Finite Group Databasebeta

← C2×#78548 #11658583

Subgroups — C2×#78548 #11658583

Order 768.

3342 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D3×M16C211yesmaximal, characteristic
2C23×ΓL(1,𝔽25)C211yesmaximal, characteristic
3C2×D3×#117C211yesmaximal, characteristic
4C2×C3×#405C211yesmaximal, characteristic, complemented
5C2×#5122C211yesmaximal, characteristic, complemented
6C2×#5101C211yesmaximal, characteristic, complemented
7C2×#4997C211yesmaximal, characteristic, complemented
8#78548C2116yesmaximal, direct product factor
9C2×#4708C212yesmaximal, complemented
10C2×#4707C212yesmaximal, complemented
11C2×#4689C212yesmaximal, complemented
12C2×C4×ΓL(1,𝔽25)C212yesmaximal, complemented
13C2×#2873—31nomaximal, Sylow
14C23×C4×D3C2211yescharacteristic
15C22×C3×M16C2211yescharacteristic
16C23×C3×C8C2211yescharacteristic
17C2×C3×#117C2211yescharacteristic
18C2×#926C2211yescharacteristic
19C22×#208C2211yescharacteristic
20C23×C3⋊C8C2211yescharacteristic
21C2×D3×M16C4116yescomplemented
22C2×D3×M16C2218yes—
23C22×ΓL(1,𝔽25)C4116yescomplemented
24C22×ΓL(1,𝔽25)C2211yescharacteristic
25C22×ΓL(1,𝔽25)C2211yescharacteristic
26C22×ΓL(1,𝔽25)C2212yes—
27C22×ΓL(1,𝔽25)C2211yescharacteristic
28C22×ΓL(1,𝔽25)C2211yescharacteristic
29C22×ΓL(1,𝔽25)C2218yes—
30C22×D3×C8C2212yes—
31D3×#117C418yes—
32D3×#117C2218yes—
33C2×#892C412yes—
34C2×D3×C4⋊C4C412yes—
35C2×D3×C22⋊C4C412yes—
36C2×#877C412yes—
37C2×#872C2212yes—
38C2×C42×D3C2212yes—
39C3×#405C2218yescomplemented
40C2×C3×#97C2212yescomplemented
41C2×C3×C4×C8C2212yescomplemented
42#5122C2218yescomplemented
43#5101C2218yescomplemented
44C2×#815C2212yescomplemented
45C2×#815C2212yescomplemented
46C2×C8×Dic3C2212yescomplemented
47C2×C8×Dic3C2212yescomplemented
48#4997C2218yescomplemented
49C2×#803C2212yescomplemented
50C2×C4×C3⋊C8C2212yescomplemented
51#4708C22116yescomplemented
52#4707C22116yescomplemented
53#4689C22116yescomplemented
54C4×ΓL(1,𝔽25)C22116yescomplemented
55C23×M16—31no—
56C23×M16—31no—
57C22×#117—31no—
58#2873—316no—
59C2×#405—32no—
60C2×#405—32no—
61C2×#405—31no—
62C2×#405—31no—
63C2×#405—31no—
64C2×#405D311yescharacteristic, complemented
65C2×C4×M16—32no—
66C2×C4×M16—32no—
67C24×D3C2×C411yescharacteristic
68C23×C3×C4C2311yescharacteristic
69C23×Dic3C2×C411yescharacteristic
70C22×C4×D3C2×C414yes—
71C22×C4×D3C2×C411yescharacteristic
72C22×C4×D3C2×C411yescharacteristic
73C22×C4×D3C2×C414yes—
