Finite Group Databasebeta

← #10844654

Subgroups — #10844654

Order 768.

760 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2546C211yesmaximal, characteristic
2C2×#5082C211yesmaximal, characteristic
3C2×#5022C211yesmaximal, characteristic
4#73659C211yesmaximal, characteristic
5#71781C211yesmaximal, characteristic
6#71781C211yesmaximal, characteristic
7#71778C211yesmaximal, characteristic
8#71778C211yesmaximal, characteristic
9#71778C211yesmaximal, characteristic
10#71631C211yesmaximal, characteristic
11#69683C211yesmaximal, characteristic
12#69682C211yesmaximal, characteristic
13#69667C211yesmaximal, characteristic
14#69666C211yesmaximal, characteristic
15#69485C211yesmaximal, characteristic
16#25290—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×C3×C4⋊Q8C2211yescharacteristic
20C2×C3×C4×Q8C2211yescharacteristic
21C2×#870C2211yescharacteristic
22C3×#398—21no—
23C3×#398—21no—
24C3×#391C2211yescharacteristic
25C3×#391C2211yescharacteristic
26C3×#389—21no—
27C3×#389—21no—
28C3×C4×C4⋊C4C2211yescharacteristic
29C2×#835C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#835C2211yescharacteristic
32C2×#835C2211yescharacteristic
33C2×#835C2211yescharacteristic
34C2×#835C2211yescharacteristic
35C2×#835C2211yescharacteristic
36C2×#835C2211yescharacteristic
37C2×#835C2211yescharacteristic
38C2×#835C2211yescharacteristic
39C2×#835C2211yescharacteristic
40C2×#835C2211yescharacteristic
41#5082—24no—
42#5022—24no—
43#4978—21no—
44#4976—21no—
45C2×#808C2211yescharacteristic
46C2×#807C2211yescharacteristic
47C4×C4⋊Dic3C2211yescharacteristic
48C4×C4⋊Dic3C2211yescharacteristic
49C4×C4⋊Dic3C2211yescharacteristic
50#4928C2211yescharacteristic
51#4928C2211yescharacteristic
52C2×C4⋊(C3⋊C8)C2211yescharacteristic
53C2×C4⋊(C3⋊C8)C2211yescharacteristic
54C2×C4⋊(C3⋊C8)C2211yescharacteristic
55C2×#803C2211yescharacteristic
56#4652—21no—
57#4649—21no—
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4524C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#491—31no—
63C2×#479—31no—
64#2546D311yescharacteristic
65#2370—31no—
66#2068—31no—
67#2066—31no—
68#2043—31no—
69#2043—31no—
70#1981—31no—
71#1981—31no—
72#1981—31no—
73#1940—31no—
74#1928—31no—
75#1927—31no—
76#1763—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×Q8C2311yescharacteristic
79C22×C3×Q8C2311yescharacteristic
80C3×C4⋊Q8—22no—
81C3×C4⋊Q8—22no—
82C3×C4⋊Q8—22no—
83C3×C4⋊Q8—22no—
84C3×C4⋊Q8—41no—
85C3×C4⋊Q8—41no—
86C3×C4×Q8—22no—
87C3×C4×Q8—22no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4D411yescharacteristic
96C2×C3×C4⋊C4D411yescharacteristic
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C42—21no—
99C2×C3×C42C2311yescharacteristic
100C2×C3×C42C2311yescharacteristic
101C2×C3×C42C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×C4×Dic3—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3—21no—
113C22×C3⋊C8C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117#870—22no—
118#870D414yes—
119C3×#95—21no—
120C3×#95—21no—
121C3×#95—21no—
122C3×#95—21no—
123C3×#95D411yescharacteristic
124C3×#95D411yescharacteristic
125#835—22no—
126#835—22no—
127#835—22no—
128#835—22no—
129#835—22no—
130#835—22no—
131#835—22no—
132#835—22no—
133#835—22no—
134#835—22no—
135#835—41no—
136#835—41no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142#808—22no—
143#807—22no—
144C4⋊(C3⋊C8)—22no—
145C4⋊(C3⋊C8)—41no—
146C4⋊(C3⋊C8)—41no—
147#803—22no—
148C2×C4⋊Q8C2×D311yescharacteristic
149C2×C4⋊Q8C2×D311yescharacteristic
150C2×C4⋊Q8C2×D311yescharacteristic
151C2×#125—31no—
152C2×C4×Q8C2×D311yescharacteristic
153#491—64no—
154#479—64no—
155C2×#107—31no—
156C2×#106—31no—
