Finite Group Databasebeta

← #10804875

Subgroups — #10804875

Order 768.

1166 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5624C211yesmaximal, characteristic
2C2×#4977C211yesmaximal, characteristic, complemented
3C2×#4718C211yesmaximal, characteristic, complemented
4C3×#1872C211yesmaximal, characteristic, complemented
5#71692C211yesmaximal, characteristic, complemented
6#71560C211yesmaximal, characteristic
7#70226C212yesmaximal, complemented
8#70222C212yesmaximal, complemented
9#69932C211yesmaximal, characteristic
10#68893C212yesmaximal, complemented
11#68804C212yesmaximal, complemented
12#21861—31nomaximal, Sylow
13C2×C3×C22⋊Q8C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C2×#947C2211yescharacteristic
16C2×Q8×Dic3C2211yescharacteristic
17C2×#930C2211yescharacteristic
18C22×C4⋊Dic3C2211yescharacteristic
19C22×C4×Dic3C2211yescharacteristic
20#5624—24no—
21#5624C418yescomplemented
22C2×#898—21no—
23C2×#893C2211yescharacteristic
24C2×#892—21no—
25C2×#890C2211yescharacteristic
26C2×#886—21no—
27C2×#67⋊C4—21no—
28C2×C3×#103C2212yes—
29C2×C3×C22⋊C8C2211yescharacteristic
30C2×#835C2212yes—
31C2×C3⋊#95—21no—
32C2×C3⋊#95—21no—
33C2×#830C2211yescharacteristic
34C2×#821C2212yes—
35C2×Dic3⋊C8C2212yes—
36C2×C8×Dic3C2212yes—
37#4977—21no—
38#4977—21no—
39#4977C2212yes—
40#4977C2212yes—
41C2×#809C2212yes—
42#4718—24no—
43C3×#340C2212yes—
44#4551C2212yes—
45#4529C2212yes—
46#4529C2212yes—
47C2×#529—31no—
48C2×#431—31no—
49C2×#382—31no—
50#1991—32no—
51#1958—32no—
52#1874—31no—
53#1874—31no—
54#1872—31no—
55#1872D311yescharacteristic, complemented
56#1805—32no—
57#1804—32no—
58C22×C3×Q8C2311yescharacteristic
59C23×C3×C4—21no—
60C23×C3×C4C2311yescharacteristic
61C22×#67—21no—
62C23×Dic3—21no—
63C23×Dic3D411yescharacteristic
64C23×Dic3D411yescharacteristic
65C22×D12C2311yescharacteristic
66C22×C4×D3—21no—
67C22×C3×C8C2312yes—
68C3×C22⋊Q8—22no—
69C3×C22⋊Q8C2×C414yes—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—42no—
73C2×C3×C4⋊C4C2312yes—
74C2×C3×C4⋊C4C2311yescharacteristic
75C2×C3×C22⋊C4—21no—
76#947—22no—
77#947C2×C414yescomplemented
78Q8×Dic3—22no—
79Q8×Dic3C2×C414yescomplemented
80#930—22no—
81#930C2×C414yes—
82C2×(C2×D3)⋊C4—21no—
83C2×(C2×D3)⋊C4—21no—
84C2×(C2×D3)⋊C4—21no—
85C2×(C2×D3)⋊C4—21no—
86C2×C4⋊Dic3—21no—
87C2×C4⋊Dic3—21no—
88C2×C4⋊Dic3—21no—
89C2×C4⋊Dic3—42no—
90C2×C4⋊Dic3C2312yes—
91C2×C4⋊Dic3C2311yescharacteristic
92C2×Dic3⋊C4—21no—
93C2×C4×Dic3—21no—
94C2×C4×Dic3C2×C414yes—
95C2×C4×Dic3D411yescharacteristic
96C2×C4×Dic3D411yescharacteristic
97C2×C4×Dic3D414yes—
98C2×C4×Dic3C2311yescharacteristic
99C2×C4×Dic3C2311yescharacteristic
100C22×C3⋊C8C2312yes—
101#898—28no—
102#893—22no—
103#893C2×C414yescomplemented
104#892—28no—
105#890—22no—
106#890C2×C414yescomplemented
107#886—28no—
108#67⋊C4—28no—
109C3×#103—42no—
110C3×C22⋊C8—22no—
111#835—42no—
112C3⋊#95—22no—
113C3⋊#95—22no—
114C3⋊#95—22no—
115C3⋊#95—22no—
116#830—22no—
117#821—42no—
118Dic3⋊C8—24no—
119C8×Dic3—24no—
120#809—42no—
121#529—38no—
122#529—64no—
123C2×#126—61no—
