Finite Group Databasebeta

← #10855128

Subgroups — #10855128

Order 768.

896 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2352C211yesmaximal, characteristic, complemented
2#82350C211yesmaximal, characteristic
3C2×#5003C211yesmaximal, characteristic, complemented
4C2×#4964C211yesmaximal, characteristic, complemented
5#71611C211yesmaximal, characteristic, complemented
6#71576C211yesmaximal, characteristic, complemented
7#71560C211yesmaximal, characteristic, complemented
8#69558C211yesmaximal, characteristic, complemented
9#69541C211yesmaximal, characteristic, complemented
10#69523C211yesmaximal, characteristic, complemented
11#69517C211yesmaximal, characteristic, complemented
12#69507C211yesmaximal, characteristic, complemented
13#69506C211yesmaximal, characteristic, complemented
14#69463C211yesmaximal, characteristic, complemented
15#69458C211yesmaximal, characteristic, complemented
16#26156—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C22×C3×C4⋊C4C2211yescharacteristic
20C2×#930C2211yescharacteristic
21C2×#930C2211yescharacteristic
22C2×#926C2211yescharacteristic
23C22×#208C2211yescharacteristic
24C3×#388—21no—
25C3×#387—21no—
26C3×#386—21no—
27C3×#382C2211yescharacteristic
28C3×#380C2211yescharacteristic
29C3×#380C2211yescharacteristic
30C3×C4×C4⋊C4C2211yescharacteristic
31C3×C4×C22⋊C4—21no—
32#5223—21no—
33#5212C2211yescharacteristic
34C2×#830C2211yescharacteristic
35C2×#830C2211yescharacteristic
36C2×#830C2211yescharacteristic
37#5003—24no—
38#4990—21no—
39C2×#809C2211yescharacteristic
40C2×#809C2211yescharacteristic
41C2×#809C2211yescharacteristic
42C2×#809C2211yescharacteristic
43#4964—24no—
44C2×#808C2211yescharacteristic
45C2×#808C2211yescharacteristic
46C2×#808C2211yescharacteristic
47C2×#807C2211yescharacteristic
48C2×#807C2211yescharacteristic
49C2×#807C2211yescharacteristic
50#4959—21no—
51#4957—21no—
52#4939C2211yescharacteristic
53#4928C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4524C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#482—31no—
63C2×#428—31no—
64#2414—31no—
65#2352D311yescharacteristic, complemented
66#2092—31no—
67#2085—31no—
68#2077—31no—
69#2033—31no—
70#1926—31no—
71#1918—31no—
72#1895—31no—
73#1874—31no—
74#1870—31no—
75#1854—31no—
76#1854—31no—
77C22×C3×D4D411yescharacteristic
78C23×C3×C4—21no—
79C23×C3×C4C2311yescharacteristic
80C23×C3×C4C2311yescharacteristic
81C22×#67—21no—
82C22×#67—21no—
83C22×D12C2311yescharacteristic
84C3×C4×D4—22no—
85C3×C4×D4—22no—
86C3×#117—22no—
87C3×#117—41no—
88C2×C3×C4⋊C4—22no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—41no—
91C2×C3×C4⋊C4D411yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—21no—
98C2×C3×C22⋊C4—21no—
99C2×C3×C22⋊C4D411yescharacteristic
100C2×C3×C22⋊C4D411yescharacteristic
101C2×C3×C42—21no—
102C2×C3×C42—21no—
103C2×C3×C42C2311yescharacteristic
104C2×#217—21no—
105C2×#217—21no—
106#930—22no—
107#930—22no—
108#930—22no—
109#930—41no—
110C2×(C2×D3)⋊C4—21no—
111C2×(C2×D3)⋊C4—21no—
112C2×(C2×D3)⋊C4—21no—
113C2×(C2×D3)⋊C4—21no—
114#926—22no—
115#926—41no—
116C2×C4⋊Dic3C2311yescharacteristic
117C2×C4⋊Dic3C2311yescharacteristic
118C2×C4⋊Dic3C2311yescharacteristic
119C2×Dic3⋊C4—21no—
120C2×C4×Dic3—21no—
121C2×#208—22no—
122C2×#208—22no—
123C2×#208—22no—
124C22×C3⋊C8C2311yescharacteristic
125C22×C3⋊C8C2311yescharacteristic
126C22×C3⋊C8C2311yescharacteristic
