Finite Group Databasebeta

← #10896390

Subgroups — #10896390

Order 768.

790 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82329C211yesmaximal, characteristic
2#75972C211yesmaximal, characteristic, complemented
3C3×#1864C211yesmaximal, characteristic, complemented
4#71607C211yesmaximal, characteristic, complemented
5#70587C211yesmaximal, characteristic, complemented
6#70564C211yesmaximal, characteristic, complemented
7#70183C211yesmaximal, characteristic, complemented
8#70164C211yesmaximal, characteristic, complemented
9#70161C211yesmaximal, characteristic
10#70161C211yesmaximal, characteristic
11#69625C211yesmaximal, characteristic, complemented
12#68899C211yesmaximal, characteristic, complemented
13#68798C211yesmaximal, characteristic, complemented
14#68670C211yesmaximal, characteristic, complemented
15#68606C211yesmaximal, characteristic, complemented
16#29265—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C2×#940C2211yescharacteristic
19C2×#890C2211yescharacteristic
20C2×C4×D12C2211yescharacteristic
21C2×C3×#106C2211yescharacteristic
22C2×C3×#102C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C3×#389C2211yescharacteristic
25#5221C2211yescharacteristic
26C2×#832C2211yescharacteristic
27C2×#832C2211yescharacteristic
28#5213—21no—
29#5213—21no—
30C2×#821C2211yescharacteristic
31C2×#821C2211yescharacteristic
32C2×#818C2211yescharacteristic
33C2×Dic3⋊C8C2211yescharacteristic
34C2×Dic3⋊C8C2211yescharacteristic
35#4988C2211yescharacteristic
36#4977C2211yescharacteristic
37#4976—21no—
38#4975C2211yescharacteristic
39Dic3×C4⋊C4C2211yescharacteristic
40Dic3×C4⋊C4C2211yescharacteristic
41#4972—21no—
42C2×#809C2211yescharacteristic
43C2×#809C2211yescharacteristic
44C2×#808C2211yescharacteristic
45C2×#807C2211yescharacteristic
46#4956—21no—
47#4956—21no—
48#4932C2211yescharacteristic
49#4928C2211yescharacteristic
50#4653—21no—
51#4652—21no—
52C3×#340C2211yescharacteristic
53C3×#336C2211yescharacteristic
54#4551C2211yescharacteristic
55#4550C2211yescharacteristic
56#4550C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2330—31no—
61#2328—31no—
62#2094—31no—
63#2093—31no—
64#2025—31no—
65#2025—31no—
66#1982—31no—
67#1982—31no—
68#1949—31no—
69#1949—31no—
70#1864—31no—
71#1864D311yescharacteristic, complemented
72#1862—31no—
73#1862—31no—
74#1832—31no—
75C22×C3×D4C2311yescharacteristic
76C23×C3×C4—21no—
77C22×#67—21no—
78C22×#67—21no—
79C22×D12C2311yescharacteristic
80C22×C4×D3—21no—
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C3×C4×D4—41no—
84C3×C4×D4—41no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C22⋊C4—21no—
90C2×C3×C42C2311yescharacteristic
91C2×#217—21no—
92C2×#217—21no—
93#940—22no—
94#940Q814yes—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×C4×Dic3—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3C2311yescharacteristic
109C2×C4×Dic3C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111C22×C3⋊C8C2311yescharacteristic
112#890—22no—
113#890—22no—
114C4×D12—41no—
115C4×D12—41no—
116C3×#106—41no—
117C3×#102—22no—
118C3×#102—41no—
119C3×#95—21no—
120#832—22no—
121#832—41no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95D411yescharacteristic
