Finite Group Databasebeta

← #10814352

Subgroups — #10814352

Order 768.

874 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4758C211yesmaximal, characteristic
2#76302C211yesmaximal, characteristic
3#75981C211yesmaximal, characteristic, complemented
4C3×#1936C211yesmaximal, characteristic
5#70213C211yesmaximal, characteristic
6#70187C211yesmaximal, characteristic
7#70153C211yesmaximal, characteristic
8#69833C211yesmaximal, characteristic, complemented
9#69813C211yesmaximal, characteristic, complemented
10#69660C211yesmaximal, characteristic
11#69601C211yesmaximal, characteristic, complemented
12#68904C211yesmaximal, characteristic, complemented
13#68847C211yesmaximal, characteristic, complemented
14#68817C211yesmaximal, characteristic, complemented
15#68783C211yesmaximal, characteristic, complemented
16#22759—31nomaximal, Sylow
17C2×D4×Dic3C2211yescharacteristic
18C22×#214C2211yescharacteristic
19C2×#887C2211yescharacteristic
20C2×#887C2211yescharacteristic
21C2×#877—21no—
22C2×C4×Dic6C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C2×C3×#97C2211yescharacteristic
26C3×#390C2211yescharacteristic
27C3×#389C2211yescharacteristic
28#5221C2211yescharacteristic
29C2×#832C2211yescharacteristic
30C2×#816C2211yescharacteristic
31C2×#816C2211yescharacteristic
32C2×#815C2211yescharacteristic
33C2×Dic3⋊C8C2211yescharacteristic
34C2×C8×Dic3C2211yescharacteristic
35#4984C2211yescharacteristic
36#4976—21no—
37#4975—21no—
38#4975C2211yescharacteristic
39#4975C2211yescharacteristic
40Dic3×C4⋊C4C2211yescharacteristic
41C2×#810C2211yescharacteristic
42C2×#808C2211yescharacteristic
43#4945—21no—
44#4944—21no—
45#4942—21no—
46C4×C4⋊Dic3C2211yescharacteristic
47C4×Dic3⋊C4C2211yescharacteristic
48#4758C418yescomplemented
49#4649—21no—
50#4647—21no—
51C3×#340C2211yescharacteristic
52C3×#340C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4550C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4524C2211yescharacteristic
61C2×C4×SD16—31no—
62#2354—31no—
63Q8×C4⋊C4—31no—
64#1936—31no—
65#1936D311yescharacteristic
66#1919—31no—
67#1919—31no—
68#1880—31no—
69#1880—31no—
70#1834—31no—
71#1803—31no—
72#1802—31no—
73#1770—31no—
74#1763—31no—
75#1763—31no—
76C22×C3×D4C2311yescharacteristic
77C23×Dic3—21no—
78C22×Dic6C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C22⋊C4—21no—
86C2×C3×C22⋊C4—21no—
87C2×C3×C42C2311yescharacteristic
88C2×#217—21no—
89C2×#217—21no—
90C2×#217—21no—
91D4×Dic3—22no—
92D4×Dic3C2×C414yes—
93C2×#214—22no—
94C2×#214—22no—
95C2×#214C2×C414yes—
96C2×C4⋊Dic3—21no—
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4D411yescharacteristic
108C2×Dic3⋊C4D411yescharacteristic
109C2×C4×Dic3—21no—
110C2×C4×Dic3—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3—21no—
113C2×C4×Dic3—21no—
114C2×C4×Dic3C2311yescharacteristic
115C2×C4×Dic3C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118#887—22no—
119#887—22no—
120#887—41no—
121#887C2×C414yes—
122#877—44no—
123C4×Dic6—22no—
124C4×Dic6—41no—
125C3×#102—41no—
126C3×#102C2×C414yes—
127C3×#97—41no—
128C3×#95—21no—