124C2×#125—31no—
125C2×#124—31no—
126C2×#123—61no—
127C2×C22⋊Q8C2×D311yescharacteristic
128C2×#121—31no—
129C2×C4×Q8—31no—
130C2×C4×D4—31no—
131C2×C4×D4—61no—
132C2×#117—61no—
133C22×C4⋊C4—31no—
134C22×C4⋊C4C2×D311yescharacteristic
135C22×C42—31no—
136#431—64no—
137C2×C4⋊C8—32no—
138C2×#103—32no—
139C2×#103C2×D312yes—
140C2×#102—32no—
141C2×#102—32no—
142C2×C22⋊C8—31no—
143C2×C22⋊C8C2×D311yescharacteristic
144C2×C4×C8—32no—
145#382—32no—
146#382—32no—
147#382—61no—
148#382—61no—
149C2×#95—61no—
150C2×#95—61no—
151#340—32no—
152#340—32no—
153#340—32no—
154#340C2×D312yes—
155C24×C3C2×D411yescharacteristic
156C23×D3—21no—
157C2×C3×Q8—21no—
158C2×C3×Q8—22no—
159C2×C3×Q8—22no—
160C2×C3×Q8C22×C412yes—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—22no—
166C22×C3×C4—42no—
167C22×C3×C4—42no—
168C22×C3×C4—42no—
169C22×C3×C4C22×C412yes—
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2411yescharacteristic
173C2×#67—24no—
174C2×#67—42no—
175C2×#67—42no—
176C22×Dic3—22no—
177C22×Dic3—22no—
178C22×Dic3—21no—
179C22×Dic3—22no—
180C22×Dic3—41no—
181C22×Dic3—41no—
182C22×Dic3—42no—
183C22×Dic3—42no—
184C22×Dic3—42no—
185C22×Dic3C22⋊C414yes—
186C22×Dic3C22⋊C414yes—
187C22×Dic3C2×D411yescharacteristic
188C22×Dic3C2×D411yescharacteristic
189C22×Dic3C2×D411yescharacteristic
190C22×Dic3C2×D411yescharacteristic
191C2×D12—22no—
192C2×D12—22no—
193C2×D12—21no—
194C2×D12C22×C412yes—
195C2×C4×D3—24no—
196C2×C4×D3—42no—
197C2×C4×D3—42no—
198C2×C3×C8—22no—
199C2×C3×C8—22no—
200C2×C3×C8—22no—
201C3×C4⋊C4—24no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—42no—
206C3×C4⋊C4C22×C412yes—
207C3×C22⋊C4—24no—
208(C2×D3)⋊C4—24no—
209(C2×D3)⋊C4—24no—
210(C2×D3)⋊C4—24no—
211(C2×D3)⋊C4—24no—
212C4⋊Dic3—21no—
213C4⋊Dic3—24no—
214C4⋊Dic3—24no—
215C4⋊Dic3—22no—
216C4⋊Dic3—22no—
217C4⋊Dic3—42no—
218C4⋊Dic3C22×C412yes—
219Dic3⋊C4—24no—
220C4×Dic3—24no—
221C4×Dic3C22⋊C414yes—
222C4×Dic3C22⋊C414yes—
223C4×Dic3C22×C412yes—
224C4×Dic3C22×C412yes—
225C4×Dic3C2×D412yes—
226C4×Dic3C2×D412yes—
227C2×C3⋊C8—22no—
228C2×C3⋊C8—22no—
229C2×C3⋊C8—22no—
230C22×Q8C22×D311yescharacteristic
231C22×D4—31no—
232C22×D4—61no—
233C23×C4—21no—
234C23×C4—31no—
235C23×C4—31no—
236C23×C4—61no—
237C23×C4—61no—
238C23×C4C22×D311yescharacteristic
239C22×C8—32no—
240C22×C8C22×D312yes—
241#126—68no—
242#125—34no—
243#125—62no—
244#124—34no—
245#124—62no—
246#123—68no—
247C22⋊Q8—22no—
248C22⋊Q8C4×D314yes—
249#121—34no—
250#121—62no—
251C4×Q8—34no—
252C4×Q8—62no—
253C4×D4—34no—
254C4×D4—62no—
255C4×D4—68no—
256#117—68no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—31no—
260C2×C4⋊C4—32no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—61no—
263C2×C4⋊C4—61no—
264C2×C4⋊C4—61no—
265C2×C4⋊C4—61no—
266C2×C4⋊C4—122no—
267C2×C4⋊C4C22×D312yes—
268C2×C4⋊C4C22×D311yescharacteristic
269C2×C22⋊C4—21no—
270C2×C22⋊C4—61no—
271C2×C22⋊C4—61no—
272C2×C22⋊C4—61no—
273C2×C22⋊C4—61no—
274C2×C42—34no—
275C2×C42—31no—
276C2×C42—31no—
277C2×C42—34no—
278C2×C42—31no—
279C2×C42—31no—
280C2×C42—61no—
281C4⋊C8—64no—
282#103—42no—
