Finite Group Databasebeta

← #10824854

Subgroups — #10824854

Order 768.

876 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82396C211yesmaximal, characteristic
2C2×#5018C211yesmaximal, characteristic, complemented
3C2×#4690C211yesmaximal, characteristic, complemented
4#73762C211yesmaximal, characteristic, complemented
5C3×C4×#102C211yesmaximal, characteristic, complemented
6#70217C211yesmaximal, characteristic, complemented
7#70206C211yesmaximal, characteristic
8#70205C211yesmaximal, characteristic
9#70193C211yesmaximal, characteristic, complemented
10C4×#816C211yesmaximal, characteristic, complemented
11#69481C211yesmaximal, characteristic, complemented
12#68913C211yesmaximal, characteristic, complemented
13#68890C211yesmaximal, characteristic, complemented
14#68824C211yesmaximal, characteristic, complemented
15#68807C211yesmaximal, characteristic, complemented
16#23715—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C2×#938C2211yescharacteristic
19C22×#214C2211yescharacteristic
20C2×#897C2211yescharacteristic
21C2×#872C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C2×C3×C4×C8C2211yescharacteristic
26C3×C4×C4⋊C4C2211yescharacteristic
27#5221C2211yescharacteristic
28C2×#832C2211yescharacteristic
29#5213—21no—
30#5210—21no—
31C2×(C2×D3)⋊C8C2211yescharacteristic
32C2×(C2×D3)⋊C8C2211yescharacteristic
33C2×#816C2211yescharacteristic
34C2×#816C2211yescharacteristic
35C2×Dic3⋊C8C2211yescharacteristic
36C2×Dic3⋊C8C2211yescharacteristic
37#5018—24no—
38#4987C2211yescharacteristic
39#4987C2211yescharacteristic
40#4984C2211yescharacteristic
41#4973C2211yescharacteristic
42#4972C2211yescharacteristic
43C2×#810C2211yescharacteristic
44C2×#808C2211yescharacteristic
45C2×#807C2211yescharacteristic
46C2×#807C2211yescharacteristic
47#4957—21no—
48#4945—21no—
49#4940—21no—
50C4×C4⋊Dic3C2211yescharacteristic
51#4928—21no—
52C2×#803C2211yescharacteristic
53#4690—24no—
54#4670—21no—
55#4653—21no—
56C3×#340C2211yescharacteristic
57C3×#340C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#440—31no—
63C2×#433—31no—
64#2324—31no—
65#2323—31no—
66#2086—31no—
67#2076—31no—
68#2068—31no—
69#2065—31no—
70#2055—31no—
71#2035—31no—
72#2033—31no—
73#2000—31no—
74#1776—31no—
75C4×#103—31no—
76C4×#102D311yescharacteristic, complemented
77C22×C3×D4C2311yescharacteristic
78C23×C3×C4—21no—
79C22×#67—21no—
80C22×C4×D3C2311yescharacteristic
81C22×Dic6C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C3×C4×D4—22no—
85C3×C4×D4—22no—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C22⋊C4—21no—
90C2×C3×C42—21no—
91C2×C3×C42C2311yescharacteristic
92C2×#217—21no—
93C2×#217—21no—
94C2×#217—21no—
95#938—22no—
96#938—22no—
97C2×#214—22no—
98C2×#214—22no—
99C2×#214—22no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4D411yescharacteristic
104C2×(C2×D3)⋊C4D411yescharacteristic
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4D411yescharacteristic
113C2×Dic3⋊C4D411yescharacteristic
114C2×C4×Dic3—21no—
115C2×C4×Dic3—21no—
116C2×C4×Dic3C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119#897—22no—
120#897—22no—
121#872—22no—
122#872—22no—
123C4×Dic6—22no—
124C4×Dic6—22no—
125C3×#102—22no—
126C3×#102—22no—
127C3×C4×C8—22no—
128C3×#95—21no—
