Finite Group Databasebeta

← C2×#79730 #11670344

Subgroups — C2×#79730 #11670344

Order 768.

2762 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#918C212yesmaximal, complemented
2C2×#5576C212yesmaximal, complemented
3C2×C3×C4×Q16C211yesmaximal, characteristic, complemented
4C2×#5028C212yesmaximal, complemented
5#79730C2116yesmaximal, direct product factor
6C2×#4886C212yesmaximal, complemented
7C2×#4810C212yesmaximal, complemented
8C2×#4809C212yesmaximal, complemented
9C2×#4696C211yesmaximal, characteristic, complemented
10C2×#4688C211yesmaximal, characteristic, complemented
11C2×#3277—31nomaximal, Sylow
12C22×#239C2212yes—
13C22×D3×Q8C2212yes—
14C22×C3×Q16C2211yescharacteristic
15C2×C3×C4×Q8C2212yes—
16C22×#216C2212yes—
17C22×#215C2212yes—
18C2×#918—216no—
19C2×#918C22116yescomplemented
20C22×Dic12C2211yescharacteristic
21C22×D24C2211yescharacteristic
22C22×#204C2212yes—
23C22×ΓL(1,𝔽25)C2212yes—
24#5576—28no—
25#5576C22116yescomplemented
26C2×#897—22no—
27C2×#897C2212yes—
28C2×#895—22no—
29C2×#895C2212yes—
30C2×#875—22no—
31C2×#875C2212yes—
32C2×C4×D12—22no—
33C2×C4×D12C2211yescharacteristic
34C3×C4×Q16C2218yescomplemented
35C2×C3×#107C2211yescharacteristic
36C2×C3×#103C2212yes—
37C2×C3×C4×C8C2211yescharacteristic
38C2×(C2×D3)⋊C8C2212yes—
39#5028C22116yescomplemented
40C2×#810C2212yes—
41C2×#809C2212yes—
42C2×C4⋊(C3⋊C8)C2211yescharacteristic
43#4886C22116yescomplemented
44#4810C22116yescomplemented
45#4809C22116yescomplemented
46#4696C2218yescomplemented
47#4688C2218yescomplemented
48C22×#137—32no—
49C2×#548—32no—
50#3277—316no—
51C2×#495—31no—
52C2×#468—32no—
53C2×#464—32no—
54C2×#449—32no—
55C2×#451—32no—
56C2×C4×Q16D311yescharacteristic, complemented
57C2×#436—31no—
58C22×C3×Q8C2312yes—
59C2×#239—24no—
60C2×#239—28no—
61C2×#239—44no—
62C2×#239C2318yes—
63C2×D3×Q8—24no—
64C2×D3×Q8—28no—
65C2×D3×Q8—44no—
66C2×D3×Q8C2318yes—
67C22×D12—22no—
68C22×D12—22no—
69C22×D12D411yescharacteristic
70C22×D12C2312yes—
71C22×C4×D3—22no—
72C22×C4×D3—22no—
73C22×C4×D3C2312yes—
74C22×Dic6—22no—
75C22×Dic6C2312yes—
76C2×C3×Q16D418yescomplemented
77C2×C3×Q16C2314yes—
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C3×C4×Q8—24no—
81C3×C4×Q8C2318yes—
82C2×C3×C4⋊C4—22no—
83C2×C3×C4⋊C4C2312yes—
84C2×C3×C42—22no—
85C2×C3×C42C2311yescharacteristic
86C2×#216—28no—
87C2×#216C2318yes—
88C2×#215—28no—
89C2×#215C2318yes—
90C2×(C2×D3)⋊C4—22no—
91C2×(C2×D3)⋊C4—22no—
92C2×(C2×D3)⋊C4—22no—
93C2×(C2×D3)⋊C4—22no—
94C2×(C2×D3)⋊C4D412yes—
95C2×C4⋊Dic3—22no—
96C2×C4⋊Dic3D411yescharacteristic
97C22×C3⋊C8C2312yes—
98#918—264no—
99C2×Dic12—42no—
100C2×Dic12C2314yes—
101C2×D24—42no—
102C2×D24C2314yes—
103C2×#204—28no—
104C2×#204C2318yes—
105C2×ΓL(1,𝔽25)—28no—
106C2×ΓL(1,𝔽25)C2318yes—
107#897—24no—
108#897—216no—
109#897C2318yes—
110#895—216no—
111#895—24no—
112#895C2318yes—
