Finite Group Databasebeta

← C2×#81598 #11664391

Subgroups — C2×#81598 #11664391

Order 768.

2222 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5728C211yesmaximal, characteristic, complemented
2C2×#5725C211yesmaximal, characteristic
3C2×C3×#487C211yesmaximal, characteristic, complemented
4C2×#5079C211yesmaximal, characteristic, complemented
5C2×#5088C211yesmaximal, characteristic, complemented
6#81598C2116yesmaximal, direct product factor
7C2×#4804C214yesmaximal, complemented
8C2×#4800C214yesmaximal, complemented
9C2×#4694C211yesmaximal, characteristic, complemented
10C2×C4×D3×C8C211yesmaximal, characteristic, complemented
11C2×#3131—31nomaximal, Sylow
12C22×#239C2212yes—
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×#947C2212yes—
16C2×#946—21no—
17C2×Q8×Dic3—21no—
18C22×D3×C8C2212yes—
19#5728—24no—
20#5728C2218yescomplemented
21#5725—24no—
22#5725C2218yes—
23C2×#897—21no—
24C2×#895—21no—
25C2×#895—21no—
26C2×#893—21no—
27C2×#893—21no—
28C2×#892C2212yes—
29C2×#890C2212yes—
30C2×#887—21no—
31C2×#874C2211yescharacteristic
32C2×C42×D3C2211yescharacteristic
33C2×#869C2211yescharacteristic
34C3×#487C2218yescomplemented
35C2×C3×#103C2214yes—
36C2×C3×C4×C8C2211yescharacteristic
37C2×#835C2214yes—
38C2×#821C2214yes—
39C2×C8×Dic3C2212yes—
40#5088C2218yescomplemented
41#5079C2218yescomplemented
42C2×#809C2214yes—
43C2×C4×C3⋊C8C2211yescharacteristic
44#4804C22132yescomplemented
45#4800C22132yescomplemented
46#4694C2218yescomplemented
47C4×D3×C8C2218yescomplemented
48C2×#533—31no—
49C2×#532—31no—
50#3131—316no—
51C2×#488—34no—
52C2×#487—31no—
53C2×#487D311yescharacteristic, complemented
54C2×#486—31no—
55C2×#486—31no—
56C2×#416—34no—
57C22×C4×C8—31no—
58C22×C3×Q8—21no—
59C22×C3×Q8C2312yes—
60C2×#239—24no—
61C2×#239—216no—
62C2×#239C2318yes—
63C22×D12—22no—
64C22×D12—21no—
65C22×D12—21no—
66C22×D12C2312yes—
67C22×C4×D3—22no—
68C22×C4×D3D411yescharacteristic
69C22×C4×D3D411yescharacteristic
70C22×C4×D3C2311yescharacteristic
71C22×Dic6—21no—
72C22×C3×C8C2312yes—
73C3×C4⋊Q8—22no—
74C3×C4⋊Q8—22no—
75C3×C4⋊Q8C2314yes—
76C3×C4⋊Q8C2314yes—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C42C2311yescharacteristic
82#947—24no—
83#947C2318yes—
84#946—28no—
85Q8×Dic3—28no—
86C2×(C2×D3)⋊C4—22no—
87C2×(C2×D3)⋊C4—22no—
88C2×(C2×D3)⋊C4—22no—
89C2×C4⋊Dic3—21no—
90C2×C4⋊Dic3—22no—
91C2×C4⋊Dic3C2312yes—
92C2×Dic3⋊C4—22no—
93C2×C4×Dic3—22no—
94C2×C4×Dic3D411yescharacteristic
95C2×C4×Dic3D411yescharacteristic
96C2×C4×Dic3C2311yescharacteristic
97C22×C3⋊C8C2312yes—
98C2×D3×C8—28no—
99C2×D3×C8C2318yes—
100#897—28no—
101#895—28no—
102#895—28no—
103#893—28no—
104#893—28no—
105#892—24no—
106#892C2318yes—
107#890—24no—
