Finite Group Databasebeta

← C2×#78345 #10965007

Subgroups — C2×#78345 #10965007

Order 768.

2918 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D3×C4⋊C4C211yesmaximal, characteristic
2C22×#892C211yesmaximal, characteristic
3C2×C3×#400C211yesmaximal, characteristic, complemented
4C2×#4985C211yesmaximal, characteristic, complemented
5C2×#4981C211yesmaximal, characteristic, complemented
6#78345C2116yesmaximal, direct product factor
7C2×#4668C212yesmaximal
8C2×#4664C212yesmaximal
9C2×#4664C211yesmaximal, characteristic
10C2×#4661C211yesmaximal, characteristic, complemented
11C2×#4655C212yesmaximal, complemented
12C2×#4654C212yesmaximal, complemented
13C2×#2484—31nomaximal, Sylow
14C23×C4×D3C2211yescharacteristic
15C23×C4×D3C2211yescharacteristic
16C23×C4×D3C2211yescharacteristic
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C4⋊C4C2211yescharacteristic
19C22×(C2×D3)⋊C4C2212yes—
20C22×(C2×D3)⋊C4C2212yes—
21C22×C4⋊Dic3C2212yes—
22C22×C4⋊Dic3C2211yescharacteristic
23C22×C4⋊Dic3C2211yescharacteristic
24C22×Dic3⋊C4C2211yescharacteristic
25C22×Dic3⋊C4C2212yes—
26C22×Dic3⋊C4C2211yescharacteristic
27C22×C4×Dic3C2212yes—
28C2×#892—28no—
29C2×#892C2218yes—
30C2×D3×C4⋊C4—28no—
31C2×D3×C4⋊C4C2218yes—
32C3×#400C2218yescomplemented
33C2×C3×#95C2212yes—
34C2×C3×#95C2212yes—
35C2×C3×#95C2211yescharacteristic
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2212yes—
38C2×C3⋊#95C2212yes—
39C2×C3⋊#95C2212yes—
40C2×C3⋊#95C2212yes—
41C2×C3⋊#95C2211yescharacteristic
42#4985C2218yescomplemented
43#4981C2218yescomplemented
44#4668C22116yes—
45#4664C2218yes—
46#4664C22116yes—
47#4661C2218yescomplemented
48#4655C22116yescomplemented
49#4654C22116yescomplemented
50C22×#117—31no—
51C23×C4⋊C4—31no—
52#2484—316no—
53C2×#400—31no—
54C2×#400D311yescharacteristic, complemented
55C2×#398—31no—
56C2×#398—31no—
57C2×#387—32no—
58C2×#387—32no—
59C2×#385—32no—
60C2×#381—32no—
61C2×#381—31no—
62C24×D3C2311yescharacteristic
63C23×C3×C4C2311yescharacteristic
64C23×C3×C4C2312yes—
65C23×C3×C4C2312yes—
66C23×C3×C4C2311yescharacteristic
67C23×C3×C4C2311yescharacteristic
68C23×Dic3C2311yescharacteristic
69C23×Dic3C2312yes—
70C23×Dic3C2312yes—
71C23×Dic3C2311yescharacteristic
72C23×Dic3C2311yescharacteristic
73C22×C4×D3—24no—
74C22×C4×D3—28no—
75C22×C4×D3—24no—
76C22×C4×D3—24no—
77C22×C4×D3—24no—
78C22×C4×D3—28no—
79C22×C4×D3D418yes—
80C22×C4×D3C2314yes—
81C22×C4×D3C2314yes—
82C22×C4×D3C2314yes—
83C2×C3×C4⋊C4—24no—
84C2×C3×C4⋊C4—24no—
85C2×C3×C4⋊C4C2314yes—
86C2×C3×C4⋊C4C2314yes—
87C2×(C2×D3)⋊C4—28no—
88C2×(C2×D3)⋊C4—28no—
89C2×(C2×D3)⋊C4C2318yes—
90C2×(C2×D3)⋊C4C2318yes—
91C2×C4⋊Dic3—24no—
92C2×C4⋊Dic3—24no—
93C2×C4⋊Dic3—28no—
94C2×C4⋊Dic3C2318yes—
95C2×C4⋊Dic3C2314yes—
96C2×C4⋊Dic3C2314yes—
97C2×Dic3⋊C4—24no—
98C2×Dic3⋊C4—28no—
99C2×Dic3⋊C4—24no—
