Finite Group Databasebeta

← C2×#80740 #11664914

Subgroups — C2×#80740 #11664914

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×#5624C212yesmaximal, complemented
2C2×C3×#467C211yesmaximal, characteristic, complemented
3C2×#5191C211yesmaximal, characteristic, complemented
4C2×Dic3×Q16C211yesmaximal, characteristic, complemented
5C2×#5115C211yesmaximal, characteristic
6C22×C8×Dic3C211yesmaximal, characteristic, complemented
7C2×#5048C212yesmaximal, complemented
8#80740C2116yesmaximal, direct product factor
9C2×#4878C211yesmaximal, characteristic, complemented
10C2×#4872C211yesmaximal, characteristic, complemented
11C2×#4800C212yesmaximal, complemented
12C2×#4788C212yesmaximal, complemented
13C2×#3168—31nomaximal, Sylow
14C22×C3×Q16C2211yescharacteristic
15C23×C3×C8C2211yescharacteristic
16C2×C3×C22⋊Q8C2212yes—
17C2×#947C2212yes—
18C2×Q8×Dic3C2212yes—
19C22×#215C2212yes—
20C2×#930C2212yes—
21C22×C4×Dic3C2211yescharacteristic
22C23×C3⋊C8C2211yescharacteristic
23C22×D24C2211yescharacteristic
24#5624—28no—
25#5624C22116yescomplemented
26C2×#898—22no—
27C2×#893C2212yes—
28C2×#892—22no—
29C2×#890C2212yes—
30C2×#886—22no—
31C2×#67⋊C4—22no—
32C3×#467C2218yescomplemented
33C2×C3×#107C2211yescharacteristic
34C2×C3×#103C2212yes—
35C2×#835C2212yes—
36#5191C2218yescomplemented
37Dic3×Q16C2218yescomplemented
38#5115C2218yes—
39C2×#821C2212yes—
40C2×#818C2211yescharacteristic
41C2×C8×Dic3C2211yescharacteristic
42C2×C8×Dic3C2211yescharacteristic
43C2×C8×Dic3C2211yescharacteristic
44C2×C8×Dic3C2211yescharacteristic
45C2×C8×Dic3C2218yescomplemented
46#5048C22116yescomplemented
47C2×#809C2212yes—
48C2×#808C2212yes—
49#4878C2218yescomplemented
50#4872C2218yescomplemented
51#4800C22116yescomplemented
52#4788C22116yescomplemented
53C2×#529—32no—
54#3168—316no—
55C2×#499—31no—
56C2×#495—31no—
57C2×#488—32no—
58C2×#467D311yescharacteristic, complemented
59C2×#466—31no—
60C2×#465—32no—
61C2×C4×Q16—31no—
62C2×C4×SD16—32no—
63C2×C4×D8—31no—
64C22×C4×C8—31no—
65C22×C3×Q8C2312yes—
66C23×C3×C4C2311yescharacteristic
67C22×#67—22no—
68C23×Dic3D411yescharacteristic
69C23×Dic3D411yescharacteristic
70C22×D12C2312yes—
71C22×C4×D3—22no—
72C2×C3×Q16—24no—
73C2×C3×Q16C2314yes—
74C22×C3×C8—24no—
75C22×C3×C8C2311yescharacteristic
76C22×C3×C8C2314yes—
77C22×C3×C8C2311yescharacteristic
78C3×C22⋊Q8—24no—
79C3×C22⋊Q8C2318yes—
80C2×C3×C4⋊C4—22no—
81C2×C3×C4⋊C4C2312yes—
82C2×C3×C22⋊C4—22no—
83#947—24no—
84#947C2318yes—
85Q8×Dic3—24no—
86Q8×Dic3C2318yes—
87C2×#215—28no—
88C2×#215C2318yes—
89#930—24no—
90#930C2318yes—
91C2×(C2×D3)⋊C4—22no—
92C2×(C2×D3)⋊C4—22no—
93C2×(C2×D3)⋊C4—22no—
94C2×(C2×D3)⋊C4—22no—
95C2×C4⋊Dic3—22no—
96C2×C4⋊Dic3—22no—
97C2×C4⋊Dic3C2312yes—
98C2×Dic3⋊C4—22no—
