Finite Group Databasebeta

← C2×SD16×Dic6 #11675607

Subgroups — C2×SD16×Dic6 #11675607

Order 768.

2058 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×Dic6C211yesmaximal, characteristic, complemented
2C2×D4×Dic6C211yesmaximal, characteristic
3C2×C3×C4×SD16C211yesmaximal, characteristic
4C2×Dic3×SD16C212yesmaximal
5C2×#5020C211yesmaximal, characteristic, complemented
6C2×#5008C211yesmaximal, characteristic
7SD16×Dic6C2116yesmaximal, direct product factor
8C2×#4855C212yesmaximal, complemented
9C2×#4791C212yesmaximal, complemented
10C2×#4761C212yesmaximal
11C2×C8×Dic6C211yesmaximal, characteristic, complemented
12C2×#4680C211yesmaximal, characteristic, complemented
13C2×Q8×SD16—31nomaximal, Sylow
14C23×Dic6—21no—
15C22×C3×SD16C2211yescharacteristic
16C2×C3×C4×Q8C2211yescharacteristic
17C2×C3×C4×D4C2211yescharacteristic
18C2×Q8×Dic3C2212yes—
19C2×D4×Dic3C2212yes—
20C2×#924—21no—
21Q8×Dic6—24no—
22Q8×Dic6C2218yescomplemented
23D4×Dic6—24no—
24D4×Dic6C2218yes—
25C2×#888—22no—
26C2×#888C2212yes—
27C2×#888C2212yes—
28C2×#878—22no—
29C2×#869—21no—
30C2×#869C2211yescharacteristic
31C2×#869C2211yescharacteristic
32C2×C4×Dic6—21no—
33C2×C4×Dic6C2211yescharacteristic
34C3×C4×SD16C2218yes—
35C2×C3×#106C2211yescharacteristic
36C2×C3×#103C2211yescharacteristic
37C2×C3×#102C2211yescharacteristic
38C2×C3×C4×C8C2211yescharacteristic
39C2×#835C2212yes—
40C2×#832C2212yes—
41Dic3×SD16C22116yes—
42C2×#817C2212yes—
43C2×#817C2212yes—
44C2×Dic3⋊C8C2212yes—
45C2×C8×Dic3C2212yes—
46#5020C2218yescomplemented
47#5008C2218yes—
48C2×#808C2212yes—
49C2×#808C2212yes—
50C2×C4⋊(C3⋊C8)C2211yescharacteristic
51#4855C22116yescomplemented
52#4791C22116yescomplemented
53#4761C22116yes—
54#4680C2218yescomplemented
55C8×Dic6C2218yescomplemented
56C2×Q82—31no—
57C2×D4×Q8—31no—
58Q8×SD16—316no—
59C2×#498—32no—
60C2×#498—31no—
61C2×#475—32no—
62C2×#475—31no—
63C2×#476—32no—
64C2×#476—31no—
65C2×C8×Q8—31no—
66C2×C4×SD16—32no—
67C2×C4×SD16D311yescharacteristic
68C22×C3×Q8C2311yescharacteristic
69C22×C3×D4C2311yescharacteristic
70C23×C3×C4—21no—
71C23×Dic3—22no—
72C22×Dic6—21no—
73C22×Dic6—22no—
74C22×Dic6—21no—
75C22×Dic6—22no—
76C22×Dic6—28no—
77C22×Dic6—48no—
78C22×Dic6D411yescharacteristic
79C2×C3×SD16Q818yescomplemented
80C2×C3×SD16C2314yes—
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C3×C4×Q8—22no—
84C3×C4×Q8C2314yes—
85C3×C4×D4—22no—
86C3×C4×D4C2314yes—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C22⋊C4—21no—
91C2×C3×C42—21no—
92C2×C3×C42C2311yescharacteristic
93C2×#217—22no—
94Q8×Dic3—24no—
95Q8×Dic3C2318yes—
96D4×Dic3—24no—
97D4×Dic3C2318yes—
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3—22no—
101C2×C4⋊Dic3D411yescharacteristic
102C2×C4⋊Dic3C2312yes—
103C2×C4⋊Dic3C2312yes—
104#924—28no—
105C2×Dic3⋊C4—22no—
106C2×Dic3⋊C4—22no—
107C2×Dic3⋊C4D412yes—
108C2×C4×Dic3—22no—
109C2×C4×Dic3C2312yes—
110C22×C3⋊C8C2312yes—
111#888—216no—
112#888—24no—
113#888—24no—
114#888C2318yes—
115#888C2318yes—
116#878—216no—
117#869—22no—
118#869—22no—
119#869—28no—
120#869C2314yes—
121#869C2314yes—
122C4×Dic6—28no—
123C4×Dic6D414yes—
124C4×Dic6C2314yes—
125C3×#106C2314yes—
126C3×#103C2314yes—
127C3×#102C2314yes—
128C3×C4×C8C2314yes—
129#835C2318yes—
130#832C2318yes—
131#817C2318yes—
132#817C2318yes—
133Dic3⋊C8C2318yes—
134C8×Dic3C2318yes—
135#808C2318yes—
136#808C2318yes—
137C4⋊(C3⋊C8)C2314yes—
138C23×Q8—61no—
139C22×SD16C2×D311yescharacteristic
140Q82—38no—
141Q82—64no—
142D4×Q8—38no—
143D4×Q8—64no—
144C2×C4⋊Q8—32no—
145C2×C4⋊Q8—32no—
146C2×C4⋊Q8—31no—
147C2×C4⋊Q8—31no—
148C2×C4⋊Q8—61no—
149C2×C4⋊Q8—62no—
150C2×C22⋊Q8—62no—
151C2×C22⋊Q8—61no—
152C2×C4×Q8—31no—
