Finite Group Databasebeta

← #10855143

Subgroups — #10855143

Order 768.

924 conjugacy classes of subgroups.

The subgroup lattice has 722 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2354C211yesmaximal, characteristic, complemented
2#82352C211yesmaximal, characteristic
3C2×#5003C211yesmaximal, characteristic, complemented
4C2×#4995C211yesmaximal, characteristic, complemented
5#71611C211yesmaximal, characteristic, complemented
6#71586C211yesmaximal, characteristic, complemented
7#71576C211yesmaximal, characteristic, complemented
8#69586C211yesmaximal, characteristic, complemented
9#69569C211yesmaximal, characteristic, complemented
10#69523C211yesmaximal, characteristic, complemented
11#69519C211yesmaximal, characteristic, complemented
12#69517C211yesmaximal, characteristic, complemented
13#69504C211yesmaximal, characteristic, complemented
14#69463C211yesmaximal, characteristic, complemented
15#69459C211yesmaximal, characteristic, complemented
16#26157—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C2×C3×#117C2211yescharacteristic
20C2×#939—21no—
21C2×#930C2211yescharacteristic
22C2×#930C2211yescharacteristic
23C22×C4⋊Dic3C2211yescharacteristic
24C22×#208C2211yescharacteristic
25C3×#390—21no—
26C3×#388—21no—
27C3×#388—21no—
28C3×#384C2211yescharacteristic
29C3×#380C2211yescharacteristic
30C3×#380C2211yescharacteristic
31C3×C4×C4⋊C4—21no—
32C3×C4×C4⋊C4C2211yescharacteristic
33#5212C2211yescharacteristic
34C2×#830C2211yescharacteristic
35C2×#830C2211yescharacteristic
36C2×#830C2211yescharacteristic
37#5003—24no—
38#4995—24no—
39#4994—21no—
40C2×#809C2211yescharacteristic
41C2×#809C2211yescharacteristic
42C2×#809C2211yescharacteristic
43C2×#809C2211yescharacteristic
44C2×#808C2211yescharacteristic
45C2×#808C2211yescharacteristic
46C2×#807C2211yescharacteristic
47C2×#807C2211yescharacteristic
48C2×#807C2211yescharacteristic
49C2×#807C2211yescharacteristic
50#4961—21no—
51#4958—21no—
52#4939C2211yescharacteristic
53#4928C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4524C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#482—31no—
63C2×#428—31no—
64#2416—31no—
65#2354D311yescharacteristic, complemented
66#2093—31no—
67#2083—31no—
68#2076—31no—
69#2033—31no—
70#1926—31no—
71#1919—31no—
72#1895—31no—
73#1875—31no—
74#1870—31no—
75#1854—31no—
76#1852—31no—
77C22×C3×D4D411yescharacteristic
78C23×C3×C4C2311yescharacteristic
79C23×C3×C4C2311yescharacteristic
80C22×#67—21no—
81C22×#67—21no—
82C23×Dic3—21no—
83C22×D12C2311yescharacteristic
84C3×C4×D4—22no—
85C3×C4×D4—22no—
86C3×#117—22no—
87C3×#117—22no—
88C3×#117—41no—
89C3×#117—41no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4D411yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C22⋊C4—21no—
98C2×C3×C22⋊C4—21no—
99C2×C3×C22⋊C4—21no—
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4D411yescharacteristic
102C2×C3×C22⋊C4D411yescharacteristic
103C2×C3×C42—21no—
104C2×C3×C42—21no—
105C2×C3×C42—21no—
106C2×C3×C42C2311yescharacteristic
107C2×#217—21no—
108#939—44no—
109#930—22no—
110#930—22no—
111#930—22no—
112#930—41no—
113C2×(C2×D3)⋊C4—21no—
114C2×(C2×D3)⋊C4—21no—
115C2×(C2×D3)⋊C4—21no—
116C2×(C2×D3)⋊C4—21no—
117C2×C4⋊Dic3—22no—
118C2×C4⋊Dic3—21no—
119C2×C4⋊Dic3—41no—
120C2×C4⋊Dic3C2311yescharacteristic
121C2×C4⋊Dic3C2311yescharacteristic
122C2×C4⋊Dic3C2311yescharacteristic
123C2×Dic3⋊C4—21no—
124C2×#208—22no—
125C2×#208—22no—
126C2×#208—22no—
127C22×C3⋊C8C2311yescharacteristic
128C22×C3⋊C8C2311yescharacteristic