127C22×C3⋊C8C2311yescharacteristic
128C3×#95—21no—
129C3×#95—21no—
130C3×#95—21no—
131C3×#95—21no—
132C3×#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135#830—22no—
136#830—22no—
137#830—22no—
138#809—22no—
139#809—22no—
140#809—22no—
141#809—41no—
142#808—22no—
143#808—22no—
144#808—22no—
145#807—22no—
146#807—41no—
147#807D414yescomplemented
148C22×M16—31no—
149C2×#121—31no—
150C2×#121—31no—
151C2×C4×D4C2×D311yescharacteristic
152C2×#117—31no—
153C2×#117C2×D311yescharacteristic
154C22×C4⋊C4C2×D311yescharacteristic
155#482—64no—
156#428—64no—
157C2×#107—31no—
158C2×#107—31no—
159C2×#107—31no—
160C2×#106—31no—
161C2×#106—31no—
162C2×#106—31no—
163C2×#102—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×C22⋊C8—31no—
168C2×C22⋊C8—31no—
169C2×C22⋊C8—31no—
170#394—61no—
171#392—61no—
172#390—31no—
173#390—61no—
174#389—31no—
175#388—21no—
176#388—61no—
177#387—21no—
178#386—21no—
179#382C2×D311yescharacteristic
180#380—31no—
181#380C2×D311yescharacteristic
182#380C2×D311yescharacteristic
183C4×C4⋊C4C2×D311yescharacteristic
184C4×C22⋊C4—21no—
185#340—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#336—31no—
192#336—31no—
193C24×C3C2×D411yescharacteristic
194C24×C3C2×D411yescharacteristic
195C23×D3—21no—
196C2×C3×D4—22no—
197C2×C3×D4—41no—
198C2×C3×D4—41no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—42no—
214C22×C3×C4—42no—
215C22×C3×C4—81no—
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4#4112yes—
221C22×C3×C4#4112yes—
222C22×C3×C4C2411yescharacteristic
223C2×#67—42no—
224C2×#67—42no—
225C2×#67—42no—
226C2×#67—81no—
227C2×#67—81no—
228C2×#67—81no—
229C22×Dic3—21no—
230C22×Dic3—21no—
231C22×Dic3—21no—
232C2×D12—21no—
233C2×D12—21no—
234C2×D12—21no—
235C2×D12—21no—
236C2×D12—21no—
237C2×D12—21no—
238C3×C4⋊C4—22no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—42no—
247C3×C4⋊C4C2×D412yes—
248C3×C22⋊C4—22no—
249C3×C22⋊C4—22no—
250C3×C22⋊C4—42no—
251C3×C22⋊C4—42no—
252C3×C22⋊C4—81no—
253C3×C42—21no—
254C3×C42—21no—
255C3×C42—42no—
256C3×C42—81no—
257#217—42no—
258#217—81no—
259(C2×D3)⋊C4—42no—
260(C2×D3)⋊C4—42no—
261(C2×D3)⋊C4—42no—
262(C2×D3)⋊C4—81no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3C2×D412yes—
269Dic3⋊C4—81no—
270C4×Dic3—42no—
271#208—28no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8C2×D412yes—
283C2×C3⋊C8#4112yes—
284C22×D4—31no—
285C22×D4—61no—
286C22×D4—61no—
287C22×D4#6711yescharacteristic
288C23×C4—21no—
289C23×C4C22×D311yescharacteristic
290C23×C4C22×D311yescharacteristic
291C2×M16—62no—
292C2×M16—62no—
293C2×M16—62no—
294C22×C8—31no—
295C22×C8—31no—
296C22×C8—31no—
297C22×C8—31no—
298#121—62no—
299#121—62no—
300#121—62no—
301#121—121no—
302C4×D4—22no—
303C4×D4—22no—
304#117—22no—
305#117—41no—
306#117—62no—
307#117—121no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—41no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4#6711yescharacteristic
316C2×C4⋊C4C22×D311yescharacteristic
317C2×C4⋊C4C22×D311yescharacteristic
318C2×C4⋊C4C22×D311yescharacteristic
319C2×C4⋊C4C22×D311yescharacteristic
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—61no—
324C2×C22⋊C4—61no—
325C2×C22⋊C4—61no—
326C2×C22⋊C4—61no—
327C2×C22⋊C4—61no—
328C2×C22⋊C4—61no—
329C2×C22⋊C4#6711yescharacteristic
330C2×C22⋊C4#6711yescharacteristic
331C2×C42—21no—
332C2×C42—21no—
333C2×C42—61no—
334C2×C42C22×D311yescharacteristic
335#107—34no—
