Finite Group Databasebeta

← C2×#81557 #11664922

Subgroups — C2×#81557 #11664922

Order 768.

2566 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
1C2×#5725C211yesmaximal, characteristic, complemented
2C2×#5716C211yesmaximal, characteristic, complemented
3C2×C3×#486C211yesmaximal, characteristic, complemented
4C2×#5078C211yesmaximal, characteristic, complemented
5C2×#5073C211yesmaximal, characteristic, complemented
6#81557C2116yesmaximal, direct product factor
7C2×#4775C214yesmaximal, complemented
8C2×#4756C214yesmaximal, complemented
9C2×#4694C211yesmaximal, characteristic, complemented
10C2×C4×D3×C8C211yesmaximal, characteristic, complemented
11C2×#3169—31nomaximal, Sylow
12C22×#239—21no—
13C22×#237—21no—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×#127C2211yescharacteristic
16C2×#947—21no—
17C2×#938—21no—
18C2×#938C2212yes—
19C2×#936—21no—
20C2×D4×Dic3—21no—
21C2×D4×Dic3C2212yes—
22C22×#213C2214yes—
23C22×D3×C8C2212yes—
24#5725—24no—
25#5725C2218yescomplemented
26#5716—24no—
27#5716C2218yescomplemented
28C2×#895—21no—
29C2×#895C2212yes—
30C2×#893—21no—
31C2×#893C2212yes—
32C2×#874C2211yescharacteristic
33C2×C42×D3C2211yescharacteristic
34C2×#869C2211yescharacteristic
35C3×#486C2218yescomplemented
36C2×C3×#102C2214yes—
37C2×C3×C4×C8C2211yescharacteristic
38C2×#821C2214yes—
39C2×C8×Dic3C2212yes—
40#5078C2218yescomplemented
41#5073C2218yescomplemented
42C2×#807C2214yes—
43C2×C4×C3⋊C8C2211yescharacteristic
44#4775C22132yescomplemented
45#4756C22132yescomplemented
46#4694C2218yescomplemented
47C4×D3×C8C2218yescomplemented
48C2×#532—31no—
49C2×#532—31no—
50#3169—316no—
51C2×#500—31no—
52C2×#493—31no—
53C2×#486—31no—
54C2×#486D311yescharacteristic, complemented
55C2×#466—34no—
56C2×C4×D8—34no—
57C22×C4×C8—31no—
58C22×C3×Q8—21no—
59C22×C3×D4—21no—
60C22×C3×D4—21no—
61C22×C3×D4C2312yes—
62C22×#67—22no—
63C23×Dic3—22no—
64C2×#239—28no—
65C2×#239—48no—
66C2×#237—28no—
67C2×#237—48no—
68C22×D12—22no—
69C22×D12—21no—
70C22×D12—21no—
71C22×D12C2312yes—
72C22×C4×D3—22no—
73C22×C4×D3D411yescharacteristic
74C22×C4×D3D411yescharacteristic
75C22×C4×D3C2311yescharacteristic
76C22×Dic6—21no—
77C22×C3×C8C2312yes—
78C3×C4⋊Q8—22no—
79C3×C4⋊Q8C2314yes—
80C3×#127—22no—
81C3×#127C2314yes—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4C2312yes—
84C2×C3×C42C2311yescharacteristic
85C2×#217—22no—
86C2×#217—22no—
87#947—28no—
88#938—24no—
89#938—28no—
90#938C2318yes—
91#936—28no—
92D4×Dic3—24no—
93D4×Dic3—28no—
94D4×Dic3C2318yes—
95C2×#213—216no—
96C2×#213C23116yes—
97C2×(C2×D3)⋊C4—22no—
98C2×(C2×D3)⋊C4—22no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3C2312yes—
101C2×C4×Dic3D411yescharacteristic
102C2×C4×Dic3D411yescharacteristic
103C2×C4×Dic3C2311yescharacteristic
104C22×C3⋊C8C2312yes—
105C2×D3×C8—28no—
106C2×D3×C8C2318yes—
107#895—24no—
108#895—28no—
109#895C2318yes—
110#893—24no—
111#893—28no—