99C2×C4×Dic3—22no—
100C2×C4×Dic3D411yescharacteristic
101C2×C4×Dic3D411yescharacteristic
102C2×C4×Dic3D414yes—
103C2×C4×Dic3C2311yescharacteristic
104C2×C4×Dic3C2311yescharacteristic
105C2×C4×Dic3C2314yes—
106C22×C3⋊C8—24no—
107C22×C3⋊C8C2311yescharacteristic
108C22×C3⋊C8C2314yes—
109C22×C3⋊C8C2311yescharacteristic
110C2×D24—24no—
111C2×D24C2314yes—
112#898—216no—
113#893—24no—
114#893C2318yes—
115#892—216no—
116#890—24no—
117#890C2318yes—
118#886—216no—
119#67⋊C4—216no—
120C3×#107C2314yes—
121C3×#103C2318yes—
122#835C2318yes—
123#821C2318yes—
124#818C2314yes—
125C8×Dic3C2314yes—
126C8×Dic3C2314yes—
127C8×Dic3C2314yes—
128C8×Dic3C2314yes—
129#809C2318yes—
130#808C2318yes—
131C22×Q16C2×D311yescharacteristic
132C22×SD16—32no—
133C22×D8—31no—
134C23×C8—31no—
135C23×C8C2×D311yescharacteristic
136#529—316no—
137#529—68no—
138C2×#126—62no—
139C2×#125—32no—
140C2×#124—32no—
141C2×#123—62no—
142C2×C22⋊Q8C2×D312yes—
143C2×#121—32no—
144C2×C4×Q8—32no—
145C2×C4×D4—32no—
146C2×C4×D4—62no—
147C2×#117—62no—
148C22×C42—31no—
149#499—38no—
150#495—38no—
151#488—316no—
152#467C2×D318yescomplemented
153#466—38no—
154#465—316no—
155C4×Q16—38no—
156C4×SD16—316no—
157C4×D8—38no—
158C2×#107—31no—
159C2×#107C2×D311yescharacteristic
160C2×#106—32no—
161C2×#103—32no—
162C2×#103C2×D312yes—
163C2×#102—32no—
164C2×#102—32no—
165C2×C4×C8—31no—
166C2×C4×C8—31no—
167C2×C4×C8—38no—
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—48no—
186C22×Dic3—24no—
187C22×Dic3—24no—
188C22×Dic3—22no—
189C22×Dic3C2×D414yes—
190C22×Dic3C2×D414yes—
191C22×Dic3C2×D411yescharacteristic
192C22×Dic3C2×D411yescharacteristic
193C22×Dic3C2×D411yescharacteristic
194C22×Dic3C2×D411yescharacteristic
195C2×D12—24no—
196C2×D12—24no—
197C2×D12—22no—
198C2×D12C2414yes—
199C2×C4×D3—28no—
200C2×C4×D3—48no—
201C3×Q16—28no—
202C2×C3×C8—22no—
203C2×C3×C8—28no—
204C2×C3×C8#4114yes—
205C2×C3×C8C2412yes—
206C2×C3×C8C2412yes—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—28no—
209C3×C4⋊C4C2414yes—
210C3×C22⋊C4—28no—
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⋊Dic3—28no—
219C4⋊Dic3C2414yes—
220Dic3⋊C4—28no—
221C4×Dic3—28no—
222C4×Dic3C2×D412yes—
223C4×Dic3C2×D414yes—
224C4×Dic3C2×D412yes—
225C4×Dic3C2×D414yes—
226C4×Dic3C2412yes—
227C4×Dic3C2412yes—
228C2×C3⋊C8—22no—
229C2×C3⋊C8—28no—
230C2×C3⋊C8#4114yes—
231C2×C3⋊C8C2412yes—
232C2×C3⋊C8C2412yes—
233D24—28no—
234C22×Q8C22×D312yes—
235C22×D4—32no—
236C22×D4—62no—
237C23×C4—31no—
238C23×C4—31no—
239C23×C4—62no—
240C23×C4C22×D311yescharacteristic
241C2×Q16—24no—
242C2×Q16C22×D314yes—
243C2×SD16—38no—
244C2×SD16—68no—
245C2×D8—34no—
246C2×D8—64no—
247C22×C8—24no—
248C22×C8—31no—
249C22×C8—31no—
250C22×C8—34no—
251C22×C8—64no—
