Finite Group Databasebeta

← #10720123

Subgroups — #10720123

Order 768.

2366 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4⋊Dic3C211yesmaximal, characteristic
2C2×C3×C4×C22⋊C4C211yesmaximal, characteristic
3C2×#5212C211yesmaximal, characteristic
4C2×#4946C212yesmaximal
5C2×#4928C212yesmaximal, complemented
6#71544C212yesmaximal
7#71535C212yesmaximal, complemented
8#69420C212yesmaximal, complemented
9#69415C212yesmaximal, complemented
10#14130—31nomaximal, Sylow
11C24×C3×C4C2211yescharacteristic
12C24×Dic3C2212yes—
13C22×C3×C22⋊C4C2211yescharacteristic
14C22×C3×C22⋊C4C2211yescharacteristic
15C22×C3×C42C2211yescharacteristic
16C22×C3×C42C2211yescharacteristic
17C22×#217C2212yes—
18C22×C4⋊Dic3—24no—
19C22×C4⋊Dic3—28no—
20C22×C4⋊Dic3C2212yes—
21C22×C4⋊Dic3C2212yes—
22C22×C4⋊Dic3C2212yes—
23C22×Dic3⋊C4C2214yes—
24C3×C4×C22⋊C4—24no—
25C2×C3×#95C2211yescharacteristic
26C2×C3×#95C2211yescharacteristic
27#5212C418yescomplemented
28C2×C3⋊#95C2214yes—
29C2×C3⋊#95C2214yes—
30C2×C3⋊#95C2214yes—
31C2×C3⋊#95C2212yes—
32#4946—28no—
33#4928—28no—
34C23×C4⋊C4—31no—
35C2×#389—32no—
36C2×#385—32no—
37C2×#380—31no—
38C2×C4×C22⋊C4D311yescharacteristic
39#1444—32no—
40#1442—32no—
41C4⋊#95—32no—
42C22⋊#95—32no—
43C25×C3C2311yescharacteristic
44C23×C3×C4—24no—
45C23×C3×C4—24no—
46C23×C3×C4C2×C414yes—
47C23×C3×C4D414yes—
48C23×C3×C4Q814yes—
49C23×C3×C4C2311yescharacteristic
50C23×C3×C4C2311yescharacteristic
51C23×C3×C4C2311yescharacteristic
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2311yescharacteristic
55C23×Dic3—24no—
56C23×Dic3—28no—
57C23×Dic3—24no—
58C23×Dic3—24no—
59C23×Dic3—44no—
60C23×Dic3C2314yes—
61C23×Dic3C2314yes—
62C2×C3×C22⋊C4—22no—
63C2×C3×C22⋊C4—22no—
64C2×C3×C22⋊C4—22no—
65C2×C3×C22⋊C4—22no—
66C2×C3×C22⋊C4D414yes—
67C2×C3×C22⋊C4Q814yes—
68C2×C3×C42—24no—
69C2×C3×C42—22no—
70C2×C3×C42—22no—
71C2×C3×C42—24no—
72C2×#217—24no—
73C2×#217—24no—
74C2×#217C2×C418yes—
75C2×C4⋊Dic3—232no—
76C2×C4⋊Dic3—24no—
77C2×C4⋊Dic3—24no—
78C2×C4⋊Dic3—24no—
79C2×C4⋊Dic3—24no—
80C2×C4⋊Dic3—24no—
81C2×C4⋊Dic3—28no—
82C2×C4⋊Dic3C2×C418yescomplemented
83C2×C4⋊Dic3D418yes—
84C2×Dic3⋊C4—28no—
85C2×Dic3⋊C4—28no—
86C2×Dic3⋊C4—44no—
87C3×#95—22no—
88C3×#95—22no—
89C3⋊#95—28no—
90C3⋊#95—28no—
91C3⋊#95—44no—
92C3⋊#95C2×C418yescomplemented
93C24×C4—32no—
94C24×C4C2×D311yescharacteristic
95C22×C4⋊C4—32no—
96C22×C4⋊C4—32no—
97C22×C4⋊C4—32no—
98C22×C4⋊C4—34no—
99C22×C4⋊C4—68no—
100C22×C4⋊C4—64no—
101C22×C22⋊C4—32no—
102C22×C22⋊C4C2×D311yescharacteristic
103C22×C22⋊C4C2×D311yescharacteristic
104C22×C42C2×D311yescharacteristic
105C22×C42C2×D311yescharacteristic
106#389—68no—
107#385—68no—
108#380—38no—
109C4×C22⋊C4—24no—
110C2×#95—32no—
111C2×#95—34no—
112C2×#95—34no—
113C2×#95—34no—
114C2×#95C2×D311yescharacteristic
115C2×#95C2×D311yescharacteristic
116C24×C3—22no—
117C24×C3—22no—
118C24×C3—22no—
119C24×C3—22no—
120C24×C3C22×C412yes—
121C24×C3C2×D412yes—
122C24×C3C2×D412yes—
123C24×C3C2×Q812yes—
124C24×C3C2×Q812yes—
125C24×C3#4112yes—
126C24×C3#4112yes—
127C24×C3C2411yescharacteristic
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—216no—
