Finite Group Databasebeta

← #10858794

Subgroups — #10858794

Order 768.

818 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2440C211yesmaximal, characteristic
2#73465C211yesmaximal, characteristic
3#71794C211yesmaximal, characteristic, complemented
4#71780C211yesmaximal, characteristic, complemented
5#71559C211yesmaximal, characteristic
6C2×#4529C211yesmaximal, characteristic
7#69634C211yesmaximal, characteristic, complemented
8#69632C211yesmaximal, characteristic, complemented
9#69624C211yesmaximal, characteristic, complemented
10#69601C211yesmaximal, characteristic, complemented
11#69588C211yesmaximal, characteristic, complemented
12#69571C211yesmaximal, characteristic, complemented
13#69514C211yesmaximal, characteristic
14#69505C211yesmaximal, characteristic
15#69449C211yesmaximal, characteristic
16#26412—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×C22⋊Q8—21no—
19C22×C3×C4⋊C4C2211yescharacteristic
20C22×C4⋊Dic3C2211yescharacteristic
21C2×#924C2211yescharacteristic
22C23×C3⋊C8C2211yescharacteristic
23C2×C4×Dic6C2211yescharacteristic
24C3×#398—21no—
25C3×#395—21no—
26C3×#389C2211yescharacteristic
27C3×#389C2211yescharacteristic
28C3×#385—21no—
29C3×#384C2211yescharacteristic
30C3×#380C2211yescharacteristic
31C3×C4×C22⋊C4C2211yescharacteristic
32C2×#830C2211yescharacteristic
33C2×#810C2211yescharacteristic
34C2×#810C2211yescharacteristic
35C2×#810C2211yescharacteristic
36C2×#810C2211yescharacteristic
37C2×#808C2211yescharacteristic
38C2×#808C2211yescharacteristic
39#4946—21no—
40Dic3×C22⋊C4—21no—
41C4×C4⋊Dic3C2211yescharacteristic
42#4929C2211yescharacteristic
43#4928C2211yescharacteristic
44C2×C4⋊(C3⋊C8)C2211yescharacteristic
45C2×C4⋊(C3⋊C8)C2211yescharacteristic
46#4655—21no—
47#4647—21no—
48#4529C2211yescharacteristic
49#4529C2211yescharacteristic
50#4529C2211yescharacteristic
51#4529C2211yescharacteristic
52#4529C2211yescharacteristic
53#4529C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4524C2211yescharacteristic
59#4524C2211yescharacteristic
60#2440D311yescharacteristic
61#2360—31no—
62#2099—31no—
63#2088—31no—
64#2066—31no—
65#2024—31no—
66#1988—31no—
67#1919—31no—
68#1892—31no—
69#1872—31no—
70#1865—31no—
71#1853—31no—
72#1850—31no—
73#1803—31no—
74C2×#340—31no—
75C22×C3×Q8—21no—
76C23×C3×C4—21no—
77C23×C3×C4C2311yescharacteristic
78C23×Dic3—21no—
79C22×Dic6C2311yescharacteristic
80C3×C4⋊Q8—22no—
81C3×C4⋊Q8—41no—
82C3×C22⋊Q8—44no—
83C2×C3×C4⋊C4—22no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—41no—
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4D411yescharacteristic
97C2×C3×C22⋊C4D411yescharacteristic
98C2×C3×C42C2311yescharacteristic
99C2×C3×C42C2311yescharacteristic
100C2×#217—21no—
101C2×C4⋊Dic3—22no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3—41no—
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107#924—22no—
108#924Q814yescomplemented
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×C4×Dic3—21no—
