Finite Group Databasebeta

← C2×#81905 #11675669

Subgroups — C2×#81905 #11675669

Order 768.

2082 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5730C211yesmaximal, characteristic, complemented
2C2×#5727C211yesmaximal, characteristic, complemented
3C2×C3×#498C211yesmaximal, characteristic, complemented
4C2×#5084C211yesmaximal, characteristic, complemented
5C2×#5087C211yesmaximal, characteristic, complemented
6#81905C2116yesmaximal, direct product factor
7C2×#4871C212yesmaximal, complemented
8C2×#4870C212yesmaximal, complemented
9C2×#4856C212yesmaximal, complemented
10C2×#4869C212yesmaximal
11C2×C4×D24C211yesmaximal, characteristic
12C2×#4679C211yesmaximal, characteristic, complemented
13C2×#3384—31nomaximal, Sylow
14C22×D3×Q8—21no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×#946—21no—
18C2×#946—22no—
19C22×D24C2211yescharacteristic
20#5730—24no—
21#5730C2218yescomplemented
22#5727—24no—
23#5727C2218yescomplemented
24C2×#897—21no—
25C2×#896—22no—
26C2×#893C2212yes—
27C2×#893C2212yes—
28C2×D3×C4⋊C4—21no—
29C2×#890C2212yes—
30C2×#888C2212yes—
31C2×C4×D12C2211yescharacteristic
32C2×C4×D12C2211yescharacteristic
33C2×C4×Dic6C2211yescharacteristic
34C3×#498C2218yescomplemented
35C2×C3×#106C2212yes—
36C2×C3×#106C2212yes—
37C2×C3×C4×C8C2211yescharacteristic
38C2×#821C2211yescharacteristic
39C2×#821C2211yescharacteristic
40C2×#818C2211yescharacteristic
41C2×#815C2212yes—
42C2×Dic3⋊C8C2212yes—
43#5087C2218yescomplemented
44#5084C2218yescomplemented
45C2×#809C2212yes—
46C2×#809C2212yes—
47C2×#808C2212yes—
48C2×#807C2212yes—
49C2×C4⋊(C3⋊C8)C2211yescharacteristic
50#4871C22116yescomplemented
51#4870C22116yescomplemented
52#4869C22116yes—
53#4856C22116yescomplemented
54C4×D24C2218yes—
55#4679C2218yescomplemented
56C2×#554—31no—
57C2×D4×Q8—31no—
58#3384—316no—
59C2×#501—32no—
60C2×#498D311yescharacteristic, complemented
61C2×#478—32no—
62C2×#476—32no—
63C2×#476—31no—
64C2×#474—31no—
65C2×#446—31no—
66C2×#442—32no—
67C2×C4×D8—31no—
68C22×C3×Q8—21no—
69C22×C3×Q8—21no—
70C2×D3×Q8—28no—
71C2×D3×Q8—48no—
72C22×D12C2311yescharacteristic
73C22×D12C2311yescharacteristic
74C22×C4×D3—21no—
75C22×C4×D3—21no—
76C22×C4×D3—22no—
77C22×C4×D3—22no—
78C22×Dic6—22no—
79C22×Dic6D411yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C3×C4⋊Q8—22no—
83C3×C4⋊Q8—22no—
84C3×C4⋊Q8C2314yes—
85C3×C4⋊Q8C2314yes—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4C2312yes—
89C2×C3×C4⋊C4C2312yes—
90C2×C3×C42C2311yescharacteristic
91#946—28no—
92#946—216no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—22no—
96C2×(C2×D3)⋊C4—22no—
97C2×C4⋊Dic3—22no—
98C2×C4⋊Dic3D411yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×Dic3⋊C4—22no—
102C2×Dic3⋊C4—22no—
103C2×Dic3⋊C4D412yes—
104C2×C4×Dic3C2312yes—
105C22×C3⋊C8C2312yes—
106C2×D24Q818yescomplemented
107C2×D24C2314yes—
108#897—28no—
109#896—216no—
110#893—24no—
111#893—24no—
112#893C2318yes—
113#893C2318yes—
114D3×C4⋊C4—28no—
115#890—24no—
116#890C2318yes—
117#888—24no—
118#888C2318yes—
119C4×D12—22no—
120C4×D12—22no—
121C4×D12C2314yes—
122C4×D12C2314yes—
123C4×Dic6D414yes—
124C4×Dic6C2314yes—
125C3×#106C2318yes—
126C3×#106C2318yes—
127C3×C4×C8C2314yes—
128#821C2314yes—
129#821C2314yes—
130#818C2314yes—
131#815C2318yes—
132Dic3⋊C8C2318yes—
133#809C2318yes—
134#809C2318yes—
135#808C2318yes—
136#807C2318yes—
137C4⋊(C3⋊C8)C2314yes—
138C23×Q8—61no—
139C22×D8—31no—
140#554—38no—
141#554—64no—
142D4×Q8—38no—
143D4×Q8—64no—
144C2×C4⋊Q8—32no—
145C2×C4⋊Q8C2×D311yescharacteristic
146C2×C4⋊Q8C2×D311yescharacteristic
147C2×#125—32no—
148C2×C22⋊Q8—62no—
149C2×C22⋊Q8—61no—
150C2×C22⋊Q8—61no—
