Finite Group Databasebeta

← #87187

Subgroups — #87187

Order 384.

1424 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5964C212yesmaximal, complemented
2C2×C3×#135C211yesmaximal, characteristic, complemented
3#5822C212yesmaximal, complemented
4#5771C212yesmaximal, complemented
5#5770C214yesmaximal, complemented
6#5769C212yesmaximal, complemented
7#5768C214yesmaximal, complemented
8D3×#135C214yesmaximal, complemented
9#5766C212yesmaximal, complemented
10#5762C214yesmaximal, complemented
11#5757C212yesmaximal, complemented
12C2×#904C211yesmaximal, characteristic, complemented
13C2×#901C211yesmaximal, characteristic, complemented
14#3584—31nomaximal, Sylow
15C2×C3×#41C2212yescomplemented
16#1010—22no—
17#1010C2214yescomplemented
18#1009—22no—
19#1009C2214yescomplemented
20D3×#41—24no—
21D3×#41C2218yescomplemented
22#1007C2214yescomplemented
23#1004—24no—
24#1004C2214yescomplemented
25C2×#235—24no—
26C2×#235C2211yescharacteristic
27C2×#235C2212yescomplemented
28C3×#135C2212yescomplemented
29C3×#135C2212yescomplemented
30C3×#135C2212yescomplemented
31C3×#135C2212yescomplemented
32C2×C3×Q16C2211yescharacteristic, complemented
33C2×C3×SD16C2212yescomplemented
34C2×C3×D8C2211yescharacteristic, complemented
35C22×C3×C8C2211yescharacteristic, complemented
36#951C2214yescomplemented
37#950C2218yescomplemented
38#949C2214yescomplemented
39#942C2214yescomplemented
40#932C2214yescomplemented
41C2×#208C2211yescharacteristic, complemented
42#919C2214yescomplemented
43#918C2218yescomplemented
44D3×Q16C2214yescomplemented
45#916C2218yescomplemented
46#915C2218yescomplemented
47#914C2218yescomplemented
48D3×SD16C2218yescomplemented
49#912C2214yescomplemented
50#911C2218yescomplemented
51D3×D8C2214yescomplemented
52C2×Dic12C2211yescharacteristic, complemented
53#904C2212yescomplemented
54#904C2212yescomplemented
55#904C2212yescomplemented
56#904C2212yescomplemented
57C2×D24C2211yescharacteristic, complemented
58C2×#204C2212yescomplemented
59#901C2212yescomplemented
60#901C2212yescomplemented
61#901C2212yescomplemented
62#901C2212yescomplemented
63C2×ΓL(1,𝔽25)C2212yescomplemented
64C2×D3×C8C2212yescomplemented
65#585—32no—
66#578—32no—
67#578—32no—
68#577—34no—
69#577—34no—
70#576—32no—
71#576—32no—
72#575—32no—
73#575—34no—
74C2×#135—34no—
75C2×#135—31no—
76C2×#135D311yescharacteristic, complemented
77C2×#131—31no—
78C3×#41—22no—
79C3×#41—22no—
80C3×#41C2314yes—
81C3×#41C2314yes—
82C2×C3×Q8C2312yes—
83C2×C3×D4—22no—
84C2×C3×D4C2312yes—
85C22×C3×C4—22no—
86C22×C3×C4C2311yescharacteristic
87C2×#67—22no—
88C2×#67—28no—
89C2×#67D412yes—
90#239—24no—
91#239C2318yes—
92D3×Q8—24no—
93D3×Q8C2318yes—
94#237—28no—
95#237—28no—
96#237—24no—
97#237C2318yes—
98D3×D4—28no—
99D3×D4—28no—
100D3×D4—24no—