108#890C2318yes—
109#887—28no—
110#874—22no—
111#874C2314yes—
112C42×D3D414yes—
113C42×D3C2314yes—
114#869—22no—
115#869C2314yes—
116C3×#103C23116yes—
117C3×C4×C8C2314yes—
118#835C23116yes—
119#821C23116yes—
120C8×Dic3C2318yes—
121#809C23116yes—
122C4×C3⋊C8C2314yes—
123C22×#41—32no—
124C23×C8—32no—
125#533—38no—
126#533—64no—
127#532—38no—
128#532—64no—
129C2×C4⋊Q8—31no—
130C2×C4⋊Q8C2×D311yescharacteristic
131C2×C4⋊Q8C2×D311yescharacteristic
132C2×#127—31no—
133C2×#125—32no—
134C2×#124—32no—
135C2×C22⋊Q8—61no—
136C2×C22⋊Q8—61no—
137C2×#121—61no—
138C2×#121—61no—
139C2×C4×Q8—61no—
140C2×C4×Q8—61no—
141C2×C4×D4—61no—
142C2×C4×D4—61no—
143C2×#117—32no—
144C22×C42—31no—
145#488—332no—
146#487—38no—
147#487C2×D318yescomplemented
148#486—38no—
149#486—38no—
150#416—332no—
151C2×#103—34no—
152C2×#103C2×D314yes—
153C2×#102—34no—
154C2×#102—34no—
155C2×C4×C8—32no—
156C2×C4×C8—31no—
157C2×C4×C8—38no—
158C2×C4×C8C2×D311yescharacteristic
159C23×D3—22no—
160C23×D3C2×D411yescharacteristic
161C2×C3×Q8—24no—
162C2×C3×Q8—22no—
163C2×C3×Q8—28no—
164C2×C3×Q8—44no—
165C2×C3×Q8C2414yes—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2411yescharacteristic
171C22×Dic3—22no—
172C22×Dic3C2×D411yescharacteristic
173C22×Dic3C2×D411yescharacteristic
174C22×Dic3C2×D411yescharacteristic
175#239—232no—
176#239—416no—
177C2×D12—28no—
178C2×D12—28no—
179C2×D12—22no—
180C2×D12—24no—
181C2×D12—24no—
182C2×D12—48no—
183C2×D12—44no—
184C2×D12—44no—
185C2×D12C2414yes—
186C2×C4×D3—24no—
187C2×C4×D3—24no—
188C2×C4×D3—28no—
189C2×C4×D3—48no—
190C2×C4×D3C2×D414yes—
191C2×C4×D3C2×D414yes—
192C2×C4×D3C2×D412yes—
193C2×C4×D3#4118yes—
194C2×C4×D3C2412yes—
195C2×Dic6—24no—
196C2×Dic6—44no—
197C2×C3×C8—24no—
198C2×C3×C8C2414yes—
199C3×C4⋊C4—24no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—24no—
202C3×C4⋊C4—24no—
203C3×C4⋊C4C2414yes—
204C3×C42C2×D412yes—
205C3×C42C2412yes—
206(C2×D3)⋊C4—28no—
207(C2×D3)⋊C4—28no—
208(C2×D3)⋊C4—28no—
209C4⋊Dic3—22no—
210C4⋊Dic3—24no—
211C4⋊Dic3—28no—
212C4⋊Dic3C2414yes—
213Dic3⋊C4—28no—
214C4×Dic3—28no—
215C4×Dic3C2×D412yes—
216C4×Dic3C2×D414yes—
217C4×Dic3C2×D414yes—
218C4×Dic3C2412yes—
219C2×C3⋊C8—24no—
220C2×C3⋊C8C2414yes—
221D3×C8—216no—
222C2×#41—38no—
223C2×#41—64no—
224C2×#41—616no—
225C22×Q8—21no—
226C22×Q8—61no—
227C22×Q8C22×D312yes—
228C22×D4—32no—
229C22×D4—62no—
230C22×D4—61no—
231C22×D4—61no—
232C23×C4—31no—
233C23×C4—31no—
234C23×C4—31no—
235C23×C4—62no—
236C22×C8—32no—
237C22×C8—38no—
238C22×C8—68no—
239C22×C8C22×D312yes—
240C4⋊Q8—22no—
241C4⋊Q8—22no—
242C4⋊Q8—34no—
243C4⋊Q8—62no—
244C4⋊Q8C22×D314yes—
245C4⋊Q8C22×D314yes—
246#127—34no—
247#127—62no—
248#125—38no—
249#125—64no—
250#124—38no—
251#124—64no—
