Finite Group Databasebeta

← #10845512

Subgroups — #10845512

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×#2547C211yesmaximal, characteristic, complemented
2C2×#5084C211yesmaximal, characteristic, complemented
3#82415C211yesmaximal, characteristic
4C2×#5011C211yesmaximal, characteristic, complemented
5#71788C211yesmaximal, characteristic, complemented
6#69524C211yesmaximal, characteristic, complemented
7#69523C211yesmaximal, characteristic, complemented
8#69520C211yesmaximal, characteristic, complemented
9#69517C211yesmaximal, characteristic, complemented
10#69515C211yesmaximal, characteristic, complemented
11#69513C211yesmaximal, characteristic, complemented
12#69511C211yesmaximal, characteristic, complemented
13#69507C211yesmaximal, characteristic, complemented
14#69474C211yesmaximal, characteristic, complemented
15#69468C211yesmaximal, characteristic, complemented
16#25354—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#124—21no—
19C2×C3×C4×D4C2211yescharacteristic
20C2×#930C2211yescharacteristic
21C2×#930C2211yescharacteristic
22C22×#208C2211yescharacteristic
23C23×C3⋊C8C2211yescharacteristic
24C2×C4×D12C2211yescharacteristic
25C3×#397—21no—
26C3×#397—21no—
27C3×#391—21no—
28C3×#380C2211yescharacteristic
29C3×#380C2211yescharacteristic
30C3×#380C2211yescharacteristic
31C3×#380C2211yescharacteristic
32C3×C4×C4⋊C4C2211yescharacteristic
33#5223—21no—
34#5212C2211yescharacteristic
35#5212C2211yescharacteristic
36C2×#830C2211yescharacteristic
37C2×#830C2211yescharacteristic
38#5084—24no—
39#5011—24no—
40#4992—21no—
41C2×#809C2211yescharacteristic
42C2×#809C2211yescharacteristic
43C2×#809C2211yescharacteristic
44C2×#809C2211yescharacteristic
45C2×#808C2211yescharacteristic
46C2×#808C2211yescharacteristic
47C2×#808C2211yescharacteristic
48C2×#807C2211yescharacteristic
49#4956—21no—
50#4956—21no—
51#4936C2211yescharacteristic
52C2×C4⋊(C3⋊C8)C2211yescharacteristic
53C2×#803C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#476—31no—
63C2×#435—31no—
64#2547D311yescharacteristic, complemented
65#2535—31no—
66#2077—31no—
67#2077—31no—
68#2034—31no—
69#2033—31no—
70#1993—31no—
71#1949—31no—
72#1895—31no—
73#1893—31no—
74#1809—31no—
75#1807—31no—
76#1762—31no—
77C22×C3×Q8—21no—
78C22×C3×D4D411yescharacteristic
79C23×C3×C4C2311yescharacteristic
80C23×C3×C4C2311yescharacteristic
81C22×#67—21no—
82C22×#67—21no—
83C22×D12C2311yescharacteristic
84C22×C4×D3—21no—
85C3×C4⋊Q8—22no—
86C3×C4⋊Q8—22no—
87C3×#124—44no—
88C3×C4×D4—22no—
89C3×C4×D4—22no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4D411yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—21no—
98C2×C3×C22⋊C4—21no—
99C2×C3×C22⋊C4—21no—
100C2×C3×C22⋊C4D411yescharacteristic
101C2×C3×C22⋊C4D411yescharacteristic
102C2×C3×C42—21no—
103C2×C3×C42C2311yescharacteristic
104C2×#217—21no—
105C2×#217—21no—
106#930—22no—
107#930—22no—
108#930—41no—
109#930—41no—
110C2×(C2×D3)⋊C4—21no—
111C2×(C2×D3)⋊C4—21no—
112C2×(C2×D3)⋊C4—21no—
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116C2×#208—22no—
117C2×#208—22no—
118C2×#208—41no—
119C22×C3⋊C8—22no—
120C22×C3⋊C8—22no—
121C22×C3⋊C8—41no—
