Finite Group Databasebeta

← #10767964

Subgroups — #10767964

Order 768.

1250 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86241C211yesmaximal, characteristic
2C3×#3112C211yesmaximal, characteristic, complemented
3#82492C211yesmaximal, characteristic, complemented
4#82444C211yesmaximal, characteristic, complemented
5#71812C211yesmaximal, characteristic, complemented
6#71164C214yesmaximal, complemented
7#70972C214yesmaximal, complemented
8#70832C211yesmaximal, characteristic, complemented
9#70827C211yesmaximal, characteristic, complemented
10#18179—31nomaximal, Sylow
11C2×#1009—21no—
12C3×#532C2211yescharacteristic
13C2×C3×C4×Q8C2211yescharacteristic, complemented
14#5833—21no—
15#5726—21no—
16#5717—21no—
17#5676—22no—
18#5605C2214yes—
19D3×#121—22no—
20#5551—21no—
21#5550—22no—
22D4×D12—21no—
23#5477C2211yescharacteristic
24C2×C4×D12C2211yescharacteristic
25C3×#480C2212yescomplemented
26C3×#477—21no—
27C3×#460—21no—
28C3×#448C2212yescomplemented
29C3×#440—22no—
30C3×#432C2211yescharacteristic, complemented
31#5231C2212yescomplemented
32#5227C2212yescomplemented
33#5027—21no—
34#5026—22no—
35#5021—21no—
36#4992C2212yescomplemented
37#4991—21no—
38#4986—21no—
39#4967C2212yescomplemented
40C4×(C2×D3)⋊C4—22no—
41#4932C2211yescharacteristic, complemented
42#4926C2211yescharacteristic, complemented
43#4537C2214yescomplemented
44#4470C2214yescomplemented
45#4468C2214yescomplemented
46#3496—31no—
47#3112—31no—
48#3112D311yescharacteristic, complemented
49#2398—31no—
50#1607—34no—
51#1601—34no—
52#1505—31no—
53#1505—31no—
54#1466—31no—
55C2×C3×#41—21no—
56C22×C3×Q8C2311yescharacteristic
57#1009—48no—
58C2×#239—22no—
59C2×D3×D4—22no—
60C2×D3×D4—24no—
61C2×D3×D4—22no—
62C2×#235—22no—
63C2×#235—21no—
64C22×D12—22no—
65C22×D12D411yescharacteristic
66C22×C4×D3D412yes—
67C2×C3×SD16—22no—
68C3×C4⋊Q8D411yescharacteristic
69C3×#127D411yescharacteristic
70C3×#124—21no—
71C3×#121—21no—
72C3×#121C2312yes—
73C3×C4×Q8—21no—
74C3×C4×Q8—21no—
75C3×C4×Q8—21no—
76C3×C4×Q8—21no—
77C3×C4×D4—21no—
78C3×C4×D4D412yes—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C42—21no—
82C2×C3×C42C2311yescharacteristic
83#946—22no—
84C2×#215—22no—
85#939—24no—
86#938—22no—
87#938—22no—
88#937—24no—
89#930—22no—
90#930—21no—
91#930C2312yes—
92C2×(C2×D3)⋊C4—22no—
93C2×(C2×D3)⋊C4—22no—
94C2×(C2×D3)⋊C4C2312yes—
95C2×C4⋊Dic3D411yescharacteristic
96#924—22no—
97C2×C4×Dic3—21no—
98#895—22no—
99#894—24no—
100D3×C4⋊C4—22no—
101#884—24no—
102#883—24no—
103#882—24no—
104D3×C22⋊C4—24no—
105#875—21no—
106#874D411yescharacteristic
107C4×D12—21no—
108C4×D12—21no—
109C4×D12—21no—
110C4×D12—21no—
111C4×D12—21no—
112C4×D12D412yes—
113#869D411yescharacteristic
114C3×#107—22no—
115C3×C4⋊C8—21no—
116C3×#103—22no—
117C3×#102—22no—
118C3×C22⋊C8C2312yes—
119C3×#97—21no—
120#835—22no—
121C3⋊#95—21no—
122C3⋊#95—22no—
123C3⋊#95—22no—
124C3⋊#95C2312yes—
125#830C2312yes—
126#809—22no—
127#807—22no—
128C4⋊(C3⋊C8)—21no—
129#803—21no—