74C22×C4×D3C2×C418yes—
75C22×C4×D3C2×C418yes—
76C22×C4×D3C2311yescharacteristic
77C22×C4×D3C2311yescharacteristic
78C22×C4×D3C2314yes—
79C2×C3×M16C2×C418yescomplemented
80C2×C3×M16C2314yes—
81C22×C3×C8C2×C418yescomplemented
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2312yes—
85C22×C3×C8C2314yes—
86C3×#117C2×C414yes—
87C3×#117C2314yes—
88C2×C3×C4⋊C4C2×C412yes—
89C2×C3×C22⋊C4C2×C412yes—
90C2×C3×C42C2312yes—
91C2×#217C2×C412yes—
92C2×(C2×D3)⋊C4C2×C414yes—
93#926C2×C414yes—
94#926C2314yes—
95C2×C4⋊Dic3C2×C412yes—
96C2×Dic3⋊C4C2×C414yes—
97C2×C4×Dic3C2×C414yes—
98C2×C4×Dic3C2312yes—
99C2×#208C2×C418yescomplemented
100C2×#208C2314yes—
101C22×C3⋊C8C2×C418yescomplemented
102C22×C3⋊C8C2311yescharacteristic
103C22×C3⋊C8C2312yes—
104C22×C3⋊C8C2311yescharacteristic
105C22×C3⋊C8C2314yes—
106D3×M16C2×C4164yescomplemented
107C2×ΓL(1,𝔽25)C2×C418yes—
108C2×ΓL(1,𝔽25)C2×C418yes—
109C2×ΓL(1,𝔽25)C2×C4116yes—
110C2×ΓL(1,𝔽25)C2×C4164yescomplemented
111C2×ΓL(1,𝔽25)C2×C418yescomplemented
112C2×ΓL(1,𝔽25)C2×C418yescomplemented
113C2×ΓL(1,𝔽25)C2314yes—
114C2×ΓL(1,𝔽25)C2318yes—
115C2×ΓL(1,𝔽25)C2314yes—
116C2×ΓL(1,𝔽25)C2314yes—
117C2×ΓL(1,𝔽25)C2314yes—
118C2×D3×C8C2×C4116yescomplemented
119C2×D3×C8C2318yes—
120#892C2×C4116yes—
121D3×C4⋊C4C2×C4116yes—
122D3×C22⋊C4C2×C4116yes—
123#877C2×C4116yes—
124#872C2×C418yes—
125#872C2318yes—
126C42×D3C2×C418yes—
127C42×D3C2318yes—
128C3×#97C2318yescomplemented
129C3×C4×C8C2318yescomplemented
130#815C2318yescomplemented
131#815C2318yescomplemented
132C8×Dic3C2318yescomplemented
133C8×Dic3C2318yescomplemented
134#803C2318yescomplemented
135C4×C3⋊C8C2318yescomplemented
136C24×C4—31no—
137C22×M16—31no—
138C22×M16—316no—
139C22×M16—38no—
140C22×M16—316no—
141C22×M16—32no—
142C22×M16—31no—
143C22×M16—31no—
144C22×M16—31no—
145C22×M16—31no—
146C22×M16—38no—
147C22×M16C2×D311yescharacteristic
148C23×C8—31no—
149C23×C8—32no—
150C23×C8C2×D311yescharacteristic
151C2×#117—31no—
152C2×#117—38no—
153C2×#117—38no—
154C2×#117—32no—
155C2×#117—32no—
156C2×#117—32no—
157C2×#117C2×D311yescharacteristic
158C22×C4⋊C4—32no—
159C22×C22⋊C4—32no—
160C22×C42—32no—
161#405—38no—
162#405—38no—
163#405—38no—
164#405—316no—
165#405—316no—
166#405C2×D318yescomplemented
167C4×M16—316no—
168C4×M16—316no—
169C2×#97—32no—
170C2×#97—32no—
171C2×#97—32no—
172C2×#97C2×D312yescomplemented
173C2×C4×C8—32no—
174C2×C4×C8—32no—
175C2×C4×C8—32no—