112#893C2318yes—
113#874—22no—
114#874C2314yes—
115C42×D3D414yes—
116C42×D3C2314yes—
117#869—22no—
118#869C2314yes—
119C3×#102C23116yes—
120C3×C4×C8C2314yes—
121#821C23116yes—
122C8×Dic3C2318yes—
123#807C23116yes—
124C4×C3⋊C8C2314yes—
125C22×#41—61no—
126C22×#41—61no—
127C22×D8—34no—
128C23×C8—32no—
129#532—38no—
130#532—38no—
131#532—64no—
132#532—64no—
133C2×C4⋊Q8—31no—
134C2×C4⋊Q8C2×D311yescharacteristic
135C2×#127—31no—
136C2×#127C2×D311yescharacteristic
137C2×#124—61no—
138C2×#124—61no—
139C2×#121—32no—
140C2×#121—32no—
141C2×#121—61no—
142C2×#121—61no—
143C2×C4×D4—32no—
144C2×C4×D4—32no—
145C2×C4×D4—61no—
146C2×C4×D4—61no—
147C22×C42—31no—
148#500—38no—
149#493—38no—
150#486—38no—
151#486C2×D318yescomplemented
152#466—332no—
153C4×D8—332no—
154C2×#107—34no—
155C2×#102—34no—
156C2×#102C2×D314yes—
157C2×C4×C8—32no—
158C2×C4×C8—31no—
159C2×C4×C8—38no—
160C2×C4×C8C2×D311yescharacteristic
161C24×C3—22no—
162C23×D3—22no—
163C23×D3C2×D411yescharacteristic
164C2×C3×Q8—24no—
165C2×C3×Q8—44no—
166C2×C3×D4—22no—
167C2×C3×D4—24no—
168C2×C3×D4—24no—
169C2×C3×D4—28no—
170C2×C3×D4—44no—
171C2×C3×D4—44no—
172C2×C3×D4C2414yes—
173C22×C3×C4—22no—
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2411yescharacteristic
177C2×#67—28no—
178C2×#67—48no—
179C22×Dic3—22no—
180C22×Dic3—28no—
181C22×Dic3—48no—
182C22×Dic3C2×D411yescharacteristic
183C22×Dic3C2×D411yescharacteristic
184C22×Dic3C2×D411yescharacteristic
185#239—432no—
186#237—432no—
187C2×D12—28no—
188C2×D12—28no—
189C2×D12—22no—
190C2×D12—24no—
191C2×D12—24no—
192C2×D12—48no—
193C2×D12—44no—
194C2×D12—44no—
195C2×D12C2414yes—
196C2×C4×D3—24no—
197C2×C4×D3—24no—
198C2×C4×D3—28no—
199C2×C4×D3—48no—
200C2×C4×D3D818yescomplemented
201C2×C4×D3C2×D412yes—
202C2×C4×D3C2×D414yes—
203C2×C4×D3C2×D414yes—
204C2×C4×D3C2412yes—
205C2×Dic6—24no—
206C2×Dic6—44no—
207C2×C3×C8—24no—
208C2×C3×C8C2414yes—
209C3×C4⋊C4—24no—
210C3×C4⋊C4—22no—
211C3×C4⋊C4C2414yes—
212C3×C42C2×D412yes—
213C3×C42C2412yes—
214#217—28no—
215#217—28no—
216#213—232no—
217(C2×D3)⋊C4—28no—
218(C2×D3)⋊C4—28no—
219C4⋊Dic3—22no—
220C4⋊Dic3—24no—
221C4⋊Dic3C2414yes—
222C4×Dic3C2×D412yes—
223C4×Dic3C2×D414yes—
224C4×Dic3C2×D414yes—
225C4×Dic3C2412yes—
226C2×C3⋊C8#4118yescomplemented
227C2×C3⋊C8C2414yes—
228D3×C8—216no—
229C2×#41—68no—
230C2×#41—68no—
231C2×#41—128no—
232C2×#41—128no—
233C22×Q8—21no—
234C22×Q8—61no—
235C22×D4—21no—
236C22×D4—21no—
237C22×D4—32no—
238C22×D4—62no—
239C22×D4—62no—
240C22×D4—61no—
241C22×D4—61no—
242C22×D4C22×D312yes—
243C23×C4—31no—
244C23×C4—31no—
245C23×C4—31no—
246C23×C4—62no—
247C23×C4—62no—
248C2×D8—316no—
249C2×D8—616no—
250C22×C8—32no—
251C22×C8—38no—
252C22×C8—68no—
253C22×C8C22×D312yes—
254C4⋊Q8—22no—
255C4⋊Q8—34no—
256C4⋊Q8—62no—
257C4⋊Q8C22×D314yes—
258#127—22no—
259#127—34no—
260#127—62no—
261#127C22×D314yes—
262#124—68no—
263#124—68no—
264#121—38no—
265#121—38no—
266#121—64no—
