Finite Group Databasebeta

← #10767474

Subgroups — #10767474

Order 768.

1012 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3458C211yesmaximal, characteristic, complemented
2#83351C212yesmaximal, complemented
3#82273C211yesmaximal, characteristic
4#71902C212yesmaximal, complemented
5#71898C212yesmaximal, complemented
6#71892C212yesmaximal
7#71890C212yesmaximal, complemented
8#71835C212yesmaximal, complemented
9#71571C211yesmaximal, characteristic, complemented
10#18124—31nomaximal, Sylow
11C3×#544—22no—
12C3×#544—22no—
13C3×#544—21no—
14C3×#544C2212yes—
15C3×#543—21no—
16C3×#543C2212yes—
17C3×#533—24no—
18C3×#533—21no—
19C3×#533C2212yes—
20C2×C3×#117—21no—
21C2×C3×#117C2211yescharacteristic
22C2×C4×#67—21no—
23C2×C4×D12C2211yescharacteristic
24#5084—22no—
25#5083—22no—
26#5046C2214yes—
27#5043C2214yes—
28C4×#215—24no—
29#4994—21no—
30#4991C2211yescharacteristic
31#4981C2211yescharacteristic
32#4961—21no—
33#4960C2211yescharacteristic
34#4953C2211yescharacteristic
35C4×(C2×D3)⋊C4—21no—
36C4×C4⋊Dic3C2211yescharacteristic
37#4925C2212yes—
38#4538C2214yes—
39#4538C2214yes—
40#4528C2212yes—
41#4525C2212yes—
42#3458D311yescharacteristic, complemented
43#3101—32no—
44#2392—31no—
45#1611—32no—
46#1608—32no—
47#1599—32no—
48#1596—32no—
49#1503—32no—
50#1458—31no—
51C23×C3×C4D411yescharacteristic
52C22×#67—21no—
53C22×D12C2311yescharacteristic
54C22×C4×D3—21no—
55C3×C4⋊Q8—24no—
56C3×C4⋊Q8—21no—
57C3×C4⋊Q8—21no—
58C3×C4⋊Q8—24no—
59C3×C4⋊Q8D412yescomplemented
60C3×C4⋊Q8D412yescomplemented
61C3×#125—24no—
62C3×#125—22no—
63C3×#125—24no—
64C3×#125—22no—
65C3×C22⋊Q8—24no—
66C3×C22⋊Q8—24no—
67C3×C22⋊Q8—24no—
68C3×C22⋊Q8—22no—
69C3×C22⋊Q8—22no—
70C3×C22⋊Q8—22no—
71C3×C22⋊Q8—24no—
72C3×C22⋊Q8C2314yes—
73C3×C4×Q8—22no—
74C3×C4×Q8—22no—
75C3×C4×Q8—22no—
76C3×C4×Q8D414yescomplemented
77C3×#117—22no—
78C3×#117—22no—
79C3×#117—22no—
80C3×#117—22no—
81C3×#117—22no—
82C3×#117—22no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4D411yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C22⋊C4—22no—
88C2×C3×C22⋊C4D411yescharacteristic
89C2×C3×C22⋊C4C2311yescharacteristic
90C2×C3×C42—22no—
91C2×C3×C42D411yescharacteristic
92C2×C3×C42C2311yescharacteristic
93C2×#217—21no—
94C2×#215—24no—
95C4×#67—44no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3C2311yescharacteristic
102C2×Dic3⋊C4—21no—
103C2×C4×Dic3—21no—
104C4×D12—22no—
105C4×D12—22no—
106#835—24no—
107C3⋊#95—21no—
108C3⋊#95—21no—
109C3⋊#95C2311yescharacteristic
110C3⋊#95C2311yescharacteristic
111#830C2314yes—
112#809—24no—
113#808—24no—
114C4⋊(C3⋊C8)—22no—
115C4×C3⋊C8—22no—
116#544—22no—
117#544—22no—
118#544—21no—
119#544C2×D312yes—
120#543—21no—
121#543C2×D312yes—
122#533—24no—
123#533—21no—
124#533C2×D312yes—
125C2×C4×D4—31no—
126C2×C4×D4—61no—
127C2×#117—21no—
128C2×#117C2×D311yescharacteristic
129#483—34no—
130#476—62no—
131#461—62no—
132#450—34no—
133C4×SD16—64no—
134#431—32no—
135#398—31no—
136#397—31no—
137#396—61no—
138#394—31no—
139#394—61no—
140#392—31no—
141C4×C4⋊C4—31no—
142C4×C22⋊C4—61no—
143#328—34no—
144#328—34no—
145#324—32no—
146#323—32no—
147C24×C3C2×D411yescharacteristic
148C23×D3—21no—
149C2×C3×Q8—22no—
150C2×C3×Q8—22no—
151C2×C3×Q8—22no—
152C2×C3×Q8C2×D414yes—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4SD1614yescomplemented
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2411yescharacteristic
165C2×#67—42no—
166C2×#67—42no—
167C2×#67—42no—
168C22×Dic3—21no—
169C22×Dic3—21no—
170C22×Dic3—21no—
