Finite Group Databasebeta

← #10877315

Subgroups — #10877315

Order 768.

788 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#76198C211yesmaximal, characteristic
2#74225C211yesmaximal, characteristic, complemented
3C3×#2101C211yesmaximal, characteristic, complemented
4#69611C211yesmaximal, characteristic, complemented
5#69558C211yesmaximal, characteristic, complemented
6#68893C211yesmaximal, characteristic, complemented
7#68892C211yesmaximal, characteristic, complemented
8#68881C211yesmaximal, characteristic
9#68880C211yesmaximal, characteristic
10#68876C211yesmaximal, characteristic, complemented
11#68875C211yesmaximal, characteristic, complemented
12#68797C211yesmaximal, characteristic, complemented
13#68796C211yesmaximal, characteristic, complemented
14#68682C211yesmaximal, characteristic, complemented
15#68663C211yesmaximal, characteristic, complemented
16#27688—31nomaximal, Sylow
17C2×#895C2211yescharacteristic
18C2×#893C2211yescharacteristic
19C2×#892C2211yescharacteristic
20C2×D3×C4⋊C4C2211yescharacteristic
21C2×#872C2211yescharacteristic
22C3×#389C2211yescharacteristic
23C3×#382C2211yescharacteristic
24C2×#821C2211yescharacteristic
25C2×#821C2211yescharacteristic
26C2×(C2×D3)⋊C8C2211yescharacteristic
27C2×(C2×D3)⋊C8C2211yescharacteristic
28C2×#817C2211yescharacteristic
29C2×#817C2211yescharacteristic
30C2×Dic3⋊C8C2211yescharacteristic
31C2×Dic3⋊C8C2211yescharacteristic
32#4989—21no—
33#4988—21no—
34#4987C2211yescharacteristic
35#4977C2211yescharacteristic
36#4975C2211yescharacteristic
37Dic3×C4⋊C4C2211yescharacteristic
38C2×#809C2211yescharacteristic
39C2×#809C2211yescharacteristic
40C2×#808C2211yescharacteristic
41C2×#807C2211yescharacteristic
42#4939C2211yescharacteristic
43#4935C2211yescharacteristic
44#4669—21no—
45#4668—21no—
46#4668—21no—
47#4665—21no—
48#4647—21no—
49#4647—21no—
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54C3×#336C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2324—31no—
61#2317—31no—
62#2101D311yescharacteristic, complemented
63#2099—31no—
64#2094—31no—
65#2092—31no—
66#1949—31no—
67#1948—31no—
68#1875—31no—
69#1874—31no—
70#1854—31no—
71#1853—31no—
72#1804—31no—
73#1802—31no—
74#1798—31no—
75C22×D12—21no—
76C22×D12C2311yescharacteristic
77C22×C4×D3—21no—
78C22×C4×D3—21no—
79C22×C4×D3C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C42C2311yescharacteristic
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4D411yescharacteristic
95C2×(C2×D3)⋊C4D411yescharacteristic
96C2×C4⋊Dic3—21no—
97C2×C4⋊Dic3—21no—
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4D411yescharacteristic
105C2×Dic3⋊C4D411yescharacteristic
106C2×C4×Dic3—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111#895—22no—
112#895Q814yes—
113#893—22no—
114#893—22no—
115#892—41no—
116#892—41no—
117D3×C4⋊C4—41no—
118D3×C4⋊C4—41no—
119#872—22no—
120#872—22no—
121C3×#95—21no—
122C3×#95—21no—
123C3×#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128#821—41no—
129#821—41no—
130(C2×D3)⋊C8—22no—
131(C2×D3)⋊C8—22no—
132#817—41no—
133#817—41no—
134Dic3⋊C8—22no—