153C2×C4×Q8—32no—
154C2×C4×Q8—61no—
155C2×C4×Q8C2×D311yescharacteristic
156C2×C4×D4—32no—
157C2×C4×D4C2×D311yescharacteristic
158#498—316no—
159#498—38no—
160#476—38no—
161#476—316no—
162#475—38no—
163#475—316no—
164C8×Q8—38no—
165C4×SD16—316no—
166C4×SD16C2×D318yes—
167C2×#106—32no—
168C2×#106—32no—
169C2×#106—32no—
170C2×#106—32no—
171C2×#106C2×D311yescharacteristic
172C2×C4⋊C8—32no—
173C2×C4⋊C8—31no—
174C2×#103—32no—
175C2×#103C2×D311yescharacteristic
176C2×#102—32no—
177C2×#102C2×D311yescharacteristic
178C2×C4×C8—32no—
179C2×C4×C8C2×D311yescharacteristic
180C24×C3—21no—
181C2×C3×Q8—21no—
182C2×C3×Q8—22no—
183C2×C3×Q8C2×Q814yescomplemented
184C2×C3×Q8C2412yes—
185C2×C3×D4—21no—
186C2×C3×D4—22no—
187C2×C3×D4C2×Q814yes—
188C2×C3×D4C2412yes—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—24no—
193C22×C3×C4—44no—
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2411yescharacteristic
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3—28no—
200C22×Dic3—48no—
201C22×Dic3C2×D412yes—
202C22×Dic3C2×D412yes—
203C2×Dic6—24no—
204C2×Dic6—28no—
205C2×Dic6—24no—
206C2×Dic6—28no—
207C2×Dic6—44no—
208C2×Dic6—48no—
209C2×Dic6—432no—
210C2×Dic6—84no—
211C2×Dic6—82no—
212C2×Dic6SD1618yesdirect product factor
213C2×Dic6C2×D414yes—
214C3×SD16C2×Q8116yescomplemented
215C2×C3×C8—22no—
216C2×C3×C8C2×Q814yescomplemented
217C2×C3×C8C2412yes—
218C2×C3×C8C2412yes—
219C3×C4⋊C4—24no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4C2412yes—
223C3×C4⋊C4C2412yes—
224C3×C22⋊C4—24no—
225C3×C42—24no—
226C3×C42C2×D412yes—
227C3×C42C2412yes—
228#217—28no—
229C4⋊Dic3—24no—
230C4⋊Dic3—22no—
231C4⋊Dic3—22no—
232C4⋊Dic3—24no—
233C4⋊Dic3—28no—
234C4⋊Dic3C2×D414yes—
235C4⋊Dic3C2414yes—
236C4⋊Dic3C2414yes—
237Dic3⋊C4—28no—
238Dic3⋊C4—28no—
239Dic3⋊C4C2×D418yes—
240C4×Dic3—28no—
241C4×Dic3C2×D414yes—
242C4×Dic3C2414yes—
243C2×C3⋊C8—24no—
244C2×C3⋊C8C2414yes—
245C22×Q8—31no—
246C22×Q8—62no—
247C22×Q8—62no—
248C22×Q8—68no—
249C22×Q8—61no—
250C22×Q8—61no—
251C22×Q8—128no—
252C22×Q8C22×D311yescharacteristic
253C22×D4C22×D311yescharacteristic
254C23×C4—21no—
255C23×C4—62no—
256C2×SD16Dic618yesdirect product factor
257C2×SD16C22×D314yes—
258C22×C8—32no—
259C22×C8C22×D311yescharacteristic
260C22×C8C22×D311yescharacteristic
261C4⋊Q8—38no—
262C4⋊Q8—38no—
263C4⋊Q8—34no—
264C4⋊Q8—34no—
265C4⋊Q8—68no—
266C4⋊Q8—64no—
267C4⋊Q8—64no—
268C4⋊Q8—616no—
269C4⋊Q8—62no—
270C4⋊Q8—62no—
271C22⋊Q8—616no—
272C22⋊Q8—68no—
273C4×Q8—22no—
274C4×Q8—34no—
275C4×Q8—34no—
276C4×Q8—38no—
277C4×Q8—68no—
278C4×Q8—64no—
279C4×Q8C22×D314yes—
280C4×D4—22no—
281C4×D4—38no—
282C4×D4—64no—
283C4×D4C22×D314yes—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—32no—
286C2×C4⋊C4—31no—
287C2×C4⋊C4—32no—
288C2×C4⋊C4—32no—
289C2×C4⋊C4—62no—
290C2×C4⋊C4—62no—
291C2×C4⋊C4—62no—
292C2×C4⋊C4—61no—
293C2×C4⋊C4—61no—
294C2×C4⋊C4C22×D311yescharacteristic
295C2×C4⋊C4C22×D311yescharacteristic
296C2×C22⋊C4—21no—
297C2×C22⋊C4—62no—
298C2×C42—21no—
299C2×C42—32no—
300C2×C42—62no—
301C2×C42C22×D311yescharacteristic
302#106—38no—
303#106—38no—
304#106—38no—
305#106—38no—
306#106C22×D314yes—
307C4⋊C8—38no—
308C4⋊C8—34no—
309#103—38no—
310#103C22×D314yes—
311#102—38no—
312#102C22×D314yes—
313C4×C8—38no—
314C4×C8C22×D314yes—
315C23×C3—24no—
316C23×C3—22no—
317C23×C3—42no—
318C23×C3—42no—
319C23×C3C22×D411yescharacteristic
320C3×Q8—24no—
321C3×Q8—22no—
322C3×Q8C22×Q814yescomplemented
323C3×D4—24no—
324C3×D4—22no—
325C3×D4C22×Q814yes—
326C2×C3×C4—24no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—48no—
334C2×C3×C4C2×SD1614yes—
335C2×C3×C4C22×D412yes—
336C2×C3×C4C22×D411yescharacteristic
337C2×C3×C4C22×D412yes—
338C2×C3×C4C22×Q812yes—