171C2×D12—22no—
172C2×D12—42no—
173C2×C4×D3—41no—
174C2×C4×D3—41no—
175C2×C4×D3—42no—
176C2×C4×D3—42no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—24no—
181C3×C4⋊C4—24no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—24no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—24no—
190C3×C4⋊C4C2×D414yes—
191C3×C22⋊C4—24no—
192C3×C22⋊C4—24no—
193C3×C22⋊C4—22no—
194C3×C22⋊C4—22no—
195C3×C42—24no—
196C3×C42—24no—
197C3×C42—21no—
198C3×C42—21no—
199C3×C42C2×D412yes—
200C3×C42C2×D412yes—
201#217—42no—
202#215—48no—
203(C2×D3)⋊C4—42no—
204(C2×D3)⋊C4—42no—
205(C2×D3)⋊C4—42no—
206(C2×D3)⋊C4—42no—
207C4⋊Dic3—22no—
208C4⋊Dic3—42no—
209Dic3⋊C4—42no—
210C4×Dic3—42no—
211C2×C3⋊C8—24no—
212C22×D4—31no—
213C22×D4—61no—
214C23×C4—61no—
215C23×C4#6711yescharacteristic
216C2×SD16—64no—
217C4⋊Q8—24no—
218C4⋊Q8—21no—
219C4⋊Q8—21no—
220C4⋊Q8—24no—
221C4⋊Q8#6712yescomplemented
222C4⋊Q8#6712yescomplemented
223#125—24no—
224#125—22no—
225#125—24no—
226#125—22no—
227C22⋊Q8—24no—
228C22⋊Q8—24no—
229C22⋊Q8—24no—
230C22⋊Q8—22no—
231C22⋊Q8—22no—
232C22⋊Q8—22no—
233C22⋊Q8—24no—
234C22⋊Q8C22×D314yes—
235C4×Q8—22no—
236C4×Q8—22no—
237C4×Q8—22no—
238C4×Q8#6714yescomplemented
239C4×D4—62no—
240C4×D4—62no—
241C4×D4—124no—
242#117—22no—
243#117—22no—
244#117—22no—
245#117—22no—
246#117—22no—
247#117—22no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—31no—
251C2×C4⋊C4—61no—
252C2×C4⋊C4—61no—
253C2×C4⋊C4#6711yescharacteristic
254C2×C4⋊C4C22×D311yescharacteristic
255C2×C22⋊C4—22no—
256C2×C22⋊C4—61no—
257C2×C22⋊C4—61no—
258C2×C22⋊C4—61no—
259C2×C22⋊C4—61no—
260C2×C22⋊C4—61no—
261C2×C22⋊C4#6711yescharacteristic
262C2×C22⋊C4C22×D311yescharacteristic
263C2×C42—22no—
264C2×C42—61no—
265C2×C42#6711yescharacteristic
266C2×C42C22×D311yescharacteristic
267#106—64no—
268C4⋊C8—62no—
269#103—64no—
270#102—64no—
271C22⋊C8—34no—
272C4×C8—62no—
273#95—31no—
274#95—31no—
275#95—61no—
276#95—61no—
277C23×C3—21no—
278C23×C3—21no—
279C23×C3—41no—
280C23×C3—41no—
281C23×C3C2×SD1612yes—
282C23×C3C22×D411yescharacteristic
283C22×D3—22no—
284C22×D3—41no—
285C22×D3—41no—
286C22×D3—41no—
287C22×D3—41no—
288C22×D3—41no—
289C22×D3—41no—
290C3×Q8—44no—
291C3×Q8—82no—
292C3×Q8—82no—
293C3×Q8—82no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—24no—
297C2×C3×C4—24no—
298C2×C3×C4—24no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—24no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4—82no—
307C2×C3×C4—82no—
308C2×C3×C4#12012yes—
309C2×C3×C4#12011yescharacteristic
310C2×C3×C4#12011yescharacteristic
311C2×C3×C4C2×SD1612yes—
312C2×C3×C4C2×SD1612yes—
313C2×C3×C4#13712yes—
314C2×C3×C4C22×D412yes—
315#67—48no—
316C2×Dic3—22no—
317C2×Dic3—41no—
318C2×Dic3—41no—
319C2×Dic3—41no—
320C2×Dic3—41no—
321C2×Dic3—41no—
322C2×Dic3—41no—
323C2×Dic3—41no—
324C2×Dic3—41no—
325D12—24no—
326D12—42no—
327C4×D3—82no—
328C4×D3—82no—
329C3⋊C8—44no—
330C24—61no—
331C24C2×#6711yescharacteristic
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8C2×#6714yes—
336C2×D4—62no—
337C2×D4—122no—
338C2×D4—122no—
339C2×D4—122no—
340C2×D4—122no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—61no—
347C22×C4—61no—
348C22×C4—61no—
349C22×C4—122no—
350C22×C4—122no—
351C22×C4—121no—
352C22×C4—121no—
353C22×C4#21514yescomplemented
354C22×C4C2×#6711yescharacteristic
355C22×C4C2×#6711yescharacteristic
356C22×C4C2×#6711yescharacteristic
357C22×C4C2×#6711yescharacteristic
358C22×C4C2×#6711yescharacteristic
359C22×C4C23×D311yescharacteristic