130C2×H(5,𝔽2)—61no—
131#547—61no—
132#547—61no—
133#546—62no—
134#546—62no—
135D42—61no—
136D42—61no—
137#537—61no—
138#534—34no—
139#532—31no—
140#532C2×D311yescharacteristic
141C2×#121—62no—
142C2×C4×Q8C2×D311yescharacteristic, complemented
143C2×C4×D4—31no—
144#480—32no—
145#480C2×D312yescomplemented
146#477—21no—
147#477—61no—
148#460—21no—
149#460—61no—
150#448—32no—
151#448C2×D312yescomplemented
152#440—22no—
153#440—62no—
154#432—31no—
155#432C2×D311yescharacteristic, complemented
156#400—61no—
157#397—32no—
158#394—61no—
159#393—32no—
160C4×C22⋊C4—62no—
161#376—31no—
162#327—34no—
163#327—34no—
164#323—34no—
165C23×D3—24no—
166C23×D3C2×D412yes—
167C3×#41—44no—
168C2×C3×Q8—21no—
169C2×C3×Q8—21no—
170C2×C3×Q8—21no—
171C2×C3×Q8—42no—
172C2×C3×D4—22no—
173C2×C3×D4—21no—
174C2×C3×D4—21no—
175C2×C3×D4C2×D412yes—
176C22×C3×C4—21no—
177C22×C3×C4—22no—
178C22×C3×C4—21no—
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2411yescharacteristic
182C2×#67—24no—
183C2×#67—22no—
184C2×#67—24no—
185C22×Dic3—22no—
186C22×Dic3C2×D412yes—
187#239—48no—
188D3×D4—48no—
189D3×D4—48no—
190D3×D4—48no—
191D3×D4—84no—
192#235—48no—
193#235—44no—
194C2×D12—24no—
195C2×D12—21no—
196C2×D12—21no—
197C2×D12—24no—
198C2×D12—21no—
199C2×D12—21no—
200C2×D12—48no—
201C2×D12C2×D412yes—
202C2×C4×D3—24no—
203C2×C4×D3—22no—
204C2×C4×D3—22no—
205C2×C4×D3—22no—
206C2×C4×D3—44no—
207C2×Dic6—21no—
208C3×SD16—44no—
209C2×C3×C8—22no—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—41no—
214C3×C4⋊C4—41no—
215C3×C4⋊C4C2×D412yes—
216C3×C22⋊C4—22no—
217C3×C22⋊C4—22no—
218C3×C42—41no—
219C3×C42—41no—
220C3×C42C2×D411yescharacteristic
221C3×C42C2×D411yescharacteristic
222C3×C42C2×D411yescharacteristic
223C3×C42C2×D411yescharacteristic
224#217—24no—
225#215—44no—
226(C2×D3)⋊C4—24no—
227(C2×D3)⋊C4—22no—
228(C2×D3)⋊C4—22no—
229(C2×D3)⋊C4—22no—
230(C2×D3)⋊C4—22no—
231(C2×D3)⋊C4—44no—
232(C2×D3)⋊C4—44no—
233C4⋊Dic3—21no—
234C4⋊Dic3—21no—
235C4⋊Dic3—21no—
236C4⋊Dic3C2×D412yes—
237Dic3⋊C4—24no—
238C4×Dic3—42no—
239C2×C3⋊C8—22no—
240H(5,𝔽2)—128no—
241C2×#41—21no—
242C2×#41—62no—
243C2×#41—62no—
244C2×#41—61no—
245C22×Q8C22×D311yescharacteristic
246C22×D4—31no—
247C22×D4—62no—
248C22×D4—64no—
249C22×D4—62no—
250C22×D4—62no—
251C23×C4—32no—
252C2×SD16—22no—
253C2×SD16—62no—
254C4⋊Q8—31no—
255C4⋊Q8#6711yescharacteristic
256#127—31no—
257#127#6711yescharacteristic
258#124—21no—
259#124—61no—
260#123—64no—
261#123—64no—
262C22⋊Q8—62no—
263C22⋊Q8—62no—
264#121—21no—
265#121—32no—
266#121—64no—
267#121—62no—
268#121—62no—
269#121—64no—
270#121—62no—
271#121—62no—
272#121—61no—
273#121C22×D312yes—
274#120—64no—
275#120—64no—
276C4×Q8—21no—
277C4×Q8—21no—
278C4×Q8—21no—
279C4×Q8—21no—
280C4×D4—21no—
281C4×D4—32no—
282C4×D4—61no—
283C4×D4—61no—
284C4×D4—61no—
285C4×D4—61no—
286C4×D4—61no—
287C4×D4#6712yes—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—62no—
291C2×C4⋊C4C22×D311yescharacteristic
292C2×C22⋊C4—32no—
