Finite Group Databasebeta

← #10812486

Subgroups — #10812486

Order 768.

1010 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4771C211yesmaximal, characteristic
2#76301C211yesmaximal, characteristic
3#73767C211yesmaximal, characteristic, complemented
4C3×#1925C211yesmaximal, characteristic, complemented
5#70642C211yesmaximal, characteristic, complemented
6#70631C211yesmaximal, characteristic, complemented
7#70199C211yesmaximal, characteristic
8#70188C211yesmaximal, characteristic, complemented
9#70156C211yesmaximal, characteristic, complemented
10#69644C211yesmaximal, characteristic, complemented
11C4×#810C211yesmaximal, characteristic, complemented
12#68872C211yesmaximal, characteristic, complemented
13#68783C211yesmaximal, characteristic, complemented
14#68684C211yesmaximal, characteristic, complemented
15#68649C211yesmaximal, characteristic, complemented
16#22583—31nomaximal, Sylow
17C22×#237C2211yescharacteristic
18C22×ΓL(1,𝔽25)C2211yescharacteristic
19C2×D3×C4⋊C4C2211yescharacteristic
20C2×#887C2211yescharacteristic
21C2×#872—21no—
22C2×C4×Dic6C2211yescharacteristic
23C2×C3×#107C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C3×#390C2211yescharacteristic
27C3×C4×C4⋊C4C2211yescharacteristic
28#5221C2211yescharacteristic
29C2×#832C2211yescharacteristic
30C2×#832C2211yescharacteristic
31C2×(C2×D3)⋊C8C2211yescharacteristic
32C2×#817C2211yescharacteristic
33C2×#816C2211yescharacteristic
34C2×#816C2211yescharacteristic
35C2×#815C2211yescharacteristic
36#4987C2211yescharacteristic
37#4987C2211yescharacteristic
38Dic3×C4⋊C4C2211yescharacteristic
39#4972C2211yescharacteristic
40C2×#810C2211yescharacteristic
41C2×#810C2211yescharacteristic
42#4957—21no—
43#4957—21no—
44#4942—21no—
45Dic3×C22⋊C4—21no—
46C4×(C2×D3)⋊C4C2211yescharacteristic
47#4933—21no—
48#4928C2211yescharacteristic
49C2×C4×C3⋊C8C2211yescharacteristic
50#4771C418yescomplemented
51#4664—21no—
52#4652—21no—
53C3×#340C2211yescharacteristic
54C3×#336C2211yescharacteristic
55#4551C2211yescharacteristic
56#4550C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61C2×#417—31no—
62#2344—31no—
63#2341—31no—
64#1936—31no—
65#1934—31no—
66#1926—31no—
67#1925D311yescharacteristic, complemented
68#1865—31no—
69#1863—31no—
70#1824—31no—
71#1809—31no—
72#1788—31no—
73#1763—31no—
74C4×#103—31no—
75#1753—31no—
76C22×C3×D4C2311yescharacteristic
77C22×#67—21no—
78C23×Dic3—21no—
79C2×#237—22no—
80C2×#237—22no—
81C2×#237—22no—
82C2×#237—41no—
83C2×#237—41no—
84C2×#237C2×C414yes—
85C22×C4×D3—21no—
86C22×C4×D3C2311yescharacteristic
87C22×Dic6C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C22×C3×C8C2311yescharacteristic
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×C3×C42C2311yescharacteristic
97C2×#217—21no—
98C2×#217—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4D411yescharacteristic
102C2×(C2×D3)⋊C4D411yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3—21no—
113C2×C4×Dic3—21no—
114C2×C4×Dic3C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117C2×ΓL(1,𝔽25)—41no—
118C2×ΓL(1,𝔽25)—41no—
119C2×ΓL(1,𝔽25)C2×C414yes—
120D3×C4⋊C4—22no—
121D3×C4⋊C4C2×C414yes—
122#887—22no—
123#887—41no—
124#872—44no—
125C4×Dic6—22no—
126C4×Dic6—41no—
127C3×#107—41no—
128C3×#102—41no—
129C3×#102C2×C414yes—
130C3×#95—21no—
131#832—41no—
132#832C2×C414yes—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95D411yescharacteristic
139C3⋊#95D411yescharacteristic
140(C2×D3)⋊C8—22no—
141#817—41no—
