Finite Group Databasebeta

← C2×#81608 #11667232

Subgroups — C2×#81608 #11667232

Order 768.

2006 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×C4⋊Q8C211yesmaximal, characteristic
2C2×#5728C211yesmaximal, characteristic
3C2×C3×#487C211yesmaximal, characteristic, complemented
4C2×#5078C211yesmaximal, characteristic, complemented
5C2×#5079C211yesmaximal, characteristic, complemented
6#81608C2116yesmaximal, direct product factor
7C2×#4811C214yesmaximal
8C2×#4793C214yesmaximal, complemented
9C2×#4685C211yesmaximal, characteristic
10C2×#4682C211yesmaximal, characteristic, complemented
11C2×#3230—31nomaximal, Sylow
12C22×D3×Q8—21no—
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×#946C2212yes—
16C2×Q8×Dic3C2212yes—
17C2×#944—21no—
18#5728—24no—
19#5728C2218yes—
20D3×C4⋊Q8—24no—
21D3×C4⋊Q8C2218yes—
22C2×#897—21no—
23C2×#892—21no—
24C2×D3×C4⋊C4—21no—
25C2×D3×C4⋊C4C2212yes—
26C2×#890—21no—
27C2×#888—21no—
28C2×#888C2212yes—
29C2×#887—21no—
30C2×C42×D3C2211yescharacteristic
31C2×#869C2211yescharacteristic
32C2×#869C2211yescharacteristic
33C3×#487C2218yescomplemented
34C2×C3×#103C2214yes—
35C2×C3×C4×C8C2211yescharacteristic
36C2×#835C2214yes—
37C2×(C2×D3)⋊C8C2212yes—
38C2×#818C2214yes—
39C2×Dic3⋊C8C2212yes—
40#5079C2218yescomplemented
41#5078C2218yescomplemented
42C2×#807C2214yes—
43C2×C4×C3⋊C8C2211yescharacteristic
44#4811C22132yes—
45#4793C22132yescomplemented
46#4685C2218yes—
47#4682C2218yescomplemented
48C2×#533—31no—
49C22×C4⋊Q8—31no—
50#3230—316no—
51C2×#500—31no—
52C2×#500—31no—
53C2×#487—31no—
54C2×#487D311yescharacteristic, complemented
55C2×#484—34no—
56C2×#477—34no—
57C2×#431—31no—
58C22×C3×Q8—21no—
59C22×C3×Q8C2312yes—
60C2×D3×Q8—28no—
61C2×D3×Q8—48no—
62C22×C4×D3—22no—
63C22×C4×D3D411yescharacteristic
64C22×C4×D3D411yescharacteristic
65C22×C4×D3C2311yescharacteristic
66C22×Dic6—21no—
67C22×Dic6—21no—
68C22×Dic6—22no—
69C22×C3×C8C2312yes—
70C3×C4⋊Q8—22no—
71C3×C4⋊Q8—22no—
72C3×C4⋊Q8C2314yes—
73C3×C4⋊Q8C2314yes—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C42C2311yescharacteristic
79#946—24no—
80#946C2318yes—
81Q8×Dic3—24no—
82Q8×Dic3C2318yes—
83#944—28no—
84C2×(C2×D3)⋊C4—22no—
85C2×C4⋊Dic3—21no—
86C2×C4⋊Dic3—21no—
87C2×C4⋊Dic3—22no—
88C2×C4⋊Dic3C2312yes—
89C2×C4⋊Dic3C2312yes—
90C2×Dic3⋊C4—22no—
91C2×Dic3⋊C4—22no—
92C2×Dic3⋊C4—22no—
93C2×C4×Dic3—22no—
94C2×C4×Dic3D411yescharacteristic
95C2×C4×Dic3D411yescharacteristic
96C2×C4×Dic3C2311yescharacteristic
97C22×C3⋊C8C2312yes—
98#897—28no—
99#892—28no—
100D3×C4⋊C4—24no—
101D3×C4⋊C4—28no—
