Finite Group Databasebeta

← #10863263

Subgroups — #10863263

Order 768.

874 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82399C211yesmaximal, characteristic, complemented
2#73787C211yesmaximal, characteristic, complemented
3C3×C4×#102C211yesmaximal, characteristic, complemented
4#71613C211yesmaximal, characteristic, complemented
5#70212C211yesmaximal, characteristic, complemented
6#70207C211yesmaximal, characteristic, complemented
7#70204C211yesmaximal, characteristic
8#70164C211yesmaximal, characteristic, complemented
9#69785C211yesmaximal, characteristic, complemented
10#69768C211yesmaximal, characteristic, complemented
11#69459C211yesmaximal, characteristic, complemented
12#68880C211yesmaximal, characteristic, complemented
13#68872C211yesmaximal, characteristic, complemented
14#68802C211yesmaximal, characteristic, complemented
15#68792C211yesmaximal, characteristic, complemented
16#26723—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C2×#938C2211yescharacteristic
19C22×#213C2211yescharacteristic
20C2×C4×#67—21no—
21C2×#895C2211yescharacteristic
22C2×D3×C4⋊C4C2211yescharacteristic
23C2×#890C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C2×C3×C4×C8C2211yescharacteristic
27C3×C4×C4⋊C4C2211yescharacteristic
28#5221C2211yescharacteristic
29C2×#832C2211yescharacteristic
30#5210—21no—
31C2×#821C2211yescharacteristic
32C2×#821C2211yescharacteristic
33C2×(C2×D3)⋊C8C2211yescharacteristic
34C2×(C2×D3)⋊C8C2211yescharacteristic
35C2×Dic3⋊C8C2211yescharacteristic
36#4988C2211yescharacteristic
37#4987C2211yescharacteristic
38Dic3×C4⋊C4C2211yescharacteristic
39C2×#809C2211yescharacteristic
40C2×#808C2211yescharacteristic
41#4956—21no—
42#4945—21no—
43#4940—21no—
44#4939—21no—
45#4939C2211yescharacteristic
46C4×(C2×D3)⋊C4C2211yescharacteristic
47#4928C2211yescharacteristic
48#4674—21no—
49#4672—21no—
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4550C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2490—31no—
61#2361—31no—
62#2054—31no—
63#2034—31no—
64#2025—31no—
65#1999—31no—
66#1989—31no—
67#1934—31no—
68#1926—31no—
69#1923—31no—
70#1919—31no—
71#1879—31no—
72#1874—31no—
73#1824—31no—
74C4×#102D311yescharacteristic, complemented
75C22×C3×D4C2311yescharacteristic
76C23×C3×C4—21no—
77C22×#67—21no—
78C22×D12—21no—
79C22×D12C2311yescharacteristic
80C22×C4×D3—21no—
81C22×C4×D3C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C3×C4×D4—22no—
85C3×C4×D4—41no—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C22⋊C4—21no—
90C2×C3×C42—21no—
91C2×C3×C42C2311yescharacteristic
92C2×#217—21no—
93C2×#217—21no—
94C2×#217—21no—
95#938—22no—
96#938—22no—
97C2×#213—22no—
98C2×#213—41no—
99C2×#213—41no—
100C4×#67—44no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—21no—
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4D411yescharacteristic
108C2×(C2×D3)⋊C4D411yescharacteristic
109C2×C4⋊Dic3—21no—
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×C4×Dic3—21no—
117C2×C4×Dic3C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120#895—22no—
121#895—41no—
122D3×C4⋊C4—22no—
123D3×C4⋊C4—41no—
124#890—22no—
125#890—41no—
126C3×#102—22no—
127C3×#102—22no—
128C3×C4×C8—41no—
129C3×#95—21no—
130#832—41no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95D411yescharacteristic
135C3⋊#95D411yescharacteristic
136#821—22no—
137#821—41no—
138(C2×D3)⋊C8—41no—
139(C2×D3)⋊C8—41no—
