Finite Group Databasebeta

← #10814908

Subgroups — #10814908

Order 768.

858 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4788C211yesmaximal, characteristic
2#76489C211yesmaximal, characteristic, complemented
3#73639C211yesmaximal, characteristic
4C3×#1927C211yesmaximal, characteristic, complemented
5#70569C211yesmaximal, characteristic
6#70554C211yesmaximal, characteristic, complemented
7#70257C211yesmaximal, characteristic
8#70223C211yesmaximal, characteristic, complemented
9#70221C211yesmaximal, characteristic, complemented
10#69676C211yesmaximal, characteristic
11C4×#808C211yesmaximal, characteristic, complemented
12#68892C211yesmaximal, characteristic
13#68784C211yesmaximal, characteristic, complemented
14#68651C211yesmaximal, characteristic, complemented
15#68626C211yesmaximal, characteristic
16#22805—31nomaximal, Sylow
17C2×Q8×Dic3C2211yescharacteristic
18C22×#215C2211yescharacteristic
19C2×#893C2211yescharacteristic
20C2×#872—21no—
21C2×C3×#106C2211yescharacteristic
22C2×C3×#103C2211yescharacteristic
23C2×C3×#103C2211yescharacteristic
24C3×#391C2211yescharacteristic
25C3×C4×C4⋊C4C2211yescharacteristic
26#5229C2211yescharacteristic
27C2×#835C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#817C2211yescharacteristic
31C2×Dic3⋊C8C2211yescharacteristic
32C2×C8×Dic3C2211yescharacteristic
33C2×C8×Dic3C2211yescharacteristic
34#4988C2211yescharacteristic
35#4975C2211yescharacteristic
36#4975C2211yescharacteristic
37Dic3×C4⋊C4C2211yescharacteristic
38Dic3×C4⋊C4C2211yescharacteristic
39C2×#809C2211yescharacteristic
40C2×#808C2211yescharacteristic
41C2×#808C2211yescharacteristic
42C2×#808C2211yescharacteristic
43#4940—21no—
44#4939C2211yescharacteristic
45C4×C4⋊Dic3C2211yescharacteristic
46C4×Dic3⋊C4C2211yescharacteristic
47C2×C4×C3⋊C8C2211yescharacteristic
48#4788C418yescomplemented
49#4669—21no—
50#4667—21no—
51#4654—21no—
52#4653—21no—
53#4650—21no—
54#4649—21no—
55C3×#340C2211yescharacteristic
56C3×#336C2211yescharacteristic
57#4551C2211yescharacteristic
58#4550C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61C2×C4×SD16—31no—
62#2355—31no—
63#2354—31no—
64#1968—31no—
65#1968—31no—
66#1927—31no—
67#1927D311yescharacteristic, complemented
68#1926—31no—
69#1926—31no—
70#1844—31no—
71#1803—31no—
72#1802—31no—
73C4×#106—31no—
74C4×#106—31no—
75C8×C4⋊C4—31no—
76C22×C3×Q8C2311yescharacteristic
77C22×D12C2311yescharacteristic
78C22×C4×D3—21no—
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C42—21no—
85C2×C3×C42C2311yescharacteristic
86Q8×Dic3—22no—
87Q8×Dic3C2×C414yes—
88C2×#215—22no—
89C2×#215—22no—
90C2×#215C2×C414yes—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×C4⋊Dic3—21no—
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4D411yescharacteristic
105C2×Dic3⋊C4D411yescharacteristic
106C2×C4×Dic3—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3C2311yescharacteristic
111C2×C4×Dic3C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114#893—22no—
115#893C2×C414yes—
116#872—44no—
117C3×#106—41no—
118C3×#103—41no—
119C3×#103C2×C414yes—
120C3×#95—21no—
