Finite Group Databasebeta

← #69662

Subgroups — #69662

Order 384.

620 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C2×#1004C211yesmaximal, characteristic
2C3×#390C211yesmaximal, characteristic, complemented
3C2×#833C211yesmaximal, characteristic, complemented
4#4966C212yesmaximal, complemented
5C2×#805C212yesmaximal, complemented
6#1792—31nomaximal, Sylow
7C22×C3×D4C2211yescharacteristic
8C22×#67—22no—
9#1004—22no—
10#1004—24no—
11#1004C414yescomplemented
12C2×#237—22no—
13C2×D3×D4—22no—
14C2×#235C2212yes—
15C2×C3×C4⋊C4C2211yescharacteristic, complemented
16C2×C3×C22⋊C4—22no—
17C2×C3×C42C2211yescharacteristic, complemented
18C2×#208C2212yescomplemented
19#833—22no—
20#810—22no—
21#809—22no—
22#805—24no—
23C2×H(5,𝔽2)—31no—
24C2×#104—32no—
25#417—32no—
26C2×#100—31no—
27#390D311yescharacteristic, complemented
28C24×C3D412yes—
29C23×D3—22no—
30C2×C3×D4—22no—
31C2×C3×D4—22no—
32C2×C3×D4C2×C412yes—
33C2×C3×D4D412yes—
34C22×C3×C4—21no—
35C22×C3×C4—21no—
36C22×C3×C4C2311yescharacteristic
37C2×#67—24no—
38C2×#67—28no—
39C2×#67—28no—
40C2×#67—22no—
41C22×Dic3—22no—
42#237—28no—
43#237—28no—
44D3×D4—28no—
45D3×D4—28no—
46#235—22no—
47#235—22no—
48#235C2×C414yes—
49#235D414yes—
50C2×D12D412yes—
51C2×C4×D3—22no—
52C2×Dic6D412yes—
53C3×C4⋊C4—22no—
54C3×C22⋊C4—44no—
55C3×C42—22no—
56#208—22no—
57#208—22no—
58C2×C3⋊C8—22no—
59H(5,𝔽2)—34no—
60H(5,𝔽2)—64no—
61H(5,𝔽2)—62no—
62C2×#41—32no—
63C2×#41—62no—
64C22×D4—62no—
65C22×D4—62no—
66C22×D4C2×D311yescharacteristic
67C2×M16—32no—
68C2×C4⋊C4C2×D311yescharacteristic, complemented
69C2×C22⋊C4—22no—
70C2×C42C2×D311yescharacteristic, complemented
71#104—64no—
72#103—62no—
73#102—62no—
74#100—62no—
75C23×C3—22no—
76C23×C3—22no—
77C23×C3—42no—
78C23×C3—42no—
79C23×C3C22⋊C414yes—
80C23×C3C2×D411yescharacteristic
81C22×D3—22no—
82C22×D3—28no—
83C22×D3—82no—
84C22×D3—82no—
85C3×D4—24no—
86C3×D4—24no—
87C2×C3×C4—22no—
88C2×C3×C4—41no—
89C2×C3×C4—41no—
90C2×C3×C4—42no—
91C2×C3×C4—41no—
92C2×C3×C4C22×C411yescharacteristic
93C2×C3×C4C2×D411yescharacteristic
94C2×C3×C4C2×D412yes—
95C2×C3×C4C2×D412yes—
96#67—216no—
97#67—24no—
98#67—24no—
99C2×Dic3—28no—
100C2×Dic3—22no—
101D12—22no—
102D12C22⋊C414yescomplemented
103C4×D3—24no—
104C4×D3—24no—
105Dic6—22no—
106Dic6C22⋊C414yescomplemented
107C3⋊C8—42no—
108C24—62no—
109C24#6712yes—
110#41—34no—
111#41—34no—
112#41—68no—
113#41—68no—
114#41—62no—
115#41—62no—
116C2×Q8—32no—
117C2×D4—22no—
118C2×D4—22no—
119C2×D4—32no—
120C2×D4—68no—
121C2×D4—68no—
122C2×D4—64no—
123C2×D4—68no—
124C2×D4—68no—
125C2×D4—62no—