252C22⋊Q8—68no—
253C22⋊Q8—68no—
254#121—68no—
255#121—68no—
256C4×Q8—68no—
257C4×Q8—68no—
258C4×D4—68no—
259C4×D4—68no—
260#117—38no—
261#117—64no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—32no—
266C2×C4⋊C4—62no—
267C2×C4⋊C4—62no—
268C2×C4⋊C4—61no—
269C2×C4⋊C4C22×D312yes—
270C2×C22⋊C4—62no—
271C2×C22⋊C4—62no—
272C2×C22⋊C4—62no—
273C2×C42—31no—
274C2×C42—31no—
275C2×C42—31no—
276C2×C42—34no—
277C2×C42—34no—
278C2×C42—62no—
279C2×C42C22×D311yescharacteristic
280#103—316no—
281#103C22×D3116yes—
282#102—316no—
283#102—316no—
284C4×C8—38no—
285C4×C8—34no—
286C4×C8C22×D314yes—
287C23×C3C22×D411yescharacteristic
288C22×D3—22no—
289C22×D3—22no—
290C22×D3—24no—
291C22×D3—44no—
292C22×D3—44no—
293C22×D3—44no—
294C22×D3#12414yes—
295C22×D3C22×D412yes—
296C3×Q8—28no—
297C3×Q8—44no—
298C3×Q8—48no—
299C2×C3×C4—24no—
300C2×C3×C4—24no—
301C2×C3×C4—44no—
302C2×C3×C4—44no—
303C2×C3×C4#13514yes—
304C2×C3×C4#13514yes—
305C2×C3×C4C22×D411yescharacteristic
306C2×C3×C4C22×D412yes—
307C2×C3×C4C22×D412yes—
308C2×C3×C4C2×#4114yes—
309C2×C3×C4C2511yescharacteristic
310C2×Dic3—22no—
311C2×Dic3—22no—
312C2×Dic3—24no—
313C2×Dic3—44no—
314C2×Dic3#12414yes—
315C2×Dic3C22×D412yes—
316C2×Dic3C22×D412yes—
317C2×Dic3C22×D412yes—
318D12—28no—
319D12—416no—
320D12—44no—
321D12—48no—
322D12—48no—
323C4×D3—28no—
324C4×D3—28no—
325C4×D3—416no—
326C4×D3#12418yes—
327C4×D3C2×#4118yes—
328Dic6—48no—
329C3×C8—24no—
330C3⋊C8—24no—
331C24—31no—
332C24—62no—
333#41—632no—
334#41—1216no—
335C2×Q8—24no—
336C2×Q8—22no—
337C2×Q8—28no—
338C2×Q8—44no—
339C2×Q8—64no—
340C2×Q8—124no—
341C2×Q8C23×D314yes—
342C2×D4—34no—
343C2×D4—68no—
344C2×D4—64no—
345C2×D4—64no—
346C2×D4—62no—
347C2×D4—68no—
348C2×D4—128no—
349C2×D4—124no—
350C2×D4—124no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—31no—
354C22×C4—32no—
355C22×C4—38no—
356C22×C4—32no—
357C22×C4—31no—
358C22×C4—31no—
359C22×C4—34no—
360C22×C4—34no—
361C22×C4—68no—
362C22×C4—64no—
363C22×C4—64no—
364C22×C4—62no—
365C22×C4—128no—
366C22×C4D3×D411yescharacteristic
367C22×C4D3×D411yescharacteristic
368C22×C4C23×D311yescharacteristic
369C2×C8—24no—
370C2×C8—34no—
371C2×C8—616no—
372C2×C8—64no—
373C2×C8C23×D314yes—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—22no—
378C4⋊C4—34no—
379C4⋊C4—68no—
380C4⋊C4—68no—
381C4⋊C4—62no—
382C4⋊C4—64no—
383C4⋊C4C23×D314yes—
384C22⋊C4—68no—
385C22⋊C4—68no—
386C22⋊C4—68no—
387C42—34no—
388C42—34no—
389C42—32no—
390C42—32no—
391C42—68no—
392C42D3×D412yes—
393C42C23×D312yes—
394C22×C3C2×#12412yes—
395C22×C3C2×#13512yes—
396C22×C3C2×#13512yes—
397C22×C3C23×D411yescharacteristic
398C2×D3—28no—
399C2×D3—22no—
400C2×D3—22no—
401C2×D3—44no—
402C2×D3—44no—
