Finite Group Databasebeta

← #83395

Subgroups — #83395

Order 384.

736 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 145 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#5964C211yesmaximal, characteristic, complemented
2C3×#535C211yesmaximal, characteristic, complemented
3C2×#932C212yesmaximal, complemented
4#5237C211yesmaximal, characteristic, complemented
5#5077C214yesmaximal, complemented
6#5062C214yesmaximal, complemented
7#5005C212yesmaximal, complemented
8#3021—31nomaximal, Sylow
9C2×C3×#41C2211yescharacteristic
10C22×C3×D4C2211yescharacteristic, complemented
11#1010—22no—
12#1009—22no—
13D3×#41—24no—
14#1007C2212yescomplemented
15#1004—22no—
16#1004C2212yescomplemented
17C2×#235—22no—
18C2×#235C2212yes—
19C3×#127C2212yescomplemented
20C3×#124C2212yescomplemented
21C3×#121—22no—
22C3×#120—22no—
23C3×#117C2211yescharacteristic, complemented
24C2×#214—22no—
25#932—24no—
26#932C2218yescomplemented
27C2×#213—22no—
28C2×#208C2212yescomplemented
29#836C2212yescomplemented
30#833C2212yescomplemented
31#805C2218yescomplemented
32#585—31no—
33C2×AGL(1,ℤ/8)—32no—
34#535D311yescharacteristic, complemented
35#454—34no—
36#453—34no—
37#421—32no—
38#413—31no—
39C24×C3—21no—
40C3×#41—22no—
41C3×#41—22no—
42C2×C3×Q8C2311yescharacteristic
43C2×C3×D4—24no—
44C2×C3×D4—22no—
45C2×C3×D4—22no—
46C2×C3×D4—21no—
47C2×C3×D4D412yes—
48C2×C3×D4C2312yes—
49C2×C3×D4C2311yescharacteristic
50C22×C3×C4D412yes—
51C22×C3×C4C2311yescharacteristic
52C2×#67—24no—
53C2×#67—22no—
54#239—24no—
55D3×Q8—24no—
56#237—28no—
57#237—24no—
58D3×D4—28no—
59D3×D4—24no—
60#235—24no—
61#235—24no—
62#235—22no—
63#235—24no—
64#235—22no—
65#235D414yescomplemented
66#235C2314yes—
67C2×D12—22no—
68C2×D12D412yescomplemented
69C2×C4×D3—24no—
70C2×C4×D3—22no—
71C2×Dic6—22no—
72C2×Dic6D412yescomplemented
73C3×C4⋊C4—21no—
74C3×C22⋊C4—21no—
75C3×C22⋊C4—24no—
76C3×C42C2312yes—
77#214—28no—
78#213—28no—
79#208—22no—
80#208C2314yes—
81C2×C3⋊C8—22no—
822−1+4—32no—
832−1+4—62no—
84H(5,𝔽2)—32no—
85H(5,𝔽2)—62no—
86H(5,𝔽2)—62no—
87C2×#41—32no—
88C2×#41—64no—
89C2×#41—62no—
90C2×#41C2×D311yescharacteristic
91C22×D4C2×D311yescharacteristic, complemented
92AGL(1,ℤ/8)—38no—
93AGL(1,ℤ/8)—64no—
94C2×SD16—62no—
95C2×D8—62no—
96C2×M16—32no—
97#127C2×D312yescomplemented
98#124C2×D312yescomplemented
99#121—22no—
100#120—22no—
101#117C2×D311yescharacteristic, complemented
102#104—38no—
103#101—32no—
104#100—32no—
105C23×C3—22no—
106C23×C3—42no—
107C23×C3—82no—
108C23×C3C2×D412yes—
109C23×C3C2×D411yescharacteristic
110C22×D3—24no—
111C22×D3—22no—
112C3×Q8—22no—
113C3×D4—24no—
114C3×D4—22no—
115C3×D4—24no—
116C3×D4—41no—
117C3×D4—41no—
118C3×D4—84no—
119C2×C3×C4—22no—
120C2×C3×C4—22no—
121C2×C3×C4—22no—
122C2×C3×C4C2×D412yes—
123C2×C3×C4C2×D412yes—
124C2×C3×C4C2×D412yes—
125C2×C3×C4C2×D411yescharacteristic
126C2×C3×C4C2411yescharacteristic
127#67—28no—
128#67—24no—
129#67—24no—
130C2×Dic3—24no—
131C2×Dic3—22no—
132D12—24no—
133D12—22no—
134D12C2×D414yes—
135C4×D3—28no—
136C4×D3—24no—
137C4×D3—24no—
138Dic6—24no—
139Dic6—22no—
140Dic6C2×D414yes—
141C3⋊C8—24no—
142C24—21no—