113C2×C4×Dic3—21no—
114C22×C3⋊C8—22no—
115C22×C3⋊C8—41no—
116C22×C3⋊C8—41no—
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121C4×Dic6—22no—
122C4×Dic6—41no—
123C3×#95—21no—
124C3×#95—21no—
125C3×#95—21no—
126C3×#95D411yescharacteristic
127C3×#95D411yescharacteristic
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132#830—22no—
133#810—22no—
134#810—41no—
135#810—41no—
136#810—41no—
137#808—22no—
138#808—41no—
139C4⋊(C3⋊C8)—41no—
140C4⋊(C3⋊C8)—41no—
141C23×C8—31no—
142C2×C4⋊Q8C2×D311yescharacteristic
143C2×C22⋊Q8—21no—
144C2×C22⋊Q8—31no—
145C2×C4×Q8—31no—
146C22×C4⋊C4—31no—
147C22×C4⋊C4C2×D311yescharacteristic
148C2×#106—31no—
149C2×#106—31no—
150C2×C4⋊C8—31no—
151C2×C4⋊C8—31no—
152C2×#103—31no—
153C2×#103—31no—
154C2×#103—31no—
155C2×#103—31no—
156C2×C22⋊C8—31no—
157#398—21no—
158#395—21no—
159#391—31no—
160#389—31no—
161#389—61no—
162#389C2×D311yescharacteristic
163#389C2×D311yescharacteristic
164#387—61no—
165#385—21no—
166#385—61no—
167#384C2×D311yescharacteristic
168#380C2×D311yescharacteristic
169C4×C4⋊C4—31no—
170C4×C22⋊C4—61no—
171C4×C22⋊C4C2×D311yescharacteristic
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#340—31no—
178#340—31no—
179#340—31no—
180#340—31no—
181#340—31no—
182#336—31no—
183#336—31no—
184C24×C3C2×D411yescharacteristic
185C2×C3×Q8—42no—
186C2×C3×Q8—42no—
187C2×C3×Q8—81no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—42no—
197C22×C3×C4—42no—
198C22×C3×C4—81no—
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×Q812yes—
205C22×C3×C4#4112yes—
206C22×C3×C4C2411yescharacteristic
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3—21no—
211C22×Dic3—42no—
212C22×Dic3—42no—
213C22×Dic3—81no—
214C2×Dic6—21no—
215C2×Dic6—21no—
216C2×Dic6—21no—
217C2×Dic6—21no—
218C2×Dic6—21no—
219C2×Dic6C2×Q812yes—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—42no—
228C3×C4⋊C4—42no—
229C3×C4⋊C4—42no—
230C3×C4⋊C4—42no—
231C3×C4⋊C4—42no—
232C3×C4⋊C4—81no—
233C3×C4⋊C4#4112yes—
234C3×C22⋊C4—22no—
235C3×C22⋊C4—42no—
236C3×C22⋊C4SD1614yes—
237C3×C42—21no—
238C3×C42—21no—
239C3×C42—21no—
240C3×C42—21no—
241#217—24no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—42no—
247C4⋊Dic3C2×Q812yes—
248C4⋊Dic3#4112yes—
249Dic3⋊C4—24no—
250Dic3⋊C4—42no—
251Dic3⋊C4—81no—
252C4×Dic3—42no—
253C4×Dic3—81no—
254C2×C3⋊C8—21no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
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—44no—
267C22×Q8—21no—
268C22×Q8—31no—
269C23×C4—21no—
270C23×C4—61no—
271C23×C4C22×D311yescharacteristic
272C22×C8—31no—
273C22×C8—31no—
274C22×C8—31no—
275C22×C8—31no—
276C22×C8—62no—
277C22×C8—121no—
278C22×C8—121no—
279C4⋊Q8—22no—
280C4⋊Q8—41no—
281C22⋊Q8—34no—
282C22⋊Q8—44no—
283C22⋊Q8—62no—
284C4×Q8—62no—
285C4×Q8—121no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—31no—
294C2×C4⋊C4—31no—
295C2×C4⋊C4—31no—
296C2×C4⋊C4—41no—
297C2×C4⋊C4—62no—
298C2×C4⋊C4—61no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—121no—
