Finite Group Databasebeta

← #10968225

Subgroups — #10968225

Order 768.

2650 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×C22⋊Q8C212yesmaximal
2C3×#2857C211yesmaximal, characteristic, complemented
3#83232C212yesmaximal, complemented
4#82811C212yesmaximal, complemented
5#82076C211yesmaximal, characteristic, complemented
6C2×C4×(C2×D3)⋊C4C211yesmaximal, characteristic
7#76519C212yesmaximal, complemented
8#76499C212yesmaximal, complemented
9#76247C218yesmaximal, complemented
10#74329C212yesmaximal, complemented
11#74005C214yesmaximal
12#73957C214yesmaximal, complemented
13#35011—31nomaximal, Sylow
14C22×D3×Q8C2212yes—
15C23×C4×D3C2211yescharacteristic
16C2×C3×C4⋊Q8C2211yescharacteristic, complemented
17C2×C3×C22⋊Q8C2212yes—
18C22×C3×C42C2211yescharacteristic
19C2×#946C2212yes—
20C2×#944C2212yescomplemented
21C22×(C2×D3)⋊C4C2212yes—
22C2×#924C2212yes—
23C22×C4×Dic3C2211yescharacteristic
24D3×C22⋊Q8—216no—
25C2×#897C2212yes—
26C2×#896C2214yes—
27C2×D3×C4⋊C4C2212yes—
28C2×D3×C4⋊C4C2214yes—
29C2×D3×C22⋊C4C2214yes—
30C2×#878C2214yes—
31C2×#869C2211yescharacteristic, complemented
32C3×#391C2212yescomplemented
33C3×#389C2214yescomplemented
34C3×#384C2211yescharacteristic, complemented
35C3×#380C2214yes—
36#5229C2212yescomplemented
37#5212C2214yes—
38C2×C3⋊#95C2212yes—
39#4989C2218yes—
40#4989C2218yes—
41#4982C2218yescomplemented
42#4979C2218yescomplemented
43#4976C2218yescomplemented
44#4975C2212yescomplemented
45#4973C2212yescomplemented
46#4942C2218yes—
47C4×(C2×D3)⋊C4C2212yes—
48C4×(C2×D3)⋊C4C2212yes—
49C4×(C2×D3)⋊C4C2214yes—
50#4935C2211yescharacteristic, complemented
51#4929C2212yescomplemented
52#4928C2214yescomplemented
53#4668C2218yes—
54#4668C2218yes—
55#4657C2218yescomplemented
56#4656C2218yescomplemented
57C22×C22⋊Q8—32no—
58#2857—31no—
59#2857—32no—
60#2857D311yescharacteristic, complemented
61#2659—34no—
62#2583—38no—
63#2502—32no—
64#2502—32no—
65#2434—32no—
66#2434—32no—
67#2315—34no—
68C2×C4×C22⋊C4—31no—
69C24×D3C2311yescharacteristic
70C22×C3×Q8C2312yes—
71C23×C3×C4C2311yescharacteristic
72C23×C3×C4C2312yes—
73C23×Dic3C2311yescharacteristic
74C23×Dic3C2312yes—
75C2×D3×Q8—28no—
76C2×D3×Q8—44no—
77C2×D3×Q8—44no—
78C22×C4×D3—42no—
79C22×C4×D3—42no—
80C22×C4×D3D418yes—
81C22×C4×D3C2312yes—
82C22×C4×D3C2314yes—
83C22×C4×D3C2314yes—
84C22×C4×D3C2312yes—
85C22×Dic6C2312yes—
86C22×Dic6C2314yes—
87C3×C4⋊Q8—42no—
88C3×C22⋊Q8—28no—
89C2×C3×C4⋊C4C2312yes—
90C2×C3×C4⋊C4C2314yes—
91C2×C3×C4⋊C4C2314yes—
92C2×C3×C4⋊C4C2314yes—
93C2×C3×C22⋊C4C2314yes—
94C2×C3×C42—42no—
95C2×C3×C42C2311yescharacteristic
96C2×C3×C42C2311yescharacteristic
97C2×C3×C42C2312yes—
98C2×#217C2314yes—
99#946—28no—
100#944—44no—
101C2×(C2×D3)⋊C4Q8116yescomplemented
102C2×(C2×D3)⋊C4C2314yes—
103C2×(C2×D3)⋊C4C2314yes—
104C2×(C2×D3)⋊C4C2314yes—
105C2×(C2×D3)⋊C4C2314yes—
106C2×C4⋊Dic3C2314yes—
107C2×C4⋊Dic3C2312yes—
108C2×C4⋊Dic3C2314yes—