142#816—22no—
143#816C2×C414yes—
144#815—41no—
145#810—22no—
146#810C2×C414yes—
147C4×C3⋊C8—41no—
148C22×#41—31no—
149C22×M16—31no—
150C2×C4×Q8—31no—
151C2×C4×Q8—31no—
152C2×#117—61no—
153C22×C4⋊C4—31no—
154C2×#107C2×D311yescharacteristic
155C2×#106—31no—
156#417—38no—
157C2×#103—31no—
158C2×#103—31no—
159C2×#103—31no—
160C2×#103—31no—
161C2×#102—31no—
162C2×#102—31no—
163C2×#102C2×D311yescharacteristic
164C2×#102C2×D311yescharacteristic
165C2×C22⋊C8—31no—
166C2×#97—31no—
167C2×C4×C8—31no—
168#390—31no—
169#390—61no—
170#390—61no—
171#390C2×D311yescharacteristic
172#389—31no—
173#389—31no—
174#387—61no—
175#387—61no—
176#381—61no—
177#380—31no—
178#380—31no—
179#380—61no—
180C4×C4⋊C4—31no—
181C4×C4⋊C4C2×D311yescharacteristic
182C4×C22⋊C4—31no—
183C4×C22⋊C4—61no—
184#340—31no—
185#340—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340C2×D311yescharacteristic
190#336—31no—
191#336C2×D311yescharacteristic
192C24×C3—21no—
193C23×D3C2×D411yescharacteristic
194C2×C3×D4—21no—
195C2×C3×D4—21no—
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4C22×C412yes—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2411yescharacteristic
206C2×#67—24no—
207C2×#67—42no—
208C2×#67—42no—
209C22×Dic3—21no—
210C22×Dic3—21no—
211C22×Dic3—24no—
212C22×Dic3—21no—
213C22×Dic3—21no—
214C22×Dic3—42no—
215C22×Dic3—42no—
216C22×Dic3C2×D411yescharacteristic
217C22×Dic3C2×D411yescharacteristic
218C22×Dic3C2×D411yescharacteristic
219#237—28no—
220#237—44no—
221#237—44no—
222#237—44no—
223C2×C4×D3—21no—
224C2×C4×D3—21no—
225C2×C4×D3—21no—
226C2×C4×D3—21no—
227C2×C4×D3—24no—
228C2×C4×D3—42no—
229C2×C4×D3—42no—
230C2×C4×D3C22×C412yes—
231C2×C4×D3#4112yes—
232C2×Dic6—21no—
233C2×Dic6—21no—
234C2×Dic6—21no—
235C2×Dic6C22×C412yes—
236C2×Dic6#4112yes—
237C2×Dic6#4112yes—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8C22×C412yes—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4C22×C412yes—
250C3×C22⋊C4—42no—
251C3×C22⋊C4—81no—
252C3×C42—21no—
253C3×C42—21no—
254C3×C42—42no—
255#217—42no—
256#217—81no—
257(C2×D3)⋊C4—22no—
258(C2×D3)⋊C4—41no—
259(C2×D3)⋊C4—42no—
260(C2×D3)⋊C4—81no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3C22×C412yes—
267Dic3⋊C4—24no—
268Dic3⋊C4—42no—
269Dic3⋊C4—42no—
270Dic3⋊C4—42no—
271Dic3⋊C4—42no—
272C4×Dic3—21no—
273C4×Dic3—21no—
274C4×Dic3—42no—
275C4×Dic3—42no—
276C4×Dic3—42no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8C22×C412yes—
283ΓL(1,𝔽25)—44no—
284C2×#41—34no—
285C2×#41—62no—
286C2×#41—62no—
287C2×#41—62no—
288C2×#41—121no—
289C2×#41—121no—
290C22×Q8—31no—
291C22×D4—61no—
292C22×D4C22×D311yescharacteristic
293C23×C4—31no—
294C23×C4—61no—
295C23×C4—61no—
296C2×M16—34no—
297C2×M16—121no—
298C2×M16—121no—
299C22×C8—31no—
300C22×C8—31no—
301C22×C8C22×D311yescharacteristic
302C22×C8C22×D311yescharacteristic
303C4×Q8—62no—
304C4×Q8—62no—
305C4×Q8—121no—
306C4×Q8—121no—
307#117—124no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—34no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—62no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4—61no—
316C2×C4⋊C4—61no—
317C2×C4⋊C4—61no—
318C2×C4⋊C4C22×D311yescharacteristic
319C2×C4⋊C4C22×D311yescharacteristic
320C2×C4⋊C4C22×D311yescharacteristic
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—31no—
324C2×C22⋊C4—31no—
325C2×C22⋊C4—61no—
326C2×C22⋊C4—61no—
327C2×C22⋊C4—61no—
328C2×C22⋊C4—61no—
329C2×C42—21no—
330C2×C42—31no—
331C2×C42—61no—
332C2×C42—61no—
333C2×C42—61no—
