Finite Group Databasebeta

← C2×#80742 #11665010

Subgroups — C2×#80742 #11665010

Order 768.

1970 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5643C212yesmaximal, complemented
2C2×C3×#467C211yesmaximal, characteristic, complemented
3C2×#5191C211yesmaximal, characteristic, complemented
4C2×Dic3×Q16C211yesmaximal, characteristic, complemented
5C2×#5115C211yesmaximal, characteristic
6C2×#5101C211yesmaximal, characteristic, complemented
7C2×#5049C212yesmaximal, complemented
8#80742C2116yesmaximal, direct product factor
9C2×#4874C211yesmaximal, characteristic, complemented
10C2×#4872C211yesmaximal, characteristic, complemented
11C2×#4794C212yesmaximal, complemented
12C2×#4817C212yesmaximal, complemented
13C2×#3175—31nomaximal, Sylow
14C22×C3×Q16C2211yescharacteristic
15C23×C3×C8C2211yescharacteristic
16C2×C3×C22⋊Q8C2212yes—
17C2×#947C2212yes—
18C2×Q8×Dic3C2212yes—
19C22×#215C2212yes—
20C2×#930C2212yes—
21C2×#926C2211yescharacteristic
22C22×#208C2211yescharacteristic
23C22×D24C2211yescharacteristic
24#5643—28no—
25#5643C22116yescomplemented
26C2×#898—22no—
27C2×#897—22no—
28C2×#893C2212yes—
29C2×#888C2212yes—
30C2×#885—22no—
31C2×#883—22no—
32C3×#467C2218yescomplemented
33C2×C3×#107C2211yescharacteristic
34C2×C3×#103C2212yes—
35C2×#835C2212yes—
36#5191C2218yescomplemented
37Dic3×Q16C2218yescomplemented
38#5115C2218yes—
39C2×#821C2212yes—
40C2×#818C2211yescharacteristic
41#5101C2218yescomplemented
42C2×#815C2211yescharacteristic
43C2×#815C2211yescharacteristic
44C2×C8×Dic3C2211yescharacteristic
45C2×C8×Dic3C2211yescharacteristic
46#5049C22116yescomplemented
47C2×#809C2212yes—
48C2×#807C2212yes—
49#4874C2218yescomplemented
50#4872C2218yescomplemented
51#4817C22116yescomplemented
52#4794C22116yescomplemented
53C2×#542—32no—
54#3175—316no—
55C2×#500—31no—
56C2×#495—31no—
57C2×#489—32no—
58C2×#468—32no—
59C2×#467D311yescharacteristic, complemented
60C2×#466—31no—
61C2×#440—32no—
62C2×C4×Q16—31no—
63C2×C4×D8—31no—
64C2×#405—31no—
65C22×C3×Q8C2312yes—
66C23×C3×C4C2311yescharacteristic
67C22×#67—22no—
68C22×D12C2312yes—
69C22×C4×D3—22no—
70C22×Dic6—22no—
71C2×C3×Q16—24no—
72C2×C3×Q16C2314yes—
73C22×C3×C8—24no—
74C22×C3×C8C2311yescharacteristic
75C22×C3×C8C2314yes—
76C22×C3×C8C2311yescharacteristic
77C3×C22⋊Q8—24no—
78C3×C22⋊Q8C2318yes—
79C2×C3×C4⋊C4—22no—
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C22⋊C4—22no—
82C2×#217D411yescharacteristic
83C2×#217D411yescharacteristic
84#947—24no—
85#947C2318yes—
86Q8×Dic3—24no—
87Q8×Dic3C2318yes—
88C2×#215—28no—
89C2×#215C2318yes—
90#930—24no—
91#930C2318yes—
92C2×(C2×D3)⋊C4—22no—
93C2×(C2×D3)⋊C4—22no—
94C2×(C2×D3)⋊C4—22no—
95C2×(C2×D3)⋊C4—22no—
96#926D414yes—
97#926C2314yes—
98C2×C4⋊Dic3—22no—
99C2×C4⋊Dic3D411yescharacteristic
100C2×C4⋊Dic3D411yescharacteristic
101C2×C4⋊Dic3C2312yes—
102C2×Dic3⋊C4—22no—
103C2×C4×Dic3—22no—
