Finite Group Databasebeta

← #10876279

Subgroups — #10876279

Order 768.

842 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82186C211yesmaximal, characteristic
2#73868C211yesmaximal, characteristic, complemented
3C3×#1853C211yesmaximal, characteristic, complemented
4#71556C211yesmaximal, characteristic, complemented
5#70590C211yesmaximal, characteristic, complemented
6#70587C211yesmaximal, characteristic, complemented
7#70570C211yesmaximal, characteristic, complemented
8#70569C211yesmaximal, characteristic, complemented
9#69960C211yesmaximal, characteristic
10#69911C211yesmaximal, characteristic, complemented
11#69522C211yesmaximal, characteristic
12#68915C211yesmaximal, characteristic, complemented
13#68903C211yesmaximal, characteristic, complemented
14#68793C211yesmaximal, characteristic, complemented
15#68792C211yesmaximal, characteristic, complemented
16#27597—31nomaximal, Sylow
17C22×#235C2211yescharacteristic
18C22×C3×C4⋊C4C2211yescharacteristic
19C2×#926C2211yescharacteristic
20C2×C3×#106C2211yescharacteristic
21C2×C3×#106C2211yescharacteristic
22C2×C3×C22⋊C8C2211yescharacteristic
23C3×#380C2211yescharacteristic
24#5213—21no—
25#5213—21no—
26#5212C2211yescharacteristic
27C2×#830C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#816C2211yescharacteristic
31C2×#816C2211yescharacteristic
32#4988C2211yescharacteristic
33#4988C2211yescharacteristic
34#4987—21no—
35#4987—21no—
36#4977C2211yescharacteristic
37#4975C2211yescharacteristic
38Dic3×C4⋊C4C2211yescharacteristic
39Dic3×C4⋊C4C2211yescharacteristic
40#4973C2211yescharacteristic
41#4973C2211yescharacteristic
42C2×#810C2211yescharacteristic
43C2×#810C2211yescharacteristic
44C2×#809C2211yescharacteristic
45C2×#809C2211yescharacteristic
46C2×#808C2211yescharacteristic
47C2×#808C2211yescharacteristic
48#4945—21no—
49#4944—21no—
50#4654—21no—
51#4653—21no—
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54#4550C2211yescharacteristic
55#4550C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2314—31no—
61#2311—31no—
62#2096—31no—
63#2094—31no—
64#2074—31no—
65#2073—31no—
66#1927—31no—
67#1927—31no—
68#1926—31no—
69#1926—31no—
70#1853—31no—
71#1853D311yescharacteristic, complemented
72#1818—31no—
73#1788—31no—
74#1788—31no—
75C23×C3×C4—21no—
76C23×C3×C4C2311yescharacteristic
77C22×#67—21no—
78C2×#235—22no—
79C2×#235—22no—
80C2×#235—22no—
81C2×#235—22no—
82C2×#235—41no—
83C2×#235—41no—
84C22×D12C2311yescharacteristic
85C22×C4×D3—21no—
86C22×Dic6C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—41no—
91C2×C3×C4⋊C4—41no—
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C22⋊C4—21no—
97C2×#217—21no—
98C2×#217D411yescharacteristic
99C2×#217D411yescharacteristic
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104#926—22no—
105#926—22no—
106C2×C4⋊Dic3D411yescharacteristic
107C2×C4⋊Dic3D411yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×C4×Dic3—21no—
113C2×C4×Dic3—21no—
114C2×C4×Dic3C2311yescharacteristic
115C2×C4×Dic3C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118C3×#106—41no—
119C3×#106—41no—
120C3×C22⋊C8—22no—
121C3×#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131#830—22no—
