Finite Group Databasebeta

← #87201

Subgroups — #87201

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×#1010C211yesmaximal, characteristic, complemented
3C3×#575C211yesmaximal, characteristic, complemented
4C2×#951C211yesmaximal, characteristic, complemented
5#5822C211yesmaximal, characteristic, complemented
6#5779C214yesmaximal, complemented
7D3×#137C214yesmaximal, complemented
8#5775C214yesmaximal, complemented
9#5773C214yesmaximal, complemented
10#5771C214yesmaximal, complemented
11#5770C214yesmaximal, complemented
12C2×#909C211yesmaximal, characteristic, complemented
13#5745C211yesmaximal, characteristic, complemented
14#3586—31nomaximal, Sylow
15C2×C3×#41C2211yescharacteristic
16C2×C3×#41C2211yescharacteristic, complemented
17#1010—22no—
18#1010—22no—
19#1010—21no—
20#1010C2214yescomplemented
21#1010C2214yescomplemented
22#1010C2212yescomplemented
23#1009—21no—
24#1009C2212yescomplemented
25D3×#41—22no—
26D3×#41C2214yes—
27#1007C2212yescomplemented
28C2×D3×Q8C2212yescomplemented
29#1004—22no—
30#1004C2212yescomplemented
31C2×#237—22no—
32C2×#237C2212yescomplemented
33C2×#235—22no—
34C2×#235—21no—
35C2×#235C2211yescharacteristic
36C2×#235C2211yescharacteristic
37C22×Dic6—21no—
38C22×Dic6C2211yescharacteristic, complemented
39C3×#137C2212yescomplemented
40C3×#137C2212yescomplemented
41C3×AGL(1,ℤ/8)C2212yescomplemented
42C3×AGL(1,ℤ/8)C2212yescomplemented
43C3×#135C2214yescomplemented
44C2×C3×M16C2211yescharacteristic, complemented
45#951C2214yescomplemented
46#951C2212yescomplemented
47#951C2214yescomplemented
48#950C2214yescomplemented
49#949C2212yescomplemented
50C2×#216C2212yescomplemented
51#942C2212yescomplemented
52C2×#214C2212yescomplemented
53#932C2212yescomplemented
54C2×#208C2211yescharacteristic, complemented
55#918C2218yescomplemented
56D3×Q16C2218yescomplemented
57#916C2218yescomplemented
58#915C2218yescomplemented
59#915C2218yescomplemented
60D3×SD16C2218yescomplemented
61#912C2218yescomplemented
62#911C2218yescomplemented
63#909C2214yescomplemented
64#909C2214yescomplemented
65#907C2214yescomplemented
66D3×M16C2214yescomplemented
67C2×Dic12C2212yescomplemented
68C2×#204C2212yescomplemented
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#575D311yescharacteristic, complemented
79C2×#137—31no—
80C2×#137—34no—
81C2×#137—31no—
82#568—31no—
83C3×#41—21no—
84C3×#41—21no—
85C3×#41—21no—
86C3×#41—21no—
87C3×#41C2312yes—
88C3×#41C2312yes—
89C3×#41C2312yes—
90C3×#41C2312yes—
91C2×C3×Q8C2311yescharacteristic
92C2×C3×Q8C2311yescharacteristic
93C2×C3×D4—21no—
94C2×C3×D4—21no—
95C2×C3×D4C2311yescharacteristic
96C2×C3×D4C2311yescharacteristic
97C22×C3×C4—21no—
98C22×C3×C4—21no—
99C22×C3×C4C2311yescharacteristic
100C2×#67—21no—
101C2×#67—22no—
102C2×#67—24no—
103C2×#67D412yes—
104C22×Dic3—21no—
105C22×Dic3—22no—
106#239—22no—
107#239C2314yes—
108D3×Q8—24no—
109D3×Q8—22no—