140Dic3⋊C8—41no—
141#809—22no—
142#808—22no—
143C22×D8—31no—
144C2×#125—31no—
145C2×#121—31no—
146C2×#121—31no—
147C2×C4×D4—61no—
148C2×C4×D4C2×D311yescharacteristic
149C22×C4⋊C4—31no—
150C2×#106—31no—
151C2×C4⋊C8—31no—
152C2×#102—31no—
153C2×#102—31no—
154C2×#102—31no—
155C2×#102—31no—
156C2×#102C2×D311yescharacteristic
157C2×#102C2×D311yescharacteristic
158C2×C22⋊C8—31no—
159C2×C22⋊C8—31no—
160C2×C4×C8C2×D311yescharacteristic
161#397—61no—
162#394—61no—
163#390—31no—
164#390—31no—
165#390—31no—
166#390—61no—
167#389—31no—
168#388—61no—
169#388—61no—
170#386—61no—
171#385—61no—
172#380—31no—
173C4×C4⋊C4—31no—
174C4×C4⋊C4C2×D311yescharacteristic
175C4×C22⋊C4—31no—
176#340—31no—
177#340—31no—
178#340—31no—
179#340—31no—
180#340—31no—
181#340—31no—
182#340C2×D311yescharacteristic
183#340C2×D311yescharacteristic
184#336—31no—
185#336—31no—
186C24×C3—21no—
187C23×D3—21no—
188C23×D3C2×D411yescharacteristic
189C2×C3×D4—21no—
190C2×C3×D4—21no—
191C2×C3×D4—21no—
192C2×C3×D4—21no—
193C2×C3×D4—21no—
194C2×C3×D4#4112yes—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—42no—
199C22×C3×C4—42no—
200C22×C3×C4—81no—
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2411yescharacteristic
204C2×#67—24no—
205C2×#67—42no—
206C2×#67—42no—
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3C2×D411yescharacteristic
211C22×Dic3C2×D411yescharacteristic
212C22×Dic3C2×D411yescharacteristic
213C2×D12—21no—
214C2×D12—21no—
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12—21no—
219C2×D12—42no—
220C2×D12—42no—
221C2×D12—81no—
222C2×C4×D3—21no—
223C2×C4×D3—21no—
224C2×C4×D3—21no—
225C2×C4×D3—21no—
226C2×C4×D3—21no—
227C2×C4×D3—21no—
228C2×C4×D3—42no—
229C2×C4×D3—42no—
230C2×C4×D3—81no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4#4112yes—
242C3×C4⋊C4#4112yes—
243C3×C22⋊C4—42no—
244C3×C42—21no—
245C3×C42—21no—
246C3×C42—81no—
247#217—24no—
248#217—42no—
249#217—81no—
250#213—44no—
251(C2×D3)⋊C4—22no—
252(C2×D3)⋊C4—22no—
253(C2×D3)⋊C4—42no—
254(C2×D3)⋊C4—42no—
255(C2×D3)⋊C4—81no—
256(C2×D3)⋊C4—81no—
257(C2×D3)⋊C4—81no—
258(C2×D3)⋊C4—81no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—42no—
265C4⋊Dic3#4112yes—
266Dic3⋊C4—42no—
267Dic3⋊C4—42no—
268Dic3⋊C4—42no—
269C4×Dic3—21no—
270C4×Dic3—21no—
271C4×Dic3—42no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C22×D4—31no—
279C22×D4—61no—
280C22×D4—61no—
281C22×D4C22×D311yescharacteristic
282C23×C4—21no—
283C23×C4—31no—
284C23×C4—61no—
285C2×D8—62no—
286C2×D8—121no—
287C2×D8—121no—
288C22×C8—31no—
289C22×C8—31no—
290C22×C8C22×D311yescharacteristic
291C22×C8C22×D311yescharacteristic
292#125—62no—
293#125—121no—
294#121—62no—
295#121—62no—
296#121—62no—
297#121—121no—
298C4×D4—22no—
299C4×D4—41no—
300C4×D4—124no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—62no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4—121no—
310C2×C4⋊C4C22×D311yescharacteristic
311C2×C4⋊C4C22×D311yescharacteristic
312C2×C4⋊C4C22×D311yescharacteristic
313C2×C22⋊C4—21no—
314C2×C22⋊C4—31no—
315C2×C22⋊C4—31no—
316C2×C22⋊C4—61no—
317C2×C22⋊C4—61no—
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C22⋊C4—61no—
323C2×C22⋊C4—61no—
324C2×C22⋊C4—61no—
325C2×C42—21no—
326C2×C42—31no—
327C2×C42—61no—
328C2×C42C22×D311yescharacteristic
329#106—62no—
330C4⋊C8—121no—
331#102—22no—
332#102—22no—
333#102—62no—
334#102—62no—
335#102—121no—
336#102—121no—
