Finite Group Databasebeta

← #69454

Subgroups — #69454

Order 384.

618 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 105 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
1C22×#235C211yesmaximal, characteristic
2C3×C4×C22⋊C4C211yesmaximal, characteristic, complemented
3#5207C211yesmaximal, characteristic, complemented
4C2×#805C214yesmaximal, complemented
5#1789—31nomaximal, Sylow
6C23×C3×C4C2211yescharacteristic
7C22×#67—21no—
8C2×#235—21no—
9C2×#235—21no—
10C2×#235—28no—
11C2×#235C412yescomplemented
12C2×#235C2212yes—
13C22×D12C411yescharacteristic, complemented
14C22×C4×D3—21no—
15C22×Dic6C411yescharacteristic, complemented
16C2×C3×C22⋊C4—21no—
17C2×C3×C42C2212yescomplemented
18C2×#208C2212yescomplemented
19C3×#95—21no—
20#830—21no—
21#830—21no—
22#805—28no—
23C22×#41—31no—
24C2×#104—34no—
25#407—31no—
26C4×C22⋊C4D311yescharacteristic, complemented
27C24×C3D411yescharacteristic
28C23×D3—21no—
29C22×C3×C4—24no—
30C22×C3×C4—22no—
31C22×C3×C4C2×C411yescharacteristic
32C22×C3×C4C2×C411yescharacteristic
33C22×C3×C4D411yescharacteristic
34C22×C3×C4D411yescharacteristic
35C22×C3×C4D411yescharacteristic
36C22×C3×C4C2311yescharacteristic
37C2×#67—28no—
38C2×#67—22no—
39C2×#67—22no—
40C22×Dic3—21no—
41#235—28no—
42#235—28no—
43#235—24no—
44#235—24no—
45#235—24no—
46#235D418yes—
47C2×D12—21no—
48C2×D12—24no—
49C2×D12C2×C412yes—
50C2×C4×D3—28no—
51C2×C4×D3—22no—
52C2×C4×D3—22no—
53C2×Dic6—21no—
54C2×Dic6—24no—
55C2×Dic6C2×C412yes—
56C3×C22⋊C4—24no—
57C3×C42—24no—
58#208—24no—
59C2×C3⋊C8—22no—
60C2×#41—32no—
61C2×#41—32no—
62C2×#41—68no—
63C2×#41—61no—
64C2×#41—61no—
65C22×Q8—31no—
66C22×D4—31no—
67C22×D4—61no—
68C23×C4—61no—
69C23×C4C2×D311yescharacteristic
70C2×M16—32no—
71C2×C22⋊C4—21no—
72C2×C42C2×D312yescomplemented
73#104—68no—
74C22⋊C8—61no—
75C22⋊C8—61no—
76#95—21no—
77C23×C3—22no—
78C23×C3—22no—
79C23×C3—41no—
80C23×C3C22⋊C411yescharacteristic
81C23×C3C22⋊C411yescharacteristic
82C23×C3C2×D411yescharacteristic
83C22×D3—24no—
84C22×D3—22no—
85C22×D3—42no—
86C22×D3—42no—
87C2×C3×C4—22no—
88C2×C3×C4—22no—
89C2×C3×C4—22no—
90C2×C3×C4—22no—
91C2×C3×C4—22no—
92C2×C3×C4—24no—
93C2×C3×C4—42no—
94C2×C3×C4—42no—
95C2×C3×C4C22⋊C411yescharacteristic
96C2×C3×C4C22⋊C411yescharacteristic
97C2×C3×C4C22×C411yescharacteristic
98C2×C3×C4C2×D411yescharacteristic
99C2×C3×C4C2×D414yes—
100#67—28no—
101#67—28no—
102C2×Dic3—24no—
103C2×Dic3—22no—
104D12—24no—
105D12—22no—
106D12C22⋊C414yescomplemented
107C4×D3—28no—
108C4×D3—28no—
109Dic6—24no—
110Dic6—22no—
111Dic6C22⋊C414yescomplemented
112C3⋊C8—42no—
113C24—61no—
114C24#6711yescharacteristic
115#41—38no—
116#41—64no—
117#41—64no—
118#41—68no—
119#41—68no—
120#41—64no—
121C2×Q8—32no—
122C2×Q8—64no—
123C2×Q8—61no—
124C2×D4—32no—
125C2×D4—64no—
126C2×D4—68no—
127C2×D4—62no—
128C2×D4—62no—
