Finite Group Databasebeta

← C2×#81801 #11664682

Subgroups — C2×#81801 #11664682

Order 768.

3150 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#918C212yesmaximal, complemented
2C22×#919C211yesmaximal, characteristic
3C22×D3×Q16C211yesmaximal, characteristic, complemented
4C2×#5724C212yesmaximal, complemented
5C2×C3×#494C211yesmaximal, characteristic, complemented
6C2×#5190C212yesmaximal, complemented
7C2×#5191C211yesmaximal, characteristic, complemented
8C2×#5183C211yesmaximal, characteristic, complemented
9C2×#5086C212yesmaximal, complemented
10#81801C2116yesmaximal, direct product factor
11C2×#4696C211yesmaximal, characteristic, complemented
12C2×#4689C211yesmaximal, characteristic, complemented
13C2×#3156—31nomaximal, Sylow
14C22×#239C2212yes—
15C22×D3×Q8C2212yes—
16C22×C3×Q16C2211yescharacteristic
17C22×C3×Q16C2211yescharacteristic
18C22×C3×Q16C2212yes—
19C2×C3×C4⋊Q8C2212yes—
20C2×#947C2212yes—
21C2×#944C2212yes—
22C22×#216C2212yes—
23C22×#216C2212yes—
24C22×#215C2212yes—
25C22×#215C2212yes—
26C2×#919—28no—
27C2×#919C2218yes—
28C2×#918—216no—
29C2×#918C22116yescomplemented
30C2×D3×Q16—28no—
31C2×D3×Q16C2218yescomplemented
32C22×Dic12C2211yescharacteristic
33C22×D24C2211yescharacteristic
34C22×#204C2212yes—
35C22×ΓL(1,𝔽25)C2211yescharacteristic
36C22×D3×C8C2211yescharacteristic
37#5724—28no—
38#5724C22116yescomplemented
39C2×#896—24no—
40C2×#894—24no—
41C2×#875C2212yes—
42C2×#872C2211yescharacteristic
43C3×#494C2218yescomplemented
44C2×C3×C4×C8C2211yescharacteristic
45#5191C2218yescomplemented
46#5190C22116yescomplemented
47#5183C2218yescomplemented
48C2×#815C2211yescharacteristic
49C2×C8×Dic3C2211yescharacteristic
50#5086C22116yescomplemented
51C2×#803C2211yescharacteristic
52#4696C2218yescomplemented
53#4689C2218yescomplemented
54C22×#137—32no—
55C22×#135—31no—
56C23×Q16—31no—
57C2×#536—32no—
58#3156—316no—
59C2×#497—32no—
60C2×#497—32no—
61C2×#495—31no—
62C2×#495—31no—
63C2×#494—31no—
64C2×#494D311yescharacteristic, complemented
65C2×#405—31no—
66C22×C3×Q8C2312yes—
67C22×C3×Q8C2312yes—
68C2×#239—24no—
69C2×#239—28no—
70C2×#239—28no—
71C2×#239C2318yes—
72C2×D3×Q8—24no—
73C2×D3×Q8—28no—
74C2×D3×Q8—28no—
75C2×D3×Q8C2318yes—
76C22×D12—22no—
77C22×D12C2312yes—
78C22×C4×D3—22no—
79C22×C4×D3—22no—
80C22×C4×D3C2311yescharacteristic
81C22×Dic6—22no—
82C22×Dic6C2312yes—
83C2×C3×Q16—24no—
84C2×C3×Q16—28no—
85C2×C3×Q16—24no—
86C2×C3×Q16C2314yes—
87C2×C3×Q16C2318yes—
88C2×C3×Q16C2314yes—
89C22×C3×C8C2311yescharacteristic
90C22×C3×C8C2311yescharacteristic
91C3×C4⋊Q8—24no—
92C3×C4⋊Q8C2318yes—
93C2×C3×C4⋊C4—24no—
94C2×C3×C42C2311yescharacteristic
95#947—24no—
96#947C2318yes—
97#944—24no—
98#944C2318yes—
99C2×#216—28no—
100C2×#216—28no—
101C2×#216C2318yes—
102C2×#216C2318yes—
103C2×#215—28no—
104C2×#215—28no—
105C2×#215C2318yes—
106C2×#215C2318yes—
107C2×(C2×D3)⋊C4—24no—
108C2×(C2×D3)⋊C4—24no—
109C2×(C2×D3)⋊C4D412yes—
110C2×Dic3⋊C4—24no—
111C2×Dic3⋊C4D412yes—