131C22×C3×C4—28no—
132C22×C3×C4—216no—
133C22×C3×C4—28no—
134C22×C3×C4—22no—
135C22×C3×C4—21no—
136C22×C3×C4—21no—
137C22×C3×C4—21no—
138C22×C3×C4—21no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—21no—
143C22×C3×C4—21no—
144C22×C3×C4—21no—
145C22×C3×C4—21no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4C4⋊C4116yescomplemented
151C22×C3×C4C22×C412yes—
152C22×C3×C4C22×C412yes—
153C22×C3×C4C2×D412yes—
154C22×C3×C4C2×D412yes—
155C22×C3×C4C2×D414yes—
156C22×C3×C4C2×D412yes—
157C22×C3×C4C2×D412yes—
158C22×C3×C4C2×Q812yes—
159C22×C3×C4C2×Q812yes—
160C22×C3×C4C2×Q812yes—
161C22×C3×C4C2×Q812yes—
162C22×C3×C4#4114yes—
163C22×Dic3—216no—
164C22×Dic3—24no—
165C22×Dic3—24no—
166C22×Dic3—24no—
167C22×Dic3—24no—
168C22×Dic3—24no—
169C22×Dic3—24no—
170C22×Dic3—24no—
171C22×Dic3—24no—
172C22×Dic3—24no—
173C22×Dic3—24no—
174C22×Dic3—24no—
175C22×Dic3—24no—
176C22×Dic3—416no—
177C22×Dic3—416no—
178C22×Dic3—48no—
179C22×Dic3—48no—
180C22×Dic3C22×C418yes—
181C22×Dic3C2×D418yes—
182C3×C22⋊C4—28no—
183C3×C22⋊C4—28no—
184C3×C42—44no—
185C3×C42—44no—
186#217—216no—
187C4⋊Dic3—216no—
188C4⋊Dic3—216no—
189C4⋊Dic3—216no—
190Dic3⋊C4—416no—
191C25C22×D311yescharacteristic
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—34no—
195C23×C4—34no—
196C23×C4—68no—
197C23×C4—64no—
198C23×C4—64no—
199C23×C4—64no—
200C23×C4—124no—
201C23×C4Dic614yes—
202C23×C4C4×D314yes—
203C23×C4#6714yes—
204C23×C4C22×D311yescharacteristic
205C23×C4C22×D311yescharacteristic
206C23×C4C22×D311yescharacteristic
207C23×C4C22×D311yescharacteristic
208C23×C4C22×D311yescharacteristic
209C23×C4C22×D311yescharacteristic
210C2×C4⋊C4—38no—
211C2×C4⋊C4—38no—
212C2×C4⋊C4—68no—
213C2×C4⋊C4—632no—
214C2×C4⋊C4—64no—
215C2×C4⋊C4—64no—
216C2×C4⋊C4—64no—
217C2×C4⋊C4—64no—
218C2×C4⋊C4—64no—
219C2×C4⋊C4—68no—
220C2×C4⋊C4—68no—
221C2×C4⋊C4—124no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—38no—
227C2×C22⋊C4—64no—
228C2×C22⋊C4—64no—
229C2×C22⋊C4Dic614yes—
230C2×C22⋊C4D1214yes—
231C2×C42—24no—
232C2×C42—24no—
233C2×C42—22no—
234C2×C42—22no—
235#95—22no—
236#95—22no—
237#95—38no—
238#95—68no—
239#95—68no—
240#95—124no—
241C23×C3—24no—
242C23×C3—24no—
243C23×C3—22no—
244C23×C3—24no—
245C23×C3—22no—
246C23×C3—22no—
247C23×C3—22no—
248C23×C3—24no—
249C23×C3—24no—
250C23×C3—24no—
251C23×C3—24no—
252C23×C3—24no—
253C23×C3—24no—
254C23×C3—24no—
255C23×C3—24no—
256C23×C3—24no—
257C23×C3C2×C4⋊C414yes—
258C23×C3C4×D414yes—
259C23×C3C4×Q814yes—
260C23×C3#12114yes—
261C23×C3C22⋊Q814yes—
262C23×C3#12514yes—
263C23×C3C4⋊Q814yes—
264C23×C3C23×C411yescharacteristic
265C23×C3C22×D411yescharacteristic
266C23×C3C22×D412yes—
267C23×C3C22×D411yescharacteristic
268C23×C3C22×Q811yescharacteristic
269C23×C3C22×Q811yescharacteristic
270C23×C3C2×#4111yescharacteristic
271C23×C3C2×#4111yescharacteristic
272C23×C3C2×#4112yes—
273C23×C3H(5,𝔽2)12yes—
274C23×C32−1+412yes—
275C2×C3×C4—24no—
276C2×C3×C4—22no—
277C2×C3×C4—24no—
278C2×C3×C4—22no—
279C2×C3×C4—232no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—24no—
285C2×C3×C4—24no—
286C2×C3×C4—24no—
