Finite Group Databasebeta

← #10875957

Subgroups — #10875957

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
1C4⋊C4×D12C211yesmaximal, characteristic, complemented
2#73692C211yesmaximal, characteristic, complemented
3C3×#1917C211yesmaximal, characteristic, complemented
4#70484C211yesmaximal, characteristic, complemented
5#70479C211yesmaximal, characteristic, complemented
6#69818C211yesmaximal, characteristic, complemented
7#69812C211yesmaximal, characteristic, complemented
8#68673C211yesmaximal, characteristic
9#68673C211yesmaximal, characteristic
10#68673C211yesmaximal, characteristic
11#68670C211yesmaximal, characteristic, complemented
12#68643C211yesmaximal, characteristic, complemented
13#68643C211yesmaximal, characteristic, complemented
14#68635C211yesmaximal, characteristic, complemented
15#68624C211yesmaximal, characteristic, complemented
16#27571—31nomaximal, Sylow
17C22×D24C2211yescharacteristic
18C2×#898—21no—
19C2×D3×C4⋊C4—21no—
20C2×C4×D12C2211yescharacteristic
21C2×C4×D12C2211yescharacteristic
22C2×C4×D12C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×#97C2211yescharacteristic
25C3×C4×C4⋊C4C2211yescharacteristic
26C2×#821C2211yescharacteristic
27C2×#821C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#821C2211yescharacteristic
31C2×#821C2211yescharacteristic
32C2×#818C2211yescharacteristic
33#4994—21no—
34#4987—21no—
35#4939C2211yescharacteristic
36#4939C2211yescharacteristic
37#4939C2211yescharacteristic
38#4935C2211yescharacteristic
39#4935C2211yescharacteristic
40C4×C4⋊Dic3C2211yescharacteristic
41#4928C2211yescharacteristic
42#4928C2211yescharacteristic
43#4928C2211yescharacteristic
44#4674—21no—
45#4670—21no—
46#4668—21no—
47#4668—21no—
48C3×#336C2211yescharacteristic
49C3×#336C2211yescharacteristic
50C3×#336C2211yescharacteristic
51C3×#336C2211yescharacteristic
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
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#1917D311yescharacteristic, complemented
69#1879—31no—
70#1879—31no—
71#1879—31no—
72#1862—31no—
73#1802—31no—
74#1762—31no—
75C22×D12C2311yescharacteristic
76C22×D12C2311yescharacteristic
77C22×C4×D3—21no—
78C22×C4×D3—21no—
79C22×C4×D3—21no—
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C2×C3×C4⋊C4D411yescharacteristic
85C2×C3×C4⋊C4D411yescharacteristic
86C2×C3×C42C2311yescharacteristic
87C2×C3×C42C2311yescharacteristic
88C2×C3×C42C2311yescharacteristic
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—21no—
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×C4⋊Dic3—21no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×C4×Dic3—21no—
112C2×D24—22no—
113C2×D24—22no—
114C2×D24—22no—
115#898—44no—
116D3×C4⋊C4—44no—
117C4×D12—22no—
118C4×D12—22no—
119C4×D12—22no—
120C4×D12—22no—
121C4×D12—41no—
122C4×D12—41no—
123C3×C4⋊C8—41no—
124C3×#97—41no—
125C3×#95D411yescharacteristic
126C3×#95D411yescharacteristic
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131#821—22no—
132#821—22no—
133#821—22no—
134#821—22no—
135#821—41no—
136#821—41no—
137#818—22no—
138C22×D8—31no—
139C2×#126—61no—
140C2×C4×D4—31no—
141C2×C4×D4—31no—
142C2×C4×D4—31no—
143C22×C4⋊C4—61no—
144C2×#107—31no—
145C2×C4⋊C8C2×D311yescharacteristic
146C2×#102—31no—
147C2×#102—31no—
148C2×#102—31no—
149C2×#102—31no—
150C2×#102—31no—
151C2×#102—31no—
152C2×#97C2×D311yescharacteristic
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#384—31no—
166#380—61no—
167C4×C4⋊C4—31no—
168C4×C4⋊C4C2×D311yescharacteristic
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#336C2×D311yescharacteristic
178#336C2×D311yescharacteristic
179#336C2×D311yescharacteristic
180#336C2×D311yescharacteristic
181C23×D3—21no—
182C23×D3—21no—
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2411yescharacteristic
190C22×Dic3—21no—
191C22×Dic3—21no—
192C22×Dic3—21no—
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C2×D12—21no—
197C2×D12—21no—