102D3×C4⋊C4C2318yes—
103#890—28no—
104#888—24no—
105#888—28no—
106#888C2318yes—
107#887—28no—
108C42×D3D414yes—
109C42×D3C2314yes—
110#869—22no—
111#869—22no—
112#869C2314yes—
113#869C2314yes—
114C3×#103C23116yes—
115C3×C4×C8C2314yes—
116#835C23116yes—
117(C2×D3)⋊C8C2318yes—
118#818C23116yes—
119Dic3⋊C8C2318yes—
120#807C23116yes—
121C4×C3⋊C8C2314yes—
122C23×Q8—61no—
123#533—38no—
124#533—64no—
125C2×C4⋊Q8—38no—
126C2×C4⋊Q8—32no—
127C2×C4⋊Q8—31no—
128C2×C4⋊Q8—31no—
129C2×C4⋊Q8—61no—
130C2×C4⋊Q8—64no—
131C2×C4⋊Q8—61no—
132C2×C4⋊Q8C2×D311yescharacteristic
133C2×C4⋊Q8C2×D311yescharacteristic
134C2×#125—61no—
135C2×C22⋊Q8—32no—
136C2×C22⋊Q8—61no—
137C2×C4×Q8—32no—
138C2×C4×Q8—61no—
139C2×#117—61no—
140C22×C4⋊C4—32no—
141C22×C4⋊C4—61no—
142C22×C42—31no—
143#500—38no—
144#500—38no—
145#487—38no—
146#487C2×D318yescomplemented
147#484—332no—
148#477—332no—
149#431—38no—
150C2×#107—34no—
151C2×#107—34no—
152C2×C4⋊C8—32no—
153C2×#103—34no—
154C2×#103C2×D314yes—
155C2×C22⋊C8—32no—
156C2×C4×C8—31no—
157C2×C4×C8C2×D311yescharacteristic
158C23×D3C2×D411yescharacteristic
159C2×C3×Q8—24no—
160C2×C3×Q8—22no—
161C2×C3×Q8—28no—
162C2×C3×Q8—44no—
163C2×C3×Q8C2414yes—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2411yescharacteristic
169C22×Dic3—22no—
170C22×Dic3—22no—
171C22×Dic3C2×D411yescharacteristic
172C22×Dic3C2×D411yescharacteristic
173C22×Dic3C2×D411yescharacteristic
174D3×Q8—432no—
175C2×C4×D3—24no—
176C2×C4×D3—24no—
177C2×C4×D3—28no—
178C2×C4×D3—48no—
179C2×C4×D3Q1618yes—
180C2×C4×D3C2×D412yes—
181C2×C4×D3C2×D414yes—
182C2×C4×D3C2×D414yes—
183C2×C4×D3C2412yes—
184C2×Dic6—24no—
185C2×Dic6—24no—
186C2×Dic6—28no—
187C2×Dic6—48no—
188C2×Dic6—44no—
189C2×Dic6—44no—
190C2×C3×C8—24no—
191C2×C3×C8C2414yes—
192C3×C4⋊C4—24no—
193C3×C4⋊C4—24no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—24no—
196C3×C4⋊C4C2414yes—
197C3×C42C2×D412yes—
198C3×C42C2412yes—
199(C2×D3)⋊C4—28no—
200C4⋊Dic3—22no—
201C4⋊Dic3—22no—
202C4⋊Dic3—24no—
203C4⋊Dic3—24no—
204C4⋊Dic3—28no—
205C4⋊Dic3C2414yes—
206C4⋊Dic3C2414yes—
207Dic3⋊C4—28no—
208Dic3⋊C4—28no—
209Dic3⋊C4—28no—
210C4×Dic3—28no—
211C4×Dic3C2×D412yes—
212C4×Dic3C2×D414yes—
213C4×Dic3C2×D414yes—
214C4×Dic3C2412yes—
215C2×C3⋊C8—24no—
216C2×C3⋊C8C2414yes—
217C22×Q8—21no—
218C22×Q8—62no—
219C22×Q8—68no—
220C22×Q8—61no—
221C22×Q8—61no—
222C22×Q8—128no—
223C22×Q8C22×D312yes—
224C23×C4—31no—
225C23×C4—31no—
226C23×C4—31no—
227C23×C4—62no—
228C22×C8—32no—
229C22×C8C22×D312yes—
230C4⋊Q8—22no—
231C4⋊Q8—22no—
232C4⋊Q8—38no—
233C4⋊Q8—34no—
