Finite Group Databasebeta

← #10836601

Subgroups — #10836601

Order 768.

794 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2629C211yesmaximal, characteristic, complemented
2C2×#5066C211yesmaximal, characteristic, complemented
3#82236C211yesmaximal, characteristic
4C2×#5000C211yesmaximal, characteristic, complemented
5#71738C211yesmaximal, characteristic, complemented
6#69621C211yesmaximal, characteristic, complemented
7#69618C211yesmaximal, characteristic, complemented
8#69617C211yesmaximal, characteristic, complemented
9#69615C211yesmaximal, characteristic, complemented
10#69584C211yesmaximal, characteristic, complemented
11#69568C211yesmaximal, characteristic, complemented
12#69524C211yesmaximal, characteristic
13#69524C211yesmaximal, characteristic
14#69520C211yesmaximal, characteristic, complemented
15#69518C211yesmaximal, characteristic, complemented
16#24717—31nomaximal, Sylow
17C2×C3×#125C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C2×#930C2211yescharacteristic
20C2×C4×D12C2211yescharacteristic
21C3×#398—21no—
22C3×#398—21no—
23C3×#396—21no—
24C3×#396—21no—
25C3×#389C2211yescharacteristic
26C3×#389C2211yescharacteristic
27C3×#384C2211yescharacteristic
28C3×#380C2211yescharacteristic
29C3×#380C2211yescharacteristic
30#5212C2211yescharacteristic
31C2×#830C2211yescharacteristic
32C2×#830C2211yescharacteristic
33#5066—24no—
34#5000—24no—
35#4990—21no—
36#4989—21no—
37C2×#809C2211yescharacteristic
38C2×#809C2211yescharacteristic
39C2×#809C2211yescharacteristic
40C2×#809C2211yescharacteristic
41C2×#808C2211yescharacteristic
42C2×#808C2211yescharacteristic
43C2×#808C2211yescharacteristic
44C2×#807C2211yescharacteristic
45#4957—21no—
46#4956—21no—
47#4939C2211yescharacteristic
48C4×C4⋊Dic3C2211yescharacteristic
49#4928C2211yescharacteristic
50C2×C4⋊(C3⋊C8)C2211yescharacteristic
51C2×C4⋊(C3⋊C8)C2211yescharacteristic
52C2×#803C2211yescharacteristic
53C2×C4×C3⋊C8C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#478—31no—
63C2×#433—31no—
64#2629D311yescharacteristic, complemented
65#2324—31no—
66#2077—31no—
67#2075—31no—
68#2034—31no—
69#2034—31no—
70#1989—31no—
71#1948—31no—
72#1944—31no—
73#1941—31no—
74#1936—31no—
75#1934—31no—
76#1802—31no—
77C23×C3×C4C2311yescharacteristic
78C22×#67—21no—
79C22×D12C2311yescharacteristic
80C22×C4×D3—21no—
81C3×#125—22no—
82C3×#125—22no—
83C3×#117—22no—
84C3×#117—22no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
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⋊C4—21no—
99C2×C3×C22⋊C4D411yescharacteristic
100C2×C3×C22⋊C4D411yescharacteristic
101C2×C3×C42C2311yescharacteristic
102C2×C3×C42C2311yescharacteristic
103C2×#217—21no—
104#930—22no—
105#930—22no—
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—22no—
131#809—22no—
132#809—22no—
133#809—22no—
134#808—22no—
135#808—22no—
136#808—22no—
137#807—22no—
138C4⋊(C3⋊C8)—22no—
139C4⋊(C3⋊C8)D414yescomplemented
140#803—22no—
141C4×C3⋊C8—22no—
142C2×#125C2×D311yescharacteristic
143C2×#121—31no—
144C2×C4×D4—31no—
145C2×#117C2×D311yescharacteristic
146#478—64no—
147#433—64no—
148C2×#107—31no—
149C2×#106—31no—
150C2×#106—31no—
151C2×#106—31no—
152C2×C4⋊C8—31no—
153C2×C4⋊C8—31no—
154C2×#102—31no—
155C2×#102—31no—
156C2×#102—31no—
157C2×#102—31no—
158C2×C22⋊C8—31no—
159C2×C22⋊C8—31no—
160C2×#97—31no—
161C2×C4×C8—31no—
162#398—21no—
163#398—21no—
164#396—21no—
165#396—21no—
166#390—31no—
167#390—61no—
168#389—31no—
169#389C2×D311yescharacteristic
170#389C2×D311yescharacteristic
171#388—61no—
172#386—61no—
173#385—61no—
174#384C2×D311yescharacteristic
175#380—31no—
176#380C2×D311yescharacteristic
177#380C2×D311yescharacteristic
178C4×C4⋊C4—31no—
179#340—31no—
180#340—31no—
181#340—31no—
182#340—31no—
183#340—31no—
184#340—31no—
185#340—31no—
