Finite Group Databasebeta

← C2×#81977 #11664842

Subgroups — C2×#81977 #11664842

Order 768.

1902 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5728C211yesmaximal, characteristic
2C2×D3×#125C211yesmaximal, characteristic
3C2×C3×#501C211yesmaximal, characteristic, complemented
4C2×#5081C211yesmaximal, characteristic, complemented
5C2×#5064C211yesmaximal, characteristic, complemented
6#81977C2116yesmaximal, direct product factor
7C2×#4881C212yesmaximal
8C2×#4878C212yesmaximal, complemented
9C2×#4861C212yesmaximal
10C2×#4858C212yesmaximal, complemented
11C2×#4707C211yesmaximal, characteristic
12C2×#4702C211yesmaximal, characteristic, complemented
13C2×#3166—31nomaximal, Sylow
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×#125C2211yescharacteristic
16C2×#946—21no—
17C2×Q8×Dic3—21no—
18C22×ΓL(1,𝔽25)C2212yes—
19#5728—24no—
20#5728C2218yes—
21D3×#125—24no—
22D3×#125C2218yes—
23C2×#897—21no—
24C2×#892C2212yes—
25C2×D3×C4⋊C4—21no—
26C2×D3×C4⋊C4—21no—
27C2×D3×C4⋊C4C2211yescharacteristic
28C2×D3×C4⋊C4C2211yescharacteristic
29C2×#890C2211yescharacteristic
30C2×#890C2212yes—
31C2×#890C2211yescharacteristic
32C2×#889—21no—
33C2×#889—21no—
34C2×#887—21no—
35C2×C42×D3C2211yescharacteristic
36C2×#870C2211yescharacteristic
37C2×#869C2211yescharacteristic
38C3×#501C2218yescomplemented
39C2×C3×#107C2212yes—
40C2×C3×#106C2212yes—
41C2×C3×#97C2211yescharacteristic
42C2×#818C2212yes—
43C2×#817C2212yes—
44C2×C8×Dic3C2212yes—
45#5081C2218yescomplemented
46#5064C2218yescomplemented
47C2×#808C2212yes—
48C2×#808C2212yes—
49C2×#807C2212yes—
50C2×#807C2212yes—
51C2×#803C2211yescharacteristic
52#4881C22116yes—
53#4878C22116yescomplemented
54#4861C22116yes—
55#4858C22116yescomplemented
56#4707C2218yes—
57#4702C2218yescomplemented
58C2×#533—31no—
59C22×#125—31no—
60#3166—316no—
61C2×#501—31no—
62C2×#501—31no—
63C2×#501—31no—
64C2×#501D311yescharacteristic, complemented
65C2×#499—32no—
66C2×#499—32no—
67C2×#428—32no—
68C2×#428—32no—
69C2×C4×M16—31no—
70C22×C3×Q8—21no—
71C22×C4×D3—21no—
72C22×C4×D3—21no—
73C22×C4×D3D411yescharacteristic
74C22×C4×D3D411yescharacteristic
75C22×C4×D3C2311yescharacteristic
76C22×Dic6—21no—
77C22×C3×C8C2312yes—
78C3×C4⋊Q8—22no—
79C3×C4⋊Q8C2314yes—
80C3×#125—22no—
81C3×#125C2314yes—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2312yes—
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C42C2311yescharacteristic
89#946—28no—
90Q8×Dic3—28no—
91C2×(C2×D3)⋊C4—22no—
92C2×C4⋊Dic3—21no—
93C2×C4⋊Dic3—21no—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3—22no—
96C2×C4⋊Dic3C2312yes—
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—22no—
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×C4×Dic3—22no—
107C2×C4×Dic3D411yescharacteristic
108C2×C4×Dic3D411yescharacteristic
109C2×C4×Dic3C2311yescharacteristic
110C22×C3⋊C8C2312yes—