122C22×C3⋊C8C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124C22×C3⋊C8C2311yescharacteristic
125C22×C3⋊C8C2311yescharacteristic
126C4×D12—22no—
127C4×D12—22no—
128C3×#95—21no—
129C3×#95—21no—
130C3×#95—21no—
131C3×#95—21no—
132C3×#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137#830—22no—
138#830—22no—
139#809—22no—
140#809—22no—
141#809—22no—
142#809—22no—
143#808—22no—
144#808—22no—
145#808—41no—
146#807—41no—
147C4⋊(C3⋊C8)D414yescomplemented
148#803—22no—
149C22×M16—31no—
150C23×C8—31no—
151C2×C4⋊Q8C2×D311yescharacteristic
152C2×#124—21no—
153C2×#121—31no—
154C2×#121—31no—
155C2×C4×D4—31no—
156C2×C4×D4C2×D311yescharacteristic
157#476—64no—
158#435—64no—
159C2×#107—31no—
160C2×#106—31no—
161C2×#106—31no—
162C2×#106—31no—
163C2×C4⋊C8—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×C22⋊C8—31no—
169C2×C22⋊C8—31no—
170C2×#97—31no—
171#397—21no—
172#397—21no—
173#397—61no—
174#392—61no—
175#391—21no—
176#386—61no—
177#386—61no—
178#382—31no—
179#380—31no—
180#380—31no—
181#380C2×D311yescharacteristic
182#380C2×D311yescharacteristic
183#380C2×D311yescharacteristic
184#380C2×D311yescharacteristic
185C4×C4⋊C4C2×D311yescharacteristic
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#340—31no—
192#340—31no—
193#340—31no—
194C24×C3C2×D411yescharacteristic
195C24×C3C2×D411yescharacteristic
196C23×D3—21no—
197C2×C3×Q8—42no—
198C2×C3×Q8—42no—
199C2×C3×Q8—42no—
200C2×C3×D4—22no—
201C2×C3×D4—22no—
202C2×C3×D4—41no—
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×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4—21no—
217C22×C3×C4—21no—
218C22×C3×C4—21no—
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4C2411yescharacteristic
224C2×#67—42no—
225C2×#67—42no—
226C2×#67—42no—
227C2×#67—42no—
228C2×#67—81no—
229C2×#67—81no—
230C22×Dic3—21no—
231C22×Dic3—21no—
232C22×Dic3—21no—
233C2×D12—21no—
234C2×D12—21no—
235C2×D12—21no—
236C2×D12—21no—
237C2×D12—21no—
238C2×D12#4112yes—
239C2×C4×D3—42no—
240C2×C4×D3—81no—
241C2×C4×D3—81no—
242C3×C4⋊C4—22no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—42no—
252C3×C22⋊C4—22no—
253C3×C22⋊C4—22no—
254C3×C22⋊C4—42no—
255C3×C22⋊C4—42no—
256C3×C22⋊C4—42no—
257C3×C22⋊C4—42no—
258C3×C42—42no—
259C3×C42C2×D412yes—
260C3×C42#4112yes—
261#217—81no—
262#217—81no—
263(C2×D3)⋊C4—42no—
264(C2×D3)⋊C4—42no—
265(C2×D3)⋊C4—42no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272#208—44no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
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—44no—
283C2×C3⋊C8C2×D412yes—
284C2×C3⋊C8C2×D412yes—
285C2×C3⋊C8#4112yes—
286C22×Q8—21no—
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—121no—
297C22×C8—31no—
298C22×C8—31no—
299C22×C8—31no—
300C22×C8—31no—
301C22×C8—62no—
302C22×C8—62no—
303C22×C8—121no—
304C4⋊Q8—22no—
305C4⋊Q8—22no—
306#124—44no—
307#121—62no—
308#121—62no—
309#121—121no—
310#121—121no—
311C4×D4—22no—
312C4×D4—22no—
313C4×D4—62no—
314C4×D4—62no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—31no—
317C2×C4⋊C4—31no—
318C2×C4⋊C4—31no—
319C2×C4⋊C4#6711yescharacteristic
320C2×C4⋊C4C22×D311yescharacteristic
321C2×C4⋊C4C22×D311yescharacteristic
