—
68#1921—31no—
69#1919—31no—
70#1916—31no—
71#1864—31no—
72#1864—31no—
73#1804—31no—
74C4×#102—31no—
75C22×D12—21no—
76C22×D12—21no—
77C22×D12C2311yescharacteristic
78C22×D12C2311yescharacteristic
79C22×C4×D3—21no—
80C22×C4×D3—21no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4D411yescharacteristic
86C2×C3×C4⋊C4D411yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C42—21no—
92C2×C3×C42C2311yescharacteristic
93C2×C3×C42C2311yescharacteristic
94C2×C3×C42C2311yescharacteristic
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×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4×Dic3—21no—
104C22×C3⋊C8C2311yescharacteristic
105C22×C3⋊C8C2311yescharacteristic
106C22×C3⋊C8C2311yescharacteristic
107C22×C3⋊C8C2311yescharacteristic
108#893—44no—
109#874—22no—
110#874—22no—
111C4×D12—22no—
112C4×D12—22no—
113C4×D12—41no—
114C4×D12—41no—
115C3×#95—21no—
116C3×#95—21no—
117C3×#95—21no—
118C3×#95—21no—
119C3×#95—21no—
120C3×#95—21no—
121C3×#95—21no—
122C3×#95D411yescharacteristic
123C3×#95D411yescharacteristic
124C3⋊#95—21no—
125#809—22no—
126#809—22no—
127#809—22no—
128#809—22no—
129#809—22no—
130#809—22no—
131#809—41no—
132#809—41no—
133#808—22no—
134#807—22no—
135C4⋊(C3⋊C8)—41no—
136C4×C3⋊C8—41no—
137C2×#127—31no—
138C2×C4×D4—31no—
139C2×C4×D4—31no—
140C2×C4×D4—61no—
141C2×#107—31no—
142C2×#106—31no—
143C2×C4⋊C8—31no—
144C2×#102—31no—
145C2×#102—31no—
146C2×#102—31no—
147C2×#102—31no—
148C2×#102—31no—
149C2×#102—31no—
150C2×#102—31no—
151C2×#102—31no—
152C2×C4×C8—31no—
153#400—21no—
154#398—21no—
155#390—31no—
156#390—31no—
157#390—61no—
158#389C2×D311yescharacteristic
159#389C2×D311yescharacteristic
160#389C2×D311yescharacteristic
161#389C2×D311yescharacteristic
162#387—21no—
163#386—61no—
164#386—61no—
165#382—21no—
166#382C2×D311yescharacteristic
167C4×C4⋊C4—31no—
168C4×C4⋊C4C2×D311yescharacteristic
169C4×C4⋊C4C2×D311yescharacteristic
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#336—31no—
177#336—31no—
178#336—31no—
179#336—31no—
180C23×D3—21no—
181C23×D3—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2411yescharacteristic
193C22×Dic3—21no—
194C22×Dic3—21no—
195C2×D12—21no—
196C2×D12—21no—
197C2×D12—21no—
198C2×D12—21no—
199C2×D12—21no—
200C2×D12—21no—
201C2×D12—21no—
202C2×D12—21no—
203C2×D12—24no—
204C2×D12—21no—
205C2×D12—21no—
206C2×D12—24no—
207C2×D12—42no—
208C2×D12—42no—
209C2×D12—81no—
210C2×D12—81no—
211C2×D12#4112yes—
212C2×D12#4112yes—
213C2×C4×D3—42no—
214C2×C4×D3—42no—
215C2×C4×D3—42no—
216C2×C4×D3—42no—
217C2×C4×D3—81no—
218C2×C4×D3—81no—
219C3×C4⋊C4—22no—
220C3×C4⋊C4—22no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—42no—
229C3×C4⋊C4—81no—
230C3×C4⋊C4—81no—
231C3×C4⋊C4—81no—
232C3×C4⋊C4#4112yes—
233C3×C42—21no—
234C3×C42—21no—
235C3×C42—21no—
236C3×C42—21no—
237C3×C42—21no—
238C3×C42—21no—
239C3×C42—42no—
240(C2×D3)⋊C4—42no—
241(C2×D3)⋊C4—42no—
242(C2×D3)⋊C4—42no—
243(C2×D3)⋊C4—42no—
244(C2×D3)⋊C4—81no—
245(C2×D3)⋊C4—81no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3#4112yes—
250C4×Dic3—42no—
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—21no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8—21no—
263C22×D4—31no—
264C22×D4—31no—
265C22×D4—61no—
266C22×D4—61no—
267C23×C4—61no—
268C23×C4—61no—
269C22×C8—31no—
270C22×C8—31no—
271C22×C8—31no—
272C22×C8—31no—
273#127—62no—
274#127—62no—
275C4×D4—62no—
276C4×D4—62no—
277C4×D4—121no—
278C4×D4—121no—