121C3×#95—21no—
122C3×#95—21no—
123#835—41no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95D411yescharacteristic
130C3⋊#95D411yescharacteristic
131#821—41no—
132#821C2×C414yes—
133#817—41no—
134Dic3⋊C8—22no—
135C8×Dic3—22no—
136C8×Dic3C2×C414yes—
137#809—41no—
138#808—22no—
139#808—22no—
140#808C2×C414yes—
141C4×C3⋊C8—22no—
142C22×SD16—31no—
143C2×C4×Q8—31no—
144C2×C4×D4—31no—
145C2×#117—61no—
146C4×SD16—38no—
147C2×#106—31no—
148C2×#106—31no—
149C2×#106—31no—
150C2×#106—31no—
151C2×#106C2×D311yescharacteristic
152C2×C4⋊C8—31no—
153C2×#103—31no—
154C2×#103C2×D311yescharacteristic
155C2×#103C2×D311yescharacteristic
156C2×#102—31no—
157C2×#102—31no—
158C2×#102—31no—
159C2×C4×C8—31no—
160C2×C4×C8—31no—
161C2×C4×C8—31no—
162#391—31no—
163#391C2×D311yescharacteristic
164#390—31no—
165#390—31no—
166#388—61no—
167#388—61no—
168#387—61no—
169#387—61no—
170#384—31no—
171#384—31no—
172#382—61no—
173#380—61no—
174C4×C4⋊C4—31no—
175C4×C4⋊C4—31no—
176C4×C4⋊C4—31no—
177C4×C4⋊C4—31no—
178C4×C4⋊C4—61no—
179C4×C4⋊C4C2×D311yescharacteristic
180#340—31no—
181#340—31no—
182#340—31no—
183#340C2×D311yescharacteristic
184#336—31no—
185#336C2×D311yescharacteristic
186C23×D3—21no—
187C2×C3×Q8—21no—
188C2×C3×Q8—21no—
189C2×C3×Q8—21no—
190C2×C3×Q8—21no—
191C2×C3×Q8—21no—
192C2×C3×Q8C22×C412yes—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2411yescharacteristic
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3C2×D411yescharacteristic
204C22×Dic3C2×D411yescharacteristic
205C22×Dic3C2×D411yescharacteristic
206C22×Dic3C2×D411yescharacteristic
207C2×D12—21no—
208C2×D12—21no—
209C2×D12—21no—
210C2×D12—21no—
211C2×D12—21no—
212C2×D12C22×C412yes—
213C2×C4×D3—24no—
214C2×C4×D3—42no—
215C2×C4×D3—42no—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8C22×C412yes—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4C22×C412yes—
228C3×C42—21no—
229C3×C42—21no—
230C3×C42—42no—
231#215—28no—
232(C2×D3)⋊C4—24no—
233(C2×D3)⋊C4—42no—
234(C2×D3)⋊C4—42no—
235(C2×D3)⋊C4—81no—
236(C2×D3)⋊C4—81no—
237C4⋊Dic3—21no—
238C4⋊Dic3—24no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3C22×C412yes—
244Dic3⋊C4—22no—
245Dic3⋊C4—42no—
246Dic3⋊C4—42no—
247Dic3⋊C4—81no—
248Dic3⋊C4—81no—
249Dic3⋊C4SD1614yescomplemented
250C4×Dic3—21no—
251C4×Dic3—21no—
252C4×Dic3—24no—
253C4×Dic3—42no—
254C4×Dic3—81no—
255C4×Dic3—81no—
256C4×Dic3C22×C412yes—
257C4×Dic3#4112yes—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8C22×C412yes—
262C2×C3⋊C8#4112yes—
263C2×C3⋊C8#4112yes—
264C22×Q8C22×D311yescharacteristic
265C22×D4—31no—
266C23×C4—61no—
267C2×SD16—34no—
268C2×SD16—62no—
269C2×SD16—62no—
270C22×C8—31no—
271C22×C8—31no—
272C22×C8C22×D311yescharacteristic
273C22×C8C22×D311yescharacteristic
274C4×Q8—34no—
275C4×Q8—62no—
276C4×D4—34no—
277C4×D4—62no—
278#117—124no—
279C2×C4⋊C4—31no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4C22×D311yescharacteristic
290C2×C4⋊C4C22×D311yescharacteristic
291C2×C4⋊C4C22×D311yescharacteristic
292C2×C22⋊C4—61no—
293C2×C22⋊C4—61no—
294C2×C22⋊C4—61no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—61no—