322C2×C4⋊C4C22×D311yescharacteristic
323C2×C4⋊C4C22×D311yescharacteristic
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—61no—
329C2×C22⋊C4—61no—
330C2×C22⋊C4—61no—
331C2×C22⋊C4—61no—
332C2×C22⋊C4—61no—
333C2×C22⋊C4#6711yescharacteristic
334C2×C22⋊C4#6711yescharacteristic
335C2×C42—21no—
336C2×C42C22×D311yescharacteristic
337#107—121no—
338#106—62no—
339#106—62no—
340#106—121no—
341C4⋊C8—34no—
342#102—62no—
343#102—62no—
344#102—62no—
345#102—62no—
346C22⋊C8—62no—
347C22⋊C8—62no—
348#97—62no—
349#95—21no—
350#95—21no—
351#95—21no—
352#95—21no—
353#95—21no—
354#95—61no—
355#95—61no—
356#95—61no—
357#95—61no—
358C23×C3—21no—
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×C3AGL(1,ℤ/8)12yes—
372C23×C3C22×D411yescharacteristic
373C22×D3—22no—
374C22×D3—41no—
375C22×D3—41no—
376C22×D3—41no—
377C22×D3—41no—
378C22×D3—41no—
379C22×D3—41no—
380C3×Q8—84no—
381C3×D4—44no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
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#12112yes—
413C2×C3×C4#12112yes—
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—41no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3—41no—
426C2×Dic3—41no—
427C2×Dic3—41no—
428C2×Dic3—41no—
429C2×Dic3—41no—
430C2×Dic3—41no—
431D12—22no—
432D12—22no—
433D12—41no—
434D12—41no—
435C4×D3—84no—
436C3⋊C8—22no—
437C3⋊C8—22no—
438C3⋊C8—22no—
439C3⋊C8—41no—
440C24—61no—
441C24C2×#6711yescharacteristic
442C24C2×#6711yescharacteristic
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—32no—
449C2×D4—41no—
450C2×D4—61no—
451C2×D4—61no—
452C2×D4—61no—
453C2×D4—61no—
454C2×D4—61no—
455C2×D4—122no—
456C2×D4—122no—
457C2×D4—122no—
458C2×D4—122no—
459C2×D4—241no—
460C2×D4—241no—
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—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—61no—
478C22×C4—61no—
479C22×C4—61no—
480C22×C4—122no—
481C22×C4—241no—
482C22×C4—241no—
483C22×C4C2×#6711yescharacteristic
484C22×C4C2×#6711yescharacteristic
485C22×C4C2×#6711yescharacteristic
486C22×C4C2×#6711yescharacteristic
487C22×C4C23×D311yescharacteristic
488M16—124no—
489C2×C8—32no—
490C2×C8—32no—
491C2×C8—32no—
492C2×C8—61no—
493C2×C8—61no—
494C2×C8—61no—
495C2×C8—61no—
496C2×C8—61no—
497C2×C8—61no—
498C2×C8—61no—
499C2×C8—61no—
500C2×C8—61no—
501C2×C8—124no—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—42no—
512C4⋊C4—61no—
513C4⋊C4—61no—
514C4⋊C4—61no—
515C4⋊C4—61no—
516C4⋊C4—61no—
517C4⋊C4—61no—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—42no—
523C22⋊C4—42no—
524C22⋊C4—122no—
525C22⋊C4—122no—
526C22⋊C4—122no—
527C22⋊C4—241no—
528C22⋊C4—241no—
529C42—42no—
530C42D3×D412yes—
531C42#23712yes—
532C22×C3—22no—
533C22×C3—22no—
534C22×C3—22no—
535C22×C3—41no—
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×C3C2×#12111yescharacteristic
547C22×C3#54611yescharacteristic
548C22×C3#54711yescharacteristic
549C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
550C22×C3#57511yescharacteristic
551C22×C3#57711yescharacteristic
552C22×C3#57711yescharacteristic
553C2×D3—42no—
554C2×D3—42no—
555C2×D3—42no—
556C2×D3—81no—
557C2×D3—81no—
558C2×D3—81no—
559C2×D3—81no—
560C3×C4—22no—
