Finite Group Databasebeta

← #10823644

Subgroups — #10823644

Order 768.

1182 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3460C211yesmaximal, characteristic, complemented
2C4×#949C212yesmaximal, complemented
3#82329C211yesmaximal, characteristic, complemented
4#71903C211yesmaximal, characteristic, complemented
5#71901C211yesmaximal, characteristic, complemented
6#71879C211yesmaximal, characteristic, complemented
7#71877C211yesmaximal, characteristic, complemented
8#71856C212yesmaximal, complemented
9#71852C212yesmaximal, complemented
10#71643C212yesmaximal, complemented
11C4×#833C211yesmaximal, characteristic, complemented
12#23591—31nomaximal, Sylow
13C3×#544C2211yescharacteristic
14C3×#542—22no—
15C3×#540—22no—
16C3×#538—22no—
17C3×#537C2211yescharacteristic
18C3×#535C2212yes—
19C3×#532—22no—
20C3×#529—22no—
21C3×#525—22no—
22C3×C4×#41C2212yes—
23C2×C3×C4×D4C2211yescharacteristic
24C2×#949C2212yes—
25C2×#940C2211yescharacteristic
26C2×C4×D12C2211yescharacteristic
27C2×#837C2212yes—
28C2×#837C2212yes—
29#5232C2212yes—
30#5221C2211yescharacteristic
31C2×#833C2212yes—
32#5213—22no—
33#5026—22no—
34C4×#215—22no—
35#5014—22no—
36C4×#213—22no—
37#5001C2212yes—
38#4995C2212yes—
39#4988C2211yescharacteristic
40#4977C2211yescharacteristic
41#4956—22no—
42#4932C2211yescharacteristic
43C4×#208C2212yes—
44#4532C2212yes—
45#4532C2212yes—
46#4530C2211yescharacteristic
47#4530C2211yescharacteristic
48#3460D311yescharacteristic, complemented
49C4×AGL(1,ℤ/8)—32no—
50#2328—31no—
51#2061—31no—
52#2040—31no—
53#2037—32no—
54#2027—31no—
55#2006—31no—
56#2002—32no—
57#1759—32no—
58C4×#100—31no—
59C2×C3×#41—22no—
60C2×C3×#41C2312yes—
61C22×C3×D4C2311yescharacteristic
62C23×C3×C4D412yes—
63C22×#67—22no—
64C22×D12C2311yescharacteristic
65C22×C4×D3—21no—
66C3×C4⋊Q8—22no—
67C3×#127—22no—
68C3×#126—24no—
69C3×#125—22no—
70C3×#124—24no—
71C3×#124—22no—
72C3×#123—24no—
73C3×#123—24no—
74C3×C22⋊Q8—22no—
75C3×C22⋊Q8—22no—
76C3×C22⋊Q8—22no—
77C3×#121—22no—
78C3×#121—24no—
79C3×#121—22no—
80C3×#121—22no—
81C3×#121—22no—
82C3×#121—22no—
83C3×#120—24no—
84C3×C4×Q8D412yescomplemented
85C3×C4×Q8D412yescomplemented
86C3×C4×D4—24no—
87C3×C4×D4—22no—
88C3×C4×D4—22no—
89C3×C4×D4—22no—
90C3×C4×D4—22no—
91C3×C4×D4D412yescomplemented
92C3×C4×D4D412yescomplemented
93C3×#117—22no—
94C3×#117—22no—
95C3×#117C2312yes—
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C22⋊C4D412yes—
98C2×C3×C42—22no—
99C2×C3×C42C2311yescharacteristic
100C2×#217—22no—
101#949—24no—
102#949—24no—
103C2×#215—22no—
104#940—22no—
105#940—22no—
106C2×#213—22no—
107C2×(C2×D3)⋊C4—21no—
108C2×(C2×D3)⋊C4—22no—
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4×Dic3C2311yescharacteristic
111C2×C4×Dic3C2311yescharacteristic
112C2×#208C2312yes—
113C2×#208C2312yes—
114C4×D12—22no—
115C4×D12—22no—
116#837—24no—
117#837—24no—
118#835—22no—
119#833—24no—
