Finite Group Databasebeta

← #11864573

Subgroups — #11864573

Order 1152.

2292 conjugacy classes of subgroups.

The subgroup lattice has 537 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64366C211yesmaximal, characteristic, complemented
2#10596304C211yesmaximal, characteristic, complemented
3C2×#64117C211yesmaximal, characteristic, complemented
4C2×#64113C211yesmaximal, characteristic, complemented
5C32×#408C211yesmaximal, characteristic, complemented
6#10592832C211yesmaximal, characteristic, complemented
7C2×#63648C211yesmaximal, characteristic, complemented
8#10592433C212yesmaximal, complemented
9#10592432C212yesmaximal, complemented
10#10592431C212yesmaximal, complemented
11#10592430C212yesmaximal, complemented
12#78672—34nomaximal
13C2×C32×#41C2211yescharacteristic
14#64366—24no—
15#64366—24no—
16C2×#3832—21no—
17C2×D4×C3⋊D3—21no—
18C2×D4×C3⋊D3—22no—
19C2×#3828C2212yes—
20C2×#3828C2211yescharacteristic
21C2×#3828C2211yescharacteristic, complemented
22C22×C4⋊(C3⋊D3)C2212yes—
23C22×C4⋊(C3⋊D3)C2211yescharacteristic
24C2×C32×M16C2211yescharacteristic, complemented
25C22×C32×C8C2211yescharacteristic, complemented
26#64155—21no—
27#64148—22no—
28#64147—21no—
29#64139C2212yescomplemented
30C2×C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
31#64133C2212yescomplemented
32#64117—24no—
33C2×#3746C2211yescharacteristic, complemented
34#64113—24no—
35C2×#3745C2211yescharacteristic, complemented
36C2×#3745C2212yescomplemented
37C2×#3744C2211yescharacteristic, complemented
38C2×#3744C2211yescharacteristic, complemented
39C2×#3744C2212yescomplemented
40C32×C22⋊C8C2212yescomplemented
41C32×C22⋊C8C2212yescomplemented
42#63648C2212yescomplemented
43#63648C2212yescomplemented
44#63648C2212yescomplemented
45#63643C2212yescomplemented
46C2×#1009—34no—
47#5833—34no—
48C2×#908—34no—
49C2×#904—34no—
50C3×#408D314yescomplemented
51#5133—34no—
52C2×#821—34no—
53#4734—38no—
54#4733—38no—
55#4732—38no—
56#4731—38no—
57C23×C3⋊D3—22no—
58C23×C3⋊D3—21no—
59C32×#41—22no—
60C32×#41—22no—
61C2×C32×Q8D411yescharacteristic
62C2×C32×D4D411yescharacteristic
63C2×C32×D4D412yes—
64C22×C32×C4C2312yes—
65C22×C32×C4C2311yescharacteristic
66C2×#638—22no—
67C2×#638—21no—
68C2×#638—21no—
69C2×#638—22no—
70C22×C3⋊Dic3—21no—
71#3832—28no—
72D4×C3⋊D3—216no—
73D4×C3⋊D3—28no—
74#3828—22no—
75#3828—24no—
76#3828—22no—
77#3828—22no—
78#3828—24no—
79#3828D414yescomplemented
80C2×C4⋊(C3⋊D3)—22no—
81C2×C4⋊(C3⋊D3)—21no—
82C2×C4⋊(C3⋊D3)—22no—
83C2×C4⋊(C3⋊D3)—24no—