303C2×C4⋊C4C22×D311yescharacteristic
304C2×C4⋊C4C22×D311yescharacteristic
305C2×C4⋊C4C22×D311yescharacteristic
306C2×C4⋊C4C22×D311yescharacteristic
307C2×C22⋊C4—21no—
308C2×C22⋊C4—61no—
309C2×C22⋊C4#6711yescharacteristic
310C2×C22⋊C4#6711yescharacteristic
311C2×C42—61no—
312C2×C42—61no—
313C2×C42C22×D311yescharacteristic
314C2×C42C22×D311yescharacteristic
315#106—62no—
316#106—121no—
317C4⋊C8—121no—
318C4⋊C8—121no—
319#103—62no—
320#103—121no—
321#103—121no—
322#103—121no—
323C22⋊C8—62no—
324#95—21no—
325#95—21no—
326#95—21no—
327#95—61no—
328#95—61no—
329#95—61no—
330#95—61no—
331#95#6711yescharacteristic
332#95#6711yescharacteristic
333C23×C3—21no—
334C23×C3—21no—
335C23×C3—21no—
336C23×C3—21no—
337C23×C3C22⋊Q812yes—
338C23×C3C2×SD1612yes—
339C23×C3#13712yes—
340C23×C3C22×D411yescharacteristic
341C3×Q8—84no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—22no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—84no—
371C2×C3×C4C22⋊Q812yes—
372C2×C3×C4C2×SD1612yes—
373C2×C3×C4C2×SD1612yes—
374C2×C3×C4C22×Q811yescharacteristic
375C2×C3×C4C2×#4111yescharacteristic
376C2×C3×C4C2×#4111yescharacteristic
377C2×C3×C4C2×#4111yescharacteristic
378C2×C3×C4H(5,𝔽2)11yescharacteristic
379C2×C3×C4H(5,𝔽2)11yescharacteristic
380C2×Dic3—22no—
381C2×Dic3—22no—
382C2×Dic3—22no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—84no—
393Dic6—22no—
394Dic6—22no—
395Dic6—41no—
396Dic6—41no—
397C3⋊C8—41no—
398C3⋊C8—41no—
399C3⋊C8—41no—
400C3⋊C8—41no—
401C24C2×#6711yescharacteristic
402C2×Q8—32no—
403C2×Q8—42no—
404C2×Q8—42no—
405C2×Q8—61no—
406C2×Q8—61no—
407C2×Q8—61no—
408C2×Q8—61no—
409C2×Q8—61no—
410C2×Q8—81no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—61no—
422C22×C4—61no—
423C22×C4—61no—
424C22×C4—61no—
425C22×C4—81no—
426C22×C4—122no—
427C22×C4—122no—
428C22×C4—241no—
429C22×C4D3×Q812yes—
430C22×C4#23912yes—
431C22×C4C2×#6711yescharacteristic
432C22×C4C2×#6711yescharacteristic
433C22×C4C2×#6711yescharacteristic
434C22×C4C2×#6711yescharacteristic
435C22×C4C2×#6711yescharacteristic
436C22×C4C23×D311yescharacteristic
437C2×C8—61no—
438C2×C8—61no—
439C2×C8—61no—
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—124no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—32no—
458C4⋊C4—32no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—64no—
465C4⋊C4—61no—
466C4⋊C4—61no—
467C4⋊C4—61no—
468C4⋊C4—61no—
469C4⋊C4—81no—
470C4⋊C4—122no—
471C4⋊C4—122no—
472C4⋊C4—241no—
473C4⋊C4#23712yes—
474C22⋊C4—22no—
475C22⋊C4—42no—
476C22⋊C4—64no—
477C22⋊C4#21414yes—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—122no—
483C42—241no—
484C22×C3—22no—
485C22×C3—22no—
486C22×C3—22no—
487C22×C3—22no—
488C22×C3—22no—
489C22×C3—22no—
490C22×C3#47514yes—
491C22×C3C2×C22⋊Q811yescharacteristic
492C22×C3#54611yescharacteristic
493C22×C3#54611yescharacteristic
494C22×C3C22×SD1611yescharacteristic
495C22×C3C2×#13711yescharacteristic
496C22×C3#57611yescharacteristic