186#340—31no—
187C24×C3C2×D411yescharacteristic
188C23×D3—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2411yescharacteristic
205C2×#67—24no—
206C2×#67—81no—
207C2×#67—81no—
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3—21no—
211C2×D12—21no—
212C2×D12—21no—
213C2×D12—21no—
214C2×D12—21no—
215C2×D12—21no—
216C2×D12#4112yes—
217C2×C4×D3—42no—
218C2×C4×D3—81no—
219C2×C4×D3—81no—
220C3×C4⋊C4—22no—
221C3×C4⋊C4—22no—
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—21no—
230C3×C4⋊C4—42no—
231C3×C4⋊C4—42no—
232C3×C4⋊C4—81no—
233C3×C4⋊C4—81no—
234C3×C4⋊C4—81no—
235C3×C4⋊C4—81no—
236C3×C22⋊C4—22no—
237C3×C22⋊C4—22no—
238C3×C22⋊C4—81no—
239C3×C22⋊C4—81no—
240C3×C42—21no—
241C3×C42—21no—
242C3×C42C2×D412yes—
243C3×C42#4112yes—
244#217—81no—
245(C2×D3)⋊C4—24no—
246(C2×D3)⋊C4—42no—
247(C2×D3)⋊C4—81no—
248(C2×D3)⋊C4—81no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3#4112yes—
255Dic3⋊C4—81no—
256C4×Dic3—81no—
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—
267C2×C3⋊C8C2×D412yes—
268C2×C3⋊C8C2×D412yes—
269C22×D4—31no—
270C22×D4—61no—
271C23×C4—61no—
272C23×C4C22×D311yescharacteristic
273C22×C8—31no—
274C22×C8—31no—
275C22×C8—31no—
276C22×C8—31no—
277#125—22no—
278#125—22no—
279#121—62no—
280#121—62no—
281C4×D4—62no—
282C4×D4—62no—
283#117—22no—
284#117—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—31no—
292C2×C4⋊C4—31no—
293C2×C4⋊C4—31no—
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—21no—
303C2×C22⋊C4—61no—
304C2×C22⋊C4—61no—
305C2×C22⋊C4—61no—
306C2×C22⋊C4—61no—
307C2×C22⋊C4—61no—
308C2×C22⋊C4#6711yescharacteristic
309C2×C22⋊C4#6711yescharacteristic
310C2×C42—61no—
311C2×C42C22×D311yescharacteristic
312C2×C42C22×D311yescharacteristic
313#107—62no—
314#106—62no—
315#106—62no—
316#106—62no—
317C4⋊C8—34no—
318C4⋊C8—62no—
319#102—62no—
320#102—62no—
321#102—62no—
322#102—62no—
323C22⋊C8—62no—
324C22⋊C8—62no—
325#97—62no—
326C4×C8—62no—
327#95—21no—
328#95—21no—
329#95—21no—
330#95—21no—
331#95—61no—
332#95—61no—
333#95—61no—
334C23×C3—21no—
335C23×C3—21no—
336C23×C3—21no—
337C23×C3—21no—
338C23×C3—21no—
339C23×C3—21no—
340C23×C3—21no—
341C23×C3C22×D411yescharacteristic
342C22×D3—22no—
343C22×D3—41no—
344C22×D3—41no—
345C22×D3—41no—
346C22×D3—41no—
347C22×D3—41no—
348C22×D3—41no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
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—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—41no—
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—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4#12112yes—
379C2×C3×C4#12112yes—
380C2×C3×C4C22×D411yescharacteristic
381C2×C3×C4C2×#4111yescharacteristic
382C2×C3×C4C2×#4111yescharacteristic
383C2×C3×C4C2×#4111yescharacteristic
384C2×C3×C42−1+411yescharacteristic
385C2×C3×C42−1+411yescharacteristic
386#67—84no—
387C2×Dic3—22no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396D12—22no—
397D12—22no—
398D12—41no—
399D12—41no—
400C4×D3—84no—
401C3⋊C8—22no—
402C3⋊C8—22no—
403C3⋊C8—41no—
404C3⋊C8—41no—
405C24—61no—
406C24C2×#6711yescharacteristic
407C2×D4—32no—
408C2×D4—64no—
409C2×D4—61no—
410C2×D4—61no—
411C2×D4—61no—
412C2×D4—61no—
413C2×D4—61no—
414C2×D4—241no—
415C2×D4—241no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—61no—
427C22×C4—61no—
428C22×C4—61no—
429C22×C4—122no—
430C22×C4—241no—
431C22×C4—241no—
432C22×C4C2×#6711yescharacteristic
433C22×C4C2×#6711yescharacteristic
434C22×C4C2×#6711yescharacteristic
435C22×C4C2×#6711yescharacteristic
436C22×C4C2×#6711yescharacteristic
437C22×C4C23×D311yescharacteristic
438C2×C8—32no—
439C2×C8—32no—
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—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—32no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—61no—