157C2×C4⋊C8—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×#103—31no—
166C2×#103—31no—
167C2×#103—31no—
168C2×#103—31no—
169C2×#103—31no—
170C2×#103—31no—
171C2×#103—31no—
172C2×#97—31no—
173#398—21no—
174#398—21no—
175#391C2×D311yescharacteristic
176#391C2×D311yescharacteristic
177#389—21no—
178#389—21no—
179#389—31no—
180#389—31no—
181#389—61no—
182#387—61no—
183#387—61no—
184C4×C4⋊C4—31no—
185C4×C4⋊C4—31no—
186C4×C4⋊C4—31no—
187C4×C4⋊C4—61no—
188C4×C4⋊C4C2×D311yescharacteristic
189#340—31no—
190#340—31no—
191#336—31no—
192#336—31no—
193C2×C3×Q8—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8—21no—
204C2×C3×Q8—21no—
205C2×C3×Q8—21no—
206C2×C3×Q8—21no—
207C2×C3×Q8—21no—
208C2×C3×Q8—21no—
209C2×C3×Q8—21no—
210C2×C3×Q8#4112yes—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4C2411yescharacteristic
222C22×Dic3—21no—
223C22×Dic3—21no—
224C22×Dic3—21no—
225C22×Dic3—21no—
226C3×C4⋊C4—22no—
227C3×C4⋊C4—22no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—42no—
231C3×C4⋊C4—42no—
232C3×C4⋊C4—42no—
233C3×C4⋊C4—42no—
234C3×C4⋊C4—42no—
235C3×C4⋊C4—42no—
236C3×C4⋊C4—42no—
237C3×C42—21no—
238C3×C42—21no—
239C3×C42—21no—
240C3×C42—21no—
241C3×C42—42no—
242C3×C42C2×D412yes—
243C3×C42#4112yes—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3C2×D412yes—
250C4⋊Dic3C2×D412yes—
251C4⋊Dic3#4112yes—
252Dic3⋊C4—24no—
253Dic3⋊C4—24no—
254Dic3⋊C4—42no—
255Dic3⋊C4—81no—
256C4×Dic3—42no—
257C4×Dic3—81no—
258C4×Dic3—81no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8—21no—
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—
271C22×Q8C22×D311yescharacteristic
272C22×Q8C22×D311yescharacteristic
273C22×Q8C22×D311yescharacteristic
274C22×C8—31no—
275C22×C8—31no—
276C22×C8—31no—
277C22×C8—31no—
278C4⋊Q8—22no—
279C4⋊Q8—22no—
280C4⋊Q8—22no—
281C4⋊Q8—22no—
282C4⋊Q8—41no—
283C4⋊Q8—41no—
284#125—34no—
285#125—62no—
286C4×Q8—22no—
287C4×Q8—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—31no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4#6711yescharacteristic
304C2×C4⋊C4#6711yescharacteristic
305C2×C4⋊C4C22×D311yescharacteristic
306C2×C42—21no—
307C2×C42—61no—
308C2×C42—61no—
309C2×C42—61no—
310C2×C42C22×D311yescharacteristic
311C2×C42C22×D311yescharacteristic
312C2×C42C22×D311yescharacteristic
313#107—62no—
314#106—62no—
315C4⋊C8—62no—
316C4⋊C8—121no—
317C4⋊C8—121no—
318#103—62no—
319#103—62no—
320#103—62no—
321#103—62no—
322#103—62no—
323#103—62no—
324#103—62no—
325#103—62no—
326#103—62no—
327#103—62no—
328#103—121no—
329#103—121no—
330#97—62no—
331#95—21no—
332#95—21no—
333#95—21no—
334#95—21no—
335#95—61no—
336#95—61no—
337#95—61no—
338#95—61no—
339#95—61no—
340#95#6711yescharacteristic
341#95#6711yescharacteristic
342C23×C3C22×D411yescharacteristic
343C3×Q8—22no—
344C3×Q8—22no—
345C3×Q8—41no—
346C3×Q8—41no—
347C3×Q8—41no—
348C3×Q8—41no—
349C3×Q8—41no—
350C3×Q8—41no—
351C3×Q8—41no—
352C3×Q8—41no—
353C3×Q8—41no—
354C3×Q8—41no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—22no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4#12112yes—
382C2×C3×C4#12112yes—
383C2×C3×C4#13712yes—
384C2×C3×C4#13712yes—
385C2×C3×C4C22×D411yescharacteristic
386C2×C3×C4C2×#4111yescharacteristic
387C2×C3×C4C2×#4111yescharacteristic
388C2×C3×C4C2×#4111yescharacteristic
389C2×C3×C4H(5,𝔽2)11yescharacteristic
390C2×C3×C42−1+411yescharacteristic
391C2×Dic3—22no—
392C2×Dic3—22no—
393C2×Dic3—22no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C3⋊C8—41no—
404C3⋊C8—41no—
405C3⋊C8—41no—
406C3⋊C8—41no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8#23712yes—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—61no—
430C22×C4—61no—
431C22×C4—61no—
432C22×C4—61no—