84C2×C4⋊(C3⋊D3)—24no—
85C2×C4⋊(C3⋊D3)D414yescomplemented
86C2×C4⋊(C3⋊D3)C2311yescharacteristic
87C2×C4⋊(C3⋊D3)C2311yescharacteristic
88C2×C4⋊(C3⋊D3)C2312yes—
89C2×C4×C3⋊D3—22no—
90C2×C4×C3⋊D3—21no—
91C2×C4×C3⋊D3—21no—
92C2×#634C2311yescharacteristic
93C2×#634C2311yescharacteristic
94C32×M16—22no—
95C2×C32×C8—22no—
96C2×C32×C8C2311yescharacteristic
97C2×C32×C8C2312yes—
98C2×C32×C8C2311yescharacteristic
99#3757—22no—
100#3752—22no—
101C4⋊(C3⋊Dic3)—21no—
102C4⋊(C3⋊Dic3)C2312yes—
103#3750—22no—
104#3746—22no—
105#3745—22no—
106#3745—24no—
107#3744—22no—
108#3744—24no—
109#3744—22no—
110#3012—91noSylow
111C2×C3×#41C2×D314yes—
112#1009—616no—
113#1009—616no—
114C2×#239—64no—
115C2×D3×D4—64no—
116C2×D3×D4—68no—
117C2×#235—34no—
118C2×#235—38no—
119C2×#235—34no—
120C22×D12—34no—
121C22×D12—38no—
122C2×C3×M16C2×D314yescomplemented
123C22×C3×C8C2×D314yescomplemented
124#946—64no—
125#939—68no—
126#938—64no—
127#930—38no—
128C2×C4⋊Dic3—34no—
129#924—38no—
130#908—616no—
131C2×Dic12—34no—
132#904—616no—
133C2×D24—34no—
134C2×D24—38no—
135C2×#204—34no—
136C2×#204—34no—
137C2×#204—38no—
138C3×C22⋊C8C2×D318yescomplemented
139C3×C22⋊C8C2×D318yescomplemented
140#821—38no—
141#821—38no—
142#821—38no—
143#816—38no—
144C23×C32C2×D412yes—
145C23×C32C2×D411yescharacteristic
146C22×C3⋊D3—22no—
147C22×C3⋊D3—21no—
148C22×C3⋊D3—28no—
149C22×C3⋊D3—24no—
150C22×C3⋊D3—21no—
151C22×C3⋊D3—81no—
152C22×C3⋊D3—82no—
153C22×C3⋊D3—81no—
154C22×C3⋊D3—82no—
155C32×Q8—22no—
156C32×D4—24no—
157C32×D4—22no—
158C2×C32×C4—22no—
159C2×C32×C4—22no—
160C2×C32×C4C2×D412yes—
161C2×C32×C4C2×D411yescharacteristic
162C2×C32×C4C2×D412yes—
163C2×C32×C4C2×D412yes—
164C2×C32×C4C2411yescharacteristic
165#638—28no—
166#638—24no—
167#638—42no—
168#638—82no—
169C2×C3⋊Dic3—21no—
170C2×C3⋊Dic3—21no—
171C2×C3⋊Dic3—22no—
172C2×C3⋊Dic3—81no—
173C4⋊(C3⋊D3)—28no—
174C4⋊(C3⋊D3)—24no—
175C4⋊(C3⋊D3)—21no—
176C4⋊(C3⋊D3)—22no—
177C4⋊(C3⋊D3)—21no—
178C4⋊(C3⋊D3)—21no—
179C4⋊(C3⋊D3)C2×D412yes—
180C4⋊(C3⋊D3)C2×D414yes—
181C4×C3⋊D3—28no—
182C4×C3⋊D3—24no—
183C4×C3⋊D3—42no—
184#634—21no—
185#634—21no—
186#634—21no—
187#634C2×D412yes—
188C32×C8—21no—
189C32×C8—22no—
190C32×C8—21no—
191C2×H(5,𝔽2)—91no—
192C2×AGL(1,ℤ/8)—91no—
193C2×#135—91no—
194#537—91no—
195#452—92no—
196#450—92no—
197#449—92no—
198#447—92no—
199#417—91no—
200C2×#102—91no—
201#408C3⋊D311yescharacteristic, complemented
202C23×D3—68no—
203C23×D3—64no—
204C3×#41—28no—
205C3×#41—28no—
206C2×C3×Q8D1214yes—
207C2×C3×D4D1214yes—
208C2×C3×D4D1218yes—
