Finite Group Databasebeta

← #10877306

Subgroups — #10877306

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
1C3×#2317C211yesmaximal, characteristic, complemented
2#82236C211yesmaximal, characteristic
3#71586C211yesmaximal, characteristic
4#71583C211yesmaximal, characteristic, complemented
5#71576C211yesmaximal, characteristic, complemented
6#71560C211yesmaximal, characteristic
7#71556C211yesmaximal, characteristic, complemented
8#69587C211yesmaximal, characteristic, complemented
9#69584C211yesmaximal, characteristic, complemented
10#69571C211yesmaximal, characteristic, complemented
11#69568C211yesmaximal, characteristic, complemented
12#69558C211yesmaximal, characteristic, complemented
13#69556C211yesmaximal, characteristic, complemented
14#69542C211yesmaximal, characteristic, complemented
15#69540C211yesmaximal, characteristic, complemented
16#27688—31nomaximal, Sylow
17C2×C3×#117C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C22×C3×C4⋊C4C2211yescharacteristic
20C2×#930C2211yescharacteristic
21C2×C4×D12C2211yescharacteristic
22C3×#389—21no—
23C3×#389—21no—
24C3×#385—21no—
25C3×#385—21no—
26C3×#384C2211yescharacteristic
27C3×#384C2211yescharacteristic
28C3×#382C2211yescharacteristic
29C3×#382C2211yescharacteristic
30#5212C2211yescharacteristic
31C2×#830C2211yescharacteristic
32C2×#830C2211yescharacteristic
33#4990—21no—
34#4989—21no—
35C2×#809C2211yescharacteristic
36C2×#809C2211yescharacteristic
37C2×#809C2211yescharacteristic
38C2×#809C2211yescharacteristic
39C2×#808C2211yescharacteristic
40C2×#808C2211yescharacteristic
41C2×#808C2211yescharacteristic
42C2×#807C2211yescharacteristic
43#4957—21no—
44#4956—21no—
45#4939C2211yescharacteristic
46C4×C4⋊Dic3C2211yescharacteristic
47#4928C2211yescharacteristic
48C2×C4⋊(C3⋊C8)C2211yescharacteristic
49C2×C4⋊(C3⋊C8)C2211yescharacteristic
50#4529C2211yescharacteristic
51#4529C2211yescharacteristic
52#4529C2211yescharacteristic
53#4529C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4524C2211yescharacteristic
59#4524C2211yescharacteristic
60#2324—31no—
61#2317D311yescharacteristic, complemented
62#2101—31no—
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—
75C23×C3×C4—21no—
76C23×C3×C4C2311yescharacteristic
77C22×#67—21no—
78C22×D12C2311yescharacteristic
79C22×C4×D3—21no—
80C3×#117—22no—
81C3×#117—22no—
82C3×#117—41no—
83C3×#117—41no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—41no—
90C2×C3×C4⋊C4—41no—
91C2×C3×C4⋊C4D411yescharacteristic
92C2×C3×C4⋊C4D411yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C22⋊C4—21no—
98C2×C3×C22⋊C4D411yescharacteristic
99C2×C3×C22⋊C4D411yescharacteristic
100C2×C3×C42—21no—
101C2×C3×C42C2311yescharacteristic
102C2×C3×C42C2311yescharacteristic
103C2×#217—21no—
104#930—22no—
105#930Q814yes—
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4—21no—
108C2×(C2×D3)⋊C4—21no—
109C2×(C2×D3)⋊C4—21no—
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×Dic3⋊C4—21no—
114C2×C4×Dic3—21no—
115C22×C3⋊C8C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119C4×D12—22no—
120C4×D12—22no—
121C3×#95—21no—
122C3×#95—21no—
123C3×#95—21no—
124C3×#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128#830—22no—
129#830—22no—
130#809—41no—
131#809—41no—
132#809—41no—
133#809—41no—
134#808—41no—
135#808—41no—
136#808—41no—
137#807—41no—
138C4⋊(C3⋊C8)—22no—
139C4⋊(C3⋊C8)—22no—
140C2×#121—31no—
141C2×C4×D4—31no—
142C2×#117C2×D311yescharacteristic
143C2×#117C2×D311yescharacteristic
144C22×C4⋊C4C2×D311yescharacteristic
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—21no—
160#389—21no—
161#389—31no—
162#388—61no—
163#386—61no—
164#385—21no—
165#385—21no—
166#385—61no—
167#384C2×D311yescharacteristic
168#384C2×D311yescharacteristic
169#382C2×D311yescharacteristic
170#382C2×D311yescharacteristic
171#380—31no—
172C4×C4⋊C4—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#340—31no—
