Finite Group Databasebeta

← C2×#76854 #11001750

Subgroups — C2×#76854 #11001750

Order 768.

2370 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#944C211yesmaximal, characteristic, complemented
2C22×#879C211yesmaximal, characteristic
3C2×C3×#393C211yesmaximal, characteristic
4C2×#5230C211yesmaximal, characteristic, complemented
5C2×#4949C212yesmaximal
6C2×#4952C212yesmaximal
7C2×#4947C211yesmaximal, characteristic
8#76854C2116yesmaximal, direct product factor
9C2×#4660C212yesmaximal, complemented
10C2×#4657C212yesmaximal, complemented
11C2×#4656C211yesmaximal, characteristic, complemented
12C2×#4651C211yesmaximal, characteristic, complemented
13C2×#2821—31nomaximal, Sylow
14C23×C3×Q8C2211yescharacteristic
15C23×Dic6C2211yescharacteristic
16C22×C3×C22⋊C4C2211yescharacteristic
17C22×C3×C22⋊C4C2212yes—
18C22×#217C2211yescharacteristic
19C22×#217C2211yescharacteristic
20C22×#217C2212yes—
21C2×#944—28no—
22C2×#944C2218yescomplemented
23C22×C4⋊Dic3C2211yescharacteristic
24C22×C4⋊Dic3C2212yes—
25C22×Dic3⋊C4C2211yescharacteristic
26C22×Dic3⋊C4C2211yescharacteristic
27C22×Dic3⋊C4C2212yes—
28C22×Dic3⋊C4C2212yes—
29C22×Dic3⋊C4C2212yes—
30C22×C4×Dic3C2211yescharacteristic
31C2×#879—28no—
32C2×#879C2218yes—
33C3×#393C2218yes—
34C2×C3×#95C2211yescharacteristic
35C2×C3×#95C2212yes—
36#5230C2218yescomplemented
37C2×C3⋊#95C2211yescharacteristic
38C2×C3⋊#95C2212yes—
39C2×C3⋊#95C2212yes—
40C2×C3⋊#95C2212yes—
41C2×C3⋊#95C2212yes—
42C2×C3⋊#95C2212yes—
43#4952C22116yes—
44#4949C22116yes—
45#4947C2218yes—
46#4660C22116yescomplemented
47#4657C22116yescomplemented
48#4656C2218yescomplemented
49#4651C2218yescomplemented
50C22×C4⋊Q8—31no—
51C22×#126—31no—
52#2821—316no—
53C2×#400—32no—
54C2×#399—32no—
55C2×#398—32no—
56C2×#397—32no—
57C2×#395—31no—
58C2×#395—31no—
59C2×#393—31no—
60C2×#393D311yescharacteristic
61C2×#383—31no—
62C25×C3C2311yescharacteristic
63C22×C3×Q8—24no—
64C22×C3×Q8—44no—
65C22×C3×Q8C2314yes—
66C23×C3×C4C2312yes—
67C23×C3×C4C2312yes—
68C23×C3×C4C2311yescharacteristic
69C23×C3×C4C2311yescharacteristic
70C23×Dic3C2312yes—
71C23×Dic3C2312yes—
72C23×Dic3C2311yescharacteristic
73C23×Dic3C2311yescharacteristic
74C23×Dic3C2311yescharacteristic
75C23×Dic3C2311yescharacteristic
76C22×Dic6—24no—
77C22×Dic6—44no—
78C22×Dic6C2314yes—
79C2×C3×C22⋊C4—24no—
80C2×C3×C22⋊C4—44no—
81C2×C3×C22⋊C4C2314yes—
82C2×C3×C22⋊C4C2318yes—
83C2×#217—24no—
84C2×#217—24no—
85C2×#217—44no—
86C2×#217C2314yes—
87C2×#217C2314yes—
88C2×#217C2318yes—
89#944—232no—
90C2×C4⋊Dic3—24no—
91C2×C4⋊Dic3—44no—
92C2×C4⋊Dic3C2314yes—
93C2×C4⋊Dic3C2318yes—
94C2×Dic3⋊C4—24no—
95C2×Dic3⋊C4—24no—
96C2×Dic3⋊C4—28no—
97C2×Dic3⋊C4—28no—
98C2×Dic3⋊C4—44no—
99C2×Dic3⋊C4C2314yes—
