Finite Group Databasebeta

← C2×#78317 #11001749

Subgroups — C2×#78317 #11001749

Order 768.

2370 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#888C211yesmaximal, characteristic, complemented
2C22×#898C211yesmaximal, characteristic
3C2×C3×#400C211yesmaximal, characteristic, complemented
4C2×#4993C211yesmaximal, characteristic
5C2×#4986C211yesmaximal, characteristic, complemented
6#78317C2116yesmaximal, direct product factor
7C2×#4677C211yesmaximal, characteristic
8C2×#4675C212yesmaximal
9C2×#4673C212yesmaximal
10C2×#4657C212yesmaximal, complemented
11C2×#4656C212yesmaximal, complemented
12C2×#4651C211yesmaximal, characteristic, complemented
13C2×#2821—31nomaximal, Sylow
14C23×Dic6C2212yes—
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×(C2×D3)⋊C4C2211yescharacteristic
18C22×(C2×D3)⋊C4C2211yescharacteristic
19C22×(C2×D3)⋊C4C2211yescharacteristic
20C22×(C2×D3)⋊C4C2212yes—
21C22×(C2×D3)⋊C4C2212yes—
22C22×C4⋊Dic3C2211yescharacteristic
23C22×C4⋊Dic3C2211yescharacteristic
24C22×C4⋊Dic3C2212yes—
25C22×Dic3⋊C4C2211yescharacteristic
26C22×Dic3⋊C4C2212yes—
27C22×Dic3⋊C4C2212yes—
28C22×C4×Dic3C2211yescharacteristic
29C2×#898—28no—
30C2×#898C2218yes—
31C2×#888—28no—
32C2×#888C2218yescomplemented
33C3×#400C2218yescomplemented
34C2×C3×#95C2211yescharacteristic
35C2×C3×#95C2212yes—
36C2×C3×#95C2212yes—
37C2×C3⋊#95C2211yescharacteristic
38C2×C3⋊#95C2212yes—
39C2×C3⋊#95C2212yes—
40C2×C3⋊#95C2212yes—
41C2×C3⋊#95C2212yes—
42#4993C2218yes—
43#4986C2218yescomplemented
44#4677C2218yes—
45#4675C22116yes—
46#4673C22116yes—
47#4657C22116yescomplemented
48#4656C22116yescomplemented
49#4651C2218yescomplemented
50C22×C4⋊Q8—31no—
51C22×#126—31no—
52#2821—316no—
53C2×#400—31no—
54C2×#400D311yescharacteristic, complemented
55C2×#399—31no—
56C2×#399—31no—
57C2×#398—32no—
58C2×#397—32no—
59C2×#395—32no—
60C2×#393—32no—
61C2×#383—31no—
62C24×D3C2311yescharacteristic
63C23×C3×C4C2312yes—
64C23×C3×C4C2312yes—
65C23×C3×C4C2311yescharacteristic
66C23×C3×C4C2311yescharacteristic
67C23×C3×C4C2311yescharacteristic
68C23×Dic3C2312yes—
69C23×Dic3C2312yes—
70C23×Dic3C2311yescharacteristic
71C23×Dic3C2311yescharacteristic
72C23×Dic3C2311yescharacteristic
73C22×Dic6—28no—
74C22×Dic6—44no—
75C22×Dic6—44no—
76C22×Dic6C2318yes—
77C2×C3×C4⋊C4—24no—
78C2×C3×C4⋊C4—24no—
79C2×C3×C4⋊C4C2314yes—
80C2×C3×C4⋊C4C2314yes—
81C2×(C2×D3)⋊C4—24no—
82C2×(C2×D3)⋊C4—24no—
83C2×(C2×D3)⋊C4—24no—
84C2×(C2×D3)⋊C4—44no—
85C2×(C2×D3)⋊C4—44no—
86C2×(C2×D3)⋊C4C2314yes—
87C2×(C2×D3)⋊C4C2314yes—
88C2×(C2×D3)⋊C4C2314yes—
89C2×(C2×D3)⋊C4C2318yes—
90C2×(C2×D3)⋊C4C2318yes—
91C2×C4⋊Dic3—24no—
92C2×C4⋊Dic3—24no—
93C2×C4⋊Dic3—44no—
94C2×C4⋊Dic3C2314yes—
95C2×C4⋊Dic3C2314yes—
96C2×C4⋊Dic3C2318yes—
97C2×Dic3⋊C4—28no—