334C2×C42C22×D311yescharacteristic
335#107—41no—
336#106—121no—
337#103—34no—
338#103—34no—
339#103—62no—
340#103—62no—
341#102—34no—
342#102—41no—
343#102—121no—
344#102C4×D314yes—
345C22⋊C8—62no—
346#97—121no—
347C4×C8—121no—
348#95—21no—
349#95—31no—
350#95—31no—
351#95—61no—
352#95—61no—
353#95—61no—
354#95—61no—
355#95—61no—
356C23×C3—22no—
357C23×C3—41no—
358C23×C3—41no—
359C23×C3—41no—
360C23×C3—41no—
361C23×C3—41no—
362C23×C3—41no—
363C23×C3C22×D411yescharacteristic
364C22×D3—21no—
365C22×D3—21no—
366C22×D3—21no—
367C22×D3—21no—
368C22×D3—21no—
369C22×D3—21no—
370C22×D3C4×D412yes—
371C3×D4—22no—
372C3×D4—22no—
373C3×D4—41no—
374C3×D4—41no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—22no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4AGL(1,ℤ/8)12yes—
390C2×C3×C4C23×C411yescharacteristic
391C2×C3×C4C2×#4111yescharacteristic
392C2×C3×C4C2×#4111yescharacteristic
393C2×C3×C4C2×#4111yescharacteristic
394C2×C3×C4H(5,𝔽2)11yescharacteristic
395C2×C3×C4H(5,𝔽2)11yescharacteristic
396#67—48no—
397C2×Dic3—22no—
398C2×Dic3—22no—
399C2×Dic3—22no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—22no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—21no—
409C2×Dic3—48no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3C4×D412yes—
419C4×D3—22no—
420C4×D3—22no—
421C4×D3—22no—
422C4×D3—22no—
423C4×D3—48no—
424Dic6—22no—
425Dic6—22no—
426Dic6—22no—
427Dic6#11714yescomplemented
428C3×C8—22no—
429C3×C8—41no—
430C3⋊C8—22no—
431C3⋊C8—41no—
432C24—21no—
433C24—31no—
434#41—68no—
435#41—124no—
436#41—124no—
437#41—124no—
438C2×Q8—32no—
439C2×Q8—32no—
440C2×Q8—32no—
441C2×Q8—61no—
442C2×Q8—61no—
443C2×Q8—61no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—64no—
450C2×D4—122no—
451C2×D4—122no—
452C2×D4C2×C4×D312yes—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—31no—
457C22×C4—32no—
458C22×C4—32no—
459C22×C4—31no—
460C22×C4—31no—
461C22×C4—64no—
462C22×C4—64no—
463C22×C4—61no—
464C22×C4—61no—
465C22×C4—61no—
466C22×C4—61no—
467C22×C4—61no—
468C22×C4—61no—
469C22×C4—61no—
470C22×C4—61no—
471C22×C4—122no—
472C22×C4—122no—
473C22×C4—122no—
474C22×C4—122no—
475C22×C4D3×D411yescharacteristic
476C22×C4D3×D411yescharacteristic
477C22×C4C23×D311yescharacteristic
478M16—124no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—32no—
485C2×C8—61no—
486C2×C8—61no—
487C2×C8—61no—
488C2×C8—61no—
489C2×C8—61no—
490C2×C8C2×C4×D312yes—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—32no—
497C4⋊C4—64no—
498C4⋊C4—61no—
499C4⋊C4—61no—
500C4⋊C4—61no—
501C4⋊C4—61no—
502C4⋊C4—61no—
503C4⋊C4—122no—
504C4⋊C4—122no—
505C4⋊C4—122no—
506C4⋊C4—122no—
507C4⋊C4C2×C4×D312yes—
508C22⋊C4—42no—
509C22⋊C4—62no—
510C22⋊C4—81no—
511C22⋊C4—121no—
512C22⋊C4—122no—
513C22⋊C4—122no—
514C22⋊C4—241no—
515C22⋊C4—241no—
516C42—21no—
517C42—21no—
518C42—42no—
519C42—61no—
520C42—61no—
521C42—122no—
522C42—122no—
523C42—122no—
524C22×C3—42no—
525C22×C3—42no—
526C22×C3—42no—
527C22×C3—81no—
528C22×C3—81no—
529C22×C3—81no—
530C22×C3—81no—
531C22×C3C2×C4×D411yescharacteristic
532C22×C3#54611yescharacteristic
533C22×C3#54611yescharacteristic
534C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
535C22×C3#57511yescharacteristic
536C22×C3#57711yescharacteristic
537C22×C3#57811yescharacteristic
538C2×D3—22no—
539C2×D3—22no—
540C2×D3—22no—
541C2×D3—41no—
542C2×D3—41no—
543C2×D3—41no—
544C2×D3—41no—
545C3×C4—22no—
546C3×C4—42no—
547C3×C4—41no—
548C3×C4—81no—
549C3×C4—81no—