100C2×Dic3⋊C4C2314yes—
101C2×Dic3⋊C4C2318yes—
102C2×Dic3⋊C4C2318yes—
103C2×Dic3⋊C4C2318yes—
104C2×C4×Dic3D418yes—
105C2×C4×Dic3C2314yes—
106#879—232no—
107C3×#95C2314yes—
108C3×#95C2318yes—
109C3⋊#95C2318yes—
110C3⋊#95C2318yes—
111C3⋊#95C2314yes—
112C3⋊#95C2318yes—
113C3⋊#95C2318yes—
114C3⋊#95C2318yes—
115C23×Q8—31no—
116C23×Q8C2×D311yescharacteristic
117C2×C4⋊Q8—38no—
118C2×C4⋊Q8—68no—
119C2×#126—38no—
120C2×#126—68no—
121C22×C4⋊C4—31no—
122C22×C4⋊C4—32no—
123C22×C4⋊C4—31no—
124C22×C4⋊C4—32no—
125C22×C4⋊C4—31no—
126C22×C4⋊C4—32no—
127C22×C4⋊C4—32no—
128C22×C22⋊C4—31no—
129C22×C22⋊C4—31no—
130C22×C22⋊C4—32no—
131C22×C22⋊C4C2×D311yescharacteristic
132C22×C22⋊C4C2×D312yes—
133C22×C42—31no—
134#400—316no—
135#399—316no—
136#398—316no—
137#397—316no—
138#395—38no—
139#395—38no—
140#393—38no—
141#393C2×D318yes—
142#383—38no—
143C2×#95—31no—
144C2×#95—32no—
145C2×#95—32no—
146C2×#95—32no—
147C2×#95—32no—
148C2×#95—32no—
149C2×#95C2×D311yescharacteristic
150C2×#95C2×D312yes—
151C24×C3—22no—
152C24×C3—24no—
153C24×C3—22no—
154C24×C3—24no—
155C24×C3—22no—
156C24×C3C2412yes—
157C24×C3C2411yescharacteristic
158C2×C3×Q8—28no—
159C2×C3×Q8—48no—
160C2×C3×Q8—416no—
161C22×C3×C4—24no—
162C22×C3×C4—24no—
163C22×C3×C4—22no—
164C22×C3×C4—24no—
165C22×C3×C4—24no—
166C22×C3×C4—24no—
167C22×C3×C4—24no—
168C22×C3×C4—24no—
169C22×C3×C4—24no—
170C22×C3×C4C2×D414yes—
171C22×C3×C4C2414yes—
172C22×C3×C4C2412yes—
173C22×C3×C4C2414yes—
174C22×C3×C4C2412yes—
175C22×Dic3—22no—
176C22×Dic3—24no—
177C22×Dic3—22no—
178C22×Dic3—22no—
179C22×Dic3—24no—
180C22×Dic3—22no—
181C22×Dic3—24no—
182C22×Dic3—24no—
183C22×Dic3—24no—
184C22×Dic3—24no—
185C22×Dic3—22no—
186C22×Dic3—24no—
187C22×Dic3—22no—
188C22×Dic3—24no—
189C22×Dic3C2×D414yes—
190C22×Dic3C2×D414yes—
191C22×Dic3C2414yes—
192C22×Dic3C2412yes—
193C22×Dic3C2414yes—
194C22×Dic3C2412yes—
195C22×Dic3C2412yes—
196C22×Dic3C2412yes—
197C2×Dic6—28no—
198C2×Dic6—48no—
199C2×Dic6—416no—
200C3×C22⋊C4—28no—
201C3×C22⋊C4—48no—
202#217—28no—
203#217—28no—
204#217—48no—
205C4⋊Dic3—28no—
206C4⋊Dic3—48no—
207Dic3⋊C4—28no—
208Dic3⋊C4—28no—
209Dic3⋊C4—216no—
210Dic3⋊C4—216no—
211Dic3⋊C4—48no—
212C4×Dic3C2×D4116yes—
213C25C22×D311yescharacteristic
214C22×Q8—24no—
215C22×Q8—34no—
216C22×Q8—44no—
217C22×Q8—64no—
218C22×Q8—124no—
219C22×Q8C22×D314yes—
220C23×C4—31no—
221C23×C4—31no—
222C23×C4—31no—
223C23×C4—32no—
224C23×C4—31no—
225C23×C4—32no—
226C23×C4C22×D312yes—
227C23×C4C22×D311yescharacteristic
228C23×C4C22×D312yes—
229C23×C4C22×D311yescharacteristic
230C4⋊Q8—632no—
231#126—632no—
232C2×C4⋊C4—34no—
233C2×C4⋊C4—34no—
234C2×C4⋊C4—34no—
235C2×C4⋊C4—38no—
236C2×C4⋊C4—38no—
237C2×C4⋊C4—38no—
238C2×C4⋊C4—38no—
239C2×C4⋊C4—64no—