113#875—24no—
114#875—216no—
115#875C2318yes—
116C4×D12—216no—
117C4×D12D414yes—
118C4×D12C2314yes—
119C3×#107C2314yes—
120C3×#103C2318yes—
121C3×C4×C8C2314yes—
122(C2×D3)⋊C8C2318yes—
123#810C2318yes—
124#809C2318yes—
125C4⋊(C3⋊C8)C2314yes—
126C22×#41—32no—
127C23×Q8—32no—
128C2×#137—316no—
129C2×#137—616no—
130C22×Q16—32no—
131C22×Q16—31no—
132C22×Q16C2×D311yescharacteristic
133C22×SD16—32no—
134C22×SD16—32no—
135C22×D8—31no—
136C22×M16—32no—
137#548—316no—
138#548—68no—
139C2×#124—32no—
140C2×#124—62no—
141C2×C22⋊Q8—32no—
142C2×C22⋊Q8—62no—
143C2×#121—32no—
144C2×#121—62no—
145C2×C4×Q8C2×D312yes—
146C2×C4×D4—31no—
147C2×C4×D4—62no—
148#495—38no—
149#468—316no—
150#464—316no—
151#451—316no—
152#449—316no—
153C4×Q16C2×D318yescomplemented
154#436—38no—
155C2×#107C2×D311yescharacteristic
156C2×C4⋊C8—31no—
157C2×#103—32no—
158C2×#103C2×D312yes—
159C2×#102—32no—
160C2×C22⋊C8—32no—
161C2×C4×C8C2×D311yescharacteristic
162C23×D3—22no—
163C23×D3C2×D412yes—
164C2×C3×Q8—22no—
165C2×C3×Q8—24no—
166C2×C3×Q8C2×D418yes—
167C2×C3×Q8C2414yes—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2411yescharacteristic
173C22×Dic3—22no—
174C22×Dic3C2×D412yes—
175#239—216no—
176#239—48no—
177#239—416no—
178D3×Q8—216no—
179D3×Q8—48no—
180D3×Q8—416no—
181C2×D12—24no—
182C2×D12—24no—
183C2×D12—28no—
184C2×D12—28no—
185C2×D12—22no—
186C2×D12—42no—
187C2×D12—48no—
188C2×D12—48no—
189C2×D12C2×D414yes—
190C2×D12C2414yes—
191C2×C4×D3—24no—
192C2×C4×D3—24no—
193C2×C4×D3—28no—
194C2×C4×D3—28no—
195C2×C4×D3—48no—
196C2×C4×D3—48no—
197C2×C4×D3C2×D414yes—
198C2×C4×D3C2414yes—
199C2×Dic6—24no—
200C2×Dic6—24no—
201C2×Dic6—22no—
202C2×Dic6—28no—
203C2×Dic6—48no—
204C2×Dic6C2414yes—
205C3×Q16C2×D4116yescomplemented
206C2×C3×C8—22no—
207C2×C3×C8C2×D414yes—
208C2×C3×C8C2412yes—
209C2×C3×C8C2412yes—
210C3×C4⋊C4—28no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4C2414yes—
213C3×C42—28no—
214C3×C42C2×D412yes—
215C3×C42C2412yes—
216#216—216no—
217#215—216no—
218(C2×D3)⋊C4—28no—
219(C2×D3)⋊C4—28no—
220(C2×D3)⋊C4—28no—
221(C2×D3)⋊C4—28no—
222(C2×D3)⋊C4C2×D418yes—
223C4⋊Dic3—28no—
224C4⋊Dic3C2×D414yes—
225C2×C3⋊C8—24no—
226C2×C3⋊C8C2414yes—
227Dic12—44no—
228D24—44no—
229#204—216no—
230ΓL(1,𝔽25)—216no—
231C2×#41—38no—
232C2×#41—64no—
233C2×#41—68no—
234C2×#41—124no—
235C22×Q8—38no—
236C22×Q8—32no—
237C22×Q8—62no—
238C22×Q8—64no—
239C22×Q8—68no—
240C22×Q8—124no—
241C22×Q8C22×D312yes—
242C22×D4—31no—
243C22×D4—32no—
244C22×D4—62no—
245C22×D4—62no—
246C23×C4—32no—
247C23×C4—62no—
248C23×C4—62no—
249#137—664no—
250C2×Q16—38no—
251C2×Q16—34no—
252C2×Q16—68no—
253C2×Q16—122no—
254C2×Q16D1218yescomplemented
255C2×Q16C22×D314yes—
256C2×SD16—38no—
257C2×SD16—38no—
258C2×SD16—68no—
259C2×SD16—68no—
260C2×D8—34no—
261C2×D8—122no—
262C2×M16—38no—
263C2×M16—68no—
264C22×C8—32no—
265C22×C8C22×D311yescharacteristic
