Finite Group Databasebeta

← #10876203

Subgroups — #10876203

Order 768.

872 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#76478C211yesmaximal, characteristic
2#74147C211yesmaximal, characteristic, complemented
3C3×#2097C211yesmaximal, characteristic, complemented
4#70633C211yesmaximal, characteristic, complemented
5#70623C211yesmaximal, characteristic, complemented
6#70580C211yesmaximal, characteristic, complemented
7#70569C211yesmaximal, characteristic, complemented
8#70267C211yesmaximal, characteristic
9#70267C211yesmaximal, characteristic
10#70242C211yesmaximal, characteristic, complemented
11#70241C211yesmaximal, characteristic, complemented
12#69671C211yesmaximal, characteristic, complemented
13#69558C211yesmaximal, characteristic, complemented
14#68664C211yesmaximal, characteristic, complemented
15#68637C211yesmaximal, characteristic, complemented
16#27590—31nomaximal, Sylow
17C22×#239C2211yescharacteristic
18C2×#892C2211yescharacteristic
19C2×C42×D3C2211yescharacteristic
20C2×C3×#107C2211yescharacteristic
21C2×C3×#106C2211yescharacteristic
22C2×C3×#103C2211yescharacteristic
23C2×C3×#103C2211yescharacteristic
24C3×#391C2211yescharacteristic
25C3×#382C2211yescharacteristic
26#5229C2211yescharacteristic
27C2×#835C2211yescharacteristic
28C2×#835C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#821C2211yescharacteristic
31C2×(C2×D3)⋊C8C2211yescharacteristic
32C2×(C2×D3)⋊C8C2211yescharacteristic
33#4988C2211yescharacteristic
34#4987C2211yescharacteristic
35#4987C2211yescharacteristic
36Dic3×C4⋊C4C2211yescharacteristic
37Dic3×C4⋊C4C2211yescharacteristic
38C2×#809C2211yescharacteristic
39C2×#809C2211yescharacteristic
40C2×#808C2211yescharacteristic
41C2×#807C2211yescharacteristic
42#4939C2211yescharacteristic
43#4936C2211yescharacteristic
44#4666—21no—
45#4666—21no—
46#4666—21no—
47#4665—21no—
48#4665—21no—
49#4664—21no—
50#4651—21no—
51#4648—21no—
52C3×#336C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4550C2211yescharacteristic
56#4550C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2297—31no—
61#2283—31no—
62#2097D311yescharacteristic, complemented
63#2092—31no—
64#2074—31no—
65#2073—31no—
66#1928—31no—
67#1927—31no—
68#1926—31no—
69#1925—31no—
70#1854—31no—
71#1854—31no—
72#1817—31no—
73#1788—31no—
74#1788—31no—
75C22×C3×Q8C2311yescharacteristic
76C2×#239—22no—
77C2×#239—22no—
78C2×#239—22no—
79C2×#239—22no—
80C2×#239—41no—
81C2×#239—41no—
82C22×D12—21no—
83C22×D12C2311yescharacteristic
84C22×C4×D3—21no—
85C22×C4×D3D411yescharacteristic
86C22×C4×D3D411yescharacteristic
87C22×C4×D3C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C22×C3×C8C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C42C2311yescharacteristic
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4×Dic3—21no—
105C2×C4×Dic3—21no—
106C2×C4×Dic3—21no—
107C2×C4×Dic3D411yescharacteristic
108C2×C4×Dic3D411yescharacteristic
109C2×C4×Dic3C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111C22×C3⋊C8C2311yescharacteristic
112#892—41no—
113#892—41no—
114C42×D3—22no—
115C42×D3—22no—
116C3×#107—41no—
117C3×#106—41no—
118C3×#103—41no—
119C3×#103—41no—
120C3×#95—21no—
121C3×#95—21no—
122C3×#95—21no—
123C3×#95—21no—
124#835—22no—