143#41—22no—
144#41—22no—
145#41—34no—
146#41—34no—
147#41—64no—
148#41—64no—
149#41—64no—
150#41—68no—
151#41—64no—
152#41—62no—
153#41—62no—
154#41—64no—
155C2×Q8—32no—
156C2×Q8—62no—
157C2×Q8—64no—
158C2×Q8C22×D311yescharacteristic
159C2×D4—24no—
160C2×D4—22no—
161C2×D4—22no—
162C2×D4—21no—
163C2×D4—32no—
164C2×D4—62no—
165C2×D4—64no—
166C2×D4—68no—
167C2×D4—62no—
168C2×D4—64no—
169C2×D4#6712yes—
170C2×D4C22×D312yes—
171C2×D4C22×D311yescharacteristic
172C22×C4—64no—
173C22×C4—62no—
174C22×C4#6712yes—
175C22×C4C22×D311yescharacteristic
176SD16—68no—
177D8—68no—
178M16—34no—
179M16—62no—
180C2×C8—62no—
181C4⋊C4—21no—
182C22⋊C4—21no—
183C22⋊C4—24no—
184C42C22×D312yes—
185C22×C3—22no—
186C22×C3—41no—
187C22×C3—42no—
188C22×C3—44no—
189C22×C3—42no—
190C22×C3—84no—
191C22×C3#12012yes—
192C22×C3C22×D411yescharacteristic
193C2×D3—24no—
194C2×D3—82no—
195C2×D3—84no—
196C3×C4—22no—
197C3×C4—42no—
198C3×C4#12011yescharacteristic
199C3×C4#12011yescharacteristic
200C3×C4C22×D412yes—
201Dic3—24no—
202C23—22no—
203C23—42no—
204C23—64no—
205C23—62no—
206C23—82no—
207C23C2×#6712yes—
208C23C2×#6711yescharacteristic
209Q8—22no—
210Q8—34no—
211Q8—64no—
212Q8—62no—
213D4—24no—
214D4—22no—
215D4—24no—
216D4—34no—
217D4—41no—
218D4—41no—
219D4—64no—
220D4—68no—
221D4—64no—
222D4—64no—
223D4—62no—
224D4—84no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—68no—
229C2×C4—64no—
230C2×C4—62no—
231C2×C4—64no—
232C2×C4—64no—
233C2×C4D3×D412yes—
234C2×C4D3×D412yes—
235C2×C4C2×#6712yes—
236C2×C4C2×#6711yescharacteristic
237C2×C4C23×D311yescharacteristic
238C8—64no—
239C2×C3—21no—
240C2×C3—22no—
241C2×C3—42no—
242C2×C3—82no—
243C2×C3C2×#12011yescharacteristic
244D3—44no—
245C22—22no—
246C22—41no—
247C22—42no—
248C22—44no—
249C22—42no—
250C22—64no—
251C22—84no—
252C22—242no—
253C22—244no—
254C22#93712yes—
255C22C22×#6711yescharacteristic
256C4—22no—
257C4—42no—
258C4—64no—
259C4#93711yescharacteristic
260C4#93711yescharacteristic
261C4C2×D3×D412yes—
262C3#302111yescharacteristic, complemented, Sylow
263C2—21no—
264C2—22no—
265C2—42no—
266C2—82no—
267C2—124no—
268C2C2×#93711yescharacteristic
subgroupG/H
centerC2C2×#937
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#535C2
socleC2×C3C2×#120
hypercenter#535D3
hypocenterC3#3021
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #5964C2semidirect
2. C3×#535C2semidirect
3. C2×#932C2semidirect
4. #5237C2semidirect
5. #5077C2semidirect
6. #5062C2semidirect
7. #5005C2semidirect
10. C22×C3×D4C22semidirect
14. #1007C22semidirect
16. #1004C22semidirect
19. C3×#127C22semidirect
20. C3×#124C22semidirect
23. C3×#117C22semidirect
26. #932C22semidirect
28. C2×#208C22semidirect
29. #836C22semidirect
30. #833C22semidirect
31. #805C22semidirect
34. #535D3semidirect
65. #235D4semidirect
68. C2×D12D4semidirect
72. C2×Dic6D4semidirect
91. C22×D4C2×D3semidirect
97. #127C2×D3semidirect
98. #124C2×D3semidirect
101. #117C2×D3semidirect
262. C3#3021semidirect

Maximal subgroups

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

subgroupindex
#59642
C3×#5352
C2×#9322
#52372
#50772
#50622
#50052
#30213