Finite Group Databasebeta

← #10824384

Subgroups — #10824384

Order 768.

880 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
1C3×C4×#124C211yesmaximal, characteristic
2C2×C4×#216C211yesmaximal, characteristic, complemented
3C2×C4×#214C211yesmaximal, characteristic
4#73300C211yesmaximal, characteristic, complemented
5#71729C211yesmaximal, characteristic
6#71722C211yesmaximal, characteristic
7C4×#835C211yesmaximal, characteristic, complemented
8C4×#832C211yesmaximal, characteristic
9#69682C211yesmaximal, characteristic, complemented
10#69681C211yesmaximal, characteristic, complemented
11#69653C211yesmaximal, characteristic
12#69651C211yesmaximal, characteristic
13#69560C211yesmaximal, characteristic, complemented
14#69541C211yesmaximal, characteristic, complemented
15C4×C4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
16#23681—31nomaximal, Sylow
17C2×C3×#124C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20C22×#216C2211yescharacteristic
21C22×#214C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C3×#391C2211yescharacteristic
24C3×#390C2211yescharacteristic
25C3×#388—21no—
26C3×#388—21no—
27C3×#382C2211yescharacteristic
28C3×C4×C22⋊C4—21no—
29C3×C4×C22⋊C4—21no—
30C3×C43C2211yescharacteristic
31C2×#835C2211yescharacteristic
32C2×#835C2211yescharacteristic
33C2×#835C2211yescharacteristic
34C2×#832C2211yescharacteristic
35C2×#832C2211yescharacteristic
36C2×#832C2211yescharacteristic
37C4×#216—24no—
38C4×#214—24no—
39C2×#810C2211yescharacteristic
40C2×#810C2211yescharacteristic
41C2×#808C2211yescharacteristic
42C2×#807C2211yescharacteristic
43C4×C4⋊Dic3C2211yescharacteristic
44#4933—21no—
45#4933—21no—
46#4932—21no—
47C4×Dic3⋊C4—21no—
48#4929C2211yescharacteristic
49#4929C2211yescharacteristic
50#4928C2211yescharacteristic
51#4928C2211yescharacteristic
52C2×C4⋊(C3⋊C8)C2211yescharacteristic
53C2×C4⋊(C3⋊C8)C2211yescharacteristic
54C2×C4×C3⋊C8C2211yescharacteristic
55C2×C4×C3⋊C8C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4524C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×C4×Q16—31no—
63C2×C4×SD16—31no—
64C4×#124D311yescharacteristic
65#2295—31no—
66#2085—31no—
67#2074—31no—
68#2067—31no—
69#2066—31no—
70#2056—31no—
71#2055—31no—
72#2042—31no—
73#2015—31no—
74C4×C4⋊C8—31no—
75C4×#103—31no—
76C4×#102—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×D4C2311yescharacteristic
79C23×C3×C4—21no—
80C22×Dic6C2311yescharacteristic
81C3×#124—22no—
82C3×#124—22no—
83C3×C4×Q8—22no—
84C3×C4×Q8—22no—
85C3×C4×D4—22no—
86C3×C4×D4—22no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C22⋊C4—21no—
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C42—21no—
96C2×C3×C42D411yescharacteristic
97C2×C3×C42D411yescharacteristic
98C2×C3×C42D411yescharacteristic
99C2×C3×C42D411yescharacteristic
100C2×C3×C42C2311yescharacteristic
101C2×C3×C42C2311yescharacteristic
102C2×C3×C42C2311yescharacteristic
103C2×#216—22no—
104C2×#216—22no—
105C2×#216—22no—
106C2×#214—22no—
107C2×#214—22no—
108C2×#214—22no—
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×C4×Dic3—21no—
116C2×C4×Dic3—21no—
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121C4×Dic6—22no—
122C4×Dic6D414yes—
123C3×#95—21no—
124C3×#95—21no—
125C3×#95—21no—
126C3×#95—21no—
127#835—22no—
128#835—22no—
129#835—22no—
130#832—22no—
131#832—22no—
132#832—22no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140#810—22no—
141#810—22no—
142#808—22no—
143#807—22no—
144C4⋊(C3⋊C8)—22no—
145C4⋊(C3⋊C8)—22no—
146C4×C3⋊C8—22no—
147C4×C3⋊C8—22no—
148C22×Q16—31no—
149C22×SD16—31no—
150C2×#124C2×D311yescharacteristic
151C2×C4×Q8—31no—
152C2×C4×Q8C2×D311yescharacteristic
153C2×C4×D4C2×D311yescharacteristic
154C4×Q16—64no—
155C4×SD16—64no—
156C2×#107—31no—
157C2×#106—31no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8—31no—
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×C4×C8—31no—
169C2×C4×C8—31no—
170#391—31no—
171#391—31no—
172#391C2×D311yescharacteristic
