Finite Group Databasebeta

← #10875997

Subgroups — #10875997

Order 768.

800 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#76047C211yesmaximal, characteristic, complemented
2#74228C211yesmaximal, characteristic, complemented
3C3×#2101C211yesmaximal, characteristic, complemented
4#69626C211yesmaximal, characteristic, complemented
5#69557C211yesmaximal, characteristic, complemented
6#68900C211yesmaximal, characteristic, complemented
7#68897C211yesmaximal, characteristic
8#68897C211yesmaximal, characteristic
9#68892C211yesmaximal, characteristic, complemented
10#68856C211yesmaximal, characteristic, complemented
11#68810C211yesmaximal, characteristic, complemented
12#68788C211yesmaximal, characteristic, complemented
13#68782C211yesmaximal, characteristic, complemented
14#68673C211yesmaximal, characteristic
15#68612C211yesmaximal, characteristic, complemented
16#27574—31nomaximal, Sylow
17C22×D24C2211yescharacteristic
18C2×#898—21no—
19C2×#893C2211yescharacteristic
20C2×#893C2211yescharacteristic
21C2×C4×D12C2211yescharacteristic
22C3×#389C2211yescharacteristic
23C3×#382C2211yescharacteristic
24C2×#821C2211yescharacteristic
25C2×#821C2211yescharacteristic
26C2×#815C2211yescharacteristic
27C2×Dic3⋊C8C2211yescharacteristic
28#4994—21no—
29#4988C2211yescharacteristic
30#4988C2211yescharacteristic
31#4975C2211yescharacteristic
32Dic3×C4⋊C4C2211yescharacteristic
33#4972C2211yescharacteristic
34#4972C2211yescharacteristic
35C2×#809C2211yescharacteristic
36C2×#809C2211yescharacteristic
37C2×#809C2211yescharacteristic
38C2×#809C2211yescharacteristic
39C2×#808C2211yescharacteristic
40#4939C2211yescharacteristic
41C4×C4⋊Dic3C2211yescharacteristic
42#4675—21no—
43#4669—21no—
44#4668—21no—
45#4668—21no—
46#4667—21no—
47D3×#95—21no—
48C3×#340C2211yescharacteristic
49C3×#340C2211yescharacteristic
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52C3×#336C2211yescharacteristic
53#4551C2211yescharacteristic
54#4550C2211yescharacteristic
55#4550C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2512—31no—
61#2349—31no—
62#2101D311yescharacteristic, complemented
63#2099—31no—
64#2073—31no—
65#2065—31no—
66#1936—31no—
67#1919—31no—
68#1917—31no—
69#1879—31no—
70#1879—31no—
71#1879—31no—
72#1864—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
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C42C2311yescharacteristic
88C2×(C2×D3)⋊C4—21no—
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×C4⋊Dic3D411yescharacteristic
98C2×C4⋊Dic3D411yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×C4×Dic3—21no—
106C2×C4×Dic3C2311yescharacteristic
107C2×C4×Dic3C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110C2×D24—22no—
111C2×D24—22no—
112C2×D24—22no—
113#898—44no—
114#893—22no—
115#893—22no—
116#893—41no—
117#893—41no—
118C4×D12—22no—
119C4×D12—22no—
120C3×#95—21no—
121C3×#95—21no—
122C3×#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95D411yescharacteristic
127C3⋊#95D411yescharacteristic
128#821—22no—
129#821—22no—
130#815—41no—
131Dic3⋊C8—41no—
132#809—22no—
133#809—22no—
134#809—41no—
135#809—41no—
136#808—22no—
137C22×D8—31no—
138C2×#126—61no—
139C2×C4×D4—31no—
140C2×C4×D4—31no—
141C2×C4×D4—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×#97—31no—
151#397—61no—
152#396—61no—
153#390—31no—
154#390—31no—
155#390—31no—
156#389—31no—
157#389—31no—
158#389C2×D311yescharacteristic
159#388—61no—
160#385—61no—
161#385—61no—
162#384—31no—
163#382C2×D311yescharacteristic
164#380—61no—
165C4×C4⋊C4—31no—
166C4×C4⋊C4—31no—
167C2×#95—61no—
168#340—31no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340C2×D311yescharacteristic
173#340C2×D311yescharacteristic
174#340C2×D311yescharacteristic
175#340C2×D311yescharacteristic
