Finite Group Databasebeta

← #10855151

Subgroups — #10855151

Order 768.

924 conjugacy classes of subgroups.

The subgroup lattice has 722 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82705C211yesmaximal, characteristic
2#82241C211yesmaximal, characteristic, complemented
3C2×#4995C211yesmaximal, characteristic, complemented
4C2×#4784C211yesmaximal, characteristic, complemented
5C3×#1875C211yesmaximal, characteristic, complemented
6#71682C211yesmaximal, characteristic, complemented
7#70206C211yesmaximal, characteristic, complemented
8#70183C211yesmaximal, characteristic, complemented
9#70174C211yesmaximal, characteristic, complemented
10#69953C211yesmaximal, characteristic, complemented
11#69927C211yesmaximal, characteristic, complemented
12#68890C211yesmaximal, characteristic, complemented
13#68877C211yesmaximal, characteristic, complemented
14#68834C211yesmaximal, characteristic, complemented
15#68788C211yesmaximal, characteristic, complemented
16#26157—31nomaximal, Sylow
17C2×C3×#121C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C2×#939—21no—
20C2×#938C2211yescharacteristic
21C2×C4×#67C2211yescharacteristic
22C22×C4⋊Dic3C2211yescharacteristic
23C22×#208C2211yescharacteristic
24C2×#892C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C2×C3×C22⋊C8C2211yescharacteristic
28#5221C2211yescharacteristic
29#5220—21no—
30C2×#832C2211yescharacteristic
31C2×#832C2211yescharacteristic
32#5212C2211yescharacteristic
33C2×(C2×D3)⋊C8C2211yescharacteristic
34C2×(C2×D3)⋊C8C2211yescharacteristic
35C2×#818C2211yescharacteristic
36C2×#818C2211yescharacteristic
37#4995—24no—
38#4987C2211yescharacteristic
39#4987C2211yescharacteristic
40#4975C2211yescharacteristic
41Dic3×C4⋊C4C2211yescharacteristic
42#4972C2211yescharacteristic
43C2×#808C2211yescharacteristic
44C2×#808C2211yescharacteristic
45C2×#807C2211yescharacteristic
46C2×#807C2211yescharacteristic
47#4961—21no—
48#4957—21no—
49#4948—21no—
50#4944—21no—
51#4940—21no—
52C4×Dic3⋊C4—21no—
53#4784—24no—
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#482—31no—
63C2×#428—31no—
64#2416—31no—
65#2354—31no—
66#2093—31no—
67#2083—31no—
68#2076—31no—
69#2033—31no—
70#1926—31no—
71#1919—31no—
72#1895—31no—
73#1875D311yescharacteristic, complemented
74#1870—31no—
75#1854—31no—
76#1852—31no—
77C22×C3×D4—21no—
78C22×C3×D4C2311yescharacteristic
79C23×C3×C4C2311yescharacteristic
80C22×#67—21no—
81C22×#67D411yescharacteristic
82C23×Dic3—21no—
83C22×C4×D3C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C3×#121—22no—
87C3×#121—22no—
88C3×#117—22no—
89C3×#117—41no—
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C42—21no—
96C2×#217—21no—
97C2×#217—21no—
98C2×#217—21no—
99C2×#217—21no—
100C2×#217D411yescharacteristic
101#939—44no—
102#938—22no—
103#938—41no—
104C4×#67—22no—
105C4×#67—22no—
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4—21no—
108C2×(C2×D3)⋊C4—21no—
109C2×(C2×D3)⋊C4D411yescharacteristic
110C2×C4⋊Dic3—22no—
111C2×C4⋊Dic3—21no—
112C2×C4⋊Dic3—41no—
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116C2×C4⋊Dic3C2311yescharacteristic
117C2×Dic3⋊C4—21no—
118C2×Dic3⋊C4—21no—
119C2×Dic3⋊C4—21no—
120C2×Dic3⋊C4D411yescharacteristic
121C2×C4×Dic3—21no—
122C2×C4×Dic3—21no—
123C2×C4×Dic3C2311yescharacteristic
124C2×#208—22no—
125C2×#208—22no—
126C2×#208—22no—
127C22×C3⋊C8C2311yescharacteristic
128C22×C3⋊C8C2311yescharacteristic
129#892—22no—
130#892—41no—
131C3×#102—22no—
132C3×#102—41no—
133C3×C22⋊C8—22no—
134#832—22no—
135#832—22no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141(C2×D3)⋊C8—22no—