112C2×C4×Dic3C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115#919—232no—
116#918—264no—
117D3×Q16—232no—
118C2×Dic12—24no—
119C2×Dic12C2314yes—
120C2×D24—24no—
121C2×D24C2314yes—
122C2×#204—28no—
123C2×#204C2318yes—
124C2×ΓL(1,𝔽25)D418yes—
125C2×ΓL(1,𝔽25)C2314yes—
126C2×D3×C8D418yes—
127C2×D3×C8C2314yes—
128#896—232no—
129#894—232no—
130#875—24no—
131#875C2318yes—
132#872D414yes—
133#872C2314yes—
134C3×C4×C8C2314yes—
135#815C2314yes—
136C8×Dic3C2314yes—
137#803C2314yes—
138C22×#41—32no—
139C23×Q8—32no—
140C2×#137—316no—
141C2×#137—616no—
142C2×#135—38no—
143C2×#135—68no—
144C22×Q16—32no—
145C22×Q16—32no—
146C22×Q16—38no—
147C22×Q16—31no—
148C22×Q16—68no—
149C22×Q16C2×D311yescharacteristic
150C22×Q16C2×D311yescharacteristic
151C22×Q16C2×D312yes—
152C22×SD16—32no—
153C22×SD16—32no—
154C22×SD16—32no—
155C22×D8—31no—
156C22×M16—31no—
157C23×C8—31no—
158#536—316no—
159#536—68no—
160C2×C4⋊Q8—32no—
161C2×C4⋊Q8C2×D312yes—
162C2×#124—32no—
163C2×#124—32no—
164C2×#123—64no—
165C2×C22⋊Q8—64no—
166C2×#117—31no—
167#497—316no—
168#497—316no—
169#495—38no—
170#495—38no—
171#494—38no—
172#494C2×D318yescomplemented
173#405—38no—
174C2×#97—31no—
175C2×#97—31no—
176C2×C4×C8—31no—
177C2×C4×C8C2×D311yescharacteristic
178C23×D3—22no—
179C23×D3C2×D411yescharacteristic
180C2×C3×Q8—22no—
181C2×C3×Q8—22no—
182C2×C3×Q8—24no—
183C2×C3×Q8—24no—
184C2×C3×Q8—24no—
185C2×C3×Q8—24no—
186C2×C3×Q8C2414yes—
187C2×C3×Q8C2414yes—
188C22×C3×C4—22no—
189C22×C3×C4—22no—
190C22×C3×C4C2×D412yes—
191C22×C3×C4C2411yescharacteristic
192C22×Dic3—22no—
193C22×Dic3C2×D412yes—
194C22×Dic3C2×D411yescharacteristic
195#239—216no—
196#239—216no—
197#239—416no—
198D3×Q8—216no—
199D3×Q8—216no—
200D3×Q8—416no—
201C2×D12—24no—
202C2×D12—24no—
203C2×D12—28no—
204C2×D12—22no—
205C2×D12—48no—
206C2×D12C2414yes—
207C2×C4×D3—28no—
208C2×C4×D3—28no—
209C2×C4×D3—48no—
210C2×C4×D3—48no—
211C2×C4×D3C2×D412yes—
212C2×C4×D3C2×D414yes—
213C2×C4×D3C2×D414yes—
214C2×C4×D3C2412yes—
215C2×Dic6—24no—
216C2×Dic6—24no—
217C2×Dic6—22no—
218C2×Dic6—28no—
219C2×Dic6—48no—
220C2×Dic6C2414yes—
221C3×Q16—28no—
222C3×Q16—216no—
223C3×Q16—28no—
224C2×C3×C8C2×D414yes—
225C2×C3×C8C2×D414yes—
226C2×C3×C8C2412yes—
227C2×C3×C8C2412yes—
228C3×C4⋊C4—216no—
229C3×C42C2×D412yes—
230C3×C42C2412yes—
231#216—216no—
232#216—216no—
233#215—216no—
234#215—216no—
235(C2×D3)⋊C4—216no—
236(C2×D3)⋊C4—216no—
237(C2×D3)⋊C4C2×D418yes—
238Dic3⋊C4—216no—
239Dic3⋊C4C2×D418yes—
240C4×Dic3C2×D412yes—
241C4×Dic3C2412yes—
242C2×C3⋊C8C2×D414yes—
243C2×C3⋊C8C2×D414yes—
244C2×C3⋊C8C2412yes—
245C2×C3⋊C8C2412yes—
246Dic12—28no—
247D24—28no—
248#204—216no—
249ΓL(1,𝔽25)C2×D4116yes—
250D3×C8C2×D4116yes—
251C2×#41—38no—
252C2×#41—64no—
253C2×#41—68no—
254C2×#41—68no—
255C22×Q8—38no—
256C22×Q8—32no—
257C22×Q8—62no—
258C22×Q8—64no—
259C22×Q8—68no—
260C22×Q8—68no—
261C22×Q8C22×D312yes—
262C22×Q8C22×D312yes—
263C22×D4—32no—
264C22×D4—62no—
265C23×C4—31no—
266C23×C4—62no—
267C23×C4—62no—