293C2×C22⋊C4—64no—
294C2×C22⋊C4—62no—
295C2×C22⋊C4—62no—
296C2×C42—21no—
297C2×C42—61no—
298C2×C42C22×D311yescharacteristic
299#107—22no—
300#107—62no—
301C4⋊C8—21no—
302C4⋊C8—61no—
303#103—22no—
304#103—62no—
305#102—22no—
306#102—62no—
307C22⋊C8—32no—
308C22⋊C8C22×D312yes—
309#97—21no—
310#97—61no—
311#95—32no—
312#95—62no—
313#95—62no—
314#95—61no—
315C23×C3—22no—
316C23×C3C22×D411yescharacteristic
317C22×D3—22no—
318C22×D3—22no—
319C22×D3—22no—
320C22×D3—48no—
321C22×D3—42no—
322C22×D3—42no—
323C22×D3—84no—
324C22×D3—84no—
325C22×D3AGL(1,ℤ/8)14yes—
326C3×Q8—22no—
327C3×Q8—22no—
328C3×Q8—42no—
329C3×Q8—41no—
330C3×Q8—41no—
331C3×D4—42no—
332C3×D4—42no—
333C3×D4—44no—
334C3×D4—42no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—41no—
340C2×C3×C4—44no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—42no—
345C2×C3×C4#12011yescharacteristic
346C2×C3×C4#12011yescharacteristic
347C2×C3×C4#12011yescharacteristic
348C2×C3×C4#12011yescharacteristic
349C2×C3×C4C22×D411yescharacteristic
350C2×C3×C4C22×D411yescharacteristic
351#67—48no—
352#67—44no—
353#67—84no—
354C2×Dic3—22no—
355C2×Dic3—22no—
356C2×Dic3—42no—
357C2×Dic3—42no—
358C2×Dic3—42no—
359C2×Dic3—42no—
360D12—24no—
361D12—24no—
362D12—42no—
363D12—42no—
364D12—48no—
365D12—42no—
366C4×D3—44no—
367C4×D3—48no—
368C4×D3—44no—
369C4×D3—82no—
370Dic6—42no—
371C3×C8—42no—
372C3⋊C8—42no—
373C24—32no—
374C24—64no—
375#41—44no—
376#41—124no—
377#41—128no—
378#41—128no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—42no—
383C2×Q8—61no—
384C2×D4—22no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—32no—
388C2×D4—64no—
389C2×D4—64no—
390C2×D4—64no—
391C2×D4—62no—
392C2×D4—64no—
393C2×D4—61no—
394C2×D4—61no—
395C2×D4—61no—
396C2×D4—61no—
397C2×D4—128no—
398C2×D4—128no—
399C2×D4—128no—
400C2×D4—128no—
401C2×D4—244no—
402C2×D4C2×#6712yes—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—32no—
407C22×C4—64no—
408C22×C4—62no—
409C22×C4—62no—
410C22×C4—62no—
411C22×C4—62no—
412C22×C4—124no—
413C22×C4D3×D411yescharacteristic
414C22×C4D3×D411yescharacteristic
415C22×C4C23×D311yescharacteristic
416SD16—44no—
417SD16—124no—
418C2×C8—22no—
419C2×C8—62no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—32no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—64no—
427C4⋊C4—61no—
428C4⋊C4—61no—
429C4⋊C4—61no—
430C4⋊C4C2×#6712yes—
431C22⋊C4—22no—
432C22⋊C4—22no—
433C22⋊C4—64no—
434C22⋊C4—64no—
435C22⋊C4—62no—
436C22⋊C4—62no—
437C22⋊C4—62no—
438C22⋊C4—62no—
439C22⋊C4—124no—
440C22⋊C4—124no—
441C42—41no—
442C42—41no—
443C42—122no—
444C42D3×D411yescharacteristic
445C42D3×D411yescharacteristic
446C42C2×#6711yescharacteristic
447C42C2×#6711yescharacteristic
448C22×C3—21no—
449C22×C3—21no—
450C22×C3—42no—
451C22×C3—42no—
452C22×C3—42no—
453C22×C3C2×#12011yescharacteristic
454C22×C3C2×AGL(1,ℤ/8)12yes—
455C2×D3—24no—
456C2×D3—42no—
457C2×D3—42no—
458C2×D3—44no—
459C2×D3—42no—
460C2×D3—42no—
461C2×D3—42no—
462C2×D3—82no—