151C2×C22⋊Q8—62no—
152C2×C4×Q8—31no—
153C2×C4×D4—32no—
154C2×C4×D4—32no—
155C2×C4×D4—31no—
156C2×C4×D4—31no—
157C22×C4⋊C4—61no—
158#501—316no—
159#498C2×D318yescomplemented
160#478—316no—
161#476—38no—
162#476—316no—
163#474—38no—
164#446—38no—
165#442—316no—
166C4×D8—38no—
167C2×#107—31no—
168C2×#107—32no—
169C2×#106—32no—
170C2×#106C2×D312yes—
171C2×#106C2×D312yes—
172C2×C4⋊C8—32no—
173C2×C4⋊C8—31no—
174C2×#102—31no—
175C2×#102—31no—
176C2×#102—32no—
177C2×#102—32no—
178C2×#97—32no—
179C2×C4×C8C2×D311yescharacteristic
180C23×D3—21no—
181C23×D3—21no—
182C2×C3×Q8—24no—
183C2×C3×Q8—24no—
184C2×C3×Q8—44no—
185C2×C3×Q8—82no—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2411yescharacteristic
191C22×Dic3—21no—
192C22×Dic3—21no—
193C22×Dic3C2×D412yes—
194C22×Dic3C2×D412yes—
195D3×Q8—432no—
196C2×D12—21no—
197C2×D12—21no—
198C2×D12—22no—
199C2×D12—22no—
200C2×D12C2×Q814yes—
201C2×D12C2×Q814yescomplemented
202C2×D12C2412yes—
203C2×D12C2412yes—
204C2×C4×D3—24no—
205C2×C4×D3—24no—
206C2×C4×D3—28no—
207C2×C4×D3—28no—
208C2×C4×D3—48no—
209C2×C4×D3—44no—
210C2×C4×D3—44no—
211C2×C4×D3—84no—
212C2×Dic6—24no—
213C2×Dic6—28no—
214C2×Dic6—48no—
215C2×Dic6C2×D414yes—
216C2×C3×C8—22no—
217C2×C3×C8C2×Q814yescomplemented
218C2×C3×C8C2412yes—
219C2×C3×C8C2412yes—
220C3×C4⋊C4—22no—
221C3×C4⋊C4—22no—
222C3×C4⋊C4—24no—
223C3×C4⋊C4—24no—
224C3×C4⋊C4C2414yes—
225C3×C4⋊C4C2414yes—
226C3×C42C2×D412yes—
227C3×C42C2412yes—
228(C2×D3)⋊C4—24no—
229(C2×D3)⋊C4—24no—
230(C2×D3)⋊C4—28no—
231(C2×D3)⋊C4—28no—
232C4⋊Dic3—21no—
233C4⋊Dic3—21no—
234C4⋊Dic3—28no—
235C4⋊Dic3C2×D414yes—
236C4⋊Dic3C2412yes—
237C4⋊Dic3C2412yes—
238Dic3⋊C4—28no—
239Dic3⋊C4—28no—
240Dic3⋊C4C2×D418yes—
241C4×Dic3C2×D414yes—
242C4×Dic3C2414yes—
243C2×C3⋊C8—24no—
244C2×C3⋊C8C2414yes—
245D24C2×Q8116yescomplemented
246C22×Q8—21no—
247C22×Q8—21no—
248C22×Q8—31no—
249C22×Q8—62no—
250C22×Q8—68no—
251C22×Q8—128no—
252C22×D4—31no—
253C22×D4—31no—
254C23×C4—62no—
255C23×C4—62no—
256C23×C4—61no—
257C23×C4—61no—
258C2×D8—38no—
259C2×D8—34no—
260C22×C8—32no—
261C22×C8C22×D311yescharacteristic
262C22×C8C22×D311yescharacteristic
263C4⋊Q8—22no—
264C4⋊Q8—22no—
265C4⋊Q8—38no—
266C4⋊Q8—64no—
267C4⋊Q8C22×D314yes—
268C4⋊Q8C22×D314yes—
269#125—38no—
270#125—64no—
271C22⋊Q8—616no—
272C22⋊Q8—68no—
273C22⋊Q8—68no—
274C22⋊Q8—616no—
275C4×Q8—34no—
276C4×Q8—34no—
277C4×D4—38no—
278C4×D4—38no—
279C4×D4—34no—
280C4×D4—34no—
281C4×D4—64no—
282C4×D4—64no—
283C4×D4—62no—
284C4×D4—62no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—31no—
288C2×C4⋊C4—32no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4—68no—
292C2×C4⋊C4—62no—
293C2×C4⋊C4—62no—
294C2×C4⋊C4—62no—
295C2×C4⋊C4C22×D312yes—
296C2×C4⋊C4C22×D312yes—
297C2×C22⋊C4—62no—
298C2×C22⋊C4—62no—
299C2×C22⋊C4—61no—
300C2×C22⋊C4—61no—
301C2×C42—32no—
302C2×C42C22×D311yescharacteristic
303#107—34no—
304#107—38no—
305#106—38no—
306#106C22×D318yes—
307#106C22×D318yes—
308C4⋊C8—38no—
309C4⋊C8—34no—
310#102—34no—
311#102—34no—
312#102—38no—
313#102—38no—
314#97—38no—
315C4×C8C22×D314yes—
316C23×C3C22×D411yescharacteristic
317C22×D3—24no—
318C22×D3—24no—
319C22×D3—22no—
320C22×D3—22no—
321C22×D3—42no—
322C22×D3—42no—
323C22×D3—42no—
324C22×D3—42no—
325C3×Q8—48no—
326C3×Q8—84no—
327C2×C3×C4—22no—
328C2×C3×C4—24no—
329C2×C3×C4—24no—
330C2×C3×C4—44no—
331C2×C3×C4—44no—
332C2×C3×C4AGL(1,ℤ/8)14yes—
333C2×C3×C4C22×D412yes—
334C2×C3×C4C22×D411yescharacteristic
335C2×C3×C4C22×D412yes—
336C2×C3×C4C22×Q812yes—
337C2×C3×C42−1+412yes—
338C2×C3×C4C2511yescharacteristic
339C2×Dic3—24no—
340C2×Dic3—24no—
341C2×Dic3—22no—
342C2×Dic3—22no—
343C2×Dic3—42no—
344C2×Dic3—42no—