178#340—31no—
179#340—31no—
180#340—31no—
181#336—31no—
182#336—31no—
183C24×C3C2×D411yescharacteristic
184C23×D3—21no—
185C22×C3×C4—21no—
186C22×C3×C4—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—42no—
194C22×C3×C4—81no—
195C22×C3×C4—81no—
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×Q812yes—
202C22×C3×C4#4112yes—
203C22×C3×C4C2411yescharacteristic
204C2×#67—24no—
205C2×#67—81no—
206C2×#67—81no—
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3—21no—
210C2×D12—21no—
211C2×D12—21no—
212C2×D12—21no—
213C2×D12—21no—
214C2×D12—21no—
215C2×D12C2×Q812yes—
216C2×C4×D3—42no—
217C2×C4×D3—81no—
218C2×C4×D3—81no—
219C3×C4⋊C4—22no—
220C3×C4⋊C4—22no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—42no—
230C3×C4⋊C4—81no—
231C3×C4⋊C4—81no—
232C3×C4⋊C4—81no—
233C3×C4⋊C4—81no—
234C3×C22⋊C4—22no—
235C3×C22⋊C4—22no—
236C3×C22⋊C4—42no—
237C3×C42—21no—
238C3×C42—21no—
239C3×C42—42no—
240C3×C42#4112yes—
241C3×C42#4112yes—
242#217—81no—
243(C2×D3)⋊C4—24no—
244(C2×D3)⋊C4—42no—
245(C2×D3)⋊C4—81no—
246(C2×D3)⋊C4—81no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3C2×Q812yes—
253Dic3⋊C4—81no—
254C4×Dic3—81no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
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—21no—
270C23×C4—61no—
271C23×C4C22×D311yescharacteristic
272C22×C8—31no—
273C22×C8—31no—
274C22×C8—31no—
275C22×C8—31no—
276#121—34no—
277#121—62no—
278C4×D4—62no—
279C4×D4—62no—
280#117—22no—
281#117—22no—
282#117—41no—
283#117—41no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4—31no—
292C2×C4⋊C4—41no—
293C2×C4⋊C4—41no—
294C2×C4⋊C4—61no—
295C2×C4⋊C4#6711yescharacteristic
296C2×C4⋊C4#6711yescharacteristic
297C2×C4⋊C4C22×D311yescharacteristic
298C2×C4⋊C4C22×D311yescharacteristic
299C2×C4⋊C4C22×D311yescharacteristic
300C2×C4⋊C4C22×D311yescharacteristic
301C2×C22⋊C4—21no—
302C2×C22⋊C4—61no—
303C2×C22⋊C4—61no—
304C2×C22⋊C4—61no—
305C2×C22⋊C4—61no—
306C2×C22⋊C4—61no—
307C2×C22⋊C4#6711yescharacteristic
308C2×C22⋊C4#6711yescharacteristic
309C2×C42—21no—
310C2×C42—61no—
311C2×C42C22×D311yescharacteristic
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—21no—
329#95—61no—
330#95—61no—
331#95—61no—
332C23×C3—21no—
333C23×C3—21no—
334C23×C3—21no—
335C23×C3—21no—
336C23×C3—21no—
337C23×C3—21no—
338C23×C3C22⋊Q812yes—
339C23×C3C22×D411yescharacteristic
340C22×D3—22no—
341C22×D3—41no—
342C22×D3—41no—
343C22×D3—41no—
344C22×D3—41no—
345C22×D3—41no—
346C22×D3—41no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
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—41no—
375C2×C3×C4—84no—
376C2×C3×C4C22⋊Q812yes—
377C2×C3×C4#12312yes—
378C2×C3×C4#12312yes—
379C2×C3×C4C22×Q811yescharacteristic
380C2×C3×C4C2×#4111yescharacteristic
381C2×C3×C4C2×#4111yescharacteristic
382C2×C3×C4C2×#4111yescharacteristic
383C2×C3×C4H(5,𝔽2)11yescharacteristic
384C2×C3×C4H(5,𝔽2)11yescharacteristic
385#67—84no—
386C2×Dic3—22no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395D12—22no—
396D12—22no—
397D12—41no—
398D12—41no—
399C4×D3—84no—
400C3⋊C8—41no—
401C3⋊C8—41no—
402C3⋊C8—41no—
403C3⋊C8—41no—
404C24—61no—
405C24C2×#6711yescharacteristic
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—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—42no—
424C22×C4—61no—
425C22×C4—61no—
426C22×C4—61no—
427C22×C4—81no—
428C22×C4—81no—
429C22×C4—122no—
430C22×C4—241no—
431C22×C4—241no—
432C22×C4D3×Q812yes—
433C22×C4#23912yes—
434C22×C4C2×#6711yescharacteristic
435C22×C4C2×#6711yescharacteristic
436C22×C4C2×#6711yescharacteristic
437C22×C4C2×#6711yescharacteristic
438C22×C4C2×#6711yescharacteristic
439C22×C4C23×D311yescharacteristic
440C2×C8—61no—
441C2×C8—61no—
442C2×C8—61no—
443C2×C8—61no—
444C2×C8—61no—
445C2×C8—61no—
446C2×C8—61no—
447C2×C8—61no—
448C2×C8—61no—
449C2×C8—61no—
450C2×C8—61no—
451C2×C8—61no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—32no—