287C2×C3×C4—24no—
288C2×C3×C4—41no—
289C2×C3×C4—41no—
290C2×C3×C4—41no—
291C2×C3×C4—41no—
292C2×C3×C4—41no—
293C2×C3×C4—41no—
294C2×C3×C4—41no—
295C2×C3×C4—41no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4C2×C4⋊C414yes—
301C2×C3×C4C2×C4⋊C414yes—
302C2×C3×C4C4×D418yes—
303C2×C3×C4#12014yes—
304C2×C3×C4C22⋊Q818yes—
305C2×C3×C4#12314yes—
306C2×Dic3—28no—
307C2×Dic3—28no—
308C2×Dic3—28no—
309C2×Dic3—28no—
310C2×Dic3—28no—
311C2×Dic3—28no—
312C2×Dic3—432no—
313C2×Dic3—44no—
314C2×Dic3—44no—
315C2×Dic3—44no—
316C2×Dic3—44no—
317C2×Dic3—44no—
318C2×Dic3—44no—
319C2×Dic3—44no—
320C2×Dic3—44no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24C2×Dic612yes—
326C24C2×Dic612yes—
327C24C2×C4×D312yes—
328C24C2×D1212yes—
329C24#23512yes—
330C24#23512yes—
331C24C2×#6712yes—
332C24C23×D311yescharacteristic
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—216no—
336C22×C4—216no—
337C22×C4—28no—
338C22×C4—28no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—38no—
356C22×C4—38no—
357C22×C4—616no—
358C22×C4—64no—
359C22×C4—64no—
360C22×C4—64no—
361C22×C4—64no—
362C22×C4—64no—
363C22×C4—64no—
364C22×C4—64no—
365C22×C4—64no—
366C22×C4—64no—
367C22×C4—64no—
368C22×C4—64no—
369C22×C4—64no—
370C22×C4—1216no—
371C22×C4—1216no—
372C22×C4—128no—
373C22×C4—128no—
374C22×C4Dic3⋊C4116yescomplemented
375C22×C4C2×Dic612yes—
376C22×C4C2×Dic612yes—
377C22×C4C2×Dic612yes—
378C22×C4C2×Dic612yes—
379C22×C4C2×C4×D312yes—
380C22×C4C2×C4×D312yes—
381C22×C4C2×D1212yes—
382C22×C4C2×D1212yes—
383C22×C4D3×D414yes—
384C22×C4#23714yes—
385C22×C4C2×#6712yes—
386C22×C4C2×#6712yes—
387C4⋊C4—616no—
388C4⋊C4—616no—
389C4⋊C4—616no—
390C4⋊C4—1216no—
391C22⋊C4—28no—
392C22⋊C4—28no—
393C22⋊C4—616no—
394C42—44no—
395C42—44no—
396C22×C3—24no—
397C22×C3—28no—
398C22×C3—28no—
399C22×C3—24no—
400C22×C3—24no—
401C22×C3—28no—
402C22×C3—28no—
403C22×C3—24no—
404C22×C3—24no—
405C22×C3—24no—
406C22×C3#38918yescomplemented
407C22×C3C22×C4⋊C411yescharacteristic
408C22×C3C2×C4×D411yescharacteristic
409C22×C3C2×C4×D412yes—
410C22×C3C2×C4×Q811yescharacteristic
411C22×C3#52012yes—
412C22×C3C2×#12011yescharacteristic
413C22×C3C2×#12111yescharacteristic
414C22×C3C2×C22⋊Q812yes—
415C22×C3C2×C22⋊Q811yescharacteristic
416C22×C3C2×#12311yescharacteristic
417C22×C3C2×#12511yescharacteristic
418C22×C3C2×C4⋊Q811yescharacteristic
419C22×C3#53511yescharacteristic
420C22×C3#53611yescharacteristic
421C22×C3#54412yes—
422C22×C3D4212yes—
423C22×C3#54612yes—
424C22×C3#54712yes—
425C22×C3#54812yes—
426C22×C3D4×Q812yes—
427C22×C3#55212yes—
428C22×C3#55312yes—
429C22×C3#55412yes—
430C3×C4—24no—
431C3×C4—42no—
432C3×C4—42no—
433C3×C4—42no—
434C3×C4—42no—
435C3×C4#38518yescomplemented
436Dic3—48no—
437Dic3—48no—
438C23—24no—
439C23—24no—
440C23—24no—
441C23—24no—
442C23—24no—
443C23—22no—
444C23—24no—
445C23—24no—
446C23—22no—
447C23—24no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—22no—
453C23—22no—
454C23C4×Dic614yes—
455C23#86914yes—
456C23#87014yes—
457C23C4×D1214yes—
458C23C2×Dic3⋊C414yes—
459C23#92414yes—
460C23#93014yes—
461C23C22×Dic611yescharacteristic
