Finite Group Databasebeta

← C2×#81039 #11675608

Subgroups — C2×#81039 #11675608

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×#5727C211yesmaximal, characteristic
2C2×Q8×Dic6C211yesmaximal, characteristic, complemented
3C2×C3×#475C211yesmaximal, characteristic, complemented
4C2×#5080C211yesmaximal, characteristic, complemented
5C2×C4×#215C211yesmaximal, characteristic
6#81039C2116yesmaximal, direct product factor
7C2×#4870C212yesmaximal
8C2×#4855C212yesmaximal, complemented
9C2×#4842C211yesmaximal, characteristic
10C2×#4830C211yesmaximal, characteristic, complemented
11C2×#4791C212yesmaximal, complemented
12C2×#4788C212yesmaximal
13C2×Q8×SD16—31nomaximal, Sylow
14C22×D3×Q8—21no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×C4×Q8C2211yescharacteristic
17C2×#946—21no—
18C2×Q8×Dic3C2212yes—
19C22×#215C2211yescharacteristic
20#5727—24no—
21#5727C2218yes—
22Q8×Dic6—24no—
23Q8×Dic6C2218yescomplemented
24C2×#896—22no—
25C2×#893C2212yes—
26C2×#888—22no—
27C2×#888C2212yes—
28C2×#888C2212yes—
29C2×C4×D12C2211yescharacteristic
30C2×#869—21no—
31C2×#869C2211yescharacteristic
32C2×C4×Dic6—21no—
33C2×C4×Dic6C2211yescharacteristic
34C3×#475C2218yescomplemented
35C2×C3×#106C2212yes—
36C2×C3×C4⋊C8C2211yescharacteristic
37C2×C3×#103C2212yes—
38C2×#835C2211yescharacteristic
39C2×#821C2212yes—
40C2×#817C2212yes—
41C2×Dic3⋊C8C2212yes—
42C2×C8×Dic3C2212yes—
43#5080C2218yescomplemented
44C4×#215C2218yes—
45C2×#809C2211yescharacteristic
46C2×#808C2211yescharacteristic
47C2×#808C2212yes—
48C2×#808C2212yes—
49C2×C4×C3⋊C8C2211yescharacteristic
50#4870C22116yes—
51#4855C22116yescomplemented
52#4842C2218yes—
53#4830C2218yescomplemented
54#4791C22116yescomplemented
55#4788C22116yes—
56C2×Q82—31no—
57C2×D4×Q8—31no—
58Q8×SD16—316no—
59C2×#498—32no—
60C2×#498—31no—
61C2×#475—32no—
62C2×#475D311yescharacteristic, complemented
63C2×#476—32no—
64C2×#476—31no—
65C2×C8×Q8—31no—
66C2×C4×SD16—32no—
67C2×C4×SD16—31no—
68C22×C3×Q8—21no—
69C22×C3×Q8C2311yescharacteristic
70C2×D3×Q8—28no—
71C2×D3×Q8—48no—
72C22×D12C2311yescharacteristic
73C22×C4×D3—21no—
74C22×C4×D3—22no—
75C22×Dic6—22no—
76C22×Dic6—21no—
77C22×Dic6—22no—
78C22×Dic6D411yescharacteristic
79C22×C3×C8C2312yes—
80C3×C4⋊Q8—22no—
81C3×C4⋊Q8C2314yes—
82C3×C4×Q8—22no—
83C3×C4×Q8C2314yes—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4C2312yes—
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C42—21no—
89C2×C3×C42C2311yescharacteristic
90#946—28no—
91Q8×Dic3—24no—
92Q8×Dic3C2318yes—
93C2×#215Q818yescomplemented
94C2×#215C2314yes—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—22no—
97C2×C4⋊Dic3—21no—
98C2×C4⋊Dic3—22no—
99C2×C4⋊Dic3D411yescharacteristic
100C2×C4⋊Dic3C2312yes—
101C2×C4⋊Dic3C2311yescharacteristic
102C2×Dic3⋊C4—22no—
103C2×Dic3⋊C4—22no—
104C2×Dic3⋊C4D412yes—
105C2×C4×Dic3—22no—
106C2×C4×Dic3C2312yes—
