Finite Group Databasebeta

← #78718

Subgroups — #78718

Order 384.

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

normal subgroups only

This group has 704 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
2#5825C211yesmaximal, characteristic, complemented
3C2×#909C212yesmaximal, complemented
4C3×#413C211yesmaximal, characteristic, complemented
5#5139C212yesmaximal, complemented
6#4752C214yesmaximal, complemented
7#4746C214yesmaximal, complemented
8#3022—31nomaximal, Sylow
9C2×C3×#41C2211yescharacteristic
10#1010—22no—
11#1009—22no—
12D3×#41—24no—
13#1007C2212yes—
14#1004—22no—
15#1004C2212yes—
16C2×#235—22no—
17C2×#235C2212yes—
18C22×Dic6C2211yescharacteristic, complemented
19C2×C3×M16C2212yescomplemented
20#944C2212yescomplemented
21#936C2212yescomplemented
22#935—22no—
23#926C2211yescharacteristic, complemented
24#924—22no—
25#909—24no—
26#909C2218yescomplemented
27C2×Dic12—22no—
28C2×#204—22no—
29C3×#101C2212yescomplemented
30C3×#100C2212yescomplemented
31#825C2218yescomplemented
32#585—31no—
33C2×#137—32no—
34#536—31no—
35#456—34no—
36#455—34no—
37#421—32no—
38#413D311yescharacteristic, complemented
39C3×#41—22no—
40C3×#41—22no—
41C2×C3×Q8C2311yescharacteristic
42C2×C3×D4D412yes—
43C2×C3×D4C2311yescharacteristic
44C22×C3×C4D412yes—
45C22×C3×C4C2311yescharacteristic
46C2×#67—24no—
47C2×#67—22no—
48C22×Dic3—21no—
49#239—24no—
50D3×Q8—24no—
51#237—28no—
52#237—24no—
53D3×D4—28no—
54D3×D4—24no—
55#235—24no—
56#235—24no—
57#235—22no—
58#235—24no—
59#235—22no—
60#235D414yes—
61#235C2314yes—
62C2×D12—22no—
63C2×D12D412yes—
64C2×C4×D3—24no—
65C2×C4×D3—22no—
66C2×Dic6—22no—
67C2×Dic6—22no—
68C2×Dic6—22no—
69C2×Dic6—21no—
70C2×Dic6D412yes—
71C2×Dic6C2312yes—
72C3×M16—22no—
73C3×M16C2314yes—
74C2×C3×C8—22no—
75#217—24no—
76#217—21no—
77C4⋊Dic3—21no—
78Dic3⋊C4—24no—
79C4×Dic3C2312yes—
80Dic12—28no—
81#204—28no—
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×Q8—31no—
92#137—38no—
93#137—64no—
94C2×Q16—62no—
95C2×SD16—62no—
96C2×M16C2×D312yescomplemented
97C4⋊Q8—32no—
98#124—32no—
99#123—62no—
100C22⋊Q8—62no—
101#117—31no—
102#104—38no—
103#101C2×D312yescomplemented
104#100C2×D312yescomplemented
105C23×C3C2×D412yes—
106C23×C3C2×D411yescharacteristic
107C22×D3—24no—
108C22×D3—22no—
109C3×Q8—22no—
110C3×D4—24no—
111C3×D4—22no—
112C2×C3×C4—22no—
113C2×C3×C4—22no—
114C2×C3×C4C2×D412yes—
115C2×C3×C4C2×D411yescharacteristic
116C2×C3×C4C2×D412yes—
117C2×C3×C4C2×D412yes—
118C2×C3×C4C2411yescharacteristic
119#67—28no—
120#67—24no—
121#67—24no—
122C2×Dic3—24no—
123C2×Dic3—22no—
124C2×Dic3—22no—
125C2×Dic3—22no—
126C2×Dic3—42no—
127D12—24no—
128D12—22no—
129D12C2×D414yes—
130C4×D3—28no—
131C4×D3—24no—
132C4×D3—24no—
133Dic6—24no—
134Dic6—22no—
135Dic6—24no—
136Dic6—41no—
137Dic6—41no—
138Dic6C2×D414yes—
139C3×C8—24no—
140#41—22no—
141#41—22no—
142#41—34no—