234C4⋊Q8—34no—
235C4⋊Q8—64no—
236C4⋊Q8—68no—
237C4⋊Q8—68no—
238C4⋊Q8—62no—
239C4⋊Q8—62no—
240C4⋊Q8C22×D314yes—
241C4⋊Q8C22×D314yes—
242#125—68no—
243C22⋊Q8—38no—
244C22⋊Q8—68no—
245C22⋊Q8—64no—
246C4×Q8—38no—
247C4×Q8—68no—
248C4×Q8—64no—
249#117—68no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—32no—
254C2×C4⋊C4—32no—
255C2×C4⋊C4—38no—
256C2×C4⋊C4—64no—
257C2×C4⋊C4—68no—
258C2×C4⋊C4—62no—
259C2×C4⋊C4—62no—
260C2×C4⋊C4—62no—
261C2×C4⋊C4—62no—
262C2×C4⋊C4—61no—
263C2×C4⋊C4—61no—
264C2×C4⋊C4C22×D312yes—
265C2×C22⋊C4—62no—
266C2×C42—31no—
267C2×C42—31no—
268C2×C42—31no—
269C2×C42—34no—
270C2×C42—34no—
271C2×C42—62no—
272C2×C42C22×D311yescharacteristic
273#107—316no—
274#107—316no—
275C4⋊C8—38no—
276#103—316no—
277#103C22×D3116yes—
278C22⋊C8—38no—
279C4×C8—34no—
280C4×C8C22×D314yes—
281C23×C3C22×D411yescharacteristic
282C22×D3—22no—
283C22×D3—22no—
284C22×D3C2×Q1614yes—
285C22×D3C22×D412yes—
286C3×Q8—28no—
287C3×Q8—44no—
288C3×Q8—48no—
289C2×C3×C4—24no—
290C2×C3×C4—24no—
291C2×C3×C4—44no—
292C2×C3×C4—44no—
293C2×C3×C4C2×Q1614yes—
294C2×C3×C4AGL(1,ℤ/8)14yes—
295C2×C3×C4C22×D411yescharacteristic
296C2×C3×C4C22×D412yes—
297C2×C3×C4C22×D412yes—
298C2×C3×C42−1+414yes—
299C2×C3×C4C2511yescharacteristic
300C2×Dic3—22no—
301C2×Dic3—22no—
302C2×Dic3—24no—
303C2×Dic3—24no—
304C2×Dic3—44no—
305C2×Dic3—44no—
306C2×Dic3C2×Q1614yes—
307C2×Dic3C22×D412yes—
308C2×Dic3C22×D412yes—
309C2×Dic3C22×D412yes—
310C4×D3—24no—
311C4×D3—28no—
312C4×D3—24no—
313C4×D3—416no—
314C4×D3C2×Q16116yes—
315Dic6—416no—
316Dic6—48no—
317Dic6—48no—
318C3×C8—24no—
319C3⋊C8—24no—
320C24—31no—
321C2×Q8—22no—
322C2×Q8—24no—
323C2×Q8—28no—
324C2×Q8—44no—
325C2×Q8—68no—
326C2×Q8—64no—
327C2×Q8—64no—
328C2×Q8—128no—
329C2×Q8—1232no—
330C2×Q8—124no—
331C2×Q8—124no—
332C2×Q8C23×D314yes—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—31no—
336C22×C4—38no—
337C22×C4—34no—
338C22×C4—31no—
339C22×C4—31no—
340C22×C4—32no—
341C22×C4—34no—
342C22×C4—32no—
343C22×C4—68no—
344C22×C4—64no—
345C22×C4—64no—
346C22×C4—62no—
347C22×C4—62no—
348C22×C4—128no—
349C22×C4D3×D411yescharacteristic
350C22×C4D3×D411yescharacteristic
351C22×C4C23×D311yescharacteristic
352C2×C8—24no—
353C2×C8—34no—
354C2×C8—64no—
355C2×C8C23×D314yes—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—22no—
360C4⋊C4—34no—
361C4⋊C4—34no—
362C4⋊C4—68no—
363C4⋊C4—68no—
364C4⋊C4—68no—
365C4⋊C4—68no—
366C4⋊C4—62no—
367C4⋊C4—62no—
368C4⋊C4—64no—
369C4⋊C4—64no—
370C4⋊C4C23×D314yes—
371C22⋊C4—68no—
372C42—34no—
373C42—32no—
374C42—34no—
375C42—32no—
