Finite Group Databasebeta

← C2×#78469 #11664913

Subgroups — C2×#78469 #11664913

Order 768.

2070 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5479C212yesmaximal, complemented
2C22×C3×C4×C8C211yesmaximal, characteristic, complemented
3C2×#5116C211yesmaximal, characteristic, complemented
4C2×#5115C211yesmaximal, characteristic
5C2×#5114C212yesmaximal, complemented
6C2×C4×Dic12C211yesmaximal, characteristic, complemented
7C2×C4×#204C212yesmaximal, complemented
8#78469C2116yesmaximal, direct product factor
9C2×#4696C211yesmaximal, characteristic, complemented
10C2×#4697C212yesmaximal, complemented
11C2×C4×D24C211yesmaximal, characteristic, complemented
12C2×#4683C211yesmaximal, characteristic, complemented
13C2×#3168—31nomaximal, Sylow
14C23×C3×C8C2211yescharacteristic
15C23×C3×C8C2211yescharacteristic
16C22×C3×C42C2211yescharacteristic
17C2×#930C2212yes—
18C2×#929—22no—
19C2×C4×#67—22no—
20C2×#924C2212yes—
21C22×Dic12C2211yescharacteristic
22C22×D24C2211yescharacteristic
23C22×#204C2212yes—
24#5479—28no—
25#5479C22116yescomplemented
26C2×#876—22no—
27C2×#875C2212yes—
28C2×C4×D12C2212yes—
29C2×#872—22no—
30C2×#870C2212yes—
31C2×C4×Dic6C2212yes—
32C2×C3×C4×C8C2211yescharacteristic
33C2×C3×C4×C8C2211yescharacteristic
34C2×C3×C4×C8C2211yescharacteristic
35C2×C3×C4×C8C2211yescharacteristic
36C2×C3×C4×C8C2218yescomplemented
37#5116C2218yescomplemented
38#5115C2218yes—
39#5114C22116yescomplemented
40C2×#821C2212yes—
41C2×#821C2212yes—
42C2×#818C2211yescharacteristic
43C2×#818C2211yescharacteristic
44C2×#817C2212yes—
45C2×#816C2212yes—
46C2×#816C2212yes—
47C4×Dic12C2218yescomplemented
48#4697C22116yescomplemented
49#4696C2218yescomplemented
50C4×D24C2218yescomplemented
51C4×#204C22116yescomplemented
52#4683C2218yescomplemented
53C2×#529—32no—
54#3168—316no—
55C2×#499—31no—
56C2×#495—31no—
57C2×#488—32no—
58C2×#467—31no—
59C2×#466—31no—
60C2×#465—32no—
61C2×C4×Q16—31no—
62C2×C4×SD16—32no—
63C2×C4×D8—31no—
64C22×C4×C8D311yescharacteristic, complemented
65C23×C3×C4D411yescharacteristic
66C23×C3×C4D411yescharacteristic
67C23×C3×C4C2311yescharacteristic
68C22×#67—22no—
69C22×D12C2312yes—
70C22×C4×D3—22no—
71C22×Dic6C2312yes—
72C22×C3×C8—24no—
73C22×C3×C8—24no—
74C22×C3×C8C2311yescharacteristic
75C22×C3×C8C2311yescharacteristic
76C22×C3×C8C2314yes—
77C22×C3×C8C2314yes—
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C2×C3×C42D411yescharacteristic
81C2×C3×C42D411yescharacteristic
82C2×C3×C42D414yes—
83C2×C3×C42C2311yescharacteristic
84C2×C3×C42C2311yescharacteristic
85C2×C3×C42C2314yes—
86C2×#217—22no—
87#930—24no—
88#930C2318yes—
89#929—216no—
90C4×#67—216no—
91C2×(C2×D3)⋊C4—22no—
92C2×(C2×D3)⋊C4—22no—
93C2×(C2×D3)⋊C4—22no—
94C2×(C2×D3)⋊C4—22no—
95C2×C4⋊Dic3C2312yes—
96C2×C4⋊Dic3C2312yes—
97#924—24no—
98#924C2318yes—
99C2×Dic3⋊C4—22no—
100C2×Dic3⋊C4—22no—
101C2×Dic3⋊C4—22no—
102C2×Dic3⋊C4—22no—
103C2×C4×Dic3—22no—
104C2×Dic12—24no—
105C2×Dic12C2314yes—
106C2×D24—24no—
107C2×D24C2314yes—