100C2×Dic3⋊C4C2314yes—
101C2×Dic3⋊C4C2318yes—
102C2×Dic3⋊C4C2314yes—
103C2×C4×Dic3—28no—
104C2×C4×Dic3C2318yes—
105#892—232no—
106D3×C4⋊C4—232no—
107C3×#95C2318yes—
108C3×#95C2318yes—
109C3×#95C2314yes—
110C3⋊#95C2318yes—
111C3⋊#95C2318yes—
112C3⋊#95C2318yes—
113C3⋊#95C2318yes—
114C3⋊#95C2318yes—
115C3⋊#95C2314yes—
116C24×C4—31no—
117C24×C4—31no—
118C24×C4—31no—
119C2×#117—38no—
120C2×#117—68no—
121C22×C4⋊C4—31no—
122C22×C4⋊C4—31no—
123C22×C4⋊C4—32no—
124C22×C4⋊C4—32no—
125C22×C4⋊C4—31no—
126C22×C4⋊C4—31no—
127C22×C4⋊C4—38no—
128C22×C4⋊C4—68no—
129C22×C4⋊C4C2×D311yescharacteristic
130C22×C4⋊C4C2×D311yescharacteristic
131C22×C22⋊C4—32no—
132C22×C22⋊C4—32no—
133C22×C42—32no—
134#400—38no—
135#400C2×D318yescomplemented
136#398—38no—
137#398—38no—
138#387—316no—
139#387—316no—
140#385—316no—
141#381—38no—
142#381—316no—
143C2×#95—31no—
144C2×#95—32no—
145C2×#95—32no—
146C2×#95—32no—
147C2×#95—32no—
148C2×#95—32no—
149C2×#95C2×D311yescharacteristic
150C2×#95C2×D312yes—
151C2×#95C2×D312yes—
152C24×C3C2411yescharacteristic
153C23×D3—22no—
154C23×D3—22no—
155C23×D3—24no—
156C23×D3C2×D414yes—
157C23×D3#4118yes—
158C23×D3C2412yes—
159C22×C3×C4—22no—
160C22×C3×C4—24no—
161C22×C3×C4—22no—
162C22×C3×C4—24no—
163C22×C3×C4—22no—
164C22×C3×C4—24no—
165C22×C3×C4—24no—
166C22×C3×C4—24no—
167C22×C3×C4—28no—
168C22×C3×C4—24no—
169C22×C3×C4C2×D414yes—
170C22×C3×C4#4114yes—
171C22×C3×C4C2414yes—
172C22×C3×C4C2414yes—
173C22×C3×C4C2412yes—
174C22×C3×C4C2412yes—
175C22×C3×C4C2412yes—
176C22×Dic3—22no—
177C22×Dic3—22no—
178C22×Dic3—24no—
179C22×Dic3—24no—
180C22×Dic3—22no—
181C22×Dic3—24no—
182C22×Dic3—24no—
183C22×Dic3—24no—
184C22×Dic3—24no—
185C22×Dic3—24no—
186C22×Dic3—24no—
187C22×Dic3C2×D414yes—
188C22×Dic3#4114yes—
189C22×Dic3C2414yes—
190C22×Dic3C2412yes—
191C22×Dic3C2414yes—
192C22×Dic3C2412yes—
193C22×Dic3C2412yes—
194C2×C4×D3—232no—
195C2×C4×D3—232no—
196C2×C4×D3—28no—
197C2×C4×D3—216no—
198C2×C4×D3—28no—
199C2×C4×D3—28no—
200C2×C4×D3—28no—
201C2×C4×D3—216no—
202C2×C4×D3—416no—
203C2×C4×D3C2×D4116yes—
204C3×C4⋊C4—28no—
205C3×C4⋊C4—28no—
206(C2×D3)⋊C4—216no—
207(C2×D3)⋊C4—216no—
208C4⋊Dic3—28no—
209C4⋊Dic3—28no—
210C4⋊Dic3—216no—
211Dic3⋊C4—28no—
212Dic3⋊C4—216no—
213Dic3⋊C4—28no—
214C4×Dic3—216no—
215C25—31no—
216C23×C4—31no—
217C23×C4—31no—
218C23×C4—38no—
219C23×C4—34no—
220C23×C4—32no—
221C23×C4—32no—
222C23×C4—31no—
223C23×C4—34no—
224C23×C4—34no—
225C23×C4—64no—
226C23×C4—64no—
227C23×C4—68no—
228C23×C4—64no—
229C23×C4—68no—
230C23×C4—64no—
231C23×C4C22×D312yes—
232C23×C4C22×D312yes—
233C23×C4C22×D311yescharacteristic
234C23×C4C22×D311yescharacteristic
235C23×C4C22×D311yescharacteristic
236#117—632no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—34no—
240C2×C4⋊C4—34no—
241C2×C4⋊C4—34no—
242C2×C4⋊C4—34no—