135Dic3⋊C8—22no—
136#809—41no—
137#809—41no—
138#808—41no—
139#807—41no—
140C2×#121—31no—
141C2×C4×D4—31no—
142C2×#117—31no—
143C2×#117—31no—
144C22×C4⋊C4—31no—
145C2×#107—31no—
146C2×#106—31no—
147C2×#106—31no—
148C2×#106—31no—
149C2×C4⋊C8—31no—
150C2×C4⋊C8—31no—
151C2×#102—31no—
152C2×#102—31no—
153C2×#102—31no—
154C2×#102—31no—
155C2×C22⋊C8—31no—
156C2×C22⋊C8—31no—
157#390—31no—
158#390—61no—
159#389—61no—
160#389—61no—
161#389C2×D311yescharacteristic
162#388—61no—
163#386—61no—
164#385—61no—
165#385—61no—
166#385—61no—
167#384—31no—
168#384—31no—
169#382—31no—
170#382C2×D311yescharacteristic
171#380—31no—
172C4×C4⋊C4—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#340C2×D311yescharacteristic
178#340C2×D311yescharacteristic
179#340C2×D311yescharacteristic
180#340C2×D311yescharacteristic
181#336—31no—
182#336C2×D311yescharacteristic
183C23×D3—21no—
184C23×D3C2×D411yescharacteristic
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2411yescharacteristic
192C22×Dic3—21no—
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3C2×D411yescharacteristic
196C22×Dic3C2×D411yescharacteristic
197C22×Dic3C2×D411yescharacteristic
198C2×D12—21no—
199C2×D12—21no—
200C2×D12—21no—
201C2×D12—21no—
202C2×D12—21no—
203C2×D12—24no—
204C2×D12—81no—
205C2×D12—81no—
206C2×D12C2×Q812yes—
207C2×C4×D3—21no—
208C2×C4×D3—21no—
209C2×C4×D3—21no—
210C2×C4×D3—21no—
211C2×C4×D3—42no—
212C2×C4×D3—42no—
213C2×C4×D3—81no—
214C2×C4×D3—81no—
215C2×C4×D3—81no—
216C2×C4×D3—81no—
217C2×C4×D3C2×Q812yes—
218C2×C4×D3#4112yes—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—81no—
233C3×C4⋊C4C2×Q812yes—
234C3×C42—21no—
235C3×C42—21no—
236(C2×D3)⋊C4—24no—
237(C2×D3)⋊C4—22no—
238(C2×D3)⋊C4—22no—
239(C2×D3)⋊C4—42no—
240(C2×D3)⋊C4—42no—
241(C2×D3)⋊C4—81no—
242(C2×D3)⋊C4—81no—
243(C2×D3)⋊C4—81no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—81no—
251C4⋊Dic3—81no—
252Dic3⋊C4—22no—
253Dic3⋊C4—22no—
254Dic3⋊C4—42no—
255Dic3⋊C4—81no—
256Dic3⋊C4—81no—
257C4×Dic3—42no—
258C4×Dic3—81no—
259C4×Dic3#4112yes—
260C4×Dic3#4112yes—
261C2×C3⋊C8—21no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C22×D4—31no—
268C22×D4—61no—
269C23×C4—31no—
270C23×C4—61no—
271C23×C4—61no—
272C22×C8—31no—
273C22×C8—31no—
274C22×C8C22×D311yescharacteristic
275C22×C8C22×D311yescharacteristic
276#121—34no—
277#121—62no—
278C4×D4—62no—
279C4×D4—62no—
280#117—62no—
281#117—62no—
282#117—121no—
283#117—121no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—31no—
286C2×C4⋊C4—31no—
287C2×C4⋊C4—31no—
288C2×C4⋊C4—31no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—61no—
291C2×C4⋊C4—61no—
292C2×C4⋊C4—61no—
293C2×C4⋊C4—61no—
294C2×C4⋊C4—61no—
295C2×C4⋊C4—121no—
296C2×C4⋊C4—121no—
297C2×C4⋊C4C22×D311yescharacteristic
298C2×C4⋊C4C22×D311yescharacteristic
299C2×C4⋊C4C22×D311yescharacteristic
300C2×C4⋊C4C22×D311yescharacteristic
301C2×C22⋊C4—31no—
302C2×C22⋊C4—31no—
303C2×C22⋊C4—61no—
304C2×C22⋊C4—61no—
305C2×C22⋊C4—61no—
306C2×C22⋊C4—61no—
307C2×C22⋊C4—61no—
308C2×C22⋊C4—61no—
309C2×C42—31no—
310C2×C42—61no—
311C2×C42—61no—
312C2×C42C22×D311yescharacteristic