240C2×C4⋊C4—64no—
241C2×C4⋊C4—68no—
242C2×C4⋊C4—68no—
243C2×C4⋊C4—64no—
244C2×C4⋊C4—124no—
245C2×C4⋊C4—124no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—34no—
248C2×C22⋊C4—34no—
249C2×C22⋊C4—38no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—64no—
252C2×C22⋊C4—64no—
253C2×C22⋊C4—124no—
254C2×C22⋊C4C22×D314yes—
255C2×C22⋊C4C22×D318yes—
256C2×C42—38no—
257C2×C42—34no—
258#95—34no—
259#95—38no—
260#95—38no—
261#95—38no—
262#95—38no—
263#95—38no—
264#95C22×D314yes—
265#95C22×D318yes—
266C23×C3—22no—
267C23×C3—24no—
268C23×C3—22no—
269C23×C3—22no—
270C23×C3—24no—
271C23×C3—44no—
272C23×C3—44no—
273C23×C3—44no—
274C23×C3—48no—
275C23×C3—48no—
276C23×C3C22×D412yes—
277C23×C3H(5,𝔽2)14yes—
278C23×C32−1+412yes—
279C23×C32−1+414yes—
280C23×C32−1+412yes—
281C23×C3C2511yescharacteristic
282C3×Q8—416no—
283C2×C3×C4—28no—
284C2×C3×C4—24no—
285C2×C3×C4—24no—
286C2×C3×C4—22no—
287C2×C3×C4—24no—
288C2×C3×C4—24no—
289C2×C3×C4—24no—
290C2×C3×C4—24no—
291C2×C3×C4—24no—
292C2×C3×C4—24no—
293C2×C3×C4—48no—
294C2×C3×C4—44no—
295C2×C3×C4—48no—
296C2×C3×C4C22×D414yes—
297C2×Dic3—28no—
298C2×Dic3—28no—
299C2×Dic3—22no—
300C2×Dic3—24no—
301C2×Dic3—22no—
302C2×Dic3—22no—
303C2×Dic3—24no—
304C2×Dic3—22no—
305C2×Dic3—24no—
306C2×Dic3—24no—
307C2×Dic3—24no—
308C2×Dic3—24no—
309C2×Dic3—22no—
310C2×Dic3—24no—
311C2×Dic3—22no—
312C2×Dic3—24no—
313C2×Dic3—48no—
314C2×Dic3—44no—
315C2×Dic3—44no—
316C2×Dic3—48no—
317C2×Dic3C22×D414yes—
318C2×Dic3C22×D414yes—
319Dic6—416no—
320C24—22no—
321C24—24no—
322C24—22no—
323C24—24no—
324C24—22no—
325C24C23×D312yes—
326C24C23×D311yescharacteristic
327C2×Q8—28no—
328C2×Q8—48no—
329C2×Q8—416no—
330C2×Q8—68no—
331C2×Q8—128no—
332C2×Q8—1216no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—34no—
343C22×C4—34no—
344C22×C4—32no—
345C22×C4—32no—
346C22×C4—34no—
347C22×C4—32no—
348C22×C4—34no—
349C22×C4—32no—
350C22×C4—62no—
351C22×C4—64no—
352C22×C4—62no—
353C22×C4—62no—
354C22×C4—62no—
355C22×C4—64no—
356C22×C4—64no—
357C22×C4—64no—
358C22×C4—64no—
359C22×C4—64no—
360C22×C4—64no—
361C22×C4—62no—
362C22×C4—64no—
363C22×C4—62no—
364C22×C4D3×D414yes—
365C22×C4C23×D314yes—
366C22×C4C23×D312yes—
367C22×C4C23×D314yes—
368C22×C4C23×D312yes—
369C4⋊C4—68no—
370C4⋊C4—68no—
371C4⋊C4—616no—
372C4⋊C4—616no—
373C4⋊C4—68no—
374C4⋊C4—128no—
375C4⋊C4—128no—
376C22⋊C4—28no—
377C22⋊C4—48no—
378C22⋊C4—68no—
379C22⋊C4—68no—
380C22⋊C4—128no—
381C42—316no—
382C22×C3—42no—
383C22×C3—42no—
384C22×C3—42no—
385C22×C3—44no—
386C22×C3—44no—
387C22×C3—88no—
388C22×C3#53514yes—
389C22×C3#53614yes—
390C22×C3#53614yes—
391C22×C3#56318yes—
392C22×C3#56318yes—
393C22×C3C23×D411yescharacteristic
394C22×C3C2×H(5,𝔽2)12yes—
395C22×C3C2×2−1+411yescharacteristic