173#390C2×D311yescharacteristic
174#389—31no—
175#389—31no—
176#388—21no—
177#388—21no—
178#387—61no—
179#387—61no—
180#382C2×D311yescharacteristic
181C4×C4⋊C4—31no—
182C4×C4⋊C4—61no—
183C4×C22⋊C4—21no—
184C4×C22⋊C4—21no—
185#376—61no—
186C43C2×D311yescharacteristic
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#336—31no—
192#336—31no—
193C24×C3—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8#4112yes—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4#4112yes—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—42no—
210C22×C3×C4—42no—
211C22×C3×C4—42no—
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2411yescharacteristic
219C22×Dic3—21no—
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3—21no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6C2×D412yes—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—42no—
234C3×C22⋊C4—24no—
235C3×C22⋊C4—24no—
236C3×C22⋊C4—42no—
237C3×C22⋊C4—42no—
238C3×C22⋊C4—42no—
239C3×C42—22no—
240C3×C42—22no—
241C3×C42—22no—
242C3×C42—21no—
243C3×C42—21no—
244C3×C42—21no—
245C3×C42—21no—
246C3×C42—41no—
247C3×C42—42no—
248C3×C42C2×D412yes—
249C3×C42#4112yes—
250#216—28no—
251#214—28no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3C2×D412yes—
258Dic3⋊C4—24no—
259Dic3⋊C4—42no—
260Dic3⋊C4—42no—
261C4×Dic3—24no—
262C4×Dic3—42no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8#4112yes—
274C2×C3⋊C8#4112yes—
275C22×Q8—31no—
276C22×Q8C22×D311yescharacteristic
277C22×D4C22×D311yescharacteristic
278C23×C4—21no—
279C2×Q16—62no—
280C2×Q16—62no—
281C2×Q16—62no—
282C2×SD16—62no—
283C2×SD16—62no—
284C2×SD16—62no—
285C22×C8—31no—
286C22×C8—31no—
287C22×C8—31no—
288C22×C8—31no—
289#124—22no—
290#124—22no—
291C4×Q8—22no—
292C4×Q8—22no—
293C4×Q8—34no—
294C4×Q8—62no—
295C4×D4—22no—
296C4×D4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4C22×D311yescharacteristic
305C2×C4⋊C4C22×D311yescharacteristic
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C42—21no—
312C2×C42—61no—
313C2×C42—61no—
314C2×C42#6711yescharacteristic
315C2×C42#6711yescharacteristic
316C2×C42#6711yescharacteristic
317C2×C42#6711yescharacteristic
318C2×C42C22×D311yescharacteristic
319C2×C42C22×D311yescharacteristic
320C2×C42C22×D311yescharacteristic
321#107—62no—
322#106—62no—
323C4⋊C8—62no—
324C4⋊C8—62no—
325#103—62no—
326#103—62no—
327#103—62no—
328#103—62no—
329#103—62no—
330#102—62no—
331#102—62no—
332#102—62no—
333C4×C8—62no—
334C4×C8—62no—
335#95—21no—
336#95—21no—
337#95—21no—
338#95—21no—
339#95—61no—
340#95—61no—
341#95—61no—
342#95—61no—
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
354C3×Q8—22no—
355C3×Q8—22no—
356C3×Q8—41no—
357C3×Q8—41no—
358C3×D4—22no—
359C3×D4—22no—
360C3×D4—41no—
361C3×D4—41no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—22no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—84no—
385C2×C3×C4#12112yes—
386C2×C3×C4#12112yes—
387C2×C3×C4#13512yes—
388C2×C3×C4#13512yes—
389C2×C3×C4#13712yes—
390C2×C3×C4C22×D411yescharacteristic
391C2×C3×C4C2×#4111yescharacteristic
392C2×C3×C4C2×#4111yescharacteristic
393C2×C3×C4C2×#4111yescharacteristic
394C2×C3×C4C2×#4111yescharacteristic
395C2×C3×C4C2×#4111yescharacteristic
396C2×Dic3—22no—
397C2×Dic3—22no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408Dic6—22no—
409Dic6—22no—
410Dic6—41no—
411Dic6—41no—
412C3⋊C8—22no—
413C3⋊C8—22no—
414C3⋊C8—41no—
415C3⋊C8—41no—
416C24—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—32no—
423C2×Q8—61no—
424C2×Q8—61no—
425C2×Q8—61no—
426C2×Q8—61no—
427C2×Q8—61no—
428C2×Q8#23512yes—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4#23512yes—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—61no—
442C22×C4—61no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4C2×#6711yescharacteristic
446C22×C4C2×#6711yescharacteristic
447C22×C4C2×#6711yescharacteristic
448C22×C4C2×#6711yescharacteristic
449C22×C4C2×#6711yescharacteristic
450C22×C4C2×#6711yescharacteristic