376C42—68no—
377C42D3×D412yes—
378C42C23×D312yes—
379C22×C3#53612yes—
380C22×C3C22×Q1612yes—
381C22×C3C2×AGL(1,ℤ/8)12yes—
382C22×C3C23×D411yescharacteristic
383C2×D3—22no—
384C2×D3—28no—
385C2×D3—22no—
386C2×D3C22×Q1614yes—
387C3×C4—44no—
388C3×C4—44no—
389C3×C4#53612yes—
390C3×C4C22×Q1614yes—
391C3×C4C2×AGL(1,ℤ/8)14yes—
392C3×C4C2×2−1+412yes—
393Dic3—22no—
394Dic3—22no—
395Dic3—44no—
396Dic3—44no—
397Dic3C22×Q1614yes—
398C23—32no—
399C23—34no—
400C23—62no—
401C23—62no—
402C23C2×D3×D411yescharacteristic
403Q8—28no—
404Q8—44no—
405Q8—48no—
406Q8—1216no—
407Q8—128no—
408Q8—128no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—32no—
412C2×C4—34no—
413C2×C4—316no—
414C2×C4—32no—
415C2×C4—32no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—62no—
419C2×C4—62no—
420C2×C4—64no—
421C2×C4—64no—
422C2×C4—68no—
423C2×C4—64no—
424C2×C4—64no—
425C2×C4—1216no—
426C2×C4—124no—
427C2×C4—124no—
428C2×C4#91414yes—
429C2×C4D3×Q1614yes—
430C2×C4C2×D3×D411yescharacteristic
431C2×C4C2×D3×D412yes—
432C2×C4C2×D3×D412yes—
433C2×C4#101014yes—
434C2×C4C24×D311yescharacteristic
435C8—24no—
436C8—64no—
437C2×C3#323014yescomplemented
438C2×C3C2×#53611yescharacteristic
439C2×C3C23×Q1611yescharacteristic
440C2×C3C22×AGL(1,ℤ/8)11yescharacteristic
441D3—24no—
442C22—34no—
443C22—62no—
444C22—62no—
445C22—68no—
446C22#567512yes—
447C22C2×#91412yes—
448C22C2×D3×Q1612yes—
449C22C22×D3×D411yescharacteristic
450C4—34no—
451C4—44no—
452C4—44no—
453C4—62no—
454C4—62no—
455C4—124no—
456C4—124no—
457C4#567512yes—
458C4C2×#91414yes—
459C4C2×D3×Q1614yes—
460C4C2×#101012yes—
461C3C2×#323011yescharacteristic, complemented, Sylow
462C2—64no—
463C2#8160814yesdirect product factor
464C2C2×#567511yescharacteristic
465C2C22×#91411yescharacteristic
466C2C22×D3×Q1611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#487C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenterC2×#487D3
hypocenterC3C2×#3230
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
3. C2×C3×#487C2semidirect
4. C2×#5078C2semidirect
5. C2×#5079C2semidirect
6. #81608C2direct (N × H)
8. C2×#4793C2semidirect
10. C2×#4682C2semidirect
33. C3×#487C22semidirect
40. #5079C22semidirect
41. #5078C22semidirect
45. #4793C22semidirect
47. #4682C22semidirect
54. C2×#487D3semidirect
146. #487C2×D3semidirect
437. C2×C3#3230semidirect
461. C3C2×#3230semidirect
463. C2#81608direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D3×C4⋊Q82
C2×#57282
C2×C3×#4872
C2×#50782
C2×#50792
#816082
C2×#48112
C2×#47932
C2×#46852
C2×#46822
C2×#32303