109C2×C4⋊Dic3C2314yes—
110#924—28no—
111C2×Dic3⋊C4C2314yes—
112C2×Dic3⋊C4C2314yes—
113C2×Dic3⋊C4C2314yes—
114C2×Dic3⋊C4C2318yes—
115C2×Dic3⋊C4C2318yes—
116C2×C4×Dic3—42no—
117C2×C4×Dic3C2312yes—
118C2×C4×Dic3C2312yes—
119#897—28no—
120#896—216no—
121D3×C4⋊C4—216no—
122D3×C4⋊C4—28no—
123D3×C22⋊C4—216no—
124#878—216no—
125#869—42no—
126C3×#95C2318yes—
127C3⋊#95C2314yes—
128C3⋊#95C2318yes—
129C3⋊#95C2318yes—
130C3⋊#95C2318yes—
131C3⋊#95C2314yes—
132C23×Q8—32no—
133C24×C4—31no—
134C2×C4⋊Q8—32no—
135C2×C4⋊Q8—31no—
136C2×C4⋊Q8C2×D311yescharacteristic, complemented
137C2×C22⋊Q8—32no—
138C2×C22⋊Q8—32no—
139C2×C22⋊Q8—34no—
140C2×C22⋊Q8—32no—
141C2×C22⋊Q8—34no—
142C2×C22⋊Q8—616no—
143C2×C22⋊Q8C2×D312yes—
144C22×C4⋊C4—34no—
145C22×C4⋊C4—32no—
146C22×C22⋊C4—32no—
147C22×C22⋊C4—34no—
148C22×C42—31no—
149C22×C42C2×D311yescharacteristic
150#398—38no—
151#398—38no—
152#395—38no—
153#395—38no—
154#391—32no—
155#391—32no—
156#391—32no—
157#391C2×D312yescomplemented
158#389—38no—
159#389—34no—
160#389C2×D314yescomplemented
161#385—38no—
162#385—38no—
163#385—38no—
164#385—38no—
165#384—32no—
166#384—31no—
167#384C2×D311yescharacteristic, complemented
168#380—34no—
169#380—38no—
170#380C2×D314yes—
171C4×C22⋊C4—34no—
172C4×C22⋊C4—32no—
173C4×C22⋊C4—32no—
174C2×#95—32no—
175C24×C3C2411yescharacteristic
176C23×D3—41no—
177C23×D3—41no—
178C23×D3—42no—
179C23×D3C2×D414yes—
180C23×D3C2×Q818yes—
181C23×D3C2412yes—
182C2×C3×Q8—24no—
183C2×C3×Q8—24no—
184C2×C3×Q8—24no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—42no—
188C22×C3×C4—42no—
189C22×C3×C4C2×D414yes—
190C22×C3×C4C2×Q818yescomplemented
191C22×C3×C4C2414yes—
192C22×C3×C4C2414yes—
193C22×C3×C4C2411yescharacteristic
194C22×C3×C4C2412yes—
195C22×C3×C4C2412yes—
196C22×C3×C4C2411yescharacteristic
197C22×Dic3—42no—
198C22×Dic3—41no—
199C22×Dic3—41no—
200C22×Dic3—42no—
201C22×Dic3C2×D414yes—
202C22×Dic3C2×Q818yescomplemented
203C22×Dic3C2414yes—
204C22×Dic3C2414yes—
205C22×Dic3C2412yes—
206C22×Dic3C2414yes—
207D3×Q8—432no—
208C2×C4×D3—24no—
209C2×C4×D3—24no—
210C2×C4×D3—28no—
211C2×C4×D3—28no—
212C2×C4×D3—416no—
213C2×C4×D3—42no—
214C2×C4×D3—42no—
215C2×C4×D3—48no—
216C2×C4×D3—44no—
217C2×C4×D3—44no—
218C2×C4×D3C2×D418yes—
219C2×C4×D3C2×D418yes—
220C2×Dic6—24no—
221C2×Dic6—24no—
222C2×Dic6—24no—
223C2×Dic6—28no—
224C2×Dic6—44no—
225C2×Dic6—44no—
226C3×C4⋊C4—28no—
227C3×C4⋊C4—24no—
228C3×C4⋊C4—44no—
229C3×C4⋊C4—44no—
230C3×C22⋊C4—28no—
231C3×C42—22no—
232C3×C42—22no—
233C3×C42—42no—
234#217—28no—
235(C2×D3)⋊C4—28no—
236(C2×D3)⋊C4—28no—
237(C2×D3)⋊C4C2×Q8116yescomplemented
238(C2×D3)⋊C4C2×Q8116yescomplemented
239C4⋊Dic3—24no—
240C4⋊Dic3—28no—
241C4⋊Dic3—44no—
242C4⋊Dic3—44no—
243Dic3⋊C4—28no—
244Dic3⋊C4—28no—
245Dic3⋊C4—28no—
246Dic3⋊C4—48no—
247Dic3⋊C4—48no—
248C4×Dic3—24no—
249C4×Dic3—42no—
250C25—31no—
251C22×Q8—32no—
252C22×Q8—34no—
253C22×Q8—68no—
254C22×Q8—124no—
