Finite Group Databasebeta

← C2×#74333 #10964321

Subgroups — C2×#74333 #10964321

Order 768.

2726 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#898C212yesmaximal, complemented
2C2×C3×#384C211yesmaximal, characteristic, complemented
3C2×#4992C212yesmaximal, complemented
4C2×#4994C212yesmaximal, complemented
5C2×#4988C211yesmaximal, characteristic
6C2×#4988C211yesmaximal, characteristic
7C2×#4984C211yesmaximal, characteristic, complemented
8C2×#4990C212yesmaximal, complemented
9C2×Dic3×C4⋊C4C211yesmaximal, characteristic, complemented
10C2×#4939C211yesmaximal, characteristic
11C2×C4×Dic3⋊C4C211yesmaximal, characteristic, complemented
12#74333C2116yesmaximal, direct product factor
13C2×#2512—31nomaximal, Sylow
14C23×D12C2211yescharacteristic
15C22×C3×C4⋊C4C2212yes—
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C4⋊C4C2212yes—
19C22×C3×C42C2211yescharacteristic
20C22×(C2×D3)⋊C4C2212yes—
21C22×(C2×D3)⋊C4C2212yes—
22C22×(C2×D3)⋊C4C2212yes—
23C22×(C2×D3)⋊C4C2212yes—
24C22×(C2×D3)⋊C4C2212yes—
25C22×(C2×D3)⋊C4C2212yes—
26C22×C4⋊Dic3C2212yes—
27C22×C4⋊Dic3C2211yescharacteristic
28C22×Dic3⋊C4C2211yescharacteristic
29C22×Dic3⋊C4C2211yescharacteristic
30C22×C4×Dic3C2211yescharacteristic
31C22×C4×Dic3C2211yescharacteristic
32C22×C4×Dic3C2212yes—
33C2×#898—216no—
34C2×#898C22116yescomplemented
35C3×#384C2218yescomplemented
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2211yescharacteristic
38C2×C3⋊#95C2212yes—
39C2×C3⋊#95C2211yescharacteristic
40#4994C22116yescomplemented
41#4992C22116yescomplemented
42#4990C22116yescomplemented
43#4988C2218yes—
44#4988C2218yes—
45#4984C2218yescomplemented
46Dic3×C4⋊C4C2218yescomplemented
47#4939C2218yes—
48C4×Dic3⋊C4C2218yescomplemented
49C22×#126—32no—
50#2512—316no—
51C2×#397—32no—
52C2×#396—32no—
53C2×#390—31no—
54C2×#390—31no—
55C2×#390—31no—
56C2×#389—31no—
57C2×#388—32no—
58C2×#384D311yescharacteristic, complemented
59C2×C4×C4⋊C4—31no—
60C2×C4×C4⋊C4—31no—
61C24×D3C2312yes—
62C23×C3×C4C2312yes—
63C23×C3×C4C2312yes—
64C23×C3×C4C2311yescharacteristic
65C23×C3×C4C2311yescharacteristic
66C23×C3×C4C2311yescharacteristic
67C23×Dic3C2311yescharacteristic
68C23×Dic3C2311yescharacteristic
69C23×Dic3C2312yes—
70C23×Dic3C2311yescharacteristic
71C23×Dic3C2311yescharacteristic
72C22×D12—24no—
73C22×D12—24no—
74C22×D12—24no—
75C22×D12C2314yes—
76C2×C3×C4⋊C4—28no—
77C2×C3×C4⋊C4—24no—
78C2×C3×C4⋊C4—24no—
79C2×C3×C4⋊C4—44no—
80C2×C3×C4⋊C4C2314yes—
81C2×C3×C4⋊C4C2314yes—
82C2×C3×C4⋊C4C2318yes—
83C2×C3×C4⋊C4C2318yes—
84C2×C3×C42—24no—
