Finite Group Databasebeta

← #10798243

Subgroups — #10798243

Order 768.

922 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×#4889C211yesmaximal, characteristic
2#75961C211yesmaximal, characteristic
3C4×#887C211yesmaximal, characteristic, complemented
4C3×#1918C211yesmaximal, characteristic, complemented
5#70610C211yesmaximal, characteristic, complemented
6#69632C211yesmaximal, characteristic, complemented
7#69462C211yesmaximal, characteristic
8#68909C211yesmaximal, characteristic
9#68906C211yesmaximal, characteristic, complemented
10#68792C211yesmaximal, characteristic
11#68786C211yesmaximal, characteristic, complemented
12#68783C211yesmaximal, characteristic, complemented
13#68782C211yesmaximal, characteristic
14#68686C211yesmaximal, characteristic
15#68597C211yesmaximal, characteristic, complemented
16#21124—31nomaximal, Sylow
17C22×#204C2211yescharacteristic
18C2×#893C2211yescharacteristic
19C2×#892—21no—
20C2×#887C2211yescharacteristic
21C2×#887C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C2×C3×#107C2211yescharacteristic
24C3×#389C2211yescharacteristic
25C3×C4×C4⋊C4C2211yescharacteristic
26C2×#821C2211yescharacteristic
27C2×#816C2211yescharacteristic
28C2×#815C2211yescharacteristic
29C2×#815C2211yescharacteristic
30#4990—21no—
31#4988C2211yescharacteristic
32#4984C2211yescharacteristic
33Dic3×C4⋊C4C2211yescharacteristic
34Dic3×C4⋊C4C2211yescharacteristic
35Dic3×C4⋊C4C2211yescharacteristic
36C2×#810C2211yescharacteristic
37C2×#810C2211yescharacteristic
38C2×#809C2211yescharacteristic
39C2×#809C2211yescharacteristic
40C2×#808C2211yescharacteristic
41C2×#807C2211yescharacteristic
42#4939C2211yescharacteristic
43C4×Dic3⋊C4—21no—
44C42×Dic3—21no—
45C42×Dic3C2211yescharacteristic
46#4889C418yescomplemented
47#4670—21no—
48#4666—21no—
49#4649—21no—
50#4649—21no—
51C3×#340C2211yescharacteristic
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54C3×#336C2211yescharacteristic
55#4551C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4524C2211yescharacteristic
61C2×#440—31no—
62#2293—31no—
63C42×Q8—31no—
64#1926—31no—
65#1925—31no—
66#1921—31no—
67#1918D311yescharacteristic, complemented
68#1880—31no—
69#1880—31no—
70#1865—31no—
71#1863—31no—
72#1776—31no—
73#1763—31no—
74#1762—31no—
75#1745—31no—
76C22×D12C2311yescharacteristic
77C22×C4×D3—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×C4⋊C4C2311yescharacteristic
86C2×C3×C42—21no—
87C2×C3×C42C2311yescharacteristic
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×C4⋊Dic3—21no—
94C2×C4⋊Dic3C2311yescharacteristic
95C2×C4⋊Dic3C2311yescharacteristic
96C2×Dic3⋊C4—21no—
97C2×Dic3⋊C4—21no—
98C2×Dic3⋊C4—21no—
99C2×C4×Dic3—21no—
100C2×C4×Dic3—21no—
101C2×C4×Dic3—21no—
102C2×C4×Dic3—21no—
103C2×C4×Dic3—21no—
104C2×C4×Dic3—21no—
105C2×C4×Dic3D411yescharacteristic
106C2×C4×Dic3D411yescharacteristic
107C2×C4×Dic3D411yescharacteristic
108C2×C4×Dic3D411yescharacteristic
109C2×C4×Dic3C2311yescharacteristic
110C2×C4×Dic3C2311yescharacteristic