125#835—22no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134#821—22no—
135#821—22no—
136(C2×D3)⋊C8—22no—
137(C2×D3)⋊C8—22no—
138#809—41no—
139#809—41no—
140#808—41no—
141#807—41no—
142C22×#41—31no—
143C2×#117—31no—
144C22×C42—31no—
145C2×#107—31no—
146C2×#107C2×D311yescharacteristic
147C2×#106—31no—
148C2×#106C2×D311yescharacteristic
149C2×#103—31no—
150C2×#103—31no—
151C2×#103C2×D311yescharacteristic
152C2×#103C2×D311yescharacteristic
153C2×#102—31no—
154C2×#102—31no—
155C2×#102—31no—
156C2×#102—31no—
157C2×C22⋊C8—31no—
158C2×C22⋊C8—31no—
159#391—31no—
160#391C2×D311yescharacteristic
161#390—31no—
162#390—31no—
163#386—61no—
164#386—61no—
165#383—61no—
166#382—31no—
167#382C2×D311yescharacteristic
168#381—61no—
169#380—31no—
170#380—31no—
171C4×C4⋊C4—31no—
172C4×C4⋊C4—31no—
173C4×C22⋊C4—61no—
174C4×C22⋊C4—61no—
175C4×C22⋊C4—61no—
176#376—61no—
177#340—31no—
178#340—31no—
179#340—31no—
180#340—31no—
181#336—31no—
182#336—31no—
183#336—31no—
184#336C2×D311yescharacteristic
185C23×D3—21no—
186C23×D3C2×D411yescharacteristic
187C2×C3×Q8—21no—
188C2×C3×Q8—21no—
189C2×C3×Q8—21no—
190C2×C3×Q8—21no—
191C2×C3×Q8—21no—
192C2×C3×Q8#4112yes—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2411yescharacteristic
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3C2×D411yescharacteristic
204C22×Dic3C2×D411yescharacteristic
205C22×Dic3C2×D411yescharacteristic
206#239—44no—
207#239—44no—
208#239—44no—
209#239—44no—
210C2×D12—21no—
211C2×D12—21no—
212C2×D12—21no—
213C2×D12—21no—
214C2×D12—24no—
215C2×D12—21no—
216C2×D12—42no—
217C2×D12—42no—
218C2×D12#4112yes—
219C2×C4×D3—21no—
220C2×C4×D3—21no—
221C2×C4×D3—21no—
222C2×C4×D3—21no—
223C2×C4×D3—24no—
224C2×C4×D3—22no—
225C2×C4×D3—22no—
226C2×C4×D3—22no—
227C2×C4×D3—41no—
228C2×C4×D3—41no—
229C2×C4×D3—41no—
230C2×C4×D3—42no—
231C2×C4×D3—42no—
232C2×C4×D3#4112yes—
233C2×C4×D3#4112yes—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C42—21no—
247C3×C42—21no—
248(C2×D3)⋊C4—42no—
249(C2×D3)⋊C4—42no—
250(C2×D3)⋊C4—42no—
251(C2×D3)⋊C4—42no—
252(C2×D3)⋊C4—42no—
253(C2×D3)⋊C4—42no—
254(C2×D3)⋊C4—42no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4×Dic3—21no—
262C4×Dic3—21no—
263C4×Dic3—22no—
264C4×Dic3—22no—
265C4×Dic3—42no—
266C4×Dic3—42no—
267C4×Dic3—42no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×#41—62no—
275C2×#41—62no—
276C2×#41—62no—
277C2×#41—62no—
278C2×#41—121no—
279C2×#41—121no—
280C22×Q8C22×D311yescharacteristic
281C22×D4—31no—
282C22×D4—61no—
283C23×C4—31no—
284C23×C4—31no—
285C23×C4—31no—
286C23×C4—61no—
287C22×C8—31no—
288C22×C8—31no—
289C22×C8C22×D311yescharacteristic
290C22×C8C22×D311yescharacteristic
291#117—121no—
292#117—121no—
293C2×C4⋊C4—31no—
294C2×C4⋊C4—31no—
295C2×C4⋊C4—31no—
296C2×C4⋊C4C22×D311yescharacteristic
297C2×C4⋊C4C22×D311yescharacteristic
298C2×C4⋊C4C22×D311yescharacteristic
299C2×C22⋊C4—61no—
300C2×C22⋊C4—61no—
301C2×C22⋊C4—61no—
302C2×C22⋊C4—61no—
303C2×C22⋊C4—61no—
304C2×C22⋊C4—61no—
305C2×C22⋊C4—61no—
306C2×C42—31no—
307C2×C42—31no—
308C2×C42—31no—
309C2×C42—62no—
310C2×C42—62no—
311C2×C42—61no—
312C2×C42—61no—
313C2×C42—61no—
314C2×C42C22×D311yescharacteristic
315#107—41no—
316#107—121no—
317#106—41no—
318#106—121no—
319#103—41no—
320#103—41no—
321#103—62no—
