Finite Group Databasebeta

← C2×#81979 #11667331

Subgroups — C2×#81979 #11667331

Order 768.

2038 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5725C211yesmaximal, characteristic
2C2×#5688C211yesmaximal, characteristic, complemented
3C2×C3×#501C211yesmaximal, characteristic, complemented
4C2×#5081C211yesmaximal, characteristic, complemented
5C2×#5067C211yesmaximal, characteristic, complemented
6#81979C2116yesmaximal, direct product factor
7C2×#4890C212yesmaximal, complemented
8C2×#4885C212yesmaximal, complemented
9C2×#4870C212yesmaximal, complemented
10C2×#4866C212yesmaximal, complemented
11C2×#4713C211yesmaximal, characteristic, complemented
12C2×#4705C211yesmaximal, characteristic, complemented
13C2×#3235—31nomaximal, Sylow
14C22×#239—21no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#125C2211yescharacteristic
17C2×#947—21no—
18#5725—24no—
19#5725C2218yes—
20#5688—24no—
21#5688C2218yescomplemented
22C2×#896—21no—
23C2×#896—21no—
24C2×#895—21no—
25C2×#895C2212yes—
26C2×#893—21no—
27C2×#893C2212yes—
28C2×#892C2211yescharacteristic
29C2×#892C2211yescharacteristic
30C2×#888C2211yescharacteristic
31C2×#888C2211yescharacteristic
32C2×#887—21no—
33C2×#887—21no—
34C2×#874C2211yescharacteristic
35C2×C42×D3C2211yescharacteristic
36C2×#870C2211yescharacteristic
37C3×#501C2218yescomplemented
38C2×C3×#107C2212yes—
39C2×C3×#106C2212yes—
40C2×C3×#97C2211yescharacteristic
41C2×#821C2212yes—
42C2×#821C2212yes—
43C2×(C2×D3)⋊C8C2212yes—
44C2×Dic3⋊C8C2212yes—
45#5081C2218yescomplemented
46#5067C2218yescomplemented
47C2×#809C2212yes—
48C2×#809C2212yes—
49C2×#808C2212yes—
50C2×#807C2212yes—
51C2×#803C2211yescharacteristic
52#4890C22116yescomplemented
53#4885C22116yescomplemented
54#4870C22116yescomplemented
55#4866C22116yescomplemented
56#4713C2218yescomplemented
57#4705C2218yescomplemented
58C2×#533—31no—
59C2×#532—31no—
60#3235—316no—
61C2×#501—31no—
62C2×#501D311yescharacteristic, complemented
63C2×#490—31no—
64C2×#490—31no—
65C2×#482—32no—
66C2×#482—32no—
67C2×#476—32no—
68C2×#474—32no—
69C2×#432—31no—
70C22×C3×Q8—21no—
71C2×#239—28no—
72C2×#239—48no—
73C22×D12—22no—
74C22×D12—21no—
75C22×D12—21no—
76C22×D12C2312yes—
77C22×C4×D3—22no—
78C22×C4×D3D411yescharacteristic
79C22×C4×D3D411yescharacteristic
80C22×C4×D3C2311yescharacteristic
81C22×Dic6—21no—
82C22×Dic6—21no—
83C22×C3×C8C2312yes—
84C3×C4⋊Q8—22no—
85C3×C4⋊Q8C2314yes—
86C3×#125—22no—
87C3×#125C2314yes—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4C2312yes—
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C42C2311yescharacteristic
95#947—28no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—22no—
99C2×(C2×D3)⋊C4—22no—
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×C4×Dic3—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3D411yescharacteristic
109C2×C4×Dic3D411yescharacteristic
110C2×C4×Dic3C2311yescharacteristic
111C22×C3⋊C8C2312yes—
112#896—28no—
113#896—28no—
114#895—24no—
115#895—28no—
116#895C2318yes—
117#893—24no—