360SD16—128no—
361C2×C8—64no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—24no—
373C4⋊C4—22no—
374C4⋊C4—24no—
375C4⋊C4—62no—
376C4⋊C4—122no—
377C4⋊C4—122no—
378C4⋊C4C2×#6714yes—
379C22⋊C4—24no—
380C22⋊C4—24no—
381C22⋊C4—22no—
382C22⋊C4—22no—
383C22⋊C4—122no—
384C22⋊C4—122no—
385C22⋊C4—122no—
386C22⋊C4—122no—
387C22⋊C4—122no—
388C42—24no—
389C42—21no—
390C42—24no—
391C42—21no—
392C42—122no—
393C42C2×#6712yes—
394C42C2×#6712yes—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—22no—
398C22×C3—42no—
399C22×C3—41no—
400C22×C3—41no—
401C22×C3—81no—
402C22×C3C2×#12011yescharacteristic
403C22×C3C22×SD1611yescharacteristic
404C22×C3C2×#13711yescharacteristic
405C2×D3—42no—
406C2×D3—42no—
407C2×D3—42no—
408C2×D3—81no—
409C2×D3—81no—
410C2×D3—81no—
411C2×D3—81no—
412C3×C4—22no—
413C3×C4—22no—
414C3×C4—22no—
415C3×C4—41no—
416C3×C4—44no—
417C3×C4—41no—
418C3×C4—44no—
419C3×C4#44812yes—
420C3×C4#44812yes—
421Dic3—42no—
422Dic3—81no—
423Dic3—81no—
424C23—21no—
425C23—21no—
426C23—41no—
427C23—41no—
428C23—62no—
429C23—121no—
430C23—121no—
431C23—121no—
432C23—121no—
433C23—121no—
434C23—121no—
435C23C2×#21512yes—
436C23C22×#6711yescharacteristic
437Q8—44no—
438Q8—82no—
439Q8—82no—
440Q8—82no—
441D4—64no—
442D4—122no—
443D4—128no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—24no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—62no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—121no—
460C2×C4—121no—
461C2×C4—121no—
462C2×C4—121no—
463C2×C4—121no—
464C2×C4—121no—
465C2×C4—121no—
466C2×C4—121no—
467C2×C4—242no—
468C2×C4—242no—
469C2×C4C2×#21512yes—
470C2×C4C2×#21512yes—
471C2×C4#94212yes—
472C2×C4#95312yes—
473C2×C4#95311yescharacteristic
474C2×C4#95311yescharacteristic
475C2×C4C22×#6712yes—
476C8—124no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—42no—
480C2×C3C2×#44811yescharacteristic
481C2×C3#301811yescharacteristic
482C2×C3#302511yescharacteristic
483D3—82no—
484C22—21no—
485C22—21no—
486C22—22no—
487C22—42no—
488C22—41no—
489C22—41no—
490C22—81no—
491C22—122no—
492C22—122no—
493C22—122no—
494C22—241no—
495C22—241no—
496C22—241no—
497C22—241no—
498C22C22×#21511yescharacteristic
499C22C2×#94211yescharacteristic
500C22C2×#95311yescharacteristic
501C4—22no—
502C4—22no—
503C4—22no—
504C4—41no—
505C4—44no—
506C4—41no—
507C4—44no—
508C4—122no—
509C4—241no—
510C4—241no—
511C4#522712yes—
512C4#522712yes—
513C3#1812411yescharacteristic, complemented, Sylow
514C2—21no—
515C2—21no—
516C2—42no—
517C2—242no—
518C2C2×#522711yescharacteristic
519C2#8412111yescharacteristic
520C2#8412511yescharacteristic
subgroupG/H
centerC2×C4#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3458C2
socleC22×C3C2×#120
prime radical#4960C22
hypercenter#3458D3
hypocenterC3#18124
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3458

Complements

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

NH ≅ G/Ntype
1. C3×#3458C2semidirect
2. #83351C2semidirect
4. #71902C2semidirect
5. #71898C2semidirect
7. #71890C2semidirect
8. #71835C2semidirect
9. #71571C2semidirect
42. #3458D3semidirect
59. C3×C4⋊Q8D4semidirect
60. C3×C4⋊Q8D4semidirect
76. C3×C4×Q8D4semidirect
158. C22×C3×C4SD16semidirect
221. C4⋊Q8#67semidirect
222. C4⋊Q8#67semidirect
238. C4×Q8#67semidirect
353. C22×C4#215semidirect
513. C3#18124semidirect

Maximal subgroups

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

subgroupindex
C3×#34582
#833512
#822732
#719022
#718982
#718922
#718902
#718352
#715712
#181243