297C2×C42—21no—
298C2×C42—31no—
299C2×C42—31no—
300C2×C42—61no—
301C2×C42—61no—
302C2×C42—61no—
303C2×C42—61no—
304C2×C42C22×D311yescharacteristic
305#106—34no—
306#106—41no—
307#106—62no—
308#106—62no—
309#106—121no—
310C4⋊C8—62no—
311#103—41no—
312#103—121no—
313#103C4×D314yes—
314#102—34no—
315#102—121no—
316#102—121no—
317C4×C8—34no—
318C4×C8—62no—
319C4×C8—62no—
320#95—21no—
321#95—21no—
322#95—21no—
323#95—31no—
324#95—31no—
325#95—61no—
326#95—61no—
327#95—61no—
328#95—61no—
329#95—61no—
330C23×C3C22×D411yescharacteristic
331C22×D3—22no—
332C22×D3—41no—
333C22×D3—41no—
334C22×D3—41no—
335C22×D3—41no—
336C22×D3—41no—
337C22×D3—41no—
338C3×Q8—22no—
339C3×Q8—22no—
340C3×Q8—41no—
341C3×Q8—41no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4C2×SD1612yes—
360C2×C3×C4C23×C411yescharacteristic
361C2×C3×C4C2×#4111yescharacteristic
362C2×C3×C4C2×#4111yescharacteristic
363C2×C3×C4C2×#4111yescharacteristic
364C2×C3×C4H(5,𝔽2)11yescharacteristic
365C2×C3×C42−1+411yescharacteristic
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—22no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3C4×D412yes—
383C2×Dic3C4×D412yes—
384C2×Dic3C2×SD1612yes—
385C2×Dic3C2×SD1612yes—
386C2×Dic3#13512yes—
387D12—22no—
388D12—22no—
389D12—41no—
390D12—41no—
391C4×D3—48no—
392C3×C8—22no—
393C3×C8—41no—
394C3⋊C8—41no—
395C3⋊C8#11714yescomplemented
396C24—61no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8C2×C4×D312yes—
403C2×D4—32no—
404C2×D4—61no—
405C2×D4—61no—
406C2×D4—61no—
407C2×D4—61no—
408C2×D4—61no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—31no—
414C22×C4—31no—
415C22×C4—31no—
416C22×C4—31no—
417C22×C4—64no—
418C22×C4—61no—
419C22×C4—61no—
420C22×C4—61no—
421C22×C4—122no—
422C22×C4—122no—
423C22×C4D3×D411yescharacteristic
424C22×C4D3×D411yescharacteristic
425C22×C4C23×D311yescharacteristic
426SD16—68no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—32no—
433C2×C8—32no—
434C2×C8—32no—
435C2×C8—61no—
436C2×C8—61no—
437C2×C8—61no—
438C2×C8C2×C4×D312yes—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—34no—
445C4⋊C4—32no—
446C4⋊C4—62no—
447C4⋊C4—64no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C4⋊C4—61no—
451C4⋊C4—61no—
452C4⋊C4—61no—
453C4⋊C4—122no—
454C4⋊C4—122no—
455C4⋊C4—241no—
456C4⋊C4—241no—
457C4⋊C4C2×C4×D312yes—
458C22⋊C4—64no—
459C22⋊C4—122no—
460C22⋊C4—122no—
461C22⋊C4—241no—
462C22⋊C4—241no—
463C42—21no—
464C42—21no—
465C42—32no—
466C42—32no—
467C42—42no—
468C42—64no—
469C42—61no—
470C42—61no—
471C42—122no—
472C42—241no—
473C42—241no—
474C22×C3C2×C4×D411yescharacteristic
475C22×C3#54711yescharacteristic
476C22×C3#55011yescharacteristic
477C22×C3C22×SD1611yescharacteristic
478C22×C3C2×#13511yescharacteristic
479C22×C3#57711yescharacteristic
480C22×C3#57711yescharacteristic
481C2×D3—42no—
482C2×D3—42no—
483C2×D3—42no—
484C2×D3—81no—
485C2×D3—81no—
486C2×D3—81no—
487C2×D3—81no—
488C3×C4—22no—
489C3×C4—41no—
490C3×C4—42no—
491C3×C4—42no—
492C3×C4—81no—
493C3×C4—81no—
494C3×C4C2×#11711yescharacteristic
495C3×C4#52011yescharacteristic