111C2×ΓL(1,𝔽25)Q8116yescomplemented
112C2×ΓL(1,𝔽25)C2318yes—
113#897—28no—
114#892—24no—
115#892C2318yes—
116D3×C4⋊C4—22no—
117D3×C4⋊C4—22no—
118D3×C4⋊C4—28no—
119D3×C4⋊C4—28no—
120D3×C4⋊C4C2314yes—
121D3×C4⋊C4C2314yes—
122#890—22no—
123#890—22no—
124#890—24no—
125#890C2314yes—
126#890C2318yes—
127#890C2314yes—
128#889—28no—
129#889—28no—
130#887—28no—
131C42×D3D414yes—
132C42×D3C2314yes—
133#870—22no—
134#870C2314yes—
135#869—22no—
136#869C2314yes—
137C3×#107C2318yes—
138C3×#106C2318yes—
139C3×#97C2314yes—
140#818C2318yes—
141#817C2318yes—
142C8×Dic3C2318yes—
143#808C2318yes—
144#808C2318yes—
145#807C2318yes—
146#807C2318yes—
147#803C2314yes—
148C22×M16—32no—
149#533—38no—
150#533—64no—
151C2×C4⋊Q8—31no—
152C2×C4⋊Q8C2×D311yescharacteristic
153C2×#125—38no—
154C2×#125—31no—
155C2×#125—31no—
156C2×#125—32no—
157C2×#125—31no—
158C2×#125—61no—
159C2×#125—61no—
160C2×#125—64no—
161C2×#125C2×D311yescharacteristic
162C2×C22⋊Q8—61no—
163C2×C22⋊Q8—61no—
164C2×C4×Q8—61no—
165C2×C4×Q8—61no—
166C2×#117—32no—
167C22×C4⋊C4—31no—
168C22×C4⋊C4—31no—
169C22×C4⋊C4—61no—
170C22×C4⋊C4—61no—
171C22×C42—31no—
172#501—38no—
173#501—38no—
174#501—38no—
175#501C2×D318yescomplemented
176#499—316no—
177#499—316no—
178#428—316no—
179#428—316no—
180C2×#107—32no—
181C2×#107—32no—
182C2×#107—32no—
183C2×#107C2×D312yes—
184C2×#106—32no—
185C2×#106—32no—
186C2×#106—32no—
187C2×#106C2×D312yes—
188C4×M16—38no—
189C2×#97—31no—
190C2×#97C2×D311yescharacteristic
191C2×C4×C8—32no—
192C23×D3C2×D411yescharacteristic
193C2×C3×Q8—24no—
194C2×C3×Q8—44no—
195C22×C3×C4—22no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2411yescharacteristic
201C22×Dic3—22no—
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3C2×D411yescharacteristic
205C22×Dic3C2×D411yescharacteristic
206C22×Dic3C2×D411yescharacteristic
207C2×C4×D3—24no—
208C2×C4×D3—24no—
209C2×C4×D3—42no—
210C2×C4×D3—42no—
211C2×C4×D3—44no—
212C2×C4×D3—44no—
213C2×C4×D3C2×D414yes—
214C2×C4×D3C2×D414yes—
215C2×C4×D3C2×D412yes—
216C2×C4×D3C2×Q818yes—
217C2×C4×D3C2412yes—
218C2×Dic6—24no—
219C2×Dic6—44no—
220C2×C3×C8C2×Q818yes—
221C2×C3×C8C2414yes—
222C3×C4⋊C4—24no—
223C3×C4⋊C4—24no—
224C3×C4⋊C4—24no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—22no—
228C3×C4⋊C4C2412yes—
229C3×C4⋊C4C2414yes—
230C3×C4⋊C4C2412yes—
231C3×C42C2×D412yes—
232C3×C42C2412yes—
233(C2×D3)⋊C4—28no—
234C4⋊Dic3—22no—
235C4⋊Dic3—24no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—24no—
239C4⋊Dic3—24no—
240C4⋊Dic3—28no—
241C4⋊Dic3C2412yes—
242C4⋊Dic3C2414yes—
243C4⋊Dic3C2412yes—
244Dic3⋊C4—24no—
245Dic3⋊C4—24no—
246Dic3⋊C4—28no—
247Dic3⋊C4—24no—
248Dic3⋊C4—24no—
249Dic3⋊C4—24no—
250Dic3⋊C4—24no—
251C4×Dic3—28no—
252C4×Dic3C2×D412yes—
253C4×Dic3C2×D414yes—
254C4×Dic3C2×D414yes—
255C4×Dic3C2412yes—
256C2×C3⋊C8C2×Q818yes—
257C2×C3⋊C8C2414yes—
258ΓL(1,𝔽25)C2×Q8132yescomplemented
259C22×Q8—21no—
