Finite Group Databasebeta

← #84010

Subgroups — #84010

Order 384.

476 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#576C211yesmaximal, characteristic, complemented
2#5835C212yesmaximal, complemented
3#5769C211yesmaximal, characteristic, complemented
4#5241C212yesmaximal, complemented
5#5199C212yesmaximal, complemented
6#5198C212yesmaximal, complemented
7#5196C211yesmaximal, characteristic, complemented
8#5160C211yesmaximal, characteristic, complemented
9#5151C211yesmaximal, characteristic, complemented
10#5144C211yesmaximal, characteristic, complemented
11#5142C211yesmaximal, characteristic, complemented
12#3292—31nomaximal, Sylow
13C3×H(5,𝔽2)C2212yescomplemented
14#1009C2212yescomplemented
15C3×AGL(1,ℤ/8)—22no—
16C3×AGL(1,ℤ/8)C2212yescomplemented
17C3×#135—21no—
18C3×#135C2211yescharacteristic
19C2×C3×D8—21no—
20C2×C3×D8C2211yescharacteristic, complemented
21C3×#131C2211yescharacteristic, complemented
22#950—22no—
23#950C2211yescharacteristic
24#949—22no—
25#949C2212yescomplemented
26#949C2212yescomplemented
27#948C2212yescomplemented
28#940C2212yescomplemented
29#932—22no—
30#932C2212yescomplemented
31C2×#213—22no—
32C2×#213C2212yescomplemented
33#919—21no—
34#914—22no—
35D3×D8—21no—
36C2×D24C2211yescharacteristic, complemented
37#901C2211yescharacteristic, complemented
38#837C2211yescharacteristic, complemented
39#837C2212yescomplemented
40#833C2212yescomplemented
41#825C2211yescharacteristic, complemented
42#823C2212yescomplemented
43#822C2211yescharacteristic, complemented
44#819C2211yescharacteristic, complemented
45C8×Dic3C2211yescharacteristic, complemented
46#576—32no—
47#576—31no—
48#576D311yescharacteristic, complemented
49#493—31no—
50#472—32no—
51#472—31no—
52#472—31no—
53#453—32no—
54#453—32no—
55#443—31no—
56#443—31no—
57C3×#41—22no—
58C3×#41—22no—
59C3×#41C2312yes—
60C3×#41C2311yescharacteristic
61C2×C3×D4—22no—
62C2×C3×D4—22no—
63C2×C3×D4—42no—
64C2×C3×D4C2312yes—
65C2×#67—42no—
66#239—22no—
67D3×D4—22no—
68D3×D4—42no—
69#235—22no—
70#235C2311yescharacteristic
71C2×D12—22no—
72C2×D12C2312yes—
73C3×Q16D411yescharacteristic, complemented
74C3×SD16—22no—
75C3×SD16D412yescomplemented
76C3×D8—22no—
77C3×D8—21no—
78C3×D8—21no—
79C3×D8D411yescharacteristic, complemented
80C3×M16—21no—
81C3×M16C2311yescharacteristic
82C2×C3×C8—21no—
83C2×C3×C8C2311yescharacteristic
84#216D411yescharacteristic, complemented
85#215—22no—
86#215—22no—
87#215D411yescharacteristic, complemented
88#214—22no—
89#214D411yescharacteristic, complemented
90#213—22no—
91#213—22no—
92#213—22no—
93#213—22no—
94#213—22no—
95#213D411yescharacteristic, complemented
96C4×Dic3C2311yescharacteristic
97#208—22no—
98#208C2312yes—
99#208C2311yescharacteristic
100C2×C3⋊C8—22no—
101C2×C3⋊C8C2311yescharacteristic
102D24—21no—
103D24—21no—
104ΓL(1,𝔽25)—21no—
105D3×C8—21no—
106H(5,𝔽2)—32no—
107H(5,𝔽2)C2×D312yescomplemented
108AGL(1,ℤ/8)—22no—
109AGL(1,ℤ/8)—32no—
110AGL(1,ℤ/8)—32no—
111AGL(1,ℤ/8)—32no—
112AGL(1,ℤ/8)—62no—
113AGL(1,ℤ/8)—62no—
114AGL(1,ℤ/8)—62no—
115AGL(1,ℤ/8)C2×D312yescomplemented
116#135—21no—
117#135—31no—
118#135—62no—
119#135—61no—
120#135C2×D311yescharacteristic
121C2×D8—21no—
122C2×D8—32no—
123C2×D8—31no—
124C2×D8—62no—
125C2×D8—61no—
126C2×D8C2×D311yescharacteristic, complemented
127#131—32no—
128#131—31no—
129#131C2×D311yescharacteristic, complemented
130#127—32no—
131#108—31no—
132#108—31no—
133#104—31no—
134#104—32no—
135#104—31no—
136#100—32no—
137#100—32no—
138C4×C8—31no—
139C23×C3—22no—
140C23×C3—42no—
141C22×D3—22no—
142C22×D3—42no—
143C3×Q8C2×D411yescharacteristic
144C3×Q8C2×D412yes—
145C3×D4—22no—
146C3×D4—21no—
147C3×D4—22no—
148C3×D4—22no—
149C3×D4—42no—
150C3×D4—42no—
151C3×D4C2×D411yescharacteristic
152C3×D4C2×D412yes—
153C2×C3×C4—22no—
154C2×C3×C4—21no—
155C2×C3×C4—42no—
156C2×C3×C4C2411yescharacteristic
157#67—21no—
158#67—42no—
159#67—84no—
160C2×Dic3—21no—
161D12—22no—
162D12—22no—
163D12—42no—
164D12C2×D411yescharacteristic
165C4×D3—21no—
166C4×D3—42no—
167Dic6C2×D411yescharacteristic
168C3×C8—21no—
169C3×C8C2×D411yescharacteristic
170C3×C8C2×D411yescharacteristic
171C3⋊C8—21no—
172C3⋊C8—22no—
173C3⋊C8C2×D411yescharacteristic
174C3⋊C8C2×D411yescharacteristic
175#41—22no—
176#41—22no—
177#41—31no—
178#41—62no—
179#41—62no—
180#41C22×D311yescharacteristic
181#41C22×D312yes—
182C2×D4—22no—
183C2×D4—22no—
184C2×D4—32no—
185C2×D4—42no—
186C2×D4—62no—
187C2×D4—62no—
188C2×D4—122no—
189C2×D4—122no—
190C2×D4C22×D312yes—
191Q16—31no—
192Q16#6711yescharacteristic, complemented
193SD16—22no—
194SD16—31no—
195SD16—31no—
196SD16—62no—
197SD16—62no—
198SD16—62no—
199SD16#6712yescomplemented
200D8—22no—
201D8—21no—
202D8—21no—
203D8—31no—
204D8—62no—
205D8—62no—
206D8—62no—
207D8—62no—
208D8—62no—
209D8—61no—
210D8—61no—
211D8#6711yescharacteristic, complemented
212M16—21no—
213M16—32no—
214M16—31no—
215M16—62no—
216M16—61no—
217M16C22×D311yescharacteristic
218C2×C8—21no—
219C2×C8—31no—
220C2×C8—62no—
221C2×C8—61no—
222C2×C8C22×D311yescharacteristic
223C42—31no—
224C22×C3—22no—
225C22×C3—21no—
226C22×C3—42no—
227C22×C3—82no—
228C22×C3—162no—
229C22×C3H(5,𝔽2)11yescharacteristic
230C2×D3—21no—
231C2×D3—42no—
232C2×D3—82no—
233C2×D3—162no—
234C3×C4—22no—
235C3×C4—21no—
236C3×C4C22×D411yescharacteristic
237C3×C4C22×D411yescharacteristic
238Dic3—22no—
239Dic3—21no—
240C23—22no—
241C23—42no—
242C23—62no—
243C23—122no—
244Q8—31no—
245Q8D3×D411yescharacteristic
246Q8C2×#6712yes—
247D4—21no—
248D4—22no—
249D4—22no—
250D4—22no—
251D4—31no—
252D4—42no—
253D4—42no—
254D4—61no—
255D4—62no—
256D4—62no—
257D4—122no—
258D4—122no—
259D4—244no—
260D4D3×D411yescharacteristic
261D4C2×#6712yes—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—42no—
265C2×C4—61no—
266C2×C4—61no—
267C2×C4—122no—
268C2×C4C23×D311yescharacteristic
269C8—21no—
270C8—31no—
271C8—31no—
272C8—62no—
273C8—61no—
274C8C2×#6711yescharacteristic
275C8C2×#6711yescharacteristic
276C2×C3—21no—
277C2×C3—42no—
278C2×C3—41no—
279C2×C3—82no—
280C2×C3D4211yescharacteristic
281D3—41no—
282D3—82no—
283C22—21no—
284C22—22no—
285C22—42no—
286C22—61no—
287C22—82no—
288C22—122no—
289C22—162no—
290C22—242no—
291C22—482no—
292C22#100411yescharacteristic
293C4—21no—
294C4—22no—
295C4—62no—
296C4—61no—
297C4C2×D3×D411yescharacteristic
298C4C22×#6711yescharacteristic
299C3#329211yescharacteristic, complemented, Sylow
300C2—21no—
301C2—41no—
302C2—42no—
303C2—82no—
304C2—121no—
305C2—242no—
306C2D4×#6711yescharacteristic
subgroupG/H
centerC2D4×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#576C2
socleC2×C3D42
hypercenter#576D3
hypocenterC3#3292
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ #576