132#821—22no—
133#821—41no—
134#816—22no—
135#816—41no—
136#810—22no—
137#810—41no—
138#809—22no—
139#809—41no—
140#808—41no—
141#808—41no—
142C22×#41—31no—
143C2×#117—31no—
144C22×C4⋊C4C2×D311yescharacteristic
145C2×#106—31no—
146C2×#106—31no—
147C2×#106C2×D311yescharacteristic
148C2×#106C2×D311yescharacteristic
149C2×#103—31no—
150C2×#103—31no—
151C2×#103—31no—
152C2×#103—31no—
153C2×#102—31no—
154C2×#102—31no—
155C2×#102—31no—
156C2×#102—31no—
157C2×C22⋊C8—31no—
158C2×C22⋊C8C2×D311yescharacteristic
159#391—31no—
160#391—31no—
161#390—31no—
162#390—31no—
163#388—61no—
164#388—61no—
165#387—61no—
166#387—61no—
167#386—61no—
168#386—61no—
169#384—31no—
170#382—31no—
171#380—31no—
172#380—61no—
173#380—61no—
174#380C2×D311yescharacteristic
175C4×C4⋊C4—31no—
176C4×C4⋊C4—31no—
177#340—31no—
178#340—31no—
179#340C2×D311yescharacteristic
180#340C2×D311yescharacteristic
181#336—31no—
182#336—31no—
183#336—31no—
184#336—31no—
185C24×C3C2×D411yescharacteristic
186C23×D3—21no—
187C22×C3×C4—21no—
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—42no—
196C22×C3×C4—81no—
197C22×C3×C4—81no—
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4#4112yes—
200C22×C3×C4#4112yes—
201C22×C3×C4C2411yescharacteristic
202C2×#67—24no—
203C2×#67—42no—
204C2×#67—42no—
205C22×Dic3—21no—
206C22×Dic3—21no—
207C22×Dic3—21no—
208C22×Dic3C2×D411yescharacteristic
209C22×Dic3C2×D411yescharacteristic
210C22×Dic3C2×D411yescharacteristic
211C22×Dic3C2×D411yescharacteristic
212#235—44no—
213#235—44no—
214#235—44no—
215#235—44no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12—21no—
219C2×D12—21no—
220C2×D12—21no—
221C2×D12#4112yes—
222C2×C4×D3—24no—
223C2×C4×D3—42no—
224C2×C4×D3—42no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6#4112yes—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—42no—
246C3×C22⋊C4—81no—
247#217—22no—
248#217—22no—
249#217—81no—
250(C2×D3)⋊C4—81no—
251(C2×D3)⋊C4—81no—
252(C2×D3)⋊C4—81no—
253(C2×D3)⋊C4—81no—
254C4⋊Dic3—22no—
255C4⋊Dic3—22no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260Dic3⋊C4—81no—
261Dic3⋊C4—81no—
262C4×Dic3—21no—
263C4×Dic3—21no—
264C4×Dic3—21no—
265C4×Dic3—21no—
266C4×Dic3—81no—
267C4×Dic3—81no—
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×Q8—31no—
281C22×D4—31no—
282C22×D4—61no—
283C23×C4—21no—
284C23×C4—61no—
285C23×C4C22×D311yescharacteristic
286C22×C8—31no—
287C22×C8—31no—
288C22×C8C22×D311yescharacteristic
289C22×C8C22×D311yescharacteristic
290#117—62no—
291#117—62no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—31no—
294C2×C4⋊C4—31no—
295C2×C4⋊C4—31no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—41no—
298C2×C4⋊C4—41no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4C22×D311yescharacteristic
302C2×C4⋊C4C22×D311yescharacteristic
303C2×C4⋊C4C22×D311yescharacteristic
304C2×C4⋊C4C22×D311yescharacteristic
305C2×C22⋊C4—21no—
306C2×C22⋊C4—31no—
307C2×C22⋊C4—31no—
308C2×C22⋊C4—61no—
309C2×C22⋊C4—61no—
310C2×C22⋊C4—61no—
311C2×C22⋊C4—61no—
312C2×C22⋊C4—61no—
313C2×C42—31no—
314C2×C42—31no—
315C2×C42—61no—
316C2×C42—61no—
317#106—41no—
318#106—41no—
319#106—121no—