433C22×C4C2×#6711yescharacteristic
434C22×C4C2×#6711yescharacteristic
435C22×C4C2×#6711yescharacteristic
436C22×C4C2×#6711yescharacteristic
437C22×C4C2×#6711yescharacteristic
438C22×C4C2×#6711yescharacteristic
439C22×C4C23×D311yescharacteristic
440C2×C8—61no—
441C2×C8—61no—
442C2×C8—61no—
443C2×C8—61no—
444C2×C8—61no—
445C2×C8—61no—
446C2×C8—61no—
447C2×C8—61no—
448C2×C8—61no—
449C2×C8—61no—
450C2×C8—61no—
451C2×C8—61no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—32no—
457C4⋊C4—32no—
458C4⋊C4—32no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—64no—
467C4⋊C4—64no—
468C4⋊C4—61no—
469C4⋊C4—61no—
470C4⋊C4—61no—
471C4⋊C4—61no—
472C4⋊C4—61no—
473C4⋊C4—122no—
474C4⋊C4—241no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—42no—
480C42—122no—
481C42—241no—
482C42—241no—
483C42D3×D412yes—
484C42#23712yes—
485C22×C3C2×#12111yescharacteristic
486C22×C3#54611yescharacteristic
487C22×C3#54811yescharacteristic
488C22×C3C2×#13711yescharacteristic
489C22×C3C2×#13711yescharacteristic
490C22×C3#57711yescharacteristic
491C22×C3#57811yescharacteristic
492C3×C4—22no—
493C3×C4—22no—
494C3×C4—22no—
495C3×C4—22no—
496C3×C4—42no—
497C3×C4—41no—
498C3×C4—41no—
499C3×C4—81no—
500C3×C4—81no—
501C3×C4—81no—
502C3×C4#52911yescharacteristic
503C3×C4#54211yescharacteristic
504C3×C4#54811yescharacteristic
505C3×C4#55011yescharacteristic
506Dic3—42no—
507Dic3—42no—
508Dic3—42no—
509Dic3—81no—
510C23C22×#6711yescharacteristic
511Q8—22no—
512Q8—22no—
513Q8—41no—
514Q8—41no—
515Q8—41no—
516Q8—41no—
517Q8—41no—
518Q8—41no—
519Q8—41no—
520Q8—41no—
521Q8—41no—
522Q8—41no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—62no—
550C2×C4—62no—
551C2×C4—62no—
552C2×C4—121no—
553C2×C4—121no—
554C2×C4—121no—
555C2×C4—121no—
556C2×C4—121no—
557C2×C4—121no—
558C2×C4—121no—
559C2×C4—121no—
560C2×C4—121no—
561C2×C4#93812yes—
562C2×C4#93812yes—
563C2×C4#94212yes—
564C2×C4#94212yes—
565C2×C4C2×D3×D411yescharacteristic
566C2×C4C2×#23711yescharacteristic
567C2×C4C2×#23711yescharacteristic
568C2×C4#100411yescharacteristic
569C2×C4D3×#4111yescharacteristic
570C2×C4#101011yescharacteristic
571C8—121no—
572C8—121no—
573C8—121no—
574C8—121no—
575C2×C3#242611yescharacteristic
576C2×C3#303911yescharacteristic
577C2×C3#304611yescharacteristic
578C2×C3#331211yescharacteristic
579C2×C3#332011yescharacteristic
580C2×C3#337211yescharacteristic
581C2×C3#337411yescharacteristic
582C22C2×#93811yescharacteristic
583C22#580611yescharacteristic
584C22C2×#94211yescharacteristic
585C22C2×#94211yescharacteristic
586C22#583111yescharacteristic
587C22#583711yescharacteristic
588C22#583811yescharacteristic
589C4—22no—
590C4—22no—
591C4—22no—
592C4—22no—
593C4—41no—
594C4—41no—
595C4—42no—
596C4—81no—
597C4—81no—
598C4—81no—
599C4—122no—
600C4—122no—
601C4—122no—
602C4—241no—
603C4#559811yescharacteristic
604C4#559911yescharacteristic
605C4#560011yescharacteristic
606C4#560311yescharacteristic
607C3#2529011yescharacteristic, complemented, Sylow
608C2#8264611yescharacteristic
609C2#8320911yescharacteristic
610C2#8340611yescharacteristic
611C2#8355211yescharacteristic
612C2#8355311yescharacteristic
613C2#8369211yescharacteristic
614C2#8369911yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2546C2
socle, prime radicalC23×C3C22×D4
hypercenter#2546D3
hypocenterC3#25290
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2546

Complements

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

NH ≅ G/Ntype
607. C3#25290semidirect

Maximal subgroups

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

subgroupindex
C3×#25462
C2×#50822
C2×#50222
#736592
#717812
#717782
#716312
#696832
#696822
#696672
#696662
#694852
#252903