345C2×Dic3C22×D414yes—
346C2×Dic3C22×D414yes—
347D12—24no—
348D12—24no—
349D12—22no—
350D12—22no—
351D12C22×Q814yes—
352D12C22×Q814yescomplemented
353C4×D3—416no—
354C4×D3—48no—
355C4×D3—48no—
356C4×D3—88no—
357Dic6—28no—
358Dic6—416no—
359C3×C8—22no—
360C3×C8C22×Q814yescomplemented
361C3⋊C8—24no—
362C24—61no—
363C24—61no—
364C2×Q8—24no—
365C2×Q8—24no—
366C2×Q8—34no—
367C2×Q8—44no—
368C2×Q8—68no—
369C2×Q8—64no—
370C2×Q8—82no—
371C2×Q8—128no—
372C2×Q8—1232no—
373C2×D4—34no—
374C2×D4—34no—
375C2×D4—32no—
376C2×D4—32no—
377C2×D4—61no—
378C2×D4—61no—
379C2×D4—62no—
380C2×D4—62no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—32no—
384C22×C4—32no—
385C22×C4—68no—
386C22×C4—68no—
387C22×C4—61no—
388C22×C4—61no—
389C22×C4—64no—
390C22×C4—64no—
391C22×C4—128no—
392C22×C4—124no—
393C22×C4—124no—
394C22×C4—244no—
395C22×C4D3×D411yescharacteristic
396C22×C4D3×D411yescharacteristic
397C22×C4C23×D311yescharacteristic
398D8—316no—
399C2×C8—22no—
400C2×C8—34no—
401C2×C8—64no—
402C2×C8D3×Q814yescomplemented
403C2×C8C23×D312yes—
404C2×C8C23×D312yes—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—34no—
410C4⋊C4—38no—
411C4⋊C4—32no—
412C4⋊C4—32no—
413C4⋊C4—68no—
414C4⋊C4—68no—
415C4⋊C4—68no—
416C4⋊C4—61no—
417C4⋊C4—61no—
418C4⋊C4C23×D314yes—
419C4⋊C4C23×D314yes—
420C22⋊C4—68no—
421C22⋊C4—68no—
422C22⋊C4—64no—
423C22⋊C4—64no—
424C42—34no—
425C42—34no—
426C42D3×D412yes—
427C42C23×D312yes—
428C22×C3D4×Q812yes—
429C22×C3C2×AGL(1,ℤ/8)12yes—
430C22×C3#57712yes—
431C22×C3C23×D411yescharacteristic
432C2×D3—24no—
433C2×D3—24no—
434C2×D3—48no—
435C2×D3—48no—
436C2×D3—42no—
437C2×D3—42no—
438C2×D3—42no—
439C2×D3—42no—
440C3×C4—22no—
441C3×C4—44no—
442C3×C4—44no—
443C3×C4D4×Q812yes—
444C3×C4C2×AGL(1,ℤ/8)14yes—
445C3×C4C23×Q811yescharacteristic
446C3×C4C2×2−1+411yescharacteristic
447Dic3—24no—
448Dic3—24no—
449Dic3—42no—
450Dic3—42no—
451C23—62no—
452C23—62no—
453C23—64no—
454C23—64no—
455C23—122no—
456C23—122no—
457C23—122no—
458C23—122no—
459C23C2×D3×D411yescharacteristic
460Q8—48no—
461Q8—68no—
462Q8—84no—
463Q8—1216no—
464D4—34no—
465D4—34no—
466D4—62no—
467D4—62no—
468D4—64no—
469D4—64no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—34no—
474C2×C4—34no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—64no—
478C2×C4—64no—
479C2×C4—62no—
480C2×C4—62no—
481C2×C4—1216no—
482C2×C4—122no—
483C2×C4—122no—
484C2×C4—128no—
485C2×C4—128no—
486C2×C4—248no—
487C2×C4#91414yes—
488C2×C4C2×D3×D412yes—
489C2×C4C2×D3×D411yescharacteristic
490C2×C4C2×D3×D412yes—
491C2×C4C2×D3×Q812yes—
492C2×C4#100712yes—
493C2×C4C24×D311yescharacteristic
494C8—22no—
495C8—64no—
496C8C2×D3×Q814yescomplemented
497C2×C3#338414yescomplemented
498C2×C3C2×D4×Q811yescharacteristic
499C2×C3C22×AGL(1,ℤ/8)11yescharacteristic
500C2×C3C2×#57711yescharacteristic
501D3—44no—
502D3—44no—
503C22—64no—
504C22—64no—
505C22—122no—
506C22—122no—
507C22—122no—
508C22—122no—
509C22—128no—
510C22—128no—
511C22#572712yes—
512C22C2×#91412yes—
513C22#576212yes—
514C22C22×D3×D411yescharacteristic
515C4—22no—
516C4—44no—
517C4—44no—
518C4—64no—
519C4—64no—
520C4—122no—
521C4—122no—
522C4#572712yes—
523C4C2×#91414yes—
524C4C22×D3×Q811yescharacteristic
525C4C2×#100711yescharacteristic
526C3C2×#338411yescharacteristic, complemented, Sylow
527C2—124no—
528C2—124no—
529C2#8190514yesdirect product factor
530C2C2×#572711yescharacteristic
531C2C22×#91411yescharacteristic
532C2C2×#576211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#498C2
socleC23×C3C22×D4
prime radicalC22×D24C22
hypercenterC2×#498D3
hypocenterC3C2×#3384
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#498