129C22×C3⋊C8C2311yescharacteristic
130C22×C3⋊C8C2311yescharacteristic
131C3×#95—21no—
132C3×#95—21no—
133C3×#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136#830—22no—
137#830—22no—
138#830—22no—
139#809—22no—
140#809—22no—
141#809—22no—
142#809—41no—
143#808—22no—
144#808D414yescomplemented
145#807—22no—
146#807—22no—
147#807—22no—
148#807—41no—
149C22×M16—31no—
150C2×#121—31no—
151C2×#121—31no—
152C2×#121—61no—
153C2×C4×D4C2×D311yescharacteristic
154C2×#117C2×D311yescharacteristic
155C2×#117C2×D311yescharacteristic
156C22×C4⋊C4—31no—
157#482—64no—
158#428—64no—
159C2×#107—31no—
160C2×#107—31no—
161C2×#107—31no—
162C2×#107—31no—
163C2×#106—31no—
164C2×#106—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#102—31no—
169C2×C22⋊C8—31no—
170C2×C22⋊C8—31no—
171C2×C22⋊C8—31no—
172#396—61no—
173#394—61no—
174#390—21no—
175#390—31no—
176#389—31no—
177#388—21no—
178#388—21no—
179#386—61no—
180#384C2×D311yescharacteristic
181#380—31no—
182#380C2×D311yescharacteristic
183#380C2×D311yescharacteristic
184C4×C4⋊C4—21no—
185C4×C4⋊C4C2×D311yescharacteristic
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#340—31no—
192#336—31no—
193#336—31no—
194C24×C3C2×D411yescharacteristic
195C24×C3C2×D411yescharacteristic
196C23×D3—21no—
197C2×C3×D4—22no—
198C2×C3×D4—22no—
199C2×C3×D4—41no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4#4112yes—
219C22×C3×C4#4112yes—
220C22×C3×C4C2411yescharacteristic
221C2×#67—42no—
222C2×#67—42no—
223C2×#67—42no—
224C2×#67—42no—
225C2×#67—42no—
226C2×#67—81no—
227C22×Dic3—21no—
228C22×Dic3—21no—
229C22×Dic3—21no—
230C22×Dic3—42no—
231C22×Dic3—42no—
232C22×Dic3—81no—
233C2×D12—21no—
234C2×D12—21no—
235C2×D12—21no—
236C2×D12—21no—
237C2×D12—21no—
238C2×D12—21no—
239C3×C4⋊C4—22no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—42no—
248C3×C4⋊C4—81no—
249C3×C4⋊C4C2×D412yes—
250C3×C22⋊C4—22no—
251C3×C22⋊C4—22no—
252C3×C22⋊C4—42no—
253C3×C22⋊C4—42no—
254C3×C22⋊C4—81no—
255C3×C22⋊C4—81no—
256C3×C42—21no—
257C3×C42—21no—
258C3×C42—42no—
259C3×C42—42no—
260C3×C42—42no—
261#217—42no—
262(C2×D3)⋊C4—42no—
263(C2×D3)⋊C4—42no—
264(C2×D3)⋊C4—42no—
265(C2×D3)⋊C4—81no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—42no—
272C4⋊Dic3C2×D412yes—
273Dic3⋊C4—42no—
274#208—28no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8C2×D412yes—
286C2×C3⋊C8#4112yes—
287C22×D4—31no—
288C22×D4—61no—
289C22×D4—61no—
290C22×D4#6711yescharacteristic
291C23×C4—61no—
292C23×C4C22×D311yescharacteristic
293C23×C4C22×D311yescharacteristic
294C2×M16—62no—
295C2×M16—62no—
296C2×M16—62no—
297C22×C8—31no—
298C22×C8—31no—
299C22×C8—31no—
300C22×C8—31no—
301#121—62no—
302#121—62no—
303#121—62no—
304#121—121no—
305#121—124no—
306C4×D4—22no—
307C4×D4—22no—
308#117—22no—
309#117—22no—
310#117—41no—
311#117—41no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—31no—
315C2×C4⋊C4—31no—
316C2×C4⋊C4—31no—
317C2×C4⋊C4—62no—
318C2×C4⋊C4—61no—
319C2×C4⋊C4—61no—
320C2×C4⋊C4—121no—
321C2×C4⋊C4#6711yescharacteristic
322C2×C4⋊C4C22×D311yescharacteristic
323C2×C4⋊C4C22×D311yescharacteristic
324C2×C4⋊C4C22×D311yescharacteristic
325C2×C4⋊C4C22×D311yescharacteristic
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—61no—
331C2×C22⋊C4—61no—
332C2×C22⋊C4—61no—
333C2×C22⋊C4—61no—
334C2×C22⋊C4—61no—
335C2×C22⋊C4#6711yescharacteristic
336C2×C22⋊C4#6711yescharacteristic
337C2×C42—21no—
338C2×C42—21no—
339C2×C42—21no—
340C2×C42C22×D311yescharacteristic
341#107—62no—
342#107—62no—
343#107—62no—
344#107—121no—
345#106—34no—
346#106—62no—