108C2×#204—28no—
109C2×#204C2318yes—
110#876—216no—
111#875—24no—
112#875C2318yes—
113C4×D12—24no—
114C4×D12C2318yes—
115#872—216no—
116#870—24no—
117#870C2318yes—
118C4×Dic6—24no—
119C4×Dic6C2318yes—
120C3×C4×C8C2314yes—
121C3×C4×C8C2314yes—
122C3×C4×C8C2314yes—
123C3×C4×C8C2314yes—
124#821C2318yes—
125#821C2318yes—
126#818C2314yes—
127#818C2314yes—
128#817C2318yes—
129#816C2318yes—
130#816C2318yes—
131C22×Q16—31no—
132C22×SD16—32no—
133C22×D8—31no—
134C23×C8C2×D311yescharacteristic
135C23×C8C2×D311yescharacteristic
136#529—316no—
137#529—68no—
138C2×#126—62no—
139C2×#125—32no—
140C2×#124—32no—
141C2×#123—62no—
142C2×C22⋊Q8—32no—
143C2×#121—32no—
144C2×C4×Q8—32no—
145C2×C4×D4—32no—
146C2×C4×D4—62no—
147C2×#117—62no—
148C22×C42C2×D311yescharacteristic
149#499—38no—
150#495—38no—
151#488—316no—
152#467—38no—
153#466—38no—
154#465—316no—
155C4×Q16—38no—
156C4×SD16—316no—
157C4×D8—38no—
158C2×#107—31no—
159C2×#107—31no—
160C2×#106—32no—
161C2×#103—32no—
162C2×#103—32no—
163C2×#102—32no—
164C2×#102—32no—
165C2×C4×C8C2×D311yescharacteristic
166C2×C4×C8C2×D311yescharacteristic
167C2×C4×C8C2×D311yescharacteristic
168C2×C4×C8C2×D311yescharacteristic
169C2×C4×C8C2×D318yescomplemented
170C24×C3C2×D411yescharacteristic
171C23×D3—22no—
172C22×C3×C4—24no—
173C22×C3×C4—24no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2×D412yes—
181C22×C3×C4C2×D414yes—
182C22×C3×C4C2×D414yes—
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2412yes—
185C22×C3×C4C2411yescharacteristic
186C2×#67—28no—
187C2×#67—48no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—22no—
191C2×D12—24no—
192C2×D12—24no—
193C2×D12—22no—
194C2×D12C2414yes—
195C2×C4×D3—28no—
196C2×C4×D3—48no—
197C2×Dic6—24no—
198C2×Dic6—24no—
199C2×Dic6—22no—
200C2×Dic6C2414yes—
201C2×C3×C8—22no—
202C2×C3×C8—22no—
203C2×C3×C8—28no—
204C2×C3×C8—28no—
205C2×C3×C8#4114yes—
206C2×C3×C8#4114yes—
207C2×C3×C8C2412yes—
208C2×C3×C8C2412yes—
209C2×C3×C8C2412yes—
210C2×C3×C8C2412yes—
211C3×C42C2×D412yes—
212C3×C42C2×D412yes—
213C3×C42C2×D414yes—
214C3×C42C2×D414yes—
215C3×C42C2412yes—
216C3×C42C2412yes—
217#217—28no—
218(C2×D3)⋊C4—28no—
219(C2×D3)⋊C4—28no—
220(C2×D3)⋊C4—28no—
221(C2×D3)⋊C4—28no—
222C4⋊Dic3—22no—
223C4⋊Dic3—22no—
224C4⋊Dic3C2414yes—
225C4⋊Dic3C2414yes—
226Dic3⋊C4—28no—
227Dic3⋊C4—28no—
228Dic3⋊C4—28no—
229Dic3⋊C4—28no—
230C4×Dic3—28no—
231Dic12—28no—
232D24—28no—
233#204—216no—
234C22×Q8—32no—
235C22×D4—32no—
236C22×D4—62no—
237C23×C4—62no—
238C23×C4D1211yescharacteristic
239C23×C4D1211yescharacteristic
240C23×C4C22×D311yescharacteristic
241C2×Q16—34no—
242C2×Q16—64no—
243C2×SD16—38no—
244C2×SD16—68no—
245C2×D8—34no—
246C2×D8—64no—
247C22×C8—24no—
248C22×C8—24no—
249C22×C8C22×D311yescharacteristic
250C22×C8C22×D311yescharacteristic
251C22×C8C22×D314yes—
252C22×C8C22×D314yes—
253C22×C8C22×D311yescharacteristic
254C22×C8C22×D311yescharacteristic
255#126—616no—
256#125—38no—
257#125—64no—
258#124—38no—
259#124—64no—
260#123—616no—
261C22⋊Q8—38no—