120#832—22no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—22no—
124#809—22no—
125#809—22no—
126#808—22no—
127#807—22no—
128#803—22no—
129C4×C3⋊C8—22no—
130C2×AGL(1,ℤ/8)—32no—
131#544C2×D311yescharacteristic
132#542—22no—
133#540—22no—
134#538—22no—
135#537C2×D311yescharacteristic
136#535C2×D312yes—
137#532—22no—
138C2×#127—31no—
139#529—22no—
140#525—22no—
141C4×#41C2×D312yes—
142C2×C4×D4—31no—
143C2×C4×D4C2×D311yescharacteristic
144#442—62no—
145#440—62no—
146C4×SD16—62no—
147C4×D8—62no—
148#428—32no—
149C2×#104—32no—
150C2×#104—32no—
151#418—32no—
152#417—32no—
153C2×#100—32no—
154C4×M16—32no—
155#390—31no—
156#390—31no—
157#386—62no—
158#386—62no—
159#382—31no—
160#376—31no—
161#343—31no—
162#343—31no—
163#339—32no—
164#339—32no—
165C24×C3C2×D412yes—
166C23×D3—21no—
167C3×#41—22no—
168C3×#41—22no—
169C3×#41—22no—
170C3×#41—22no—
171C3×#41—84no—
172C2×C3×Q8—22no—
173C2×C3×Q8C2×D412yes—
174C2×C3×D4—22no—
175C2×C3×D4—24no—
176C2×C3×D4—24no—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—21no—
181C2×C3×D4—21no—
182C2×C3×D4C2×D412yes—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—82no—
189C22×C3×C4C2×D412yes—
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2411yescharacteristic
193C2×#67—28no—
194C2×#67—82no—
195C2×#67—82no—
196C22×Dic3—21no—
197C22×Dic3—21no—
198C22×Dic3—21no—
199C2×D12—22no—
200C2×D12—22no—
201C2×D12—21no—
202C2×D12#4112yescomplemented
203C2×C4×D3—42no—
204C2×C4×D3—82no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—24no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—22no—
210C3×C4⋊C4—22no—
211C3×C4⋊C4C2×D412yes—
212C3×C4⋊C4C2×D412yes—
213C3×C22⋊C4—24no—
214C3×C22⋊C4—24no—
215C3×C22⋊C4—24no—
216C3×C22⋊C4—22no—
217C3×C22⋊C4—24no—
218C3×C22⋊C4—22no—
219C3×C22⋊C4—22no—
220C3×C42—22no—
221C3×C42—22no—
222C3×C42—22no—
223C3×C42C2×D412yes—
224C3×C42C2×D412yes—
225#217—82no—
226#215—28no—
227#213—28no—
228(C2×D3)⋊C4—41no—
229(C2×D3)⋊C4—41no—
230(C2×D3)⋊C4—82no—
231C4⋊Dic3—22no—
232C4×Dic3—21no—
233C4×Dic3—22no—
234C4×Dic3#4112yescomplemented
235#208—22no—
236#208—22no—
237#208—22no—
238#208#4114yescomplemented
239C2×C3⋊C8—22no—
240C2×C3⋊C8—22no—
241C2×#41—22no—
242C2×#41C22×D312yes—
243C22×D4—31no—
244C22×D4—62no—
245C22×D4C22×D311yescharacteristic
246C23×C4—61no—
247C23×C4#6712yes—
248AGL(1,ℤ/8)—64no—
249AGL(1,ℤ/8)—64no—
250C2×SD16—62no—
251C2×D8—62no—
252C2×M16—32no—
253C2×M16—32no—
254C4⋊Q8—22no—
255#127—22no—
256#127—62no—
257#127—62no—
258#126—24no—
259#125—22no—
260#124—24no—
261#124—22no—
262#123—24no—
263#123—24no—
264C22⋊Q8—22no—
265C22⋊Q8—22no—
266C22⋊Q8—22no—
267#121—22no—
268#121—24no—
269#121—22no—
270#121—22no—
271#121—22no—
272#121—22no—
273#120—24no—
274C4×Q8#6712yescomplemented
275C4×Q8#6712yescomplemented
276C4×D4—24no—
277C4×D4—22no—
278C4×D4—22no—
279C4×D4—22no—
280C4×D4—22no—
281C4×D4—62no—
282C4×D4—62no—
283C4×D4#6712yescomplemented
284C4×D4#6712yescomplemented
285#117—22no—
286#117—22no—