267#121—68no—
268#121—64no—
269#121—68no—
270C4×D4—38no—
271C4×D4—38no—
272C4×D4—64no—
273C4×D4—68no—
274C4×D4—64no—
275C4×D4—68no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—32no—
278C2×C4⋊C4—61no—
279C2×C4⋊C4C22×D312yes—
280C2×C22⋊C4—62no—
281C2×C22⋊C4—62no—
282C2×C22⋊C4—62no—
283C2×C22⋊C4—62no—
284C2×C42—31no—
285C2×C42—31no—
286C2×C42—31no—
287C2×C42—34no—
288C2×C42—34no—
289C2×C42C22×D311yescharacteristic
290#107—316no—
291#102—316no—
292#102C22×D3116yes—
293C4×C8—38no—
294C4×C8—34no—
295C4×C8C22×D314yes—
296C23×C3—24no—
297C23×C3—44no—
298C23×C3—44no—
299C23×C3—44no—
300C23×C3C22×D411yescharacteristic
301C22×D3—22no—
302C22×D3—22no—
303C22×D3—24no—
304C22×D3—44no—
305C22×D3—44no—
306C22×D3—44no—
307C22×D3C2×D814yes—
308C22×D3C22×D412yes—
309C3×Q8—48no—
310C3×D4—28no—
311C3×D4—44no—
312C3×D4—48no—
313C3×D4—48no—
314C2×C3×C4—24no—
315C2×C3×C4—44no—
316C2×C3×C4C2×D814yes—
317C2×C3×C4#13514yes—
318C2×C3×C4C22×D411yescharacteristic
319C2×C3×C4C22×D412yes—
320C2×C3×C4C22×D412yes—
321C2×C3×C4C2×#4114yes—
322C2×C3×C4C2511yescharacteristic
323#67—416no—
324C2×Dic3—22no—
325C2×Dic3—22no—
326C2×Dic3—24no—
327C2×Dic3—44no—
328C2×Dic3—416no—
329C2×Dic3C2×D814yes—
330C2×Dic3C22×D412yes—
331C2×Dic3C22×D412yes—
332C2×Dic3C22×D412yes—
333D12—28no—
334D12—416no—
335D12—44no—
336D12—48no—
337D12—48no—
338C4×D3—24no—
339C4×D3—28no—
340C4×D3—24no—
341C4×D3—416no—
342C4×D3C2×D8116yescomplemented
343Dic6—48no—
344C3×C8—24no—
345C3⋊C8C2×#4118yescomplemented
346C24—22no—
347C24—31no—
348C24—62no—
349#41—1232no—
350#41—1232no—
351C2×Q8—24no—
352C2×Q8—44no—
353C2×Q8—64no—
354C2×Q8—124no—
355C2×D4—22no—
356C2×D4—24no—
357C2×D4—24no—
358C2×D4—28no—
359C2×D4—34no—
360C2×D4—44no—
361C2×D4—44no—
362C2×D4—68no—
363C2×D4—68no—
364C2×D4—64no—
365C2×D4—62no—
366C2×D4—64no—
367C2×D4—68no—
368C2×D4—128no—
369C2×D4—128no—
370C2×D4—124no—
371C2×D4—124no—
372C2×D4C23×D314yes—
373C22×C4—22no—
374C22×C4—31no—
375C22×C4—38no—
376C22×C4—34no—
377C22×C4—31no—
378C22×C4—31no—
379C22×C4—32no—
380C22×C4—34no—
381C22×C4—32no—
382C22×C4—68no—
383C22×C4—68no—
384C22×C4—64no—
385C22×C4—64no—
386C22×C4—62no—
387C22×C4—128no—
388C22×C4—128no—
389C22×C4D3×D411yescharacteristic
390C22×C4D3×D411yescharacteristic
391C22×C4C23×D311yescharacteristic
392D8—632no—
393C2×C8—24no—
394C2×C8—38no—
395C2×C8—34no—
396C2×C8—616no—
397C2×C8C23×D314yes—
398C4⋊C4—24no—
399C4⋊C4—22no—
400C4⋊C4—34no—
401C4⋊C4—62no—
402C4⋊C4—64no—
403C4⋊C4C23×D314yes—
404C22⋊C4—68no—
405C22⋊C4—68no—
406C22⋊C4—68no—
407C22⋊C4—68no—
408C42—32no—
409C42—34no—
410C42—34no—
411C42—32no—
412C42D3×D412yes—
413C42C23×D312yes—
414C22×C3—44no—
415C22×C3—44no—
416C22×C3—44no—
417C22×C3—88no—
418C22×C3#53212yes—
419C22×C3C22×D812yes—
420C22×C3C2×#13512yes—
421C22×C3C23×D411yescharacteristic
422C2×D3—28no—
423C2×D3—22no—
424C2×D3—22no—
425C2×D3—44no—
426C2×D3—44no—
427C2×D3—44no—
428C2×D3—88no—