268#137—664no—
269#135—632no—
270C2×Q16—24no—
271C2×Q16—24no—
272C2×Q16—28no—
273C2×Q16—38no—
274C2×Q16—38no—
275C2×Q16—34no—
276C2×Q16—632no—
277C2×Q16—68no—
278C2×Q16—68no—
279C2×Q16—64no—
280C2×Q16C22×D314yes—
281C2×Q16C22×D314yes—
282C2×Q16C22×D318yes—
283C2×SD16—38no—
284C2×SD16—38no—
285C2×SD16—38no—
286C2×SD16—68no—
287C2×SD16—68no—
288C2×SD16—68no—
289C2×D8—34no—
290C2×D8—64no—
291C2×M16—38no—
292C2×M16—34no—
293C22×C8—31no—
294C22×C8—31no—
295C22×C8—38no—
296C22×C8—34no—
297C22×C8C22×D311yescharacteristic
298C22×C8C22×D311yescharacteristic
299C4⋊Q8—24no—
300C4⋊Q8—38no—
301C4⋊Q8—64no—
302C4⋊Q8C22×D318yes—
303#124—38no—
304#124—38no—
305#124—64no—
306#124—64no—
307#123—632no—
308C22⋊Q8—632no—
309#117—34no—
310#117—34no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—32no—
313C2×C4⋊C4—64no—
314C2×C22⋊C4—32no—
315C2×C22⋊C4—64no—
316C2×C22⋊C4—64no—
317C2×C42—31no—
318C2×C42C22×D311yescharacteristic
319#97—34no—
320#97—34no—
321C4×C8—34no—
322C4×C8C22×D314yes—
323C23×C3C22×D411yescharacteristic
324C22×D3—24no—
325C22×D3—24no—
326C22×D3—44no—
327C22×D3—48no—
328C22×D3#12714yes—
329C22×D3C22×D412yes—
330C3×Q8—24no—
331C3×Q8—24no—
332C3×Q8—24no—
333C3×Q8—24no—
334C3×Q8—44no—
335C3×Q8—44no—
336C2×C3×C4—24no—
337C2×C3×C4—24no—
338C2×C3×C4—24no—
339C2×C3×C4—44no—
340C2×C3×C4—44no—
341C2×C3×C4C22×D414yes—
342C2×C3×C4C22×D411yescharacteristic
343C2×C3×C4C22×D412yes—
344C2×C3×C4C22×D412yes—
345C2×C3×C4C2511yescharacteristic
346C2×Dic3—24no—
347C2×Dic3—24no—
348C2×Dic3—44no—
349C2×Dic3#12714yes—
350C2×Dic3C22×D414yes—
351C2×Dic3C22×D412yes—
352D12—24no—
353D12—24no—
354D12—416no—
355D12—44no—
356C4×D3—416no—
357C4×D3—416no—
358C4×D3#12718yes—
359C4×D3C22×D414yes—
360C4×D3C22×D414yes—
361Dic6—24no—
362Dic6—24no—
363Dic6—44no—
364Dic6—416no—
365C3×C8C22×D414yes—
366C3×C8C22×D414yes—
367C3⋊C8C22×D414yes—
368C3⋊C8C22×D414yes—
369C24—31no—
370C24—62no—
371#41—616no—
372#41—616no—
373#41—1216no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—24no—
377C2×Q8—24no—
378C2×Q8—24no—
379C2×Q8—24no—
380C2×Q8—34no—
381C2×Q8—68no—
382C2×Q8—616no—
383C2×Q8—616no—
384C2×Q8—62no—
385C2×Q8—64no—
386C2×Q8—64no—
387C2×Q8—128no—
388C2×Q8—1216no—
389C2×Q8C23×D314yes—
390C2×Q8C23×D314yes—
391C2×D4—34no—
392C2×D4—68no—
393C2×D4—64no—
394C2×D4—64no—
395C2×D4—62no—
396C2×D4—128no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—31no—
400C22×C4—32no—
401C22×C4—34no—
402C22×C4—34no—
403C22×C4—32no—
404C22×C4—32no—
405C22×C4—68no—
406C22×C4—68no—
407C22×C4—62no—
408C22×C4—128no—
409C22×C4—128no—
410C22×C4D3×D412yes—
411C22×C4C23×D311yescharacteristic
412Q16—28no—
413Q16—28no—
414Q16—216no—
415Q16—616no—
416Q16—616no—
417Q16—68no—
418SD16—616no—
419SD16—616no—
420SD16—616no—
421D8—68no—
422M16—316no—
423C2×C8—34no—
424C2×C8—34no—
425C2×C8—32no—
426C2×C8—32no—
427C2×C8—316no—
428C2×C8D3×D414yes—
429C2×C8D3×D414yes—
430C2×C8C23×D312yes—
431C2×C8C23×D312yes—
432C4⋊C4—216no—
433C4⋊C4—38no—
