Finite Group Databasebeta

← C2×#81973 #11664800

Subgroups — C2×#81973 #11664800

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×#5688C211yesmaximal, characteristic
3C2×C3×#501C211yesmaximal, characteristic, complemented
4C2×#5081C211yesmaximal, characteristic, complemented
5C2×#5064C211yesmaximal, characteristic, complemented
6#81973C2116yesmaximal, direct product factor
7C2×#4880C212yesmaximal
8C2×#4876C212yesmaximal, complemented
9C2×#4860C212yesmaximal
10C2×#4856C212yesmaximal, complemented
11C2×D3×#97C211yesmaximal, characteristic
12C2×#4702C211yesmaximal, characteristic, complemented
13C2×#3162—31nomaximal, Sylow
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×#125C2211yescharacteristic
16C2×#946—21no—
17C2×Q8×Dic3—21no—
18C22×D3×C8C2212yes—
19#5728—24no—
20#5728C2218yes—
21#5688—24no—
22#5688C2218yes—
23C2×#897—21no—
24C2×#896—21no—
25C2×#896—21no—
26C2×#892C2211yescharacteristic
27C2×#892C2212yes—
28C2×#892C2211yescharacteristic
29C2×#890C2212yes—
30C2×#888C2211yescharacteristic
31C2×#888C2211yescharacteristic
32C2×#887—21no—
33C2×#887—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×#815C2212yes—
45#5081C2218yescomplemented
46#5064C2218yescomplemented
47C2×#808C2212yes—
48C2×#808C2212yes—
49C2×#807C2212yes—
50C2×#807C2212yes—
51C2×#803C2211yescharacteristic
52#4880C22116yes—
53#4876C22116yescomplemented
54#4860C22116yes—
55#4856C22116yescomplemented
56D3×#97C2218yes—
57#4702C2218yescomplemented
58C2×#533—31no—
59C2×#533—31no—
60#3162—316no—
61C2×#501—32no—
62C2×#501—32no—
63C2×#501—31no—
64C2×#501—31no—
65C2×#501—31no—
66C2×#501D311yescharacteristic, complemented
67C2×#427—32no—
68C2×#427—32no—
69C22×#97—31no—
70C22×C3×Q8—21no—
71C22×C4×D3D411yescharacteristic
72C22×C4×D3D411yescharacteristic
73C22×C4×D3C2311yescharacteristic
74C22×Dic6—21no—
75C22×Dic6—21no—
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—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—22no—
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—21no—
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—22no—
104C2×C4×Dic3—21no—
105C2×C4×Dic3—21no—
106C2×C4×Dic3—22no—
107C2×C4×Dic3D411yescharacteristic
108C2×C4×Dic3D411yescharacteristic
109C2×C4×Dic3C2311yescharacteristic
110C22×C3⋊C8C2312yes—
111C2×D3×C8Q8116yescomplemented
112C2×D3×C8C2318yes—
113#897—28no—
114#896—28no—
115#896—28no—
116#892—22no—
117#892—22no—
118#892—24no—
119#892C2314yes—
120#892C2318yes—
121#892C2314yes—
122#890—24no—
123#890C2318yes—
124#888—22no—
125#888—22no—
126#888C2314yes—
127#888C2314yes—
128#887—28no—
129#887—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—
142#815C2318yes—
143#808C2318yes—
144#808C2318yes—
145#807C2318yes—
146#807C2318yes—
147#803C2314yes—
148C23×C8—32no—
149#533—38no—
150#533—38no—
151#533—64no—
152#533—64no—
153C2×C4⋊Q8—31no—
154C2×C4⋊Q8—31no—
155C2×C4⋊Q8—31no—