243C2×C4⋊C4—38no—
244C2×C4⋊C4—38no—
245C2×C4⋊C4—632no—
246C2×C4⋊C4—64no—
247C2×C4⋊C4—64no—
248C2×C4⋊C4—68no—
249C2×C4⋊C4—64no—
250C2×C4⋊C4—64no—
251C2×C4⋊C4—68no—
252C2×C4⋊C4C22×D314yes—
253C2×C4⋊C4C22×D314yes—
254C2×C22⋊C4—38no—
255C2×C22⋊C4—38no—
256C2×C22⋊C4—68no—
257C2×C22⋊C4—68no—
258C2×C42—38no—
259C2×C42—68no—
260#95—38no—
261#95—38no—
262#95—38no—
263#95—38no—
264#95—34no—
265#95—38no—
266#95C22×D318yes—
267#95C22×D314yes—
268#95C22×D318yes—
269C23×C3C22×D412yes—
270C23×C3C2×#4112yes—
271C23×C3C2×#4114yes—
272C23×C3C2×#4112yes—
273C23×C32−1+414yes—
274C23×C3C2511yescharacteristic
275C22×D3—28no—
276C22×D3—28no—
277C22×D3—216no—
278C22×D3—216no—
279C22×D3—22no—
280C22×D3—22no—
281C22×D3—24no—
282C22×D3#123116yes—
283C22×D3C22×D414yes—
284C22×D3C2×#4118yes—
285C2×C3×C4—28no—
286C2×C3×C4—28no—
287C2×C3×C4—22no—
288C2×C3×C4—24no—
289C2×C3×C4—22no—
290C2×C3×C4—24no—
291C2×C3×C4—22no—
292C2×C3×C4—24no—
293C2×C3×C4—24no—
294C2×C3×C4—24no—
295C2×C3×C4—28no—
296C2×C3×C4—24no—
297C2×C3×C4—44no—
298C2×C3×C4—48no—
299C2×C3×C4—48no—
300C2×C3×C4C22×D414yes—
301C2×C3×C4C2×#4114yes—
302C2×Dic3—28no—
303C2×Dic3—28no—
304C2×Dic3—22no—
305C2×Dic3—22no—
306C2×Dic3—24no—
307C2×Dic3—24no—
308C2×Dic3—22no—
309C2×Dic3—24no—
310C2×Dic3—24no—
311C2×Dic3—24no—
312C2×Dic3—24no—
313C2×Dic3—24no—
314C2×Dic3—24no—
315C2×Dic3—44no—
316C2×Dic3—48no—
317C2×Dic3—48no—
318C2×Dic3C22×D414yes—
319C2×Dic3C2×#4114yes—
320C4×D3—232no—
321C4×D3—232no—
322C4×D3—416no—
323C24—34no—
324C24—38no—
325C24—32no—
326C24—62no—
327C24—62no—
328C24—64no—
329C24C23×D311yescharacteristic
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—28no—
339C22×C4—24no—
340C22×C4—34no—
341C22×C4—34no—
342C22×C4—32no—
343C22×C4—32no—
344C22×C4—316no—
345C22×C4—34no—
346C22×C4—32no—
347C22×C4—34no—
348C22×C4—68no—
349C22×C4—68no—
350C22×C4—632no—
351C22×C4—616no—
352C22×C4—632no—
353C22×C4—68no—
354C22×C4—616no—
355C22×C4—62no—
356C22×C4—62no—
357C22×C4—64no—
358C22×C4—68no—
359C22×C4—64no—
360C22×C4—62no—
361C22×C4—64no—
362C22×C4—64no—
363C22×C4—64no—
364C22×C4—64no—
365C22×C4—64no—
366C22×C4—64no—
367C22×C4—1216no—
368C22×C4D3×D414yes—
369C22×C4#23914yes—
370C22×C4C23×D314yes—
371C22×C4C23×D314yes—
372C22×C4C23×D312yes—
373C22×C4C23×D312yes—
374C22×C4C23×D312yes—
375C4⋊C4—28no—
376C4⋊C4—28no—
377C4⋊C4—68no—
378C4⋊C4—68no—
379C4⋊C4—616no—
380C4⋊C4—68no—
381C4⋊C4—68no—
382C4⋊C4—616no—
383C22⋊C4—616no—
384C22⋊C4—616no—
385C42—616no—
386C22×C3C2×#12314yes—
387C22×C3#52514yes—
388C22×C3#53614yes—
389C22×C3#55218yes—
390C22×C3#55218yes—
391C22×C3C23×D411yescharacteristic
392C22×C3C22×#4111yescharacteristic
393C22×C3C22×#4112yes—
394C22×C3C22×#4111yescharacteristic
395C22×C3C2×2−1+412yes—
396C2×D3—24no—
397C2×D3—24no—
398C2×D3—28no—
399C2×D3—28no—
400C2×D3—232no—
401C2×D3C2×#12318yes—