497C22×C3#57811yescharacteristic
498C3×C4—22no—
499C3×C4—22no—
500C3×C4—22no—
501C3×C4—42no—
502C3×C4—41no—
503C3×C4—41no—
504C3×C4—81no—
505C3×C4—81no—
506C3×C4—81no—
507C3×C4#52911yescharacteristic
508C3×C4#53811yescharacteristic
509C3×C4#55411yescharacteristic
510C3×C4#55411yescharacteristic
511Dic3—42no—
512Dic3—42no—
513Dic3—42no—
514Dic3—81no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23C2×#21412yes—
520C23#94212yes—
521C23#94612yes—
522C23C22×#6711yescharacteristic
523Q8—62no—
524Q8—62no—
525Q8—84no—
526Q8—121no—
527Q8—121no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—62no—
557C2×C4—62no—
558C2×C4—62no—
559C2×C4—84no—
560C2×C4—121no—
561C2×C4—121no—
562C2×C4—121no—
563C2×C4—121no—
564C2×C4—121no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4—121no—
569C2×C4—244no—
570C2×C4C2×#21412yes—
571C2×C4C2×#21412yes—
572C2×C4#94612yes—
573C2×C4C2×#23711yescharacteristic
574C2×C4#100411yescharacteristic
575C2×C4C2×D3×Q811yescharacteristic
576C2×C4C2×#23911yescharacteristic
577C2×C4D3×#4111yescharacteristic
578C2×C4#100911yescharacteristic
579C8—121no—
580C8—121no—
581C8—121no—
582C8—121no—
583C2×C3—24no—
584C2×C3#245011yescharacteristic
585C2×C3C2×#47511yescharacteristic
586C2×C3#308311yescharacteristic
587C2×C3#329511yescharacteristic
588C2×C3#329811yescharacteristic
589C2×C3#331211yescharacteristic
590C2×C3#331411yescharacteristic
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22#509414yes—
598C22C22×#21411yescharacteristic
599C22#580611yescharacteristic
600C22C2×#94211yescharacteristic
601C22C2×#94611yescharacteristic
602C22#582911yescharacteristic
603C22#583511yescharacteristic
604C22#583811yescharacteristic
605C4—22no—
606C4—22no—
607C4—22no—
608C4—42no—
609C4—41no—
610C4—41no—
611C4—81no—
612C4—81no—
613C4—81no—
614C4—122no—
615C4—122no—
616C4—122no—
617C4—241no—
618C4#562411yescharacteristic
619C4#563511yescharacteristic
620C4#563611yescharacteristic
621C4#564411yescharacteristic
622C3#2641211yescharacteristic, complemented, Sylow
623C2—24no—
624C2#8276511yescharacteristic
625C2C2×#509411yescharacteristic
626C2#8347511yescharacteristic
627C2#8348311yescharacteristic
628C2#8353111yescharacteristic
629C2#8355211yescharacteristic
630C2#8363511yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2440C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#2440D3
hypocenterC3#26412
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2440

Complements

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

NH ≅ G/Ntype
3. #71794C2semidirect
4. #71780C2semidirect
7. #69634C2semidirect
8. #69632C2semidirect
9. #69624C2semidirect
10. #69601C2semidirect
11. #69588C2semidirect
12. #69571C2semidirect
108. #924Q8semidirect
622. C3#26412semidirect

Maximal subgroups

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

subgroupindex
C3×#24402
#734652
#717942
#717802
#715592
C2×#45292
#696342
#696322
#696242
#696012
#695882
#695712
#695142
#695052
#694492
#264123