101D3×D4C2318yes—
102#235—22no—
103#235—22no—
104#235—28no—
105#235—24no—
106#235—28no—
107#235—24no—
108#235D412yes—
109#235D412yes—
110#235C2312yes—
111#235C2312yes—
112#235C2314yes—
113#235C2314yes—
114C2×D12—24no—
115C2×D12D411yescharacteristic
116C2×D12C2312yes—
117C2×C4×D3—22no—
118C2×C4×D3—28no—
119C2×C4×D3C2312yes—
120C2×Dic6—24no—
121C2×Dic6D411yescharacteristic
122C2×Dic6C2312yes—
123C3×Q16C2312yes—
124C3×Q16C2312yes—
125C3×SD16C2314yes—
126C3×SD16C2314yes—
127C3×D8C2312yes—
128C3×D8C2312yes—
129C2×C3×C8C2311yescharacteristic
130C2×C3×C8C2311yescharacteristic
131C2×C3×C8C2311yescharacteristic
132C2×C3×C8C2311yescharacteristic
133C2×C3×C8C2311yescharacteristic
134C2×C3×C8C2311yescharacteristic
135#216C2318yes—
136#215C2318yes—
137#214C2318yes—
138#213C2318yes—
139#208C2312yes—
140#208C2312yes—
141C2×C3⋊C8C2312yes—
142Dic12C2312yes—
143Dic12C2312yes—
144D24C2312yes—
145D24C2312yes—
146#204C2314yes—
147#204C2314yes—
148ΓL(1,𝔽25)C2314yes—
149ΓL(1,𝔽25)C2314yes—
150D3×C8C2314yes—
151D3×C8C2314yes—
1522−1+4—34no—
1532−1+4—34no—
1542−1+4—62no—
155H(5,𝔽2)—34no—
156H(5,𝔽2)—34no—
157H(5,𝔽2)—64no—
158H(5,𝔽2)—62no—
159C2×#41—31no—
160C2×#41—38no—
161C2×#41—32no—
162C2×#41—64no—
163C2×#41—64no—
164C2×#41C2×D312yescomplemented
165#137—34no—
166#137—34no—
167#137—38no—
168#137—38no—
169AGL(1,ℤ/8)—34no—
170AGL(1,ℤ/8)—34no—
171AGL(1,ℤ/8)—38no—
172AGL(1,ℤ/8)—38no—
173#135—38no—
174#135—34no—
175#135—38no—
176#135—34no—
177#135—32no—
178#135—32no—
179#135—32no—
180#135—32no—
181#135C2×D312yescomplemented
182#135C2×D312yescomplemented
183#135C2×D312yescomplemented
184#135C2×D312yescomplemented
185C2×Q16—34no—
186C2×Q16—31no—
187C2×Q16C2×D311yescharacteristic, complemented
188C2×SD16—38no—
189C2×SD16—32no—
190C2×SD16C2×D312yescomplemented
191C2×D8—34no—
192C2×D8—31no—
193C2×D8C2×D311yescharacteristic, complemented
194#131—32no—
195#131—32no—
196#131—32no—
197#131—32no—
198C2×M16—31no—
199C2×M16—32no—
200C22×C8—32no—
201C22×C8C2×D311yescharacteristic, complemented
202C23×C3—22no—
203C23×C3C2×D411yescharacteristic
204C22×D3—22no—
205C22×D3—28no—
206C22×D3C2×D412yes—
207C3×Q8—22no—
208C3×Q8C2414yes—
209C3×D4—24no—
210C3×D4—24no—
211C3×D4—22no—
212C3×D4C2414yes—
213C2×C3×C4—22no—
214C2×C3×C4—24no—
215C2×C3×C4—24no—
216C2×C3×C4C2×D411yescharacteristic
217C2×C3×C4C2×D411yescharacteristic
218C2×C3×C4C2×D411yescharacteristic
219C2×C3×C4C2411yescharacteristic
220C2×C3×C4C2411yescharacteristic
221C2×C3×C4C2411yescharacteristic
222#67—24no—
223#67—24no—
224#67—28no—
225#67—28no—
226#67C2×D414yes—
227#67C2×D414yes—
228C2×Dic3—22no—
229C2×Dic3—28no—
230C2×Dic3C2×D412yes—
231D12—28no—
232D12—22no—
233D12C2×D412yes—
234D12C2×D412yes—
235D12C2414yes—
236C4×D3—28no—
237C4×D3—24no—
238C4×D3—24no—
239C4×D3—28no—
