Finite Group Databasebeta

← #87197

Subgroups — #87197

Order 384.

1412 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
1#5964C211yesmaximal, characteristic, complemented
2C2×#1007C211yesmaximal, characteristic, complemented
3C2×C3×#137C211yesmaximal, characteristic, complemented
4#5822C211yesmaximal, characteristic, complemented
5C2×#942C211yesmaximal, characteristic, complemented
6#5779C214yesmaximal, complemented
7#5778C214yesmaximal, complemented
8#5777C214yesmaximal, complemented
9D3×#137C214yesmaximal, complemented
10#5766C214yesmaximal, complemented
11#5762C214yesmaximal, complemented
12#5748C211yesmaximal, characteristic, complemented
13#5745C211yesmaximal, characteristic, complemented
14#3586—31nomaximal, Sylow
15C2×C3×#41C2211yescharacteristic
16C22×C3×Q8C2211yescharacteristic, complemented
17#1010—21no—
18#1010C2212yescomplemented
19#1009—21no—
20#1009C2212yescomplemented
21D3×#41—22no—
22D3×#41C2214yes—
23#1007—22no—
24#1007—22no—
25#1007C2214yescomplemented
26#1007C2214yescomplemented
27#1007C2212yescomplemented
28C2×#239—21no—
29C2×#239C2212yescomplemented
30C2×D3×Q8—21no—
31C2×D3×Q8C2212yescomplemented
32#1004—22no—
33#1004C2212yescomplemented
34C2×#235—22no—
35C2×#235—22no—
36C2×#235C2211yescharacteristic
37C2×#235C2211yescharacteristic
38C2×#235C2211yescharacteristic, complemented
39C3×#137C2214yescomplemented
40C3×#137C2214yescomplemented
41C2×C3×Q16C2212yescomplemented
42C2×C3×SD16C2212yescomplemented
43C2×C3×M16C2211yescharacteristic, complemented
44#951C2212yescomplemented
45#950C2214yescomplemented
46#949C2212yescomplemented
47C2×#216C2212yescomplemented
48#942C2214yescomplemented
49#942C2212yescomplemented
50#942C2214yescomplemented
51C2×#215C2212yescomplemented
52#932C2212yescomplemented
53C2×#208C2211yescharacteristic, complemented
54#919C2218yescomplemented
55#918C2218yescomplemented
56#918C2218yescomplemented
57D3×Q16C2218yescomplemented
58#916C2218yescomplemented
59#915C2218yescomplemented
60#914C2218yescomplemented
61D3×SD16C2218yescomplemented
62#909C2212yescomplemented
63#909C2212yescomplemented
64#908C2212yescomplemented
65#908C2212yescomplemented
66#907C2214yescomplemented
67D3×M16C2214yescomplemented
68#904C2214yescomplemented
69#901C2214yescomplemented
70#585—31no—
71C2×2−1+4—31no—
72#578—34no—
73#578—34no—
74#577—34no—
75#577—34no—
76#575—31no—
77#575—34no—
78#575—31no—
79C2×#137—31no—
80C2×#137—34no—
81C2×#137D311yescharacteristic, complemented
82#568—31no—
83C3×#41—21no—
84C3×#41—21no—
85C3×#41C2312yes—
86C3×#41C2312yes—
87C2×C3×Q8—21no—
88C2×C3×Q8—21no—
89C2×C3×Q8—21no—
90C2×C3×Q8C2312yes—
91C2×C3×Q8C2312yes—
92C2×C3×Q8C2311yescharacteristic
93C2×C3×Q8C2312yes—
94C2×C3×D4—21no—
95C2×C3×D4C2311yescharacteristic
96C22×C3×C4—21no—
97C22×C3×C4—21no—
98C22×C3×C4C2311yescharacteristic
99C2×#67—21no—
100C2×#67—21no—
101C2×#67—24no—
102C2×#67D412yes—
103#239—24no—
104#239—24no—
105#239—22no—
106#239—24no—
107#239C2318yes—
108#239C2314yes—
109D3×Q8—24no—
110D3×Q8—24no—
111D3×Q8—24no—
112D3×Q8—22no—