111C22×C3⋊C8C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113C2×#204—22no—
114C2×#204—22no—
115C2×#204C2×C414yes—
116#893—22no—
117#893C2×C414yes—
118#892—44no—
119#887—22no—
120#887—22no—
121#887—22no—
122#887C2×C414yes—
123C4×Dic6—22no—
124C4×Dic6—22no—
125C3×#107C2×C414yes—
126C3×#95—21no—
127C3×#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132#821—22no—
133#816—22no—
134#815—22no—
135#815C2×C414yes—
136#810—22no—
137#810C2×C414yes—
138#809—22no—
139#809C2×C414yes—
140#808—22no—
141#807—22no—
142C22×SD16—31no—
143C2×C4×Q8—31no—
144C2×C4×Q8—31no—
145C2×C4×Q8—31no—
146C2×C4×D4—31no—
147C2×#117—61no—
148#440—38no—
149C2×#107—31no—
150C2×#107C2×D311yescharacteristic
151C2×#106—31no—
152C2×#103—31no—
153C2×#103—31no—
154C2×#103—31no—
155C2×#102—31no—
156C2×#102—31no—
157C2×#102—31no—
158C2×#97—31no—
159C2×#97—31no—
160#390—31no—
161#390—31no—
162#389—31no—
163#389C2×D311yescharacteristic
164#388—61no—
165#388—61no—
166C4×C4⋊C4—31no—
167C4×C4⋊C4—31no—
168C4×C4⋊C4—31no—
169C4×C4⋊C4—61no—
170C4×C4⋊C4—61no—
171C4×C4⋊C4—61no—
172C4×C4⋊C4C2×D311yescharacteristic
173C4×C22⋊C4—61no—
174C43—31no—
175C43—61no—
176#340—31no—
177#340—31no—
178#340—31no—
179#340C2×D311yescharacteristic
180#340C2×D311yescharacteristic
181#340C2×D311yescharacteristic
182#336—31no—
183#336—31no—
184#336—31no—
185#336C2×D311yescharacteristic
186C23×D3—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2411yescharacteristic
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3—21no—
197C22×Dic3C2×D411yescharacteristic
198C22×Dic3C2×D411yescharacteristic
199C22×Dic3C2×D411yescharacteristic
200C22×Dic3C2×D411yescharacteristic
201C2×D12—21no—
202C2×D12—21no—
203C2×D12—21no—
204C2×D12—21no—
205C2×D12—21no—
206C2×D12C22×C412yes—
207C2×C4×D3—24no—
208C2×C4×D3—42no—
209C2×C4×D3—42no—
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—42no—
228C3×C4⋊C4C22×C412yes—
229C3×C4⋊C4C22×C412yes—
230C3×C4⋊C4#4112yes—
231C3×C42—21no—
232C3×C42—21no—
233C3×C42—42no—
234(C2×D3)⋊C4—24no—
235(C2×D3)⋊C4—42no—
236(C2×D3)⋊C4—42no—
237(C2×D3)⋊C4—42no—
238(C2×D3)⋊C4—42no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—42no—
243C4⋊Dic3#4112yes—
244Dic3⋊C4—24no—
245Dic3⋊C4—24no—
246Dic3⋊C4—24no—
247C4×Dic3—24no—
248C4×Dic3—24no—
249C4×Dic3—22no—
250C4×Dic3—21no—
251C4×Dic3—21no—
252C4×Dic3—24no—
253C4×Dic3—22no—
254C4×Dic3—41no—
255C4×Dic3—41no—
256C4×Dic3—42no—
257C4×Dic3—42no—
258C4×Dic3—42no—
259C4×Dic3C22×C412yes—
260C4×Dic3#4112yes—
261C2×C3⋊C8—21no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8C22×C412yes—
267#204—28no—
268C22×Q8—31no—
269C22×D4—31no—
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—62no—
283C4×Q8—62no—
284C4×D4—34no—
285C4×D4—62no—
286#117—124no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—31no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—61no—
291C2×C4⋊C4—61no—
292C2×C4⋊C4—61no—
293C2×C4⋊C4—61no—
294C2×C4⋊C4C22×D311yescharacteristic
295C2×C4⋊C4C22×D311yescharacteristic
296C2×C4⋊C4C22×D311yescharacteristic