496C3×C4#55511yescharacteristic
497C3×C4#55611yescharacteristic
498Dic3—22no—
499Dic3—22no—
500Dic3—41no—
501Dic3—42no—
502Dic3—81no—
503Dic3—81no—
504Dic3C4×SD1614yes—
505C23—62no—
506C23—121no—
507C23—121no—
508C23—121no—
509C23—121no—
510C23—121no—
511C23—121no—
512C23C2×D3×D411yescharacteristic
513Q8—22no—
514Q8—22no—
515Q8—41no—
516Q8—41no—
517D4—62no—
518D4—62no—
519D4—121no—
520D4—121no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—32no—
529C2×C4—32no—
530C2×C4—32no—
531C2×C4—32no—
532C2×C4—32no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—61no—
544C2×C4—61no—
545C2×C4—61no—
546C2×C4—61no—
547C2×C4—61no—
548C2×C4—61no—
549C2×C4—61no—
550C2×C4—62no—
551C2×C4—128no—
552C2×C4—121no—
553C2×C4—121no—
554C2×C4—121no—
555C2×C4—121no—
556C2×C4—121no—
557C2×C4—121no—
558C2×C4—121no—
559C2×C4—121no—
560C2×C4D3×SD1612yes—
561C2×C4C22×C4×D311yescharacteristic
562C2×C4#100411yescharacteristic
563C2×C4D3×#4111yescharacteristic
564C2×C4D3×#4111yescharacteristic
565C2×C4D3×#4111yescharacteristic
566C2×C4#101011yescharacteristic
567C8—22no—
568C8—34no—
569C8—41no—
570C8—121no—
571C2×C3#235411yescharacteristic
572C2×C3C2×C4×SD1611yescharacteristic
573C2×C3#295211yescharacteristic
574C2×C3#333511yescharacteristic
575C2×C3#334011yescharacteristic
576C2×C3#339211yescharacteristic
577C2×C3#339411yescharacteristic
578D3—82no—
579C22—122no—
580C22—122no—
581C22—122no—
582C22—241no—
583C22—241no—
584C22—241no—
585C22—241no—
586C22C4×D3×D411yescharacteristic
587C22#559911yescharacteristic
588C22#565511yescharacteristic
589C22C2×D3×SD1611yescharacteristic
590C22D3×#13511yescharacteristic
591C22#577811yescharacteristic
592C22#577811yescharacteristic
593C4—22no—
594C4—34no—
595C4—42no—
596C4—41no—
597C4—42no—
598C4—62no—
599C4—62no—
600C4—81no—
601C4—81no—
602C4—121no—
603C4—122no—
604C4—241no—
605C4—241no—
606C4D3×#11711yescharacteristic
607C4#554611yescharacteristic
608C4#567011yescharacteristic
609C4#570211yescharacteristic
610C3#2280511yescharacteristic, complemented, Sylow
611C2—242no—
612C2#7547511yescharacteristic
613C2C4×D3×SD1611yescharacteristic
614C2#7983211yescharacteristic
615C2#8042011yescharacteristic
616C2#8058911yescharacteristic
617C2#8143511yescharacteristic
618C2#8153211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1927C2
socleC23×C3C22×D4
prime radicalC22×D12C23
cosocleC22×C3×C4C24
hypercenter#1927D3
hypocenterC3#22805
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1927

Complements

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

NH ≅ G/Ntype
2. #76489C2semidirect
4. C3×#1927C2semidirect
6. #70554C2semidirect
8. #70223C2semidirect
9. #70221C2semidirect
11. C4×#808C2semidirect
13. #68784C2semidirect
14. #68651C2semidirect
48. #4788C4semidirect
67. #1927D3semidirect
249. Dic3⋊C4SD16semidirect
395. C3⋊C8#117semidirect
610. C3#22805semidirect

Maximal subgroups

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

subgroupindex
C2×#47882
#764892
#736392
C3×#19272
#705692
#705542
#702572
#702232
#702212
#696762
C4×#8082
#688922
#687842
#686512
#686262
#228053