113D3×Q8C2318yes—
114D3×Q8C2314yes—
115#237—24no—
116#237—24no—
117#237—22no—
118#237C2314yes—
119D3×D4—24no—
120D3×D4—24no—
121D3×D4—22no—
122D3×D4C2314yes—
123#235—21no—
124#235—21no—
125#235—21no—
126#235—21no—
127#235—24no—
128#235—28no—
129#235—28no—
130#235—22no—
131#235—24no—
132#235—22no—
133#235D412yes—
134#235D412yes—
135#235C2312yes—
136#235C2312yes—
137#235C2312yes—
138#235C2312yes—
139#235C2312yes—
140#235C2312yes—
141C2×D12—22no—
142C2×D12—22no—
143C2×D12D411yescharacteristic
144C2×D12C2311yescharacteristic
145C2×D12C2311yescharacteristic
146C2×C4×D3—22no—
147C2×C4×D3—21no—
148C2×C4×D3—21no—
149C2×C4×D3—22no—
150C2×C4×D3—24no—
151C2×C4×D3C2312yes—
152C2×Dic6—22no—
153C2×Dic6—22no—
154C2×Dic6D411yescharacteristic
155C2×Dic6C2311yescharacteristic
156C2×Dic6C2311yescharacteristic
157C3×Q16C2318yes—
158C3×SD16C2318yes—
159C3×M16C2312yes—
160C3×M16C2312yes—
161C2×C3×C8C2312yes—
162#216C2318yes—
163#216C2314yes—
164#215C2318yes—
165#215C2314yes—
166#214C2314yes—
167#213C2314yes—
168#208C2312yes—
169#208C2312yes—
170C2×C3⋊C8C2312yes—
171Dic12C2314yes—
172D24C2314yes—
173#204C2314yes—
174#204C2314yes—
175ΓL(1,𝔽25)C2318yes—
176D3×C8C2318yes—
1772−1+4—32no—
1782−1+4—34no—
1792−1+4—34no—
1802−1+4—32no—
1812−1+4—62no—
1822−1+4—62no—
1832−1+4—61no—
184H(5,𝔽2)—32no—
185H(5,𝔽2)—32no—
186H(5,𝔽2)—62no—
187H(5,𝔽2)—61no—
188C2×#41—31no—
189C2×#41—34no—
190C2×#41—32no—
191C2×#41—31no—
192C2×#41—31no—
193C2×#41—62no—
194C2×#41—62no—
195C2×#41—61no—
196C2×#41—62no—
197C2×#41C2×D311yescharacteristic
198C22×Q8—32no—
199C22×Q8—61no—
200C22×Q8C2×D311yescharacteristic, complemented
201#137—32no—
202#137—34no—
203#137—34no—
204#137—32no—
205#137—38no—
206#137—38no—
207#137—38no—
208#137—32no—
209#137—32no—
210#137C2×D314yescomplemented
211#137C2×D314yescomplemented
212AGL(1,ℤ/8)—32no—
213AGL(1,ℤ/8)—32no—
214AGL(1,ℤ/8)—38no—
215AGL(1,ℤ/8)—32no—
216AGL(1,ℤ/8)—32no—
217#135—34no—
218#135—38no—
219#135—38no—
220#135—34no—
221C2×Q16—32no—
222C2×Q16—38no—
223C2×Q16C2×D312yescomplemented
224C2×SD16—32no—
225C2×SD16—38no—
226C2×SD16C2×D312yescomplemented
227#131—34no—
228#131—34no—
229C2×M16—31no—
230C2×M16—34no—
231C2×M16C2×D311yescharacteristic, complemented
232C23×C3—21no—
233C23×C3C2×D411yescharacteristic
234C22×D3—21no—
235C22×D3—21no—
236C22×D3—24no—
237C22×D3C2×D412yes—
238C3×Q8—22no—
239C3×Q8—22no—
240C3×Q8—22no—
241C3×Q8—21no—
242C3×Q8C2414yes—
243C3×Q8C2412yes—
244C3×D4—22no—
245C3×D4—22no—
246C3×D4—21no—
247C3×D4C2412yes—
248C2×C3×C4—22no—
249C2×C3×C4—21no—
250C2×C3×C4—22no—
251C2×C3×C4—22no—
252C2×C3×C4—22no—
253C2×C3×C4—22no—
254C2×C3×C4C2×D411yescharacteristic
255C2×C3×C4C2×D411yescharacteristic
256C2×C3×C4C2×D411yescharacteristic
257C2×C3×C4C2411yescharacteristic
258C2×C3×C4C2411yescharacteristic
259C2×C3×C4C2411yescharacteristic
260#67—22no—
261#67—22no—
262#67—22no—
263#67—22no—
264#67—24no—
265#67—24no—
266#67C2×D414yes—
267#67C2×D414yes—
268C2×Dic3—21no—
269C2×Dic3—21no—
270C2×Dic3—24no—
271C2×Dic3C2×D412yes—
272D12—24no—
273D12—28no—
