Finite Group Databasebeta

← C2×#77765 #10964364

Subgroups — C2×#77765 #10964364

Order 768.

3094 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#896C211yesmaximal, characteristic
2C22×D3×C4⋊C4C211yesmaximal, characteristic
3C2×C3×#398C211yesmaximal, characteristic, complemented
4C2×#4989C212yesmaximal
5C2×#4981C211yesmaximal, characteristic, complemented
6C2×#4983C211yesmaximal, characteristic, complemented
7C2×#4975C212yesmaximal, complemented
8#77765C2116yesmaximal, direct product factor
9C2×#4668C212yesmaximal
10C2×#4661C211yesmaximal, characteristic, complemented
11C2×#4646C212yesmaximal, complemented
12C2×#4666C211yesmaximal, characteristic
13C2×#2502—31nomaximal, Sylow
14C23×C4×D3C2211yescharacteristic
15C23×C4×D3C2212yes—
16C23×Dic6C2211yescharacteristic
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C4⋊C4C2211yescharacteristic
19C22×C3×C4⋊C4C2212yes—
20C22×(C2×D3)⋊C4C2211yescharacteristic
21C22×(C2×D3)⋊C4C2211yescharacteristic
22C22×(C2×D3)⋊C4C2212yes—
23C22×C4⋊Dic3C2211yescharacteristic
24C22×C4⋊Dic3C2212yes—
25C22×Dic3⋊C4C2212yes—
26C22×Dic3⋊C4C2211yescharacteristic
27C22×Dic3⋊C4C2211yescharacteristic
28C22×Dic3⋊C4C2212yes—
29C22×Dic3⋊C4C2212yes—
30C22×C4×Dic3C2212yes—
31C2×#896—28no—
32C2×#896C2218yes—
33C2×D3×C4⋊C4—28no—
34C2×D3×C4⋊C4C2218yes—
35C3×#398C2218yescomplemented
36C2×C3×#95C2211yescharacteristic
37C2×C3×#95C2212yes—
38C2×C3⋊#95C2212yes—
39C2×C3⋊#95C2212yes—
40C2×C3⋊#95C2211yescharacteristic
41C2×C3⋊#95C2212yes—
42#4989C22116yes—
43#4983C2218yescomplemented
44#4981C2218yescomplemented
45#4975C22116yescomplemented
46#4668C22116yes—
47#4666C2218yes—
48#4661C2218yescomplemented
49#4646C22116yescomplemented
50C22×C22⋊Q8—31no—
51C23×C4⋊C4—31no—
52#2502—316no—
53C2×#398—31no—
54C2×#398—31no—
55C2×#398—31no—
56C2×#398D311yescharacteristic, complemented
57C2×#391—32no—
58C2×#385—32no—
59C2×#385—32no—
60C2×#384—32no—
61C2×C4×C22⋊C4—31no—
62C24×D3C2311yescharacteristic
63C23×C3×C4C2311yescharacteristic
64C23×C3×C4C2312yes—
65C23×C3×C4C2311yescharacteristic
66C23×C3×C4C2311yescharacteristic
67C23×C3×C4C2312yes—
68C23×Dic3C2311yescharacteristic
69C23×Dic3C2312yes—
70C23×Dic3C2311yescharacteristic
71C23×Dic3C2312yes—
72C23×Dic3C2311yescharacteristic
73C22×C4×D3—28no—
74C22×C4×D3—28no—
75C22×C4×D3—44no—
76C22×C4×D3—44no—
77C22×C4×D3D418yes—
78C22×C4×D3C2314yes—
79C22×C4×D3C2318yes—
80C22×Dic6—24no—
81C22×Dic6—28no—
82C22×Dic6C2314yes—
83C2×C3×C4⋊C4—24no—
84C2×C3×C4⋊C4—24no—
85C2×C3×C4⋊C4—44no—
86C2×C3×C4⋊C4C2314yes—
87C2×C3×C4⋊C4C2314yes—
88C2×C3×C4⋊C4C2318yes—
89C2×(C2×D3)⋊C4—24no—
90C2×(C2×D3)⋊C4—24no—
91C2×(C2×D3)⋊C4Q8116yescomplemented
92C2×(C2×D3)⋊C4C2314yes—
93C2×(C2×D3)⋊C4C2314yes—
94C2×(C2×D3)⋊C4C2318yes—
95C2×C4⋊Dic3—24no—
96C2×C4⋊Dic3—44no—
97C2×C4⋊Dic3C2314yes—
98C2×C4⋊Dic3C2318yes—
99C2×Dic3⋊C4—24no—
100C2×Dic3⋊C4—28no—