320#106—121no—
321#103—62no—
322#103—62no—
323#103—121no—
324#103—121no—
325#102—62no—
326#102—62no—
327#102—121no—
328#102—121no—
329C22⋊C8—22no—
330C22⋊C8—62no—
331#95—21no—
332#95—61no—
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×C3—21no—
342C23×C3—21no—
343C23×C3—21no—
344C23×C3—21no—
345C23×C3—21no—
346C23×C3—21no—
347C23×C3#12312yes—
348C23×C3C22×D411yescharacteristic
349C22×D3—22no—
350C22×D3—41no—
351C22×D3—41no—
352C22×D3—41no—
353C22×D3—41no—
354C22×D3—41no—
355C22×D3—41no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
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—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—84no—
375C2×C3×C4#12312yes—
376C2×C3×C4C2×#4111yescharacteristic
377C2×C3×C4C2×#4111yescharacteristic
378C2×C3×C4C2×#4111yescharacteristic
379C2×C3×C4C2×#4111yescharacteristic
380C2×C3×C4H(5,𝔽2)11yescharacteristic
381C2×C3×C4H(5,𝔽2)11yescharacteristic
382#67—48no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—22no—
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404D12—22no—
405D12—22no—
406D12—41no—
407D12—41no—
408C4×D3—48no—
409Dic6—22no—
410Dic6—22no—
411Dic6—41no—
412Dic6—41no—
413C3×C8—41no—
414C3×C8—41no—
415C3⋊C8—41no—
416C3⋊C8—41no—
417C24—61no—
418C24D3×D411yescharacteristic
419#41—124no—
420#41—124no—
421#41—124no—
422#41—124no—
423C2×Q8—32no—
424C2×Q8—61no—
425C2×Q8—61no—
426C2×Q8—61no—
427C2×Q8—61no—
428C2×Q8—61no—
429C2×D4—32no—
430C2×D4—64no—
431C2×D4—61no—
432C2×D4—61no—
433C2×D4—61no—
434C2×D4—61no—
435C2×D4—61no—
436C2×D4—122no—
437C2×D4—122no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—31no—
447C22×C4—31no—
448C22×C4—31no—
449C22×C4—31no—
450C22×C4—42no—
451C22×C4—64no—
452C22×C4—61no—
453C22×C4—61no—
454C22×C4—61no—
455C22×C4—81no—
456C22×C4—81no—
457C22×C4—122no—
458C22×C4—122no—
459C22×C4#23512yes—
460C22×C4D3×D411yescharacteristic
461C22×C4#23912yes—
462C22×C4C23×D311yescharacteristic
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—61no—
470C2×C8—61no—
471C2×C8—61no—
472C2×C8—61no—
473C2×C8—61no—
474C2×C8—61no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—42no—
484C4⋊C4—62no—
485C4⋊C4—62no—
486C4⋊C4—61no—
487C4⋊C4—61no—
488C4⋊C4—61no—
489C4⋊C4—61no—
490C4⋊C4—241no—
491C4⋊C4—241no—
492C22⋊C4—62no—
493C22⋊C4—62no—
494C22⋊C4—81no—
495C22⋊C4—241no—
496C22⋊C4—241no—
497C22⋊C4—241no—
498C22⋊C4—241no—
499C22⋊C4—241no—
500C42—61no—
501C42—61no—
502C42—61no—
503C42—61no—
504C42—241no—
505C42—241no—
506C22×C3—22no—
507C22×C3—22no—
508C22×C3—22no—
509C22×C3—41no—
510C22×C3—41no—
511C22×C3—41no—
512C22×C3—41no—
513C22×C3C2×#12311yescharacteristic
514C22×C3#52511yescharacteristic
515C22×C3#53511yescharacteristic
516C22×C3#57711yescharacteristic
517C22×C3#57711yescharacteristic
518C22×C3#57711yescharacteristic
519C22×C3#57711yescharacteristic
520C2×D3—42no—
521C2×D3—42no—
522C2×D3—42no—
523C2×D3—81no—
524C2×D3—81no—
525C2×D3—81no—
526C2×D3—81no—
527C3×C4—22no—
528C3×C4—81no—
529C3×C4—81no—
530C3×C4—81no—
531C3×C4—81no—
532C3×C4#55111yescharacteristic
533C3×C4#55111yescharacteristic
534C3×C4#55111yescharacteristic
535C3×C4#55111yescharacteristic
536Dic3—41no—