283#103—122no—
284#102—122no—
285#102—122no—
286C22⋊C8—22no—
287C22⋊C8—62no—
288C4×C8—64no—
289#95—62no—
290#95—62no—
291#95—62no—
292#95—62no—
293C23×C3—21no—
294C23×C3—21no—
295C23×C3—21no—
296C23×C3—21no—
297C23×C3C2×C22⋊C412yes—
298C23×C3#13512yes—
299C23×C3#13512yes—
300C23×C3C22×D411yescharacteristic
301C22×D3—22no—
302C22×D3—41no—
303C22×D3—41no—
304C22×D3—41no—
305C22×D3—41no—
306C22×D3—41no—
307C22×D3—41no—
308C3×Q8—24no—
309C3×Q8—42no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—24no—
313C2×C3×C4—24no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—41no—
317C2×C3×C4—41no—
318C2×C3×C4—41no—
319C2×C3×C4—41no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—84no—
324C2×C3×C4C2×C22⋊C412yes—
325C2×C3×C4C23×C411yescharacteristic
326C2×C3×C4C22×D411yescharacteristic
327C2×C3×C4H(5,𝔽2)12yes—
328C2×C3×C42−1+412yes—
329#67—48no—
330C2×Dic3—21no—
331C2×Dic3—21no—
332C2×Dic3—21no—
333C2×Dic3—21no—
334C2×Dic3—21no—
335C2×Dic3—21no—
336C2×Dic3—21no—
337C2×Dic3—21no—
338C2×Dic3—28no—
339C2×Dic3—22no—
340C2×Dic3—44no—
341C2×Dic3—41no—
342C2×Dic3—41no—
343C2×Dic3—42no—
344C2×Dic3—42no—
345C2×Dic3—42no—
346C2×Dic3—84no—
347C2×Dic3C2×C22⋊C412yes—
348C2×Dic3C2×C22⋊C412yes—
349C2×Dic3C2×C22⋊C412yes—
350C2×Dic3C2×C22⋊C412yes—
351D12—24no—
352D12—42no—
353C4×D3—48no—
354C3×C8—42no—
355C3⋊C8—42no—
356C24—61no—
357C24D3×D411yescharacteristic
358C2×Q8—21no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8C2×C4×D312yes—
362C2×D4—32no—
363C2×D4—64no—
364C2×D4—61no—
365C2×D4—62no—
366C2×D4—62no—
367C2×D4—122no—
368C2×D4—122no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—34no—
375C22×C4—31no—
376C22×C4—31no—
377C22×C4—31no—
378C22×C4—31no—
379C22×C4—34no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—64no—
384C22×C4—62no—
385C22×C4—62no—
386C22×C4—61no—
387C22×C4—62no—
388C22×C4—122no—
389C22×C4—122no—
390C22×C4—121no—
391C22×C4—121no—
392C22×C4—122no—
393C22×C4—122no—
394C22×C4—122no—
395C22×C4C2×C4×D312yes—
396C22×C4D3×D411yescharacteristic
397C22×C4D3×D412yes—
398C22×C4C23×D311yescharacteristic
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—62no—
403C2×C8—62no—
404C2×C8—62no—
405C4⋊C4—24no—
406C4⋊C4—21no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—32no—
410C4⋊C4—42no—
411C4⋊C4—64no—
412C4⋊C4—64no—
413C4⋊C4—64no—
414C4⋊C4—61no—
415C4⋊C4—62no—
416C4⋊C4—62no—
417C4⋊C4—122no—
418C4⋊C4C2×C4×D312yes—
419C22⋊C4—24no—
420C22⋊C4—64no—
421C22⋊C4—64no—
422C22⋊C4—64no—
423C22⋊C4—64no—
424C42—34no—
425C42—34no—
426C42—32no—
427C42—32no—
428C42—32no—
429C42—32no—
430C42—64no—
431C22×C3—22no—
432C22×C3—22no—
433C22×C3—22no—
434C22×C3—22no—
435C22×C3—22no—
436C22×C3—22no—
437C22×C3#41614yes—
438C22×C3C22×C22⋊C411yescharacteristic
439C22×C3#53511yescharacteristic
440C22×C3#53611yescharacteristic
441C22×C3C2×#13511yescharacteristic
442C22×C3C2×#13511yescharacteristic
443C22×C3#57511yescharacteristic
444C22×C3#57511yescharacteristic
445C2×D3—42no—
446C2×D3—42no—
447C2×D3—42no—
448C2×D3—81no—
449C2×D3—81no—
450C2×D3—81no—
451C2×D3—81no—