101C2×Dic3⋊C4—24no—
102C2×Dic3⋊C4—44no—
103C2×Dic3⋊C4—44no—
104C2×Dic3⋊C4C2318yes—
105C2×Dic3⋊C4C2314yes—
106C2×Dic3⋊C4C2314yes—
107C2×Dic3⋊C4C2318yes—
108C2×Dic3⋊C4C2318yes—
109C2×C4×Dic3—28no—
110C2×C4×Dic3C2318yes—
111#896—232no—
112D3×C4⋊C4—232no—
113C3×#95C2318yes—
114C3×#95C2314yes—
115C3⋊#95C2318yes—
116C3⋊#95C2318yes—
117C3⋊#95C2318yes—
118C3⋊#95C2314yes—
119C23×Q8—31no—
120C24×C4—31no—
121C24×C4—32no—
122C2×C22⋊Q8—38no—
123C2×C22⋊Q8—68no—
124C22×C4⋊C4—31no—
125C22×C4⋊C4—31no—
126C22×C4⋊C4—32no—
127C22×C4⋊C4—31no—
128C22×C4⋊C4—32no—
129C22×C4⋊C4—32no—
130C22×C4⋊C4—32no—
131C22×C4⋊C4—38no—
132C22×C4⋊C4—68no—
133C22×C4⋊C4C2×D311yescharacteristic
134C22×C4⋊C4C2×D311yescharacteristic
135C22×C4⋊C4C2×D312yes—
136C22×C22⋊C4—32no—
137C22×C22⋊C4—31no—
138C22×C22⋊C4—31no—
139C22×C42—32no—
140#398—38no—
141#398—38no—
142#398—38no—
143#398C2×D318yescomplemented
144#391—316no—
145#385—316no—
146#385—316no—
147#384—316no—
148C4×C22⋊C4—38no—
149C2×#95—31no—
150C2×#95—32no—
151C2×#95—32no—
152C2×#95—32no—
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—
162C22×C3×C4—24no—
163C22×C3×C4—24no—
164C22×C3×C4—24no—
165C22×C3×C4—22no—
166C22×C3×C4—24no—
167C22×C3×C4—22no—
168C22×C3×C4—24no—
169C22×C3×C4—22no—
170C22×C3×C4—24no—
171C22×C3×C4—22no—
172C22×C3×C4—24no—
173C22×C3×C4C2×D414yes—
174C22×C3×C4C2×Q818yes—
175C22×C3×C4C2412yes—
176C22×C3×C4C2414yes—
177C22×C3×C4C2412yes—
178C22×C3×C4C2412yes—
179C22×C3×C4C2414yes—
180C22×Dic3—24no—
181C22×Dic3—24no—
182C22×Dic3—22no—
183C22×Dic3—24no—
184C22×Dic3—22no—
185C22×Dic3—24no—
186C22×Dic3—22no—
187C22×Dic3—24no—
188C22×Dic3—22no—
189C22×Dic3—24no—
190C22×Dic3C2×D414yes—
191C22×Dic3C2×Q818yes—
192C22×Dic3#4118yes—
193C22×Dic3C2412yes—
194C22×Dic3C2414yes—
195C22×Dic3C2412yes—
196C22×Dic3C2414yes—
197C22×Dic3C2412yes—
198C2×C4×D3—232no—
199C2×C4×D3—216no—
200C2×C4×D3—216no—
201C2×C4×D3—48no—
202C2×C4×D3—48no—
203C2×C4×D3—432no—
204C2×C4×D3C2×D4116yes—
205C2×Dic6—28no—
206C2×Dic6—216no—
207C2×Dic6—416no—
208C3×C4⋊C4—28no—
209C3×C4⋊C4—28no—
210C3×C4⋊C4—48no—
211(C2×D3)⋊C4—28no—
212(C2×D3)⋊C4—28no—
213(C2×D3)⋊C4C2×Q8132yescomplemented
214C4⋊Dic3—28no—
215C4⋊Dic3—48no—
216Dic3⋊C4—28no—
217Dic3⋊C4—216no—
218Dic3⋊C4—28no—
219Dic3⋊C4—48no—
220Dic3⋊C4—48no—
221C4×Dic3—216no—
222C25—31no—
223C22×Q8—34no—
224C22×Q8—64no—
225C22×Q8—68no—
226C23×C4—31no—
227C23×C4—32no—
228C23×C4—38no—
229C23×C4—34no—
230C23×C4—31no—
231C23×C4—31no—
232C23×C4—32no—
233C23×C4—38no—
234C23×C4—68no—
235C23×C4—68no—
236C23×C4—124no—
237C23×C4—124no—
238C23×C4C22×D311yescharacteristic
239C23×C4C22×D311yescharacteristic
240C23×C4C22×D312yes—
241C23×C4C22×D311yescharacteristic
242C23×C4C22×D312yes—
243C22⋊Q8—632no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—38no—
247C2×C4⋊C4—34no—
248C2×C4⋊C4—34no—
249C2×C4⋊C4—34no—
250C2×C4⋊C4—38no—
251C2×C4⋊C4—38no—
