Finite Group Databasebeta

← C2×#77258 #10961731

Subgroups — C2×#77258 #10961731

Order 768.

2982 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#946C211yesmaximal, characteristic
2C22×#896C211yesmaximal, characteristic
3C2×C3×#395C211yesmaximal, characteristic, complemented
4C2×#5230C211yesmaximal, characteristic, complemented
5C2×#4987C212yesmaximal
6C2×#4981C211yesmaximal, characteristic, complemented
7C2×#4979C212yesmaximal, complemented
8#77258C2116yesmaximal, direct product factor
9C2×#4667C212yesmaximal
10C2×D3×#95C211yesmaximal, characteristic
11C2×#4661C211yesmaximal, characteristic, complemented
12C2×#4657C212yesmaximal, complemented
13C2×#2659—31nomaximal, Sylow
14C23×C3×Q8C2211yescharacteristic
15C23×C4×D3C2211yescharacteristic
16C23×C4×D3C2212yes—
17C23×Dic6C2211yescharacteristic
18C22×C3×C4⋊C4C2211yescharacteristic
19C22×C3×C4⋊C4C2212yes—
20C2×#946—28no—
21C2×#946C2218yes—
22C22×(C2×D3)⋊C4C2211yescharacteristic
23C22×(C2×D3)⋊C4C2212yes—
24C22×(C2×D3)⋊C4C2211yescharacteristic
25C22×C4⋊Dic3C2211yescharacteristic
26C22×C4⋊Dic3C2212yes—
27C22×Dic3⋊C4C2211yescharacteristic
28C22×Dic3⋊C4C2211yescharacteristic
29C22×Dic3⋊C4C2212yes—
30C22×Dic3⋊C4C2212yes—
31C22×Dic3⋊C4C2212yes—
32C2×#896—28no—
33C2×#896C2218yes—
34C3×#395C2218yescomplemented
35C2×C3×#95C2212yes—
36C2×C3×#95C2211yescharacteristic
37#5230C2218yescomplemented
38C2×C3⋊#95C2212yes—
39C2×C3⋊#95C2212yes—
40C2×C3⋊#95C2211yescharacteristic
41C2×C3⋊#95C2212yes—
42C2×C3⋊#95C2212yes—
43#4987C22116yes—
44#4981C2218yescomplemented
45#4979C22116yescomplemented
46#4667C22116yes—
47D3×#95C2218yes—
48#4661C2218yescomplemented
49#4657C22116yescomplemented
50C22×C22⋊Q8—31no—
51C22×C22⋊Q8—31no—
52#2659—316no—
53C2×#398—32no—
54C2×#398—31no—
55C2×#398—31no—
56C2×#395—31no—
57C2×#395—32no—
58C2×#395D311yescharacteristic, complemented
59C2×#380—32no—
60C2×#380—32no—
61C22×#95—31no—
62C24×D3C2311yescharacteristic
63C22×C3×Q8—24no—
64C22×C3×Q8—44no—
65C22×C3×Q8C2314yes—
66C23×C3×C4C2311yescharacteristic
67C23×C3×C4C2312yes—
68C23×C3×C4C2311yescharacteristic
69C23×C3×C4C2311yescharacteristic
70C23×C3×C4C2312yes—
71C23×Dic3C2311yescharacteristic
72C23×Dic3C2311yescharacteristic
73C23×Dic3C2312yes—
74C23×Dic3C2311yescharacteristic
75C23×Dic3C2312yes—
76C22×C4×D3—44no—
77C22×C4×D3—44no—
78C22×C4×D3—44no—
79C22×C4×D3D418yes—
80C22×C4×D3Q8116yescomplemented
81C22×C4×D3C2314yes—
82C22×C4×D3C2318yes—
83C22×Dic6—24no—
84C22×Dic6—44no—
85C22×Dic6C2314yes—
86C2×C3×C4⋊C4—24no—
87C2×C3×C4⋊C4—44no—
88C2×C3×C4⋊C4C2314yes—
89C2×C3×C4⋊C4C2318yes—
90#946—232no—
91C2×(C2×D3)⋊C4—24no—
92C2×(C2×D3)⋊C4—24no—
93C2×(C2×D3)⋊C4—28no—
94C2×(C2×D3)⋊C4C2314yes—
95C2×(C2×D3)⋊C4C2318yes—
96C2×(C2×D3)⋊C4C2314yes—
97C2×C4⋊Dic3—24no—
98C2×C4⋊Dic3—44no—
99C2×C4⋊Dic3C2314yes—
100C2×C4⋊Dic3C2318yes—
101C2×Dic3⋊C4—24no—
102C2×Dic3⋊C4—24no—
103C2×Dic3⋊C4—28no—
104C2×Dic3⋊C4—44no—
105C2×Dic3⋊C4—44no—