262C22⋊Q8—64no—
263#121—38no—
264#121—64no—
265C4×Q8—38no—
266C4×Q8—64no—
267C4×D4—38no—
268C4×D4—616no—
269C4×D4—64no—
270#117—616no—
271C2×C4⋊C4—32no—
272C2×C4⋊C4—32no—
273C2×C4⋊C4—62no—
274C2×C4⋊C4—62no—
275C2×C4⋊C4—62no—
276C2×C4⋊C4—62no—
277C2×C22⋊C4—62no—
278C2×C22⋊C4—62no—
279C2×C22⋊C4—62no—
280C2×C22⋊C4—62no—
281C2×C22⋊C4—62no—
282C2×C42—62no—
283C2×C42D1211yescharacteristic
284C2×C42D1214yes—
285C2×C42D1211yescharacteristic
286C2×C42C22×D311yescharacteristic
287C2×C42C22×D314yes—
288C2×C42C22×D311yescharacteristic
289#107—34no—
290#107—34no—
291#106—38no—
292#103—38no—
293#103—38no—
294#102—38no—
295#102—38no—
296C4×C8C22×D314yes—
297C4×C8C22×D314yes—
298C4×C8C22×D314yes—
299C4×C8C22×D314yes—
300C23×C3—22no—
301C23×C3—22no—
302C23×C3#13514yes—
303C23×C3C22×D412yes—
304C23×C3C22×D411yescharacteristic
305C22×D3—24no—
306C22×D3—44no—
307C22×D3—44no—
308C22×D3—44no—
309C2×C3×C4—22no—
310C2×C3×C4—28no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—24no—
315C2×C3×C4—28no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4#13514yes—
319C2×C3×C4C22×D412yes—
320C2×C3×C4C22×D412yes—
321C2×C3×C4C22×D412yes—
322C2×C3×C4C22×D412yes—
323C2×C3×C4C22×D411yescharacteristic
324C2×C3×C4C22×D412yes—
325C2×C3×C4C2×#4112yes—
326C2×C3×C4C2×#4112yes—
327C2×C3×C4C2511yescharacteristic
328#67—416no—
329C2×Dic3—24no—
330C2×Dic3—24no—
331C2×Dic3—24no—
332C2×Dic3—44no—
333C2×Dic3—44no—
334C2×Dic3—44no—
335D12—24no—
336D12—24no—
337D12—44no—
338C4×D3—416no—
339Dic6—24no—
340Dic6—24no—
341Dic6—44no—
342C3×C8—22no—
343C3×C8—22no—
344C3×C8C2×#4114yes—
345C3×C8C2×#4114yes—
346C24—62no—
347C24C2×D1211yescharacteristic
348C2×Q8—34no—
349C2×Q8—62no—
350C2×Q8—64no—
351C2×Q8—64no—
352C2×D4—34no—
353C2×D4—68no—
354C2×D4—62no—
355C2×D4—64no—
356C2×D4—64no—
357C2×D4—128no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—62no—
363C22×C4—62no—
364C22×C4—62no—
365C22×C4—68no—
366C22×C4—128no—
367C22×C4C2×D1211yescharacteristic
368C22×C4C2×D1212yes—
369C22×C4C2×D1211yescharacteristic
370C22×C4C2×D1211yescharacteristic
371C22×C4C2×D1211yescharacteristic
372C22×C4C2×D1214yes—
373C22×C4C2×D1214yes—
374C22×C4C2×D1211yescharacteristic
375C22×C4C23×D312yes—
376C22×C4C23×D311yescharacteristic
377Q16—68no—
378SD16—616no—
379D8—68no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—28no—
383C2×C8—28no—
384C2×C8#23514yes—
385C2×C8#23514yes—
386C2×C8C23×D312yes—
387C2×C8C23×D312yes—
388C2×C8C23×D312yes—
389C2×C8C23×D312yes—
390C4⋊C4—34no—
391C4⋊C4—34no—
392C4⋊C4—62no—
393C4⋊C4—62no—
394C4⋊C4—68no—
395C4⋊C4—68no—
396C4⋊C4—68no—
397C4⋊C4—68no—
398C22⋊C4—68no—
399C22⋊C4—68no—
400C22⋊C4—68no—
401C22⋊C4—68no—
402C22⋊C4—68no—
403C42—68no—
404C42C2×D1214yes—
405C42C2×D1212yes—
406C42C2×D1212yes—
407C42C2×D1214yes—
408C42C23×D312yes—
409C42C23×D312yes—
410C22×C3—28no—
411C22×C3—22no—
412C22×C3—22no—
413C22×C3#53212yes—
414C22×C3C2×#13514yes—
415C22×C3C2×#13512yes—
416C22×C3C2×#13512yes—
417C22×C3C23×D411yescharacteristic
418C2×D3—44no—