129C2×D4—61no—
130C22×C4—24no—
131C22×C4—22no—
132C22×C4—68no—
133C22×C4—61no—
134C22×C4—62no—
135C22×C4—62no—
136C22×C4C4×D311yescharacteristic
137C22×C4C4×D311yescharacteristic
138C22×C4D1211yescharacteristic
139C22×C4D1211yescharacteristic
140C22×C4#6711yescharacteristic
141C22×C4C22×D311yescharacteristic
142M16—64no—
143C2×C8—62no—
144C22⋊C4—24no—
145C42—24no—
146C22×C3—21no—
147C22×C3—21no—
148C22×C3—21no—
149C22×C3—21no—
150C22×C3—21no—
151C22×C3—21no—
152C22×C3—21no—
153C22×C3—44no—
154C22×C3#10412yescomplemented
155C22×C3C2×C22⋊C411yescharacteristic
156C22×C3#12012yes—
157C2×D3—24no—
158C2×D3—48no—
159C2×D3—42no—
160C2×D3—42no—
161C3×C4—21no—
162C3×C4—21no—
163C3×C4—44no—
164C3×C4C2×C22⋊C412yes—
165C3×C4#12012yes—
166Dic3—24no—
167C23—22no—
168C23—22no—
169C23—41no—
170C23—64no—
171C23—62no—
172C23—122no—
173C23—122no—
174C23(C2×D3)⋊C411yescharacteristic
175C23(C2×D3)⋊C411yescharacteristic
176C23C2×#6711yescharacteristic
177Q8—34no—
178Q8—64no—
179Q8—62no—
180D4—34no—
181D4—64no—
182D4—68no—
183D4—68no—
184D4—62no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—24no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—68no—
194C2×C4—64no—
195C2×C4—62no—
196C2×C4—68no—
197C2×C4(C2×D3)⋊C411yescharacteristic
198C2×C4(C2×D3)⋊C411yescharacteristic
199C2×C4C2×C4×D311yescharacteristic
200C2×C4C2×D1211yescharacteristic
201C2×C4D3×D414yes—
202C8—122no—
203C2×C3—22no—
204C2×C3—22no—
205C2×C3—41no—
206C2×C3#37911yescharacteristic
207C2×C3C2×#10411yescharacteristic
208C2×C3#42111yescharacteristic
209D3—44no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—21no—
217C22—44no—
218C22—64no—
219C22—128no—
220C22—122no—
221C22—122no—
222C22#80512yescomplemented
223C22C2×(C2×D3)⋊C411yescharacteristic
224C22#93712yes—
225C4—21no—
226C4—21no—
227C4—44no—
228C4—64no—
229C4D3×C22⋊C412yes—
230C4#88212yes—
231C3#178911yescharacteristic, complemented, Sylow
232C2—22no—
233C2—22no—
234C2—41no—
235C2—124no—
236C2C2×#80511yescharacteristic
237C2#495511yescharacteristic
238C2#500511yescharacteristic
subgroupG/H
centerC2×C4C2×D12
commutatorC2×C3×C4C22×C4
FrattiniC22×C4C22×D3
FittingC3×C4×C22⋊C4C2
socleC22×C3C2×C22⋊C4
prime radicalC22×#235C2
cosocleC22×C3×C4C23
hypercenterC4×C22⋊C4D3
hypocenterC3#1789
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×C4×C22⋊C4C2semidirect
3. #5207C2semidirect
4. C2×#805C2semidirect
11. C2×#235C4semidirect
13. C22×D12C4semidirect
15. C22×Dic6C4semidirect
17. C2×C3×C42C22semidirect
18. C2×#208C22semidirect
26. C4×C22⋊C4D3semidirect
72. C2×C42C2×D3semidirect
106. D12C22⋊C4semidirect
111. Dic6C22⋊C4semidirect
154. C22×C3#104semidirect
222. C22#805semidirect
231. C3#1789semidirect

Maximal subgroups

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

subgroupindex
C22×#2352
C3×C4×C22⋊C42
#52072
C2×#8052
#17893