106C2×Dic3⋊C4C2314yes—
107C2×Dic3⋊C4C2318yes—
108C2×Dic3⋊C4C2314yes—
109C2×Dic3⋊C4C2318yes—
110C2×Dic3⋊C4C2318yes—
111#896—232no—
112C3×#95C2318yes—
113C3×#95C2314yes—
114C3⋊#95C2318yes—
115C3⋊#95C2318yes—
116C3⋊#95C2318yes—
117C3⋊#95C2314yes—
118C3⋊#95C2318yes—
119C23×Q8—31no—
120C23×Q8C2×D311yescharacteristic
121C24×C4—31no—
122C24×C4—32no—
123C2×C22⋊Q8—38no—
124C2×C22⋊Q8—38no—
125C2×C22⋊Q8—68no—
126C2×C22⋊Q8—68no—
127C22×C4⋊C4—32no—
128C22×C4⋊C4—31no—
129C22×C4⋊C4—31no—
130C22×C4⋊C4—31no—
131C22×C4⋊C4—32no—
132C22×C4⋊C4—32no—
133C22×C4⋊C4—32no—
134C22×C4⋊C4C2×D311yescharacteristic
135C22×C4⋊C4C2×D312yes—
136C22×C22⋊C4—32no—
137C22×C22⋊C4—31no—
138C22×C22⋊C4—31no—
139#398—38no—
140#398—316no—
141#398—38no—
142#395—38no—
143#395—316no—
144#395C2×D318yescomplemented
145#380—316no—
146#380—316no—
147C2×#95—32no—
148C2×#95—31no—
149C2×#95—32no—
150C2×#95—32no—
151C2×#95—32no—
152C2×#95—38no—
153C2×#95C2×D311yescharacteristic
154C2×#95C2×D312yes—
155C24×C3C2411yescharacteristic
156C23×D3—24no—
157C23×D3—22no—
158C23×D3—22no—
159C23×D3C2×D414yes—
160C23×D3C2×Q818yes—
161C23×D3C2412yes—
162C2×C3×Q8—28no—
163C2×C3×Q8—48no—
164C2×C3×Q8—416no—
165C22×C3×C4—24no—
166C22×C3×C4—24no—
167C22×C3×C4—24no—
168C22×C3×C4—22no—
169C22×C3×C4—24no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—24no—
173C22×C3×C4—24no—
174C22×C3×C4—24no—
175C22×C3×C4—22no—
176C22×C3×C4C2×D414yes—
177C22×C3×C4C2×Q818yes—
178C22×C3×C4C2412yes—
179C22×C3×C4C2414yes—
180C22×C3×C4C2412yes—
181C22×C3×C4C2412yes—
182C22×C3×C4C2414yes—
183C22×Dic3—24no—
184C22×Dic3—24no—
185C22×Dic3—24no—
186C22×Dic3—22no—
187C22×Dic3—24no—
188C22×Dic3—22no—
189C22×Dic3—24no—
190C22×Dic3—24no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—24no—
194C22×Dic3C2×D414yes—
195C22×Dic3C2×Q818yes—
196C22×Dic3C2412yes—
197C22×Dic3C2412yes—
198C22×Dic3C2414yes—
199C22×Dic3C2412yes—
200C22×Dic3C2414yes—
201C2×C4×D3—416no—
202C2×C4×D3—432no—
203C2×C4×D3—48no—
204C2×C4×D3—48no—
205C2×C4×D3—48no—
206C2×C4×D3C2×D4116yes—
207C2×C4×D3C2×Q8132yescomplemented
208C2×Dic6—28no—
209C2×Dic6—48no—
210C2×Dic6—416no—
211C3×C4⋊C4—28no—
212C3×C4⋊C4—48no—
213(C2×D3)⋊C4—28no—
214(C2×D3)⋊C4—28no—
215(C2×D3)⋊C4—216no—
216C4⋊Dic3—28no—
217C4⋊Dic3—48no—
218Dic3⋊C4—28no—
219Dic3⋊C4—28no—
220Dic3⋊C4—216no—
221Dic3⋊C4—48no—
222Dic3⋊C4—48no—
223C25—31no—
224C22×Q8—24no—
225C22×Q8—34no—
226C22×Q8—44no—
227C22×Q8—64no—
228C22×Q8—124no—
229C22×Q8C22×D314yes—
230C23×C4—31no—
231C23×C4—32no—
232C23×C4—38no—
233C23×C4—316no—
234C23×C4—34no—
235C23×C4—38no—
236C23×C4—31no—
237C23×C4—32no—
238C23×C4—31no—
239C23×C4—124no—
240C23×C4—124no—
241C23×C4—124no—
242C23×C4C22×D311yescharacteristic
243C23×C4C22×D312yes—
244C23×C4C22×D311yescharacteristic
245C23×C4C22×D311yescharacteristic
246C23×C4C22×D312yes—
247C22⋊Q8—632no—
248C22⋊Q8—632no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—38no—