176C2×C4×C8C2×D312yescomplemented
177C24×C3C22×C411yescharacteristic
178C23×D3—44no—
179C23×D3C4218yes—
180C23×D3C22×C411yescharacteristic
181C23×D3C22×C411yescharacteristic
182C23×D3C22×C414yes—
183C22×C3×C4C22×C414yes—
184C22×C3×C4C22×C411yescharacteristic
185C22×C3×C4C22×C412yes—
186C22×C3×C4C22×C414yes—
187C22×C3×C4C22×C414yes—
188C22×C3×C4C2412yes—
189C22×C3×C4C2411yescharacteristic
190C22×Dic3C4218yes—
191C22×Dic3C22×C414yes—
192C22×Dic3C22×C414yes—
193C22×Dic3C22×C411yescharacteristic
194C22×Dic3C22×C411yescharacteristic
195C2×C4×D3—48no—
196C2×C4×D3C4218yes—
197C2×C4×D3C4218yes—
198C2×C4×D3C42132yes—
199C2×C4×D3C22×C4116yes—
200C2×C4×D3C22×C412yes—
201C2×C4×D3C22×C412yes—
202C2×C4×D3C22×C414yes—
203C2×C4×D3C22×C414yes—
204C2×C4×D3C22×C414yes—
205C2×C4×D3C22×C414yes—
206C2×C4×D3C22×C414yes—
207C2×C4×D3C22×C4116yes—
208C2×C4×D3C22×C414yes—
209C2×C4×D3C22×C4116yes—
210C2×C4×D3C2412yes—
211C2×C4×D3C2412yes—
212C3×M16C22×C4116yescomplemented
213C2×C3×C8C22×C418yes—
214C2×C3×C8C22×C414yes—
215C2×C3×C8C22×C414yes—
216C2×C3×C8C22×C4116yescomplemented
217C2×C3×C8C2412yes—
218C2×C3×C8C2414yes—
219C2×C3×C8C2412yes—
220C3×C4⋊C4C22×C418yes—
221C3×C22⋊C4C22×C418yes—
222C3×C42C22×C414yes—
223C3×C42C2414yes—
224#217C22×C418yes—
225(C2×D3)⋊C4C22×C4116yes—
226C4⋊Dic3C22×C418yes—
227Dic3⋊C4C22×C4116yes—
228C4×Dic3C22×C414yes—
229C4×Dic3C22×C4116yes—
230C4×Dic3C2414yes—
231#208C22×C4116yescomplemented
232C2×C3⋊C8C22×C418yes—
233C2×C3⋊C8C22×C414yes—
234C2×C3⋊C8C22×C414yes—
235C2×C3⋊C8C22×C4116yescomplemented
236C2×C3⋊C8C2412yes—
237C2×C3⋊C8C2414yes—
238C2×C3⋊C8C2412yes—
239ΓL(1,𝔽25)C22×C4116yes—
240ΓL(1,𝔽25)C22×C4116yes—
241ΓL(1,𝔽25)C22×C4132yes—
242ΓL(1,𝔽25)C22×C4116yescomplemented
243ΓL(1,𝔽25)C22×C4116yescomplemented
244D3×C8C22×C4132yescomplemented
245C25—31no—
246C23×C4—31no—
247C23×C4—31no—
248C23×C4—31no—
249C23×C4—34no—
250C23×C4—38no—
251C23×C4—38no—
252C23×C4—34no—
253C23×C4—31no—
254C23×C4—31no—
255C23×C4—34no—
256C23×C4C22×D311yescharacteristic
257C2×M16—38no—
258C2×M16—34no—
259C2×M16—364no—
260C2×M16—316no—
261C2×M16—38no—
262C2×M16—38no—
263C2×M16—38no—
264C2×M16—38no—
265C2×M16—364no—
266C2×M16—34no—
267C2×M16—38no—
268C2×M16—34no—
269C2×M16—34no—
270C2×M16—34no—
271C2×M16C4×D318yescomplemented
272C2×M16C22×D314yes—
273C22×C8—38no—
274C22×C8—34no—
275C22×C8—31no—
276C22×C8—31no—
277C22×C8—32no—
278C22×C8—316no—
279C22×C8—38no—