85C2×C3×C42C2314yes—
86C2×(C2×D3)⋊C4—28no—
87C2×(C2×D3)⋊C4—28no—
88C2×(C2×D3)⋊C4—28no—
89C2×(C2×D3)⋊C4—28no—
90C2×(C2×D3)⋊C4—44no—
91C2×(C2×D3)⋊C4—44no—
92C2×(C2×D3)⋊C4C2318yes—
93C2×(C2×D3)⋊C4C2318yes—
94C2×(C2×D3)⋊C4C2318yes—
95C2×(C2×D3)⋊C4C2318yes—
96C2×(C2×D3)⋊C4C2318yes—
97C2×(C2×D3)⋊C4C2318yes—
98C2×C4⋊Dic3—24no—
99C2×C4⋊Dic3—28no—
100C2×C4⋊Dic3C2314yes—
101C2×C4⋊Dic3C2318yes—
102C2×Dic3⋊C4—24no—
103C2×Dic3⋊C4D418yescomplemented
104C2×Dic3⋊C4C2314yes—
105C2×Dic3⋊C4C2314yes—
106C2×C4×Dic3—24no—
107C2×C4×Dic3—24no—
108C2×C4×Dic3—28no—
109C2×C4×Dic3C2314yes—
110C2×C4×Dic3C2314yes—
111C2×C4×Dic3C2318yes—
112#898—264no—
113C3⋊#95C2314yes—
114C3⋊#95C2314yes—
115C3⋊#95C2318yes—
116C3⋊#95C2318yes—
117C23×D4—31no—
118C2×#126—316no—
119C2×#126—616no—
120C22×C4⋊C4—31no—
121C22×C4⋊C4—31no—
122C22×C4⋊C4—32no—
123C22×C4⋊C4—31no—
124C22×C4⋊C4C2×D312yes—
125C22×C4⋊C4C2×D311yescharacteristic
126C22×C4⋊C4C2×D311yescharacteristic
127C22×C4⋊C4C2×D312yes—
128C22×C22⋊C4—32no—
129C22×C22⋊C4—32no—
130C22×C22⋊C4—32no—
131C22×C22⋊C4—32no—
132C22×C22⋊C4—32no—
133C22×C22⋊C4—32no—
134C22×C42—32no—
135C22×C42—31no—
136C22×C42—31no—
137C22×C42C2×D311yescharacteristic
138#397—316no—
139#396—316no—
140#390—38no—
141#390—38no—
142#390—38no—
143#389—38no—
144#388—316no—
145#384C2×D318yescomplemented
146C4×C4⋊C4—38no—
147C4×C4⋊C4—38no—
148C2×#95—32no—
149C2×#95—31no—
150C2×#95—32no—
151C2×#95—31no—
152C24×C3C2411yescharacteristic
153C23×D3—24no—
154C23×D3—24no—
155C23×D3—24no—
156C23×D3—24no—
157C23×D3—24no—
158C23×D3—24no—
159C23×D3—24no—
160C23×D3C2414yes—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—24no—
166C22×C3×C4—24no—
167C22×C3×C4—24no—
168C22×C3×C4—24no—
169C22×C3×C4—24no—
170C22×C3×C4—24no—
171C22×C3×C4C2×D414yes—
172C22×C3×C4#4114yes—
173C22×C3×C4#4114yes—
174C22×C3×C4#4114yes—
175C22×C3×C4#4114yes—
176C22×C3×C4C2414yes—
177C22×C3×C4C2414yes—
178C22×C3×C4C2412yes—
179C22×C3×C4C2412yes—
180C22×C3×C4C2412yes—
181C22×Dic3—22no—
182C22×Dic3—22no—
183C22×Dic3—22no—
184C22×Dic3—22no—
185C22×Dic3—24no—
186C22×Dic3—22no—
187C22×Dic3—22no—
188C22×Dic3—22no—
189C22×Dic3—24no—
190C22×Dic3—24no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3C2×D414yes—
195C22×Dic3C2×D414yes—
196C22×Dic3C2412yes—
197C22×Dic3C2412yes—
198C22×Dic3C2414yes—
199C22×Dic3C2412yes—
200C22×Dic3C2412yes—
201C2×D12—28no—
202C2×D12—28no—
203C2×D12—28no—
204C2×D12—416no—
205C3×C4⋊C4—216no—
206C3×C4⋊C4—28no—
207C3×C4⋊C4—28no—
208C3×C4⋊C4—48no—
209C3×C42—28no—
210(C2×D3)⋊C4—216no—
211(C2×D3)⋊C4—216no—