550C3×C4C2×#11711yescharacteristic
551C3×C4#51811yescharacteristic
552C3×C4#55111yescharacteristic
553C3×C4#55111yescharacteristic
554Dic3—24no—
555Dic3—22no—
556Dic3—41no—
557Dic3—42no—
558Dic3—41no—
559Dic3—81no—
560Dic3—81no—
561C23—22no—
562C23—32no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—61no—
570C23—61no—
571C23—61no—
572C23—61no—
573C23—61no—
574C23—61no—
575C23C2×D3×D411yescharacteristic
576Q8—34no—
577Q8—62no—
578Q8—62no—
579Q8—62no—
580D4—22no—
581D4—22no—
582D4—41no—
583D4—41no—
584D4—128no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—32no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—62no—
601C2×C4—61no—
602C2×C4—62no—
603C2×C4—62no—
604C2×C4—61no—
605C2×C4—62no—
606C2×C4—62no—
607C2×C4—62no—
608C2×C4—62no—
609C2×C4—61no—
610C2×C4—61no—
611C2×C4—61no—
612C2×C4—61no—
613C2×C4—62no—
614C2×C4—61no—
615C2×C4—61no—
616C2×C4—128no—
617C2×C4—128no—
618C2×C4—121no—
619C2×C4—121no—
620C2×C4—121no—
621C2×C4—121no—
622C2×C4—121no—
623C2×C4—121no—
624C2×C4—121no—
625C2×C4—121no—
626C2×C4#91112yes—
627C2×C4C22×C4×D311yescharacteristic
628C2×C4#100411yescharacteristic
629C2×C4D3×#4111yescharacteristic
630C2×C4D3×#4111yescharacteristic
631C2×C4D3×#4111yescharacteristic
632C2×C4#100911yescharacteristic
633C8—22no—
634C8—41no—
635C8—62no—
636C8—121no—
637C2×C3—82no—
638C2×C3#234211yescharacteristic
639C2×C3C4×AGL(1,ℤ/8)11yescharacteristic
640C2×C3#294711yescharacteristic
641C2×C3#330811yescharacteristic
642C2×C3#331711yescharacteristic
643C2×C3#332711yescharacteristic
644C2×C3#333211yescharacteristic
645D3—42no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—62no—
650C22—62no—
651C22—62no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22—121no—
657C22—121no—
658C22—121no—
659C22—121no—
660C22C4×D3×D411yescharacteristic
661C22#561211yescharacteristic
662C22#565411yescharacteristic
663C22C2×#91111yescharacteristic
664C22#576811yescharacteristic
665C22#577511yescharacteristic
666C22#577911yescharacteristic
667C4—22no—
668C4—41no—
669C4—42no—
670C4—62no—
671C4—64no—
672C4—81no—
673C4—81no—
674C4—122no—
675C4—121no—
676C4—121no—
677C4—241no—
678C4—241no—
679C4D3×#11711yescharacteristic
680C4#554511yescharacteristic
681C4#567911yescharacteristic
682C4#570511yescharacteristic
683C3#2258311yescharacteristic, complemented, Sylow
684C2—82no—
685C2—122no—
686C2#7583211yescharacteristic
687C2C4×#91111yescharacteristic
688C2#7977611yescharacteristic
689C2#8035711yescharacteristic
690C2#8062211yescharacteristic
691C2#8130611yescharacteristic
692C2#8140911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1925C2
socleC23×C3C22×D4
prime radicalC22×#237C22
cosocleC22×C3×C4C24
hypercenter#1925D3
hypocenterC3#22583
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1925

Complements

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

NH ≅ G/Ntype
3. #73767C2semidirect
4. C3×#1925C2semidirect
5. #70642C2semidirect
6. #70631C2semidirect
8. #70188C2semidirect
9. #70156C2semidirect
10. #69644C2semidirect
11. C4×#810C2semidirect
12. #68872C2semidirect
13. #68783C2semidirect
14. #68684C2semidirect
15. #68649C2semidirect
50. #4771C4semidirect
67. #1925D3semidirect
427. Dic6#117semidirect
683. C3#22583semidirect

Maximal subgroups

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

subgroupindex
C2×#47712
#763012
#737672
C3×#19252
#706422
#706312
#701992
#701882
#701562
#696442
C4×#8102
#688722
#687832
#686842
#686492
#225833