110D3×Q8C2318yes—
111D3×Q8C2314yes—
112#237—28no—
113#237—28no—
114#237—24no—
115#237—24no—
116#237—24no—
117#237—22no—
118#237C2318yes—
119#237C2314yes—
120D3×D4—24no—
121D3×D4—24no—
122D3×D4—22no—
123D3×D4C2314yes—
124#235—21no—
125#235—21no—
126#235—24no—
127#235—22no—
128#235—24no—
129#235—24no—
130#235—24no—
131#235—22no—
132#235D412yes—
133#235D412yes—
134#235C2312yes—
135#235C2312yes—
136#235C2312yes—
137#235C2312yes—
138C2×D12—22no—
139C2×D12D411yescharacteristic
140C2×D12C2311yescharacteristic
141C2×C4×D3—22no—
142C2×C4×D3—21no—
143C2×C4×D3—24no—
144C2×C4×D3C2312yes—
145C2×Dic6—21no—
146C2×Dic6—21no—
147C2×Dic6—21no—
148C2×Dic6—22no—
149C2×Dic6—24no—
150C2×Dic6—24no—
151C2×Dic6—22no—
152C2×Dic6D411yescharacteristic
153C2×Dic6C2312yes—
154C2×Dic6C2311yescharacteristic
155C2×Dic6C2312yes—
156C2×Dic6C2312yes—
157C3×Q16C2314yes—
158C3×SD16C2314yes—
159C3×SD16C2314yes—
160C3×D8C2314yes—
161C3×M16C2312yes—
162C3×M16C2312yes—
163C2×C3×C8C2312yes—
164#216C2314yes—
165#216C2318yes—
166#215C2314yes—
167#214C2318yes—
168#214C2314yes—
169#213C2314yes—
170#208C2312yes—
171#208C2312yes—
172C2×C3⋊C8C2312yes—
173Dic12C2318yes—
174#204C2318yes—
175ΓL(1,𝔽25)C2318yes—
176D3×C8C2318yes—
1772−1+4—34no—
1782−1+4—34no—
1792−1+4—32no—
1802−1+4—32no—
1812−1+4—61no—
1822−1+4—62no—
1832−1+4—62no—
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—61no—
193C2×#41—62no—
194C2×#41—62no—
195C2×#41—62no—
196C2×#41C2×D311yescharacteristic
197C2×#41C2×D311yescharacteristic, complemented
198C22×Q8—32no—
199C22×Q8—31no—
200C22×Q8—61no—
201#137—34no—
202#137—34no—
203#137—32no—
204#137—32no—
205#137—38no—
206#137—38no—
207#137—38no—
208#137—34no—
209#137—34no—
210#137C2×D312yescomplemented
211#137C2×D312yescomplemented
212AGL(1,ℤ/8)—32no—
213AGL(1,ℤ/8)—32no—
214AGL(1,ℤ/8)—38no—
215AGL(1,ℤ/8)C2×D312yescomplemented
216AGL(1,ℤ/8)C2×D312yescomplemented
217#135—34no—
218#135—38no—
219#135—38no—
220#135C2×D314yescomplemented
221C2×Q16—32no—
222C2×Q16—38no—
223C2×Q16—32no—
224C2×SD16—32no—
225C2×SD16—38no—
226C2×SD16—32no—
227#131—34no—
228#131—34no—
229C2×M16—31no—
230C2×M16—34no—
231C2×M16C2×D311yescharacteristic, complemented
232C23×C3—21no—
233C23×C3—21no—
234C23×C3C2×D411yescharacteristic
235C22×D3—21no—
236C22×D3—24no—
237C22×D3C2×D412yes—
238C3×Q8—21no—
239C3×Q8—21no—
240C3×Q8C2412yes—
241C3×Q8C2412yes—
242C3×D4—22no—
243C3×D4—22no—
244C3×D4—22no—
245C3×D4—22no—
246C3×D4—21no—
247C3×D4—21no—
248C3×D4C2412yes—
249C3×D4C2412yes—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—22no—
253C2×C3×C4—22no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4C2×D411yescharacteristic
257C2×C3×C4C2×D411yescharacteristic
258C2×C3×C4C2×D411yescharacteristic
259C2×C3×C4C2411yescharacteristic
260C2×C3×C4C2411yescharacteristic
261C2×C3×C4C2411yescharacteristic
262#67—22no—
263#67—22no—
264#67—24no—
265#67—28no—
266#67—24no—
267#67C2×D414yes—