274D12—21no—
275D12—21no—
276D12C2×D412yes—
277D12C2×D412yes—
278D12C2412yes—
279D12C2412yes—
280C4×D3—24no—
281C4×D3—28no—
282C4×D3—22no—
283C4×D3—22no—
284C4×D3—22no—
285C4×D3—22no—
286C4×D3—24no—
287C4×D3—28no—
288C4×D3C2×D414yes—
289C4×D3C2414yes—
290Dic6—21no—
291Dic6—21no—
292Dic6—24no—
293Dic6—28no—
294Dic6C2×D412yes—
295Dic6C2×D412yes—
296Dic6C2412yes—
297Dic6C2412yes—
298C3×C8C2414yes—
299C3⋊C8C2414yes—
300#41—21no—
301#41—21no—
302#41—32no—
303#41—32no—
304#41—32no—
305#41—32no—
306#41—38no—
307#41—34no—
308#41—34no—
309#41—32no—
310#41—32no—
311#41—32no—
312#41—32no—
313#41—64no—
314#41—64no—
315#41—62no—
316#41—68no—
317#41—68no—
318#41—62no—
319#41—64no—
320#41—64no—
321#41—64no—
322#41—62no—
323#41—62no—
324#41—64no—
325#41—64no—
326#41—61no—
327#41—61no—
328#41—61no—
329#41—61no—
330#41C22×D312yes—
331#41C22×D312yes—
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8—31no—
336C2×Q8—38no—
337C2×Q8—34no—
338C2×Q8—31no—
339C2×Q8—31no—
340C2×Q8—62no—
341C2×Q8—62no—
342C2×Q8—64no—
343C2×Q8—64no—
344C2×Q8—64no—
345C2×Q8—62no—
346C2×Q8C22×D312yes—
347C2×Q8C22×D312yes—
348C2×Q8C22×D311yescharacteristic
349C2×Q8C22×D312yes—
350C2×D4—21no—
351C2×D4—32no—
352C2×D4—34no—
353C2×D4—31no—
354C2×D4—31no—
355C2×D4—31no—
356C2×D4—62no—
357C2×D4—62no—
358C2×D4—64no—
359C2×D4—62no—
360C2×D4—64no—
361C2×D4—64no—
362C2×D4—61no—
363C2×D4—61no—
364C2×D4C22×D311yescharacteristic
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—32no—
368C22×C4—62no—
369C22×C4—64no—
370C22×C4—62no—
371C22×C4—61no—
372C22×C4—61no—
373C22×C4C22×D311yescharacteristic
374Q16—34no—
375Q16—38no—
376Q16—34no—
377Q16C22×D318yes—
378SD16—38no—
379SD16—34no—
380SD16—34no—
381SD16—34no—
382SD16—34no—
383SD16C22×D318yes—
384D8—34no—
385D8—34no—
386M16—32no—
387M16—32no—
388M16—38no—
389M16C22×D312yes—
390M16C22×D312yes—
391C2×C8—32no—
392C2×C8—38no—
393C2×C8C22×D312yes—
394C22×C3—22no—
395C22×C3—41no—
396C22×C3—81no—
397C22×C3C22×D411yescharacteristic
398C22×C3C22×D411yescharacteristic
399C22×C3C22×D411yescharacteristic
400C2×D3—22no—
401C2×D3—22no—
402C2×D3—42no—
403C2×D3—42no—
404C2×D3—84no—
405C2×D3—81no—
406C2×D3C22×D414yes—
407C3×C4—22no—
408C3×C4—24no—
409C3×C4C22×D411yescharacteristic
410C3×C4C22×D411yescharacteristic
411C3×C4C22×D411yescharacteristic
412C3×C4C2511yescharacteristic
413Dic3—22no—
414Dic3—22no—
415Dic3C22×D414yes—
416C23—21no—
417C23—32no—
418C23—64no—
419C23—61no—
420C23—61no—
421C23D3×D411yescharacteristic
422Q8—22no—
423Q8—22no—
424Q8—22no—
425Q8—21no—
426Q8—32no—
427Q8—32no—
428Q8—32no—
429Q8—32no—
430Q8—68no—
431Q8—64no—
432Q8—61no—
433Q8—61no—
434Q8C23×D314yes—
435Q8C23×D312yes—
436D4—22no—
437D4—22no—
438D4—21no—
439D4—34no—
440D4—34no—
441D4—32no—
442D4—32no—
443D4—32no—
444D4—32no—
445D4—64no—
446D4—68no—
447D4—64no—
448D4—64no—
449D4—62no—
450D4—62no—
451D4—62no—
452D4—62no—
453D4—61no—
454D4—61no—
455D4C23×D312yes—
456C2×C4—22no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—32no—
463C2×C4—34no—
464C2×C4—34no—
465C2×C4—64no—