212(C2×D3)⋊C4—216no—
213(C2×D3)⋊C4—216no—
214(C2×D3)⋊C4—48no—
215(C2×D3)⋊C4—48no—
216C4⋊Dic3—28no—
217C4⋊Dic3—216no—
218Dic3⋊C4—28no—
219Dic3⋊C4C2×D4116yescomplemented
220C4×Dic3—28no—
221C4×Dic3—28no—
222C4×Dic3—216no—
223C25—32no—
224C22×D4—34no—
225C22×D4—64no—
226C22×D4—64no—
227C22×D4—64no—
228C23×C4—31no—
229C23×C4—31no—
230C23×C4—32no—
231C23×C4—31no—
232C23×C4—31no—
233C23×C4C22×D312yes—
234C23×C4C22×D312yes—
235C23×C4C22×D311yescharacteristic
236C23×C4C22×D311yescharacteristic
237C23×C4C22×D311yescharacteristic
238#126—664no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—38no—
243C2×C4⋊C4—34no—
244C2×C4⋊C4—38no—
245C2×C4⋊C4—34no—
246C2×C4⋊C4—34no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—68no—
249C2×C4⋊C4—64no—
250C2×C4⋊C4—64no—
251C2×C4⋊C4C22×D318yes—
252C2×C4⋊C4C22×D314yes—
253C2×C4⋊C4C22×D314yes—
254C2×C4⋊C4C22×D318yes—
255C2×C22⋊C4—38no—
256C2×C22⋊C4—38no—
257C2×C22⋊C4—38no—
258C2×C22⋊C4—38no—
259C2×C22⋊C4—38no—
260C2×C22⋊C4—38no—
261C2×C22⋊C4—68no—
262C2×C22⋊C4—68no—
263C2×C22⋊C4—68no—
264C2×C22⋊C4—68no—
265C2×C22⋊C4—124no—
266C2×C22⋊C4—124no—
267C2×C42—24no—
268C2×C42—38no—
269C2×C42—34no—
270C2×C42—34no—
271C2×C42—68no—
272C2×C42—64no—
273C2×C42—64no—
274C2×C42C22×D314yes—
275#95—34no—
276#95—38no—
277#95—34no—
278#95—38no—
279C23×C3C22×D412yes—
280C23×C3C2×#4112yes—
281C23×C3C2×#4112yes—
282C23×C3C2×#4112yes—
283C23×C3C2×#4112yes—
284C23×C3H(5,𝔽2)12yes—
285C23×C32−1+412yes—
286C23×C3C2511yescharacteristic
287C22×D3—24no—
288C22×D3—24no—
289C22×D3—24no—
290C22×D3—24no—
291C22×D3—24no—
292C22×D3—24no—
293C22×D3—24no—
294C22×D3—48no—
295C22×D3—48no—
296C22×D3—48no—
297C22×D3—48no—
298C22×D3—48no—
299C22×D3—48no—
300C22×D3—48no—
301C2×C3×C4—28no—
302C2×C3×C4—28no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—24no—
308C2×C3×C4—24no—
309C2×C3×C4—24no—
310C2×C3×C4—24no—
311C2×C3×C4—24no—
312C2×C3×C4—24no—
313C2×C3×C4—48no—
314C2×C3×C4—48no—
315C2×C3×C4#126116yescomplemented
316C2×C3×C4C22×D414yes—
317C2×C3×C4C2×#4114yes—
318C2×C3×C4C2×#4114yes—
319C2×C3×C4C2×#4114yes—
320C2×C3×C4C2×#4114yes—
321C2×Dic3—28no—
322C2×Dic3—28no—
323C2×Dic3—22no—
324C2×Dic3—22no—
325C2×Dic3—22no—
326C2×Dic3—22no—
327C2×Dic3—24no—
328C2×Dic3—22no—
329C2×Dic3—22no—
330C2×Dic3—22no—
331C2×Dic3—24no—
332C2×Dic3—24no—
333C2×Dic3—22no—
334C2×Dic3—22no—
335C2×Dic3—22no—
336C2×Dic3—48no—
337C2×Dic3—44no—
338C2×Dic3—44no—
339C2×Dic3C22×D414yes—
340C2×Dic3C22×D414yes—
341D12—416no—
342C24—34no—
343C24—64no—
344C24—64no—
345C24—64no—
346C24—64no—
347C24—64no—