297C2×C4⋊C4C22×D311yescharacteristic
298C2×C22⋊C4—61no—
299C2×C22⋊C4—61no—
300C2×C22⋊C4—61no—
301C2×C22⋊C4—61no—
302C2×C22⋊C4—61no—
303C2×C42—21no—
304C2×C42—31no—
305C2×C42—31no—
306C2×C42—31no—
307C2×C42—31no—
308C2×C42—31no—
309C2×C42—31no—
310C2×C42—61no—
311C2×C42—61no—
312C2×C42—61no—
313C2×C42—61no—
314C2×C42—61no—
315C2×C42—61no—
316C2×C42C22×D311yescharacteristic
317#107—62no—
318#107C4×D314yes—
319#106—62no—
320#103—34no—
321#103—62no—
322#103—62no—
323#102—34no—
324#102—62no—
325#102—62no—
326#97—34no—
327#97—62no—
328#95—21no—
329#95—21no—
330#95—61no—
331#95—61no—
332#95—61no—
333#95—61no—
334C23×C3C22×D411yescharacteristic
335C22×D3—22no—
336C22×D3—41no—
337C22×D3—41no—
338C22×D3—41no—
339C22×D3—41no—
340C22×D3—41no—
341C22×D3—41no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
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×C4AGL(1,ℤ/8)12yes—
360C2×C3×C4C23×C411yescharacteristic
361C2×C3×C4C2×#4111yescharacteristic
362C2×C3×C4C2×#4111yescharacteristic
363C2×C3×C4C2×#4111yescharacteristic
364C2×C3×C4C2×#4111yescharacteristic
365C2×C3×C4C2×#4111yescharacteristic
366C2×Dic3—22no—
367C2×Dic3—21no—
368C2×Dic3—22no—
369C2×Dic3—22no—
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—22no—
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3C4×D412yes—
385C2×Dic3C4×D412yes—
386C2×Dic3#13712yes—
387D12—22no—
388D12—22no—
389D12—41no—
390D12—41no—
391C4×D3—48no—
392Dic6—22no—
393Dic6—22no—
394Dic6—22no—
395Dic6#11714yescomplemented
396C3×C8—22no—
397C3×C8—41no—
398C3⋊C8—22no—
399C3⋊C8—41no—
400C24—61no—
401C2×Q8—32no—
402C2×Q8—32no—
403C2×Q8—32no—
404C2×Q8—61no—
405C2×Q8—61no—
406C2×Q8—61no—
407C2×D4—32no—
408C2×D4—61no—
409C2×D4—61no—
410C2×D4—61no—
411C2×D4—61no—
412C2×D4—61no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—31no—
418C22×C4—31no—
419C22×C4—31no—
420C22×C4—31no—
421C22×C4—64no—
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—42no—
450C4⋊C4—64no—
451C4⋊C4—64no—
452C4⋊C4—64no—
453C4⋊C4—61no—
454C4⋊C4—61no—
455C4⋊C4—61no—
456C4⋊C4—122no—
457C4⋊C4C2×C4×D312yes—
458C4⋊C4C2×C4×D312yes—
459C4⋊C4#23912yes—
460C22⋊C4—64no—
461C22⋊C4—122no—
462C22⋊C4—122no—
463C22⋊C4—122no—
464C22⋊C4—122no—
465C42—21no—
466C42—21no—
467C42—32no—
468C42—32no—
469C42—42no—
470C42—62no—
471C42—64no—
472C42—64no—
473C42—64no—
474C42—62no—
475C42—61no—
476C42—61no—
477C42—121no—
478C42—121no—
479C42—122no—
480C42—122no—
481C42—122no—
482C22×C3C2×C4×D411yescharacteristic
483C22×C3#52511yescharacteristic
484C22×C3#52511yescharacteristic
485C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
486C22×C3C2×#13711yescharacteristic
487C22×C3#57511yescharacteristic
488C22×C3#57511yescharacteristic
489C2×D3—42no—
490C2×D3—42no—
491C2×D3—42no—
492C2×D3—81no—
493C2×D3—81no—
494C2×D3—81no—
495C2×D3—81no—
496C3×C4—22no—
497C3×C4—41no—
498C3×C4—42no—
499C3×C4—42no—
500C3×C4—42no—
501C3×C4—81no—
502C3×C4C2×#11711yescharacteristic
503C3×C4C4×#4111yescharacteristic
504C3×C4#52911yescharacteristic