463C4⋊C4—42no—
464C4⋊C4—61no—
465C4⋊C4—61no—
466C4⋊C4—61no—
467C4⋊C4—61no—
468C4⋊C4—61no—
469C4⋊C4—81no—
470C4⋊C4—81no—
471C4⋊C4—81no—
472C4⋊C4—81no—
473C4⋊C4—241no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—42no—
477C22⋊C4—64no—
478C22⋊C4—122no—
479C22⋊C4—241no—
480C22⋊C4—241no—
481C22⋊C4—241no—
482C42—21no—
483C42—21no—
484C42—42no—
485C42—241no—
486C42#23512yes—
487C42#23512yes—
488C22×C3—22no—
489C22×C3—22no—
490C22×C3—22no—
491C22×C3—41no—
492C22×C3—41no—
493C22×C3—41no—
494C22×C3—41no—
495C22×C3C2×C22⋊Q811yescharacteristic
496C22×C3C2×#12311yescharacteristic
497C22×C3#53511yescharacteristic
498C22×C3#57611yescharacteristic
499C22×C3#57711yescharacteristic
500C22×C3#57711yescharacteristic
501C22×C3#57711yescharacteristic
502C2×D3—42no—
503C2×D3—42no—
504C2×D3—42no—
505C2×D3—81no—
506C2×D3—81no—
507C2×D3—81no—
508C2×D3—81no—
509C3×C4—22no—
510C3×C4—22no—
511C3×C4—22no—
512C3×C4—41no—
513C3×C4—41no—
514C3×C4—81no—
515C3×C4—81no—
516C3×C4—81no—
517C3×C4—81no—
518C3×C4#55311yescharacteristic
519C3×C4#55311yescharacteristic
520C3×C4#55411yescharacteristic
521C3×C4#55411yescharacteristic
522Dic3—42no—
523Dic3—81no—
524Dic3—81no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—62no—
532C23—121no—
533C23—121no—
534C23—121no—
535C23—121no—
536C23—121no—
537C23—121no—
538C23#94612yes—
539C23C22×#6711yescharacteristic
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—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—62no—
574C2×C4—84no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—244no—
584C2×C4#92912yes—
585C2×C4#92912yes—
586C2×C4#94612yes—
587C2×C4C2×#23511yescharacteristic
588C2×C4C2×#23511yescharacteristic
589C2×C4#100411yescharacteristic
590C2×C4#100411yescharacteristic
591C2×C4C2×D3×Q811yescharacteristic
592C2×C4C2×#23911yescharacteristic
593C8—121no—
594C8—121no—
595C8—121no—
596C8—121no—
597C2×C3—42no—
598C2×C3#248711yescharacteristic
599C2×C3#308111yescharacteristic
600C2×C3#308411yescharacteristic
601C2×C3#309611yescharacteristic
602C2×C3#309911yescharacteristic
603C2×C3#322111yescharacteristic
604C2×C3#322411yescharacteristic
605D3—82no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—122no—
614C22—122no—
615C22—122no—
616C22—241no—
617C22—241no—
618C22—241no—
619C22—241no—
620C22C2×#92911yescharacteristic
621C22#580511yescharacteristic
622C22C2×#94611yescharacteristic
623C22#583511yescharacteristic
624C22#583611yescharacteristic
625C22#583711yescharacteristic
626C22#583711yescharacteristic
627C4—22no—
628C4—22no—
629C4—22no—
630C4—41no—
631C4—41no—
632C4—81no—
633C4—81no—
634C4—81no—
635C4—81no—
636C4—122no—
637C4—241no—
638C4—241no—
639C4#551111yescharacteristic
640C4#551111yescharacteristic
641C4#551411yescharacteristic
642C4#551411yescharacteristic
643C3#2768811yescharacteristic, complemented, Sylow
644C2—42no—
645C2—242no—
646C2#8219711yescharacteristic
647C2#8256611yescharacteristic
648C2#8257411yescharacteristic
649C2#8338911yescharacteristic
650C2#8340911yescharacteristic
651C2#8347411yescharacteristic
652C2#8347611yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2317C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenter#2317D3
hypocenterC3#27688
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2317

Complements

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

NH ≅ G/Ntype
1. C3×#2317C2semidirect
4. #71583C2semidirect
5. #71576C2semidirect
7. #71556C2semidirect
8. #69587C2semidirect
9. #69584C2semidirect
10. #69571C2semidirect
11. #69568C2semidirect
12. #69558C2semidirect
13. #69556C2semidirect
14. #69542C2semidirect
15. #69540C2semidirect
61. #2317D3semidirect
643. C3#27688semidirect

Maximal subgroups

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

subgroupindex
C3×#23172
#822362
#715862
#715832
#715762
#715602
#715562
#695872
#695842
#695712
#695682
#695582
#695562
#695422
#695402
#276883