279C4×D4—124no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—31no—
286C2×C4⋊C4#6711yescharacteristic
287C2×C4⋊C4#6711yescharacteristic
288C2×C4⋊C4C22×D311yescharacteristic
289C2×C4⋊C4C22×D311yescharacteristic
290C2×C4⋊C4C22×D311yescharacteristic
291C2×C4⋊C4C22×D311yescharacteristic
292C2×C22⋊C4—61no—
293C2×C22⋊C4—61no—
294C2×C22⋊C4—61no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—61no—
297C2×C22⋊C4—61no—
298C2×C42—21no—
299C2×C42—61no—
300C2×C42C22×D311yescharacteristic
301C2×C42C22×D311yescharacteristic
302C2×C42C22×D311yescharacteristic
303#107—62no—
304#106—62no—
305C4⋊C8—121no—
306#102—62no—
307#102—62no—
308#102—62no—
309#102—62no—
310#102—62no—
311#102—62no—
312#102—121no—
313#102—121no—
314C4×C8—121no—
315#95—21no—
316#95—21no—
317#95—21no—
318#95—21no—
319#95—21no—
320#95—21no—
321#95—21no—
322#95—61no—
323#95#6711yescharacteristic
324#95#6711yescharacteristic
325C23×C3C22×D411yescharacteristic
326C22×D3—22no—
327C22×D3—22no—
328C22×D3—41no—
329C22×D3—41no—
330C22×D3—41no—
331C22×D3—41no—
332C22×D3—41no—
333C22×D3—41no—
334C22×D3—41no—
335C22×D3—41no—
336C22×D3—41no—
337C22×D3—41no—
338C22×D3—41no—
339C22×D3—41no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—22no—
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—41no—
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#13512yes—
369C2×C3×C4AGL(1,ℤ/8)12yes—
370C2×C3×C4C2×#4111yescharacteristic
371C2×C3×C4C2×#4111yescharacteristic
372C2×C3×C4C2×#4111yescharacteristic
373C2×C3×C4C2×#4111yescharacteristic
374C2×C3×C4H(5,𝔽2)11yescharacteristic
375C2×C3×C42−1+411yescharacteristic
376C2×Dic3—22no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382D12—22no—
383D12—22no—
384D12—22no—
385D12—22no—
386D12—48no—
387D12—41no—
388D12—41no—
389D12—41no—
390D12—41no—
391D12—84no—
392C4×D3—84no—
393C4×D3—84no—
394C3⋊C8—41no—
395C3⋊C8—41no—
396C3⋊C8—41no—
397C3⋊C8—41no—
398C24—61no—
399C24—61no—
400C2×D4—32no—
401C2×D4—32no—
402C2×D4—64no—
403C2×D4—64no—
404C2×D4—61no—
405C2×D4—61no—
406C2×D4—61no—
407C2×D4—61no—
408C2×D4—61no—
409C2×D4—61no—
410C2×D4—61no—
411C2×D4—61no—
412C2×D4—61no—
413C2×D4—61no—
414C2×D4—122no—
415C2×D4—122no—
416C2×D4—241no—
417C2×D4—241no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—61no—
423C22×C4—61no—
424C22×C4—122no—
425C22×C4—122no—
426C22×C4—122no—
427C22×C4—122no—
428C22×C4—241no—
429C22×C4—241no—
430C22×C4C2×#6711yescharacteristic
431C22×C4C2×#6711yescharacteristic
432C22×C4C2×#6711yescharacteristic
433C22×C4C2×#6711yescharacteristic
434C22×C4C2×#6711yescharacteristic
435C22×C4C2×#6711yescharacteristic
436C22×C4C23×D311yescharacteristic
437C2×C8—61no—
438C2×C8—61no—
439C2×C8—61no—
440C2×C8—61no—
441C2×C8—61no—
442C2×C8—61no—
443C2×C8—61no—
444C2×C8—61no—
445C2×C8—61no—
446C2×C8—61no—
447C2×C8—61no—
448C2×C8—61no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—32no—
459C4⋊C4—42no—
460C4⋊C4—61no—
461C4⋊C4—61no—
462C4⋊C4—61no—
463C4⋊C4—81no—
464C4⋊C4—81no—
465C4⋊C4—81no—
466C4⋊C4#23912yes—
467C22⋊C4—122no—
468C22⋊C4—122no—
469C22⋊C4—122no—
470C22⋊C4—122no—
471C22⋊C4—241no—
472C22⋊C4—241no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—42no—
480C42—122no—
481C22×C3#52511yescharacteristic
482C22×C3#54611yescharacteristic
483C22×C3#54811yescharacteristic
484C22×C3C2×#13511yescharacteristic
485C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
486C22×C3#57611yescharacteristic
487C22×C3#57711yescharacteristic