313#107—121no—
314#106—121no—
315#106—121no—
316#106—121no—
317C4⋊C8—62no—
318C4⋊C8—62no—
319#102—121no—
320#102—121no—
321#102—121no—
322#102—121no—
323C22⋊C8—62no—
324C22⋊C8—62no—
325#95—21no—
326#95—21no—
327#95—21no—
328#95—61no—
329#95—61no—
330#95—61no—
331#95—61no—
332C23×C3C22×D411yescharacteristic
333C22×D3—21no—
334C22×D3—21no—
335C22×D3—21no—
336C22×D3—21no—
337C22×D3—21no—
338C22×D3—21no—
339C22×D3—22no—
340C22×D3—41no—
341C22×D3—41no—
342C22×D3—41no—
343C22×D3—41no—
344C22×D3—41no—
345C22×D3—41no—
346C22×D3C22⋊Q812yes—
347C2×C3×C4—22no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
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×C4C22×Q811yescharacteristic
366C2×C3×C4C2×#4111yescharacteristic
367C2×C3×C4C2×#4111yescharacteristic
368C2×C3×C4C2×#4111yescharacteristic
369C2×C3×C4H(5,𝔽2)11yescharacteristic
370C2×C3×C4H(5,𝔽2)11yescharacteristic
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3C22⋊Q812yes—
387C2×Dic3#12312yes—
388C2×Dic3#12312yes—
389D12—22no—
390D12—22no—
391D12—41no—
392D12—41no—
393D12—84no—
394C4×D3—22no—
395C4×D3—22no—
396C4×D3—22no—
397C4×D3—22no—
398C4×D3—84no—
399C4×D3—84no—
400C3×C8—41no—
401C3×C8—41no—
402C3⋊C8—41no—
403C3⋊C8—41no—
404C24—31no—
405C24—61no—
406C2×D4—32no—
407C2×D4—64no—
408C2×D4—61no—
409C2×D4—61no—
410C2×D4—61no—
411C2×D4—61no—
412C2×D4—61no—
413C2×D4—241no—
414C2×D4—241no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—31no—
420C22×C4—31no—
421C22×C4—31no—
422C22×C4—32no—
423C22×C4—32no—
424C22×C4—61no—
425C22×C4—61no—
426C22×C4—61no—
427C22×C4—61no—
428C22×C4—61no—
429C22×C4—61no—
430C22×C4—61no—
431C22×C4—122no—
432C22×C4—122no—
433C22×C4—241no—
434C22×C4—241no—
435C22×C4—241no—
436C22×C4—241no—
437C22×C4D3×D411yescharacteristic
438C22×C4D3×D411yescharacteristic
439C22×C4C23×D311yescharacteristic
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—61no—
447C2×C8—61no—
448C2×C8—61no—
449C2×C8—61no—
450C2×C8—61no—
451C2×C8—61no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—62no—
460C4⋊C4—62no—
461C4⋊C4—61no—
462C4⋊C4—61no—
463C4⋊C4—61no—
464C4⋊C4—61no—
465C4⋊C4—61no—
466C4⋊C4—61no—
467C4⋊C4—81no—
468C4⋊C4—122no—
469C4⋊C4—241no—
470C4⋊C4—241no—
471C4⋊C4—241no—
472C4⋊C4—241no—
473C4⋊C4D3×Q812yes—
474C22⋊C4—62no—
475C22⋊C4—62no—
476C22⋊C4—64no—
477C22⋊C4—122no—
478C22⋊C4—122no—
479C22⋊C4—241no—
480C22⋊C4—241no—
481C22⋊C4—241no—
482C42—21no—
483C42—21no—
484C42—32no—
485C42—32no—
486C42—122no—
487C42—241no—
488C22×C3C2×C22⋊Q811yescharacteristic
489C22×C3C2×#12311yescharacteristic
490C22×C3#53511yescharacteristic
491C22×C3#57611yescharacteristic
492C22×C3#57711yescharacteristic
493C22×C3#57711yescharacteristic
494C22×C3#57711yescharacteristic
495C2×D3—22no—
496C2×D3—22no—
497C2×D3—22no—
498C2×D3—41no—
499C2×D3—41no—
500C2×D3—41no—
501C2×D3—41no—
502C2×D3—42no—
503C2×D3—42no—
504C2×D3—42no—
505C2×D3—81no—
506C2×D3—81no—
507C2×D3—81no—
508C2×D3—81no—
509C3×C4—42no—
510C3×C4—41no—
511C3×C4—41no—
512C3×C4—81no—
513C3×C4—81no—
514C3×C4—81no—
515C3×C4#55311yescharacteristic
516C3×C4#55311yescharacteristic