322#103—62no—
323#102—62no—
324#102—62no—
325#102—121no—
326#102—121no—
327C22⋊C8—62no—
328C22⋊C8—62no—
329#95—21no—
330#95—21no—
331#95—21no—
332#95—21no—
333#95—61no—
334#95—61no—
335#95—61no—
336#95—61no—
337#95—61no—
338#95—61no—
339#95—61no—
340#95—61no—
341C23×C3C22×D411yescharacteristic
342C22×D3—21no—
343C22×D3—21no—
344C22×D3—21no—
345C22×D3—21no—
346C22×D3—21no—
347C22×D3—21no—
348C22×D3—22no—
349C22×D3—41no—
350C22×D3—41no—
351C22×D3—41no—
352C22×D3—41no—
353C22×D3—41no—
354C22×D3—41no—
355C22×D3#12312yes—
356C3×Q8—22no—
357C3×Q8—22no—
358C3×Q8—41no—
359C3×Q8—41no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—22no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
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×C4C2×#4111yescharacteristic
379C2×C3×C4C2×#4111yescharacteristic
380C2×C3×C4C2×#4111yescharacteristic
381C2×C3×C4C2×#4111yescharacteristic
382C2×C3×C4H(5,𝔽2)11yescharacteristic
383C2×C3×C4H(5,𝔽2)11yescharacteristic
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3#12312yes—
401C2×Dic3#13512yes—
402D12—22no—
403D12—22no—
404D12—48no—
405D12—41no—
406D12—41no—
407C4×D3—22no—
408C4×D3—22no—
409C4×D3—22no—
410C4×D3—22no—
411C4×D3—44no—
412C4×D3—48no—
413C4×D3—44no—
414C3×C8—41no—
415C3×C8—41no—
416C3⋊C8—41no—
417C3⋊C8—41no—
418C24—31no—
419C24—61no—
420#41—124no—
421#41—124no—
422#41—124no—
423#41—124no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8#23912yes—
430C2×D4—32no—
431C2×D4—64no—
432C2×D4—61no—
433C2×D4—61no—
434C2×D4—61no—
435C2×D4—61no—
436C2×D4—61no—
437C2×D4—122no—
438C2×D4—122no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—31no—
444C22×C4—31no—
445C22×C4—32no—
446C22×C4—32no—
447C22×C4—31no—
448C22×C4—62no—
449C22×C4—62no—
450C22×C4—62no—
451C22×C4—64no—
452C22×C4—61no—
453C22×C4—61no—
454C22×C4—61no—
455C22×C4—61no—
456C22×C4—61no—
457C22×C4—61no—
458C22×C4—61no—
459C22×C4—121no—
460C22×C4—121no—
461C22×C4—121no—
462C22×C4—122no—
463C22×C4—122no—
464C22×C4D3×D411yescharacteristic
465C22×C4D3×D411yescharacteristic
466C22×C4C23×D311yescharacteristic
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—61no—
474C2×C8—61no—
475C2×C8—61no—
476C2×C8—61no—
477C2×C8—61no—
478C2×C8—61no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—61no—
486C4⋊C4—61no—
487C4⋊C4—61no—
488C4⋊C4—61no—
489C4⋊C4—61no—
490C4⋊C4—61no—
491C22⋊C4—122no—
492C22⋊C4—122no—
493C22⋊C4—122no—
494C22⋊C4—122no—
495C22⋊C4—122no—
496C22⋊C4—122no—
497C22⋊C4—122no—
498C42—21no—
499C42—21no—
500C42—62no—
501C42—62no—
502C42—61no—
503C42—61no—
504C42—122no—
505C42—122no—
506C42—122no—
507C22×C3C2×#12311yescharacteristic
508C22×C3#52511yescharacteristic
509C22×C3#53511yescharacteristic
510C22×C3C2×#13511yescharacteristic
511C22×C3#57511yescharacteristic
512C22×C3#57511yescharacteristic
513C22×C3#57511yescharacteristic
514C2×D3—22no—
515C2×D3—22no—
516C2×D3—22no—
517C2×D3—41no—
518C2×D3—41no—
519C2×D3—41no—
520C2×D3—41no—
521C2×D3—42no—
522C2×D3—42no—
523C2×D3—42no—
524C2×D3—81no—
525C2×D3—81no—
526C2×D3—81no—
527C2×D3—81no—
528C3×C4—41no—
529C3×C4—41no—
530C3×C4—42no—
531C3×C4—81no—
532C3×C4—81no—
533C3×C4—81no—
534C3×C4#55111yescharacteristic
535C3×C4#55111yescharacteristic
536C3×C4#55111yescharacteristic