128C3⋊#95D411yescharacteristic
129C3⋊#95D411yescharacteristic
130C3⋊#95D411yescharacteristic
131#821—22no—
132#821—41no—
133#818—41no—
134Dic3⋊C8—22no—
135Dic3⋊C8—22no—
136#809—22no—
137#809—41no—
138#808—41no—
139#807—41no—
140C2×#127—31no—
141C2×#125—31no—
142C2×C4×D4—31no—
143C2×C4×D4C2×D311yescharacteristic
144C2×#107—31no—
145C2×#107—31no—
146C2×#106—31no—
147C2×#106C2×D311yescharacteristic
148C2×C4⋊C8—31no—
149C2×C4⋊C8—31no—
150C2×#102—31no—
151C2×#102—31no—
152C2×#102—31no—
153C2×#102—31no—
154C2×#102—31no—
155C2×#102—31no—
156C2×#102C2×D311yescharacteristic
157C2×#102C2×D311yescharacteristic
158#390—31no—
159#390—31no—
160#389—31no—
161#389—61no—
162#389—61no—
163#389C2×D311yescharacteristic
164#387—61no—
165#387—61no—
166#386—61no—
167#386—61no—
168#386—61no—
169#386—61no—
170#384—31no—
171#382—31no—
172C4×C4⋊C4—31no—
173C4×C4⋊C4—31no—
174#376—31no—
175#340—31no—
176#340—31no—
177#340—31no—
178#340C2×D311yescharacteristic
179#336—31no—
180#336—31no—
181#336—31no—
182#336C2×D311yescharacteristic
183C24×C3—21no—
184C23×D3—21no—
185C2×C3×D4—21no—
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4—21no—
189C2×C3×D4—21no—
190C2×C3×D4C2×Q812yes—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—42no—
195C22×C3×C4—81no—
196C22×C3×C4—81no—
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2411yescharacteristic
200C2×#67—24no—
201C2×#67—24no—
202C2×#67—81no—
203C2×#67—81no—
204C2×#67—81no—
205C2×#67—81no—
206C22×Dic3—21no—
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3C2×D411yescharacteristic
210C22×Dic3C2×D411yescharacteristic
211C22×Dic3C2×D411yescharacteristic
212C22×Dic3C2×D411yescharacteristic
213C2×D12—21no—
214C2×D12—21no—
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12C2×Q812yes—
219C2×C4×D3—42no—
220C2×C4×D3—81no—
221C2×C4×D3—81no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—81no—
234C3×C4⋊C4#4112yes—
235C3×C22⋊C4—42no—
236C3×C42—21no—
237C3×C42—21no—
238#217—81no—
239#217—81no—
240(C2×D3)⋊C4—42no—
241(C2×D3)⋊C4—81no—
242(C2×D3)⋊C4—81no—
243C4⋊Dic3—24no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3#4112yes—
250Dic3⋊C4—24no—
251Dic3⋊C4—81no—
252Dic3⋊C4—81no—
253Dic3⋊C4—81no—
254C4×Dic3—21no—
255C4×Dic3—21no—
256C4×Dic3—81no—
257C4×Dic3—81no—
258C4×Dic3C2×Q812yes—
259C4×Dic3#4112yes—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C22×D4—31no—
267C22×D4—61no—
268C22×D4—61no—
269C22×D4C22×D311yescharacteristic
270C23×C4—21no—
271C23×C4—61no—
272C22×C8—31no—
273C22×C8—31no—
274C22×C8C22×D311yescharacteristic
275C22×C8C22×D311yescharacteristic
276#127—34no—
277#127—62no—
278#125—62no—
279#125—62no—
280C4×D4—41no—
281C4×D4—41no—
282C4×D4—121no—
283C4×D4—121no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—31no—
286C2×C4⋊C4—31no—
287C2×C4⋊C4—31no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4—61no—
291C2×C4⋊C4—61no—
292C2×C4⋊C4—61no—
293C2×C4⋊C4C22×D311yescharacteristic
294C2×C4⋊C4C22×D311yescharacteristic
295C2×C4⋊C4C22×D311yescharacteristic
296C2×C22⋊C4—21no—
297C2×C22⋊C4—61no—
298C2×C22⋊C4—61no—
299C2×C22⋊C4—61no—
300C2×C22⋊C4—61no—
301C2×C22⋊C4—61no—
302C2×C42—31no—