403C2×D3—44no—
404C2×D3—88no—
405C2×D3C2×#12414yes—
406C3×C4—44no—
407C3×C4—44no—
408C3×C4C2×#12412yes—
409C3×C4C2×#13514yes—
410C3×C4C2×#13514yes—
411C3×C4C22×#4112yes—
412Dic3—22no—
413Dic3—22no—
414Dic3—44no—
415Dic3C2×#12414yes—
416C23—34no—
417C23—32no—
418C23—62no—
419C23—62no—
420C23—64no—
421C23—124no—
422C23—124no—
423C23—124no—
424C23C2×D3×D411yescharacteristic
425Q8—28no—
426Q8—44no—
427Q8—48no—
428Q8—128no—
429D4—68no—
430D4—1216no—
431D4—124no—
432D4—128no—
433D4—128no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—34no—
437C2×C4—32no—
438C2×C4—38no—
439C2×C4—38no—
440C2×C4—32no—
441C2×C4—32no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—62no—
445C2×C4—62no—
446C2×C4—68no—
447C2×C4—68no—
448C2×C4—64no—
449C2×C4—1216no—
450C2×C4—124no—
451C2×C4#91614yes—
452C2×C4#91914yes—
453C2×C4C2×D3×D411yescharacteristic
454C2×C4C2×D3×D412yes—
455C2×C4C2×D3×D412yes—
456C2×C4D3×#4114yes—
457C2×C4C24×D311yescharacteristic
458C8—24no—
459C8—64no—
460C2×C3#313114yescomplemented
461C2×C3C22×#12411yescharacteristic
462C2×C3C22×#13511yescharacteristic
463C2×C3C22×#13511yescharacteristic
464D3—24no—
465D3—84no—
466C22—34no—
467C22—62no—
468C22—62no—
469C22—68no—
470C22—124no—
471C22—124no—
472C22—124no—
473C22—248no—
474C22D3×#12412yes—
475C22C2×#91612yes—
476C22C2×#91912yes—
477C22C22×D3×D411yescharacteristic
478C4—34no—
479C4—44no—
480C4—44no—
481C4—62no—
482C4—62no—
483C4—124no—
484C4D3×#12412yes—
485C4C2×#91614yes—
486C4C2×#91914yes—
487C4C2×D3×#4112yes—
488C3C2×#313111yescharacteristic, complemented, Sylow
489C2—64no—
490C2—244no—
491C2#8159814yesdirect product factor
492C2C2×D3×#12411yescharacteristic
493C2C22×#91611yescharacteristic
494C2C22×#91911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#487C2
socleC23×C3C22×D4
prime radicalC2×#5725C2
hypercenterC2×#487D3
hypocenterC3C2×#3131
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#487

Complements

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

NH ≅ G/Ntype
1. C2×#5728C2semidirect
3. C2×C3×#487C2semidirect
4. C2×#5079C2semidirect
5. C2×#5088C2semidirect
6. #81598C2direct (N × H)
7. C2×#4804C2semidirect
8. C2×#4800C2semidirect
9. C2×#4694C2semidirect
10. C2×C4×D3×C8C2semidirect
20. #5728C22semidirect
34. C3×#487C22semidirect
40. #5088C22semidirect
41. #5079C22semidirect
44. #4804C22semidirect
45. #4800C22semidirect
46. #4694C22semidirect
47. C4×D3×C8C22semidirect
53. C2×#487D3semidirect
147. #487C2×D3semidirect
460. C2×C3#3131semidirect
488. C3C2×#3131semidirect
491. C2#81598direct (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×#57252
C2×C3×#4872
C2×#50792
C2×#50882
#815982
C2×#48042
C2×#48002
C2×#46942
C2×C4×D3×C82
C2×#31313