464C4⋊C4—61no—
465C4⋊C4—61no—
466C4⋊C4—61no—
467C4⋊C4—61no—
468C4⋊C4—81no—
469C4⋊C4—81no—
470C4⋊C4—81no—
471C4⋊C4—81no—
472C4⋊C4—241no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—64no—
476C22⋊C4—81no—
477C22⋊C4—81no—
478C22⋊C4—122no—
479C22⋊C4—241no—
480C22⋊C4—241no—
481C22⋊C4—241no—
482C42—21no—
483C42—21no—
484C42—241no—
485C42D3×D412yes—
486C42#23712yes—
487C22×C3—41no—
488C22×C3—41no—
489C22×C3—41no—
490C22×C3—41no—
491C22×C3—41no—
492C22×C3—41no—
493C22×C3—41no—
494C22×C3C2×#12111yescharacteristic
495C22×C3#52511yescharacteristic
496C22×C3#53611yescharacteristic
497C22×C3#57611yescharacteristic
498C22×C3#57711yescharacteristic
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—22no—
509C3×C4—22no—
510C3×C4—41no—
511C3×C4—41no—
512C3×C4—41no—
513C3×C4—81no—
514C3×C4—81no—
515C3×C4—81no—
516C3×C4—81no—
517C3×C4#54711yescharacteristic
518C3×C4#54811yescharacteristic
519C3×C4#55211yescharacteristic
520C3×C4#55611yescharacteristic
521Dic3—42no—
522Dic3—81no—
523Dic3—81no—
524C23—21no—
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—
538C23C22×#6711yescharacteristic
539D4—62no—
540D4—62no—
541D4—121no—
542D4—121no—
543D4—244no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
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—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—62no—
574C2×C4—121no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—244no—
583C2×C4#93912yes—
584C2×C4#93912yes—
585C2×C4C2×D3×D411yescharacteristic
586C2×C4C2×#23711yescharacteristic
587C2×C4#100711yescharacteristic
588C2×C4#100711yescharacteristic
589C2×C4D3×#4111yescharacteristic
590C2×C4D3×#4111yescharacteristic
591C8—62no—
592C8—62no—
593C8—121no—
594C8—121no—
595C2×C3—81no—
596C2×C3#241511yescharacteristic
597C2×C3#306011yescharacteristic
598C2×C3#306211yescharacteristic
599C2×C3#311911yescharacteristic
600C2×C3#312311yescharacteristic
601C2×C3#324111yescharacteristic
602C2×C3#324411yescharacteristic
603D3—82no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—122no—
612C22—122no—
613C22—122no—
614C22—241no—
615C22—241no—
616C22—241no—
617C22—241no—
618C22C2×#93911yescharacteristic
619C22#581611yescharacteristic
620C22#582811yescharacteristic
621C22#583511yescharacteristic
622C22#583611yescharacteristic
623C22#583711yescharacteristic
624C22#583711yescharacteristic
625C4—22no—
626C4—22no—
627C4—41no—
628C4—41no—
629C4—41no—
630C4—81no—
631C4—81no—
632C4—81no—
633C4—81no—
634C4—122no—
635C4—241no—
636C4—241no—
637C4#563111yescharacteristic
638C4#563311yescharacteristic
639C4#563911yescharacteristic
640C4#564111yescharacteristic
641C3#2471711yescharacteristic, complemented, Sylow
642C2—81no—
643C2—242no—
644C2#8281411yescharacteristic
645C2#8342011yescharacteristic
646C2#8342211yescharacteristic
647C2#8343411yescharacteristic
648C2#8344711yescharacteristic
649C2#8347711yescharacteristic
650C2#8347911yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2629C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenter#2629D3
hypocenterC3#24717
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2629

Complements

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

NH ≅ G/Ntype
1. C3×#2629C2semidirect
2. C2×#5066C2semidirect
4. C2×#5000C2semidirect
5. #71738C2semidirect
6. #69621C2semidirect
7. #69618C2semidirect
8. #69617C2semidirect
9. #69615C2semidirect
10. #69584C2semidirect
11. #69568C2semidirect
14. #69520C2semidirect
15. #69518C2semidirect
64. #2629D3semidirect
139. C4⋊(C3⋊C8)D4semidirect
641. C3#24717semidirect

Maximal subgroups

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

subgroupindex
C3×#26292
C2×#50662
#822362
C2×#50002
#717382
#696212
#696182
#696172
#696152
#695842
#695682
#695242
#695202
#695182
#247173