240C4×D3C2×D414yes—
241C4×D3C2414yes—
242Dic6—22no—
243Dic6—28no—
244Dic6C2×D412yes—
245Dic6C2×D412yes—
246Dic6C2414yes—
247C3×C8C2411yescharacteristic
248C3×C8C2411yescharacteristic
249C3×C8C2411yescharacteristic
250C3×C8C2411yescharacteristic
251C3⋊C8C2414yes—
252#41—22no—
253#41—22no—
254#41—32no—
255#41—32no—
256#41—32no—
257#41—32no—
258#41—38no—
259#41—38no—
260#41—34no—
261#41—34no—
262#41—68no—
263#41—68no—
264#41—64no—
265#41—64no—
266#41—68no—
267#41—68no—
268#41—64no—
269#41—64no—
270#41—62no—
271#41—62no—
272#41C22×D314yes—
273#41C22×D314yes—
274C2×Q8—31no—
275C2×Q8—38no—
276C2×Q8—32no—
277C2×Q8—64no—
278C2×Q8—64no—
279C2×Q8C22×D312yes—
280C2×D4—22no—
281C2×D4—32no—
282C2×D4—38no—
283C2×D4—31no—
284C2×D4—32no—
285C2×D4—64no—
286C2×D4—68no—
287C2×D4—64no—
288C2×D4—68no—
289C2×D4—68no—
290C2×D4—62no—
291C2×D4C22×D312yes—
292C22×C4—22no—
293C22×C4—32no—
294C22×C4—68no—
295C22×C4—62no—
296C22×C4C22×D311yescharacteristic
297Q16—38no—
298Q16—32no—
299Q16—32no—
300Q16C22×D312yes—
301Q16C22×D312yes—
302SD16—38no—
303SD16—38no—
304SD16—34no—
305SD16—34no—
306SD16C22×D314yes—
307SD16C22×D314yes—
308D8—38no—
309D8—32no—
310D8—32no—
311D8C22×D312yes—
312D8C22×D312yes—
313M16—32no—
314M16—32no—
315M16—34no—
316M16—34no—
317C2×C8—32no—
318C2×C8—34no—
319C2×C8—34no—
320C2×C8C22×D311yescharacteristic
321C2×C8C22×D311yescharacteristic
322C2×C8C22×D311yescharacteristic
323C2×C8C22×D311yescharacteristic
324C2×C8C22×D311yescharacteristic
325C2×C8C22×D311yescharacteristic
326C22×C3—24no—
327C22×C3—41no—
328C22×C3—82no—
329C22×C3C22×D411yescharacteristic
330C22×C3C22×D411yescharacteristic
331C22×C3C22×D411yescharacteristic
332C2×D3—24no—
333C2×D3—42no—
334C2×D3—88no—
335C2×D3—82no—
336C2×D3C22×D414yes—
337C3×C4—24no—
338C3×C4C22×D411yescharacteristic
339C3×C4C22×D411yescharacteristic
340C3×C4C22×D411yescharacteristic
341C3×C4C2511yescharacteristic
342Dic3—24no—
343Dic3C22×D414yes—
344C23—22no—
345C23—32no—
346C23—68no—
347C23—62no—
348C23D3×D411yescharacteristic
349Q8—22no—
350Q8—32no—
351Q8—32no—
352Q8—34no—
353Q8—68no—
354Q8—62no—
355Q8C23×D314yes—
356D4—24no—
357D4—24no—
358D4—22no—
359D4—34no—
360D4—34no—
361D4—32no—
362D4—32no—
363D4—34no—
364D4—68no—
365D4—68no—
366D4—68no—
367D4—64no—
368D4—64no—
369D4—62no—
370D4C23×D314yes—
371C2×C4—22no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—32no—
375C2×C4—34no—
376C2×C4—34no—
377C2×C4—68no—
378C2×C4—68no—
379C2×C4—68no—
380C2×C4—62no—
381C2×C4—64no—
382C2×C4—64no—
383C2×C4D3×D411yescharacteristic
384C2×C4D3×D411yescharacteristic
385C2×C4D3×D411yescharacteristic
386C2×C4C23×D311yescharacteristic
387C2×C4C23×D311yescharacteristic
388C2×C4C23×D311yescharacteristic
389C8—34no—
390C8C23×D311yescharacteristic
391C8C23×D311yescharacteristic
