Finite Group Databasebeta

← #81860

Subgroups — #81860

Order 384.

464 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#5762C212yesmaximal, complemented
2#5757C212yesmaximal, complemented
3C3×#496C211yesmaximal, characteristic, complemented
4#5177C212yesmaximal, complemented
5#5160C212yesmaximal, complemented
6#5077C212yesmaximal, complemented
7#5062C211yesmaximal, characteristic, complemented
8#5061C211yesmaximal, characteristic, complemented
9#4717C212yesmaximal, complemented
10#3288—31nomaximal, Sylow
11#1007C2211yescharacteristic, complemented
12#1004C2211yescharacteristic
13#1004C2212yescomplemented
14C2×C3×SD16C2212yescomplemented
15C2×C3×D8C2212yescomplemented
16C3×#127C2211yescharacteristic, complemented
17C3×#124C2211yescharacteristic, complemented
18#942C2212yescomplemented
19#932C2212yes—
20#932C2212yescomplemented
21#932C2212yescomplemented
22#916—22no—
23#915—22no—
24#914—22no—
25D3×SD16—22no—
26#912—22no—
27#911—22no—
28#911—22no—
29D3×D8—22no—
30#904C2212yes—
31#901C2212yescomplemented
32#901C2212yescomplemented
33C3×#97C2211yescharacteristic, complemented
34#836C2211yescharacteristic, complemented
35#833C2211yescharacteristic, complemented
36#833C2212yescomplemented
37#819C2212yescomplemented
38#805C2212yescomplemented
39#805C2212yescomplemented
40#577—32no—
41#576—32no—
42#496D311yescharacteristic, complemented
43#472—32no—
44#471—32no—
45#455—31no—
46#454—31no—
47#453—32no—
48#444—32no—
49C2×C3×Q8C2311yescharacteristic
50C2×C3×D4—41no—
51C2×C3×D4C2312yes—
52C2×C3×D4C2311yescharacteristic
53C2×#67—41no—
54C2×#67—42no—
55#239—22no—
56D3×Q8—22no—
57#237—22no—
58#237—22no—
59#237—22no—
60D3×D4—22no—
61D3×D4—22no—
62D3×D4—22no—
63#235—41no—
64#235C2312yes—
65#235C2312yes—
66C3×SD16—22no—
67C3×SD16—22no—
68C3×D8—22no—
69C3×D8—22no—
70C2×C3×C8C2312yes—
71C3×C22⋊C4—41no—
72C3×C42C2311yescharacteristic
73#216—22no—
74#215—22no—
75#214—22no—
76#214—22no—
77#214—22no—
78#213—22no—
79#213—22no—
80#213—22no—
81#208C2312yes—
82#208C2312yes—
83Dic12D412yescomplemented
84D24D412yescomplemented
85#204D412yescomplemented
86#204D412yescomplemented
87ΓL(1,𝔽25)—22no—
88ΓL(1,𝔽25)—22no—
89D3×C8—22no—
90D3×C8—22no—
912−1+4—31no—
92H(5,𝔽2)—31no—
93H(5,𝔽2)—32no—
94#137—32no—
95#137—62no—
96AGL(1,ℤ/8)—32no—
97AGL(1,ℤ/8)—32no—
98AGL(1,ℤ/8)—32no—
99AGL(1,ℤ/8)—62no—
100AGL(1,ℤ/8)—62no—
101AGL(1,ℤ/8)—62no—
102#135—32no—
103#135—62no—
104#135—62no—
105C2×SD16—62no—
106C2×SD16C2×D312yescomplemented
107C2×D8—62no—
108C2×D8C2×D312yescomplemented
109#131—32no—
110#131—32no—
111#127C2×D311yescharacteristic, complemented
112#124C2×D311yescharacteristic, complemented
113#108—32no—
114#104—32no—
115#104—32no—
116#101—31no—
117#100—32no—
118#100—31no—
119#97C2×D311yescharacteristic, complemented
120C23×C3—21no—
121C23×C3—22no—
122C22×D3—42no—
123C22×D3—41no—
124C3×Q8—22no—
125C3×D4—22no—
126C3×D4—22no—
127C3×D4—22no—
128C3×D4—82no—
129C2×C3×C4—21no—
130C2×C3×C4—21no—
131C2×C3×C4C2411yescharacteristic
132#67—22no—
133#67—22no—
134#67—42no—
135#67—42no—
136#67—42no—
137C2×Dic3—42no—
138C2×Dic3—41no—
139D12—41no—
140D12C2×D412yes—
141D12C2×D412yes—
142C4×D3—22no—
143C4×D3—22no—
144C4×D3—42no—