129#832—41no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95D411yescharacteristic
134C3⋊#95D411yescharacteristic
135#816—22no—
136#816C2×C414yes—
137#815—41no—
138Dic3⋊C8—22no—
139C8×Dic3C2×C414yes—
140#810—22no—
141#808C2×C414yes—
142C22×SD16—31no—
143C2×C4×Q8—31no—
144C2×C4×Q8—31no—
145C2×C4×Q8—31no—
146C2×C4×D4—31no—
147C2×#117—61no—
148C4×SD16—38no—
149C2×#106—31no—
150C2×C4⋊C8—31no—
151C2×#103—31no—
152C2×#103—31no—
153C2×#103—31no—
154C2×#102—31no—
155C2×#102C2×D311yescharacteristic
156C2×#102C2×D311yescharacteristic
157C2×#97—31no—
158C2×#97C2×D311yescharacteristic
159C2×C4×C8—31no—
160#390—31no—
161#390C2×D311yescharacteristic
162#389—31no—
163#389—61no—
164#389—61no—
165#389C2×D311yescharacteristic
166#388—61no—
167#388—61no—
168#384—31no—
169#384—31no—
170#384—61no—
171#380—61no—
172C4×C4⋊C4—31no—
173C4×C4⋊C4—31no—
174C4×C4⋊C4—31no—
175C4×C4⋊C4—61no—
176#340—31no—
177#340—31no—
178#340—31no—
179#340—31no—
180#340—31no—
181#340—31no—
182#340C2×D311yescharacteristic
183#340C2×D311yescharacteristic
184#336—31no—
185#336—31no—
186C24×C3—21no—
187C2×C3×D4—21no—
188C2×C3×D4—21no—
189C2×C3×D4—21no—
190C2×C3×D4—21no—
191C2×C3×D4—21no—
192C2×C3×D4C22×C412yes—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2411yescharacteristic
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—24no—
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3—42no—
205C22×Dic3—42no—
206C22×Dic3C2×D411yescharacteristic
207C22×Dic3C2×D411yescharacteristic
208C22×Dic3C2×D411yescharacteristic
209C22×Dic3C2×D411yescharacteristic
210C2×Dic6—21no—
211C2×Dic6—21no—
212C2×Dic6—21no—
213C2×Dic6C22×C412yes—
214C2×Dic6#4112yes—
215C2×Dic6#4112yes—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8C22×C412yes—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—81no—
228C3×C4⋊C4C22×C412yes—
229C3×C22⋊C4—42no—
230C3×C22⋊C4—81no—
231C3×C42—21no—
232C3×C42—21no—
233#217—24no—
234#217—42no—
235#217—81no—
236#214—28no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—81no—
243C4⋊Dic3C22×C412yes—
244Dic3⋊C4—24no—
245Dic3⋊C4—22no—
246Dic3⋊C4—22no—
247Dic3⋊C4—42no—
248Dic3⋊C4—42no—
249Dic3⋊C4—42no—
250Dic3⋊C4—42no—
251Dic3⋊C4—81no—
252Dic3⋊C4—81no—
253C4×Dic3—21no—
254C4×Dic3—21no—
255C4×Dic3—24no—
256C4×Dic3—42no—
257C4×Dic3—42no—
258C4×Dic3—42no—
259C4×Dic3—42no—
260C4×Dic3C22×C412yes—
261C4×Dic3#4112yes—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8C22×C412yes—
268C22×Q8—31no—
269C22×D4C22×D311yescharacteristic
270C23×C4—61no—
271C2×SD16—34no—
272C2×SD16—62no—
273C2×SD16—62no—
274C22×C8—31no—
275C22×C8—31no—
276C22×C8C22×D311yescharacteristic
277C22×C8C22×D311yescharacteristic
278C4×Q8—34no—
279C4×Q8—62no—
280C4×Q8—62no—
281C4×Q8—62no—
282C4×Q8—121no—
283C4×Q8—121no—
284C4×D4—34no—
285C4×D4—62no—
286#117—124no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—31no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4—31no—
292C2×C4⋊C4—31no—
293C2×C4⋊C4—61no—
294C2×C4⋊C4—61no—
295C2×C4⋊C4—61no—
296C2×C4⋊C4—61no—
297C2×C4⋊C4—61no—
298C2×C4⋊C4—61no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4C22×D311yescharacteristic