142(C2×D3)⋊C8—22no—
143#818—22no—
144#818—41no—
145#808—22no—
146#808D414yescomplemented
147#807—22no—
148#807—22no—
149C22×M16—31no—
150C2×#121—31no—
151C2×#121—61no—
152C2×#121C2×D311yescharacteristic
153C2×C4×D4—31no—
154C2×#117—31no—
155C2×#117C2×D311yescharacteristic
156C22×C4⋊C4—31no—
157#482—64no—
158#428—64no—
159C2×#107—31no—
160C2×#107—31no—
161C2×#107—31no—
162C2×#107—31no—
163C2×#106—31no—
164C2×#106—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102C2×D311yescharacteristic
168C2×#102C2×D311yescharacteristic
169C2×C22⋊C8—31no—
170C2×C22⋊C8—31no—
171C2×C22⋊C8C2×D311yescharacteristic
172#396—61no—
173#394—61no—
174#390—31no—
175#390—61no—
176#389—31no—
177#388—61no—
178#388—61no—
179#386—61no—
180#384—31no—
181#380—31no—
182#380—31no—
183#380—31no—
184C4×C4⋊C4—31no—
185C4×C4⋊C4—61no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340C2×D311yescharacteristic
191#340C2×D311yescharacteristic
192#336—31no—
193#336—31no—
194C24×C3—21no—
195C24×C3C2×D411yescharacteristic
196C23×D3C2×D411yescharacteristic
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4—42no—
204C2×C3×D4—42no—
205C2×C3×D4—42no—
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×C4C2×D411yescharacteristic
214C22×C3×C4#4112yes—
215C22×C3×C4#4112yes—
216C22×C3×C4C2411yescharacteristic
217C2×#67—22no—
218C2×#67—22no—
219C2×#67—41no—
220C2×#67—42no—
221C2×#67—42no—
222C2×#67—81no—
223C22×Dic3—21no—
224C22×Dic3—21no—
225C22×Dic3—21no—
226C22×Dic3—21no—
227C22×Dic3—42no—
228C22×Dic3—42no—
229C22×Dic3—81no—
230C22×Dic3C2×D411yescharacteristic
231C22×Dic3C2×D411yescharacteristic
232C22×Dic3C2×D411yescharacteristic
233C2×C4×D3—21no—
234C2×C4×D3—21no—
235C2×C4×D3—21no—
236C2×C4×D3—21no—
237C2×C4×D3—21no—
238C2×C4×D3—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4C2×D412yes—
251C3×C22⋊C4—42no—
252C3×C22⋊C4—42no—
253C3×C42—42no—
254#217—22no—
255#217—42no—
256#217—42no—
257#217—81no—
258#217—81no—
259(C2×D3)⋊C4—22no—
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⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—42no—
271C4⋊Dic3C2×D412yes—
272Dic3⋊C4—22no—
273Dic3⋊C4—42no—
274Dic3⋊C4—42no—
275Dic3⋊C4—81no—
276C4×Dic3—21no—
277C4×Dic3—21no—
278C4×Dic3—42no—
279C4×Dic3—42no—
280#208—28no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8C2×D412yes—
286C2×C3⋊C8#4112yes—
287C22×D4—21no—
288C22×D4—31no—
289C22×D4—61no—
290C22×D4C22×D311yescharacteristic
291C23×C4—31no—
292C23×C4—61no—
293C23×C4C22×D311yescharacteristic
294C2×M16—62no—
295C2×M16—62no—
296C2×M16—62no—
297C22×C8—31no—
298C22×C8—31no—
299C22×C8C22×D311yescharacteristic
300C22×C8C22×D311yescharacteristic
301#121—22no—
302#121—22no—
303#121—62no—
304#121—121no—
305#121—124no—
306C4×D4—62no—
307C4×D4—62no—
308#117—22no—
309#117—41no—
310#117—62no—
311#117—121no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—31no—
314C2×C4⋊C4—31no—
315C2×C4⋊C4—31no—
316C2×C4⋊C4—31no—
317C2×C4⋊C4—62no—
318C2×C4⋊C4—61no—
319C2×C4⋊C4—61no—
320C2×C4⋊C4—61no—
321C2×C4⋊C4—61no—
322C2×C4⋊C4—121no—
323C2×C4⋊C4C22×D311yescharacteristic
324C2×C4⋊C4C22×D311yescharacteristic
325C2×C4⋊C4C22×D311yescharacteristic
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—31no—
329C2×C22⋊C4—31no—
330C2×C22⋊C4—61no—
331C2×C22⋊C4—61no—
332C2×C22⋊C4—61no—
333C2×C22⋊C4—61no—
334C2×C22⋊C4—61no—
335C2×C22⋊C4—61no—
336C2×C22⋊C4—61no—
337C2×C42—21no—
338C2×C42—31no—
339C2×C42—61no—
340C2×C42—61no—
341#107—62no—
342#107—62no—
343#107—62no—
344#107—121no—
345#106—34no—
346#106—62no—