537C3×C4#55111yescharacteristic
538Dic3—22no—
539Dic3—22no—
540Dic3—41no—
541Dic3—81no—
542Dic3—81no—
543Dic3—81no—
544C23—32no—
545C23—61no—
546C23—61no—
547C23—61no—
548C23—61no—
549C23—61no—
550C23—61no—
551C23—62no—
552C23—121no—
553C23—121no—
554C23—121no—
555C23—121no—
556C23—121no—
557C23—121no—
558C23C2×D3×D411yescharacteristic
559Q8—22no—
560Q8—22no—
561Q8—41no—
562Q8—41no—
563D4—62no—
564D4—62no—
565D4—128no—
566D4—121no—
567D4—121no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—32no—
576C2×C4—32no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—61no—
589C2×C4—61no—
590C2×C4—61no—
591C2×C4—61no—
592C2×C4—62no—
593C2×C4—62no—
594C2×C4—62no—
595C2×C4—62no—
596C2×C4—61no—
597C2×C4—61no—
598C2×C4—61no—
599C2×C4—124no—
600C2×C4—128no—
601C2×C4—124no—
602C2×C4—121no—
603C2×C4—121no—
604C2×C4—121no—
605C2×C4—121no—
606C2×C4—121no—
607C2×C4—121no—
608C2×C4—121no—
609C2×C4—121no—
610C2×C4—121no—
611C2×C4#100411yescharacteristic
612C2×C4#100411yescharacteristic
613C2×C4C2×#23911yescharacteristic
614C2×C4D3×#4111yescharacteristic
615C2×C4D3×#4111yescharacteristic
616C2×C4D3×#4111yescharacteristic
617C8—41no—
618C8—41no—
619C8—121no—
620C8—121no—
621C2×C3#248211yescharacteristic
622C2×C3#309111yescharacteristic
623C2×C3#309311yescharacteristic
624C2×C3#310211yescharacteristic
625C2×C3#311111yescharacteristic
626C2×C3#321211yescharacteristic
627C2×C3#321911yescharacteristic
628D3—42no—
629D3—82no—
630C22—62no—
631C22—62no—
632C22—62no—
633C22—121no—
634C22—121no—
635C22—121no—
636C22—121no—
637C22—122no—
638C22—122no—
639C22—122no—
640C22—241no—
641C22—241no—
642C22—241no—
643C22—241no—
644C22#562711yescharacteristic
645C22D3×#12311yescharacteristic
646C22#571511yescharacteristic
647C22D3×#13511yescharacteristic
648C22#576811yescharacteristic
649C22#577711yescharacteristic
650C22#577711yescharacteristic
651C4—42no—
652C4—41no—
653C4—41no—
654C4—62no—
655C4—62no—
656C4—81no—
657C4—81no—
658C4—81no—
659C4—121no—
660C4—241no—
661C4—241no—
662C4—241no—
663C4#563711yescharacteristic
664C4#563711yescharacteristic
665C4#565911yescharacteristic
666C4#565911yescharacteristic
667C3#2759011yescharacteristic, complemented, Sylow
668C2—122no—
669C2—242no—
670C2#7810511yescharacteristic
671C2#8120411yescharacteristic
672C2#8120611yescharacteristic
673C2#8151511yescharacteristic
674C2#8151611yescharacteristic
675C2#8182611yescharacteristic
676C2#8187711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2097C2
socleC23×C3C22×D4
prime radicalC22×#239C22
hypercenter#2097D3
hypocenterC3#27590
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2097

Complements

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

NH ≅ G/Ntype
2. #74147C2semidirect
3. C3×#2097C2semidirect
4. #70633C2semidirect
5. #70623C2semidirect
6. #70580C2semidirect
7. #70569C2semidirect
10. #70242C2semidirect
11. #70241C2semidirect
12. #69671C2semidirect
13. #69558C2semidirect
14. #68664C2semidirect
15. #68637C2semidirect
62. #2097D3semidirect
667. C3#27590semidirect

Maximal subgroups

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

subgroupindex
#764782
#741472
C3×#20972
#706332
#706232
#705802
#705692
#702672
#702422
#702412
#696712
#695582
#686642
#686372
#275903