118#893—28no—
119#893C2318yes—
120#892—22no—
121#892—22no—
122#892C2314yes—
123#892C2314yes—
124#888—22no—
125#888—22no—
126#888C2314yes—
127#888C2314yes—
128#887—28no—
129#887—28no—
130#874—22no—
131#874C2314yes—
132C42×D3D414yes—
133C42×D3C2314yes—
134#870—22no—
135#870C2314yes—
136C3×#107C2318yes—
137C3×#106C2318yes—
138C3×#97C2314yes—
139#821C2318yes—
140#821C2318yes—
141(C2×D3)⋊C8C2318yes—
142Dic3⋊C8C2318yes—
143#809C2318yes—
144#809C2318yes—
145#808C2318yes—
146#807C2318yes—
147#803C2314yes—
148C22×#41—61no—
149#533—38no—
150#533—64no—
151#532—38no—
152#532—64no—
153C2×C4⋊Q8—31no—
154C2×C4⋊Q8—31no—
155C2×C4⋊Q8C2×D311yescharacteristic
156C2×#127—31no—
157C2×#125—31no—
158C2×#125C2×D311yescharacteristic
159C2×#124—61no—
160C2×C22⋊Q8—61no—
161C2×C22⋊Q8—61no—
162C2×#121—32no—
163C2×#121—61no—
164C2×C4×Q8—61no—
165C2×C4×Q8—61no—
166C2×C4×D4—32no—
167C2×C4×D4—61no—
168C2×#117—31no—
169C2×#117—31no—
170C22×C42—31no—
171#501—38no—
172#501C2×D318yescomplemented
173#490—38no—
174#490—38no—
175#482—316no—
176#482—316no—
177#476—316no—
178#474—316no—
179#432—38no—
180C2×#107—32no—
181C2×#107C2×D312yes—
182C2×#106—32no—
183C2×#106C2×D312yes—
184C2×C4⋊C8—32no—
185C2×#102—32no—
186C2×#102—32no—
187C2×#102—32no—
188C2×#102—32no—
189C2×C22⋊C8—32no—
190C2×#97—31no—
191C2×#97C2×D311yescharacteristic
192C23×D3—22no—
193C23×D3C2×D411yescharacteristic
194C2×C3×Q8—24no—
195C2×C3×Q8—44no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—22no—
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2411yescharacteristic
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3C2×D411yescharacteristic
205C22×Dic3C2×D411yescharacteristic
206C22×Dic3C2×D411yescharacteristic
207#239—432no—
208C2×D12—28no—
209C2×D12—28no—
210C2×D12—22no—
211C2×D12—24no—
212C2×D12—24no—
213C2×D12—44no—
214C2×D12—44no—
215C2×D12—48no—
216C2×D12C2414yes—
217C2×C4×D3—24no—
218C2×C4×D3—24no—
219C2×C4×D3—28no—
220C2×C4×D3—42no—
221C2×C4×D3—48no—
222C2×C4×D3C2×D412yes—
223C2×C4×D3C2×D414yes—
224C2×C4×D3C2×D414yes—
225C2×C4×D3C2412yes—
226C2×Dic6—24no—
227C2×Dic6—24no—
228C2×Dic6—44no—
229C2×Dic6—44no—
230C2×C3×C8—24no—
231C2×C3×C8C2414yes—
232C3×C4⋊C4—24no—
233C3×C4⋊C4—24no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—24no—
237C3×C4⋊C4—22no—
238C3×C4⋊C4C2414yes—
239C3×C4⋊C4C2412yes—
240C3×C4⋊C4C2412yes—
241C3×C42C2×D412yes—
242C3×C42C2412yes—
243(C2×D3)⋊C4—24no—
244(C2×D3)⋊C4—24no—
245(C2×D3)⋊C4—28no—
246(C2×D3)⋊C4—28no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3C2412yes—
250C4⋊Dic3C2412yes—
251Dic3⋊C4—24no—
252Dic3⋊C4—24no—
253Dic3⋊C4—24no—
254Dic3⋊C4—24no—
255C4×Dic3—24no—
256C4×Dic3—24no—
257C4×Dic3C2×D412yes—
258C4×Dic3C2×D414yes—
259C4×Dic3C2×D414yes—
260C4×Dic3C2412yes—
261C2×C3⋊C8—24no—
262C2×C3⋊C8C2414yes—
263C2×#41—68no—
264C2×#41—128no—
265C22×Q8—21no—
266C22×Q8—61no—
267C22×Q8—61no—
268C22×D4—32no—
269C22×D4—62no—
270C22×D4—61no—
271C22×D4—61no—
272C23×C4—31no—
273C23×C4—31no—
274C23×C4—31no—
275C23×C4—62no—
276C22×C8—32no—