104C2×C4×Dic3C2311yescharacteristic
105C2×C4×Dic3C2311yescharacteristic
106C2×#208—24no—
107C2×#208C2314yes—
108C22×C3⋊C8C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110C2×D24—24no—
111C2×D24C2314yes—
112#898—216no—
113#897—216no—
114#893—24no—
115#893C2318yes—
116#888—24no—
117#888C2318yes—
118#885—216no—
119#883—216no—
120C3×#107C2314yes—
121C3×#103C2318yes—
122#835C2318yes—
123#821C2318yes—
124#818C2314yes—
125#815C2314yes—
126#815C2314yes—
127C8×Dic3C2314yes—
128C8×Dic3C2314yes—
129#809C2318yes—
130#807C2318yes—
131C22×Q16C2×D311yescharacteristic
132C22×SD16—32no—
133C22×D8—31no—
134C22×M16—31no—
135C23×C8C2×D311yescharacteristic
136#542—316no—
137#542—68no—
138C2×C4⋊Q8—32no—
139C2×#126—62no—
140C2×#124—32no—
141C2×#124—62no—
142C2×#123—62no—
143C2×C22⋊Q8—62no—
144C2×C22⋊Q8C2×D312yes—
145C2×#121—32no—
146C2×C4×Q8—32no—
147C2×C4×D4—32no—
148C2×#117—31no—
149#500—38no—
150#495—38no—
151#489—316no—
152#468—316no—
153#467C2×D318yescomplemented
154#466—38no—
155#440—316no—
156C4×Q16—38no—
157C4×D8—38no—
158C2×#107—31no—
159C2×#107—32no—
160C2×#107C2×D311yescharacteristic
161C2×#103—32no—
162C2×#103C2×D312yes—
163C2×#102—32no—
164C2×#102—32no—
165#405—38no—
166C2×#97—31no—
167C2×#97—31no—
168C2×C4×C8—31no—
169C2×C4×C8—31no—
170C24×C3C2×D411yescharacteristic
171C23×D3—22no—
172C2×C3×Q8—22no—
173C2×C3×Q8—24no—
174C2×C3×Q8—24no—
175C2×C3×Q8C2414yes—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2×D412yes—
182C22×C3×C4C2412yes—
183C22×C3×C4C2411yescharacteristic
184C2×#67—28no—
185C2×#67—84no—
186C22×Dic3—22no—
187C22×Dic3C2×D411yescharacteristic
188C22×Dic3C2×D411yescharacteristic
189C22×Dic3C2×D411yescharacteristic
190C22×Dic3C2×D411yescharacteristic
191C2×D12—24no—
192C2×D12—24no—
193C2×D12—22no—
194C2×D12C2414yes—
195C2×C4×D3—28no—
196C2×C4×D3—84no—
197C2×Dic6—28no—
198C2×Dic6—84no—
199C3×Q16—28no—
200C2×C3×C8—22no—
201C2×C3×C8—28no—
202C2×C3×C8#4114yes—
203C2×C3×C8C2412yes—
204C2×C3×C8C2412yes—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—28no—
207C3×C4⋊C4C2414yes—
208C3×C22⋊C4—28no—
209#217C2×D414yes—
210#217C2×D414yes—
211#215—216no—
212(C2×D3)⋊C4—28no—
213(C2×D3)⋊C4—28no—
214(C2×D3)⋊C4—28no—
215(C2×D3)⋊C4—28no—
216C4⋊Dic3—22no—
217C4⋊Dic3—28no—
218C4⋊Dic3C2×D414yes—
219C4⋊Dic3C2×D414yes—
220C4⋊Dic3C2414yes—
221Dic3⋊C4—28no—
222C4×Dic3—28no—
223C4×Dic3C2×D412yes—
224C4×Dic3C2×D412yes—
225C4×Dic3C2412yes—
226C4×Dic3C2412yes—
227#208—28no—
228C2×C3⋊C8—22no—
229C2×C3⋊C8#4114yes—
230C2×C3⋊C8C2412yes—
231C2×C3⋊C8C2412yes—
232D24—28no—
233C22×Q8—62no—
234C22×Q8C22×D312yes—
235C22×D4—32no—
236C22×D4—62no—
237C23×C4—62no—
238C23×C4C22×D311yescharacteristic
239C2×Q16—24no—
240C2×Q16C22×D314yes—
241C2×SD16—38no—
242C2×SD16—68no—
243C2×D8—34no—
244C2×D8—64no—
245C2×M16—34no—
246C2×M16—64no—
247C22×C8—24no—
248C22×C8—31no—
249C22×C8—31no—
250C22×C8C22×D311yescharacteristic
251C22×C8C22×D314yes—