462C23C22×Dic611yescharacteristic
463C23C22×C4×D311yescharacteristic
464C23C22×D1211yescharacteristic
465C23C2×#23511yescharacteristic
466C23C2×#23511yescharacteristic
467C23C2×D3×D412yes—
468C23C2×#23712yes—
469C23#100912yes—
470C23#101012yes—
471C23C22×#6711yescharacteristic
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—232no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—68no—
498C2×C4—68no—
499C2×C4—68no—
500C2×C4—68no—
501C2×C4—68no—
502C2×C4—68no—
503C2×C4—1232no—
504C2×C4—124no—
505C2×C4—124no—
506C2×C4—124no—
507C2×C4—124no—
508C2×C4—124no—
509C2×C4—124no—
510C2×C4—124no—
511C2×C4—124no—
512C2×C4#87818yes—
513C2×C4#67⋊C418yes—
514C2×C4C2×Dic3⋊C414yes—
515C2×C4C2×Dic3⋊C414yes—
516C2×C4#93514yes—
517C2×C4#93714yes—
518C2×C3—28no—
519C2×C3C2×#38511yescharacteristic
520C2×C3C2×#38911yescharacteristic
521C2×C3#231711yescharacteristic
522C2×C3#236012yes—
523C2×C3#236112yes—
524C2×C3#240811yescharacteristic
525C2×C3#240911yescharacteristic
526C2×C3#244012yes—
527C2×C3#244212yes—
528C2×C3#248611yescharacteristic
529C2×C3#248711yescharacteristic
530C22—28no—
531C22—28no—
532C22—24no—
533C22—24no—
534C22—24no—
535C22—24no—
536C22—24no—
537C22—24no—
538C22—28no—
539C22—28no—
540C22#492818yescomplemented
541C22C2×C4×Dic611yescharacteristic
542C22C2×#86911yescharacteristic
543C22C2×#87011yescharacteristic
544C22C2×C4×D1211yescharacteristic
545C22C2×#87812yes—
546C22C2×#67⋊C412yes—
547C22#551912yes—
548C22#552312yes—
549C22D4×Dic612yes—
550C22#554112yes—
551C22#554312yes—
552C22D4×D1212yes—
553C22#555012yes—
554C22#555212yes—
555C22#555512yes—
556C22#556512yes—
557C22C22×Dic3⋊C411yescharacteristic
558C22C2×#92411yescharacteristic
559C22C2×#93011yescharacteristic
560C22C2×#93511yescharacteristic
561C22C2×#93711yescharacteristic
562C22#582511yescharacteristic
563C22#583011yescharacteristic
564C4—24no—
565C4—42no—
566C4—42no—
567C4—42no—
568C4—42no—
569C4—128no—
570C4—128no—
571C4#494618yescomplemented
572C3#1413011yescharacteristic, complemented, Sylow
573C2—28no—
574C2#7346512yes—
575C2#7346712yes—
576C2#7349312yes—
577C2#7349512yes—
578C2C2×#492811yescharacteristic
579C2C2×#494611yescharacteristic
580C2#8223411yescharacteristic
581C2#8234911yescharacteristic
582C2#8237311yescharacteristic
583C2#8237411yescharacteristic
584C2#8240311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC2×C3×C4×C22⋊C4C2
socle, cosocleC24×C3C24
prime radicalC25×C3C23
hypercenterC2×C4×C22⋊C4D3
hypocenterC3#14130
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
5. C2×#4928C2semidirect
7. #71535C2semidirect
8. #69420C2semidirect
9. #69415C2semidirect
27. #5212C4semidirect
82. C2×C4⋊Dic3C2×C4semidirect
92. C3⋊#95C2×C4semidirect
150. C22×C3×C4C4⋊C4semidirect
374. C22×C4Dic3⋊C4semidirect
406. C22×C3#389semidirect
435. C3×C4#385semidirect
540. C22#4928semidirect
571. C4#4946semidirect
572. C3#14130semidirect

Maximal subgroups

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

subgroupindex
C23×C4⋊Dic32
C2×C3×C4×C22⋊C42
C2×#52122
C2×#49462
C2×#49282
#715442
#715352
#694202
#694152
#141303