145Dic6—41no—
146Dic6C2×D412yes—
147Dic6C2×D412yes—
148C3×C8C2×D412yes—
149C3×C8C2×D412yes—
150C3⋊C8—22no—
151C3⋊C8—22no—
152#41—32no—
153#41—32no—
154#41—62no—
155#41—62no—
156#41—62no—
157#41—62no—
158#41—121no—
159C2×Q8—62no—
160C2×Q8C22×D311yescharacteristic
161C2×D4—41no—
162C2×D4—62no—
163C2×D4—62no—
164C2×D4—62no—
165C2×D4—121no—
166C2×D4—122no—
167C2×D4C22×D311yescharacteristic
168C2×D4C22×D312yes—
169Q16—32no—
170Q16—62no—
171SD16—22no—
172SD16—22no—
173SD16—32no—
174SD16—32no—
175SD16—62no—
176SD16—62no—
177SD16—62no—
178SD16—62no—
179D8—22no—
180D8—22no—
181D8—32no—
182D8—62no—
183D8—62no—
184D8—62no—
185M16—32no—
186M16—32no—
187M16—62no—
188M16—62no—
189C2×C8—62no—
190C2×C8—62no—
191C2×C8C22×D312yes—
192C22⋊C4—41no—
193C42C22×D311yescharacteristic
194C22×C3—41no—
195C22×C3—42no—
196C22×C3—82no—
197C22×C3—81no—
198C22×C3H(5,𝔽2)11yescharacteristic
199C2×D3—22no—
200C2×D3—22no—
201C2×D3—162no—
202C2×D3—161no—
203C3×C4—41no—
204C3×C4—41no—
205C3×C4C22×D412yes—
206Dic3—22no—
207Dic3—22no—
208C23—21no—
209C23—22no—
210C23—122no—
211C23—121no—
212Q8—22no—
213Q8—32no—
214Q8—32no—
215Q8—121no—
216D4—22no—
217D4—22no—
218D4—22no—
219D4—32no—
220D4—32no—
221D4—62no—
222D4—62no—
223D4—82no—
224D4—121no—
225D4—122no—
226D4—122no—
227D4—122no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—62no—
231C2×C4—62no—
232C2×C4—122no—
233C2×C4—121no—
234C2×C4—122no—
235C2×C4C23×D311yescharacteristic
236C8—62no—
237C8—62no—
238C8D3×D412yes—
239C8D3×D412yes—
240C2×C3—21no—
241C2×C3—82no—
242C2×C3—81no—
243C2×C3D4211yescharacteristic
244D3—42no—
245D3—42no—
246C22—41no—
247C22—42no—
248C22—62no—
249C22—62no—
250C22—82no—
251C22—81no—
252C22—481no—
253C22—482no—
254C22#100411yescharacteristic
255C4—41no—
256C4—41no—
257C4—62no—
258C4—62no—
259C4C2×D3×D412yes—
260C3#328811yescharacteristic, complemented, Sylow
261C2—21no—
262C2—82no—
263C2—81no—
264C2—122no—
265C2—122no—
266C2#571711yescharacteristic
subgroupG/H
centerC2#5717
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#496C2
socleC2×C3D42
hypercenter#496D3
hypocenterC3#3288
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #5762C2semidirect
2. #5757C2semidirect
3. C3×#496C2semidirect
4. #5177C2semidirect
5. #5160C2semidirect
6. #5077C2semidirect
7. #5062C2semidirect
8. #5061C2semidirect
9. #4717C2semidirect
11. #1007C22semidirect
13. #1004C22semidirect
14. C2×C3×SD16C22semidirect
15. C2×C3×D8C22semidirect
16. C3×#127C22semidirect
17. C3×#124C22semidirect
18. #942C22semidirect
20. #932C22semidirect
21. #932C22semidirect
31. #901C22semidirect
32. #901C22semidirect
33. C3×#97C22semidirect
34. #836C22semidirect
35. #833C22semidirect
36. #833C22semidirect
37. #819C22semidirect
38. #805C22semidirect
39. #805C22semidirect
42. #496D3semidirect
83. Dic12D4semidirect
84. D24D4semidirect
85. #204D4semidirect
86. #204D4semidirect
106. C2×SD16C2×D3semidirect
108. C2×D8C2×D3semidirect
111. #127C2×D3semidirect
112. #124C2×D3semidirect
119. #97C2×D3semidirect
260. C3#3288semidirect

Maximal subgroups

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

subgroupindex
#57622
#57572
C3×#4962
#51772
#51602
#50772
#50622
#50612
#47172
#32883