176#336—31no—
177#336—31no—
178#336—31no—
179#336C2×D311yescharacteristic
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×C4C2411yescharacteristic
189C22×Dic3—21no—
190C22×Dic3—21no—
191C22×Dic3C2×D411yescharacteristic
192C22×Dic3C2×D411yescharacteristic
193C22×Dic3C2×D411yescharacteristic
194C22×Dic3C2×D411yescharacteristic
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—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—81no—
213C2×C4×D3—81no—
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—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—42no—
231C3×C42—21no—
232C3×C42—21no—
233(C2×D3)⋊C4—24no—
234(C2×D3)⋊C4—42no—
235(C2×D3)⋊C4—42no—
236(C2×D3)⋊C4—42no—
237(C2×D3)⋊C4—42no—
238(C2×D3)⋊C4—42no—
239(C2×D3)⋊C4—42no—
240(C2×D3)⋊C4—81no—
241(C2×D3)⋊C4—81no—
242C4⋊Dic3—21no—
243C4⋊Dic3—22no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—41no—
247C4⋊Dic3#4112yes—
248Dic3⋊C4—42no—
249Dic3⋊C4—81no—
250Dic3⋊C4—81no—
251Dic3⋊C4—81no—
252C4×Dic3—21no—
253C4×Dic3—21no—
254C4×Dic3—21no—
255C4×Dic3—21no—
256C4×Dic3—42no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8—21no—
263D24—44no—
264C22×D4—31no—
265C22×D4—31no—
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
276#126—124no—
277C4×D4—62no—
278C4×D4—62no—
279C4×D4—62no—
280C4×D4—62no—
281C4×D4—121no—
282C4×D4—121no—
283C2×C4⋊C4—21no—
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⋊C4C22×D311yescharacteristic
293C2×C4⋊C4C22×D311yescharacteristic
294C2×C4⋊C4C22×D311yescharacteristic
295C2×C4⋊C4C22×D311yescharacteristic
296C2×C22⋊C4—61no—
297C2×C22⋊C4—61no—
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—62no—
312#102—62no—
313#102—62no—
314#102—62no—
315#102—121no—
316#102—121no—
317#97—121no—
318#95—21no—
319#95—21no—
320#95—21no—
321#95—31no—
322#95—31no—
323#95—61no—
324#95—61no—
325#95—61no—
326C23×C3C22×D411yescharacteristic
327C22×D3—22no—
328C22×D3—22no—
329C22×D3—22no—
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—
340C22×D3—41no—
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—41no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
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×C4C2×#4111yescharacteristic
360C2×C3×C4C2×#4111yescharacteristic
361C2×C3×C4C2×#4111yescharacteristic
362C2×C3×C4C2×#4111yescharacteristic
363C2×C3×C4H(5,𝔽2)11yescharacteristic
364C2×C3×C42−1+411yescharacteristic
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—22no—
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×Dic3AGL(1,ℤ/8)12yes—
383D12—22no—
384D12—22no—
385D12—22no—
386D12—41no—
387D12—41no—
388D12—41no—
389D12—41no—
390D12#12614yescomplemented
391C4×D3—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—32no—
403C2×D4—61no—
404C2×D4—61no—
405C2×D4—61no—
406C2×D4—61no—
407C2×D4—61no—
408C2×D4—61no—
409C2×D4—61no—
410C2×D4—61no—
411C2×D4—61no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—31no—
417C22×C4—31no—
418C22×C4—31no—
419C22×C4—31no—
420C22×C4—64no—
421C22×C4—61no—
422C22×C4—61no—
423C22×C4—122no—
424C22×C4—122no—
425C22×C4—122no—
426C22×C4—122no—
427C22×C4—241no—
428C22×C4—241no—
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—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—32no—
456C4⋊C4—42no—
457C4⋊C4—62no—
458C4⋊C4—61no—
459C4⋊C4—61no—
460C4⋊C4—61no—
461C4⋊C4—121no—
462C4⋊C4—122no—
463C4⋊C4—241no—
464C4⋊C4—241no—
465C4⋊C4—241no—
466C22⋊C4—64no—
467C22⋊C4—122no—
468C22⋊C4—122no—
469C22⋊C4—122no—
470C22⋊C4—122no—
471C22⋊C4—122no—
472C22⋊C4—122no—
473C22⋊C4—241no—
474C22⋊C4—241no—
475C42—21no—
476C42—21no—
477C42—61no—
478C42—61no—
479C42—61no—
480C42—61no—
481C42—122no—
482C22×C3#52511yescharacteristic
483C22×C3#54611yescharacteristic
484C22×C3#54811yescharacteristic