129#832—22no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137(C2×D3)⋊C8—22no—
138(C2×D3)⋊C8—22no—
139#816—22no—
140#816—22no—
141Dic3⋊C8—22no—
142Dic3⋊C8—22no—
143#810—22no—
144#808—22no—
145#807—22no—
146#807—22no—
147#803D414yescomplemented
148C22×SD16—31no—
149C2×C22⋊Q8—31no—
150C2×#121—31no—
151C2×C4×Q8—31no—
152C2×C4×D4C2×D311yescharacteristic
153C2×#117—31no—
154#440—64no—
155#433—64no—
156C2×#107—31no—
157C2×#107—31no—
158C2×#106—31no—
159C2×C4⋊C8—31no—
160C2×C4⋊C8—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#102—31no—
165C2×#102C2×D311yescharacteristic
166C2×#102C2×D311yescharacteristic
167C2×C22⋊C8—31no—
168C2×C22⋊C8—31no—
169C2×#97—31no—
170C2×C4×C8C2×D311yescharacteristic
171#391—31no—
172#390—31no—
173#390—61no—
174#389—31no—
175#389—31no—
176#389—61no—
177#388—61no—
178#388—61no—
179#388—61no—
180#387—61no—
181#386—61no—
182#385—61no—
183#380—31no—
184#380—31no—
185C4×C4⋊C4—31no—
186C4×C4⋊C4C2×D311yescharacteristic
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#340C2×D311yescharacteristic
192#340C2×D311yescharacteristic
193C24×C3—21no—
194C23×D3C2×D411yescharacteristic
195C2×C3×D4—21no—
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4#4112yes—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—42no—
205C22×C3×C4—81no—
206C22×C3×C4—81no—
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2411yescharacteristic
210C2×#67—24no—
211C2×#67—81no—
212C2×#67—81no—
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3—21no—
217C22×Dic3C2×D411yescharacteristic
218C22×Dic3C2×D411yescharacteristic
219C22×Dic3C2×D411yescharacteristic
220C2×C4×D3—21no—
221C2×C4×D3—21no—
222C2×C4×D3—21no—
223C2×C4×D3—21no—
224C2×C4×D3—21no—
225C2×C4×D3—21no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6#4112yes—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4#4112yes—
244C3×C22⋊C4—42no—
245C3×C42—81no—
246C3×C42C2×D412yes—
247C3×C42#4112yes—
248#217—24no—
249#217—81no—
250#217—81no—
251#214—28no—
252(C2×D3)⋊C4—24no—
253(C2×D3)⋊C4—22no—
254(C2×D3)⋊C4—22no—
255(C2×D3)⋊C4—81no—
256(C2×D3)⋊C4—81no—
257C4⋊Dic3—24no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—81no—
264C4⋊Dic3#4112yes—
265Dic3⋊C4—22no—
266Dic3⋊C4—22no—
267Dic3⋊C4—42no—
268Dic3⋊C4—81no—
269C4×Dic3—21no—
270C4×Dic3—21no—
271C4×Dic3—42no—
272C4×Dic3—81no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8C2×D412yes—
278C2×C3⋊C8C2×D412yes—
279C22×Q8—31no—
280C22×D4—61no—
281C22×D4C22×D311yescharacteristic
282C23×C4—21no—
283C23×C4—31no—
284C2×SD16—62no—
285C2×SD16—62no—
286C2×SD16—62no—
287C22×C8—31no—
288C22×C8—31no—
289C22×C8C22×D311yescharacteristic
290C22×C8C22×D311yescharacteristic
291C22⋊Q8—62no—
292C22⋊Q8—62no—
293#121—62no—
294#121—62no—
295C4×Q8—62no—
296C4×Q8—62no—
297C4×D4—22no—
298C4×D4—22no—
299#117—62no—
300#117—62no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4C22×D311yescharacteristic
311C2×C4⋊C4C22×D311yescharacteristic
312C2×C4⋊C4C22×D311yescharacteristic
313C2×C22⋊C4—21no—
314C2×C22⋊C4—31no—
315C2×C22⋊C4—31no—
316C2×C22⋊C4—61no—
317C2×C22⋊C4—61no—
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C42—21no—
323C2×C42—31no—
324C2×C42—61no—
325C2×C42—61no—
326C2×C42C22×D311yescharacteristic
327#107—62no—