287#117C22×D312yes—
288C2×C4⋊C4—31no—
289C2×C4⋊C4C22×D311yescharacteristic
290C2×C22⋊C4—61no—
291C2×C22⋊C4—62no—
292C2×C22⋊C4—62no—
293C2×C22⋊C4#6712yes—
294C2×C42—22no—
295C2×C42—31no—
296C2×C42—31no—
297C2×C42C22×D311yescharacteristic
298#107—62no—
299#106—62no—
300#104—64no—
301#104—64no—
302#103—62no—
303#102—62no—
304#102—62no—
305#102—62no—
306#100—64no—
307#97—62no—
308C4×C8—62no—
309#95—61no—
310#95—61no—
311#95—62no—
312C23×C3—22no—
313C23×C3—22no—
314C23×C3—22no—
315C23×C3—22no—
316C23×C3—42no—
317C23×C3—42no—
318C23×C3C22×D411yescharacteristic
319C22×D3—22no—
320C22×D3—42no—
321C22×D3—42no—
322C22×D3—41no—
323C22×D3—41no—
324C3×Q8—22no—
325C3×Q8—22no—
326C3×Q8—82no—
327C3×D4—22no—
328C3×D4—22no—
329C3×D4—42no—
330C3×D4—42no—
331C3×D4—42no—
332C3×D4—42no—
333C3×D4—84no—
334C3×D4—82no—
335C3×D4—84no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—44no—
343C2×C3×C4—44no—
344C2×C3×C4—42no—
345C2×C3×C4—42no—
346C2×C3×C4—42no—
347C2×C3×C4—82no—
348C2×C3×C4—82no—
349C2×C3×C4#12011yescharacteristic
350C2×C3×C4#12011yescharacteristic
351C2×C3×C4#12012yes—
352C2×C3×C4C22×D412yes—
353C2×C3×C4C2×#4111yescharacteristic
354C2×C3×C4C2×#4112yes—
355C2×C3×C4C2×#4111yescharacteristic
356#67—88no—
357C2×Dic3—22no—
358C2×Dic3—41no—
359C2×Dic3—41no—
360C2×Dic3—41no—
361C2×Dic3—41no—
362C2×Dic3—42no—
363C2×Dic3—41no—
364C2×Dic3—41no—
365D12—24no—
366D12—42no—
367C4×D3—84no—
368C3⋊C8—24no—
369C3⋊C8—42no—
370C24—61no—
371C24C2×#6712yes—
372#41—22no—
373#41—22no—
374#41—22no—
375#41—22no—
376#41—84no—
377C2×Q8—22no—
378C2×Q8C2×#6712yes—
379C2×D4—22no—
380C2×D4—24no—
381C2×D4—24no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—22no—
387C2×D4—32no—
388C2×D4—68no—
389C2×D4—62no—
390C2×D4—61no—
391C2×D4—62no—
392C2×D4—242no—
393C2×D4—242no—
394C2×D4C2×#6712yes—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—61no—
401C22×C4—61no—
402C22×C4—61no—
403C22×C4—82no—
404C22×C4—122no—
405C22×C4—242no—
406C22×C4C2×#6712yes—
407C22×C4C2×#6711yescharacteristic
408C22×C4C2×#6711yescharacteristic
409C22×C4C23×D311yescharacteristic
410SD16—68no—
411D8—68no—
412M16—34no—
413M16—62no—
414M16—62no—
415M16—62no—
416C2×C8—62no—
417C2×C8—62no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—24no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—62no—
425C4⋊C4C2×#6712yes—
426C4⋊C4C2×#6712yes—
427C22⋊C4—24no—
428C22⋊C4—24no—
429C22⋊C4—24no—
430C22⋊C4—22no—
431C22⋊C4—24no—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—121no—
435C22⋊C4—121no—
436C22⋊C4—242no—
437C22⋊C4—242no—
438C42—22no—
439C42—22no—
440C42—22no—
441C42—32no—
442C42—61no—
443C42—62no—
444C42C2×#6712yes—
445C42C2×#6712yes—
446C22×C3—21no—
447C22×C3—21no—
448C22×C3—44no—
449C22×C3—42no—
450C22×C3—42no—
451C22×C3—42no—
452C22×C3—42no—
453C22×C3—42no—
454C22×C3—42no—
455C22×C3—82no—
456C22×C3C2×#12011yescharacteristic
457C22×C3#52511yescharacteristic