302C2×C4⋊C4C22×D311yescharacteristic
303C2×C4⋊C4C22×D311yescharacteristic
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—61no—
307C2×C22⋊C4—61no—
308C2×C22⋊C4—61no—
309C2×C42—31no—
310C2×C42—31no—
311C2×C42—61no—
312C2×C42—61no—
313C2×C42—61no—
314C2×C42—61no—
315C2×C42—61no—
316C2×C42C22×D311yescharacteristic
317#106—34no—
318C4⋊C8—62no—
319#103—34no—
320#103—62no—
321#103—62no—
322#102—41no—
323#102—121no—
324#102C4×D314yes—
325#97—41no—
326#97—121no—
327C4×C8—34no—
328#95—21no—
329#95—31no—
330#95—31no—
331#95—61no—
332#95—61no—
333#95—61no—
334C23×C3—22no—
335C23×C3—41no—
336C23×C3—41no—
337C23×C3—41no—
338C23×C3—41no—
339C23×C3—41no—
340C23×C3—41no—
341C23×C3C22×D411yescharacteristic
342C3×D4—22no—
343C3×D4—22no—
344C3×D4—41no—
345C3×D4—41no—
346C2×C3×C4—22no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4C23×C411yescharacteristic
362C2×C3×C4C2×#4111yescharacteristic
363C2×C3×C4C2×#4111yescharacteristic
364C2×C3×C4C2×#4111yescharacteristic
365C2×C3×C4H(5,𝔽2)11yescharacteristic
366C2×C3×C42−1+411yescharacteristic
367C2×Dic3—22no—
368C2×Dic3—22no—
369C2×Dic3—22no—
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—22no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—48no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3C4×D412yes—
389C2×Dic3C4×D412yes—
390C2×Dic3AGL(1,ℤ/8)12yes—
391C2×Dic3#13712yes—
392Dic6—22no—
393Dic6—22no—
394Dic6—22no—
395Dic6#11714yescomplemented
396C3×C8—22no—
397C3×C8—41no—
398C3⋊C8—22no—
399C3⋊C8—41no—
400C24—21no—
401C2×Q8—32no—
402C2×Q8—32no—
403C2×Q8—32no—
404C2×Q8—61no—
405C2×Q8—61no—
406C2×Q8—61no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4C2×C4×D312yes—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—31no—
417C22×C4—31no—
418C22×C4—31no—
419C22×C4—31no—
420C22×C4—64no—
421C22×C4—61no—
422C22×C4—61no—
423C22×C4—61no—
424C22×C4—61no—
425C22×C4—122no—
426C22×C4—122no—
427C22×C4D3×D411yescharacteristic
428C22×C4D3×D411yescharacteristic
429C22×C4C23×D311yescharacteristic
430SD16—68no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—32no—
437C2×C8—61no—
438C2×C8—61no—
439C2×C8—61no—
440C2×C8—61no—
441C2×C8—61no—
442C2×C8C2×C4×D312yes—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—32no—
449C4⋊C4—62no—
450C4⋊C4—62no—
451C4⋊C4—64no—
452C4⋊C4—61no—
453C4⋊C4—61no—
454C4⋊C4—61no—
455C4⋊C4—61no—
456C4⋊C4—61no—
457C4⋊C4—81no—
458C4⋊C4—122no—
459C4⋊C4—122no—
460C4⋊C4—122no—
461C4⋊C4—122no—
462C4⋊C4—241no—
463C4⋊C4—241no—
464C4⋊C4—241no—
465C4⋊C4C2×C4×D312yes—
466C22⋊C4—42no—
467C22⋊C4—64no—
468C22⋊C4—81no—
469C22⋊C4—122no—
470C22⋊C4—241no—
471C42—21no—
472C42—21no—
473C42—32no—
474C42—32no—
475C42—64no—
476C42—61no—
477C42—61no—
478C42—122no—
479C42—122no—
480C42—122no—
481C42—122no—
482C22×C3—42no—
483C22×C3—42no—
484C22×C3—42no—
485C22×C3—81no—
486C22×C3—81no—
487C22×C3—81no—
488C22×C3—81no—
489C22×C3C2×C4×D411yescharacteristic
490C22×C3#54611yescharacteristic
491C22×C3#54811yescharacteristic
492C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
493C22×C3C2×#13711yescharacteristic
494C22×C3#57711yescharacteristic
495C22×C3#57711yescharacteristic