277C22×C8C22×D312yes—
278C4⋊Q8—22no—
279C4⋊Q8—34no—
280C4⋊Q8—34no—
281C4⋊Q8—62no—
282C4⋊Q8—62no—
283C4⋊Q8C22×D314yes—
284#127—34no—
285#127—62no—
286#125—22no—
287#125—34no—
288#125—62no—
289#125C22×D314yes—
290#124—68no—
291C22⋊Q8—68no—
292C22⋊Q8—68no—
293#121—38no—
294#121—64no—
295#121—68no—
296C4×Q8—68no—
297C4×Q8—68no—
298C4×D4—38no—
299C4×D4—64no—
300C4×D4—68no—
301#117—34no—
302#117—34no—
303#117—62no—
304#117—62no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4C22×D311yescharacteristic
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C4⋊C4C22×D312yes—
317C2×C22⋊C4—62no—
318C2×C22⋊C4—62no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C42—31no—
322C2×C42—31no—
323C2×C42—31no—
324C2×C42—34no—
325C2×C42—34no—
326C2×C42—61no—
327C2×C42—61no—
328C2×C42C22×D311yescharacteristic
329#107—38no—
330#107C22×D318yes—
331#106—38no—
332#106C22×D318yes—
333C4⋊C8—38no—
334#102—38no—
335#102—38no—
336#102—38no—
337#102—38no—
338C22⋊C8—38no—
339#97—34no—
340#97C22×D314yes—
341C23×C3C22×D411yescharacteristic
342C22×D3—22no—
343C22×D3—22no—
344C22×D3—22no—
345C22×D3—24no—
346C22×D3—44no—
347C22×D3—44no—
348C22×D3—44no—
349C22×D3C22×D412yes—
350C3×Q8—48no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—24no—
356C2×C3×C4—42no—
357C2×C3×C4—44no—
358C2×C3×C4—42no—
359C2×C3×C4C22×D412yes—
360C2×C3×C4C22×D412yes—
361C2×C3×C4C22×D411yescharacteristic
362C2×C3×C42−1+414yes—
363C2×C3×C4C2511yescharacteristic
364C2×Dic3—22no—
365C2×Dic3—22no—
366C2×Dic3—22no—
367C2×Dic3—22no—
368C2×Dic3—42no—
369C2×Dic3—42no—
370C2×Dic3AGL(1,ℤ/8)14yes—
371C2×Dic3C22×D412yes—
372C2×Dic3C22×D412yes—
373C2×Dic3C22×D412yes—
374D12—28no—
375D12—48no—
376D12—48no—
377D12—416no—
378D12—44no—
379C4×D3—24no—
380C4×D3—24no—
381C4×D3—28no—
382C4×D3—44no—
383C4×D3—416no—
384Dic6—48no—
385Dic6—48no—
386C3×C8—24no—
387C3⋊C8—24no—
388C24—31no—
389C24—62no—
390#41—1232no—
391C2×Q8—24no—
392C2×Q8—44no—
393C2×Q8—64no—
394C2×Q8—64no—
395C2×Q8—124no—
396C2×Q8—124no—
397C2×D4—34no—
398C2×D4—68no—
399C2×D4—64no—
400C2×D4—62no—
401C2×D4—64no—
402C2×D4—68no—
403C2×D4—124no—
404C2×D4—124no—
405C2×D4—128no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—31no—
410C22×C4—31no—
411C22×C4—31no—
412C22×C4—32no—
413C22×C4—34no—
414C22×C4—32no—
415C22×C4—34no—
416C22×C4—64no—
417C22×C4—68no—
418C22×C4—64no—
419C22×C4—61no—
420C22×C4—61no—
421C22×C4—122no—
422C22×C4—128no—
423C22×C4D3×D411yescharacteristic
424C22×C4D3×D411yescharacteristic
425C22×C4C23×D311yescharacteristic
426C2×C8—24no—
427C2×C8—34no—
428C2×C8—64no—
429C2×C8C23×D314yes—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—22no—
436C4⋊C4—32no—
437C4⋊C4—32no—
438C4⋊C4—64no—
439C4⋊C4—64no—
440C4⋊C4—61no—
441C4⋊C4—61no—
442C4⋊C4—64no—
443C4⋊C4—64no—
444C4⋊C4C23×D312yes—
445C4⋊C4C23×D312yes—
446C4⋊C4C23×D314yes—
447C22⋊C4—68no—
448C22⋊C4—68no—
449C22⋊C4—64no—
450C22⋊C4—64no—
451C42—34no—
452C42—32no—
453C42—34no—
454C42—32no—
455C42—64no—
456C42—64no—
457C42D3×D412yes—
458C42C23×D312yes—