339C2×C3×C42−1+412yes—
340C2×C3×C4C2511yescharacteristic
341C2×Dic3—24no—
342C2×Dic3—24no—
343C2×Dic3—24no—
344C2×Dic3—44no—
345C2×Dic3—44no—
346C2×Dic3—416no—
347C2×Dic3C2×SD1618yes—
348C2×Dic3C22×D414yes—
349C2×Dic3C22×D414yes—
350Dic6—48no—
351Dic6—416no—
352Dic6—88no—
353Dic6—84no—
354Dic6C2×SD16116yesdirect product factor
355C3×C8—22no—
356C3×C8C22×Q814yescomplemented
357C3⋊C8—24no—
358C24—21no—
359C2×Q8—21no—
360C2×Q8—22no—
361C2×Q8—38no—
362C2×Q8—34no—
363C2×Q8—68no—
364C2×Q8—68no—
365C2×Q8—64no—
366C2×Q8—64no—
367C2×Q8—128no—
368C2×Q8—1232no—
369C2×Q8—124no—
370C2×Q8—242no—
371C2×Q8—244no—
372C2×Q8C2×Dic614yescomplemented
373C2×Q8C23×D312yes—
374C2×D4—21no—
375C2×D4—22no—
376C2×D4C2×Dic614yes—
377C2×D4C23×D312yes—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—24no—
382C22×C4—32no—
383C22×C4—32no—
384C22×C4—44no—
385C22×C4—68no—
386C22×C4—62no—
387C22×C4—62no—
388C22×C4—128no—
389C22×C4D3×D411yescharacteristic
390C22×C4D3×D411yescharacteristic
391C22×C4C23×D311yescharacteristic
392SD16C2×Dic6116yesdirect product factor
393C2×C8—22no—
394C2×C8—34no—
395C2×C8—64no—
396C2×C8C2×Dic614yescomplemented
397C2×C8C23×D312yes—
398C2×C8C23×D312yes—
399C4⋊C4—24no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—38no—
403C4⋊C4—34no—
404C4⋊C4—34no—
405C4⋊C4—34no—
406C4⋊C4—68no—
407C4⋊C4—68no—
408C4⋊C4—68no—
409C4⋊C4—62no—
410C4⋊C4—62no—
411C4⋊C4—64no—
412C4⋊C4—64no—
413C4⋊C4C23×D312yes—
414C4⋊C4C23×D312yes—
415C22⋊C4—24no—
416C22⋊C4—68no—
417C42—24no—
418C42—34no—
419C42—34no—
420C42—68no—
421C42D3×D412yes—
422C42C23×D312yes—
423C22×C3—24no—
424C22×C3—48no—
425C22×C3—42no—
426C22×C3—42no—
427C22×C3D4×Q812yes—
428C22×C3C22×SD1612yes—
429C22×C3#57712yes—
430C22×C3C23×D411yescharacteristic
431C3×C4—24no—
432C3×C4—22no—
433C3×C4—42no—
434C3×C4—42no—
435C3×C4D4×Q812yes—
436C3×C4C22×SD1614yes—
437C3×C4C23×Q811yescharacteristic
438C3×C4C2×2−1+411yescharacteristic
439Dic3—24no—
440Dic3—44no—
441Dic3—44no—
442Dic3C22×SD1618yes—
443C23—22no—
444C23—24no—
445C23—42no—
446C23—42no—
447C23C2×D3×D411yescharacteristic
448Q8—22no—
449Q8—24no—
450Q8—316no—
451Q8—1216no—
452Q8—128no—
453Q8—244no—
454Q8—248no—
455Q8C22×Dic614yescomplemented
456D4—22no—
457D4—24no—
458D4C22×Dic614yes—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—34no—
465C2×C4—38no—
466C2×C4—34no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—48no—
470C2×C4—64no—
471C2×C4—64no—
472C2×C4—64no—
473C2×C4—1216no—
474C2×C4—124no—
475C2×C4—124no—
476C2×C4D3×SD1614yes—
477C2×C4C22×Dic612yes—
478C2×C4C2×D3×D412yes—
479C2×C4C2×D3×D412yes—
480C2×C4C2×D3×D411yescharacteristic
481C2×C4#101012yes—
482C2×C4C24×D311yescharacteristic
483C8—22no—
484C8—64no—
485C8C22×Dic614yescomplemented
486C2×C3—44no—
487C2×C3Q8×SD1614yescomplemented
488C2×C3C2×D4×Q811yescharacteristic
489C2×C3C23×SD1611yescharacteristic
490C2×C3C2×#57711yescharacteristic
491C22—24no—
492C22—48no—
493C22—42no—
494C22—42no—
495C22D4×Dic612yes—
496C22C2×D3×SD1612yes—
497C22#577012yes—
498C22C22×D3×D411yescharacteristic
499C4—22no—
500C4—24no—
501C4—38no—
502C4—42no—
503C4—42no—
504C4—64no—
505C4—124no—
506C4—124no—
507C4D4×Dic612yes—
508C4C2×D3×SD1614yes—
509C4C23×Dic611yescharacteristic
510C4C2×#101011yescharacteristic
511C3C2×Q8×SD1611yescharacteristic, complemented, Sylow
512C2—44no—
513C2SD16×Dic614yesdirect product factor
514C2C2×D4×Dic611yescharacteristic
515C2C22×D3×SD1611yescharacteristic
516C2C2×#577011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×C4×SD16C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenterC2×C4×SD16D3
hypocenterC3C2×Q8×SD16
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×SD16

