Finite Group Databasebeta

← #10876277

Subgroups — #10876277

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#84087C211yesmaximal, characteristic
2#73868C211yesmaximal, characteristic, complemented
3C3×#1788C211yesmaximal, characteristic
4#72135C211yesmaximal, characteristic
5#70250C211yesmaximal, characteristic, complemented
6#70241C211yesmaximal, characteristic, complemented
7#70234C211yesmaximal, characteristic, complemented
8#70223C211yesmaximal, characteristic, complemented
9#70184C211yesmaximal, characteristic, complemented
10#70174C211yesmaximal, characteristic, complemented
11#70166C211yesmaximal, characteristic, complemented
12#70156C211yesmaximal, characteristic, complemented
13#69928C211yesmaximal, characteristic, complemented
14#69911C211yesmaximal, characteristic, complemented
15#69505C211yesmaximal, characteristic, complemented
16#27597—31nomaximal, Sylow
17C22×C3×#41C2211yescharacteristic
18C2×#926C2211yescharacteristic
19C22×C4⋊Dic3C2211yescharacteristic
20C2×C3×#103C2211yescharacteristic
21C2×C3×#103C2211yescharacteristic
22C2×C3×#102C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C2×C3×C22⋊C8C2211yescharacteristic
25C3×#380C2211yescharacteristic
26#5229C2211yescharacteristic
27#5229C2211yescharacteristic
28C2×#835C2211yescharacteristic
29C2×#835C2211yescharacteristic
30#5221C2211yescharacteristic
31#5221C2211yescharacteristic
32#5220—21no—
33#5220—21no—
34C2×#832C2211yescharacteristic
35C2×#832C2211yescharacteristic
36#5212—21no—
37#5212—21no—
38#5212C2211yescharacteristic
39C2×#830C2211yescharacteristic
40C2×#817C2211yescharacteristic
41C2×#817C2211yescharacteristic
42#4977C2211yescharacteristic
43#4975C2211yescharacteristic
44Dic3×C4⋊C4C2211yescharacteristic
45Dic3×C4⋊C4C2211yescharacteristic
46C2×#808C2211yescharacteristic
47C2×#808C2211yescharacteristic
48#4945—21no—
49#4944—21no—
50#4654—21no—
51#4653—21no—
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
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#1853—31no—
72#1818—31no—
73#1788—31no—
74#1788D311yescharacteristic
75C2×C3×#41—22no—
76C2×C3×#41—22no—
77C2×C3×#41—22no—
78C2×C3×#41—22no—
79C2×C3×#41—41no—
80C2×C3×#41—41no—
81C22×C3×Q8C2311yescharacteristic
82C22×C3×D4—21no—
83C22×C3×D4C2311yescharacteristic
84C23×C3×C4—21no—
85C23×C3×C4C2311yescharacteristic
86C23×Dic3—21no—
87C22×C3×C8C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C22⋊C4—21no—
92C2×#217—21no—
93C2×#217—21no—
94C2×#217—21no—
95C2×#217—21no—
96C2×#217—21no—
97C2×#217D411yescharacteristic
98C2×#217D411yescharacteristic
99#926—22no—
100#926—22no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3—41no—
103C2×C4⋊Dic3—41no—
104C2×C4⋊Dic3D411yescharacteristic
105C2×C4⋊Dic3D411yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
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×#103—22no—
119C3×#103—41no—
120C3×#102—22no—
121C3×#102—41no—
122C3×C22⋊C8—22no—
123C3×#95—21no—
124#835—22no—
125#835—41no—
126#832—22no—
127#832—41no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137#830—22no—
138#817—41no—
139#817—41no—
140#808—41no—
141#808—41no—
142C22×#41C2×D311yescharacteristic
143C2×#117—31no—
144C22×C4⋊C4—31no—
145C2×#106—31no—
146C2×#106—31no—
147C2×#106—31no—
148C2×#106—31no—
149C2×#103—31no—
150C2×#103—31no—
151C2×#103C2×D311yescharacteristic
152C2×#103C2×D311yescharacteristic
153C2×#102—31no—
154C2×#102—31no—
155C2×#102C2×D311yescharacteristic
156C2×#102C2×D311yescharacteristic
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#340—31no—
180#340—31no—
181#336—31no—
182#336—31no—
183#336—31no—
