Finite Group Databasebeta

← #10845525

Subgroups — #10845525

Order 768.

916 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2537C211yesmaximal, characteristic
2#82383C211yesmaximal, characteristic, complemented
3C2×#5011C211yesmaximal, characteristic, complemented
4C2×#5009C211yesmaximal, characteristic, complemented
5#71682C211yesmaximal, characteristic, complemented
6#71681C211yesmaximal, characteristic, complemented
7#71680C211yesmaximal, characteristic, complemented
8#71680C211yesmaximal, characteristic, complemented
9#71609C211yesmaximal, characteristic, complemented
10#69583C211yesmaximal, characteristic, complemented
11#69568C211yesmaximal, characteristic, complemented
12#69518C211yesmaximal, characteristic, complemented
13#69513C211yesmaximal, characteristic, complemented
14#69511C211yesmaximal, characteristic, complemented
15#69507C211yesmaximal, characteristic, complemented
16#25355—31nomaximal, Sylow
17C2×C3×#121—21no—
18C2×C3×#121C2211yescharacteristic
19C2×C3×#121C2211yescharacteristic
20C2×C3×C4×D4C2211yescharacteristic
21C2×C3×C4×D4C2211yescharacteristic
22C22×#208C2211yescharacteristic
23C23×C3⋊C8C2211yescharacteristic
24C2×#870C2211yescharacteristic
25C3×#394—21no—
26C3×#394—21no—
27C3×#384C2211yescharacteristic
28C3×#380—21no—
29C3×#380C2211yescharacteristic
30C3×#380C2211yescharacteristic
31#5222—21no—
32C2×#832C2211yescharacteristic
33C2×#832C2211yescharacteristic
34C2×#832C2211yescharacteristic
35C2×#832C2211yescharacteristic
36#5212C2211yescharacteristic
37#5212C2211yescharacteristic
38#5212C2211yescharacteristic
39#5212C2211yescharacteristic
40C2×#830C2211yescharacteristic
41C2×#830C2211yescharacteristic
42#5011—24no—
43#5009—24no—
44#4985—21no—
45C2×#808C2211yescharacteristic
46C2×#808C2211yescharacteristic
47C2×#808C2211yescharacteristic
48C2×#807C2211yescharacteristic
49#4945—21no—
50#4945—21no—
51C4×C4⋊Dic3C2211yescharacteristic
52C2×C4⋊(C3⋊C8)C2211yescharacteristic
53C2×#803C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#478—31no—
63C2×#435—31no—
64#2562—31no—
65#2537D311yescharacteristic
66#2077—31no—
67#2075—31no—
68#2034—31no—
69#2034—31no—
70#1992—31no—
71#1948—31no—
72#1895—31no—
73#1893—31no—
74#1809—31no—
75#1807—31no—
76#1762—31no—
77C22×C3×D4—21no—
78C22×C3×D4—21no—
79C22×C3×D4D411yescharacteristic
80C22×C3×D4C2311yescharacteristic
81C23×C3×C4—21no—
82C23×C3×C4C2311yescharacteristic
83C23×C3×C4C2311yescharacteristic
84C3×#121—22no—
85C3×#121—22no—
86C3×#121—41no—
87C3×#121—41no—
88C3×#121—44no—
89C3×C4×D4—22no—
90C3×C4×D4—22no—
91C3×C4×D4—22no—
92C3×C4×D4—22no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4D411yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C22⋊C4—21no—
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4D411yescharacteristic
105C2×C3×C22⋊C4D411yescharacteristic
106C2×C3×C42C2311yescharacteristic
107C2×#217—21no—
108C2×#217—21no—
109C2×#217—21no—
110C2×#217—21no—
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—21no—
117C2×C4×Dic3—21no—
118C2×#208—22no—
119C2×#208—22no—
120C2×#208—41no—
121C22×C3⋊C8—22no—
122C22×C3⋊C8—22no—
123C22×C3⋊C8—41no—
124C22×C3⋊C8C2311yescharacteristic
125C22×C3⋊C8C2311yescharacteristic