Complements

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

NH ≅ G/Ntype
1. C3×#576C2semidirect
2. #5835C2semidirect
3. #5769C2semidirect
4. #5241C2semidirect
5. #5199C2semidirect
6. #5198C2semidirect
7. #5196C2semidirect
8. #5160C2semidirect
9. #5151C2semidirect
10. #5144C2semidirect
11. #5142C2semidirect
13. C3×H(5,𝔽2)C22semidirect
14. #1009C22semidirect
16. C3×AGL(1,ℤ/8)C22semidirect
20. C2×C3×D8C22semidirect
21. C3×#131C22semidirect
25. #949C22semidirect
26. #949C22semidirect
27. #948C22semidirect
28. #940C22semidirect
30. #932C22semidirect
32. C2×#213C22semidirect
36. C2×D24C22semidirect
37. #901C22semidirect
38. #837C22semidirect
39. #837C22semidirect
40. #833C22semidirect
41. #825C22semidirect
42. #823C22semidirect
43. #822C22semidirect
44. #819C22semidirect
45. C8×Dic3C22semidirect
48. #576D3semidirect
73. C3×Q16D4semidirect
75. C3×SD16D4semidirect
79. C3×D8D4semidirect
84. #216D4semidirect
87. #215D4semidirect
89. #214D4semidirect
95. #213D4semidirect
107. H(5,𝔽2)C2×D3semidirect
115. AGL(1,ℤ/8)C2×D3semidirect
126. C2×D8C2×D3semidirect
129. #131C2×D3semidirect
192. Q16#67semidirect
199. SD16#67semidirect
211. D8#67semidirect
299. C3#3292semidirect

Maximal subgroups

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

subgroupindex
C3×#5762
#58352
#57692
#52412
#51992
#51982
#51962
#51602
#51512
#51442
#51422
#32923