280C22×C8C4×D318yescomplemented
281C22×C8C22×D312yes—
282C22×C8C22×D311yescharacteristic
283C22×C8C22×D311yescharacteristic
284C22×C8C22×D314yes—
285#117—34no—
286#117—34no—
287#117—316no—
288#117—316no—
289#117—38no—
290#117—38no—
291#117C4×D314yes—
292#117C22×D314yes—
293C2×C4⋊C4—34no—
294C2×C4⋊C4—32no—
295C2×C4⋊C4—316no—
296C2×C4⋊C4C4×D312yes—
297C2×C22⋊C4—32no—
298C2×C22⋊C4—34no—
299C2×C22⋊C4—316no—
300C2×C22⋊C4C4×D312yes—
301C2×C42—34no—
302C2×C42—32no—
303C2×C42—38no—
304C2×C42—38no—
305C2×C42C22×D312yes—
306#97—38no—
307#97—38no—
308#97—38no—
309#97C22×D318yescomplemented
310C4×C8—38no—
311C4×C8—38no—
312C4×C8—38no—
313C4×C8C22×D318yescomplemented
314C23×C3—24no—
315C23×C3C2×C4214yes—
316C23×C3C23×C412yes—
317C23×C3C23×C411yescharacteristic
318C22×D3—24no—
319C22×D3—24no—
320C22×D3—416no—
321C22×D3—48no—
322C22×D3C2×C4214yes—
323C22×D3C2×C4214yes—
324C22×D3C2×C42116yes—
325C22×D3C23×C412yes—
326C22×D3C23×C412yes—
327C2×C3×C4—28no—
328C2×C3×C4C2×C4218yes—
329C2×C3×C4C2×C4214yes—
330C2×C3×C4C23×C418yes—
331C2×C3×C4C23×C412yes—
332C2×C3×C4C23×C412yes—
333C2×C3×C4C23×C414yes—
334C2×C3×C4C23×C414yes—
335C2×C3×C4C23×C411yescharacteristic
336C2×C3×C4C23×C412yes—
337C2×C3×C4C2511yescharacteristic
338C2×Dic3—28no—
339C2×Dic3C2×C42116yes—
340C2×Dic3C2×C4214yes—
341C2×Dic3C2×C4214yes—
342C2×Dic3C23×C418yes—
343C2×Dic3C23×C412yes—
344C2×Dic3C23×C412yes—
345C4×D3—416no—
346C4×D3C2×C4218yes—
347C4×D3C2×C4218yes—
348C4×D3C2×C42116yes—
349C4×D3C2×C42116yes—
350C4×D3C23×C414yes—
351C4×D3C23×C414yes—
352C4×D3C23×C414yes—
353C4×D3C23×C414yes—
354C3×C8C23×C418yes—
355C3×C8C23×C414yes—
356C3×C8C23×C414yes—
357C3⋊C8C23×C418yes—
358C3⋊C8C23×C414yes—
359C3⋊C8C23×C414yes—
360C24—31no—
361C24—31no—
362C24—38no—
363C24—34no—
364C24—124no—
365C24C2×C4×D311yescharacteristic
366C22×C4—38no—
367C22×C4—31no—
368C22×C4—31no—
369C22×C4—34no—
370C22×C4—38no—
371C22×C4—38no—
372C22×C4—332no—
373C22×C4—32no—
374C22×C4—34no—
375C22×C4—34no—
376C22×C4—32no—
377C22×C4—34no—
378C22×C4—316no—
379C22×C4—34no—
380C22×C4—34no—
381C22×C4—34no—
382C22×C4—316no—
383C22×C4—32no—
384C22×C4—32no—
385C22×C4—34no—
386C22×C4—316no—
387C22×C4—128no—
388C22×C4C2×C4×D314yes—
389C22×C4C2×C4×D311yescharacteristic
390C22×C4C2×C4×D312yes—
391C22×C4C2×C4×D314yes—
392C22×C4C2×C4×D314yes—
393C22×C4C23×D312yes—
394C22×C4C23×D311yescharacteristic
395M16—316no—
396M16—332no—
397M16—316no—