429C2×D3C22×D814yes—
430C3×C4—44no—
431C3×C4#53212yes—
432C3×C4C22×D814yes—
433C3×C4C2×#13514yes—
434C3×C4C22×#4112yes—
435Dic3—22no—
436Dic3—22no—
437Dic3—44no—
438Dic3C22×D814yes—
439C23—24no—
440C23—32no—
441C23—34no—
442C23—44no—
443C23—44no—
444C23—44no—
445C23—62no—
446C23—62no—
447C23—64no—
448C23—124no—
449C23—124no—
450C23—124no—
451C23C2×D3×D411yescharacteristic
452Q8—48no—
453Q8—128no—
454D4—28no—
455D4—44no—
456D4—48no—
457D4—48no—
458D4—68no—
459D4—1216no—
460D4—1216no—
461D4—124no—
462D4—128no—
463D4—128no—
464C2×C4—24no—
465C2×C4—32no—
466C2×C4—34no—
467C2×C4—316no—
468C2×C4—32no—
469C2×C4—32no—
470C2×C4—44no—
471C2×C4—62no—
472C2×C4—62no—
473C2×C4—64no—
474C2×C4—64no—
475C2×C4—68no—
476C2×C4—64no—
477C2×C4—1216no—
478C2×C4—1216no—
479C2×C4—124no—
480C2×C4D3×D814yes—
481C2×C4#91614yes—
482C2×C4C2×D3×D411yescharacteristic
483C2×C4C2×D3×D412yes—
484C2×C4C2×D3×D412yes—
485C2×C4D3×#4114yes—
486C2×C4C24×D311yescharacteristic
487C8—24no—
488C8—38no—
489C2×C3—84no—
490C2×C3#316914yescomplemented
491C2×C3C2×#53211yescharacteristic
492C2×C3C23×D811yescharacteristic
493C2×C3C22×#13511yescharacteristic
494D3—24no—
495D3—84no—
496C22—34no—
497C22—44no—
498C22—44no—
499C22—44no—
500C22—62no—
501C22—62no—
502C22—68no—
503C22—88no—
504C22—124no—
505C22—124no—
506C22—124no—
507C22—248no—
508C22#566812yes—
509C22C2×D3×D812yes—
510C22C2×#91612yes—
511C22C22×D3×D411yescharacteristic
512C4—34no—
513C4—44no—
514C4—62no—
515C4—62no—
516C4—124no—
517C4#566812yes—
518C4C2×D3×D814yes—
519C4C2×#91614yes—
520C4C2×D3×#4112yes—
521C3C2×#316911yescharacteristic, complemented, Sylow
522C2—64no—
523C2—84no—
524C2—244no—
525C2#8155714yesdirect product factor
526C2C2×#566811yescharacteristic
527C2C22×D3×D811yescharacteristic
528C2C22×#91611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#486C2
socleC23×C3C22×D4
hypercenterC2×#486D3
hypocenterC3C2×#3169
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#486

Complements

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

NH ≅ G/Ntype
1. C2×#5725C2semidirect
2. C2×#5716C2semidirect
3. C2×C3×#486C2semidirect
4. C2×#5078C2semidirect
5. C2×#5073C2semidirect
6. #81557C2direct (N × H)
7. C2×#4775C2semidirect
8. C2×#4756C2semidirect
9. C2×#4694C2semidirect
10. C2×C4×D3×C8C2semidirect
25. #5725C22semidirect
27. #5716C22semidirect
35. C3×#486C22semidirect
40. #5078C22semidirect
41. #5073C22semidirect
44. #4775C22semidirect
45. #4756C22semidirect
46. #4694C22semidirect
47. C4×D3×C8C22semidirect
54. C2×#486D3semidirect
151. #486C2×D3semidirect
200. C2×C4×D3D8semidirect
226. C2×C3⋊C8#41semidirect
342. C4×D3C2×D8semidirect
345. C3⋊C8C2×#41semidirect
490. C2×C3#3169semidirect
521. C3C2×#3169semidirect
525. C2#81557direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#57252
C2×#57162
C2×C3×#4862
C2×#50782
C2×#50732
#815572
C2×#47752
C2×#47562
C2×#46942
C2×C4×D3×C82
C2×#31693