419C2×D3—44no—
420C2×D3—44no—
421C2×D3—88no—
422C3×C4—22no—
423C3×C4—22no—
424C3×C4—22no—
425C3×C4—22no—
426C3×C4#53212yes—
427C3×C4C2×#13514yes—
428C3×C4C22×#4111yescharacteristic
429C3×C4C22×#4111yescharacteristic
430Dic3—44no—
431Dic3—44no—
432Dic3—44no—
433C23—22no—
434C23—22no—
435C23—64no—
436C23—124no—
437C23—124no—
438C23—124no—
439C23#90414yes—
440C23C22×D1212yes—
441C23C22×D1211yescharacteristic
442Q8—64no—
443Q8—64no—
444Q8—124no—
445D4—64no—
446D4—64no—
447D4—1216no—
448D4—124no—
449C2×C4—22no—
450C2×C4—28no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—28no—
458C2×C4—64no—
459C2×C4—64no—
460C2×C4—64no—
461C2×C4—124no—
462C2×C4—124no—
463C2×C4—124no—
464C2×C4—1216no—
465C2×C4#90414yes—
466C2×C4C22×D1212yes—
467C2×C4C22×D1212yes—
468C2×C4C22×D1212yes—
469C2×C4C22×D1212yes—
470C2×C4C22×D1211yescharacteristic
471C2×C4C22×D1212yes—
472C2×C4C2×#23512yes—
473C2×C4C2×#23512yes—
474C2×C4C24×D311yescharacteristic
475C8—22no—
476C8—22no—
477C8C2×#23514yes—
478C8C2×#23514yes—
479C2×C3—24no—
480C2×C3#316814yescomplemented
481C2×C3C2×#53211yescharacteristic
482C2×C3C22×#13511yescharacteristic
483C2×C3C22×#13511yescharacteristic
484D3—84no—
485C22—22no—
486C22—28no—
487C22—22no—
488C22—124no—
489C22—124no—
490C22—124no—
491C22—248no—
492C22#547712yes—
493C22C2×#90414yes—
494C22C2×#90412yes—
495C22C2×#90412yes—
496C22C23×D1211yescharacteristic
497C4—22no—
498C4—22no—
499C4—22no—
500C4—22no—
501C4—124no—
502C4—124no—
503C4—124no—
504C4#547712yes—
505C4C2×#90414yes—
506C4C22×#23511yescharacteristic
507C4C22×#23511yescharacteristic
508C3C2×#316811yescharacteristic, complemented, Sylow
509C2—24no—
510C2—244no—
511C2#7846914yesdirect product factor
512C2C2×#547711yescharacteristic
513C2C22×#90411yescharacteristic
514C2C22×#90411yescharacteristic
subgroupG/H
centerC22×C4C2×D12
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC22×C3×C4×C8C2
socleC23×C3C22×D4
prime radicalC2×#5115C2
hypercenterC22×C4×C8D3
hypocenterC3C2×#3168
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C22×C4×C8

Complements

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

NH ≅ G/Ntype
1. C2×#5479C2semidirect
2. C22×C3×C4×C8C2semidirect
3. C2×#5116C2semidirect
5. C2×#5114C2semidirect
6. C2×C4×Dic12C2semidirect
7. C2×C4×#204C2semidirect
8. #78469C2direct (N × H)
9. C2×#4696C2semidirect
10. C2×#4697C2semidirect
11. C2×C4×D24C2semidirect
12. C2×#4683C2semidirect
25. #5479C22semidirect
36. C2×C3×C4×C8C22semidirect
37. #5116C22semidirect
39. #5114C22semidirect
47. C4×Dic12C22semidirect
48. #4697C22semidirect
49. #4696C22semidirect
50. C4×D24C22semidirect
51. C4×#204C22semidirect
52. #4683C22semidirect
64. C22×C4×C8D3semidirect
169. C2×C4×C8C2×D3semidirect
480. C2×C3#3168semidirect
508. C3C2×#3168semidirect
511. C2#78469direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#54792
C22×C3×C4×C82
C2×#51162
C2×#51152
C2×#51142
C2×C4×Dic122
C2×C4×#2042
#784692
C2×#46962
C2×#46972
C2×C4×D242
C2×#46832
C2×#31683