Complements

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

NH ≅ G/Ntype
1. C2×#5730C2semidirect
2. C2×#5727C2semidirect
3. C2×C3×#498C2semidirect
4. C2×#5084C2semidirect
5. C2×#5087C2semidirect
6. #81905C2direct (N × H)
7. C2×#4871C2semidirect
8. C2×#4870C2semidirect
9. C2×#4856C2semidirect
12. C2×#4679C2semidirect
21. #5730C22semidirect
23. #5727C22semidirect
34. C3×#498C22semidirect
43. #5087C22semidirect
44. #5084C22semidirect
50. #4871C22semidirect
51. #4870C22semidirect
53. #4856C22semidirect
55. #4679C22semidirect
60. C2×#498D3semidirect
106. C2×D24Q8semidirect
159. #498C2×D3semidirect
201. C2×D12C2×Q8semidirect
217. C2×C3×C8C2×Q8semidirect
245. D24C2×Q8semidirect
352. D12C22×Q8semidirect
360. C3×C8C22×Q8semidirect
402. C2×C8D3×Q8semidirect
496. C8C2×D3×Q8semidirect
497. C2×C3#3384semidirect
526. C3C2×#3384semidirect
529. C2#81905direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#57302
C2×#57272
C2×C3×#4982
C2×#50842
C2×#50872
#819052
C2×#48712
C2×#48702
C2×#48562
C2×#48692
C2×C4×D242
C2×#46792
C2×#33843