303C2×C42—31no—
304C2×C42—61no—
305C2×C42—61no—
306C2×C42C22×D311yescharacteristic
307#107—121no—
308#107—121no—
309#106—41no—
310#106—121no—
311C4⋊C8—62no—
312C4⋊C8—62no—
313#102—22no—
314#102—41no—
315#102—62no—
316#102—62no—
317#102—62no—
318#102—121no—
319#102—121no—
320#102—121no—
321#95—21no—
322#95—31no—
323#95—31no—
324#95—31no—
325#95—31no—
326#95—61no—
327#95—61no—
328#95—61no—
329#95—61no—
330#95—61no—
331C23×C3—22no—
332C23×C3—41no—
333C23×C3—41no—
334C23×C3—41no—
335C23×C3—41no—
336C23×C3—41no—
337C23×C3—41no—
338C23×C3C22×D411yescharacteristic
339C22×D3—22no—
340C22×D3—41no—
341C22×D3—41no—
342C22×D3—41no—
343C22×D3—41no—
344C22×D3—41no—
345C22×D3—41no—
346C3×D4—22no—
347C3×D4—22no—
348C3×D4—41no—
349C3×D4—41no—
350C2×C3×C4—22no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—84no—
366C2×C3×C4C22×Q811yescharacteristic
367C2×C3×C4C2×#4111yescharacteristic
368C2×C3×C4C2×#4111yescharacteristic
369C2×C3×C4C2×#4111yescharacteristic
370C2×C3×C4H(5,𝔽2)11yescharacteristic
371C2×C3×C4H(5,𝔽2)11yescharacteristic
372#67—84no—
373#67—84no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—21no—
380C2×Dic3—22no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3C22⋊Q812yes—
394C2×Dic3C22⋊Q812yes—
395D12—22no—
396D12—22no—
397D12—41no—
398D12—41no—
399C4×D3—84no—
400C3×C8—41no—
401C3×C8—41no—
402C3⋊C8—41no—
403C3⋊C8—41no—
404C24—21no—
405C24—61no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—32no—
412C2×D4—64no—
413C2×D4—64no—
414C2×D4—61no—
415C2×D4—61no—
416C2×D4—61no—
417C2×D4—61no—
418C2×D4—61no—
419C2×D4—241no—
420C2×D4—241no—
421C2×D4—241no—
422C2×D4—241no—
423C2×D4D3×Q812yes—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—31no—
428C22×C4—31no—
429C22×C4—31no—
430C22×C4—31no—
431C22×C4—42no—
432C22×C4—61no—
433C22×C4—61no—
434C22×C4—61no—
435C22×C4—81no—
436C22×C4—81no—
437C22×C4—122no—
438C22×C4—241no—
439C22×C4—241no—
440C22×C4D3×D411yescharacteristic
441C22×C4D3×D411yescharacteristic
442C22×C4C23×D311yescharacteristic
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—61no—
450C2×C8—61no—
451C2×C8—61no—
452C2×C8—61no—
453C2×C8—61no—
454C2×C8—61no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—32no—
461C4⋊C4—64no—
462C4⋊C4—64no—
463C4⋊C4—61no—
464C4⋊C4—61no—
465C4⋊C4—61no—
466C4⋊C4—61no—
467C4⋊C4—61no—
468C4⋊C4—81no—
469C4⋊C4—241no—
470C4⋊C4—241no—
471C4⋊C4—241no—
472C4⋊C4#23512yes—
473C22⋊C4—42no—
474C22⋊C4—122no—
475C22⋊C4—241no—
476C22⋊C4—241no—
477C22⋊C4—241no—
478C22⋊C4—241no—
479C42—21no—
480C42—21no—
481C42—32no—
482C42—32no—
483C42—61no—
484C42—61no—
485C42—241no—
486C42—241no—
487C22×C3—42no—
488C22×C3—42no—
489C22×C3—42no—
490C22×C3—81no—
491C22×C3—81no—
492C22×C3—81no—
493C22×C3—81no—
494C22×C3C2×C22⋊Q811yescharacteristic
495C22×C3#52511yescharacteristic
496C22×C3#53511yescharacteristic
497C22×C3#57611yescharacteristic
498C22×C3#57611yescharacteristic
499C22×C3#57711yescharacteristic
500C22×C3#57711yescharacteristic
501C2×D3—42no—
502C2×D3—42no—
503C2×D3—42no—
504C2×D3—81no—
505C2×D3—81no—
506C2×D3—81no—
507C2×D3—81no—
508C3×C4—42no—
509C3×C4—41no—
510C3×C4—41no—
511C3×C4—81no—
512C3×C4—81no—
513C3×C4#55411yescharacteristic