402C3×C4—24no—
403C3×C4—24no—
404C3×C4—42no—
405C3×C4—44no—
406C3×C4—44no—
407Dic3—24no—
408Dic3—24no—
409Dic3—42no—
410Dic3—44no—
411Dic3—44no—
412C23—316no—
413C23—34no—
414C23—38no—
415C23—68no—
416C23—616no—
417C23—62no—
418C23—62no—
419C23—64no—
420C23—616no—
421C23—68no—
422C23C2×D3×D412yes—
423C23C2×#23912yes—
424C23D3×#4114yes—
425C23D3×#4112yes—
426C23#101014yes—
427C23C24×D311yescharacteristic
428C2×C4—28no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—28no—
433C2×C4—24no—
434C2×C4—22no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—28no—
439C2×C4—24no—
440C2×C4—34no—
441C2×C4—34no—
442C2×C4—44no—
443C2×C4—48no—
444C2×C4—48no—
445C2×C4—632no—
446C2×C4—632no—
447C2×C4—68no—
448C2×C4—62no—
449C2×C4—62no—
450C2×C4—64no—
451C2×C4—68no—
452C2×C4—64no—
453C2×C4—62no—
454C2×C4—64no—
455C2×C4—64no—
456C2×C4—64no—
457C2×C4—64no—
458C2×C4—64no—
459C2×C4—64no—
460C2×C4—1216no—
461C2×C4—124no—
462C2×C4—128no—
463C2×C4—128no—
464C2×C4C2×D3×D414yes—
465C2×C4C2×#23914yes—
466C2×C3#248418yescomplemented
467C2×C3C22×#12311yescharacteristic
468C2×C3C2×#52511yescharacteristic
469C2×C3C2×#53611yescharacteristic
470C2×C3C2×#55212yes—
471C2×C3C2×#55212yes—
472D3—28no—
473C22—38no—
474C22—632no—
475C22—64no—
476C22—68no—
477C22—68no—
478C22—64no—
479C22#562714yes—
480C22D3×#12314yes—
481C22#567514yes—
482C22#569418yes—
483C22#570818yes—
484C22C22×D3×D411yescharacteristic
485C22C22×#23911yescharacteristic
486C22C2×D3×#4112yes—
487C22C2×D3×#4111yescharacteristic
488C22C2×#101012yes—
489C4—24no—
490C4—24no—
491C4—44no—
492C4—44no—
493C4—42no—
494C4—64no—
495C4—64no—
496C4—124no—
497C4—124no—
498C4—122no—
499C3C2×#248411yescharacteristic, complemented, Sylow
500C2—68no—
501C2#7834518yesdirect product factor
502C2C2×#562711yescharacteristic
503C2C2×D3×#12311yescharacteristic
504C2C2×#567511yescharacteristic
505C2C2×#569412yes—
506C2C2×#570812yes—
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#400C2
socleC24×C3C24
prime radicalC24×D3C23
hypercenterC2×#400D3
hypocenterC3C2×#2484
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#400

Complements

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

NH ≅ G/Ntype
3. C2×C3×#400C2semidirect
4. C2×#4985C2semidirect
5. C2×#4981C2semidirect
6. #78345C2direct (N × H)
10. C2×#4661C2semidirect
11. C2×#4655C2semidirect
12. C2×#4654C2semidirect
32. C3×#400C22semidirect
42. #4985C22semidirect
43. #4981C22semidirect
47. #4661C22semidirect
48. #4655C22semidirect
49. #4654C22semidirect
54. C2×#400D3semidirect
135. #400C2×D3semidirect
466. C2×C3#2484semidirect
499. C3C2×#2484semidirect
501. C2#78345direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D3×C4⋊C42
C22×#8922
C2×C3×#4002
C2×#49852
C2×#49812
#783452
C2×#46682
C2×#46642
C2×#46612
C2×#46552
C2×#46542
C2×#24843