458C22×C3#53211yescharacteristic
459C2×D3—44no—
460C2×D3—42no—
461C2×D3—82no—
462C2×D3—81no—
463C2×D3—81no—
464C3×C4—21no—
465C3×C4—21no—
466C3×C4—22no—
467C3×C4—42no—
468C3×C4—42no—
469C3×C4—42no—
470C3×C4—42no—
471C3×C4#52512yes—
472C3×C4#53212yes—
473Dic3—42no—
474Dic3—81no—
475Dic3—81no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—42no—
481C23—42no—
482C23—62no—
483C23—122no—
484C23—122no—
485C23—121no—
486C23—121no—
487C23C22×#6711yescharacteristic
488Q8—22no—
489Q8—22no—
490Q8—82no—
491D4—22no—
492D4—22no—
493D4—42no—
494D4—42no—
495D4—42no—
496D4—42no—
497D4—64no—
498D4—82no—
499D4—84no—
500D4—84no—
501D4—122no—
502D4—248no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—62no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—121no—
518C2×C4—121no—
519C2×C4—121no—
520C2×C4—121no—
521C2×C4—122no—
522C2×C4—121no—
523C2×C4—121no—
524C2×C4—244no—
525C2×C4#95311yescharacteristic
526C2×C4#95312yes—
527C2×C4#95311yescharacteristic
528C2×C4D3×#4111yescharacteristic
529C2×C4D3×#4112yes—
530C2×C4D3×#4111yescharacteristic
531C2×C4C22×#6712yes—
532C8—64no—
533C8—122no—
534C2×C3—21no—
535C2×C3—21no—
536C2×C3—82no—
537C2×C3—82no—
538C2×C3#238811yescharacteristic
539C2×C3#301911yescharacteristic
540C2×C3#302111yescharacteristic
541D3—82no—
542C22—21no—
543C22—21no—
544C22—44no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—82no—
552C22—124no—
553C22—122no—
554C22—242no—
555C22—241no—
556C22—241no—
557C22#582811yescharacteristic
558C22#583211yescharacteristic
559C22C2×#95311yescharacteristic
560C4—21no—
561C4—21no—
562C4—22no—
563C4—42no—
564C4—42no—
565C4—42no—
566C4—42no—
567C4—122no—
568C4—241no—
569C4—241no—
570C4#582812yes—
571C4#583212yes—
572C3#2359111yescharacteristic, complemented, Sylow
573C2—21no—
574C2—21no—
575C2—82no—
576C2—82no—
577C2—242no—
578C2#8409511yescharacteristic
579C2#8410411yescharacteristic
580C2#8412211yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3460C2
socleC22×C3C2×#120
hypercenter#3460D3
hypocenterC3#23591
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3460

Complements

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

NH ≅ G/Ntype
1. C3×#3460C2semidirect
2. C4×#949C2semidirect
3. #82329C2semidirect
4. #71903C2semidirect
5. #71901C2semidirect
6. #71879C2semidirect
7. #71877C2semidirect
8. #71856C2semidirect
9. #71852C2semidirect
10. #71643C2semidirect
11. C4×#833C2semidirect
48. #3460D3semidirect
84. C3×C4×Q8D4semidirect
85. C3×C4×Q8D4semidirect
91. C3×C4×D4D4semidirect
92. C3×C4×D4D4semidirect
202. C2×D12#41semidirect
234. C4×Dic3#41semidirect
238. #208#41semidirect
274. C4×Q8#67semidirect
275. C4×Q8#67semidirect
283. C4×D4#67semidirect
284. C4×D4#67semidirect
572. C3#23591semidirect

Maximal subgroups

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

subgroupindex
C3×#34602
C4×#9492
#823292
#719032
#719012
#718792
#718772
#718562
#718522
#716432
C4×#8332
#235913