184#336—31no—
185C24×C3—21no—
186C24×C3C2×D411yescharacteristic
187C3×#41—44no—
188C3×#41—44no—
189C3×#41—44no—
190C3×#41—44no—
191C2×C3×Q8—21no—
192C2×C3×Q8—21no—
193C2×C3×Q8—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8#4112yes—
197C2×C3×D4—24no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4—42no—
204C2×C3×D4—42no—
205C2×C3×D4#4112yes—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—24no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4—42no—
215C22×C3×C4—42no—
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4#4112yes—
218C22×C3×C4#4112yes—
219C22×C3×C4C2411yescharacteristic
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3—21no—
223C22×Dic3—21no—
224C22×Dic3—42no—
225C22×Dic3—81no—
226C22×Dic3—81no—
227C22×Dic3C2×D411yescharacteristic
228C22×Dic3C2×D411yescharacteristic
229C22×Dic3C2×D411yescharacteristic
230C22×Dic3C2×D411yescharacteristic
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×C22⋊C4—81no—
242#217—22no—
243#217—22no—
244#217—81no—
245#217—81no—
246#217—81no—
247#217—81no—
248#217—81no—
249C4⋊Dic3—22no—
250C4⋊Dic3—22no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—42no—
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—22no—
275C2×#41—22no—
276C2×#41—22no—
277C2×#41—22no—
278C2×#41—41no—
279C2×#41—41no—
280C22×Q8C22×D311yescharacteristic
281C22×D4—21no—
282C22×D4C22×D311yescharacteristic
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—31no—
293C2×C4⋊C4—31no—
294C2×C4⋊C4—31no—
295C2×C4⋊C4—31no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—61no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—121no—
302C2×C4⋊C4—121no—
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—121no—
318#106—121no—
319#106—121no—
320#106—121no—
321#103—22no—
322#103—41no—
323#103—62no—
324#103—121no—
325#102—22no—
326#102—41no—
327#102—62no—
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—22no—
348C23×C3—41no—
349C23×C3—41no—
350C23×C3—41no—
351C23×C3—41no—
352C23×C3—41no—
353C23×C3—41no—
354C23×C3#12312yes—
355C23×C3C22×D411yescharacteristic
356C3×Q8—22no—
357C3×Q8—22no—
358C3×Q8—41no—
359C3×Q8—41no—
360C3×D4—22no—
361C3×D4—22no—
362C3×D4—48no—
363C3×D4—41no—
364C3×D4—41no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4—22no—
371C2×C3×C4—22no—
372C2×C3×C4—48no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4#12312yes—
382C2×C3×C4C2×#4111yescharacteristic
383C2×C3×C4C2×#4111yescharacteristic
384C2×C3×C4C2×#4111yescharacteristic
385C2×C3×C4C2×#4111yescharacteristic
386C2×C3×C4H(5,𝔽2)11yescharacteristic
387C2×C3×C4H(5,𝔽2)11yescharacteristic
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—21no—
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—21no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—84no—
413C3×C8—41no—
414C3×C8—41no—
415C3⋊C8—41no—
416C3⋊C8—41no—
417C24—21no—
418C24D3×D411yescharacteristic
419#41—44no—
420#41—44no—
421#41—44no—
422#41—44no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8#23712yes—
429C2×D4—24no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4#23712yes—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—24no—
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—42no—
452C22×C4—61no—
453C22×C4—61no—
454C22×C4—61no—
455C22×C4—61no—
456C22×C4—122no—
457C22×C4—241no—
458C22×C4—241no—
459C22×C4#23512yes—
460C22×C4D3×D411yescharacteristic
461C22×C4#23712yes—
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—62no—
480C4⋊C4—62no—
481C4⋊C4—61no—
482C4⋊C4—61no—
483C4⋊C4—61no—
484C4⋊C4—61no—
485C4⋊C4—61no—
486C4⋊C4—61no—
487C4⋊C4—61no—
488C4⋊C4—61no—
489C4⋊C4—122no—