126C22×C3⋊C8C2311yescharacteristic
127C22×C3⋊C8C2311yescharacteristic
128#870—22no—
129#870—22no—
130C3×#95—21no—
131C3×#95—21no—
132#832—22no—
133#832—22no—
134#832—22no—
135#832—22no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141#830—22no—
142#830—22no—
143#808—22no—
144#808—41no—
145#808—41no—
146#807—22no—
147C4⋊(C3⋊C8)D414yescomplemented
148#803—22no—
149C22×M16—31no—
150C23×C8—31no—
151C2×#125—31no—
152C2×#121—21no—
153C2×#121C2×D311yescharacteristic
154C2×#121C2×D311yescharacteristic
155C2×C4×D4C2×D311yescharacteristic
156C2×C4×D4C2×D311yescharacteristic
157#478—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#400—61no—
172#394—21no—
173#394—21no—
174#394—61no—
175#388—61no—
176#388—61no—
177#384C2×D311yescharacteristic
178#380—21no—
179#380—31no—
180#380—31no—
181#380—31no—
182#380—31no—
183#380C2×D311yescharacteristic
184#380C2×D311yescharacteristic
185C4×C4⋊C4—31no—
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×C3—21no—
195C24×C3C2×D411yescharacteristic
196C24×C3C2×D411yescharacteristic
197C2×C3×D4—22no—
198C2×C3×D4—22no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—41no—
205C2×C3×D4—42no—
206C2×C3×D4—42no—
207C2×C3×D4—42no—
208C2×C3×D4—42no—
209C2×C3×D4—81no—
210C2×C3×D4—81no—
211C2×C3×D4#4112yes—
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×C4—21no—
220C22×C3×C4—21no—
221C22×C3×C4—21no—
222C22×C3×C4—21no—
223C22×C3×C4—21no—
224C22×C3×C4—21no—
225C22×C3×C4—21no—
226C22×C3×C4—21no—
227C22×C3×C4—42no—
228C22×C3×C4—42no—
229C22×C3×C4—42no—
230C22×C3×C4C2×D411yescharacteristic
231C22×C3×C4C2×D411yescharacteristic
232C22×C3×C4C2×D411yescharacteristic
233C22×C3×C4C2×D411yescharacteristic
234C22×C3×C4C2411yescharacteristic
235C22×Dic3—21no—
236C22×Dic3—21no—
237C22×Dic3—21no—
238C22×Dic3—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—42no—
247C3×C4⋊C4—81no—
248C3×C22⋊C4—22no—
249C3×C22⋊C4—22no—
250C3×C22⋊C4—42no—
251C3×C22⋊C4—42no—
252C3×C22⋊C4—42no—
253C3×C22⋊C4—42no—
254C3×C22⋊C4—42no—
255C3×C42C2×D412yes—
256C3×C42#4112yes—
257#217—42no—
258#217—42no—
259#217—81no—
260#217—81no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269Dic3⋊C4—42no—
270Dic3⋊C4—42no—
271C4×Dic3—81no—
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×D4—21no—
287C22×D4—21no—
288C22×D4#6711yescharacteristic
289C22×D4C22×D311yescharacteristic
290C23×C4—21no—
291C23×C4C22×D311yescharacteristic
292C23×C4C22×D311yescharacteristic
293C2×M16—62no—
294C2×M16—62no—
295C2×M16—121no—
296C22×C8—31no—
297C22×C8—31no—
298C22×C8—31no—
299C22×C8—31no—
300C22×C8—62no—
301C22×C8—62no—
302C22×C8—121no—
303#125—62no—
304#125—62no—
305#121—22no—
306#121—22no—
307#121—41no—
308#121—41no—
309#121—44no—
310C4×D4—22no—
311C4×D4—22no—
312C4×D4—22no—
313C4×D4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—31no—
317C2×C4⋊C4—31no—
318C2×C4⋊C4—31no—
319C2×C4⋊C4—31no—
320C2×C4⋊C4—61no—
321C2×C4⋊C4—61no—
322C2×C4⋊C4#6711yescharacteristic
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—21no—
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—61no—
338C2×C42C22×D311yescharacteristic
339#107—62no—
340#106—62no—
341#106—121no—
342#106—121no—
343C4⋊C8—34no—