252C22×C8C22×D311yescharacteristic
253C22×C8C22×D314yes—
254C22×C8C22×D311yescharacteristic
255#126—616no—
256#125—38no—
257#125—64no—
258#124—38no—
259#124—64no—
260#123—616no—
261C22⋊Q8—24no—
262C22⋊Q8C22×D318yes—
263#121—38no—
264#121—64no—
265C4×Q8—38no—
266C4×Q8—64no—
267C4×D4—38no—
268C4×D4—64no—
269C4×D4—616no—
270#117—616no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—32no—
273C2×C4⋊C4—62no—
274C2×C4⋊C4—62no—
275C2×C4⋊C4—62no—
276C2×C4⋊C4C22×D312yes—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—62no—
279C2×C22⋊C4—62no—
280C2×C22⋊C4—62no—
281C2×C22⋊C4—62no—
282C2×C42—31no—
283C2×C42—31no—
284C2×C42—34no—
285C2×C42—31no—
286C2×C42—31no—
287C2×C42—34no—
288C2×C42—62no—
289#107—34no—
290#107C22×D314yes—
291#106—38no—
292#103—38no—
293#103C22×D318yes—
294#102—38no—
295#102—38no—
296C4×C8—34no—
297C4×C8—34no—
298C4×C8—34no—
299C4×C8—34no—
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—
309C3×Q8—24no—
310C3×Q8—24no—
311C3×Q8—44no—
312C2×C3×C4—22no—
313C2×C3×C4—24no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—24no—
317C2×C3×C4—24no—
318C2×C3×C4—44no—
319C2×C3×C4—44no—
320C2×C3×C4C22×D412yes—
321C2×C3×C4C22×D411yescharacteristic
322C2×C3×C4C2×#4112yes—
323C2×C3×C4C2×#4112yes—
324C2×C3×C4C2511yescharacteristic
325#67—416no—
326C2×Dic3—28no—
327C2×Dic3—22no—
328C2×Dic3—22no—
329C2×Dic3—22no—
330C2×Dic3—28no—
331C2×Dic3—24no—
332C2×Dic3—44no—
333C2×Dic3#13514yes—
334C2×Dic3C22×D412yes—
335C2×Dic3C22×D412yes—
336C2×Dic3C22×D412yes—
337C2×Dic3C22×D412yes—
338D12—24no—
339D12—24no—
340D12—44no—
341C4×D3—416no—
342C3×C8—22no—
343C3×C8C2×#4114yes—
344C3⋊C8—22no—
345C3⋊C8C2×#4114yes—
346C24—62no—
347C24D3×D411yescharacteristic
348C2×Q8—22no—
349C2×Q8—24no—
350C2×Q8—24no—
351C2×Q8C23×D314yes—
352C2×D4—34no—
353C2×D4—68no—
354C2×D4—62no—
355C2×D4—64no—
356C2×D4—64no—
357C2×D4—128no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—31no—
363C22×C4—34no—
364C22×C4—34no—
365C22×C4—31no—
366C22×C4—31no—
367C22×C4—31no—
368C22×C4—68no—
369C22×C4—64no—
370C22×C4—64no—
371C22×C4—62no—
372C22×C4—128no—
373C22×C4D3×D412yes—
374C22×C4D3×D411yescharacteristic
375C22×C4C23×D312yes—
376C22×C4C23×D311yescharacteristic
377Q16—28no—
378SD16—616no—
379D8—68no—
380C2×C8—22no—
381C2×C8—28no—
382C2×C8—34no—
383C2×C8—32no—
384C2×C8—32no—
385C2×C8—68no—
386C2×C8—62no—
387C2×C8#23714yes—
388C2×C8C23×D312yes—
389C2×C8C23×D312yes—
390C4⋊C4—28no—
391C4⋊C4—22no—
392C4⋊C4—34no—
393C4⋊C4—68no—
394C4⋊C4—68no—
395C4⋊C4—68no—
396C4⋊C4—62no—
397C4⋊C4C23×D314yes—
398C22⋊C4—28no—
399C22⋊C4—68no—
400C22⋊C4—68no—
401C22⋊C4—68no—
402C22⋊C4—68no—
403C42—34no—
404C42—32no—
405C42—32no—
406C42—34no—
407C42—32no—
408C42—32no—
409C42—68no—
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—44no—