251C2×C4⋊C4—34no—
252C2×C4⋊C4—34no—
253C2×C4⋊C4—34no—
254C2×C4⋊C4—38no—
255C2×C4⋊C4—38no—
256C2×C4⋊C4—38no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—64no—
259C2×C4⋊C4—68no—
260C2×C4⋊C4—64no—
261C2×C4⋊C4—64no—
262C2×C4⋊C4—124no—
263C2×C4⋊C4—124no—
264C2×C4⋊C4—124no—
265C2×C4⋊C4C22×D314yes—
266C2×C4⋊C4C22×D318yes—
267C2×C22⋊C4—34no—
268C2×C22⋊C4—38no—
269C2×C22⋊C4—34no—
270C2×C22⋊C4—64no—
271C2×C22⋊C4—64no—
272C2×C22⋊C4—68no—
273#95—34no—
274#95—38no—
275#95—38no—
276#95—38no—
277#95—38no—
278#95C22×D314yes—
279#95C22×D318yes—
280C23×C3C22×D412yes—
281C23×C3C22×Q814yes—
282C23×C3H(5,𝔽2)14yes—
283C23×C3H(5,𝔽2)12yes—
284C23×C32−1+412yes—
285C23×C3C2511yescharacteristic
286C22×D3—28no—
287C22×D3—28no—
288C22×D3—216no—
289C22×D3—216no—
290C22×D3—24no—
291C22×D3—22no—
292C22×D3—22no—
293C22×D3C4⋊Q8116yes—
294C22×D3C22×D414yes—
295C22×D3C22×Q818yes—
296C3×Q8—416no—
297C2×C3×C4—28no—
298C2×C3×C4—216no—
299C2×C3×C4—24no—
300C2×C3×C4—24no—
301C2×C3×C4—24no—
302C2×C3×C4—22no—
303C2×C3×C4—24no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—24no—
307C2×C3×C4—24no—
308C2×C3×C4—24no—
309C2×C3×C4—22no—
310C2×C3×C4—44no—
311C2×C3×C4—48no—
312C2×C3×C4—44no—
313C2×C3×C4C22×D414yes—
314C2×C3×C4C22×Q818yes—
315C2×Dic3—28no—
316C2×Dic3—216no—
317C2×Dic3—24no—
318C2×Dic3—24no—
319C2×Dic3—24no—
320C2×Dic3—22no—
321C2×Dic3—24no—
322C2×Dic3—22no—
323C2×Dic3—24no—
324C2×Dic3—24no—
325C2×Dic3—22no—
326C2×Dic3—22no—
327C2×Dic3—24no—
328C2×Dic3—44no—
329C2×Dic3—44no—
330C2×Dic3—48no—
331C2×Dic3C22×D414yes—
332C2×Dic3C22×Q818yes—
333C4×D3—416no—
334C4×D3—432no—
335Dic6—416no—
336C24—34no—
337C24—38no—
338C24—32no—
339C24—64no—
340C24—62no—
341C24—62no—
342C24C23×D311yescharacteristic
343C2×Q8—28no—
344C2×Q8—48no—
345C2×Q8—416no—
346C2×Q8—68no—
347C2×Q8—128no—
348C2×Q8—1216no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—34no—
361C22×C4—38no—
362C22×C4—34no—
363C22×C4—32no—
364C22×C4—316no—
365C22×C4—332no—
366C22×C4—32no—
367C22×C4—34no—
368C22×C4—32no—
369C22×C4—64no—
370C22×C4—64no—
371C22×C4—64no—
372C22×C4—62no—
373C22×C4—64no—
374C22×C4—62no—
375C22×C4—64no—
376C22×C4—62no—
377C22×C4—64no—
378C22×C4—64no—
379C22×C4—62no—
380C22×C4—1232no—
381C22×C4—128no—
382C22×C4—128no—
383C22×C4—128no—
384C22×C4—1216no—
385C22×C4D3×D414yes—
386C22×C4D3×Q818yes—
387C22×C4C23×D312yes—
388C22×C4C23×D314yes—
389C22×C4C23×D312yes—
390C22×C4C23×D312yes—
391C22×C4C23×D314yes—
392C4⋊C4—28no—
393C4⋊C4—48no—
394C4⋊C4—68no—
395C4⋊C4—616no—
396C4⋊C4—68no—
397C4⋊C4—68no—
398C4⋊C4—128no—
399C4⋊C4—128no—
400C4⋊C4—128no—
401C22⋊C4—68no—
402C22⋊C4—68no—
403C22⋊C4—616no—
404C22×C3C2×C4⋊Q814yes—
405C22×C3#53414yes—
406C22×C3#53714yes—
407C22×C3#54318yes—
408C22×C3#54418yes—
409C22×C3C23×D411yescharacteristic
410C22×C3C23×Q812yes—
411C22×C3C2×H(5,𝔽2)12yes—
412C22×C3C2×H(5,𝔽2)11yescharacteristic
413C22×C3C2×2−1+411yescharacteristic
414C2×D3—232no—
415C2×D3—24no—