266C22×C8C22×D311yescharacteristic
267#124—38no—
268#124—616no—
269#124—64no—
270C22⋊Q8—38no—
271C22⋊Q8—616no—
272C22⋊Q8—64no—
273#121—38no—
274#121—64no—
275#121—616no—
276C4×Q8—24no—
277C4×Q8C22×D318yes—
278C4×D4—34no—
279C4×D4—34no—
280C4×D4—616no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—62no—
284C2×C4⋊C4C22×D312yes—
285C2×C22⋊C4—32no—
286C2×C22⋊C4—62no—
287C2×C22⋊C4—62no—
288C2×C22⋊C4—62no—
289C2×C22⋊C4—62no—
290C2×C42—22no—
291C2×C42C22×D311yescharacteristic
292#107C22×D314yes—
293C4⋊C8—34no—
294#103—38no—
295#103C22×D318yes—
296#102—38no—
297C22⋊C8—38no—
298C4×C8C22×D314yes—
299C23×C3C22×D411yescharacteristic
300C22×D3—24no—
301C22×D3—24no—
302C22×D3—24no—
303C22×D3—24no—
304C22×D3—44no—
305C22×D3—44no—
306C22×D3—44no—
307C22×D3C22×D414yes—
308C3×Q8—28no—
309C3×Q8—24no—
310C3×Q8C22×D418yes—
311C2×C3×C4—28no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—24no—
315C2×C3×C4—24no—
316C2×C3×C4—44no—
317C2×C3×C4C22×D412yes—
318C2×C3×C4C22×D412yes—
319C2×C3×C4C22×D412yes—
320C2×C3×C4C22×D411yescharacteristic
321C2×C3×C4H(5,𝔽2)12yes—
322C2×C3×C4C2511yescharacteristic
323C2×Dic3—24no—
324C2×Dic3—24no—
325C2×Dic3—44no—
326C2×Dic3C22×D414yes—
327D12—24no—
328D12—24no—
329D12—44no—
330D12—416no—
331D12—416no—
332D12—44no—
333C4×D3—28no—
334C4×D3—24no—
335C4×D3—24no—
336C4×D3—416no—
337C4×D3—416no—
338Dic6—24no—
339Dic6—24no—
340Dic6—416no—
341Dic6—44no—
342C3×C8—22no—
343C3×C8C22×D414yes—
344C3⋊C8—24no—
345C24—32no—
346C24—62no—
347#41—616no—
348#41—128no—
349#41—1216no—
350C2×Q8—22no—
351C2×Q8—24no—
352C2×Q8—34no—
353C2×Q8—68no—
354C2×Q8—62no—
355C2×Q8—616no—
356C2×Q8—64no—
357C2×Q8—64no—
358C2×Q8—128no—
359C2×Q8—128no—
360C2×Q8—1216no—
361C2×Q8C2×D1218yes—
362C2×Q8C23×D314yes—
363C2×D4—34no—
364C2×D4—34no—
365C2×D4—68no—
366C2×D4—68no—
367C2×D4—64no—
368C2×D4—62no—
369C2×D4—64no—
370C2×D4—122no—
371C2×D4—128no—
372C2×D4—128no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—32no—
376C22×C4—34no—
377C22×C4—34no—
378C22×C4—68no—
379C22×C4—68no—
380C22×C4—64no—
381C22×C4—64no—
382C22×C4—62no—
383C22×C4—128no—
384C22×C4—128no—
385C22×C4D3×D411yescharacteristic
386C22×C4D3×D411yescharacteristic
387C22×C4C23×D311yescharacteristic
388Q16—616no—
389Q16—124no—
390Q16C2×D12116yescomplemented
391SD16—616no—
392SD16—616no—
393D8—124no—
394M16—616no—
395C2×C8—22no—
396C2×C8—34no—
397C2×C8—64no—
398C2×C8C2×D1214yes—
399C2×C8C23×D312yes—
400C2×C8C23×D312yes—
401C4⋊C4—28no—
402C4⋊C4—22no—
403C4⋊C4—34no—
404C4⋊C4—68no—
405C4⋊C4C23×D314yes—
406C22⋊C4—38no—
407C22⋊C4—68no—
408C22⋊C4—68no—
409C22⋊C4—68no—
410C22⋊C4—68no—
411C42—28no—
412C42D3×D412yes—
413C42C23×D312yes—
414C22×C3D4212yes—
415C22×C3#57512yes—
416C22×C3#57812yes—
417C22×C3C23×D411yescharacteristic
418C2×D3—24no—
419C2×D3—24no—
420C2×D3—24no—
421C2×D3—48no—
422C2×D3—44no—