344#102—62no—
345#102—62no—
346#102—62no—
347#102—62no—
348C22⋊C8—62no—
349C22⋊C8—62no—
350#97—62no—
351#95—21no—
352#95—21no—
353#95—61no—
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—22no—
366C23×C3—21no—
367C23×C3—21no—
368C23×C3—21no—
369C23×C3—21no—
370C23×C3—21no—
371C23×C3—21no—
372C23×C3—41no—
373C23×C3—41no—
374C23×C3—41no—
375C23×C3—41no—
376C23×C3—41no—
377C23×C3—41no—
378C23×C3AGL(1,ℤ/8)12yes—
379C23×C3C22×D411yescharacteristic
380C3×D4—22no—
381C3×D4—22no—
382C3×D4—44no—
383C3×D4—41no—
384C3×D4—41no—
385C3×D4—84no—
386C3×D4—84no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
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—41no—
413C2×C3×C4—41no—
414C2×C3×C4—84no—
415C2×C3×C4#12112yes—
416C2×C3×C4#12112yes—
417C2×C3×C4C22×D411yescharacteristic
418C2×C3×C4C2×#4111yescharacteristic
419C2×C3×C4C2×#4111yescharacteristic
420C2×C3×C4C2×#4111yescharacteristic
421C2×C3×C4H(5,𝔽2)11yescharacteristic
422C2×C3×C42−1+411yescharacteristic
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—41no—
432C2×Dic3—41no—
433C2×Dic3—41no—
434C2×Dic3—41no—
435C3⋊C8—22no—
436C3⋊C8—22no—
437C3⋊C8—22no—
438C3⋊C8—41no—
439C24—21no—
440C24C2×#6711yescharacteristic
441C24C2×#6711yescharacteristic
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—41no—
450C2×D4—42no—
451C2×D4—42no—
452C2×D4—42no—
453C2×D4—42no—
454C2×D4—81no—
455C2×D4—81no—
456C2×D4#23712yes—
457C22×C4—21no—
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—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—61no—
476C22×C4—61no—
477C22×C4—61no—
478C22×C4—61no—
479C22×C4C2×#6711yescharacteristic
480C22×C4C2×#6711yescharacteristic
481C22×C4C2×#6711yescharacteristic
482C22×C4C2×#6711yescharacteristic
483C22×C4C23×D311yescharacteristic
484M16—124no—
485C2×C8—32no—
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—124no—
498C4⋊C4—22no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—42no—
506C4⋊C4—61no—
507C4⋊C4—61no—
508C4⋊C4—61no—
509C4⋊C4—61no—
510C4⋊C4—61no—
511C4⋊C4—61no—
512C4⋊C4—61no—
513C4⋊C4—61no—
514C4⋊C4—81no—
515C4⋊C4—122no—
516C4⋊C4—122no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—42no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—42no—
523C22⋊C4—42no—
524C22⋊C4—122no—
525C22⋊C4—122no—
526C22⋊C4—241no—
527C22⋊C4—241no—
528C42—241no—
529C42D3×D412yes—
530C42#23712yes—
531C22×C3—22no—
532C22×C3—22no—
533C22×C3—22no—
534C22×C3—41no—
535C22×C3—41no—
536C22×C3—41no—
537C22×C3—41no—
538C22×C3—41no—
539C22×C3—41no—
540C22×C3—41no—
541C22×C3—42no—
542C22×C3—41no—
543C22×C3—42no—
544C22×C3—41no—
545C22×C3—42no—
546C22×C3—41no—
547C22×C3—41no—
548C22×C3—81no—
549C22×C3—81no—
550C22×C3—81no—
551C22×C3—81no—
552C22×C3C2×#12111yescharacteristic
553C22×C3#54611yescharacteristic
554C22×C3#54711yescharacteristic
555C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
556C22×C3#57511yescharacteristic
557C22×C3#57711yescharacteristic
558C22×C3#57711yescharacteristic
559C3×C4—22no—
560C3×C4—22no—
561C3×C4—41no—
562C3×C4—41no—
563C3×C4—81no—
564C3×C4—81no—
565C3×C4—81no—
566C3×C4#52911yescharacteristic
567C3×C4#54211yescharacteristic
568C3×C4#54611yescharacteristic
569C3×C4#54811yescharacteristic