452C3×C4—22no—
453C3×C4—42no—
454C3×C4—42no—
455C3×C4—82no—
456C3×C4#52012yes—
457C3×C4#53712yes—
458Dic3—22no—
459Dic3—22no—
460Dic3—22no—
461Dic3—22no—
462Dic3—42no—
463Dic3—82no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—62no—
469C23—121no—
470C23—121no—
471C23—121no—
472C23—121no—
473C23—121no—
474C23—121no—
475C23D3×C22⋊C412yes—
476C23#91612yes—
477C23#91912yes—
478C23C2×D3×D411yescharacteristic
479Q8—24no—
480Q8—42no—
481D4—64no—
482D4—128no—
483D4—122no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—32no—
491C2×C4—32no—
492C2×C4—32no—
493C2×C4—32no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—61no—
502C2×C4—61no—
503C2×C4—61no—
504C2×C4—61no—
505C2×C4—61no—
506C2×C4—61no—
507C2×C4—61no—
508C2×C4—61no—
509C2×C4—68no—
510C2×C4—62no—
511C2×C4—84no—
512C2×C4—128no—
513C2×C4—124no—
514C2×C4—121no—
515C2×C4—121no—
516C2×C4—122no—
517C2×C4—122no—
518C2×C4—122no—
519C2×C4—244no—
520C2×C4D3×C22⋊C412yes—
521C2×C4C22×C4×D311yescharacteristic
522C2×C4C2×D3×D411yescharacteristic
523C2×C4#100412yes—
524C2×C4#101012yes—
525C8—42no—
526C8—122no—
527C2×C3—24no—
528C2×C3#231111yescharacteristic
529C2×C3C2×#41611yescharacteristic
530C2×C3#289611yescharacteristic
531C2×C3#321111yescharacteristic
532C2×C3#321111yescharacteristic
533C2×C3#323211yescharacteristic
534C2×C3#323211yescharacteristic
535D3—82no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—122no—
543C22—122no—
544C22—122no—
545C22—241no—
546C22—241no—
547C22—241no—
548C22—241no—
549C22#480414yes—
550C22C2×D3×C22⋊C411yescharacteristic
551C22#559211yescharacteristic
552C22#564811yescharacteristic
553C22C2×#91611yescharacteristic
554C22C2×#91911yescharacteristic
555C22#577311yescharacteristic
556C22#577711yescharacteristic
557C4—22no—
558C4—42no—
559C4—42no—
560C4—62no—
561C4—62no—
562C4—62no—
563C4—62no—
564C4—82no—
565C4—122no—
566C4—242no—
567C4#554612yes—
568C4#559712yes—
569C3#2186111yescharacteristic, complemented, Sylow
570C2—24no—
571C2—242no—
572C2#7466211yescharacteristic
573C2C2×#480411yescharacteristic
574C2#7888111yescharacteristic
575C2#8013911yescharacteristic
576C2#8020111yescharacteristic
577C2#8141911yescharacteristic
578C2#8146211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1872C2
socleC23×C3C22×D4
prime radicalC2×#930C22
cosocleC22×C3×C4C24
hypercenter#1872D3
hypocenterC3#21861
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1872

Complements

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

NH ≅ G/Ntype
2. C2×#4977C2semidirect
3. C2×#4718C2semidirect
4. C3×#1872C2semidirect
5. #71692C2semidirect
7. #70226C2semidirect
8. #70222C2semidirect
10. #68893C2semidirect
11. #68804C2semidirect
21. #5624C4semidirect
55. #1872D3semidirect
77. #947C2×C4semidirect
79. Q8×Dic3C2×C4semidirect
103. #893C2×C4semidirect
106. #890C2×C4semidirect
569. C3#21861semidirect

Maximal subgroups

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

subgroupindex
C2×#56242
C2×#49772
C2×#47182
C3×#18722
#716922
#715602
#702262
#702222
#699322
#688932
#688042
#218613