268#67C2×D414yes—
269C2×Dic3—22no—
270C2×Dic3—22no—
271C2×Dic3—22no—
272C2×Dic3—21no—
273C2×Dic3—28no—
274C2×Dic3—24no—
275C2×Dic3C2×D412yes—
276D12—24no—
277D12—21no—
278D12C2×D412yes—
279D12C2×D412yes—
280D12C2412yes—
281C4×D3—24no—
282C4×D3—28no—
283C4×D3—22no—
284C4×D3—22no—
285C4×D3—24no—
286C4×D3C2×D414yes—
287C4×D3C2414yes—
288Dic6—22no—
289Dic6—22no—
290Dic6—21no—
291Dic6—22no—
292Dic6—28no—
293Dic6—24no—
294Dic6C2×D412yes—
295Dic6C2×D412yes—
296Dic6C2414yes—
297Dic6C2412yes—
298C3×C8C2414yes—
299C3⋊C8C2414yes—
300#41—21no—
301#41—21no—
302#41—21no—
303#41—21no—
304#41—32no—
305#41—32no—
306#41—32no—
307#41—32no—
308#41—34no—
309#41—38no—
310#41—34no—
311#41—32no—
312#41—32no—
313#41—64no—
314#41—64no—
315#41—62no—
316#41—64no—
317#41—64no—
318#41—62no—
319#41—64no—
320#41—64no—
321#41—64no—
322#41—68no—
323#41—68no—
324#41—62no—
325#41—62no—
326#41—61no—
327#41—61no—
328#41C22×D312yes—
329#41C22×D312yes—
330#41C22×D312yes—
331#41C22×D312yes—
332C2×Q8—31no—
333C2×Q8—38no—
334C2×Q8—34no—
335C2×Q8—32no—
336C2×Q8—32no—
337C2×Q8—31no—
338C2×Q8—32no—
339C2×Q8—62no—
340C2×Q8—64no—
341C2×Q8—64no—
342C2×Q8—62no—
343C2×Q8—64no—
344C2×Q8—62no—
345C2×Q8—61no—
346C2×Q8—61no—
347C2×Q8—61no—
348C2×Q8C22×D311yescharacteristic
349C2×Q8C22×D311yescharacteristic
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—32no—
353C2×D4—34no—
354C2×D4—31no—
355C2×D4—31no—
356C2×D4—62no—
357C2×D4—64no—
358C2×D4—62no—
359C2×D4—64no—
360C2×D4—64no—
361C2×D4—62no—
362C2×D4—61no—
363C2×D4C22×D311yescharacteristic
364C2×D4C22×D311yescharacteristic
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—32no—
368C22×C4—64no—
369C22×C4—62no—
370C22×C4—62no—
371C22×C4—61no—
372C22×C4—61no—
373C22×C4C22×D311yescharacteristic
374Q16—38no—
375Q16—34no—
376Q16—38no—
377Q16C22×D314yes—
378SD16—34no—
379SD16—38no—
380SD16—34no—
381SD16—38no—
382SD16C22×D314yes—
383SD16C22×D314yes—
384D8—34no—
385D8C22×D314yes—
386M16—32no—
387M16—32no—
388M16—38no—
389M16C22×D312yes—
390M16C22×D312yes—
391C2×C8—32no—
392C2×C8—38no—
393C2×C8C22×D312yes—
394C22×C3—22no—
395C22×C3—22no—
396C22×C3—41no—
397C22×C3—42no—
398C22×C3—81no—
399C22×C3C22×D411yescharacteristic
400C22×C3C22×D411yescharacteristic
401C22×C3C22×D411yescharacteristic
402C2×D3—22no—
403C2×D3—42no—
404C2×D3—84no—
405C2×D3—81no—
406C2×D3C22×D414yes—
407C3×C4—22no—
408C3×C4—22no—
409C3×C4C22×D411yescharacteristic
410C3×C4C22×D411yescharacteristic
411C3×C4C22×D411yescharacteristic
412C3×C4C2511yescharacteristic
413Dic3—24no—
414Dic3—22no—
415Dic3C22×D414yes—
416C23—21no—
417C23—21no—
418C23—32no—
419C23—64no—
420C23—61no—
421C23D3×D411yescharacteristic
422Q8—21no—
423Q8—21no—
424Q8—32no—
425Q8—32no—
426Q8—34no—
427Q8—32no—
428Q8—68no—
429Q8—64no—
430Q8—62no—
431Q8—62no—
432Q8—62no—
433Q8—61no—
434Q8C23×D312yes—
435Q8C23×D312yes—
436D4—22no—
437D4—22no—
438D4—22no—
439D4—22no—