107C22×C3⋊C8C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109#896—216no—
110#893—24no—
111#893C2318yes—
112#888—216no—
113#888—24no—
114#888—24no—
115#888C2318yes—
116#888C2318yes—
117C4×D12—22no—
118C4×D12C2314yes—
119#869—22no—
120#869—28no—
121#869C2314yes—
122C4×Dic6—28no—
123C4×Dic6D414yes—
124C4×Dic6C2314yes—
125C3×#106C2318yes—
126C3×C4⋊C8C2314yes—
127C3×#103C2318yes—
128#835C2314yes—
129#821C2318yes—
130#817C2318yes—
131Dic3⋊C8C2318yes—
132C8×Dic3C2318yes—
133#809C2314yes—
134#808C2318yes—
135#808C2314yes—
136#808C2318yes—
137C4×C3⋊C8C2314yes—
138C23×Q8—61no—
139C22×SD16—31no—
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—61no—
148C2×C4⋊Q8—62no—
149C2×C4⋊Q8C2×D311yescharacteristic
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×D4—31no—
158#498—38no—
159#498—316no—
160#476—316no—
161#476—38no—
162#475—316no—
163#475C2×D318yescomplemented
164C8×Q8—38no—
165C4×SD16—38no—
166C4×SD16—316no—
167C2×#106—32no—
168C2×#106—32no—
169C2×#106—31no—
170C2×#106—32no—
171C2×#106C2×D312yes—
172C2×C4⋊C8—32no—
173C2×C4⋊C8C2×D311yescharacteristic
174C2×#103—31no—
175C2×#103C2×D312yes—
176C2×#102—32no—
177C2×#102—31no—
178C2×C4×C8—32no—
179C2×C4×C8—31no—
180C23×D3—21no—
181C2×C3×Q8—21no—
182C2×C3×Q8—24no—
183C2×C3×Q8—22no—
184C2×C3×Q8—44no—
185C2×C3×Q8C2×Q814yescomplemented
186C2×C3×Q8C2412yes—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—22no—
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2411yescharacteristic
193C22×Dic3—22no—
194C22×Dic3—21no—
195C22×Dic3C2×D412yes—
196C22×Dic3C2×D412yes—
197D3×Q8—432no—
198C2×D12—21no—
199C2×D12—22no—
200C2×D12C2×Q814yes—
201C2×D12C2412yes—
202C2×C4×D3—24no—
203C2×C4×D3—28no—
204C2×C4×D3—44no—
205C2×C4×D3—48no—
206C2×Dic6—28no—
207C2×Dic6—24no—
208C2×Dic6—28no—
209C2×Dic6—48no—
210C2×Dic6—84no—
211C2×Dic6—82no—
212C2×Dic6SD1618yescomplemented
213C2×Dic6C2×D414yes—
214C2×C3×C8—24no—
215C2×C3×C8C2414yes—
216C3×C4⋊C4—24no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—22no—
219C3×C4⋊C4—24no—
220C3×C4⋊C4C2414yes—
221C3×C4⋊C4C2412yes—
222C3×C42—24no—
223C3×C42C2×D412yes—
224C3×C42C2412yes—
225#215C2×Q8116yescomplemented
226(C2×D3)⋊C4—24no—
227(C2×D3)⋊C4—28no—
228C4⋊Dic3—24no—
229C4⋊Dic3—22no—
230C4⋊Dic3—21no—
231C4⋊Dic3—28no—
232C4⋊Dic3C2×D414yes—
233C4⋊Dic3C2414yes—
234C4⋊Dic3C2412yes—
235Dic3⋊C4—28no—
236Dic3⋊C4—28no—
237Dic3⋊C4C2×D418yes—
238C4×Dic3—28no—
239C4×Dic3C2×D414yes—
240C4×Dic3C2414yes—
241C2×C3⋊C8—22no—
242C2×C3⋊C8C2×Q814yescomplemented
243C2×C3⋊C8C2412yes—
244C2×C3⋊C8C2412yes—
245C22×Q8—21no—
246C22×Q8—31no—
247C22×Q8—62no—
248C22×Q8—62no—
249C22×Q8—68no—
250C22×Q8—61no—
251C22×Q8—128no—
252C22×Q8C22×D311yescharacteristic
253C22×D4—31no—
254C23×C4—62no—
255C23×C4—61no—
256C2×SD16—38no—
257C2×SD16—34no—
258C22×C8—31no—
259C22×C8—31no—
260C22×C8C22×D312yes—
261C4⋊Q8—22no—
262C4⋊Q8—38no—
263C4⋊Q8—38no—
264C4⋊Q8—34no—
265C4⋊Q8—68no—