537Dic3—41no—
538Dic3—41no—
539Dic3—41no—
540Dic3—42no—
541Dic3—81no—
542Dic3—81no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—62no—
550C23—121no—
551C23—121no—
552C23—121no—
553C23—121no—
554C23—121no—
555C23—121no—
556C23#89412yes—
557C23C2×D3×D411yescharacteristic
558Q8—62no—
559Q8—62no—
560Q8—121no—
561Q8—121no—
562D4—62no—
563D4—62no—
564D4—128no—
565D4—121no—
566D4—121no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
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—62no—
586C2×C4—61no—
587C2×C4—61no—
588C2×C4—61no—
589C2×C4—61no—
590C2×C4—61no—
591C2×C4—61no—
592C2×C4—61no—
593C2×C4—61no—
594C2×C4—61no—
595C2×C4—61no—
596C2×C4—61no—
597C2×C4—61no—
598C2×C4—84no—
599C2×C4—128no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4—121no—
603C2×C4—121no—
604C2×C4—121no—
605C2×C4—121no—
606C2×C4—121no—
607C2×C4—121no—
608C2×C4#89412yes—
609C2×C4C2×#23511yescharacteristic
610C2×C4#100411yescharacteristic
611C2×C4#100411yescharacteristic
612C2×C4C2×#23911yescharacteristic
613C2×C4D3×#4111yescharacteristic
614C2×C4D3×#4111yescharacteristic
615C8—41no—
616C8—41no—
617C8—121no—
618C8—121no—
619C2×C3—42no—
620C2×C3#248211yescharacteristic
621C2×C3#309911yescharacteristic
622C2×C3#309911yescharacteristic
623C2×C3#312511yescharacteristic
624C2×C3#312511yescharacteristic
625C2×C3#322111yescharacteristic
626C2×C3#322211yescharacteristic
627D3—82no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—122no—
636C22—122no—
637C22—122no—
638C22—241no—
639C22—241no—
640C22—241no—
641C22—241no—
642C22C2×#89411yescharacteristic
643C22#559211yescharacteristic
644C22#565611yescharacteristic
645C22#576211yescharacteristic
646C22#576211yescharacteristic
647C22#577511yescharacteristic
648C22#577811yescharacteristic
649C4—22no—
650C4—81no—
651C4—81no—
652C4—81no—
653C4—81no—
654C4—121no—
655C4—121no—
656C4—122no—
657C4—121no—
658C4—121no—
659C4—241no—
660C4—241no—
661C4#555611yescharacteristic
662C4#555611yescharacteristic
663C4#563711yescharacteristic
664C4#563711yescharacteristic
665C3#2759711yescharacteristic, complemented, Sylow
666C2—42no—
667C2—242no—
668C2#7461411yescharacteristic
669C2#7911211yescharacteristic
670C2#7922211yescharacteristic
671C2#8003611yescharacteristic
672C2#8014211yescharacteristic
673C2#8131311yescharacteristic
674C2#8141711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1853C2
socleC23×C3C22×D4
prime radicalC22×#235C22
hypercenter#1853D3
hypocenterC3#27597
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1853

Complements

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

NH ≅ G/Ntype
2. #73868C2semidirect
3. C3×#1853C2semidirect
4. #71556C2semidirect
5. #70590C2semidirect
6. #70587C2semidirect
7. #70570C2semidirect
8. #70569C2semidirect
10. #69911C2semidirect
12. #68915C2semidirect
13. #68903C2semidirect
14. #68793C2semidirect
15. #68792C2semidirect
71. #1853D3semidirect
665. C3#27597semidirect

Maximal subgroups

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

subgroupindex
#821862
#738682
C3×#18532
#715562
#705902
#705872
#705702
#705692
#699602
#699112
#695222
#689152
#689032
#687932
#687922
#275973