485C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
486C22×C3#57511yescharacteristic
487C22×C3#57611yescharacteristic
488C22×C3#57711yescharacteristic
489C2×D3—24no—
490C2×D3—42no—
491C2×D3—42no—
492C2×D3—42no—
493C2×D3—42no—
494C2×D3—42no—
495C2×D3—42no—
496C2×D3—81no—
497C2×D3—81no—
498C2×D3—81no—
499C2×D3—81no—
500C2×D3—81no—
501C2×D3—81no—
502C2×D3—81no—
503C3×C4—41no—
504C3×C4—41no—
505C3×C4—81no—
506C3×C4—81no—
507C3×C4—81no—
508C3×C4—81no—
509C3×C4C2×#12611yescharacteristic
510C3×C4#53911yescharacteristic
511C3×C4#55211yescharacteristic
512C3×C4#55311yescharacteristic
513Dic3—22no—
514Dic3—42no—
515Dic3—41no—
516Dic3—41no—
517Dic3—41no—
518Dic3—81no—
519C23—62no—
520C23—62no—
521C23—62no—
522C23—121no—
523C23—121no—
524C23—121no—
525C23—121no—
526C23—121no—
527C23—121no—
528C23—121no—
529C23—121no—
530C23—121no—
531C23—121no—
532C23—121no—
533C23C2×D3×D411yescharacteristic
534D4—34no—
535D4—62no—
536D4—62no—
537D4—62no—
538D4—121no—
539D4—121no—
540D4—121no—
541D4—121no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—32no—
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—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—61no—
562C2×C4—61no—
563C2×C4—62no—
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—121no—
574C2×C4—121no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—244no—
579C2×C4—244no—
580C2×C4—244no—
581C2×C4#100711yescharacteristic
582C2×C4D3×#4111yescharacteristic
583C2×C4D3×#4111yescharacteristic
584C2×C4D3×#4111yescharacteristic
585C2×C4D3×#4111yescharacteristic
586C2×C4#100911yescharacteristic
587C8—41no—
588C8—41no—
589C8—121no—
590C8—121no—
591C2×C3#248111yescharacteristic
592C2×C3#311511yescharacteristic
593C2×C3#311811yescharacteristic
594C2×C3#330611yescharacteristic
595C2×C3#332211yescharacteristic
596C2×C3#336611yescharacteristic
597C2×C3#337611yescharacteristic
598D3—44no—
599D3—161no—
600C22—64no—
601C22—122no—
602C22—122no—
603C22—122no—
604C22—122no—
605C22—122no—
606C22—122no—
607C22—241no—
608C22—241no—
609C22—241no—
610C22—241no—
611C22—241no—
612C22—241no—
613C22—241no—
614C22#563311yescharacteristic
615C22#565611yescharacteristic
616C22#567611yescharacteristic
617C22#576811yescharacteristic
618C22#576911yescharacteristic
619C22D3×AGL(1,ℤ/8)11yescharacteristic
620C22#577811yescharacteristic
621C4—41no—
622C4—41no—
623C4—62no—
624C4—81no—
625C4—81no—
626C4—81no—
627C4—81no—
628C4—121no—
629C4—122no—
630C4—121no—
631C4—121no—
632C4—241no—
633C4#569211yescharacteristic
634C4#569311yescharacteristic
635C4D3×#12611yescharacteristic
636C4#570911yescharacteristic
637C3#2757411yescharacteristic, complemented, Sylow
638C2—124no—
639C2—481no—
640C2#7814411yescharacteristic
641C2#8115211yescharacteristic
642C2#8126511yescharacteristic
643C2#8135111yescharacteristic
644C2#8149611yescharacteristic
645C2#8165611yescharacteristic
646C2#8169611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2101C2
socleC23×C3C22×D4
prime radicalC22×D24C22
hypercenter#2101D3
hypocenterC3#27574
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2101

Complements

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

NH ≅ G/Ntype
1. #76047C2semidirect
2. #74228C2semidirect
3. C3×#2101C2semidirect
4. #69626C2semidirect
5. #69557C2semidirect
6. #68900C2semidirect
9. #68892C2semidirect
10. #68856C2semidirect
11. #68810C2semidirect
12. #68788C2semidirect
13. #68782C2semidirect
15. #68612C2semidirect
62. #2101D3semidirect
390. D12#126semidirect
637. C3#27574semidirect

Maximal subgroups

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

subgroupindex
#760472
#742282
C3×#21012
#696262
#695572
#689002
#688972
#688922
#688562
#688102
#687882
#687822
#686732
#686122
#275743