260C22×Q8—61no—
261C23×C4—31no—
262C23×C4—31no—
263C23×C4—31no—
264C23×C4—61no—
265C23×C4—61no—
266C2×M16—316no—
267C2×M16—38no—
268C22×C8—32no—
269C22×C8C22×D312yes—
270C4⋊Q8—22no—
271C4⋊Q8—34no—
272C4⋊Q8—62no—
273C4⋊Q8C22×D314yes—
274#125—22no—
275#125—34no—
276#125—34no—
277#125—38no—
278#125—34no—
279#125—68no—
280#125—68no—
281#125—62no—
282#125—62no—
283#125—64no—
284#125—62no—
285#125C22×D314yes—
286C22⋊Q8—68no—
287C22⋊Q8—68no—
288C4×Q8—68no—
289C4×Q8—68no—
290#117—38no—
291#117—64no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—31no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—32no—
298C2×C4⋊C4—34no—
299C2×C4⋊C4—34no—
300C2×C4⋊C4—62no—
301C2×C4⋊C4—62no—
302C2×C4⋊C4—68no—
303C2×C4⋊C4—68no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4—62no—
311C2×C4⋊C4—62no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C4⋊C4C22×D311yescharacteristic
317C2×C4⋊C4C22×D312yes—
318C2×C22⋊C4—62no—
319C2×C42—31no—
320C2×C42—31no—
321C2×C42—31no—
322C2×C42—34no—
323C2×C42—34no—
324C2×C42—62no—
325C2×C42C22×D311yescharacteristic
326#107—38no—
327#107—38no—
328#107—38no—
329#107C22×D318yes—
330#106—38no—
331#106—38no—
332#106—38no—
333#106C22×D318yes—
334#97—34no—
335#97C22×D314yes—
336C4×C8—38no—
337C23×C3C22×D411yescharacteristic
338C22×D3—22no—
339C22×D3—22no—
340C22×D3C4⋊Q814yes—
341C22×D3C22×D412yes—
342C3×Q8—48no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—24no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4—44no—
351C2×C3×C4C22×D412yes—
352C2×C3×C4C22×D412yes—
353C2×C3×C4C22×D411yescharacteristic
354C2×C3×C4C22×Q814yes—
355C2×C3×C4C2511yescharacteristic
356C2×Dic3—22no—
357C2×Dic3—22no—
358C2×Dic3—24no—
359C2×Dic3—22no—
360C2×Dic3—22no—
361C2×Dic3—44no—
362C2×Dic3—42no—
363C2×Dic3—42no—
364C2×Dic3C4⋊Q814yes—
365C2×Dic3C22×D412yes—
366C2×Dic3C22×D412yes—
367C2×Dic3C22×D412yes—
368C4×D3—44no—
369C4×D3—44no—
370C4×D3—48no—
371C4×D3—48no—
372C4×D3C4⋊Q818yes—
373C4×D3C22×Q818yes—
374Dic6—48no—
375C3×C8C22×Q818yes—
376C3⋊C8C22×Q818yes—
377C24—31no—
378C2×Q8—24no—
379C2×Q8—44no—
380C2×Q8—64no—
381C2×Q8—124no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—31no—
386C22×C4—32no—
387C22×C4—38no—
388C22×C4—32no—
389C22×C4—31no—
390C22×C4—31no—
391C22×C4—34no—
392C22×C4—34no—
393C22×C4—64no—
394C22×C4—64no—
395C22×C4—62no—
396C22×C4—61no—
397C22×C4—61no—
398C22×C4—122no—
399C22×C4—122no—
400C22×C4—124no—
401C22×C4—124no—
402C22×C4D3×D411yescharacteristic
403C22×C4D3×D411yescharacteristic
404C22×C4C23×D311yescharacteristic
405M16—332no—
406C2×C8—38no—
407C2×C8—34no—
408C2×C8D3×Q818yes—
409C2×C8C23×D314yes—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—22no—
416C4⋊C4—32no—
417C4⋊C4—32no—
418C4⋊C4—34no—
419C4⋊C4—64no—
420C4⋊C4—64no—
421C4⋊C4—64no—
422C4⋊C4—64no—
423C4⋊C4—64no—
424C4⋊C4—64no—
425C4⋊C4—68no—
426C4⋊C4—68no—
427C4⋊C4—61no—