255C22×Q8—124no—
256C22×Q8C22×D312yes—
257C23×C4—32no—
258C23×C4—31no—
259C23×C4—32no—
260C23×C4—38no—
261C23×C4—32no—
262C23×C4—34no—
263C23×C4—34no—
264C23×C4—122no—
265C23×C4—122no—
266C23×C4C22×D311yescharacteristic
267C23×C4C22×D312yes—
268C4⋊Q8—42no—
269C4⋊Q8—122no—
270C4⋊Q8—124no—
271C22⋊Q8—28no—
272C22⋊Q8—616no—
273C22⋊Q8—616no—
274C22⋊Q8—68no—
275C22⋊Q8—68no—
276C22⋊Q8—68no—
277C2×C4⋊C4—32no—
278C2×C4⋊C4—34no—
279C2×C4⋊C4—34no—
280C2×C4⋊C4—34no—
281C2×C4⋊C4—34no—
282C2×C4⋊C4—38no—
283C2×C4⋊C4—34no—
284C2×C4⋊C4—38no—
285C2×C4⋊C4—34no—
286C2×C4⋊C4—68no—
287C2×C4⋊C4—616no—
288C2×C4⋊C4C22×D312yes—
289C2×C4⋊C4C22×D314yes—
290C2×C4⋊C4C22×D314yes—
291C2×C4⋊C4C22×D314yes—
292C2×C22⋊C4—316no—
293C2×C22⋊C4—34no—
294C2×C22⋊C4—34no—
295C2×C22⋊C4—34no—
296C2×C22⋊C4—34no—
297C2×C22⋊C4—34no—
298C2×C22⋊C4—616no—
299C2×C22⋊C4C22×D314yes—
300C2×C42—32no—
301C2×C42—32no—
302C2×C42—42no—
303C2×C42—122no—
304C2×C42C22×D311yescharacteristic
305C2×C42C22×D311yescharacteristic
306C2×C42C22×D312yes—
307#95—34no—
308#95—34no—
309#95—38no—
310#95—38no—
311#95—38no—
312#95C22×D318yes—
313C23×C3—21no—
314C23×C3—21no—
315C23×C3—22no—
316C23×C3C22×D412yes—
317C23×C3C22×Q814yes—
318C23×C3C2511yescharacteristic
319C22×D3—22no—
320C22×D3—22no—
321C22×D3—24no—
322C22×D3—44no—
323C22×D3—44no—
324C22×D3—48no—
325C22×D3—48no—
326C22×D3C4⋊Q8116yescomplemented
327C22×D3C22×D414yes—
328C22×D3C22×Q818yes—
329C3×Q8—48no—
330C2×C3×C4—21no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—21no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—28no—
337C2×C3×C4—24no—
338C2×C3×C4—24no—
339C2×C3×C4—24no—
340C2×C3×C4—24no—
341C2×C3×C4—28no—
342C2×C3×C4—48no—
343C2×C3×C4C22×D412yes—
344C2×C3×C4C22×D412yes—
345C2×C3×C4C22×Q814yes—
346C2×C3×C4C22×Q814yes—
347C2×C3×C4H(5,𝔽2)12yes—
348C2×C3×C4H(5,𝔽2)12yes—
349C2×Dic3—24no—
350C2×Dic3—22no—
351C2×Dic3—22no—
352C2×Dic3—24no—
353C2×Dic3—24no—
354C2×Dic3—24no—
355C2×Dic3—24no—
356C2×Dic3—24no—
357C2×Dic3—28no—
358C2×Dic3—44no—
359C2×Dic3—48no—
360C2×Dic3C22×D414yes—
361C2×Dic3C22×Q818yes—
362C4×D3—216no—
363C4×D3—48no—
364C4×D3—416no—
365C4×D3—416no—
366Dic6—416no—
367Dic6—48no—
368C24—32no—
369C24—34no—
370C24—38no—
371C24—121no—
372C24—121no—
373C24—122no—
374C24C23×D311yescharacteristic
375C2×Q8—24no—
376C2×Q8—24no—
377C2×Q8—24no—
378C2×Q8—68no—
379C2×Q8—64no—
380C2×Q8—64no—
381C2×Q8—64no—
382C2×Q8—124no—
383C2×Q8—124no—
384C2×Q8—1232no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—34no—
388C22×C4—32no—
389C22×C4—34no—
390C22×C4—38no—
391C22×C4—38no—
392C22×C4—38no—
393C22×C4—34no—
394C22×C4—34no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—68no—
398C22×C4—68no—
399C22×C4—64no—
400C22×C4—64no—
401C22×C4—1216no—
402C22×C4—122no—
403C22×C4—122no—
404C22×C4—124no—
405C22×C4—124no—
406C22×C4—128no—
407C22×C4—122no—
408C22×C4—121no—
409C22×C4—121no—
410C22×C4—122no—