434C4⋊C4—616no—
435C22⋊C4—38no—
436C22⋊C4—616no—
437C22⋊C4—616no—
438C42—32no—
439C42—32no—
440C42D3×D412yes—
441C42C23×D312yes—
442C22×C3C2×#12712yes—
443C22×C3#57814yes—
444C22×C3C23×D411yescharacteristic
445C2×D3—28no—
446C2×D3—24no—
447C2×D3—44no—
448C2×D3—48no—
449C2×D3—88no—
450C2×D3C2×#12714yes—
451C3×C4—24no—
452C3×C4—44no—
453C3×C4—44no—
454C3×C4C2×#12712yes—
455C3×C4C23×D411yescharacteristic
456C3×C4C23×D411yescharacteristic
457Dic3—24no—
458Dic3—44no—
459Dic3C2×#12714yes—
460C23—32no—
461C23—34no—
462C23—64no—
463C23—64no—
464C23—124no—
465C23—128no—
466C23C2×D3×D411yescharacteristic
467Q8—24no—
468Q8—24no—
469Q8—24no—
470Q8—24no—
471Q8—44no—
472Q8—44no—
473Q8—64no—
474Q8—64no—
475Q8—1216no—
476Q8—124no—
477D4—64no—
478D4—64no—
479D4—1216no—
480D4—124no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—32no—
485C2×C4—34no—
486C2×C4—34no—
487C2×C4—34no—
488C2×C4—38no—
489C2×C4—34no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—64no—
493C2×C4—64no—
494C2×C4—1216no—
495C2×C4—1216no—
496C2×C4—124no—
497C2×C4C2×D3×D414yes—
498C2×C4C2×D3×D412yes—
499C2×C4C2×D3×D412yes—
500C2×C4C2×D3×D411yescharacteristic
501C2×C4C24×D311yescharacteristic
502C8—34no—
503C8—34no—
504C8C2×D3×D414yes—
505C8C2×D3×D414yes—
506C2×C3#315614yescomplemented
507C2×C3C22×#12711yescharacteristic
508C2×C3C2×#57812yes—
509D3—24no—
510D3—84no—
511C22—34no—
512C22—68no—
513C22—64no—
514C22—128no—
515C22—124no—
516C22—248no—
517C22D3×#12712yes—
518C22#576614yes—
519C22C22×D3×D411yescharacteristic
520C4—24no—
521C4—34no—
522C4—44no—
523C4—44no—
524C4—64no—
525C4—124no—
526C4D3×#12712yes—
527C4C22×D3×D411yescharacteristic
528C4C22×D3×D411yescharacteristic
529C3C2×#315611yescharacteristic, complemented, Sylow
530C2—64no—
531C2—244no—
532C2#8180114yesdirect product factor
533C2C2×D3×#12711yescharacteristic
534C2C2×#576612yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#494C2
socleC23×C3C22×D4
prime radicalC22×#919C2
hypercenterC2×#494D3
hypocenterC3C2×#3156
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#494

Complements

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

NH ≅ G/Ntype
1. C22×#918C2semidirect
3. C22×D3×Q16C2semidirect
4. C2×#5724C2semidirect
5. C2×C3×#494C2semidirect
6. C2×#5190C2semidirect
7. C2×#5191C2semidirect
8. C2×#5183C2semidirect
9. C2×#5086C2semidirect
10. #81801C2direct (N × H)
11. C2×#4696C2semidirect
12. C2×#4689C2semidirect
29. C2×#918C22semidirect
31. C2×D3×Q16C22semidirect
38. #5724C22semidirect
43. C3×#494C22semidirect
45. #5191C22semidirect
46. #5190C22semidirect
47. #5183C22semidirect
50. #5086C22semidirect
52. #4696C22semidirect
53. #4689C22semidirect
64. C2×#494D3semidirect
172. #494C2×D3semidirect
506. C2×C3#3156semidirect
529. C3C2×#3156semidirect
532. C2#81801direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#9182
C22×#9192
C22×D3×Q162
C2×#57242
C2×C3×#4942
C2×#51902
C2×#51912
C2×#51832
C2×#50862
#818012
C2×#46962
C2×#46892
C2×#31563