98C2×Dic3⋊C4—24no—
99C2×Dic3⋊C4—44no—
100C2×Dic3⋊C4C2314yes—
101C2×Dic3⋊C4C2318yes—
102C2×Dic3⋊C4C2318yes—
103C2×C4×Dic3D418yes—
104C2×C4×Dic3C2314yes—
105#898—232no—
106#888—232no—
107C3×#95C2314yes—
108C3×#95C2318yes—
109C3×#95C2318yes—
110C3⋊#95C2318yes—
111C3⋊#95C2318yes—
112C3⋊#95C2314yes—
113C3⋊#95C2318yes—
114C3⋊#95C2318yes—
115C23×Q8—32no—
116C2×C4⋊Q8—38no—
117C2×C4⋊Q8—68no—
118C2×#126—38no—
119C2×#126—68no—
120C22×C4⋊C4—31no—
121C22×C4⋊C4—32no—
122C22×C4⋊C4—31no—
123C22×C4⋊C4—31no—
124C22×C4⋊C4—32no—
125C22×C4⋊C4—32no—
126C22×C4⋊C4C2×D311yescharacteristic
127C22×C4⋊C4C2×D311yescharacteristic
128C22×C22⋊C4—31no—
129C22×C22⋊C4—31no—
130C22×C22⋊C4—31no—
131C22×C22⋊C4—32no—
132C22×C22⋊C4—32no—
133C22×C42—31no—
134#400—38no—
135#400C2×D318yescomplemented
136#399—38no—
137#399—38no—
138#398—316no—
139#397—316no—
140#395—316no—
141#393—316no—
142#383—38no—
143C2×#95—32no—
144C2×#95—32no—
145C2×#95—32no—
146C2×#95—32no—
147C2×#95—31no—
148C2×#95C2×D312yes—
149C2×#95C2×D312yes—
150C2×#95C2×D311yescharacteristic
151C24×C3C2411yescharacteristic
152C23×D3—22no—
153C23×D3—24no—
154C23×D3—22no—
155C23×D3—24no—
156C23×D3—22no—
157C23×D3C2412yes—
158C22×C3×C4—24no—
159C22×C3×C4—22no—
160C22×C3×C4—24no—
161C22×C3×C4—22no—
162C22×C3×C4—24no—
163C22×C3×C4—24no—
164C22×C3×C4—24no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—24no—
168C22×C3×C4—24no—
169C22×C3×C4—24no—
170C22×C3×C4C2×D414yes—
171C22×C3×C4C2414yes—
172C22×C3×C4C2412yes—
173C22×C3×C4C2412yes—
174C22×C3×C4C2414yes—
175C22×C3×C4C2412yes—
176C22×Dic3—24no—
177C22×Dic3—22no—
178C22×Dic3—22no—
179C22×Dic3—22no—
180C22×Dic3—24no—
181C22×Dic3—24no—
182C22×Dic3—24no—
183C22×Dic3—24no—
184C22×Dic3—24no—
185C22×Dic3—24no—
186C22×Dic3—24no—
187C22×Dic3C2×D414yes—
188C22×Dic3C2×D414yes—
189C22×Dic3C2414yes—
190C22×Dic3C2414yes—
191C22×Dic3C2412yes—
192C22×Dic3C2412yes—
193C22×Dic3C2412yes—
194C2×Dic6—216no—
195C2×Dic6—48no—
196C2×Dic6—48no—
197C2×Dic6—432no—
198C3×C4⋊C4—28no—
199C3×C4⋊C4—28no—
200(C2×D3)⋊C4—28no—
201(C2×D3)⋊C4—28no—
202(C2×D3)⋊C4—28no—
203(C2×D3)⋊C4—48no—
204(C2×D3)⋊C4—48no—
205C4⋊Dic3—28no—
206C4⋊Dic3—28no—
207C4⋊Dic3—48no—
208Dic3⋊C4—216no—
209Dic3⋊C4—28no—
210Dic3⋊C4—48no—
211C4×Dic3C2×D4116yes—
212C25—31no—
213C22×Q8—38no—
214C22×Q8—68no—
215C22×Q8—124no—
216C22×Q8—124no—
217C23×C4—31no—
218C23×C4—31no—
219C23×C4—31no—
220C23×C4—32no—
221C23×C4—32no—
222C23×C4C22×D312yes—
223C23×C4C22×D311yescharacteristic
224C23×C4C22×D311yescharacteristic
225C23×C4C22×D312yes—
226C23×C4C22×D311yescharacteristic
227C4⋊Q8—632no—
228#126—632no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—34no—
232C2×C4⋊C4—34no—
233C2×C4⋊C4—38no—
234C2×C4⋊C4—34no—
235C2×C4⋊C4—38no—
236C2×C4⋊C4—38no—