424C3×C4—44no—
425C3×C4#53212yes—
426C3×C4C22×#4111yescharacteristic
427C3×C4C22×#4111yescharacteristic
428Dic3—22no—
429Dic3—22no—
430Dic3—22no—
431Dic3—44no—
432Dic3C2×#13514yes—
433C23—22no—
434C23—22no—
435C23—64no—
436C23—124no—
437C23—124no—
438C23—124no—
439C23#91914yes—
440C23C2×D3×D412yes—
441C23C2×D3×D411yescharacteristic
442Q8—24no—
443Q8—24no—
444Q8—44no—
445D4—64no—
446D4—64no—
447D4—1216no—
448D4—124no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—32no—
456C2×C4—34no—
457C2×C4—32no—
458C2×C4—32no—
459C2×C4—32no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—68no—
463C2×C4—68no—
464C2×C4—62no—
465C2×C4—62no—
466C2×C4—62no—
467C2×C4—64no—
468C2×C4—1216no—
469C2×C4—124no—
470C2×C4C2×D3×D412yes—
471C2×C4C2×D3×D411yescharacteristic
472C2×C4C2×#23712yes—
473C2×C4D3×#4112yes—
474C2×C4C24×D311yescharacteristic
475C8—22no—
476C8—34no—
477C8—62no—
478C8C2×#23714yes—
479C2×C3—24no—
480C2×C3#316814yescomplemented
481C2×C3C2×#53211yescharacteristic
482C2×C3C22×#13511yescharacteristic
483C2×C3C22×#13511yescharacteristic
484D3—84no—
485C22—28no—
486C22—22no—
487C22—22no—
488C22—124no—
489C22—124no—
490C22—124no—
491C22—248no—
492C22#559612yes—
493C22C2×#91914yes—
494C22C2×#91912yes—
495C22D3×#13512yes—
496C22C22×D3×D411yescharacteristic
497C4—22no—
498C4—34no—
499C4—44no—
500C4—44no—
501C4—62no—
502C4—62no—
503C4—62no—
504C4—124no—
505C4#559612yes—
506C4C22×#23711yescharacteristic
507C4C2×D3×#4111yescharacteristic
508C3C2×#316811yescharacteristic, complemented, Sylow
509C2—24no—
510C2—244no—
511C2#8074014yesdirect product factor
512C2C2×#559611yescharacteristic
513C2C22×#91911yescharacteristic
514C2C2×D3×#13511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#467C2
socleC23×C3C22×D4
prime radicalC2×#5115C2
hypercenterC2×#467D3
hypocenterC3C2×#3168
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×#5624C2semidirect
2. C2×C3×#467C2semidirect
3. C2×#5191C2semidirect
4. C2×Dic3×Q16C2semidirect
6. C22×C8×Dic3C2semidirect
7. C2×#5048C2semidirect
8. #80740C2direct (N × H)
9. C2×#4878C2semidirect
10. C2×#4872C2semidirect
11. C2×#4800C2semidirect
12. C2×#4788C2semidirect
25. #5624C22semidirect
32. C3×#467C22semidirect
36. #5191C22semidirect
37. Dic3×Q16C22semidirect
45. C2×C8×Dic3C22semidirect
46. #5048C22semidirect
49. #4878C22semidirect
50. #4872C22semidirect
51. #4800C22semidirect
52. #4788C22semidirect
58. C2×#467D3semidirect
152. #467C2×D3semidirect
480. C2×C3#3168semidirect
508. C3C2×#3168semidirect
511. C2#80740direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#56242
C2×C3×#4672
C2×#51912
C2×Dic3×Q162
C2×#51152
C22×C8×Dic32
C2×#50482
#807402
C2×#48782
C2×#48722
C2×#48002
C2×#47882
C2×#31683