266C4⋊Q8—64no—
267C4⋊Q8—64no—
268C4⋊Q8—616no—
269C4⋊Q8—62no—
270C4⋊Q8C22×D314yes—
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—38no—
281C4×D4—34no—
282C4×D4—64no—
283C4×D4—62no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—32no—
287C2×C4⋊C4—31no—
288C2×C4⋊C4—31no—
289C2×C4⋊C4—32no—
290C2×C4⋊C4—62no—
291C2×C4⋊C4—62no—
292C2×C4⋊C4—62no—
293C2×C4⋊C4—61no—
294C2×C4⋊C4C22×D312yes—
295C2×C4⋊C4C22×D311yescharacteristic
296C2×C22⋊C4—62no—
297C2×C22⋊C4—61no—
298C2×C42—21no—
299C2×C42—32no—
300C2×C42—62no—
301C2×C42C22×D311yescharacteristic
302#106—38no—
303#106—38no—
304#106—34no—
305#106—38no—
306#106C22×D318yes—
307C4⋊C8—38no—
308C4⋊C8C22×D314yes—
309#103—34no—
310#103C22×D318yes—
311#102—38no—
312#102—34no—
313C4×C8—38no—
314C4×C8—34no—
315C23×C3C22×D411yescharacteristic
316C22×D3—24no—
317C22×D3—22no—
318C22×D3—42no—
319C22×D3—42no—
320C3×Q8—24no—
321C3×Q8—22no—
322C3×Q8—48no—
323C3×Q8C22×Q814yescomplemented
324C2×C3×C4—24no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—24no—
329C2×C3×C4—44no—
330C2×C3×C4—42no—
331C2×C3×C4C2×SD1614yes—
332C2×C3×C4C22×D412yes—
333C2×C3×C4C22×D411yescharacteristic
334C2×C3×C4C22×D412yes—
335C2×C3×C4C22×Q812yes—
336C2×C3×C42−1+412yes—
337C2×C3×C4C2511yescharacteristic
338C2×Dic3—24no—
339C2×Dic3—24no—
340C2×Dic3—22no—
341C2×Dic3—42no—
342C2×Dic3—44no—
343C2×Dic3C2×SD1618yes—
344C2×Dic3C22×D414yes—
345C2×Dic3C22×D414yes—
346D12—24no—
347D12—22no—
348D12C22×Q814yes—
349C4×D3—48no—
350C4×D3—416no—
351Dic6—416no—
352Dic6—88no—
353Dic6—84no—
354Dic6C2×SD16116yescomplemented
355C3×C8—24no—
356C3⋊C8—22no—
357C3⋊C8C22×Q814yescomplemented
358C24—61no—
359C2×Q8—21no—
360C2×Q8—24no—
361C2×Q8—22no—
362C2×Q8—38no—
363C2×Q8—34no—
364C2×Q8—44no—
365C2×Q8—68no—
366C2×Q8—68no—
367C2×Q8—64no—
368C2×Q8—128no—
369C2×Q8—1232no—
370C2×Q8—242no—
371C2×Q8—244no—
372C2×Q8D3×Q814yescomplemented
373C2×Q8C23×D312yes—
374C2×D4—34no—
375C2×D4—32no—
376C2×D4—61no—
377C2×D4—62no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—32no—
382C22×C4—32no—
383C22×C4—68no—
384C22×C4—62no—
385C22×C4—61no—
386C22×C4—64no—
387C22×C4—128no—
388C22×C4—124no—
389C22×C4D3×D411yescharacteristic
390C22×C4D3×D411yescharacteristic
391C22×C4C23×D311yescharacteristic
392SD16—316no—
393C2×C8—24no—
394C2×C8—34no—
395C2×C8—32no—
396C2×C8—32no—
397C2×C8—62no—
398C2×C8C23×D314yes—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—22no—
402C4⋊C4—21no—
403C4⋊C4—38no—
404C4⋊C4—34no—
405C4⋊C4—34no—
406C4⋊C4—32no—
407C4⋊C4—68no—
408C4⋊C4—68no—
409C4⋊C4—68no—
410C4⋊C4—61no—
411C4⋊C4—62no—
412C4⋊C4—64no—
413C4⋊C4C23×D312yes—
414C4⋊C4C23×D314yes—
415C22⋊C4—68no—
416C22⋊C4—64no—
417C42—24no—
418C42—34no—
419C42—34no—
420C42—68no—
421C42D3×D412yes—
422C42C23×D312yes—
423C22×C3D4×Q812yes—
424C22×C3C22×SD1612yes—
425C22×C3#57712yes—
426C22×C3C23×D411yescharacteristic
427C2×D3—24no—
428C2×D3—48no—
429C2×D3—42no—