570Dic3—81no—
571Dic3—81no—
572Dic3—81no—
573Dic3—81no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—21no—
581C23—21no—
582C23—22no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—21no—
587C23—21no—
588C23—41no—
589C23—41no—
590C23—41no—
591C23—41no—
592C23—41no—
593C23—41no—
594C23#93212yes—
595C23C22×#6711yescharacteristic
596D4—22no—
597D4—22no—
598D4—44no—
599D4—41no—
600D4—41no—
601D4—84no—
602D4—84no—
603C2×C4—21no—
604C2×C4—21no—
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—41no—
614C2×C4—41no—
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—84no—
631C2×C4—121no—
632C2×C4—121no—
633C2×C4—121no—
634C2×C4—121no—
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#93912yes—
644C2×C4#93912yes—
645C2×C4C2×D3×D411yescharacteristic
646C2×C4C2×#23711yescharacteristic
647C2×C4C2×#23711yescharacteristic
648C2×C4#100411yescharacteristic
649C2×C4D3×#4111yescharacteristic
650C2×C4#101011yescharacteristic
651C8—62no—
652C8—62no—
653C8—62no—
654C8—121no—
655C2×C3—42no—
656C2×C3—81no—
657C2×C3—82no—
658C2×C3#242711yescharacteristic
659C2×C3#305511yescharacteristic
660C2×C3#306011yescharacteristic
661C2×C3#330711yescharacteristic
662C2×C3#331011yescharacteristic
663C2×C3#334511yescharacteristic
664C2×C3#334811yescharacteristic
665C22—22no—
666C22—22no—
667C22—22no—
668C22—41no—
669C22—41no—
670C22—41no—
671C22—41no—
672C22—41no—
673C22—42no—
674C22—41no—
675C22—42no—
676C22—41no—
677C22—41no—
678C22—41no—
679C22—41no—
680C22—42no—
681C22—41no—
682C22—81no—
683C22—81no—
684C22—81no—
685C22—81no—
686C22C2×#93211yescharacteristic
687C22C2×#93911yescharacteristic
688C22#580611yescharacteristic
689C22#582211yescharacteristic
690C22#582411yescharacteristic
691C22#583611yescharacteristic
692C22#583711yescharacteristic
693C4—22no—
694C4—22no—
695C4—41no—
696C4—41no—
697C4—81no—
698C4—81no—
699C4—81no—
700C4—241no—
701C4—241no—
702C4—241no—
703C4—241no—
704C4#559811yescharacteristic
705C4#560011yescharacteristic
706C4#560311yescharacteristic
707C4#561211yescharacteristic
708C3#2535511yescharacteristic, complemented, Sylow
709C2—42no—
710C2—81no—
711C2—82no—
712C2#8262111yescharacteristic
713C2#8264711yescharacteristic
714C2#8342011yescharacteristic
715C2#8348411yescharacteristic
716C2#8351211yescharacteristic
717C2#8351911yescharacteristic
718C2#8364911yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#2537C2
socleC23×C3C22×D4
hypercenter#2537D3
hypocenterC3#25355
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2537

Complements

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

NH ≅ G/Ntype
2. #82383C2semidirect
3. C2×#5011C2semidirect
4. C2×#5009C2semidirect
5. #71682C2semidirect
6. #71681C2semidirect
7. #71680C2semidirect
8. #71680C2semidirect
9. #71609C2semidirect
10. #69583C2semidirect
11. #69568C2semidirect
12. #69518C2semidirect
13. #69513C2semidirect
14. #69511C2semidirect
15. #69507C2semidirect
147. C4⋊(C3⋊C8)D4semidirect
708. C3#25355semidirect

Maximal subgroups

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

subgroupindex
C3×#25372
#823832
C2×#50112
C2×#50092
#716822
#716812
#716802
#716092
#695832
#695682
#695182
#695132
#695112
#695072
#253553