209C22×C3×C4C22×D318yes—
210C22×C3×C4C22×D314yes—
211C2×#67—68no—
212C2×#67—64no—
213C2×#67—64no—
214C2×#67—68no—
215C22×Dic3—64no—
216#239—632no—
217D3×D4—664no—
218D3×D4—632no—
219#235—316no—
220#235—616no—
221#235—616no—
222#235—68no—
223#235—68no—
224#235—68no—
225C2×D12—316no—
226C2×D12—38no—
227C2×D12—34no—
228C2×D12—34no—
229C2×D12—616no—
230C2×D12—616no—
231C2×D12—68no—
232C2×D12—64no—
233C2×D12—68no—
234C2×C4×D3—68no—
235C2×C4×D3—64no—
236C2×C4×D3—64no—
237C2×Dic6—34no—
238C2×Dic6—34no—
239C3×M16—28no—
240C2×C3×C8—28no—
241C2×C3×C8C22×D318yes—
242C2×C3×C8C22×D314yes—
243C2×C3×C8C22×D314yes—
244#217—68no—
245(C2×D3)⋊C4—68no—
246C4⋊Dic3—38no—
247C4⋊Dic3—64no—
248Dic3⋊C4—68no—
249Dic12—68no—
250D24—68no—
251D24—616no—
252#204—68no—
253#204—68no—
254#204—616no—
255C22×C32—21no—
256C22×C32—21no—
257C22×C32—22no—
258C22×C32—22no—
259C22×C32—22no—
260C22×C32#12012yes—
261C22×C32C22×D411yescharacteristic
262C2×C3⋊D3—24no—
263C2×C3⋊D3—22no—
264C2×C3⋊D3—42no—
265C2×C3⋊D3—41no—
266C2×C3⋊D3—42no—
267C2×C3⋊D3—41no—
268C2×C3⋊D3—41no—
269C2×C3⋊D3—41no—
270C2×C3⋊D3—41no—
271C2×C3⋊D3—84no—
272C2×C3⋊D3—88no—
273C32×C4—22no—
274C32×C4#12012yes—
275C32×C4C22×D411yescharacteristic
276C32×C4C22×D411yescharacteristic
277C3⋊Dic3—22no—
278C3⋊Dic3—41no—
279C3⋊Dic3—42no—
280H(5,𝔽2)—184no—
281H(5,𝔽2)—184no—
282C2×#41—91no—
283C2×#41—91no—
284C2×#41—92no—
285C2×#41—181no—
286C2×#41C2×C3⋊D311yescharacteristic
287C22×D4—91no—
288C22×D4—92no—
289C22×D4—182no—
290C22×D4—181no—
291AGL(1,ℤ/8)—184no—
292#135—184no—
293C2×Q16—91no—
294C2×SD16—91no—
295C2×SD16—91no—
296C2×SD16—92no—
297C2×D8—91no—
298C2×D8—92no—
299C2×M16C2×C3⋊D311yescharacteristic, complemented
300C22×C8C2×C3⋊D311yescharacteristic, complemented
301C22⋊Q8—92no—
302C22⋊Q8—181no—
303#121—92no—
304#121—181no—
305#121—182no—
306C2×C4⋊C4—91no—
307#103—92no—
308#102—92no—
309#102—92no—
310#102—92no—
311C22⋊C8C2×C3⋊D312yescomplemented
312C22⋊C8C2×C3⋊D312yescomplemented
313C23×C3C2×D1218yes—
314C23×C3C2×D1214yes—
315C22×D3—68no—
316C22×D3—64no—
317C22×D3—632no—
318C22×D3—616no—
319C22×D3—64no—
320C22×D3—244no—
321C22×D3—248no—
322C22×D3—244no—
323C22×D3—248no—
324C3×Q8—28no—
325C3×D4—216no—
326C3×D4—28no—
327C2×C3×C4—28no—
328C2×C3×C4—28no—
329C2×C3×C4C2×D1218yes—
330C2×C3×C4C2×D1214yes—
331C2×C3×C4D3×D418yes—
332C2×C3×C4D3×D418yes—
333C2×C3×C4C23×D314yes—
334#67—632no—
335#67—616no—
336#67—128no—
337#67—248no—
338C2×Dic3—64no—
339C2×Dic3—64no—
340C2×Dic3—68no—
341C2×Dic3—244no—
342D12—316no—