430C2×D3—42no—
431C3×C4—24no—
432C3×C4—22no—
433C3×C4—44no—
434C3×C4—42no—
435C3×C4D4×Q812yes—
436C3×C4C22×SD1614yes—
437C3×C4C23×Q811yescharacteristic
438C3×C4C2×2−1+411yescharacteristic
439Dic3—24no—
440Dic3—42no—
441Dic3—44no—
442Dic3C22×SD1618yes—
443C23—64no—
444C23—62no—
445C23—122no—
446C23—122no—
447C23C2×D3×D411yescharacteristic
448Q8—24no—
449Q8—22no—
450Q8—316no—
451Q8—48no—
452Q8—1216no—
453Q8—244no—
454Q8—248no—
455Q8C2×D3×Q814yescomplemented
456D4—34no—
457D4—64no—
458D4—62no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—34no—
465C2×C4—38no—
466C2×C4—34no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—64no—
470C2×C4—64no—
471C2×C4—62no—
472C2×C4—1216no—
473C2×C4—122no—
474C2×C4—124no—
475C2×C4—128no—
476C2×C4D3×SD1614yes—
477C2×C4C2×D3×D412yes—
478C2×C4C2×D3×D411yescharacteristic
479C2×C4C2×D3×D412yes—
480C2×C4C2×D3×Q812yes—
481C2×C4#101012yes—
482C2×C4C24×D311yescharacteristic
483C8—24no—
484C8—34no—
485C8—62no—
486C2×C3Q8×SD1614yescomplemented
487C2×C3C2×D4×Q811yescharacteristic
488C2×C3C23×SD1611yescharacteristic
489C2×C3C2×#57711yescharacteristic
490D3—44no—
491C22—64no—
492C22—122no—
493C22—128no—
494C22—122no—
495C22#563212yes—
496C22C2×D3×SD1612yes—
497C22#577812yes—
498C22C22×D3×D411yescharacteristic
499C4—22no—
500C4—24no—
501C4—38no—
502C4—44no—
503C4—42no—
504C4—64no—
505C4—122no—
506C4—124no—
507C4#563212yes—
508C4C2×D3×SD1614yes—
509C4C22×D3×Q811yescharacteristic
510C4C2×#101011yescharacteristic
511C3C2×Q8×SD1611yescharacteristic, complemented, Sylow
512C2—124no—
513C2#8103914yesdirect product factor
514C2C2×#563211yescharacteristic
515C2C22×D3×SD1611yescharacteristic
516C2C2×#577811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#475C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenterC2×#475D3
hypocenterC3C2×Q8×SD16
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#475

Complements

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

NH ≅ G/Ntype
2. C2×Q8×Dic6C2semidirect
3. C2×C3×#475C2semidirect
4. C2×#5080C2semidirect
6. #81039C2direct (N × H)
8. C2×#4855C2semidirect
10. C2×#4830C2semidirect
11. C2×#4791C2semidirect
23. Q8×Dic6C22semidirect
34. C3×#475C22semidirect
43. #5080C22semidirect
51. #4855C22semidirect
53. #4830C22semidirect
54. #4791C22semidirect
62. C2×#475D3semidirect
93. C2×#215Q8semidirect
163. #475C2×D3semidirect
185. C2×C3×Q8C2×Q8semidirect
212. C2×Dic6SD16semidirect
225. #215C2×Q8semidirect
242. C2×C3⋊C8C2×Q8semidirect
323. C3×Q8C22×Q8semidirect
354. Dic6C2×SD16semidirect
357. C3⋊C8C22×Q8semidirect
372. C2×Q8D3×Q8semidirect
455. Q8C2×D3×Q8semidirect
486. C2×C3Q8×SD16semidirect
511. C3C2×Q8×SD16semidirect
513. C2#81039direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#57272
C2×Q8×Dic62
C2×C3×#4752
C2×#50802
C2×C4×#2152
#810392
C2×#48702
C2×#48552
C2×#48422
C2×#48302
C2×#47912
C2×#47882
C2×Q8×SD163