126C2×D4C4×D312yes—
127C2×D4D1212yes—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—62no—
131C22×C4—62no—
132C22×C4C22×D311yescharacteristic
133M16—62no—
134M16—62no—
135C2×C8—62no—
136C4⋊C4—22no—
137C22⋊C4—44no—
138C42—22no—
139C22×C3—21no—
140C22×C3—21no—
141C22×C3—24no—
142C22×C3—44no—
143C22×C3—42no—
144C22×C3—42no—
145C22×C3—82no—
146C22×C3C2×C22⋊C411yescharacteristic
147C22×C3#12011yescharacteristic
148C22×C3#12011yescharacteristic
149C2×D3—24no—
150C2×D3—42no—
151C2×D3—42no—
152C2×D3—88no—
153C3×C4—42no—
154C3×C4—81no—
155C3×C4C2×C22⋊C412yes—
156C3×C4#12012yes—
157Dic3—24no—
158C23—22no—
159C23—22no—
160C23—42no—
161C23—42no—
162C23—68no—
163C23—62no—
164C23—242no—
165C23—242no—
166C23(C2×D3)⋊C414yes—
167C23C2×#6711yescharacteristic
168Q8—34no—
169Q8—62no—
170D4—24no—
171D4—24no—
172D4—34no—
173D4—616no—
174D4—64no—
175D4—64no—
176D4—62no—
177C2×C4—22no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—42no—
181C2×C4—41no—
182C2×C4—68no—
183C2×C4—62no—
184C2×C4—64no—
185C2×C4—64no—
186C2×C4C2×C4×D311yescharacteristic
187C2×C4C2×D1211yescharacteristic
188C2×C4D3×D412yes—
189C2×C4D3×D412yes—
190C8—122no—
191C2×C3—21no—
192C2×C3—21no—
193C2×C3—44no—
194C2×C3#37911yescharacteristic
195C2×C3#45311yescharacteristic
196C2×C3#45511yescharacteristic
197D3—44no—
198C22—21no—
199C22—21no—
200C22—24no—
201C22—44no—
202C22—42no—
203C22—42no—
204C22—64no—
205C22—82no—
206C22—122no—
207C22—122no—
208C22—248no—
209C22C2×(C2×D3)⋊C411yescharacteristic
210C22#93711yescharacteristic
211C22#93711yescharacteristic
212C4—42no—
213C4—64no—
214C4—81no—
215C4D3×C22⋊C412yes—
216C4#88212yes—
217C3#179211yescharacteristic, complemented, Sylow
218C2—21no—
219C2—21no—
220C2—44no—
221C2—124no—
222C2#495511yescharacteristic
223C2#506111yescharacteristic
224C2#507711yescharacteristic
subgroupG/H
centerC22#937
commutatorC2×C3×C4C22×C4
FrattiniC22×C4C22×D3
FittingC3×#390C2
socleC22×C3#120
prime radicalC2×#1004C2
cosocleC22×C3×C4C23
hypercenter#390D3
hypocenterC3#1792
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C3×#390C2semidirect
3. C2×#833C2semidirect
4. #4966C2semidirect
5. C2×#805C2semidirect
11. #1004C4semidirect
15. C2×C3×C4⋊C4C22semidirect
17. C2×C3×C42C22semidirect
18. C2×#208C22semidirect
27. #390D3semidirect
68. C2×C4⋊C4C2×D3semidirect
70. C2×C42C2×D3semidirect
102. D12C22⋊C4semidirect
106. Dic6C22⋊C4semidirect
217. C3#1792semidirect

Maximal subgroups

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

subgroupindex
C2×#10042
C3×#3902
C2×#8332
#49662
C2×#8052
#17923