428C4⋊C4—61no—
429C4⋊C4—62no—
430C4⋊C4—64no—
431C4⋊C4—64no—
432C4⋊C4—64no—
433C4⋊C4C23×D312yes—
434C4⋊C4C23×D314yes—
435C4⋊C4C23×D312yes—
436C22⋊C4—68no—
437C42—34no—
438C42—34no—
439C42—32no—
440C42—32no—
441C42—68no—
442C42D3×D412yes—
443C42C23×D312yes—
444C22×C3C2×C4⋊Q812yes—
445C22×C3#57512yes—
446C22×C3#57512yes—
447C22×C3C23×D411yescharacteristic
448C2×D3—28no—
449C2×D3—22no—
450C2×D3—22no—
451C2×D3C2×C4⋊Q814yes—
452C3×C4—22no—
453C3×C4—22no—
454C3×C4—42no—
455C3×C4—42no—
456C3×C4—44no—
457C3×C4C2×C4⋊Q812yes—
458C3×C4C23×Q812yes—
459Dic3—22no—
460Dic3—22no—
461Dic3—44no—
462Dic3—42no—
463Dic3—42no—
464Dic3C2×C4⋊Q814yes—
465C23—32no—
466C23—34no—
467C23—62no—
468C23—62no—
469C23C2×D3×D411yescharacteristic
470Q8—48no—
471Q8—128no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—32no—
478C2×C4—34no—
479C2×C4—38no—
480C2×C4—38no—
481C2×C4—32no—
482C2×C4—32no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—62no—
487C2×C4—62no—
488C2×C4—64no—
489C2×C4—62no—
490C2×C4—62no—
491C2×C4—124no—
492C2×C4—124no—
493C2×C4—128no—
494C2×C4—128no—
495C2×C4—122no—
496C2×C4—122no—
497C2×C4—124no—
498C2×C4C2×D3×D412yes—
499C2×C4C2×D3×D412yes—
500C2×C4C2×D3×D411yescharacteristic
501C2×C4C2×D3×Q814yes—
502C2×C4C24×D311yescharacteristic
503C8—38no—
504C8C2×D3×Q818yes—
505C2×C3#316614yescomplemented
506C2×C3C22×C4⋊Q811yescharacteristic
507C2×C3C2×#57511yescharacteristic
508C2×C3C2×#57511yescharacteristic
509D3—24no—
510C22—34no—
511C22—68no—
512C22—62no—
513C22—62no—
514C22D3×C4⋊Q812yes—
515C22#577312yes—
516C22#577712yes—
517C22C22×D3×D411yescharacteristic
518C4—22no—
519C4—22no—
520C4—34no—
521C4—42no—
522C4—44no—
523C4—42no—
524C4—62no—
525C4—62no—
526C4—122no—
527C4—122no—
528C4—124no—
529C4D3×C4⋊Q812yes—
530C4C22×D3×Q812yes—
531C3C2×#316611yescharacteristic, complemented, Sylow
532C2—64no—
533C2#8197714yesdirect product factor
534C2C2×D3×C4⋊Q811yescharacteristic
535C2C2×#577311yescharacteristic
536C2C2×#577711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#501C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenterC2×#501D3
hypocenterC3C2×#3166
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
3. C2×C3×#501C2semidirect
4. C2×#5081C2semidirect
5. C2×#5064C2semidirect
6. #81977C2direct (N × H)
8. C2×#4878C2semidirect
10. C2×#4858C2semidirect
12. C2×#4702C2semidirect
38. C3×#501C22semidirect
45. #5081C22semidirect
46. #5064C22semidirect
53. #4878C22semidirect
55. #4858C22semidirect
57. #4702C22semidirect
64. C2×#501D3semidirect
111. C2×ΓL(1,𝔽25)Q8semidirect
175. #501C2×D3semidirect
258. ΓL(1,𝔽25)C2×Q8semidirect
505. C2×C3#3166semidirect
531. C3C2×#3166semidirect
533. C2#81977direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#57282
C2×D3×#1252
C2×C3×#5012
C2×#50812
C2×#50642
#819772
C2×#48812
C2×#48782
C2×#48612
C2×#48582
C2×#47072
C2×#47022
C2×#31663