514C3×C4#55411yescharacteristic
515C3×C4#55911yescharacteristic
516C3×C4#55911yescharacteristic
517Dic3—22no—
518Dic3—22no—
519Dic3—41no—
520Dic3—41no—
521Dic3—42no—
522Dic3—81no—
523Dic3—81no—
524C23—22no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—62no—
532C23—121no—
533C23—121no—
534C23—121no—
535C23—121no—
536C23—121no—
537C23—121no—
538C23C2×D3×D411yescharacteristic
539D4—22no—
540D4—22no—
541D4—41no—
542D4—41no—
543D4—62no—
544D4—62no—
545D4—121no—
546D4—121no—
547D4—244no—
548D4—244no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—32no—
557C2×C4—32no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—61no—
567C2×C4—61no—
568C2×C4—61no—
569C2×C4—61no—
570C2×C4—61no—
571C2×C4—61no—
572C2×C4—61no—
573C2×C4—61no—
574C2×C4—61no—
575C2×C4—61no—
576C2×C4—62no—
577C2×C4—84no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—121no—
586C2×C4—244no—
587C2×C4C2×#23511yescharacteristic
588C2×C4#100411yescharacteristic
589C2×C4#100411yescharacteristic
590C2×C4C2×D3×Q811yescharacteristic
591C2×C4D3×#4111yescharacteristic
592C2×C4D3×#4111yescharacteristic
593C8—41no—
594C8—41no—
595C8—121no—
596C8—121no—
597C2×C3—82no—
598C2×C3#258011yescharacteristic
599C2×C3#308111yescharacteristic
600C2×C3#308111yescharacteristic
601C2×C3#311911yescharacteristic
602C2×C3#311911yescharacteristic
603C2×C3#322011yescharacteristic
604C2×C3#322211yescharacteristic
605D3—82no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—122no—
614C22—122no—
615C22—122no—
616C22—241no—
617C22—241no—
618C22—241no—
619C22—241no—
620C22#554711yescharacteristic
621C22#559211yescharacteristic
622C22D3×C22⋊Q811yescharacteristic
623C22#576211yescharacteristic
624C22#576911yescharacteristic
625C22#577411yescharacteristic
626C22#577511yescharacteristic
627C4—42no—
628C4—41no—
629C4—41no—
630C4—62no—
631C4—62no—
632C4—81no—
633C4—81no—
634C4—121no—
635C4—121no—
636C4—122no—
637C4—241no—
638C4—241no—
639C4#551111yescharacteristic
640C4#555911yescharacteristic
641C4#563611yescharacteristic
642C4#566411yescharacteristic
643C3#2926511yescharacteristic, complemented, Sylow
644C2—82no—
645C2—242no—
646C2#7448711yescharacteristic
647C2#7969711yescharacteristic
648C2#7987311yescharacteristic
649C2#8011811yescharacteristic
650C2#8014211yescharacteristic
651C2#8106211yescharacteristic
652C2#8116911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1864C2
socleC23×C3C22×D4
prime radicalC2×#940C22
hypercenter#1864D3
hypocenterC3#29265
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1864

Complements

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

NH ≅ G/Ntype
2. #75972C2semidirect
3. C3×#1864C2semidirect
4. #71607C2semidirect
5. #70587C2semidirect
6. #70564C2semidirect
7. #70183C2semidirect
8. #70164C2semidirect
11. #69625C2semidirect
12. #68899C2semidirect
13. #68798C2semidirect
14. #68670C2semidirect
15. #68606C2semidirect
71. #1864D3semidirect
643. C3#29265semidirect

Maximal subgroups

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

subgroupindex
#823292
#759722
C3×#18642
#716072
#705872
#705642
#701832
#701642
#701612
#696252
#688992
#687982
#686702
#686062
#292653