398M16—316no—
399M16—316no—
400M16—316no—
401M16C2×C4×D3116yescomplemented
402C2×C8—34no—
403C2×C8—34no—
404C2×C8—38no—
405C2×C8—316no—
406C2×C8—32no—
407C2×C8—34no—
408C2×C8—32no—
409C2×C8—332no—
410C2×C8C2×C4×D318yes—
411C2×C8C2×C4×D314yes—
412C2×C8C2×C4×D314yes—
413C2×C8C2×C4×D3116yescomplemented
414C2×C8C23×D312yes—
415C2×C8C23×D314yes—
416C2×C8C23×D312yes—
417C4⋊C4—316no—
418C4⋊C4—38no—
419C4⋊C4C2×C4×D318yes—
420C22⋊C4—38no—
421C22⋊C4—316no—
422C22⋊C4C2×C4×D318yes—
423C42—316no—
424C42—34no—
425C42—34no—
426C42C2×C4×D314yes—
427C42C23×D314yes—
428C22×C3—28no—
429C22×C3—24no—
430C22×C3C22×C4214yes—
431C22×C3C22×C4212yes—
432C22×C3#56814yes—
433C22×C3C24×C411yescharacteristic
434C2×D3—28no—
435C2×D3—28no—
436C2×D3—24no—
437C2×D3—24no—
438C2×D3—416no—
439C2×D3C22×C4214yes—
440C2×D3C22×C4214yes—
441C3×C4—28no—
442C3×C4C22×C4212yes—
443C3×C4C22×C4214yes—
444C3×C4C24×C411yescharacteristic
445C3×C4C24×C411yescharacteristic
446Dic3—28no—
447Dic3C22×C4214yes—
448Dic3C22×C4214yes—
449C23—24no—
450C23—32no—
451C23—32no—
452C23—34no—
453C23—34no—
454C23—316no—
455C23—64no—
456C23—64no—
457C23—128no—
458C23—1216no—
459C23C42×D314yes—
460C23C22×C4×D312yes—
461C23C22×C4×D311yescharacteristic
462C2×C4—28no—
463C2×C4—32no—
464C2×C4—32no—
465C2×C4—316no—
466C2×C4—34no—
467C2×C4—34no—
468C2×C4—34no—
469C2×C4—34no—
470C2×C4—34no—
471C2×C4—34no—
472C2×C4—38no—
473C2×C4—316no—
474C2×C4—38no—
475C2×C4—316no—
476C2×C4—38no—
477C2×C4—68no—
478C2×C4—1216no—
479C2×C4C42×D318yes—
480C2×C4C42×D314yes—
481C2×C4C22×C4×D318yes—
482C2×C4C22×C4×D312yes—
483C2×C4C22×C4×D312yes—
484C2×C4C22×C4×D314yes—
485C2×C4C22×C4×D314yes—
486C2×C4C22×C4×D311yescharacteristic
487C2×C4C22×C4×D312yes—
488C2×C4C24×D311yescharacteristic
489C8—34no—
490C8—34no—
491C8—38no—
492C8C22×C4×D318yes—
493C8C22×C4×D314yes—
494C8C22×C4×D314yes—
495C2×C3—24no—
496C2×C3#287314yescomplemented
497C2×C3C23×C4211yescharacteristic
498C2×C3C2×#56812yes—
499D3—24no—
500D3—24no—
501C22—24no—
502C22—28no—
503C22—34no—
504C22—34no—
505C22—64no—
506C22—64no—
507C22—68no—
508C22—68no—
509C22—1216no—
510C22C2×C42×D314yes—
511C22C2×C42×D312yes—
512C22#575014yes—
513C22C23×C4×D311yescharacteristic
514C4—28no—
515C4—34no—
516C4—34no—
517C4—68no—
518C4C2×C42×D312yes—
519C4C2×C42×D314yes—
520C4C23×C4×D311yescharacteristic
521C4C23×C4×D311yescharacteristic
522C3C2×#287311yescharacteristic, complemented, Sylow