328#107—62no—
329#106—62no—
330C4⋊C8—62no—
331C4⋊C8—62no—
332#103—62no—
333#103—62no—
334#103—62no—
335#102—22no—
336#102—22no—
337#102—62no—
338C22⋊C8—62no—
339C22⋊C8—62no—
340#97—34no—
341C4×C8—22no—
342#95—21no—
343#95—61no—
344#95—61no—
345#95—61no—
346#95—61no—
347#95—61no—
348#95—61no—
349#95—61no—
350C23×C3—22no—
351C23×C3—41no—
352C23×C3—41no—
353C23×C3—41no—
354C23×C3—41no—
355C23×C3—41no—
356C23×C3—41no—
357C23×C3C22×D411yescharacteristic
358C22×D3—21no—
359C22×D3—21no—
360C22×D3—21no—
361C22×D3—21no—
362C22×D3—21no—
363C22×D3—21no—
364C22×D3—21no—
365C3×D4—22no—
366C3×D4—22no—
367C3×D4—41no—
368C3×D4—41no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—22no—
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—41no—
382C2×C3×C4—84no—
383C2×C3×C4#12112yes—
384C2×C3×C4#12112yes—
385C2×C3×C4C22×D411yescharacteristic
386C2×C3×C4C2×#4111yescharacteristic
387C2×C3×C4C2×#4111yescharacteristic
388C2×C3×C4C2×#4111yescharacteristic
389C2×C3×C4C2×#4111yescharacteristic
390C2×C3×C4C2×#4111yescharacteristic
391#67—84no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—22no—
397C2×Dic3—22no—
398C2×Dic3—21no—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—21no—
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—41no—
413C4×D3—41no—
414C4×D3—41no—
415C4×D3—41no—
416C4×D3—41no—
417Dic6—22no—
418Dic6—22no—
419Dic6—41no—
420Dic6—41no—
421C3×C8—41no—
422C3×C8—41no—
423C3⋊C8—22no—
424C3⋊C8—22no—
425C24—21no—
426C24—31no—
427C2×Q8—32no—
428C2×Q8—61no—
429C2×Q8—61no—
430C2×Q8—61no—
431C2×Q8—61no—
432C2×Q8—61no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—64no—
439C2×D4—241no—
440C2×D4—241no—
441C2×D4#23512yes—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—31no—
446C22×C4—31no—
447C22×C4—31no—
448C22×C4—42no—
449C22×C4—61no—
450C22×C4—61no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4—61no—
454C22×C4—61no—
455C22×C4—61no—
456C22×C4—61no—
457C22×C4—61no—
458C22×C4—61no—
459C22×C4—81no—
460C22×C4—81no—
461C22×C4D3×D411yescharacteristic
462C22×C4D3×D411yescharacteristic
463C22×C4C23×D311yescharacteristic
464SD16—68no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—32no—
472C2×C8—32no—
473C2×C8—61no—
474C2×C8—61no—
475C2×C8—61no—
476C2×C8—61no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—32no—
483C4⋊C4—64no—
484C4⋊C4—62no—
485C4⋊C4—62no—
486C4⋊C4—61no—
487C4⋊C4—61no—
488C4⋊C4—61no—
489C4⋊C4—61no—
490C4⋊C4—61no—
491C4⋊C4—122no—
492C4⋊C4—241no—
493C4⋊C4—241no—
494C4⋊C4#23512yes—
495C22⋊C4—42no—
496C22⋊C4—64no—
497C22⋊C4—64no—
498C22⋊C4—62no—
499C22⋊C4—62no—
500C22⋊C4—241no—
501C22⋊C4—241no—
502C22⋊C4—241no—
503C22⋊C4—241no—
504C42—61no—
505C42—61no—
506C42—81no—
507C42—122no—
508C42—241no—
509C42#23512yes—
510C42D3×D412yes—
511C22×C3—42no—
512C22×C3—42no—
513C22×C3—42no—
514C22×C3—81no—
515C22×C3—81no—
516C22×C3—81no—
517C22×C3—81no—
518C22×C3C2×#12111yescharacteristic
519C22×C3#52511yescharacteristic
520C22×C3#52511yescharacteristic
521C22×C3#57611yescharacteristic
522C22×C3#57711yescharacteristic
523C22×C3#57711yescharacteristic
524C22×C3#57811yescharacteristic
525C2×D3—41no—
526C2×D3—41no—
527C2×D3—41no—
528C2×D3—41no—
529C2×D3—41no—
530C2×D3—41no—
531C2×D3—41no—
532C3×C4—22no—
533C3×C4—22no—