252C22×C8C22×D311yescharacteristic
253C4⋊Q8—38no—
254C4⋊Q8—64no—
255#126—616no—
256#124—38no—
257#124—64no—
258#124—616no—
259#123—616no—
260C22⋊Q8—24no—
261C22⋊Q8—616no—
262C22⋊Q8C22×D318yes—
263#121—38no—
264#121—64no—
265C4×Q8—38no—
266C4×Q8—64no—
267C4×D4—38no—
268C4×D4—64no—
269#117—34no—
270#117—34no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—31no—
273C2×C4⋊C4—31no—
274C2×C4⋊C4—32no—
275C2×C4⋊C4—62no—
276C2×C4⋊C4—62no—
277C2×C4⋊C4C22×D312yes—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—31no—
280C2×C22⋊C4—31no—
281C2×C22⋊C4—62no—
282C2×C22⋊C4—62no—
283C2×C22⋊C4—62no—
284C2×C22⋊C4—62no—
285C2×C42—31no—
286C2×C42—31no—
287C2×C42—62no—
288#107—34no—
289#107—38no—
290#107C22×D314yes—
291#103—38no—
292#103C22×D318yes—
293#102—38no—
294#102—38no—
295#97—34no—
296#97—34no—
297C4×C8—34no—
298C4×C8—34no—
299C23×C3—22no—
300C23×C3—22no—
301C23×C3—22no—
302C23×C3C22×D412yes—
303C23×C3C22×D411yescharacteristic
304C22×D3—24no—
305C22×D3—44no—
306C22×D3—44no—
307C22×D3—44no—
308C3×Q8—24no—
309C3×Q8—24no—
310C3×Q8—44no—
311C2×C3×C4—22no—
312C2×C3×C4—24no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—24no—
316C2×C3×C4—24no—
317C2×C3×C4—44no—
318C2×C3×C4—44no—
319C2×C3×C4C22×D412yes—
320C2×C3×C4C22×D411yescharacteristic
321C2×C3×C4C2×#4112yes—
322C2×C3×C4C2×#4112yes—
323C2×C3×C4C2511yescharacteristic
324#67—88no—
325C2×Dic3—22no—
326C2×Dic3—22no—
327C2×Dic3—22no—
328C2×Dic3—22no—
329C2×Dic3—24no—
330C2×Dic3—44no—
331C2×Dic3C22×D412yes—
332C2×Dic3C22×D412yes—
333C2×Dic3C22×D412yes—
334C2×Dic3C22×D412yes—
335D12—24no—
336D12—24no—
337D12—44no—
338C4×D3—88no—
339Dic6—88no—
340C3×C8—22no—
341C3×C8C2×#4114yes—
342C3⋊C8—22no—
343C3⋊C8C2×#4114yes—
344C24—62no—
345C24D3×D411yescharacteristic
346C2×Q8—22no—
347C2×Q8—24no—
348C2×Q8—24no—
349C2×Q8—68no—
350C2×Q8—244no—
351C2×Q8C23×D314yes—
352C2×D4—34no—
353C2×D4—68no—
354C2×D4—62no—
355C2×D4—64no—
356C2×D4—64no—
357C2×D4—244no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—31no—
363C22×C4—31no—
364C22×C4—31no—
365C22×C4—31no—
366C22×C4—68no—
367C22×C4—62no—
368C22×C4—244no—
369C22×C4D3×D412yes—
370C22×C4D3×D411yescharacteristic
371C22×C4C23×D312yes—
372C22×C4C23×D311yescharacteristic
373Q16—28no—
374SD16—616no—
375D8—68no—
376M16—68no—
377C2×C8—22no—
378C2×C8—28no—
379C2×C8—34no—
380C2×C8—32no—
381C2×C8—32no—
382C2×C8—62no—
383C2×C8#23714yes—
384C2×C8C23×D312yes—
385C2×C8C23×D312yes—
386C4⋊C4—28no—
387C4⋊C4—22no—
388C4⋊C4—34no—
389C4⋊C4—34no—
390C4⋊C4—34no—
391C4⋊C4—68no—
392C4⋊C4—68no—
393C4⋊C4—62no—
394C4⋊C4C23×D314yes—
395C22⋊C4—28no—
396C22⋊C4—34no—
397C22⋊C4—34no—
398C22⋊C4—68no—
399C22⋊C4—68no—
400C22⋊C4—68no—
401C22⋊C4—68no—
402C42—32no—
403C42—32no—
404C42—32no—
405C42—32no—
406C42—68no—
407C22×C3—22no—
408C22×C3—22no—
409C22×C3—22no—
410C22×C3—44no—
411C22×C3#53212yes—
412C22×C3#57612yes—
413C22×C3#57812yes—