198C2×D12—21no—
199C2×D12—21no—
200C2×D12—21no—
201C2×D12—21no—
202C2×D12—21no—
203C2×D12—21no—
204C2×D12—21no—
205C2×D12#4112yes—
206C2×D12#4112yes—
207C2×D12#4112yes—
208C2×C4×D3—24no—
209C2×C4×D3—42no—
210C2×C4×D3—42no—
211C2×C4×D3—42no—
212C2×C4×D3—42no—
213C2×C4×D3—42no—
214C2×C4×D3—42no—
215C2×C4×D3—81no—
216C2×C4×D3—81no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C3×C4⋊C4—22no—
230C3×C4⋊C4—22no—
231C3×C42—21no—
232C3×C42—21no—
233C3×C42—21no—
234C3×C42—21no—
235C3×C42—21no—
236C3×C42—21no—
237(C2×D3)⋊C4—24no—
238(C2×D3)⋊C4—42no—
239(C2×D3)⋊C4—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—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—42no—
258C4⋊Dic3—42no—
259C4⋊Dic3—81no—
260C4⋊Dic3—81no—
261C4⋊Dic3#4112yes—
262Dic3⋊C4—42no—
263Dic3⋊C4—42no—
264Dic3⋊C4—42no—
265C4×Dic3—42no—
266D24—44no—
267C22×D4—31no—
268C22×D4—31no—
269C23×C4—61no—
270C23×C4—61no—
271C23×C4—61no—
272C2×D8—62no—
273C2×D8—62no—
274C2×D8—62no—
275C22×C8C22×D311yescharacteristic
276C22×C8C22×D311yescharacteristic
277C22×C8C22×D311yescharacteristic
278C22×C8C22×D311yescharacteristic
279#126—124no—
280C4×D4—62no—
281C4×D4—62no—
282C4×D4—62no—
283C4×D4—62no—
284C4×D4—121no—
285C4×D4—121no—
286C2×C4⋊C4—31no—
287C2×C4⋊C4—31no—
288C2×C4⋊C4—31no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4—31no—
292C2×C4⋊C4—61no—
293C2×C4⋊C4—61no—
294C2×C4⋊C4—61no—
295C2×C4⋊C4—61no—
296C2×C4⋊C4—61no—
297C2×C4⋊C4—61no—
298C2×C4⋊C4—61no—
299C2×C4⋊C4—124no—
300C2×C4⋊C4D1211yescharacteristic
301C2×C4⋊C4D1211yescharacteristic
302C2×C22⋊C4—61no—
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—61no—
312C2×C42C22×D311yescharacteristic
313C2×C42C22×D311yescharacteristic
314C2×C42C22×D311yescharacteristic
315#107—62no—
316C4⋊C8—41no—
317#102—62no—
318#102—62no—
319#102—62no—
320#102—62no—
321#102—121no—
322#102—121no—
323#97—41no—
324#95—61no—
325#95—61no—
326#95—61no—
327#95—61no—
328#95D1211yescharacteristic
329#95D1211yescharacteristic
330C23×C3C22×D411yescharacteristic
331C22×D3—22no—
332C22×D3—22no—
333C22×D3—22no—
334C22×D3—41no—
335C22×D3—41no—
336C22×D3—41no—
337C22×D3—41no—
338C22×D3—41no—
339C22×D3—41no—
340C22×D3—41no—
341C22×D3—41no—
342C22×D3—41no—
343C22×D3—41no—
344C22×D3—41no—
345C2×C3×C4—21no—
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—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4AGL(1,ℤ/8)12yes—
362C2×C3×C4AGL(1,ℤ/8)12yes—
363C2×C3×C4C2×#4111yescharacteristic
364C2×C3×C4C2×#4111yescharacteristic
365C2×C3×C4C2×#4111yescharacteristic
366C2×C3×C4C2×#4111yescharacteristic
367C2×C3×C4H(5,𝔽2)11yescharacteristic
368C2×C3×C42−1+411yescharacteristic
369C2×Dic3—22no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—41no—
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×Dic3—41no—
387D12—22no—
388D12—22no—
389D12—22no—
390D12—41no—
391D12—41no—
392D12—41no—
393D12—41no—
394D12#12614yescomplemented
395C4×D3—48no—
396C4×D3—84no—
397C4×D3—84no—
398C3×C8—41no—
399C3×C8—41no—
400C3×C8—41no—
401C3×C8—41no—
402C24—61no—
403C24—61no—
404C2×D4—32no—
405C2×D4—32no—
406C2×D4—32no—
407C2×D4—61no—
408C2×D4—61no—
409C2×D4—61no—
410C2×D4—61no—
411C2×D4—61no—
412C2×D4—61no—
413C2×D4—61no—
414C2×D4—61no—
415C2×D4—61no—
416C22×C4—64no—
417C22×C4—61no—
418C22×C4—61no—
419C22×C4—61no—
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×C4C2×D1211yescharacteristic
432C22×C4C2×D1211yescharacteristic
433C22×C4C2×D1211yescharacteristic
434C22×C4C2×D1211yescharacteristic
435C22×C4C2×D1211yescharacteristic
436C22×C4C2×D1211yescharacteristic
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—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—32no—
454C4⋊C4—61no—
455C4⋊C4—61no—
456C4⋊C4—61no—
457C4⋊C4—61no—
458C4⋊C4—61no—
459C4⋊C4—61no—
460C4⋊C4—61no—
461C4⋊C4—61no—
462C4⋊C4—61no—
463C4⋊C4—61no—
464C4⋊C4—61no—
465C4⋊C4—122no—
466C4⋊C4—122no—
467C4⋊C4—122no—
468C4⋊C4—122no—
469C4⋊C4—122no—
470C4⋊C4—241no—
471C4⋊C4—241no—