156C2×C4⋊Q8C2×D311yescharacteristic
157C2×#125—32no—
158C2×#125—31no—
159C2×#125C2×D311yescharacteristic
160C2×C22⋊Q8—61no—
161C2×C22⋊Q8—61no—
162C2×C22⋊Q8—61no—
163C2×C22⋊Q8—61no—
164C2×C4×Q8—61no—
165C2×C4×Q8—61no—
166C2×C4×Q8—61no—
167C2×C4×Q8—61no—
168C2×#117—32no—
169C2×#117—31no—
170C2×#117—31no—
171C22×C42—31no—
172#501—38no—
173#501—38no—
174#501—316no—
175#501—316no—
176#501—38no—
177#501C2×D318yescomplemented
178#427—316no—
179#427—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—
188C2×#97—32no—
189C2×#97—31no—
190C2×#97—38no—
191C2×#97C2×D311yescharacteristic
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—21no—
202C22×Dic3—21no—
203C22×Dic3—22no—
204C22×Dic3C2×D411yescharacteristic
205C22×Dic3C2×D411yescharacteristic
206C22×Dic3C2×D411yescharacteristic
207C2×C4×D3—42no—
208C2×C4×D3—42no—
209C2×C4×D3C2×D414yes—
210C2×C4×D3C2×D414yes—
211C2×C4×D3C2×D412yes—
212C2×C4×D3C2×Q818yes—
213C2×C4×D3C2412yes—
214C2×Dic6—24no—
215C2×Dic6—24no—
216C2×Dic6—24no—
217C2×Dic6—44no—
218C2×Dic6—44no—
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—24no—
234(C2×D3)⋊C4—24no—
235(C2×D3)⋊C4—28no—
236C4⋊Dic3—22no—
237C4⋊Dic3—24no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—28no—
241C4⋊Dic3C2412yes—
242C4⋊Dic3C2414yes—
243C4⋊Dic3C2412yes—
244Dic3⋊C4—24no—
245Dic3⋊C4—24no—
246Dic3⋊C4—24no—
247Dic3⋊C4—24no—
248Dic3⋊C4—28no—
249C4×Dic3—24no—
250C4×Dic3—24no—
251C4×Dic3—28no—
252C4×Dic3C2×D412yes—
253C4×Dic3C2×D414yes—
254C4×Dic3C2×D414yes—
255C4×Dic3C2412yes—
256C2×C3⋊C8C2×Q818yes—
257C2×C3⋊C8C2414yes—
258D3×C8C2×Q8132yescomplemented
259C22×Q8—21no—
260C22×Q8—61no—
261C22×Q8—61no—
262C22×Q8—61no—
263C23×C4—31no—
264C23×C4—31no—
265C23×C4—31no—
266C22×C8—32no—
267C22×C8—316no—
268C22×C8—38no—
269C22×C8C22×D312yes—
270C4⋊Q8—22no—
271C4⋊Q8—34no—
272C4⋊Q8—34no—
273C4⋊Q8—34no—
274C4⋊Q8—62no—
275C4⋊Q8—62no—
276C4⋊Q8—62no—
277C4⋊Q8C22×D314yes—
278#125—22no—
279#125—38no—
280#125—34no—
281#125—64no—
282#125—62no—
283#125C22×D314yes—
284C22⋊Q8—68no—
285C22⋊Q8—68no—
286C22⋊Q8—68no—
287C22⋊Q8—68no—
288C4×Q8—68no—
289C4×Q8—68no—
290C4×Q8—68no—
291C4×Q8—68no—
292#117—34no—
293#117—38no—
294#117—34no—
295#117—62no—
296#117—62no—
297#117—64no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—32no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—62no—
305C2×C4⋊C4—62no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4C22×D312yes—
312C2×C4⋊C4C22×D311yescharacteristic
313C2×C4⋊C4C22×D311yescharacteristic
314C2×C22⋊C4—61no—
315C2×C22⋊C4—61no—
316C2×C22⋊C4—62no—
317C2×C42—31no—
318C2×C42—31no—
319C2×C42—31no—
320C2×C42—34no—
321C2×C42—34no—
322C2×C42—61no—
323C2×C42—61no—
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—38no—
335#97—34no—
336#97C22×D314yes—
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—22no—
359C2×Dic3—22no—
360C2×Dic3—24no—
361C2×Dic3—42no—
362C2×Dic3—42no—