392C8C23×D311yescharacteristic
393C8C23×D311yescharacteristic
394C2×C3—21no—
395C2×C3—21no—
396C2×C3—21no—
397C2×C3—44no—
398C2×C3C23×D411yescharacteristic
399D3—24no—
400D3—44no—
401C22—24no—
402C22—34no—
403C22—41no—
404C22—64no—
405C22—82no—
406C22—122no—
407C22—242no—
408C22—248no—
409C22C2×D3×D411yescharacteristic
410C22C2×D3×D411yescharacteristic
411C22C2×D3×D411yescharacteristic
412C4—24no—
413C4—34no—
414C4—64no—
415C4C2×D3×D411yescharacteristic
416C4C2×D3×D411yescharacteristic
417C4C2×D3×D411yescharacteristic
418C4C24×D311yescharacteristic
419C3#358411yescharacteristic, complemented, Sylow
420C2—21no—
421C2—21no—
422C2—21no—
423C2—44no—
424C2—64no—
425C2—124no—
426C2C22×D3×D411yescharacteristic
subgroupG/H
centerC4C2×D3×D4
commutator, cosocleC3×C4C25
FrattiniC4C24×D3
FittingC2×C3×#135C2
socleC2×C3C23×D4
hypercenterC2×#135D3
hypocenterC3#3584
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. #5964C2semidirect
2. C2×C3×#135C2semidirect
3. #5822C2semidirect
4. #5771C2semidirect
5. #5770C2semidirect
6. #5769C2semidirect
7. #5768C2semidirect
8. D3×#135C2semidirect
9. #5766C2semidirect
10. #5762C2semidirect
11. #5757C2semidirect
12. C2×#904C2semidirect
13. C2×#901C2semidirect
15. C2×C3×#41C22semidirect
17. #1010C22semidirect
19. #1009C22semidirect
21. D3×#41C22semidirect
22. #1007C22semidirect
24. #1004C22semidirect
27. C2×#235C22semidirect
28. C3×#135C22semidirect
29. C3×#135C22semidirect
30. C3×#135C22semidirect
31. C3×#135C22semidirect
32. C2×C3×Q16C22semidirect
33. C2×C3×SD16C22semidirect
34. C2×C3×D8C22semidirect
35. C22×C3×C8C22semidirect
36. #951C22semidirect
37. #950C22semidirect
38. #949C22semidirect
39. #942C22semidirect
40. #932C22semidirect
41. C2×#208C22semidirect
42. #919C22semidirect
43. #918C22semidirect
44. D3×Q16C22semidirect
45. #916C22semidirect
46. #915C22semidirect
47. #914C22semidirect
48. D3×SD16C22semidirect
49. #912C22semidirect
50. #911C22semidirect
51. D3×D8C22semidirect
52. C2×Dic12C22semidirect
53. #904C22semidirect
54. #904C22semidirect
55. #904C22semidirect
56. #904C22semidirect
57. C2×D24C22semidirect
58. C2×#204C22semidirect
59. #901C22semidirect
60. #901C22semidirect
61. #901C22semidirect
62. #901C22semidirect
63. C2×ΓL(1,𝔽25)C22semidirect
64. C2×D3×C8C22semidirect
76. C2×#135D3semidirect
164. C2×#41C2×D3semidirect
181. #135C2×D3semidirect
182. #135C2×D3semidirect
183. #135C2×D3semidirect
184. #135C2×D3semidirect
187. C2×Q16C2×D3semidirect
190. C2×SD16C2×D3semidirect
193. C2×D8C2×D3semidirect
201. C22×C8C2×D3semidirect
419. C3#3584semidirect

Maximal subgroups

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

subgroupindex
#59642
C2×C3×#1352
#58222
#57712
#57702
#57692
#57682
D3×#1352
#57662
#57622
#57572
C2×#9042
C2×#9012
#35843