336#107—62no—
337#107—121no—
338#106—62no—
339#106—62no—
340#106—62no—
341#102—62no—
342#102—62no—
343#102—62no—
344#102—121no—
345C22⋊C8—62no—
346C22⋊C8—62no—
347C22⋊C8—62no—
348#95—21no—
349#95—21no—
350#95—21no—
351#95—21no—
352#95—21no—
353#95—61no—
354#95—61no—
355C23×C3—21no—
356C23×C3—21no—
357C23×C3—21no—
358C23×C3—21no—
359C23×C3—21no—
360C23×C3—21no—
361C23×C3—21no—
362C23×C3—21no—
363C23×C3—21no—
364C23×C3—21no—
365C23×C3—21no—
366C23×C3—21no—
367C23×C3—21no—
368C23×C3#12312yes—
369C23×C3C22×D411yescharacteristic
370C22×D3—41no—
371C22×D3—41no—
372C22×D3—41no—
373C22×D3—41no—
374C22×D3—41no—
375C22×D3—41no—
376C22×D3—41no—
377C3×D4—44no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—22no—
381C2×C3×C4—22no—
382C2×C3×C4—22no—
383C2×C3×C4—22no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—22no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4—41no—
402C2×C3×C4—41no—
403C2×C3×C4—41no—
404C2×C3×C4—41no—
405C2×C3×C4—41no—
406C2×C3×C4—41no—
407C2×C3×C4—41no—
408C2×C3×C4—41no—
409C2×C3×C4—84no—
410C2×C3×C4#12312yes—
411C2×C3×C4C22×D411yescharacteristic
412C2×C3×C4C2×#4111yescharacteristic
413C2×C3×C4C2×#4111yescharacteristic
414C2×C3×C4C2×#4111yescharacteristic
415C2×C3×C4H(5,𝔽2)11yescharacteristic
416C2×C3×C42−1+411yescharacteristic
417#67—84no—
418#67—84no—
419C2×Dic3—22no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3—41no—
426C2×Dic3—41no—
427C2×Dic3—41no—
428D12—41no—
429D12—41no—
430D12—41no—
431D12—41no—
432C3⋊C8—22no—
433C3⋊C8—22no—
434C3⋊C8—41no—
435C3⋊C8—41no—
436C24—61no—
437C24C2×#6711yescharacteristic
438C24C2×#6711yescharacteristic
439C2×D4—22no—
440C2×D4—41no—
441C2×D4—41no—
442C2×D4—61no—
443C2×D4—61no—
444C2×D4—61no—
445C2×D4—61no—
446C2×D4—61no—
447C2×D4—61no—
448C2×D4—122no—
449C2×D4—122no—
450C2×D4—122no—
451C2×D4—241no—
452C2×D4—241no—
453C2×D4—241no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—61no—
471C22×C4—61no—
472C22×C4—61no—
473C22×C4—81no—
474C22×C4#23512yes—
475C22×C4#23512yes—
476C22×C4C2×#6711yescharacteristic
477C22×C4C2×#6711yescharacteristic
478C22×C4C2×#6711yescharacteristic
479C22×C4C2×#6711yescharacteristic
480C22×C4C23×D311yescharacteristic
481M16—68no—
482C2×C8—32no—
483C2×C8—32no—
484C2×C8—61no—
485C2×C8—61no—
486C2×C8—61no—
487C2×C8—61no—
488C2×C8—61no—
489C2×C8—61no—
490C2×C8—61no—
491C2×C8—61no—
492C2×C8—61no—
493C2×C8—61no—
494C4⋊C4—22no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—32no—
503C4⋊C4—42no—
504C4⋊C4—61no—
505C4⋊C4—61no—
506C4⋊C4—61no—
507C4⋊C4—61no—
508C4⋊C4—61no—
509C4⋊C4—241no—
510C4⋊C4D3×D412yes—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—42no—
514C22⋊C4—42no—
515C22⋊C4—81no—
516C22⋊C4—122no—
517C22⋊C4—122no—
518C22⋊C4—122no—
519C22⋊C4—122no—
520C22⋊C4—241no—
521C22⋊C4—241no—
522C42—21no—
523C42—21no—
524C42—42no—
525C42—81no—
526C42—122no—
527C22×C3—22no—
528C22×C3—22no—
529C22×C3—22no—
530C22×C3—41no—
531C22×C3—41no—
532C22×C3—41no—
533C22×C3—41no—
534C22×C3—41no—
535C22×C3—41no—
536C22×C3—41no—
537C22×C3—41no—
538C22×C3—41no—
539C22×C3—41no—
540C22×C3—41no—
541C22×C3C2×#12311yescharacteristic
542C22×C3D4211yescharacteristic
543C22×C3#54711yescharacteristic
544C22×C3#57511yescharacteristic
545C22×C3#57511yescharacteristic
546C22×C3#57611yescharacteristic
547C22×C3#57711yescharacteristic
548C2×D3—81no—
549C2×D3—81no—
550C2×D3—81no—
551C2×D3—81no—
552C2×D3—81no—
553C2×D3—81no—