534C3×C4—42no—
535C3×C4—81no—
536C3×C4—81no—
537C3×C4#52511yescharacteristic
538C3×C4#52911yescharacteristic
539C3×C4#52911yescharacteristic
540C3×C4#53211yescharacteristic
541Dic3—42no—
542Dic3—42no—
543Dic3—41no—
544Dic3—41no—
545Dic3—41no—
546Dic3—81no—
547Dic3—81no—
548C23—22no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—61no—
556C23—61no—
557C23—61no—
558C23—61no—
559C23—61no—
560C23—61no—
561C23—61no—
562C23C2×D3×D411yescharacteristic
563Q8—62no—
564Q8—62no—
565Q8—121no—
566Q8—121no—
567D4—22no—
568D4—22no—
569D4—41no—
570D4—41no—
571D4—244no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
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—61no—
586C2×C4—61no—
587C2×C4—61no—
588C2×C4—61no—
589C2×C4—61no—
590C2×C4—61no—
591C2×C4—61no—
592C2×C4—61no—
593C2×C4—62no—
594C2×C4—62no—
595C2×C4—61no—
596C2×C4—84no—
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—121no—
608C2×C4—121no—
609C2×C4—121no—
610C2×C4—121no—
611C2×C4#88412yes—
612C2×C4#88412yes—
613C2×C4C2×#23511yescharacteristic
614C2×C4C2×#23511yescharacteristic
615C2×C4C2×#23511yescharacteristic
616C2×C4C2×D3×D411yescharacteristic
617C2×C4D3×#4111yescharacteristic
618C2×C4D3×#4111yescharacteristic
619C8—41no—
620C8—41no—
621C8—62no—
622C8—62no—
623C2×C3—82no—
624C2×C3#239211yescharacteristic
625C2×C3#306211yescharacteristic
626C2×C3#306311yescharacteristic
627C2×C3#311811yescharacteristic
628C2×C3#312011yescharacteristic
629C2×C3#312411yescharacteristic
630C2×C3#312611yescharacteristic
631D3—81no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22—121no—
640C22—121no—
641C22—121no—
642C22—121no—
643C22—121no—
644C22—121no—
645C22—121no—
646C22C2×#88411yescharacteristic
647C22#554711yescharacteristic
648C22#554711yescharacteristic
649C22#575711yescharacteristic
650C22#576211yescharacteristic
651C22#577011yescharacteristic
652C22#577111yescharacteristic
653C4—22no—
654C4—22no—
655C4—42no—
656C4—81no—
657C4—81no—
658C4—121no—
659C4—122no—
660C4—121no—
661C4—122no—
662C4—121no—
663C4—241no—
664C4—241no—
665C4#547911yescharacteristic
666C4#552811yescharacteristic
667C4#554711yescharacteristic
668C4#554811yescharacteristic
669C3#2371511yescharacteristic, complemented, Sylow
670C2—82no—
671C2—241no—
672C2#7358111yescharacteristic
673C2#7875211yescharacteristic
674C2#7884111yescharacteristic
675C2#7959411yescharacteristic
676C2#7959711yescharacteristic
677C2#7965711yescharacteristic
678C2#7966111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×#102C2
socleC23×C3C22×D4
prime radicalC2×#938C22
hypercenterC4×#102D3
hypocenterC3#23715
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
2. C2×#5018C2semidirect
3. C2×#4690C2semidirect
4. #73762C2semidirect
5. C3×C4×#102C2semidirect
6. #70217C2semidirect
9. #70193C2semidirect
10. C4×#816C2semidirect
11. #69481C2semidirect
12. #68913C2semidirect
13. #68890C2semidirect
14. #68824C2semidirect
15. #68807C2semidirect
76. C4×#102D3semidirect
147. #803D4semidirect
669. C3#23715semidirect

Maximal subgroups

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

subgroupindex
#823962
C2×#50182
C2×#46902
#737622
C3×C4×#1022
#702172
#702062
#702052
#701932
C4×#8162
#694812
#689132
#688902
#688242
#688072
#237153