143#41—34no—
144#41—64no—
145#41—64no—
146#41—64no—
147#41—68no—
148#41—64no—
149#41—62no—
150#41—64no—
151#41—62no—
152C2×Q8—32no—
153C2×Q8—32no—
154C2×Q8—62no—
155C2×Q8—64no—
156C2×Q8—61no—
157C2×Q8—62no—
158C2×Q8—62no—
159C2×Q8C22×D311yescharacteristic
160C2×D4—32no—
161C2×D4—62no—
162C2×D4—64no—
163C2×D4—68no—
164C2×D4—62no—
165C2×D4—64no—
166C2×D4D1212yes—
167C2×D4C22×D311yescharacteristic
168C22×C4—64no—
169C22×C4—62no—
170C22×C4—61no—
171C22×C4D1212yes—
172C22×C4C22×D311yescharacteristic
173Q16—68no—
174SD16—68no—
175M16—22no—
176M16C22×D314yes—
177C2×C8—22no—
178C4⋊C4—61no—
179C4⋊C4—64no—
180C22⋊C4—61no—
181C22⋊C4—64no—
182C42—32no—
183C22×C3—22no—
184C22×C3—41no—
185C22×C3—42no—
186C22×C3#12012yes—
187C22×C3C22×D411yescharacteristic
188C2×D3—24no—
189C2×D3—82no—
190C2×D3—84no—
191C3×C4—22no—
192C3×C4#12011yescharacteristic
193C3×C4#12011yescharacteristic
194C3×C4C22×D412yes—
195Dic3—24no—
196Dic3—42no—
197Dic3—42no—
198C23—64no—
199C23—62no—
200C23C2×D1212yes—
201C23C2×D1211yescharacteristic
202Q8—22no—
203Q8—34no—
204Q8—64no—
205Q8—62no—
206Q8—64no—
207Q8—121no—
208Q8—121no—
209D4—24no—
210D4—22no—
211D4—34no—
212D4—64no—
213D4—68no—
214D4—64no—
215D4—64no—
216D4—62no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—68no—
220C2×C4—64no—
221C2×C4—62no—
222C2×C4—64no—
223C2×C4—64no—
224C2×C4—62no—
225C2×C4—62no—
226C2×C4—122no—
227C2×C4C2×D1212yes—
228C2×C4C2×D1211yescharacteristic
229C2×C4D3×D412yes—
230C2×C4D3×D412yes—
231C2×C4C23×D311yescharacteristic
232C8—24no—
233C2×C3—21no—
234C2×C3—22no—
235C2×C3—42no—
236C2×C3C2×#12011yescharacteristic
237D3—44no—
238C22—22no—
239C22—41no—
240C22—42no—
241C22—64no—
242C22—242no—
243C22—244no—
244C22#88212yes—
245C22C22×D1211yescharacteristic
246C4—22no—
247C4—64no—
248C4—122no—
249C4—122no—
250C4#88211yescharacteristic
251C4#88211yescharacteristic
252C4C2×D3×D412yes—
253C3#302211yescharacteristic, complemented, Sylow
254C2—21no—
255C2—22no—
256C2—42no—
257C2—124no—
258C2C2×#88211yescharacteristic
subgroupG/H
centerC2C2×#882
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#413C2
socleC2×C3C2×#120
prime radical#5964C2
hypercenter#413D3
hypocenterC3#3022
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
2. #5825C2semidirect
3. C2×#909C2semidirect
4. C3×#413C2semidirect
5. #5139C2semidirect
6. #4752C2semidirect
7. #4746C2semidirect
18. C22×Dic6C22semidirect
19. C2×C3×M16C22semidirect
20. #944C22semidirect
21. #936C22semidirect
23. #926C22semidirect
26. #909C22semidirect
29. C3×#101C22semidirect
30. C3×#100C22semidirect
31. #825C22semidirect
38. #413D3semidirect
96. C2×M16C2×D3semidirect
103. #101C2×D3semidirect
104. #100C2×D3semidirect
253. C3#3022semidirect

Maximal subgroups

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

subgroupindex
#59642
#58252
C2×#9092
C3×#4132
#51392
#47522
#47462
#30223