347#102—62no—
348#102—62no—
349#102—62no—
350#102—121no—
351C22⋊C8—62no—
352C22⋊C8—62no—
353C22⋊C8—62no—
354#95—21no—
355#95—21no—
356#95—21no—
357#95—61no—
358#95—61no—
359C23×C3—21no—
360C23×C3—21no—
361C23×C3—21no—
362C23×C3—21no—
363C23×C3—21no—
364C23×C3—21no—
365C23×C3—21no—
366C23×C3—21no—
367C23×C3—21no—
368C23×C3—21no—
369C23×C3—21no—
370C23×C3—21no—
371C23×C3—21no—
372C23×C3#12312yes—
373C23×C3C22×D411yescharacteristic
374C22×D3—41no—
375C22×D3—41no—
376C22×D3—41no—
377C22×D3—41no—
378C22×D3—41no—
379C22×D3—41no—
380C22×D3—41no—
381C3×D4—44no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—22no—
387C2×C3×C4—22no—
388C2×C3×C4—22no—
389C2×C3×C4—22no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—22no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—21no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4—41no—
402C2×C3×C4—41no—
403C2×C3×C4—41no—
404C2×C3×C4—41no—
405C2×C3×C4—41no—
406C2×C3×C4—41no—
407C2×C3×C4—41no—
408C2×C3×C4—41no—
409C2×C3×C4—41no—
410C2×C3×C4—41no—
411C2×C3×C4—41no—
412C2×C3×C4#12312yes—
413C2×C3×C4AGL(1,ℤ/8)12yes—
414C2×C3×C4C22×D411yescharacteristic
415C2×C3×C4C2×#4111yescharacteristic
416C2×C3×C4C2×#4111yescharacteristic
417C2×C3×C4C2×#4111yescharacteristic
418C2×C3×C4H(5,𝔽2)11yescharacteristic
419C2×C3×C42−1+411yescharacteristic
420#67—84no—
421#67—84no—
422C2×Dic3—22no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3—41no—
426C2×Dic3—41no—
427C2×Dic3—41no—
428C2×Dic3—41no—
429C2×Dic3—41no—
430C2×Dic3—41no—
431C2×Dic3—84no—
432D12—41no—
433D12—41no—
434D12—41no—
435D12—41no—
436C3⋊C8—22no—
437C3⋊C8—22no—
438C3⋊C8—41no—
439C3⋊C8—41no—
440C24—61no—
441C24C2×#6711yescharacteristic
442C24C2×#6711yescharacteristic
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—41no—
446C2×D4—61no—
447C2×D4—61no—
448C2×D4—61no—
449C2×D4—61no—
450C2×D4—61no—
451C2×D4—61no—
452C2×D4—122no—
453C2×D4—122no—
454C2×D4—122no—
455C2×D4—122no—
456C2×D4—122no—
457C2×D4—241no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—61no—
473C22×C4—61no—
474C22×C4—61no—
475C22×C4—122no—
476C22×C4—122no—
477C22×C4—241no—
478C22×C4#23512yes—
479C22×C4#23512yes—
480C22×C4C2×#6711yescharacteristic
481C22×C4C2×#6711yescharacteristic
482C22×C4C2×#6711yescharacteristic
483C22×C4C2×#6711yescharacteristic
484C22×C4C23×D311yescharacteristic
485M16—68no—
486C2×C8—32no—
487C2×C8—32no—
488C2×C8—61no—
489C2×C8—61no—
490C2×C8—61no—
491C2×C8—61no—
492C2×C8—61no—
493C2×C8—61no—
494C2×C8—61no—
495C2×C8—61no—
496C2×C8—61no—
497C2×C8—61no—
498C4⋊C4—22no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—32no—
507C4⋊C4—42no—
508C4⋊C4—61no—
509C4⋊C4—61no—
510C4⋊C4—61no—
511C4⋊C4—61no—
512C4⋊C4—61no—
513C4⋊C4—81no—
514C4⋊C4—122no—
515C4⋊C4—122no—
516C4⋊C4D3×D412yes—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—42no—
520C22⋊C4—42no—
521C22⋊C4—81no—
522C22⋊C4—81no—
523C22⋊C4—122no—
524C22⋊C4—122no—
525C22⋊C4—122no—
526C22⋊C4—122no—
527C22⋊C4—241no—
528C42—21no—
529C42—21no—
530C42—42no—
531C42—42no—
532C42—42no—
533C22×C3—22no—
534C22×C3—22no—
535C22×C3—22no—
536C22×C3—41no—
537C22×C3—41no—
538C22×C3—41no—
539C22×C3—41no—
540C22×C3—41no—
541C22×C3—41no—
542C22×C3—41no—
543C22×C3—41no—
544C22×C3—41no—
545C22×C3—41no—
546C22×C3—41no—
547C22×C3C2×#12311yescharacteristic
548C22×C3D4211yescharacteristic
549C22×C3#54711yescharacteristic
550C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
551C22×C3#57511yescharacteristic
552C22×C3#57611yescharacteristic
553C22×C3#57811yescharacteristic
554C2×D3—81no—