337C22⋊C8—121no—
338C22⋊C8—121no—
339C4×C8—41no—
340#95—21no—
341#95—31no—
342#95—31no—
343#95—61no—
344#95—61no—
345#95—61no—
346C23×C3—22no—
347C23×C3—41no—
348C23×C3—41no—
349C23×C3—41no—
350C23×C3—41no—
351C23×C3—41no—
352C23×C3—41no—
353C23×C3C22×D411yescharacteristic
354C22×D3—21no—
355C22×D3—21no—
356C22×D3—21no—
357C22×D3—21no—
358C22×D3—21no—
359C22×D3—21no—
360C22×D3—41no—
361C22×D3—41no—
362C22×D3—41no—
363C22×D3—41no—
364C22×D3—41no—
365C22×D3—41no—
366C22×D3—41no—
367C22×D3#13512yes—
368C3×D4—22no—
369C3×D4—22no—
370C3×D4—41no—
371C3×D4—41no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—22no—
375C2×C3×C4—22no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—84no—
387C2×C3×C4AGL(1,ℤ/8)12yes—
388C2×C3×C4C2×#4111yescharacteristic
389C2×C3×C4C2×#4111yescharacteristic
390C2×C3×C4C2×#4111yescharacteristic
391C2×C3×C4C2×#4111yescharacteristic
392C2×C3×C4H(5,𝔽2)11yescharacteristic
393C2×C3×C4H(5,𝔽2)11yescharacteristic
394#67—48no—
395C2×Dic3—22no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413D12—41no—
414D12—41no—
415D12—41no—
416D12—41no—
417D12—84no—
418C4×D3—41no—
419C4×D3—41no—
420C4×D3—41no—
421C4×D3—41no—
422C4×D3—84no—
423C3×C8—41no—
424C3×C8—41no—
425C3⋊C8—41no—
426C3⋊C8—41no—
427C24—21no—
428C24—31no—
429C24—61no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—64no—
436C2×D4—61no—
437C2×D4—61no—
438C2×D4—61no—
439C2×D4—61no—
440C2×D4—61no—
441C2×D4—61no—
442C2×D4—122no—
443C2×D4—122no—
444C2×D4—122no—
445C2×D4—122no—
446C2×D4—241no—
447C2×D4#23512yes—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—31no—
452C22×C4—31no—
453C22×C4—31no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—61no—
457C22×C4—61no—
458C22×C4—61no—
459C22×C4—61no—
460C22×C4—61no—
461C22×C4—61no—
462C22×C4—61no—
463C22×C4—61no—
464C22×C4—61no—
465C22×C4—81no—
466C22×C4—122no—
467C22×C4—122no—
468C22×C4—241no—
469C22×C4D3×D411yescharacteristic
470C22×C4D3×D411yescharacteristic
471C22×C4C23×D311yescharacteristic
472D8—124no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—61no—
480C2×C8—61no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8—61no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—32no—
490C4⋊C4—61no—
491C4⋊C4—61no—
492C4⋊C4—61no—
493C4⋊C4—61no—
494C4⋊C4—61no—
495C4⋊C4—122no—
496C4⋊C4—122no—
497C4⋊C4—122no—
498C4⋊C4—122no—
499C4⋊C4#23512yes—
500C4⋊C4#23512yes—
501C22⋊C4—42no—
502C22⋊C4—62no—
503C22⋊C4—64no—
504C22⋊C4—62no—
505C22⋊C4—122no—
506C22⋊C4—122no—
507C22⋊C4—122no—
508C22⋊C4—241no—
509C22⋊C4—241no—
510C22⋊C4—241no—
511C22⋊C4—241no—
512C22⋊C4—241no—
513C42—21no—
514C42—21no—
515C42—61no—
516C42—61no—
517C42—81no—
518C42—122no—
519C22×C3—42no—
520C22×C3—42no—
521C22×C3—42no—
522C22×C3—81no—
523C22×C3—81no—
524C22×C3—81no—
525C22×C3—81no—
526C22×C3#52511yescharacteristic
527C22×C3#54611yescharacteristic
528C22×C3#54611yescharacteristic
529C22×C3C2×#13511yescharacteristic
530C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
531C22×C3#57611yescharacteristic
532C22×C3#57711yescharacteristic
533C2×D3—22no—
534C2×D3—22no—
535C2×D3—22no—
536C2×D3—41no—
537C2×D3—41no—
538C2×D3—41no—
539C2×D3—41no—
540C2×D3—81no—
541C2×D3—81no—
542C2×D3—81no—
543C2×D3—81no—
544C2×D3—81no—
545C2×D3—81no—
546C2×D3—81no—
547C3×C4—22no—
548C3×C4—42no—
549C3×C4—41no—
550C3×C4—42no—
551C3×C4—81no—
552C3×C4#52911yescharacteristic
553C3×C4#53811yescharacteristic