348C24—64no—
349C24—64no—
350C24C23×D311yescharacteristic
351C2×D4—68no—
352C2×D4—68no—
353C2×D4—68no—
354C2×D4—1216no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—34no—
366C22×C4—34no—
367C22×C4—32no—
368C22×C4—32no—
369C22×C4—32no—
370C22×C4—32no—
371C22×C4—34no—
372C22×C4—62no—
373C22×C4—62no—
374C22×C4—62no—
375C22×C4—62no—
376C22×C4—62no—
377C22×C4—64no—
378C22×C4—62no—
379C22×C4—62no—
380C22×C4—62no—
381C22×C4—64no—
382C22×C4—64no—
383C22×C4—62no—
384C22×C4—62no—
385C22×C4#23514yes—
386C22×C4D3×D414yes—
387C22×C4#23714yes—
388C22×C4#23914yes—
389C22×C4#23914yes—
390C22×C4C23×D314yes—
391C22×C4C23×D314yes—
392C22×C4C23×D312yes—
393C22×C4C23×D312yes—
394C22×C4C23×D312yes—
395C4⋊C4—216no—
396C4⋊C4—28no—
397C4⋊C4—28no—
398C4⋊C4—316no—
399C4⋊C4—48no—
400C4⋊C4—616no—
401C4⋊C4—68no—
402C4⋊C4—68no—
403C22⋊C4—616no—
404C22⋊C4—616no—
405C22⋊C4—616no—
406C22⋊C4—616no—
407C22⋊C4—128no—
408C22⋊C4—128no—
409C42—28no—
410C42—616no—
411C42—68no—
412C42—68no—
413C22×C3C2×#12614yes—
414C22×C3#53214yes—
415C22×C3#54214yes—
416C22×C3#54714yes—
417C22×C3#55014yes—
418C22×C3#55314yes—
419C22×C3#55614yes—
420C22×C3C23×D411yescharacteristic
421C22×C3C22×#4111yescharacteristic
422C22×C3C22×#4111yescharacteristic
423C22×C3C22×#4111yescharacteristic
424C22×C3C22×#4111yescharacteristic
425C22×C3C2×H(5,𝔽2)11yescharacteristic
426C22×C3C2×2−1+411yescharacteristic
427C2×D3—44no—
428C2×D3—44no—
429C2×D3—44no—
430C2×D3—44no—
431C2×D3—44no—
432C2×D3—44no—
433C2×D3—44no—
434C2×D3—816no—
435C3×C4—24no—
436C3×C4—24no—
437C3×C4—44no—
438C3×C4—44no—
439C3×C4C2×#12618yescomplemented
440Dic3—24no—
441Dic3—24no—
442Dic3—44no—
443Dic3—42no—
444Dic3—42no—
445C23—64no—
446C23—64no—
447C23—64no—
448C23—64no—
449C23—64no—
450C23—64no—
451C23—64no—
452C23—128no—
453C23—128no—
454C23—128no—
455C23—128no—
456C23—128no—
457C23—128no—
458C23—128no—
459C23C2×#23512yes—
460C23C2×D3×D412yes—
461C23C2×#23712yes—
462C23#100412yes—
463C23C2×#23912yes—
464C23C2×#23912yes—
465C23#100712yes—
466C23C24×D311yescharacteristic
467D4—1216no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—28no—
473C2×C4—28no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—34no—
481C2×C4—34no—
482C2×C4—48no—
483C2×C4—48no—
484C2×C4—62no—
485C2×C4—62no—
486C2×C4—62no—
487C2×C4—62no—
488C2×C4—68no—
489C2×C4—68no—
490C2×C4—62no—
491C2×C4—64no—
492C2×C4—62no—
493C2×C4—62no—
494C2×C4—62no—
495C2×C4—64no—
496C2×C4—64no—
497C2×C4—62no—
498C2×C4—62no—
499C2×C4—128no—
500C2×C4—124no—
501C2×C4—124no—
502C2×C4#898116yescomplemented