343D12—38no—
344D12—632no—
345D12—616no—
346D12—68no—
347D12—64no—
348D12—64no—
349D12—64no—
350C4×D3—632no—
351C4×D3—616no—
352C4×D3—128no—
353Dic6—38no—
354Dic6—64no—
355Dic6—64no—
356Dic6—64no—
357C3×C8—24no—
358C3×C8—28no—
359C3×C8—24no—
360C2×C32—22no—
361C2×C32—42no—
362C2×C32C2×#12011yescharacteristic
363C2×C32#57611yescharacteristic
364C2×C32#57711yescharacteristic
365C3⋊D3—44no—
366C3⋊D3—42no—
367C3⋊D3—81no—
368C24—182no—
369C24—181no—
370#41—22no—
371#41—22no—
372#41—94no—
373#41—184no—
374#41—184no—
375#41—188no—
376#41—182no—
377#41—182no—
378#41—182no—
379C2×Q8—91no—
380C2×Q8—91no—
381C2×Q8C4⋊(C3⋊D3)11yescharacteristic
382C2×D4—92no—
383C2×D4—91no—
384C2×D4—91no—
385C2×D4—94no—
386C2×D4—184no—
387C2×D4—184no—
388C2×D4—1816no—
389C2×D4—188no—
390C2×D4—182no—
391C2×D4—181no—
392C2×D4—181no—
393C2×D4—182no—
394C2×D4—182no—
395C2×D4—181no—
396C2×D4—182no—
397C2×D4C4⋊(C3⋊D3)12yes—
398C2×D4C4⋊(C3⋊D3)11yescharacteristic
399C22×C4—182no—
400C22×C4—181no—
401C22×C4—181no—
402C22×C4—181no—
403C22×C4C22×C3⋊D312yes—
404C22×C4C22×C3⋊D311yescharacteristic
405Q16—182no—
406SD16—182no—
407SD16—184no—
408SD16—182no—
409D8—182no—
410D8—184no—
411M16—22no—
412C2×C8—22no—
413C2×C8C22×C3⋊D311yescharacteristic
414C2×C8C22×C3⋊D311yescharacteristic
415C2×C8C22×C3⋊D312yes—
416C4⋊C4—92no—
417C4⋊C4—181no—
418C4⋊C4—182no—
419C22⋊C4—182no—
420C22⋊C4—182no—
421C22×C3—24no—
422C22×C3—24no—
423C22×C3—28no—
424C22×C3—28no—
425C22×C3—28no—
426C22×C3#88218yes—
427C22×C3C22×D1214yes—
428C2×D3—616no—
429C2×D3—68no—
430C2×D3—128no—
431C2×D3—124no—
432C2×D3—128no—
433C2×D3—124no—
434C2×D3—124no—
435C2×D3—124no—
436C2×D3—124no—
437C2×D3—2416no—
438C2×D3—2432no—
439C3×C4—28no—
440C3×C4#88218yes—
441C3×C4C2×D3×D414yes—
442C3×C4C2×D3×D414yes—
443Dic3—68no—
444Dic3—124no—
445Dic3—128no—
446C32#301211yescharacteristic, complemented, Sylow
447C23—188no—
448C23—182no—
449C23—181no—
450C23—184no—
451C23—181no—
452C23—721no—
453C23—721no—
454C23—722no—
455C23—722no—
456C23C2×C4⋊(C3⋊D3)12yes—
457C23C2×C4⋊(C3⋊D3)11yescharacteristic
458Q8—22no—
459Q8—92no—
460Q8—181no—
461Q8—181no—
462Q8—181no—
463D4—24no—
464D4—22no—
465D4—94no—
466D4—92no—
467D4—188no—
468D4—188no—
469D4—184no—
470D4—184no—
471D4—181no—
472D4—182no—
473D4—181no—
474D4—181no—
475D4—362no—
476D4—722no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—188no—
480C2×C4—181no—
481C2×C4—184no—
482C2×C4—182no—
483C2×C4—181no—
484C2×C4—362no—
485C2×C4—721no—
486C2×C4C2×C4⋊(C3⋊D3)12yes—