523C2—24no—
524C2—64no—
525C2—64no—
526C2#7854814yesdirect product factor
527C2C22×C42×D311yescharacteristic
528C2C2×#575012yes—
subgroupG/H
centerC22×C4C23×D3
commutatorC2×C3C23×C42
FrattiniC2×C4C24×D3
FittingC2×C3×#405C2
socleC23×C3C23×C4
prime radicalC24×D3C2×C4
cosocleC2×C3×C4C25
hypercenterC2×#405D3
hypocenterC3C2×#2873
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C2×#405

Complements

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

NH ≅ G/Ntype
4. C2×C3×#405C2semidirect
5. C2×#5122C2semidirect
6. C2×#5101C2semidirect
7. C2×#4997C2semidirect
8. #78548C2direct (N × H)
9. C2×#4708C2semidirect
10. C2×#4707C2semidirect
11. C2×#4689C2semidirect
12. C2×C4×ΓL(1,𝔽25)C2semidirect
21. C2×D3×M16C4semidirect
23. C22×ΓL(1,𝔽25)C4semidirect
39. C3×#405C22semidirect
40. C2×C3×#97C22semidirect
41. C2×C3×C4×C8C22semidirect
42. #5122C22semidirect
43. #5101C22semidirect
44. C2×#815C22semidirect
45. C2×#815C22semidirect
46. C2×C8×Dic3C22semidirect
47. C2×C8×Dic3C22semidirect
48. #4997C22semidirect
49. C2×#803C22semidirect
50. C2×C4×C3⋊C8C22semidirect
51. #4708C22semidirect
52. #4707C22semidirect
53. #4689C22semidirect
54. C4×ΓL(1,𝔽25)C22semidirect
64. C2×#405D3semidirect
79. C2×C3×M16C2×C4semidirect
81. C22×C3×C8C2×C4semidirect
99. C2×#208C2×C4semidirect
101. C22×C3⋊C8C2×C4semidirect
106. D3×M16C2×C4semidirect
110. C2×ΓL(1,𝔽25)C2×C4semidirect
111. C2×ΓL(1,𝔽25)C2×C4semidirect
112. C2×ΓL(1,𝔽25)C2×C4semidirect
118. C2×D3×C8C2×C4semidirect
128. C3×#97C23semidirect
129. C3×C4×C8C23semidirect
130. #815C23semidirect
131. #815C23semidirect
132. C8×Dic3C23semidirect
133. C8×Dic3C23semidirect
134. #803C23semidirect
135. C4×C3⋊C8C23semidirect
166. #405C2×D3semidirect
172. C2×#97C2×D3semidirect
176. C2×C4×C8C2×D3semidirect
212. C3×M16C22×C4semidirect
216. C2×C3×C8C22×C4semidirect
231. #208C22×C4semidirect
235. C2×C3⋊C8C22×C4semidirect
242. ΓL(1,𝔽25)C22×C4semidirect
243. ΓL(1,𝔽25)C22×C4semidirect
244. D3×C8C22×C4semidirect
271. C2×M16C4×D3semidirect
280. C22×C8C4×D3semidirect
309. #97C22×D3semidirect
313. C4×C8C22×D3semidirect
401. M16C2×C4×D3semidirect
413. C2×C8C2×C4×D3semidirect
496. C2×C3#2873semidirect
522. C3C2×#2873semidirect
526. C2#78548direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D3×M162
C23×ΓL(1,𝔽25)2
C2×D3×#1172
C2×C3×#4052
C2×#51222
C2×#51012
C2×#49972
#785482
C2×#47082
C2×#47072
C2×#46892
C2×C4×ΓL(1,𝔽25)2
C2×#28733