Complements

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

NH ≅ G/Ntype
1. C2×Q8×Dic6C2semidirect
5. C2×#5020C2semidirect
7. SD16×Dic6C2direct (N × H)
8. C2×#4855C2semidirect
9. C2×#4791C2semidirect
11. C2×C8×Dic6C2semidirect
12. C2×#4680C2semidirect
22. Q8×Dic6C22semidirect
46. #5020C22semidirect
51. #4855C22semidirect
52. #4791C22semidirect
54. #4680C22semidirect
55. C8×Dic6C22semidirect
79. C2×C3×SD16Q8semidirect
183. C2×C3×Q8C2×Q8semidirect
212. C2×Dic6SD16direct (N × H)
214. C3×SD16C2×Q8semidirect
216. C2×C3×C8C2×Q8semidirect
256. C2×SD16Dic6direct (N × H)
322. C3×Q8C22×Q8semidirect
354. Dic6C2×SD16direct (N × H)
356. C3×C8C22×Q8semidirect
372. C2×Q8C2×Dic6semidirect
392. SD16C2×Dic6direct (N × H)
396. C2×C8C2×Dic6semidirect
455. Q8C22×Dic6semidirect
485. C8C22×Dic6semidirect
487. C2×C3Q8×SD16semidirect
511. C3C2×Q8×SD16semidirect
513. C2SD16×Dic6direct (N × H)

Maximal subgroups

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

subgroupindex
C2×Q8×Dic62
C2×D4×Dic62
C2×C3×C4×SD162
C2×Dic3×SD162
C2×#50202
C2×#50082
SD16×Dic62
C2×#48552
C2×#47912
C2×#47612
C2×C8×Dic62
C2×#46802
C2×Q8×SD163