459C22×C3#53612yes—
460C22×C3C2×AGL(1,ℤ/8)12yes—
461C22×C3#57512yes—
462C22×C3C23×D411yescharacteristic
463C2×D3—22no—
464C2×D3—22no—
465C2×D3—22no—
466C2×D3—44no—
467C2×D3—44no—
468C2×D3—44no—
469C2×D3—44no—
470C2×D3—88no—
471C3×C4—22no—
472C3×C4—22no—
473C3×C4—42no—
474C3×C4—44no—
475C3×C4—42no—
476C3×C4#53612yes—
477C3×C4C2×2−1+412yes—
478Dic3—22no—
479Dic3—22no—
480Dic3—42no—
481Dic3—42no—
482Dic3C2×AGL(1,ℤ/8)14yes—
483C23—32no—
484C23—62no—
485C23—62no—
486C23—62no—
487C23—64no—
488C23—124no—
489C23—124no—
490C23—124no—
491C23C2×D3×D411yescharacteristic
492Q8—48no—
493Q8—128no—
494Q8—128no—
495D4—68no—
496D4—128no—
497D4—128no—
498D4—1216no—
499D4—124no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—32no—
506C2×C4—34no—
507C2×C4—32no—
508C2×C4—32no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—68no—
513C2×C4—62no—
514C2×C4—62no—
515C2×C4—64no—
516C2×C4—64no—
517C2×C4—62no—
518C2×C4—62no—
519C2×C4—124no—
520C2×C4—1216no—
521C2×C4—122no—
522C2×C4—122no—
523C2×C4C2×D3×D412yes—
524C2×C4C2×D3×D412yes—
525C2×C4C2×D3×D411yescharacteristic
526C2×C4#100714yes—
527C2×C4C24×D311yescharacteristic
528C8—24no—
529C8—64no—
530C2×C3#323514yescomplemented
531C2×C3C2×#53611yescharacteristic
532C2×C3C22×AGL(1,ℤ/8)11yescharacteristic
533C2×C3C2×#57511yescharacteristic
534D3—42no—
535D3—84no—
536C22—62no—
537C22—62no—
538C22—62no—
539C22—124no—
540C22—124no—
541C22—124no—
542C22—124no—
543C22—248no—
544C22#572412yes—
545C22D3×AGL(1,ℤ/8)12yes—
546C22#577712yes—
547C22C22×D3×D411yescharacteristic
548C4—22no—
549C4—22no—
550C4—34no—
551C4—44no—
552C4—42no—
553C4—42no—
554C4—62no—
555C4—62no—
556C4—122no—
557C4—122no—
558C4#572412yes—
559C4C2×#100712yes—
560C3C2×#323511yescharacteristic, complemented, Sylow
561C2—122no—
562C2—244no—
563C2#8197914yesdirect product factor
564C2C2×#572411yescharacteristic
565C2C2×D3×AGL(1,ℤ/8)11yescharacteristic
566C2C2×#577711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#501C2
socleC23×C3C22×D4
prime radicalC2×#5725C2
hypercenterC2×#501D3
hypocenterC3C2×#3235
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#501

Complements

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

NH ≅ G/Ntype
2. C2×#5688C2semidirect
3. C2×C3×#501C2semidirect
4. C2×#5081C2semidirect
5. C2×#5067C2semidirect
6. #81979C2direct (N × H)
7. C2×#4890C2semidirect
8. C2×#4885C2semidirect
9. C2×#4870C2semidirect
10. C2×#4866C2semidirect
11. C2×#4713C2semidirect
12. C2×#4705C2semidirect
21. #5688C22semidirect
37. C3×#501C22semidirect
45. #5081C22semidirect
46. #5067C22semidirect
52. #4890C22semidirect
53. #4885C22semidirect
54. #4870C22semidirect
55. #4866C22semidirect
56. #4713C22semidirect
57. #4705C22semidirect
62. C2×#501D3semidirect
172. #501C2×D3semidirect
530. C2×C3#3235semidirect
560. C3C2×#3235semidirect
563. C2#81979direct (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×#56882
C2×C3×#5012
C2×#50812
C2×#50672
#819792
C2×#48902
C2×#48852
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