487C2×C4C2×C4⋊(C3⋊D3)11yescharacteristic
488C2×C4D4×C3⋊D312yes—
489C2×C4D4×C3⋊D312yes—
490C2×C4C23×C3⋊D311yescharacteristic
491C8—21no—
492C8—21no—
493C8—22no—
494C2×C3—28no—
495C2×C3—48no—
496C2×C3C2×#88214yes—
497C2×C3#575114yes—
498C2×C3#575214yes—
499D3—1216no—
500D3—128no—
501D3—244no—
502C22—21no—
503C22—21no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—184no—
508C22—182no—
509C22—362no—
510C22—361no—
511C22—362no—
512C22—361no—
513C22—361no—
514C22—361no—
515C22—361no—
516C22—724no—
517C22—728no—
518C22#6409112yes—
519C22C22×C4⋊(C3⋊D3)11yescharacteristic
520C4—22no—
521C4—182no—
522C4—361no—
523C4—362no—
524C4#6409112yes—
525C4C2×D4×C3⋊D311yescharacteristic
526C4C2×D4×C3⋊D311yescharacteristic
527C3#7867214yescomplemented
528C2—22no—
529C2—42no—
530C2—364no—
531C2—362no—
532C2—721no—
533C2C2×#6409111yescharacteristic
534C2#1059622211yescharacteristic
535C2#1059622311yescharacteristic
subgroupG/H
centerC22C22×C4⋊(C3⋊D3)
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
FittingC32×#408C2
socleC22×C32C22×D4
hypercenter#408C3⋊D3
hypocenterC32#3012
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
1. C2×#64366C2semidirect
2. #10596304C2semidirect
3. C2×#64117C2semidirect
4. C2×#64113C2semidirect
5. C32×#408C2semidirect
6. #10592832C2semidirect
7. C2×#63648C2semidirect
8. #10592433C2semidirect
9. #10592432C2semidirect
10. #10592431C2semidirect
11. #10592430C2semidirect
21. C2×#3828C22semidirect
24. C2×C32×M16C22semidirect
25. C22×C32×C8C22semidirect
29. #64139C22semidirect
30. C2×C4⋊(C3⋊Dic3)C22semidirect
31. #64133C22semidirect
33. C2×#3746C22semidirect
35. C2×#3745C22semidirect
36. C2×#3745C22semidirect
37. C2×#3744C22semidirect
38. C2×#3744C22semidirect
39. C2×#3744C22semidirect
40. C32×C22⋊C8C22semidirect
41. C32×C22⋊C8C22semidirect
42. #63648C22semidirect
43. #63648C22semidirect
44. #63648C22semidirect
45. #63643C22semidirect
50. C3×#408D3semidirect
79. #3828D4semidirect
85. C2×C4⋊(C3⋊D3)D4semidirect
122. C2×C3×M16C2×D3semidirect
123. C22×C3×C8C2×D3semidirect
138. C3×C22⋊C8C2×D3semidirect
139. C3×C22⋊C8C2×D3semidirect
201. #408C3⋊D3semidirect
299. C2×M16C2×C3⋊D3semidirect
300. C22×C8C2×C3⋊D3semidirect
311. C22⋊C8C2×C3⋊D3semidirect
312. C22⋊C8C2×C3⋊D3semidirect
446. C32#3012semidirect
527. C3#78672semidirect

Maximal subgroups

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

subgroupindex
C2×#643662
#105963042
C2×#641172
C2×#641132
C32×#4082
#105928322
C2×#636482
#105924332
#105924322
#105924312
#105924302
#786723