554C2×D3—81no—
555C3×C4—22no—
556C3×C4—41no—
557C3×C4—41no—
558C3×C4—41no—
559C3×C4—42no—
560C3×C4—81no—
561C3×C4—81no—
562C3×C4—81no—
563C3×C4#52511yescharacteristic
564C3×C4#53911yescharacteristic
565C3×C4#54811yescharacteristic
566C3×C4#55311yescharacteristic
567Dic3—42no—
568Dic3—81no—
569Dic3—81no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—121no—
584C23—121no—
585C23—121no—
586C23—121no—
587C23—121no—
588C23—121no—
589C23—121no—
590C23#92912yes—
591C23C22×#6711yescharacteristic
592D4—44no—
593D4—121no—
594D4—121no—
595D4—121no—
596D4—121no—
597D4—244no—
598D4—244no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—22no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—62no—
631C2×C4—84no—
632C2×C4—121no—
633C2×C4—121no—
634C2×C4—121no—
635C2×C4—121no—
636C2×C4—121no—
637C2×C4—121no—
638C2×C4—121no—
639C2×C4—121no—
640C2×C4#92912yes—
641C2×C4C2×#23511yescharacteristic
642C2×C4C2×#23511yescharacteristic
643C2×C4C2×D3×D411yescharacteristic
644C2×C4#100411yescharacteristic
645C2×C4D3×#4111yescharacteristic
646C2×C4#101011yescharacteristic
647C8—62no—
648C8—62no—
649C8—121no—
650C8—121no—
651C2×C3—42no—
652C2×C3—81no—
653C2×C3#244311yescharacteristic
654C2×C3#309311yescharacteristic
655C2×C3#309611yescharacteristic
656C2×C3#327111yescharacteristic
657C2×C3#327311yescharacteristic
658C2×C3#334711yescharacteristic
659C2×C3#335111yescharacteristic
660D3—161no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—41no—
665C22—41no—
666C22—41no—
667C22—41no—
668C22—41no—
669C22—41no—
670C22—41no—
671C22—41no—
672C22—41no—
673C22—41no—
674C22—41no—
675C22—241no—
676C22—241no—
677C22—241no—
678C22—241no—
679C22—241no—
680C22—241no—
681C22—241no—
682C22C2×#92911yescharacteristic
683C22D4×#6711yescharacteristic
684C22#582211yescharacteristic
685C22#582211yescharacteristic
686C22#582411yescharacteristic
687C22#583511yescharacteristic
688C22#583711yescharacteristic
689C4—22no—
690C4—41no—
691C4—42no—
692C4—41no—
693C4—41no—
694C4—81no—
695C4—81no—
696C4—81no—
697C4—122no—
698C4—241no—
699C4—241no—
700C4#554011yescharacteristic
701C4#554711yescharacteristic
702C4#555211yescharacteristic
703C4#556411yescharacteristic
704C3#2615611yescharacteristic, complemented, Sylow
705C2—42no—
706C2—81no—
707C2—481no—
708C2#8221611yescharacteristic
709C2#8235611yescharacteristic
710C2#8256611yescharacteristic
711C2#8350411yescharacteristic
712C2#8354711yescharacteristic
713C2#8359611yescharacteristic
714C2#8361011yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2352C2
socleC23×C3C22×D4
prime radical#82350C2
hypercenter#2352D3
hypocenterC3#26156
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2352

Complements

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

NH ≅ G/Ntype
1. C3×#2352C2semidirect
3. C2×#5003C2semidirect
4. C2×#4964C2semidirect
5. #71611C2semidirect
6. #71576C2semidirect
7. #71560C2semidirect
8. #69558C2semidirect
9. #69541C2semidirect
10. #69523C2semidirect
11. #69517C2semidirect
12. #69507C2semidirect
13. #69506C2semidirect
14. #69463C2semidirect
15. #69458C2semidirect
65. #2352D3semidirect
147. #807D4semidirect
704. C3#26156semidirect

Maximal subgroups

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

subgroupindex
C3×#23522
#823502
C2×#50032
C2×#49642
#716112
#715762
#715602
#695582
#695412
#695232
#695172
#695072
#695062
#694632
#694582
#261563