411C22×C4D3×D414yes—
412C22×C4D3×Q818yescomplemented
413C22×C4C23×D314yes—
414C22×C4C23×D314yes—
415C22×C4C23×D311yescharacteristic
416C22×C4C23×D312yes—
417C22×C4C23×D312yes—
418C22×C4C23×D311yescharacteristic
419C4⋊C4—24no—
420C4⋊C4—28no—
421C4⋊C4—44no—
422C4⋊C4—44no—
423C4⋊C4—68no—
424C4⋊C4—68no—
425C4⋊C4—68no—
426C4⋊C4—68no—
427C4⋊C4—64no—
428C4⋊C4—128no—
429C4⋊C4—124no—
430C4⋊C4—128no—
431C4⋊C4—124no—
432C22⋊C4—28no—
433C22⋊C4—316no—
434C22⋊C4—316no—
435C22⋊C4—68no—
436C22⋊C4—68no—
437C22⋊C4—68no—
438C42—22no—
439C42—22no—
440C42—42no—
441C42—64no—
442C42—122no—
443C22×C3—22no—
444C22×C3—22no—
445C22×C3—24no—
446C22×C3—24no—
447C22×C3C2×C4⋊Q814yes—
448C22×C3C23×D411yescharacteristic
449C22×C3C23×Q812yes—
450C22×C3C2×H(5,𝔽2)11yescharacteristic
451C22×C3C2×H(5,𝔽2)11yescharacteristic
452C22×C3#58512yes—
453C2×D3—24no—
454C2×D3—24no—
455C2×D3—28no—
456C2×D3—28no—
457C2×D3—416no—
458C2×D3C2×C4⋊Q818yes—
459C3×C4—22no—
460C3×C4—24no—
461C3×C4—24no—
462C3×C4—44no—
463C3×C4—44no—
464C3×C4#53414yes—
465Dic3—24no—
466Dic3—28no—
467Dic3—44no—
468Dic3—44no—
469C23—21no—
470C23—21no—
471C23—22no—
472C23—34no—
473C23—38no—
474C23—316no—
475C23—62no—
476C23—62no—
477C23—64no—
478C23—124no—
479C23—128no—
480C23—124no—
481C23—128no—
482C23C2×D3×D412yes—
483C23C2×D3×Q814yes—
484C23C24×D311yescharacteristic
485Q8—48no—
486Q8—1216no—
487Q8—128no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—28no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—28no—
500C2×C4—34no—
501C2×C4—38no—
502C2×C4—48no—
503C2×C4—62no—
504C2×C4—64no—
505C2×C4—62no—
506C2×C4—64no—
507C2×C4—616no—
508C2×C4—64no—
509C2×C4—64no—
510C2×C4—64no—
511C2×C4—64no—
512C2×C4—68no—
513C2×C4—128no—
514C2×C4—1216no—
515C2×C4—1216no—
516C2×C4—124no—
517C2×C4—128no—
518C2×C4C2×D3×D412yes—
519C2×C4C2×D3×D412yes—
520C2×C4#100412yes—
521C2×C4C2×D3×Q814yes—
522C2×C4C2×D3×Q814yes—
523C2×C4#100912yes—
524C2×C3—24no—
525C2×C3C22×C4⋊Q811yescharacteristic
526C2×C3C2×#53411yescharacteristic
527C2×C3#344911yescharacteristic
528C2×C3#350112yes—
529C2×C3#350112yes—
530D3—28no—
531C22—24no—
532C22—22no—
533C22—22no—
534C22—24no—
535C22—38no—
536C22—68no—
537C22—64no—
538C22—64no—
539C22—68no—
540C22—1216no—
541C22D3×C4⋊Q814yes—
542C22C22×D3×D411yescharacteristic
543C22C2×#100411yescharacteristic
544C22C22×D3×Q812yes—
545C22C2×#100911yescharacteristic
546C22#596412yes—
547C4—22no—
548C4—24no—
549C4—24no—
550C4—44no—
551C4—44no—
552C4—68no—
553C4—64no—
554C4—124no—
555C4—124no—
556C4#560514yes—
557C3#3501111yescharacteristic, complemented, Sylow
558C2—24no—
559C2—68no—
560C2C2×#560511yescharacteristic
561C2#8598612yes—
562C2C2×D3×C4⋊Q811yescharacteristic
563C2#8623511yescharacteristic
564C2#8628012yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2857C2
prime radicalC24×D3C23
hypercenter#2857D3
hypocenterC3#35011
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2857