451C22×C4C23×D311yescharacteristic
452Q16—68no—
453SD16—68no—
454C2×C8—32no—
455C2×C8—32no—
456C2×C8—61no—
457C2×C8—61no—
458C2×C8—61no—
459C2×C8—61no—
460C2×C8—61no—
461C2×C8—61no—
462C2×C8—61no—
463C2×C8—61no—
464C2×C8—61no—
465C2×C8—61no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—32no—
471C4⋊C4—42no—
472C4⋊C4—64no—
473C4⋊C4—61no—
474C4⋊C4—61no—
475C4⋊C4—61no—
476C4⋊C4—61no—
477C4⋊C4—61no—
478C4⋊C4—122no—
479C4⋊C4—122no—
480C22⋊C4—24no—
481C22⋊C4—24no—
482C22⋊C4—42no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—41no—
493C42—42no—
494C42—64no—
495C42—122no—
496C42D3×D412yes—
497C42#23712yes—
498C22×C3—42no—
499C22×C3—42no—
500C22×C3—42no—
501C22×C3—81no—
502C22×C3—81no—
503C22×C3—81no—
504C22×C3—81no—
505C22×C3C2×#12111yescharacteristic
506C22×C3#52511yescharacteristic
507C22×C3#53211yescharacteristic
508C22×C3C2×#13511yescharacteristic
509C22×C3C2×#13511yescharacteristic
510C22×C3C2×#13711yescharacteristic
511C22×C3#57511yescharacteristic
512C3×C4—22no—
513C3×C4—22no—
514C3×C4—41no—
515C3×C4—41no—
516C3×C4—42no—
517C3×C4—41no—
518C3×C4—81no—
519C3×C4—81no—
520C3×C4#46414yes—
521C3×C4#52511yescharacteristic
522C3×C4#52511yescharacteristic
523C3×C4#52911yescharacteristic
524C3×C4#52911yescharacteristic
525Dic3—42no—
526Dic3—42no—
527Dic3—81no—
528Dic3—81no—
529C23—22no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23C22×#6711yescharacteristic
537Q8—22no—
538Q8—22no—
539Q8—41no—
540Q8—41no—
541Q8—62no—
542Q8—62no—
543Q8—121no—
544Q8—121no—
545D4—22no—
546D4—22no—
547D4—41no—
548D4—41no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—62no—
572C2×C4—62no—
573C2×C4—84no—
574C2×C4—121no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4#93812yes—
585C2×C4#93812yes—
586C2×C4#95012yes—
587C2×C4#95012yes—
588C2×C4#95112yes—
589C2×C4C2×#23511yescharacteristic
590C2×C4C2×#23511yescharacteristic
591C2×C4C2×D3×D411yescharacteristic
592C2×C4C2×#23711yescharacteristic
593C2×C4D3×#4111yescharacteristic
594C2×C4D3×#4111yescharacteristic
595C8—62no—
596C8—62no—
597C8—121no—
598C8—121no—
599C2×C3—82no—
600C2×C3#239211yescharacteristic
601C2×C3C2×#46411yescharacteristic
602C2×C3#303511yescharacteristic
603C2×C3#310211yescharacteristic
604C2×C3#310711yescharacteristic
605C2×C3#316811yescharacteristic
606C2×C3#317211yescharacteristic
607C22—42no—
608C22—42no—
609C22—42no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22#579111yescharacteristic
615C22C2×#93811yescharacteristic
616C22C2×#95011yescharacteristic
617C22C2×#95011yescharacteristic
618C22#582211yescharacteristic
619C22C2×#95111yescharacteristic
620C22#583211yescharacteristic
621C4—22no—
622C4—22no—
623C4—41no—
624C4—41no—
625C4—41no—
626C4—42no—
627C4—81no—
628C4—81no—
629C4—122no—
630C4—122no—
631C4—241no—
632C4—241no—
633C4#505414yes—
634C4#553811yescharacteristic
635C4#553811yescharacteristic
636C4#554711yescharacteristic
637C4#554711yescharacteristic
638C3#2368111yescharacteristic, complemented, Sylow
639C2—82no—
640C2#8240111yescharacteristic
641C2#8252811yescharacteristic
642C2#8253411yescharacteristic
643C2C2×#505411yescharacteristic
644C2#8313411yescharacteristic
645C2#8313611yescharacteristic
646C2#8329611yescharacteristic
subgroupG/H
centerC22×C4C2×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×#124C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenterC4×#124D3
hypocenterC3#23681
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C43 ◁ C4×#124

Complements

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

NH ≅ G/Ntype
2. C2×C4×#216C2semidirect
4. #73300C2semidirect
7. C4×#835C2semidirect
9. #69682C2semidirect
10. #69681C2semidirect
13. #69560C2semidirect
14. #69541C2semidirect
15. C4×C4⋊(C3⋊C8)C2semidirect
638. C3#23681semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#1242
C2×C4×#2162
C2×C4×#2142
#733002
#717292
#717222
C4×#8352
C4×#8322
#696822
#696812
#696532
#696512
#695602
#695412
C4×C4⋊(C3⋊C8)2
#236813