496C3×C4—41no—
497C3×C4—41no—
498C3×C4—42no—
499C3×C4—81no—
500C3×C4—81no—
501C3×C4C2×#11711yescharacteristic
502C3×C4#52011yescharacteristic
503C3×C4#55211yescharacteristic
504C3×C4#55311yescharacteristic
505Dic3—24no—
506Dic3—22no—
507Dic3—41no—
508Dic3—41no—
509Dic3—42no—
510Dic3—81no—
511Dic3—81no—
512Dic3#44014yes—
513C23—22no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23C2×D3×D411yescharacteristic
521Q8—34no—
522Q8—62no—
523Q8—62no—
524Q8—62no—
525D4—22no—
526D4—22no—
527D4—41no—
528D4—41no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—32no—
537C2×C4—32no—
538C2×C4—32no—
539C2×C4—32no—
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—62no—
549C2×C4—61no—
550C2×C4—61no—
551C2×C4—62no—
552C2×C4—62no—
553C2×C4—61no—
554C2×C4—61no—
555C2×C4—61no—
556C2×C4—61no—
557C2×C4—62no—
558C2×C4—61no—
559C2×C4—61no—
560C2×C4—128no—
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×C4C22×C4×D311yescharacteristic
570C2×C4#100411yescharacteristic
571C2×C4D3×#4111yescharacteristic
572C2×C4D3×#4111yescharacteristic
573C2×C4D3×#4111yescharacteristic
574C2×C4#101011yescharacteristic
575C8—22no—
576C8—41no—
577C8—62no—
578C8—121no—
579C2×C3—82no—
580C2×C3#235211yescharacteristic
581C2×C3C2×#44011yescharacteristic
582C2×C3#295211yescharacteristic
583C2×C3#332911yescharacteristic
584C2×C3#333211yescharacteristic
585C2×C3#336811yescharacteristic
586C2×C3#336911yescharacteristic
587C22—42no—
588C22—42no—
589C22—42no—
590C22—81no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22C4×D3×D411yescharacteristic
595C22#561011yescharacteristic
596C22#567711yescharacteristic
597C22D3×AGL(1,ℤ/8)11yescharacteristic
598C22#577511yescharacteristic
599C22#577511yescharacteristic
600C22D3×#13711yescharacteristic
601C4—34no—
602C4—41no—
603C4—41no—
604C4—42no—
605C4—64no—
606C4—62no—
607C4—81no—
608C4—81no—
609C4—122no—
610C4—121no—
611C4—121no—
612C4—241no—
613C4—241no—
614C4D3×#11711yescharacteristic
615C4#554611yescharacteristic
616C4#566111yescharacteristic
617C4#570811yescharacteristic
618C3#2275911yescharacteristic, complemented, Sylow
619C2—82no—
620C2#7583411yescharacteristic
621C2D3×#44011yescharacteristic
622C2#7986811yescharacteristic
623C2#8084611yescharacteristic
624C2#8088711yescharacteristic
625C2#8167111yescharacteristic
626C2#8170911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1936C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
cosocleC22×C3×C4C24
hypercenter#1936D3
hypocenterC3#22759
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1936

Complements

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

NH ≅ G/Ntype
3. #75981C2semidirect
8. #69833C2semidirect
9. #69813C2semidirect
11. #69601C2semidirect
12. #68904C2semidirect
13. #68847C2semidirect
14. #68817C2semidirect
15. #68783C2semidirect
48. #4758C4semidirect
395. Dic6#117semidirect
618. C3#22759semidirect

Maximal subgroups

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

subgroupindex
C2×#47582
#763022
#759812
C3×#19362
#702132
#701872
#701532
#698332
#698132
#696602
#696012
#689042
#688472
#688172
#687832
#227593