472C22⋊C4—64no—
473C22⋊C4—122no—
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—21no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—122no—
488C22×C3#52511yescharacteristic
489C22×C3#54611yescharacteristic
490C22×C3#54811yescharacteristic
491C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
492C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
493C22×C3#57611yescharacteristic
494C22×C3#57811yescharacteristic
495C2×D3—24no—
496C2×D3—42no—
497C2×D3—42no—
498C2×D3—42no—
499C2×D3—42no—
500C2×D3—42no—
501C2×D3—42no—
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—22no—
511C3×C4—41no—
512C3×C4—41no—
513C3×C4—41no—
514C3×C4—41no—
515C3×C4C2×#12611yescharacteristic
516C3×C4#53911yescharacteristic
517C3×C4#55211yescharacteristic
518C3×C4#55311yescharacteristic
519Dic3—42no—
520Dic3—81no—
521Dic3—81no—
522Dic3—81no—
523Dic3—81no—
524Dic3—81no—
525C23—62no—
526C23—62no—
527C23—62no—
528C23—121no—
529C23—121no—
530C23—121no—
531C23—121no—
532C23—121no—
533C23—121no—
534C23—121no—
535C23—121no—
536C23—121no—
537C23—121no—
538C23—121no—
539C23C22×D1211yescharacteristic
540D4—34no—
541D4—62no—
542D4—62no—
543D4—62no—
544D4—121no—
545D4—121no—
546D4—121no—
547D4—121no—
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—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—62no—
565C2×C4—128no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4—121no—
569C2×C4—121no—
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—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—244no—
584C2×C4—244no—
585C2×C4#90812yes—
586C2×C4#90812yes—
587C2×C4#100411yescharacteristic
588C2×C4#100711yescharacteristic
589C2×C4D3×#4111yescharacteristic
590C2×C4D3×#4111yescharacteristic
591C2×C4D3×#4111yescharacteristic
592C2×C4D3×#4111yescharacteristic
593C8—41no—
594C8—41no—
595C8—41no—
596C8—41no—
597C2×C3#248111yescharacteristic
598C2×C3#310911yescharacteristic
599C2×C3#312411yescharacteristic
600C2×C3#331811yescharacteristic
601C2×C3#333011yescharacteristic
602C2×C3#336611yescharacteristic
603C2×C3#336711yescharacteristic
604D3—44no—
605D3—161no—
606C22—64no—
607C22—122no—
608C22—122no—
609C22—122no—
610C22—122no—
611C22—122no—
612C22—122no—
613C22—241no—
614C22—241no—
615C22—241no—
616C22—241no—
617C22—241no—
618C22—241no—
619C22—241no—
620C22#552411yescharacteristic
621C22#555411yescharacteristic
622C22#557611yescharacteristic
623C22C2×#90811yescharacteristic
624C22C2×#90811yescharacteristic
625C22#575211yescharacteristic
626C22#575311yescharacteristic
627C4—22no—
628C4—22no—
629C4—41no—
630C4—41no—
631C4—41no—
632C4—41no—
633C4—122no—
634C4—241no—
635C4—241no—
636C4—241no—
637C4—241no—
638C4—241no—
639C4#563811yescharacteristic
640C4#566111yescharacteristic
641C4#569311yescharacteristic
642C4D3×#12611yescharacteristic
643C3#2757111yescharacteristic, complemented, Sylow
644C2—124no—
645C2—481no—
646C2#7502011yescharacteristic
647C2#7934311yescharacteristic
648C2#7938811yescharacteristic
649C2#7950511yescharacteristic
650C2#7955711yescharacteristic
651C2#7996911yescharacteristic
652C2#7997011yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1917C2
socleC23×C3C22×D4
prime radicalC22×D24C22
hypercenter#1917D3
hypocenterC3#27571
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1917

Complements

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

NH ≅ G/Ntype
1. C4⋊C4×D12C2semidirect
2. #73692C2semidirect
3. C3×#1917C2semidirect
4. #70484C2semidirect
5. #70479C2semidirect
6. #69818C2semidirect
7. #69812C2semidirect
11. #68670C2semidirect
12. #68643C2semidirect
13. #68643C2semidirect
14. #68635C2semidirect
15. #68624C2semidirect
68. #1917D3semidirect
394. D12#126semidirect
643. C3#27571semidirect

Maximal subgroups

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

subgroupindex
C4⋊C4×D122
#736922
C3×#19172
#704842
#704792
#698182
#698122
#686732
#686702
#686432
#686352
#686242
#275713