396C22×C3C2×2−1+411yescharacteristic
397C22×C3C2×2−1+412yes—
398C3×C4—24no—
399C3×C4—44no—
400C3×C4—42no—
401C3×C4—44no—
402Dic3—24no—
403Dic3—24no—
404Dic3—44no—
405Dic3—42no—
406Dic3—42no—
407Dic3—44no—
408C23—22no—
409C23—24no—
410C23—22no—
411C23—22no—
412C23—24no—
413C23—48no—
414C23—48no—
415C23—44no—
416C23—44no—
417C23—44no—
418C23C2×D3×D412yes—
419C23#100414yes—
420C23#100712yes—
421C23#101014yes—
422C23#101012yes—
423C23C24×D311yescharacteristic
424Q8—416no—
425Q8—1216no—
426C2×C4—24no—
427C2×C4—28no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—34no—
437C2×C4—34no—
438C2×C4—48no—
439C2×C4—44no—
440C2×C4—48no—
441C2×C4—68no—
442C2×C4—62no—
443C2×C4—64no—
444C2×C4—62no—
445C2×C4—68no—
446C2×C4—62no—
447C2×C4—62no—
448C2×C4—64no—
449C2×C4—64no—
450C2×C4—64no—
451C2×C4—64no—
452C2×C4—64no—
453C2×C4—64no—
454C2×C4—62no—
455C2×C4—64no—
456C2×C4—62no—
457C2×C4—128no—
458C2×C4—128no—
459C2×C4—124no—
460C2×C4—124no—
461C2×C4C2×D3×D414yes—
462C2×C3—82no—
463C2×C3#282118yescomplemented
464C2×C3C2×#53511yescharacteristic
465C2×C3C2×#53611yescharacteristic
466C2×C3C2×#53611yescharacteristic
467C2×C3C2×#56312yes—
468C2×C3C2×#56312yes—
469C22—44no—
470C22—42no—
471C22—42no—
472C22—44no—
473C22—42no—
474C22—88no—
475C22#559214yes—
476C22#562814yes—
477C22#564618yes—
478C22#567218yes—
479C22#567514yes—
480C22C22×D3×D411yescharacteristic
481C22C2×#100412yes—
482C22C2×#100711yescharacteristic
483C22C2×#101011yescharacteristic
484C22C2×#101012yes—
485C4—24no—
486C4—44no—
487C4—42no—
488C4—44no—
489C4—64no—
490C4—64no—
491C4—124no—
492C4—122no—
493C4—124no—
494C4—122no—
495C3C2×#282111yescharacteristic, complemented, Sylow
496C2—82no—
497C2#7685418yesdirect product factor
498C2C2×#559211yescharacteristic
499C2C2×#564612yes—
500C2C2×#562811yescharacteristic
501C2C2×#567212yes—
502C2C2×#567511yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#393C2
socleC24×C3C24
prime radicalC25×C3C23
hypercenterC2×#393D3
hypocenterC3C2×#2821
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#393

Complements

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

NH ≅ G/Ntype
1. C22×#944C2semidirect
4. C2×#5230C2semidirect
8. #76854C2direct (N × H)
9. C2×#4660C2semidirect
10. C2×#4657C2semidirect
11. C2×#4656C2semidirect
12. C2×#4651C2semidirect
22. C2×#944C22semidirect
36. #5230C22semidirect
46. #4660C22semidirect
47. #4657C22semidirect
48. #4656C22semidirect
49. #4651C22semidirect
463. C2×C3#2821semidirect
495. C3C2×#2821semidirect
497. C2#76854direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#9442
C22×#8792
C2×C3×#3932
C2×#52302
C2×#49492
C2×#49522
C2×#49472
#768542
C2×#46602
C2×#46572
C2×#46562
C2×#46512
C2×#28213