347#102—22no—
348#102—41no—
349#102—62no—
350#102—62no—
351C22⋊C8—22no—
352C22⋊C8—62no—
353C22⋊C8—62no—
354#95—61no—
355#95—61no—
356#95—61no—
357#95—61no—
358#95—61no—
359C23×C3—21no—
360C23×C3—21no—
361C23×C3—21no—
362C23×C3—21no—
363C23×C3—21no—
364C23×C3—21no—
365C23×C3—41no—
366C23×C3—41no—
367C23×C3—41no—
368C23×C3—41no—
369C23×C3—41no—
370C23×C3—41no—
371C23×C3—41no—
372C23×C3#12312yes—
373C23×C3C22×D411yescharacteristic
374C22×D3—21no—
375C22×D3—21no—
376C22×D3—21no—
377C22×D3—21no—
378C22×D3—21no—
379C22×D3—21no—
380C22×D3—21no—
381C3×D4—41no—
382C3×D4—41no—
383C3×D4—41no—
384C3×D4—41no—
385C3×D4—84no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—22no—
389C2×C3×C4—22no—
390C2×C3×C4—22no—
391C2×C3×C4—22no—
392C2×C3×C4—22no—
393C2×C3×C4—41no—
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#12312yes—
402C2×C3×C4C22×D411yescharacteristic
403C2×C3×C4C2×#4111yescharacteristic
404C2×C3×C4C2×#4111yescharacteristic
405C2×C3×C4C2×#4111yescharacteristic
406C2×C3×C4H(5,𝔽2)11yescharacteristic
407C2×C3×C42−1+411yescharacteristic
408#67—44no—
409#67—84no—
410C2×Dic3—21no—
411C2×Dic3—21no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—22no—
416C2×Dic3—21no—
417C2×Dic3—21no—
418C2×Dic3—21no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3—41no—
426C2×Dic3—41no—
427C2×Dic3—41no—
428C2×Dic3—41no—
429C2×Dic3—41no—
430C2×Dic3—84no—
431C2×Dic3AGL(1,ℤ/8)12yes—
432C4×D3—41no—
433C4×D3—41no—
434C4×D3—41no—
435C4×D3—41no—
436C3×C8—41no—
437C3×C8—41no—
438C3⋊C8—22no—
439C3⋊C8—22no—
440C24—21no—
441C24—31no—
442C24D3×D411yescharacteristic
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—42no—
450C2×D4—42no—
451C2×D4—42no—
452C2×D4—62no—
453C2×D4—62no—
454C2×D4—121no—
455C2×D4—122no—
456C2×D4—122no—
457C2×D4—241no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—31no—
466C22×C4—31no—
467C22×C4—31no—
468C22×C4—61no—
469C22×C4—61no—
470C22×C4—61no—
471C22×C4—61no—
472C22×C4—61no—
473C22×C4—61no—
474C22×C4—61no—
475C22×C4—61no—
476C22×C4—61no—
477C22×C4—61no—
478C22×C4—122no—
479C22×C4—122no—
480C22×C4—241no—
481C22×C4#23512yes—
482C22×C4D3×D411yescharacteristic
483C22×C4#23712yes—
484C22×C4C23×D311yescharacteristic
485M16—68no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—32no—
493C2×C8—32no—
494C2×C8—61no—
495C2×C8—61no—
496C2×C8—61no—
497C2×C8—61no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—32no—
504C4⋊C4—62no—
505C4⋊C4—61no—
506C4⋊C4—61no—
507C4⋊C4—61no—
508C4⋊C4—61no—
509C4⋊C4—61no—
510C4⋊C4—61no—
511C4⋊C4—61no—
512C4⋊C4—122no—
513C4⋊C4—122no—
514C4⋊C4—122no—
515C4⋊C4—241no—
516C4⋊C4D3×D412yes—
517C22⋊C4—42no—
518C22⋊C4—42no—
519C22⋊C4—62no—
520C22⋊C4—62no—
521C22⋊C4—122no—
522C22⋊C4—122no—
523C22⋊C4—122no—
524C22⋊C4—122no—
525C22⋊C4—241no—
526C22⋊C4—241no—
527C22⋊C4—241no—
528C42—42no—
529C42—61no—
530C42—61no—
531C42—122no—
532C42—122no—
533C22×C3—22no—
534C22×C3—22no—
535C22×C3—22no—
536C22×C3—41no—
537C22×C3—41no—
538C22×C3—41no—
539C22×C3—41no—
540C22×C3—81no—
541C22×C3—81no—
542C22×C3—81no—
543C22×C3—81no—
544C22×C3—81no—
545C22×C3—81no—
546C22×C3—81no—
547C22×C3C2×#12311yescharacteristic
548C22×C3D4211yescharacteristic
549C22×C3#54711yescharacteristic
550C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
551C22×C3#57511yescharacteristic
552C22×C3#57611yescharacteristic
553C22×C3#57811yescharacteristic
554C2×D3—41no—
555C2×D3—41no—
556C2×D3—41no—
557C2×D3—41no—
558C2×D3—41no—
559C2×D3—41no—
560C2×D3—41no—
561C3×C4—22no—
562C3×C4—42no—
563C3×C4—81no—
564C3×C4—81no—
565C3×C4#52511yescharacteristic