423C2×D3—44no—
424C2×D3—44no—
425C2×D3—88no—
426C3×C4—28no—
427C3×C4—22no—
428C3×C4—22no—
429C3×C4—44no—
430C3×C4D4212yes—
431C3×C4C23×D411yescharacteristic
432C3×C4C2×H(5,𝔽2)11yescharacteristic
433Dic3—24no—
434Dic3—44no—
435C23—34no—
436C23—64no—
437C23—64no—
438C23—64no—
439C23—64no—
440C23—124no—
441C23—124no—
442C23—124no—
443C23C2×D3×D411yescharacteristic
444Q8—24no—
445Q8—28no—
446Q8—64no—
447Q8—64no—
448Q8—1216no—
449Q8—124no—
450Q8C22×D1218yes—
451D4—64no—
452D4—64no—
453D4—124no—
454D4—1216no—
455D4—1216no—
456D4—124no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—28no—
461C2×C4—24no—
462C2×C4—34no—
463C2×C4—44no—
464C2×C4—64no—
465C2×C4—64no—
466C2×C4—64no—
467C2×C4—68no—
468C2×C4—64no—
469C2×C4—1216no—
470C2×C4—1216no—
471C2×C4—124no—
472C2×C4C22×D1212yes—
473C2×C4C2×D3×D412yes—
474C2×C4C2×D3×D412yes—
475C2×C4C2×D3×D411yescharacteristic
476C2×C4#100912yes—
477C2×C4C24×D311yescharacteristic
478C8—22no—
479C8—64no—
480C8C22×D1214yes—
481C2×C3#327714yescomplemented
482C2×C3C2×D4211yescharacteristic
483C2×C3C2×#57511yescharacteristic
484C2×C3C2×#57811yescharacteristic
485D3—44no—
486D3—84no—
487C22—64no—
488C22—64no—
489C22—64no—
490C22—128no—
491C22—124no—
492C22—124no—
493C22—124no—
494C22—248no—
495C22D4×D1212yes—
496C22#576612yes—
497C22#576812yes—
498C22C22×D3×D411yescharacteristic
499C4—28no—
500C4—22no—
501C4—22no—
502C4—44no—
503C4—64no—
504C4—124no—
505C4D4×D1212yes—
506C4C23×D1211yescharacteristic
507C4C2×#100911yescharacteristic
508C3C2×#327711yescharacteristic, complemented, Sylow
509C2—124no—
510C2—244no—
511C2#7973014yesdirect product factor
512C2C2×D4×D1211yescharacteristic
513C2C2×#576611yescharacteristic
514C2C2×#576811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×C4×Q16C2
socleC23×C3C22×D4
hypercenterC2×C4×Q16D3
hypocenterC3C2×#3277
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×Q16

Complements

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

NH ≅ G/Ntype
1. C22×#918C2semidirect
2. C2×#5576C2semidirect
3. C2×C3×C4×Q16C2semidirect
4. C2×#5028C2semidirect
5. #79730C2direct (N × H)
6. C2×#4886C2semidirect
7. C2×#4810C2semidirect
8. C2×#4809C2semidirect
9. C2×#4696C2semidirect
10. C2×#4688C2semidirect
19. C2×#918C22semidirect
25. #5576C22semidirect
34. C3×C4×Q16C22semidirect
39. #5028C22semidirect
43. #4886C22semidirect
44. #4810C22semidirect
45. #4809C22semidirect
46. #4696C22semidirect
47. #4688C22semidirect
56. C2×C4×Q16D3semidirect
76. C2×C3×Q16D4semidirect
153. C4×Q16C2×D3semidirect
205. C3×Q16C2×D4semidirect
254. C2×Q16D12semidirect
390. Q16C2×D12semidirect
481. C2×C3#3277semidirect
508. C3C2×#3277semidirect
511. C2#79730direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#9182
C2×#55762
C2×C3×C4×Q162
C2×#50282
#797302
C2×#48862
C2×#48102
C2×#48092
C2×#46962
C2×#46882
C2×#32773