363C2×Dic3—44no—
364C2×Dic3C4⋊Q814yes—
365C2×Dic3C22×D412yes—
366C2×Dic3C22×D412yes—
367C2×Dic3C22×D412yes—
368C4×D3—44no—
369C4×D3—44no—
370C4×D3C4⋊Q818yes—
371C4×D3C22×Q818yes—
372Dic6—48no—
373Dic6—48no—
374Dic6—48no—
375C3×C8C22×Q818yes—
376C3⋊C8C22×Q818yes—
377C24—31no—
378C2×Q8—24no—
379C2×Q8—44no—
380C2×Q8—64no—
381C2×Q8—64no—
382C2×Q8—64no—
383C2×Q8—124no—
384C2×Q8—124no—
385C2×Q8—124no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—31no—
390C22×C4—32no—
391C22×C4—38no—
392C22×C4—32no—
393C22×C4—31no—
394C22×C4—31no—
395C22×C4—34no—
396C22×C4—34no—
397C22×C4—61no—
398C22×C4—62no—
399C22×C4—61no—
400C22×C4—122no—
401C22×C4—122no—
402C22×C4D3×D411yescharacteristic
403C22×C4D3×D411yescharacteristic
404C22×C4C23×D311yescharacteristic
405C2×C8—38no—
406C2×C8—34no—
407C2×C8—332no—
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—34no—
418C4⋊C4—32no—
419C4⋊C4—68no—
420C4⋊C4—68no—
421C4⋊C4—64no—
422C4⋊C4—64no—
423C4⋊C4—62no—
424C4⋊C4—61no—
425C4⋊C4—61no—
426C4⋊C4—64no—
427C4⋊C4—64no—
428C4⋊C4—64no—
429C4⋊C4C23×D312yes—
430C4⋊C4C23×D314yes—
431C4⋊C4C23×D312yes—
432C22⋊C4—64no—
433C22⋊C4—64no—
434C22⋊C4—68no—
435C42—34no—
436C42—34no—
437C42—32no—
438C42—32no—
439C42—64no—
440C42—64no—
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—42no—
462Dic3—42no—
463Dic3—44no—
464Dic3C2×C4⋊Q814yes—
465C23—32no—
466C23—34no—
467C23—62no—
468C23—62no—
469C23C2×D3×D411yescharacteristic
470Q8—48no—
471Q8—128no—
472Q8—128no—
473Q8—128no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—32no—
480C2×C4—34no—
481C2×C4—38no—
482C2×C4—38no—
483C2×C4—32no—
484C2×C4—32no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—62no—
489C2×C4—62no—
490C2×C4—62no—
491C2×C4—64no—
492C2×C4—62no—
493C2×C4—124no—
494C2×C4—124no—
495C2×C4—122no—
496C2×C4—124no—
497C2×C4—122no—
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#316214yescomplemented
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—44no—
522C4—42no—
523C4—42no—
524C4—62no—
525C4—62no—
526C4—122no—
527C4—124no—
528C4—122no—
529C4D3×C4⋊Q812yes—
530C4C22×D3×Q812yes—
531C3C2×#316211yescharacteristic, complemented, Sylow
532C2—64no—
533C2#8197314yesdirect 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×#3162
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. #81973C2direct (N × H)
8. C2×#4876C2semidirect
10. C2×#4856C2semidirect
12. C2×#4702C2semidirect
38. C3×#501C22semidirect
45. #5081C22semidirect
46. #5064C22semidirect
53. #4876C22semidirect
55. #4856C22semidirect
57. #4702C22semidirect
66. C2×#501D3semidirect
111. C2×D3×C8Q8semidirect
177. #501C2×D3semidirect
258. D3×C8C2×Q8semidirect
505. C2×C3#3162semidirect
531. C3C2×#3162semidirect
533. C2#81973direct (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×#56882
C2×C3×#5012
C2×#50812
C2×#50642
#819732
C2×#48802
C2×#48762
C2×#48602
C2×#48562
C2×D3×#972
C2×#47022
C2×#31623