561C3×C4—22no—
562C3×C4—41no—
563C3×C4—41no—
564C3×C4—81no—
565C3×C4—81no—
566C3×C4—81no—
567C3×C4—81no—
568C3×C4#52911yescharacteristic
569C3×C4#54211yescharacteristic
570C3×C4#54611yescharacteristic
571C3×C4#54811yescharacteristic
572Dic3—81no—
573Dic3—81no—
574Dic3—81no—
575C23—21no—
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—62no—
589C23—121no—
590C23—121no—
591C23—121no—
592C23—121no—
593C23—121no—
594C23—121no—
595C23#93212yes—
596C23C22×#6711yescharacteristic
597Q8—84no—
598D4—44no—
599D4—62no—
600D4—62no—
601D4—121no—
602D4—121no—
603D4—244no—
604D4—244no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
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—121no—
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#93912yes—
646C2×C4#93912yes—
647C2×C4C2×D3×D411yescharacteristic
648C2×C4C2×#23711yescharacteristic
649C2×C4D3×#4111yescharacteristic
650C2×C4D3×#4111yescharacteristic
651C2×C4#100911yescharacteristic
652C2×C4#101011yescharacteristic
653C8—62no—
654C8—62no—
655C8—62no—
656C8—121no—
657C2×C3—42no—
658C2×C3—81no—
659C2×C3#242711yescharacteristic
660C2×C3#305511yescharacteristic
661C2×C3#305711yescharacteristic
662C2×C3#330611yescharacteristic
663C2×C3#331111yescharacteristic
664C2×C3#334511yescharacteristic
665C2×C3#335011yescharacteristic
666D3—82no—
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—122no—
682C22—122no—
683C22—122no—
684C22—241no—
685C22—241no—
686C22—241no—
687C22—241no—
688C22C2×#93211yescharacteristic
689C22C2×#93911yescharacteristic
690C22#582211yescharacteristic
691C22#582411yescharacteristic
692C22#582911yescharacteristic
693C22#583711yescharacteristic
694C22#583711yescharacteristic
695C4—22no—
696C4—22no—
697C4—41no—
698C4—41no—
699C4—81no—
700C4—81no—
701C4—81no—
702C4—81no—
703C4—241no—
704C4—241no—
705C4—241no—
706C4#567611yescharacteristic
707C4#567711yescharacteristic
708C4#568011yescharacteristic
709C4#568211yescharacteristic
710C3#2535411yescharacteristic, complemented, Sylow
711C2—42no—
712C2—81no—
713C2—242no—
714C2#8262111yescharacteristic
715C2#8299611yescharacteristic
716C2#8339811yescharacteristic
717C2#8351911yescharacteristic
718C2#8352611yescharacteristic
719C2#8366111yescharacteristic
720C2#8366511yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2547C2
socleC23×C3C22×D4
prime radical#82415C2
hypercenter#2547D3
hypocenterC3#25354
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2547

Complements

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

NH ≅ G/Ntype
1. C3×#2547C2semidirect
2. C2×#5084C2semidirect
4. C2×#5011C2semidirect
5. #71788C2semidirect
6. #69524C2semidirect
7. #69523C2semidirect
8. #69520C2semidirect
9. #69517C2semidirect
10. #69515C2semidirect
11. #69513C2semidirect
12. #69511C2semidirect
13. #69507C2semidirect
14. #69474C2semidirect
15. #69468C2semidirect
64. #2547D3semidirect
147. C4⋊(C3⋊C8)D4semidirect
710. C3#25354semidirect

Maximal subgroups

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

subgroupindex
C3×#25472
C2×#50842
#824152
C2×#50112
#717882
#695242
#695232
#695202
#695172
#695152
#695132
#695112
#695072
#694742
#694682
#253543