414C22×C3C23×D411yescharacteristic
415C2×D3—44no—
416C2×D3—44no—
417C2×D3—44no—
418C2×D3—88no—
419C3×C4—22no—
420C3×C4—44no—
421C3×C4—44no—
422C3×C4#53212yes—
423C3×C4C22×#4111yescharacteristic
424C3×C4C22×#4111yescharacteristic
425Dic3—22no—
426Dic3—22no—
427Dic3—22no—
428Dic3—22no—
429Dic3—44no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—64no—
434C23—124no—
435C23—124no—
436C23—124no—
437C23C2×D3×D412yes—
438C23C2×D3×D411yescharacteristic
439Q8—24no—
440Q8—24no—
441Q8—44no—
442Q8—248no—
443D4—64no—
444D4—64no—
445D4—124no—
446D4—248no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—32no—
454C2×C4—32no—
455C2×C4—32no—
456C2×C4—32no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—62no—
460C2×C4—62no—
461C2×C4—62no—
462C2×C4—62no—
463C2×C4—64no—
464C2×C4—124no—
465C2×C4—248no—
466C2×C4C2×D3×D412yes—
467C2×C4C2×D3×D411yescharacteristic
468C2×C4C2×#23712yes—
469C2×C4D3×#4112yes—
470C2×C4C24×D311yescharacteristic
471C8—22no—
472C8—34no—
473C8—62no—
474C8C2×#23714yes—
475C2×C3—42no—
476C2×C3#317514yescomplemented
477C2×C3C2×#53211yescharacteristic
478C2×C3C2×#57611yescharacteristic
479C2×C3C2×#57811yescharacteristic
480D3—84no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—44no—
485C22—124no—
486C22—124no—
487C22—124no—
488C22—248no—
489C22#559612yes—
490C22#576612yes—
491C22#576912yes—
492C22C22×D3×D411yescharacteristic
493C4—22no—
494C4—44no—
495C4—44no—
496C4—62no—
497C4—62no—
498C4—62no—
499C4—62no—
500C4—124no—
501C4#559612yes—
502C4C22×#23711yescharacteristic
503C4C2×D3×#4111yescharacteristic
504C3C2×#317511yescharacteristic, complemented, Sylow
505C2—42no—
506C2—244no—
507C2#8074214yesdirect product factor
508C2C2×#559611yescharacteristic
509C2C2×#576611yescharacteristic
510C2C2×#576911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#467C2
socleC23×C3C22×D4
prime radicalC2×#5115C2
hypercenterC2×#467D3
hypocenterC3C2×#3175
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#467

Complements

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

NH ≅ G/Ntype
1. C2×#5643C2semidirect
2. C2×C3×#467C2semidirect
3. C2×#5191C2semidirect
4. C2×Dic3×Q16C2semidirect
6. C2×#5101C2semidirect
7. C2×#5049C2semidirect
8. #80742C2direct (N × H)
9. C2×#4874C2semidirect
10. C2×#4872C2semidirect
11. C2×#4794C2semidirect
12. C2×#4817C2semidirect
25. #5643C22semidirect
32. C3×#467C22semidirect
36. #5191C22semidirect
37. Dic3×Q16C22semidirect
41. #5101C22semidirect
46. #5049C22semidirect
49. #4874C22semidirect
50. #4872C22semidirect
51. #4817C22semidirect
52. #4794C22semidirect
59. C2×#467D3semidirect
153. #467C2×D3semidirect
476. C2×C3#3175semidirect
504. C3C2×#3175semidirect
507. C2#80742direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#56432
C2×C3×#4672
C2×#51912
C2×Dic3×Q162
C2×#51152
C2×#51012
C2×#50492
#807422
C2×#48742
C2×#48722
C2×#47942
C2×#48172
C2×#31753