237C2×C4⋊C4—64no—
238C2×C4⋊C4—64no—
239C2×C4⋊C4—68no—
240C2×C4⋊C4—64no—
241C2×C4⋊C4—124no—
242C2×C4⋊C4—124no—
243C2×C4⋊C4C22×D314yes—
244C2×C4⋊C4C22×D314yes—
245C2×C22⋊C4—34no—
246C2×C22⋊C4—34no—
247C2×C22⋊C4—34no—
248C2×C22⋊C4—38no—
249C2×C22⋊C4—38no—
250C2×C22⋊C4—64no—
251C2×C22⋊C4—64no—
252C2×C22⋊C4—64no—
253C2×C22⋊C4—124no—
254C2×C22⋊C4—124no—
255C2×C42—38no—
256C2×C42—34no—
257#95—38no—
258#95—38no—
259#95—34no—
260#95—38no—
261#95—38no—
262#95C22×D314yes—
263#95C22×D318yes—
264#95C22×D318yes—
265C23×C3C22×D412yes—
266C23×C3H(5,𝔽2)14yes—
267C23×C32−1+412yes—
268C23×C32−1+414yes—
269C23×C32−1+412yes—
270C23×C3C2511yescharacteristic
271C22×D3—22no—
272C22×D3—24no—
273C22×D3—22no—
274C22×D3—24no—
275C22×D3—22no—
276C22×D3—44no—
277C22×D3—44no—
278C22×D3—44no—
279C22×D3—48no—
280C22×D3—48no—
281C2×C3×C4—28no—
282C2×C3×C4—24no—
283C2×C3×C4—22no—
284C2×C3×C4—24no—
285C2×C3×C4—22no—
286C2×C3×C4—24no—
287C2×C3×C4—24no—
288C2×C3×C4—24no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—24no—
292C2×C3×C4—24no—
293C2×C3×C4—24no—
294C2×C3×C4—48no—
295C2×C3×C4—44no—
296C2×C3×C4—44no—
297C2×C3×C4—48no—
298C2×C3×C4C22×D414yes—
299C2×Dic3—28no—
300C2×Dic3—28no—
301C2×Dic3—24no—
302C2×Dic3—22no—
303C2×Dic3—22no—
304C2×Dic3—22no—
305C2×Dic3—24no—
306C2×Dic3—24no—
307C2×Dic3—24no—
308C2×Dic3—24no—
309C2×Dic3—24no—
310C2×Dic3—24no—
311C2×Dic3—24no—
312C2×Dic3—48no—
313C2×Dic3—44no—
314C2×Dic3—48no—
315C2×Dic3C22×D414yes—
316C2×Dic3C22×D414yes—
317Dic6—432no—
318C24—32no—
319C24—62no—
320C24—62no—
321C24—64no—
322C24—64no—
323C24—62no—
324C24C23×D311yescharacteristic
325C2×Q8—616no—
326C2×Q8—128no—
327C2×Q8—128no—
328C2×Q8—1232no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—34no—
342C22×C4—34no—
343C22×C4—32no—
344C22×C4—32no—
345C22×C4—34no—
346C22×C4—34no—
347C22×C4—32no—
348C22×C4—62no—
349C22×C4—64no—
350C22×C4—62no—
351C22×C4—62no—
352C22×C4—64no—
353C22×C4—64no—
354C22×C4—64no—
355C22×C4—64no—
356C22×C4—64no—
357C22×C4—64no—
358C22×C4—64no—
359C22×C4D3×D414yes—
360C22×C4C23×D314yes—
361C22×C4C23×D312yes—
362C22×C4C23×D312yes—
363C22×C4C23×D314yes—
364C22×C4C23×D312yes—
365C4⋊C4—28no—
366C4⋊C4—28no—
367C4⋊C4—68no—
368C4⋊C4—68no—
369C4⋊C4—616no—
370C4⋊C4—68no—
371C4⋊C4—128no—
372C4⋊C4—128no—
373C22⋊C4—68no—
374C22⋊C4—68no—
375C22⋊C4—68no—
376C22⋊C4—128no—
377C22⋊C4—128no—
378C42—316no—
379C22×C3#53514yes—
380C22×C3#53614yes—
381C22×C3#53614yes—
382C22×C3#56318yes—
383C22×C3#56318yes—
384C22×C3C23×D411yescharacteristic
385C22×C3C2×H(5,𝔽2)12yes—
386C22×C3C2×2−1+411yescharacteristic
387C22×C3C2×2−1+412yes—
388C22×C3C2×2−1+411yescharacteristic
389C2×D3—42no—
390C2×D3—42no—
391C2×D3—42no—
392C2×D3—44no—
393C2×D3—44no—
394C2×D3—88no—
395C3×C4—24no—
396C3×C4—44no—