416C2×D3—24no—
417C2×D3—28no—
418C2×D3—28no—
419C2×D3C2×C4⋊Q818yes—
420C3×C4—24no—
421C3×C4—28no—
422C3×C4—42no—
423C3×C4—44no—
424C3×C4—42no—
425Dic3—24no—
426Dic3—28no—
427Dic3—42no—
428Dic3—42no—
429Dic3—44no—
430C23—34no—
431C23—38no—
432C23—316no—
433C23—616no—
434C23—64no—
435C23—62no—
436C23—62no—
437C23—616no—
438C23—68no—
439C23—68no—
440C23C2×D3×D412yes—
441C23#100414yes—
442C23C2×D3×Q814yes—
443C23#100712yes—
444C23#100912yes—
445C23C24×D311yescharacteristic
446Q8—416no—
447Q8—1216no—
448C2×C4—216no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—28no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—34no—
462C2×C4—38no—
463C2×C4—48no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—616no—
467C2×C4—64no—
468C2×C4—64no—
469C2×C4—64no—
470C2×C4—68no—
471C2×C4—62no—
472C2×C4—64no—
473C2×C4—62no—
474C2×C4—64no—
475C2×C4—62no—
476C2×C4—64no—
477C2×C4—64no—
478C2×C4—62no—
479C2×C4—1216no—
480C2×C4—1232no—
481C2×C4—124no—
482C2×C4—124no—
483C2×C4—128no—
484C2×C4C2×D3×D414yes—
485C2×C4C2×D3×Q818yes—
486C2×C3#265918yescomplemented
487C2×C3C22×C4⋊Q811yescharacteristic
488C2×C3C2×#53411yescharacteristic
489C2×C3C2×#53711yescharacteristic
490C2×C3C2×#54312yes—
491C2×C3C2×#54412yes—
492D3—28no—
493C22—38no—
494C22—64no—
495C22—68no—
496C22—632no—
497C22—64no—
498C22—68no—
499C22#558914yes—
500C22#562914yes—
501C22#563418yes—
502C22D3×C4⋊Q814yes—
503C22#572918yes—
504C22C22×D3×D411yescharacteristic
505C22C2×#100412yes—
506C22C22×D3×Q812yes—
507C22C2×#100711yescharacteristic
508C22C2×#100911yescharacteristic
509C4—24no—
510C4—28no—
511C4—44no—
512C4—42no—
513C4—42no—
514C4—64no—
515C4—68no—
516C4—122no—
517C4—122no—
518C4—124no—
519C3C2×#265911yescharacteristic, complemented, Sylow
520C2—68no—
521C2#7725818yesdirect product factor
522C2C2×#558911yescharacteristic
523C2C2×#562911yescharacteristic
524C2C2×#563412yes—
525C2C2×D3×C4⋊Q811yescharacteristic
526C2C2×#572912yes—
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#395C2
socleC24×C3C24
prime radicalC24×D3C23
hypercenterC2×#395D3
hypocenterC3C2×#2659
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#395

Complements

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

NH ≅ G/Ntype
3. C2×C3×#395C2semidirect
4. C2×#5230C2semidirect
6. C2×#4981C2semidirect
7. C2×#4979C2semidirect
8. #77258C2direct (N × H)
11. C2×#4661C2semidirect
12. C2×#4657C2semidirect
34. C3×#395C22semidirect
37. #5230C22semidirect
44. #4981C22semidirect
45. #4979C22semidirect
48. #4661C22semidirect
49. #4657C22semidirect
58. C2×#395D3semidirect
80. C22×C4×D3Q8semidirect
144. #395C2×D3semidirect
207. C2×C4×D3C2×Q8semidirect
486. C2×C3#2659semidirect
519. C3C2×#2659semidirect
521. C2#77258direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#9462
C22×#8962
C2×C3×#3952
C2×#52302
C2×#49872
C2×#49812
C2×#49792
#772582
C2×#46672
C2×D3×#952
C2×#46612
C2×#46572
C2×#26593