440D4—21no—
441D4—21no—
442D4—34no—
443D4—34no—
444D4—32no—
445D4—32no—
446D4—32no—
447D4—64no—
448D4—64no—
449D4—64no—
450D4—68no—
451D4—62no—
452D4—62no—
453D4—61no—
454D4C23×D312yes—
455D4C23×D312yes—
456C2×C4—21no—
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—64no—
467C2×C4—68no—
468C2×C4—64no—
469C2×C4—68no—
470C2×C4—62no—
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×C3—42no—
489C2×C3C23×D411yescharacteristic
490D3—24no—
491D3—42no—
492C22—22no—
493C22—22no—
494C22—34no—
495C22—41no—
496C22—42no—
497C22—62no—
498C22—81no—
499C22—122no—
500C22—241no—
501C22—244no—
502C22C2×D3×D411yescharacteristic
503C22C2×D3×D411yescharacteristic
504C22C2×D3×D411yescharacteristic
505C4—22no—
506C4—22no—
507C4—34no—
508C4—62no—
509C4—64no—
510C4C2×D3×D411yescharacteristic
511C4C2×D3×D411yescharacteristic
512C4C2×D3×D411yescharacteristic
513C4C24×D311yescharacteristic
514C3#358611yescharacteristic, complemented, Sylow
515C2—21no—
516C2—21no—
517C2—21no—
518C2—42no—
519C2—42no—
520C2—64no—
521C2—122no—
522C2C22×D3×D411yescharacteristic
subgroupG/H
centerC2C22×D3×D4
commutator, cosocleC3×C4C25
FrattiniC4C24×D3
FittingC3×#575C2
socleC2×C3C23×D4
hypercenter#575D3
hypocenterC3#3586
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #5964C2semidirect
2. C2×#1010C2semidirect
3. C3×#575C2semidirect
4. C2×#951C2semidirect
5. #5822C2semidirect
6. #5779C2semidirect
7. D3×#137C2semidirect
8. #5775C2semidirect
9. #5773C2semidirect
10. #5771C2semidirect
11. #5770C2semidirect
12. C2×#909C2semidirect
13. #5745C2semidirect
16. C2×C3×#41C22semidirect
20. #1010C22semidirect
21. #1010C22semidirect
22. #1010C22semidirect
24. #1009C22semidirect
27. #1007C22semidirect
28. C2×D3×Q8C22semidirect
30. #1004C22semidirect
32. C2×#237C22semidirect
38. C22×Dic6C22semidirect
39. C3×#137C22semidirect
40. C3×#137C22semidirect
41. C3×AGL(1,ℤ/8)C22semidirect
42. C3×AGL(1,ℤ/8)C22semidirect
43. C3×#135C22semidirect
44. C2×C3×M16C22semidirect
45. #951C22semidirect
46. #951C22semidirect
47. #951C22semidirect
48. #950C22semidirect
49. #949C22semidirect
50. C2×#216C22semidirect
51. #942C22semidirect
52. C2×#214C22semidirect
53. #932C22semidirect
54. C2×#208C22semidirect
55. #918C22semidirect
56. D3×Q16C22semidirect
57. #916C22semidirect
58. #915C22semidirect
59. #915C22semidirect
60. D3×SD16C22semidirect
61. #912C22semidirect
62. #911C22semidirect
63. #909C22semidirect
64. #909C22semidirect
65. #907C22semidirect
66. D3×M16C22semidirect
67. C2×Dic12C22semidirect
68. C2×#204C22semidirect
69. #901C22semidirect
78. #575D3semidirect
197. C2×#41C2×D3semidirect
210. #137C2×D3semidirect
211. #137C2×D3semidirect
215. AGL(1,ℤ/8)C2×D3semidirect
216. AGL(1,ℤ/8)C2×D3semidirect
220. #135C2×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×#10102
C3×#5752
C2×#9512
#58222
#57792
D3×#1372
#57752
#57732
#57712
#57702
C2×#9092
#57452
#35863