505C3×C4#52911yescharacteristic
506Dic3—22no—
507Dic3—22no—
508Dic3—22no—
509Dic3—24no—
510Dic3—42no—
511Dic3—41no—
512Dic3—81no—
513C23—62no—
514C23—121no—
515C23—121no—
516C23—121no—
517C23—121no—
518C23—121no—
519C23—121no—
520C23C2×D3×D411yescharacteristic
521Q8—34no—
522Q8—62no—
523Q8—62no—
524Q8—62no—
525D4—62no—
526D4—62no—
527D4—121no—
528D4—121no—
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—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—41no—
549C2×C4—62no—
550C2×C4—61no—
551C2×C4—61no—
552C2×C4—61no—
553C2×C4—61no—
554C2×C4—61no—
555C2×C4—62no—
556C2×C4—61no—
557C2×C4—62no—
558C2×C4—62no—
559C2×C4—61no—
560C2×C4—61no—
561C2×C4—61no—
562C2×C4—128no—
563C2×C4—121no—
564C2×C4—121no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4#91112yes—
569C2×C4C22×C4×D311yescharacteristic
570C2×C4C2×#23911yescharacteristic
571C2×C4D3×#4111yescharacteristic
572C2×C4D3×#4111yescharacteristic
573C2×C4D3×#4111yescharacteristic
574C2×C4D3×#4111yescharacteristic
575C8—22no—
576C8—41no—
577C8—62no—
578C8—121no—
579C2×C3#229011yescharacteristic
580C2×C3C4×AGL(1,ℤ/8)11yescharacteristic
581C2×C3C4×#13711yescharacteristic
582C2×C3#311111yescharacteristic
583C2×C3#311211yescharacteristic
584C2×C3#311511yescharacteristic
585C2×C3#311711yescharacteristic
586D3—82no—
587C22—122no—
588C22—122no—
589C22—122no—
590C22—241no—
591C22—241no—
592C22—241no—
593C22—241no—
594C22C4×D3×D411yescharacteristic
595C22#562711yescharacteristic
596C22#565611yescharacteristic
597C22C2×#91111yescharacteristic
598C22#576811yescharacteristic
599C22D3×#13711yescharacteristic
600C22#577711yescharacteristic
601C4—22no—
602C4—42no—
603C4—41no—
604C4—42no—
605C4—42no—
606C4—62no—
607C4—64no—
608C4—62no—
609C4—62no—
610C4—81no—
611C4—121no—
612C4—122no—
613C4—241no—
614C4D3×#11711yescharacteristic
615C4C4×#23911yescharacteristic
616C4#569111yescharacteristic
617C4#570611yescharacteristic
618C3#2112411yescharacteristic, complemented, Sylow
619C2—242no—
620C2#7493711yescharacteristic
621C2C4×#91111yescharacteristic
622C2#7982911yescharacteristic
623C2#8095111yescharacteristic
624C2#8120711yescharacteristic
625C2#8127011yescharacteristic
626C2#8149411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1918C2
socleC23×C3C22×D4
prime radicalC22×D12C23
cosocleC22×C3×C4C24
hypercenter#1918D3
hypocenterC3#21124
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1918

Complements

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

NH ≅ G/Ntype
3. C4×#887C2semidirect
4. C3×#1918C2semidirect
5. #70610C2semidirect
6. #69632C2semidirect
9. #68906C2semidirect
11. #68786C2semidirect
12. #68783C2semidirect
15. #68597C2semidirect
46. #4889C4semidirect
67. #1918D3semidirect
395. Dic6#117semidirect
618. C3#21124semidirect

Maximal subgroups

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

subgroupindex
C2×#48892
#759612
C4×#8872
C3×#19182
#706102
#696322
#694622
#689092
#689062
#687922
#687862
#687832
#687822
#686862
#685972
#211243