566C3×C4#53911yescharacteristic
567C3×C4#54811yescharacteristic
568C3×C4#55311yescharacteristic
569Dic3—22no—
570Dic3—41no—
571Dic3—41no—
572Dic3—42no—
573Dic3—81no—
574Dic3—81no—
575Dic3—81no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—21no—
581C23—21no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23—41no—
586C23—41no—
587C23—41no—
588C23—41no—
589C23—61no—
590C23—61no—
591C23—61no—
592C23—61no—
593C23—61no—
594C23—61no—
595C23—61no—
596C23#88312yes—
597C23C2×D3×D411yescharacteristic
598D4—41no—
599D4—41no—
600D4—41no—
601D4—41no—
602D4—84no—
603D4—124no—
604D4—244no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—32no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—61no—
622C2×C4—61no—
623C2×C4—61no—
624C2×C4—61no—
625C2×C4—61no—
626C2×C4—61no—
627C2×C4—61no—
628C2×C4—61no—
629C2×C4—62no—
630C2×C4—121no—
631C2×C4—121no—
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—121no—
641C2×C4—121no—
642C2×C4—121no—
643C2×C4—121no—
644C2×C4—121no—
645C2×C4—244no—
646C2×C4#88312yes—
647C2×C4C2×#23511yescharacteristic
648C2×C4C2×D3×D411yescharacteristic
649C2×C4C2×#23711yescharacteristic
650C2×C4#100411yescharacteristic
651C2×C4D3×#4111yescharacteristic
652C2×C4#101011yescharacteristic
653C8—41no—
654C8—41no—
655C8—62no—
656C8—62no—
657C2×C3—42no—
658C2×C3—161no—
659C2×C3#244311yescharacteristic
660C2×C3#309411yescharacteristic
661C2×C3#309811yescharacteristic
662C2×C3#326911yescharacteristic
663C2×C3#327411yescharacteristic
664C2×C3#334711yescharacteristic
665C2×C3#335311yescharacteristic
666D3—81no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—41no—
671C22—41no—
672C22—41no—
673C22—41no—
674C22—81no—
675C22—81no—
676C22—81no—
677C22—81no—
678C22—81no—
679C22—81no—
680C22—81no—
681C22—121no—
682C22—121no—
683C22—121no—
684C22—121no—
685C22—121no—
686C22—121no—
687C22—121no—
688C22C2×#88311yescharacteristic
689C22#559011yescharacteristic
690C22#565511yescharacteristic
691C22#576911yescharacteristic
692C22#577111yescharacteristic
693C22D3×AGL(1,ℤ/8)11yescharacteristic
694C22#577311yescharacteristic
695C4—22no—
696C4—42no—
697C4—62no—
698C4—81no—
699C4—81no—
700C4—121no—
701C4—121no—
702C4—122no—
703C4—241no—
704C4—241no—
705C4—241no—
706C4#554711yescharacteristic
707C4#556411yescharacteristic
708C4#559811yescharacteristic
709C4#560111yescharacteristic
710C3#2615711yescharacteristic, complemented, Sylow
711C2—42no—
712C2—161no—
713C2—241no—
714C2#7470211yescharacteristic
715C2#7884311yescharacteristic
716C2#7893311yescharacteristic
717C2#8009711yescharacteristic
718C2#8012511yescharacteristic
719C2#8136711yescharacteristic
720C2#8139311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1875C2
socleC23×C3C22×D4
prime radical#82705C2
hypercenter#1875D3
hypocenterC3#26157
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1875

Complements

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

NH ≅ G/Ntype
2. #82241C2semidirect
3. C2×#4995C2semidirect
4. C2×#4784C2semidirect
5. C3×#1875C2semidirect
6. #71682C2semidirect
7. #70206C2semidirect
8. #70183C2semidirect
9. #70174C2semidirect
10. #69953C2semidirect
11. #69927C2semidirect
12. #68890C2semidirect
13. #68877C2semidirect
14. #68834C2semidirect
15. #68788C2semidirect
73. #1875D3semidirect
146. #808D4semidirect
710. C3#26157semidirect

Maximal subgroups

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

subgroupindex
#827052
#822412
C2×#49952
C2×#47842
C3×#18752
#716822
#702062
#701832
#701742
#699532
#699272
#688902
#688772
#688342
#687882
#261573