Complements

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

NH ≅ G/Ntype
2. C3×#2857C2semidirect
3. #83232C2semidirect
4. #82811C2semidirect
5. #82076C2semidirect
7. #76519C2semidirect
8. #76499C2semidirect
9. #76247C2semidirect
10. #74329C2semidirect
12. #73957C2semidirect
16. C2×C3×C4⋊Q8C22semidirect
20. C2×#944C22semidirect
31. C2×#869C22semidirect
32. C3×#391C22semidirect
33. C3×#389C22semidirect
34. C3×#384C22semidirect
36. #5229C22semidirect
41. #4982C22semidirect
42. #4979C22semidirect
43. #4976C22semidirect
44. #4975C22semidirect
45. #4973C22semidirect
50. #4935C22semidirect
51. #4929C22semidirect
52. #4928C22semidirect
55. #4657C22semidirect
56. #4656C22semidirect
60. #2857D3semidirect
101. C2×(C2×D3)⋊C4Q8semidirect
136. C2×C4⋊Q8C2×D3semidirect
157. #391C2×D3semidirect
160. #389C2×D3semidirect
167. #384C2×D3semidirect
190. C22×C3×C4C2×Q8semidirect
202. C22×Dic3C2×Q8semidirect
237. (C2×D3)⋊C4C2×Q8semidirect
238. (C2×D3)⋊C4C2×Q8semidirect
326. C22×D3C4⋊Q8semidirect
412. C22×C4D3×Q8semidirect
557. C3#35011semidirect

Maximal subgroups

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

subgroupindex
C2×D3×C22⋊Q82
C3×#28572
#832322
#828112
#820762
C2×C4×(C2×D3)⋊C42
#765192
#764992
#762472
#743292
#740052
#739572
#350113