466C2×C4—68no—
467C2×C4—64no—
468C2×C4—68no—
469C2×C4—64no—
470C2×C4—61no—
471C2×C4—61no—
472C2×C4—62no—
473C2×C4—62no—
474C2×C4—62no—
475C2×C4—62no—
476C2×C4D3×D411yescharacteristic
477C2×C4D3×D411yescharacteristic
478C2×C4D3×D411yescharacteristic
479C2×C4C23×D311yescharacteristic
480C2×C4C23×D311yescharacteristic
481C2×C4C23×D311yescharacteristic
482C8—34no—
483C8C23×D314yes—
484C2×C3—21no—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3—42no—
488C2×C3C23×D411yescharacteristic
489D3—24no—
490D3—42no—
491D3—42no—
492C22—22no—
493C22—34no—
494C22—41no—
495C22—62no—
496C22—62no—
497C22—81no—
498C22—122no—
499C22—122no—
500C22—241no—
501C22—244no—
502C22C2×D3×D411yescharacteristic
503C22C2×D3×D411yescharacteristic
504C22C2×D3×D411yescharacteristic
505C4—22no—
506C4—24no—
507C4—34no—
508C4—62no—
509C4—62no—
510C4C2×D3×D411yescharacteristic
511C4C2×D3×D411yescharacteristic
512C4C2×D3×D411yescharacteristic
513C4C24×D311yescharacteristic
514C3#358611yescharacteristic, complemented, Sylow
515C2—21no—
516C2—21no—
517C2—21no—
518C2—42no—
519C2—64no—
520C2—122no—
521C2—122no—
522C2C22×D3×D411yescharacteristic
subgroupG/H
centerC2C22×D3×D4
commutator, cosocleC3×C4C25
FrattiniC4C24×D3
FittingC2×C3×#137C2
socleC2×C3C23×D4
hypercenterC2×#137D3
hypocenterC3#3586
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. #5964C2semidirect
2. C2×#1007C2semidirect
3. C2×C3×#137C2semidirect
4. #5822C2semidirect
5. C2×#942C2semidirect
6. #5779C2semidirect
7. #5778C2semidirect
8. #5777C2semidirect
9. D3×#137C2semidirect
10. #5766C2semidirect
11. #5762C2semidirect
12. #5748C2semidirect
13. #5745C2semidirect
16. C22×C3×Q8C22semidirect
18. #1010C22semidirect
20. #1009C22semidirect
25. #1007C22semidirect
26. #1007C22semidirect
27. #1007C22semidirect
29. C2×#239C22semidirect
31. C2×D3×Q8C22semidirect
33. #1004C22semidirect
38. C2×#235C22semidirect
39. C3×#137C22semidirect
40. C3×#137C22semidirect
41. C2×C3×Q16C22semidirect
42. C2×C3×SD16C22semidirect
43. C2×C3×M16C22semidirect
44. #951C22semidirect
45. #950C22semidirect
46. #949C22semidirect
47. C2×#216C22semidirect
48. #942C22semidirect
49. #942C22semidirect
50. #942C22semidirect
51. C2×#215C22semidirect
52. #932C22semidirect
53. C2×#208C22semidirect
54. #919C22semidirect
55. #918C22semidirect
56. #918C22semidirect
57. D3×Q16C22semidirect
58. #916C22semidirect
59. #915C22semidirect
60. #914C22semidirect
61. D3×SD16C22semidirect
62. #909C22semidirect
63. #909C22semidirect
64. #908C22semidirect
65. #908C22semidirect
66. #907C22semidirect
67. D3×M16C22semidirect
68. #904C22semidirect
69. #901C22semidirect
81. C2×#137D3semidirect
200. C22×Q8C2×D3semidirect
210. #137C2×D3semidirect
211. #137C2×D3semidirect
223. C2×Q16C2×D3semidirect
226. C2×SD16C2×D3semidirect
231. C2×M16C2×D3semidirect
514. C3#3586semidirect

Maximal subgroups

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

subgroupindex
#59642
C2×#10072
C2×C3×#1372
#58222
C2×#9422
#57792
#57782
#57772
D3×#1372
#57662
#57622
#57482
#57452
#35863