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×C3—42no—
514C22×C3—42no—
515C22×C3—42no—
516C22×C3—81no—
517C22×C3—81no—
518C22×C3—81no—
519C22×C3—81no—
520C22×C3C2×#12311yescharacteristic
521C22×C3#52511yescharacteristic
522C22×C3#53511yescharacteristic
523C22×C3#57711yescharacteristic
524C22×C3#57711yescharacteristic
525C22×C3#57711yescharacteristic
526C22×C3#57711yescharacteristic
527C3×C4—22no—
528C3×C4—42no—
529C3×C4—81no—
530C3×C4—81no—
531C3×C4#55111yescharacteristic
532C3×C4#55111yescharacteristic
533C3×C4#55111yescharacteristic
534C3×C4#55111yescharacteristic
535Dic3—41no—
536Dic3—41no—
537Dic3—41no—
538Dic3—41no—
539Dic3—81no—
540Dic3—81no—
541Dic3—81no—
542Dic3—81no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—22no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23#88312yes—
557C23C2×D3×D411yescharacteristic
558Q8—22no—
559Q8—22no—
560Q8—41no—
561Q8—41no—
562D4—22no—
563D4—22no—
564D4—48no—
565D4—41no—
566D4—41no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—48no—
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—61no—
584C2×C4—61no—
585C2×C4—61no—
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—121no—
596C2×C4—121no—
597C2×C4—121no—
598C2×C4—121no—
599C2×C4—121no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4—121no—
603C2×C4—121no—
604C2×C4—121no—
605C2×C4—121no—
606C2×C4—121no—
607C2×C4—244no—
608C2×C4#88312yes—
609C2×C4C2×#23511yescharacteristic
610C2×C4C2×#23711yescharacteristic
611C2×C4C2×#23711yescharacteristic
612C2×C4C2×#23711yescharacteristic
613C2×C4#100411yescharacteristic
614C2×C4#100411yescharacteristic
615C8—41no—
616C8—41no—
617C8—121no—
618C8—121no—
619C2×C3—42no—
620C2×C3—82no—
621C2×C3#248211yescharacteristic
622C2×C3#309911yescharacteristic
623C2×C3#309911yescharacteristic
624C2×C3#312511yescharacteristic
625C2×C3#312511yescharacteristic
626C2×C3#322111yescharacteristic
627C2×C3#322211yescharacteristic
628C22—22no—
629C22—22no—
630C22—22no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22C2×#88311yescharacteristic
643C22#558611yescharacteristic
644C22#559211yescharacteristic
645C22#577011yescharacteristic
646C22#577011yescharacteristic
647C22#577511yescharacteristic
648C22#577811yescharacteristic
649C4—22no—
650C4—42no—
651C4—81no—
652C4—81no—
653C4—121no—
654C4—121no—
655C4—121no—
656C4—121no—
657C4—241no—
658C4—241no—
659C4—241no—
660C4—241no—
661C4#549511yescharacteristic
662C4#549511yescharacteristic
663C4#558711yescharacteristic
664C4#558711yescharacteristic
665C3#2759711yescharacteristic, complemented, Sylow
666C2—42no—
667C2—82no—
668C2#7383211yescharacteristic
669C2#7884411yescharacteristic
670C2#7887511yescharacteristic
671C2#8007511yescharacteristic
672C2#8012711yescharacteristic
673C2#8021911yescharacteristic
674C2#8026211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1788C2
socleC23×C3C22×D4
prime radicalC22×C3×#41C22
hypercenter#1788D3
hypocenterC3#27597
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1788

Complements

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

NH ≅ G/Ntype
2. #73868C2semidirect
5. #70250C2semidirect
6. #70241C2semidirect
7. #70234C2semidirect
8. #70223C2semidirect
9. #70184C2semidirect
10. #70174C2semidirect
11. #70166C2semidirect
12. #70156C2semidirect
13. #69928C2semidirect
14. #69911C2semidirect
15. #69505C2semidirect
665. C3#27597semidirect

Maximal subgroups

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

subgroupindex
#840872
#738682
C3×#17882
#721352
#702502
#702412
#702342
#702232
#701842
#701742
#701662
#701562
#699282
#699112
#695052
#275973