555C2×D3—81no—
556C2×D3—81no—
557C2×D3—81no—
558C2×D3—81no—
559C2×D3—81no—
560C2×D3—81no—
561C3×C4—22no—
562C3×C4—22no—
563C3×C4—41no—
564C3×C4—41no—
565C3×C4—42no—
566C3×C4—81no—
567C3×C4—81no—
568C3×C4—81no—
569C3×C4#52511yescharacteristic
570C3×C4#53911yescharacteristic
571C3×C4#54811yescharacteristic
572C3×C4#55311yescharacteristic
573Dic3—42no—
574Dic3—81no—
575Dic3—81no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—21no—
587C23—21no—
588C23—21no—
589C23—121no—
590C23—121no—
591C23—121no—
592C23—121no—
593C23—121no—
594C23—121no—
595C23—121no—
596C23#92912yes—
597C23C22×#6711yescharacteristic
598D4—44no—
599D4—121no—
600D4—121no—
601D4—121no—
602D4—121no—
603D4—244no—
604D4—244no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—22no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—62no—
636C2×C4—121no—
637C2×C4—121no—
638C2×C4—121no—
639C2×C4—121no—
640C2×C4—121no—
641C2×C4—121no—
642C2×C4—121no—
643C2×C4—121no—
644C2×C4—244no—
645C2×C4#92912yes—
646C2×C4#94912yes—
647C2×C4C2×#23511yescharacteristic
648C2×C4C2×#23511yescharacteristic
649C2×C4C2×D3×D411yescharacteristic
650C2×C4#100411yescharacteristic
651C2×C4D3×#4111yescharacteristic
652C2×C4#101011yescharacteristic
653C8—62no—
654C8—62no—
655C8—121no—
656C8—121no—
657C2×C3—42no—
658C2×C3—81no—
659C2×C3#244311yescharacteristic
660C2×C3#309411yescharacteristic
661C2×C3#309811yescharacteristic
662C2×C3#326911yescharacteristic
663C2×C3#327411yescharacteristic
664C2×C3#334711yescharacteristic
665C2×C3#335311yescharacteristic
666D3—161no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—41no—
671C22—41no—
672C22—41no—
673C22—41no—
674C22—41no—
675C22—41no—
676C22—41no—
677C22—41no—
678C22—41no—
679C22—41no—
680C22—41no—
681C22—241no—
682C22—241no—
683C22—241no—
684C22—241no—
685C22—241no—
686C22—241no—
687C22—241no—
688C22C2×#92911yescharacteristic
689C22D4×#6711yescharacteristic
690C22C2×#94911yescharacteristic
691C22#582211yescharacteristic
692C22#582411yescharacteristic
693C22#583511yescharacteristic
694C22#583811yescharacteristic
695C4—22no—
696C4—22no—
697C4—42no—
698C4—41no—
699C4—41no—
700C4—81no—
701C4—81no—
702C4—81no—
703C4—122no—
704C4—241no—
705C4—241no—
706C4#554011yescharacteristic
707C4#554711yescharacteristic
708C4#555211yescharacteristic
709C4#556411yescharacteristic
710C3#2615711yescharacteristic, complemented, Sylow
711C2—42no—
712C2—81no—
713C2—481no—
714C2#8221711yescharacteristic
715C2#8235611yescharacteristic
716C2#8256711yescharacteristic
717C2#8352811yescharacteristic
718C2#8357811yescharacteristic
719C2#8364811yescharacteristic
720C2#8366211yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2354C2
socleC23×C3C22×D4
prime radical#82352C2
hypercenter#2354D3
hypocenterC3#26157
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2354

Complements

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

NH ≅ G/Ntype
1. C3×#2354C2semidirect
3. C2×#5003C2semidirect
4. C2×#4995C2semidirect
5. #71611C2semidirect
6. #71586C2semidirect
7. #71576C2semidirect
8. #69586C2semidirect
9. #69569C2semidirect
10. #69523C2semidirect
11. #69519C2semidirect
12. #69517C2semidirect
13. #69504C2semidirect
14. #69463C2semidirect
15. #69459C2semidirect
65. #2354D3semidirect
144. #808D4semidirect
710. C3#26157semidirect

Maximal subgroups

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

subgroupindex
C3×#23542
#823522
C2×#50032
C2×#49952
#716112
#715862
#715762
#695862
#695692
#695232
#695192
#695172
#695042
#694632
#694592
#261573