488C2×D3—42no—
489C2×D3—42no—
490C2×D3—42no—
491C2×D3—42no—
492C2×D3—42no—
493C2×D3—42no—
494C2×D3—81no—
495C2×D3—81no—
496C2×D3—81no—
497C2×D3—81no—
498C2×D3—81no—
499C2×D3—81no—
500C2×D3—81no—
501C2×D3—81no—
502C3×C4—22no—
503C3×C4—22no—
504C3×C4—42no—
505C3×C4—41no—
506C3×C4—41no—
507C3×C4—41no—
508C3×C4—41no—
509C3×C4—81no—
510C3×C4—81no—
511C3×C4—81no—
512C3×C4#52911yescharacteristic
513C3×C4#53911yescharacteristic
514C3×C4#55211yescharacteristic
515C3×C4#55311yescharacteristic
516Dic3—42no—
517Dic3—81no—
518C23—62no—
519C23—62no—
520C23—121no—
521C23—121no—
522C23—121no—
523C23—121no—
524C23—121no—
525C23—121no—
526C23—121no—
527C23—121no—
528C23—121no—
529C23—121no—
530C23—121no—
531C23—121no—
532C23C22×#6711yescharacteristic
533D4—62no—
534D4—62no—
535D4—62no—
536D4—62no—
537D4—128no—
538D4—121no—
539D4—121no—
540D4—121no—
541D4—121no—
542D4—244no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—62no—
572C2×C4—121no—
573C2×C4—121no—
574C2×C4—121no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—244no—
578C2×C4—244no—
579C2×C4#94912yes—
580C2×C4#95012yes—
581C2×C4C2×#23911yescharacteristic
582C2×C4#100711yescharacteristic
583C2×C4D3×#4111yescharacteristic
584C2×C4D3×#4111yescharacteristic
585C2×C4D3×#4111yescharacteristic
586C2×C4#100911yescharacteristic
587C8—121no—
588C8—121no—
589C8—121no—
590C8—121no—
591C2×C3#246011yescharacteristic
592C2×C3#310511yescharacteristic
593C2×C3#311811yescharacteristic
594C2×C3#330511yescharacteristic
595C2×C3#331011yescharacteristic
596C2×C3#336511yescharacteristic
597C2×C3#336611yescharacteristic
598D3—82no—
599D3—82no—
600C22—122no—
601C22—122no—
602C22—122no—
603C22—122no—
604C22—122no—
605C22—122no—
606C22—241no—
607C22—241no—
608C22—241no—
609C22—241no—
610C22—241no—
611C22—241no—
612C22—241no—
613C22—241no—
614C22#581711yescharacteristic
615C22C2×#94911yescharacteristic
616C22C2×#95011yescharacteristic
617C22#582811yescharacteristic
618C22#582911yescharacteristic
619C22#583511yescharacteristic
620C22#583711yescharacteristic
621C4—22no—
622C4—22no—
623C4—41no—
624C4—41no—
625C4—42no—
626C4—41no—
627C4—41no—
628C4—81no—
629C4—81no—
630C4—81no—
631C4—122no—
632C4—241no—
633C4#569111yescharacteristic
634C4#569211yescharacteristic
635C4#569311yescharacteristic
636C4#569511yescharacteristic
637C3#2675311yescharacteristic, complemented, Sylow
638C2—242no—
639C2—242no—
640C2#8305911yescharacteristic
641C2#8314111yescharacteristic
642C2#8346011yescharacteristic
643C2#8359911yescharacteristic
644C2#8363811yescharacteristic
645C2#8365511yescharacteristic
646C2#8369711yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2508C2
socleC23×C3C22×D4
prime radicalC2×#874C22
hypercenter#2508D3
hypocenterC3#26753
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2508

Complements

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

NH ≅ G/Ntype
1. C3×#2508C2semidirect
3. #69630C2semidirect
6. #69626C2semidirect
7. #69626C2semidirect
8. #69615C2semidirect
9. #69603C2semidirect
10. #69601C2semidirect
11. #69557C2semidirect
12. #69556C2semidirect
13. #69482C2semidirect
14. C4×#809C2semidirect
15. #69462C2semidirect
60. #2508D3semidirect
637. C3#26753semidirect

Maximal subgroups

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

subgroupindex
C3×#25082
#736532
#696302
#696272
#696262
#696152
#696032
#696012
#695572
#695562
#694822
C4×#8092
#694622
#267533