463C2×D3—168no—
464C3×C4—21no—
465C3×C4—21no—
466C3×C4—21no—
467C3×C4—21no—
468C3×C4—21no—
469C3×C4—42no—
470C3×C4—42no—
471C3×C4—81no—
472C3×C4#45512yes—
473Dic3—42no—
474Dic3—42no—
475Dic3—82no—
476C23—22no—
477C23—34no—
478C23—62no—
479C23—62no—
480C23—62no—
481C23—122no—
482C23—122no—
483C23—128no—
484C23—244no—
485C23—244no—
486C23C2×D3×D411yescharacteristic
487Q8—22no—
488Q8—22no—
489Q8—41no—
490Q8—41no—
491Q8—42no—
492Q8—122no—
493D4—42no—
494D4—42no—
495D4—44no—
496D4—42no—
497D4—64no—
498D4—64no—
499D4—122no—
500D4—122no—
501D4—128no—
502D4—128no—
503D4—124no—
504D4—122no—
505D4—244no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—41no—
511C2×C4—44no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—62no—
517C2×C4—62no—
518C2×C4—124no—
519C2×C4—128no—
520C2×C4—124no—
521C2×C4—122no—
522C2×C4—122no—
523C2×C4—122no—
524C2×C4—122no—
525C2×C4—242no—
526C2×C4#93711yescharacteristic
527C2×C4#93711yescharacteristic
528C2×C4#93711yescharacteristic
529C2×C4#93711yescharacteristic
530C2×C4C2×D3×D411yescharacteristic
531C2×C4C22×#6711yescharacteristic
532C8—42no—
533C8—122no—
534C2×C3—22no—
535C2×C3—82no—
536C2×C3#300311yescharacteristic
537C2×C3C2×#45511yescharacteristic
538C2×C3#302511yescharacteristic
539D3—44no—
540D3—82no—
541C22—21no—
542C22—21no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—64no—
547C22—124no—
548C22—122no—
549C22—122no—
550C22—122no—
551C22—122no—
552C22—122no—
553C22—242no—
554C22—488no—
555C22D3×AGL(1,ℤ/8)12yes—
556C22C2×#93711yescharacteristic
557C4—21no—
558C4—21no—
559C4—21no—
560C4—21no—
561C4—21no—
562C4—42no—
563C4—42no—
564C4—81no—
565C4—122no—
566C4—122no—
567C4—242no—
568C4#520212yes—
569C3#1817911yescharacteristic, complemented, Sylow
570C2—22no—
571C2—82no—
572C2—124no—
573C2—242no—
574C2#8293611yescharacteristic
575C2#8394411yescharacteristic
576C2C2×#520211yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3112C2
socleC22×C3C2×#120
prime radical#86241C2
hypercenter#3112D3
hypocenterC3#18179
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3112

Complements

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

NH ≅ G/Ntype
2. C3×#3112C2semidirect
3. #82492C2semidirect
4. #82444C2semidirect
5. #71812C2semidirect
6. #71164C2semidirect
7. #70972C2semidirect
8. #70832C2semidirect
9. #70827C2semidirect
13. C2×C3×C4×Q8C22semidirect
25. C3×#480C22semidirect
28. C3×#448C22semidirect
30. C3×#432C22semidirect
31. #5231C22semidirect
32. #5227C22semidirect
36. #4992C22semidirect
39. #4967C22semidirect
41. #4932C22semidirect
42. #4926C22semidirect
43. #4537C22semidirect
44. #4470C22semidirect
45. #4468C22semidirect
48. #3112D3semidirect
142. C2×C4×Q8C2×D3semidirect
145. #480C2×D3semidirect
151. #448C2×D3semidirect
155. #432C2×D3semidirect
569. C3#18179semidirect

Maximal subgroups

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

subgroupindex
#862412
C3×#31122
#824922
#824442
#718122
#711642
#709722
#708322
#708272
#181793