554C3×C4#55511yescharacteristic
555C3×C4#55911yescharacteristic
556Dic3—41no—
557Dic3—42no—
558Dic3—41no—
559Dic3—41no—
560Dic3—81no—
561Dic3—81no—
562C23—22no—
563C23—32no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—61no—
571C23—61no—
572C23—61no—
573C23—61no—
574C23—61no—
575C23—61no—
576C23—121no—
577C23—121no—
578C23—121no—
579C23—121no—
580C23—121no—
581C23—121no—
582C23—121no—
583C23C2×D3×D411yescharacteristic
584D4—22no—
585D4—22no—
586D4—41no—
587D4—41no—
588D4—128no—
589D4—121no—
590D4—121no—
591D4—121no—
592D4—121no—
593D4—244no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—61no—
609C2×C4—61no—
610C2×C4—61no—
611C2×C4—61no—
612C2×C4—61no—
613C2×C4—61no—
614C2×C4—62no—
615C2×C4—61no—
616C2×C4—61no—
617C2×C4—61no—
618C2×C4—84no—
619C2×C4—121no—
620C2×C4—121no—
621C2×C4—121no—
622C2×C4—121no—
623C2×C4—121no—
624C2×C4—121no—
625C2×C4—121no—
626C2×C4—121no—
627C2×C4—121no—
628C2×C4—121no—
629C2×C4—121no—
630C2×C4—121no—
631C2×C4—244no—
632C2×C4#91412yes—
633C2×C4C2×#23511yescharacteristic
634C2×C4C2×#23511yescharacteristic
635C2×C4C2×#23511yescharacteristic
636C2×C4#100411yescharacteristic
637C2×C4D3×#4111yescharacteristic
638C2×C4#100911yescharacteristic
639C8—41no—
640C8—41no—
641C8—121no—
642C8—121no—
643C2×C3—82no—
644C2×C3#245911yescharacteristic
645C2×C3#310511yescharacteristic
646C2×C3#311911yescharacteristic
647C2×C3#329711yescharacteristic
648C2×C3#330111yescharacteristic
649C2×C3#331811yescharacteristic
650C2×C3#333211yescharacteristic
651D3—42no—
652D3—161no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—62no—
657C22—62no—
658C22—62no—
659C22—81no—
660C22—81no—
661C22—81no—
662C22—81no—
663C22—121no—
664C22—121no—
665C22—121no—
666C22—121no—
667C22—241no—
668C22—241no—
669C22—241no—
670C22—241no—
671C22—241no—
672C22—241no—
673C22—241no—
674C22#549011yescharacteristic
675C22#554711yescharacteristic
676C22#555011yescharacteristic
677C22#575711yescharacteristic
678C22C2×#91411yescharacteristic
679C22D3×#13511yescharacteristic
680C22#577011yescharacteristic
681C4—22no—
682C4—41no—
683C4—42no—
684C4—42no—
685C4—81no—
686C4—122no—
687C4—121no—
688C4—121no—
689C4—121no—
690C4—241no—
691C4—241no—
692C4#547911yescharacteristic
693C4#553211yescharacteristic
694C4#555911yescharacteristic
695C4#556011yescharacteristic
696C3#2672311yescharacteristic, complemented, Sylow
697C2—82no—
698C2—122no—
699C2—481no—
700C2#7359011yescharacteristic
701C2#7876411yescharacteristic
702C2#7884811yescharacteristic
703C2#7959511yescharacteristic
704C2#7960911yescharacteristic
705C2#7965411yescharacteristic
706C2#7968711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×#102C2
socleC23×C3C22×D4
prime radical#70204C2
hypercenterC4×#102D3
hypocenterC3#26723
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
1. #82399C2semidirect
2. #73787C2semidirect
3. C3×C4×#102C2semidirect
4. #71613C2semidirect
5. #70212C2semidirect
6. #70207C2semidirect
8. #70164C2semidirect
9. #69785C2semidirect
10. #69768C2semidirect
11. #69459C2semidirect
12. #68880C2semidirect
13. #68872C2semidirect
14. #68802C2semidirect
15. #68792C2semidirect
74. C4×#102D3semidirect
696. C3#26723semidirect

Maximal subgroups

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

subgroupindex
#823992
#737872
C3×C4×#1022
#716132
#702122
#702072
#702042
#701642
#697852
#697682
#694592
#688802
#688722
#688022
#687922
#267233