252C2×C4⋊C4—38no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—632no—
255C2×C4⋊C4—68no—
256C2×C4⋊C4—64no—
257C2×C4⋊C4—64no—
258C2×C4⋊C4—64no—
259C2×C4⋊C4—124no—
260C2×C4⋊C4—124no—
261C2×C4⋊C4—124no—
262C2×C4⋊C4C22×D314yes—
263C2×C4⋊C4C22×D314yes—
264C2×C4⋊C4C22×D318yes—
265C2×C22⋊C4—316no—
266C2×C22⋊C4—38no—
267C2×C22⋊C4—34no—
268C2×C22⋊C4—34no—
269C2×C22⋊C4—64no—
270C2×C22⋊C4—64no—
271C2×C42—38no—
272C2×C42—68no—
273#95—34no—
274#95—38no—
275#95—38no—
276#95—38no—
277#95C22×D314yes—
278#95C22×D318yes—
279C23×C3C22×D412yes—
280C23×C3C22×Q814yes—
281C23×C3C2×#4114yes—
282C23×C3H(5,𝔽2)12yes—
283C23×C3H(5,𝔽2)12yes—
284C23×C3C2511yescharacteristic
285C22×D3—28no—
286C22×D3—28no—
287C22×D3—216no—
288C22×D3—216no—
289C22×D3—24no—
290C22×D3—22no—
291C22×D3—22no—
292C22×D3C4⋊Q8116yes—
293C22×D3C22×D414yes—
294C22×D3C22×Q818yes—
295C2×C3×C4—28no—
296C2×C3×C4—216no—
297C2×C3×C4—24no—
298C2×C3×C4—24no—
299C2×C3×C4—24no—
300C2×C3×C4—22no—
301C2×C3×C4—24no—
302C2×C3×C4—22no—
303C2×C3×C4—24no—
304C2×C3×C4—22no—
305C2×C3×C4—24no—
306C2×C3×C4—22no—
307C2×C3×C4—24no—
308C2×C3×C4—44no—
309C2×C3×C4—48no—
310C2×C3×C4—44no—
311C2×C3×C4C22×D414yes—
312C2×C3×C4C22×Q818yes—
313C2×Dic3—216no—
314C2×Dic3—24no—
315C2×Dic3—24no—
316C2×Dic3—22no—
317C2×Dic3—24no—
318C2×Dic3—22no—
319C2×Dic3—24no—
320C2×Dic3—22no—
321C2×Dic3—24no—
322C2×Dic3—22no—
323C2×Dic3—24no—
324C2×Dic3—44no—
325C2×Dic3—48no—
326C2×Dic3—44no—
327C2×Dic3#123116yes—
328C2×Dic3C22×D414yes—
329C2×Dic3C22×Q818yes—
330C2×Dic3C2×#4118yes—
331C4×D3—232no—
332C4×D3—432no—
333Dic6—416no—
334C24—34no—
335C24—38no—
336C24—32no—
337C24—62no—
338C24—64no—
339C24—62no—
340C24C23×D311yescharacteristic
341C2×Q8—68no—
342C2×Q8—616no—
343C2×Q8—1216no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—34no—
356C22×C4—38no—
357C22×C4—32no—
358C22×C4—38no—
359C22×C4—34no—
360C22×C4—316no—
361C22×C4—32no—
362C22×C4—34no—
363C22×C4—32no—
364C22×C4—632no—
365C22×C4—616no—
366C22×C4—616no—
367C22×C4—64no—
368C22×C4—64no—
369C22×C4—62no—
370C22×C4—62no—
371C22×C4—64no—
372C22×C4—62no—
373C22×C4—64no—
374C22×C4—64no—
375C22×C4—62no—
376C22×C4—64no—
377C22×C4—128no—
378C22×C4—128no—
379C22×C4—1232no—
380C22×C4D3×D414yes—
381C22×C4D3×Q818yes—
382C22×C4C23×D312yes—
383C22×C4C23×D314yes—
384C22×C4C23×D312yes—
385C22×C4C23×D312yes—
386C22×C4C23×D314yes—
387C4⋊C4—28no—
388C4⋊C4—28no—
389C4⋊C4—48no—
390C4⋊C4—616no—
391C4⋊C4—68no—
392C4⋊C4—68no—
393C4⋊C4—68no—
394C4⋊C4—128no—
395C4⋊C4—128no—
396C4⋊C4—128no—
397C22⋊C4—332no—
398C22⋊C4—68no—
399C22⋊C4—68no—
400C42—616no—
401C22×C3C2×#12314yes—
402C22×C3C2×C4⋊Q814yes—
403C22×C3#53514yes—
404C22×C3#55418yes—
405C22×C3#55418yes—
406C22×C3C23×D411yescharacteristic
407C22×C3C23×Q812yes—
408C22×C3C22×#4112yes—
409C22×C3C2×H(5,𝔽2)11yescharacteristic
410C22×C3C2×H(5,𝔽2)11yescharacteristic
411C2×D3—24no—