503C2×C4C2×#23514yes—
504C2×C4C2×D3×D414yes—
505C2×C4C2×#23714yes—
506C2×C4C2×#23914yes—
507C2×C4C2×#23914yes—
508C2×C3#251218yescomplemented
509C2×C3C22×#12611yescharacteristic
510C2×C3C2×#53211yescharacteristic
511C2×C3C2×#54211yescharacteristic
512C2×C3C2×#54711yescharacteristic
513C2×C3C2×#55011yescharacteristic
514C2×C3C2×#55311yescharacteristic
515C2×C3C2×#55611yescharacteristic
516D3—84no—
517C22—124no—
518C22—124no—
519C22—124no—
520C22—124no—
521C22—124no—
522C22—124no—
523C22—124no—
524C22—2416no—
525C22#550714yes—
526C22C2×#89814yes—
527C22#551414yes—
528C22#571814yes—
529C22#572514yes—
530C22#573214yes—
531C22#573314yes—
532C22C22×#23511yescharacteristic
533C22C22×D3×D411yescharacteristic
534C22C22×#23711yescharacteristic
535C22C2×#100411yescharacteristic
536C22C22×#23911yescharacteristic
537C22C22×#23911yescharacteristic
538C22C2×#100711yescharacteristic
539C4—24no—
540C4—24no—
541C4—44no—
542C4—44no—
543C4—64no—
544C4—64no—
545C4—124no—
546C4—122no—
547C4—122no—
548C4C2×#89818yescomplemented
549C3C2×#251211yescharacteristic, complemented, Sylow
550C2—244no—
551C2#7433318yesdirect product factor
552C2C22×#89811yescharacteristic
553C2C2×#550711yescharacteristic
554C2C2×#551411yescharacteristic
555C2C2×#571811yescharacteristic
556C2C2×#572511yescharacteristic
557C2C2×#573211yescharacteristic
558C2C2×#573311yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#384C2
socleC24×C3C24
prime radicalC23×D12C22
hypercenterC2×#384D3
hypocenterC3C2×#2512
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#384

Complements

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

NH ≅ G/Ntype
1. C22×#898C2semidirect
2. C2×C3×#384C2semidirect
3. C2×#4992C2semidirect
4. C2×#4994C2semidirect
7. C2×#4984C2semidirect
8. C2×#4990C2semidirect
9. C2×Dic3×C4⋊C4C2semidirect
11. C2×C4×Dic3⋊C4C2semidirect
12. #74333C2direct (N × H)
34. C2×#898C22semidirect
35. C3×#384C22semidirect
40. #4994C22semidirect
41. #4992C22semidirect
42. #4990C22semidirect
45. #4984C22semidirect
46. Dic3×C4⋊C4C22semidirect
48. C4×Dic3⋊C4C22semidirect
58. C2×#384D3semidirect
103. C2×Dic3⋊C4D4semidirect
145. #384C2×D3semidirect
219. Dic3⋊C4C2×D4semidirect
315. C2×C3×C4#126semidirect
439. C3×C4C2×#126semidirect
502. C2×C4#898semidirect
508. C2×C3#2512semidirect
548. C4C2×#898semidirect
549. C3C2×#2512semidirect
551. C2#74333direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#8982
C2×C3×#3842
C2×#49922
C2×#49942
C2×#49882
C2×#49842
C2×#49902
C2×Dic3×C4⋊C42
C2×#49392
C2×C4×Dic3⋊C42
#743332
C2×#25123