517C3×C4#55411yescharacteristic
518C3×C4#55411yescharacteristic
519Dic3—22no—
520Dic3—22no—
521Dic3—22no—
522Dic3—81no—
523Dic3—81no—
524Dic3—81no—
525C23—32no—
526C23—61no—
527C23—61no—
528C23—61no—
529C23—61no—
530C23—61no—
531C23—61no—
532C23—62no—
533C23—121no—
534C23—121no—
535C23—121no—
536C23—121no—
537C23—121no—
538C23—121no—
539C23C2×D3×D411yescharacteristic
540D4—62no—
541D4—62no—
542D4—121no—
543D4—121no—
544D4—244no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—32no—
553C2×C4—32no—
554C2×C4—32no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
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—62no—
573C2×C4—62no—
574C2×C4—62no—
575C2×C4—62no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—244no—
586C2×C4—244no—
587C2×C4C2×D3×Q811yescharacteristic
588C2×C4D3×#4111yescharacteristic
589C2×C4D3×#4111yescharacteristic
590C2×C4D3×#4111yescharacteristic
591C2×C4#100911yescharacteristic
592C2×C4#100911yescharacteristic
593C8—41no—
594C8—41no—
595C8—121no—
596C8—121no—
597C2×C3#248711yescharacteristic
598C2×C3#308111yescharacteristic
599C2×C3#308411yescharacteristic
600C2×C3#309611yescharacteristic
601C2×C3#309911yescharacteristic
602C2×C3#322111yescharacteristic
603C2×C3#322411yescharacteristic
604D3—42no—
605D3—82no—
606C22—62no—
607C22—62no—
608C22—62no—
609C22—121no—
610C22—121no—
611C22—121no—
612C22—121no—
613C22—122no—
614C22—122no—
615C22—122no—
616C22—241no—
617C22—241no—
618C22—241no—
619C22—241no—
620C22D3×C22⋊Q811yescharacteristic
621C22D3×#12311yescharacteristic
622C22#567411yescharacteristic
623C22#577011yescharacteristic
624C22#577011yescharacteristic
625C22#577411yescharacteristic
626C22#577811yescharacteristic
627C4—42no—
628C4—41no—
629C4—41no—
630C4—62no—
631C4—62no—
632C4—62no—
633C4—81no—
634C4—81no—
635C4—81no—
636C4—241no—
637C4—241no—
638C4—241no—
639C4#569011yescharacteristic
640C4#569011yescharacteristic
641C4#570911yescharacteristic
642C4#570911yescharacteristic
643C3#2768811yescharacteristic, complemented, Sylow
644C2—122no—
645C2—242no—
646C2#7814611yescharacteristic
647C2#8116811yescharacteristic
648C2#8123511yescharacteristic
649C2#8137511yescharacteristic
650C2#8152111yescharacteristic
651C2#8164511yescharacteristic
652C2#8168811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2101C2
socleC23×C3C22×D4
prime radicalC2×#895C22
hypercenter#2101D3
hypocenterC3#27688
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2101

Complements

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

NH ≅ G/Ntype
2. #74225C2semidirect
3. C3×#2101C2semidirect
4. #69611C2semidirect
5. #69558C2semidirect
6. #68893C2semidirect
7. #68892C2semidirect
10. #68876C2semidirect
11. #68875C2semidirect
12. #68797C2semidirect
13. #68796C2semidirect
14. #68682C2semidirect
15. #68663C2semidirect
62. #2101D3semidirect
643. C3#27688semidirect

Maximal subgroups

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

subgroupindex
#761982
#742252
C3×#21012
#696112
#695582
#688932
#688922
#688812
#688802
#688762
#688752
#687972
#687962
#686822
#686632
#276883