412C2×D3—24no—
413C2×D3—28no—
414C2×D3—28no—
415C2×D3—232no—
416C2×D3C2×C4⋊Q818yes—
417C3×C4—24no—
418C3×C4—28no—
419C3×C4—42no—
420C3×C4—44no—
421C3×C4—42no—
422Dic3—28no—
423Dic3—42no—
424Dic3—44no—
425Dic3—42no—
426Dic3C2×#12318yes—
427C23—34no—
428C23—38no—
429C23—316no—
430C23—616no—
431C23—68no—
432C23—616no—
433C23—62no—
434C23—64no—
435C23—62no—
436C23—68no—
437C23C2×D3×D412yes—
438C23#100412yes—
439C23C2×D3×Q814yes—
440C23D3×#4114yes—
441C23#100912yes—
442C23C24×D311yescharacteristic
443Q8—1216no—
444C2×C4—216no—
445C2×C4—28no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—316no—
458C2×C4—34no—
459C2×C4—38no—
460C2×C4—38no—
461C2×C4—44no—
462C2×C4—48no—
463C2×C4—44no—
464C2×C4—632no—
465C2×C4—616no—
466C2×C4—64no—
467C2×C4—64no—
468C2×C4—62no—
469C2×C4—62no—
470C2×C4—64no—
471C2×C4—62no—
472C2×C4—64no—
473C2×C4—64no—
474C2×C4—62no—
475C2×C4—64no—
476C2×C4—1232no—
477C2×C4—124no—
478C2×C4—128no—
479C2×C4—124no—
480C2×C4C2×D3×D414yes—
481C2×C4C2×D3×Q818yes—
482C2×C3#250218yescomplemented
483C2×C3C22×#12311yescharacteristic
484C2×C3C22×C4⋊Q811yescharacteristic
485C2×C3C2×#53511yescharacteristic
486C2×C3C2×#55412yes—
487C2×C3C2×#55412yes—
488D3—28no—
489C22—38no—
490C22—68no—
491C22—64no—
492C22—632no—
493C22—68no—
494C22—64no—
495C22#560714yes—
496C22#563618yes—
497C22D3×#12314yes—
498C22#569018yes—
499C22D3×C4⋊Q814yes—
500C22C22×D3×D411yescharacteristic
501C22C2×#100411yescharacteristic
502C22C22×D3×Q812yes—
503C22C2×D3×#4112yes—
504C22C2×#100911yescharacteristic
505C4—28no—
506C4—24no—
507C4—38no—
508C4—44no—
509C4—42no—
510C4—42no—
511C4—68no—
512C4—124no—
513C4—122no—
514C4—122no—
515C3C2×#250211yescharacteristic, complemented, Sylow
516C2—68no—
517C2#7776518yesdirect product factor
518C2C2×#560711yescharacteristic
519C2C2×#563612yes—
520C2C2×D3×#12311yescharacteristic
521C2C2×#569012yes—
522C2C2×D3×C4⋊Q811yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#398C2
socleC24×C3C24
prime radicalC24×D3C23
hypercenterC2×#398D3
hypocenterC3C2×#2502
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#398

Complements

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

NH ≅ G/Ntype
3. C2×C3×#398C2semidirect
5. C2×#4981C2semidirect
6. C2×#4983C2semidirect
7. C2×#4975C2semidirect
8. #77765C2direct (N × H)
10. C2×#4661C2semidirect
11. C2×#4646C2semidirect
35. C3×#398C22semidirect
43. #4983C22semidirect
44. #4981C22semidirect
45. #4975C22semidirect
48. #4661C22semidirect
49. #4646C22semidirect
56. C2×#398D3semidirect
91. C2×(C2×D3)⋊C4Q8semidirect
143. #398C2×D3semidirect
213. (C2×D3)⋊C4C2×Q8semidirect
482. C2×C3#2502semidirect
515. C3C2×#2502semidirect
517. C2#77765direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#8962
C22×D3×C4⋊C42
C2×C3×#3982
C2×#49892
C2×#49812
C2×#49832
C2×#49752
#777652
C2×#46682
C2×#46612
C2×#46462
C2×#46662
C2×#25023