397C3×C4—42no—
398C3×C4—42no—
399C3×C4—44no—
400Dic3—24no—
401Dic3—24no—
402Dic3—44no—
403Dic3—42no—
404Dic3—44no—
405C23—62no—
406C23—62no—
407C23—64no—
408C23—64no—
409C23—62no—
410C23—124no—
411C23—124no—
412C23—128no—
413C23—128no—
414C23—124no—
415C23C2×D3×D412yes—
416C23#100712yes—
417C23#100914yes—
418C23#101014yes—
419C23#101012yes—
420C23C24×D311yescharacteristic
421Q8—1232no—
422C2×C4—28no—
423C2×C4—24no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—24no—
427C2×C4—22no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—22no—
434C2×C4—24no—
435C2×C4—34no—
436C2×C4—34no—
437C2×C4—44no—
438C2×C4—48no—
439C2×C4—48no—
440C2×C4—44no—
441C2×C4—68no—
442C2×C4—62no—
443C2×C4—64no—
444C2×C4—62no—
445C2×C4—68no—
446C2×C4—62no—
447C2×C4—64no—
448C2×C4—64no—
449C2×C4—64no—
450C2×C4—64no—
451C2×C4—64no—
452C2×C4—64no—
453C2×C4—64no—
454C2×C4—128no—
455C2×C4—124no—
456C2×C4—128no—
457C2×C4C2×D3×D414yes—
458C2×C3#282118yescomplemented
459C2×C3C2×#53511yescharacteristic
460C2×C3C2×#53611yescharacteristic
461C2×C3C2×#53611yescharacteristic
462C2×C3C2×#56312yes—
463C2×C3C2×#56312yes—
464D3—82no—
465C22—122no—
466C22—124no—
467C22—124no—
468C22—122no—
469C22—122no—
470C22—248no—
471C22#562814yes—
472C22#564814yes—
473C22#567414yes—
474C22#569918yes—
475C22#571218yes—
476C22C22×D3×D411yescharacteristic
477C22C2×#100711yescharacteristic
478C22C2×#100912yes—
479C22C2×#101012yes—
480C22C2×#101011yescharacteristic
481C4—24no—
482C4—44no—
483C4—44no—
484C4—42no—
485C4—42no—
486C4—64no—
487C4—64no—
488C4—124no—
489C4—122no—
490C4—124no—
491C3C2×#282111yescharacteristic, complemented, Sylow
492C2—242no—
493C2#7831718yesdirect product factor
494C2C2×#562811yescharacteristic
495C2C2×#564811yescharacteristic
496C2C2×#567411yescharacteristic
497C2C2×#569912yes—
498C2C2×#571212yes—
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#400C2
socleC24×C3C24
prime radicalC24×D3C23
hypercenterC2×#400D3
hypocenterC3C2×#2821
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#400

Complements

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

NH ≅ G/Ntype
1. C22×#888C2semidirect
3. C2×C3×#400C2semidirect
5. C2×#4986C2semidirect
6. #78317C2direct (N × H)
10. C2×#4657C2semidirect
11. C2×#4656C2semidirect
12. C2×#4651C2semidirect
32. C2×#888C22semidirect
33. C3×#400C22semidirect
43. #4986C22semidirect
47. #4657C22semidirect
48. #4656C22semidirect
49. #4651C22semidirect
54. C2×#400D3semidirect
135. #400C2×D3semidirect
458. C2×C3#2821semidirect
491. C3C2×#2821semidirect
493. C2#78317direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#8882
C22×#8982
C2×C3×#4002
C2×#49932
C2×#49862
#783172
C2×#46772
C2×#46752
C2×#46732
C2×#46572
C2×#46562
C2×#46512
C2×#28213