Finite Group Databasebeta

← #69692

Subgroups — #69692

Order 384.

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

normal subgroups only

This group has 552 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×#1007C211yesmaximal, characteristic
2C3×#391C211yesmaximal, characteristic, complemented
3C2×#836C211yesmaximal, characteristic, complemented
4#4966C212yesmaximal, complemented
5C2×#805C212yesmaximal, complemented
6#1793—31nomaximal, Sylow
7C22×C3×Q8C2211yescharacteristic
8#1007—22no—
9#1007—24no—
10#1007C414yescomplemented
11C2×#239—22no—
12C2×D3×Q8—22no—
13C2×#235—22no—
14C2×#235C2212yes—
15C2×C3×C4⋊C4C2211yescharacteristic
16C2×C3×C42C2211yescharacteristic
17C2×#208C2212yes—
18C3×#95—22no—
19#836—22no—
20#810—22no—
21#809—22no—
22#805—24no—
23C2×2−1+4—31no—
24C2×#104—32no—
25#417—32no—
26C2×#101—31no—
27#391D311yescharacteristic, complemented
28C2×C3×Q8—22no—
29C2×C3×Q8—22no—
30C2×C3×Q8C2×C412yes—
31C2×C3×Q8D412yes—
32C22×C3×C4—21no—
33C22×C3×C4—21no—
34C22×C3×C4D412yes—
35C22×C3×C4C2311yescharacteristic
36C2×#67—22no—
37#239—28no—
38#239—28no—
39D3×Q8—28no—
40D3×Q8—28no—
41#235—28no—
42#235—28no—
43#235—22no—
44#235—22no—
45#235C2×C414yes—
46#235D414yes—
47C2×D12—22no—
48C2×D12D412yes—
49C2×C4×D3—24no—
50C2×C4×D3—22no—
51C2×Dic6—22no—
52C2×Dic6D412yes—
53C3×C4⋊C4—22no—
54C3×C42—22no—
55#208—22no—
56#208—22no—
57C2×C3⋊C8—22no—
582−1+4—34no—
592−1+4—64no—
602−1+4—62no—
61C2×#41—32no—
62C2×#41—62no—
63C2×#41—62no—
64C22×Q8—62no—
65C22×Q8C2×D311yescharacteristic
66C2×M16—32no—
67C2×C4⋊C4C2×D311yescharacteristic
68C2×C42C2×D311yescharacteristic
69#104—64no—
70#103—62no—
71#102—62no—
72#101—62no—
73#95—22no—
74C23×C3C2×D411yescharacteristic
75C22×D3—22no—
76C3×Q8—24no—
77C3×Q8—24no—
78C2×C3×C4—22no—
79C2×C3×C4—22no—
80C2×C3×C4—22no—
81C2×C3×C4—41no—
82C2×C3×C4—41no—
83C2×C3×C4—42no—
84C2×C3×C4—41no—
85C2×C3×C4C22⋊C414yes—
86C2×C3×C4C22×C411yescharacteristic
87C2×C3×C4C2×D411yescharacteristic
88C2×C3×C4C2×D412yes—
89C2×C3×C4C2×D412yes—
90#67—24no—
91#67—24no—
92C2×Dic3—22no—
93D12—28no—
94D12—22no—
95D12C22⋊C414yes—
96C4×D3—216no—
97C4×D3—24no—
98C4×D3—24no—
99Dic6—28no—
100Dic6—22no—
101Dic6C22⋊C414yes—
102C3⋊C8—42no—
103#41—34no—
104#41—34no—
105#41—68no—
106#41—68no—
107#41—68no—
108#41—68no—
109#41—62no—
110#41—62no—
111C2×Q8—22no—
112C2×Q8—22no—
113C2×Q8—32no—
114C2×Q8—62no—
115C2×Q8—68no—
116C2×Q8—68no—
117C2×Q8C4×D312yes—
118C2×Q8D1212yes—
119C2×D4—32no—
120C2×D4—62no—
121C2×D4—62no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—64no—
125C22×C4—62no—
126C22×C4#6712yes—
127C22×C4C22×D311yescharacteristic
128M16—62no—
129M16—62no—
130C2×C8—62no—
131C4⋊C4—22no—
132C42—22no—
133C22×C3—21no—
134C22×C3—21no—
135C22×C3C2×C22⋊C411yescharacteristic
136C22×C3#12011yescharacteristic
137C22×C3#12011yescharacteristic
138C2×D3—24no—
139C2×D3—42no—
140C2×D3—42no—
141C3×C4—24no—
142C3×C4—42no—
143C3×C4—81no—
144C3×C4C2×C22⋊C412yes—
145C3×C4#12012yes—
146Dic3—24no—
147C23—62no—
148C23C2×#6711yescharacteristic
149Q8—24no—
150Q8—24no—
151Q8—34no—
152Q8—68no—
153Q8—62no—
154D4—34no—
155D4—68no—
156D4—64no—
157D4—64no—
158D4—62no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—42no—
165C2×C4—41no—
166C2×C4—616no—
167C2×C4—62no—
168C2×C4—64no—
169C2×C4—64no—
170C2×C4(C2×D3)⋊C414yes—
171C2×C4C2×C4×D311yescharacteristic
172C2×C4C2×D1211yescharacteristic
173C2×C4D3×D412yes—
174C2×C4D3×D412yes—
175C8—122no—
176C2×C3—21no—
177C2×C3—21no—
178C2×C3#37911yescharacteristic
179C2×C3#45411yescharacteristic
180C2×C3#45611yescharacteristic
181D3—44no—
182C22—21no—
183C22—21no—
184C22—64no—
185C22—122no—
186C22—122no—
187C22C2×(C2×D3)⋊C411yescharacteristic
188C22#93711yescharacteristic
189C22#93711yescharacteristic
190C4—24no—
191C4—42no—
192C4—64no—
193C4—81no—
194C4D3×C22⋊C412yes—
195C4#88212yes—
196C3#179311yescharacteristic, complemented, Sylow
197C2—21no—
198C2—21no—
199C2—124no—
200C2#495511yescharacteristic
201C2#506211yescharacteristic
202C2#509511yescharacteristic
subgroupG/H
centerC22#937
commutatorC2×C3×C4C22×C4
FrattiniC22×C4C22×D3
FittingC3×#391C2
socleC22×C3#120
prime radicalC2×#1007C2
cosocleC22×C3×C4C23
hypercenter#391D3
hypocenterC3#1793
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #391

Complements

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

NH ≅ G/Ntype
2. C3×#391C2semidirect
3. C2×#836C2semidirect
4. #4966C2semidirect
5. C2×#805C2semidirect
10. #1007C4semidirect
27. #391D3semidirect
196. C3#1793semidirect

Maximal subgroups

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

subgroupindex
C2×#10072
C3×#3912
C2×#8362
#49662
C2×#8052
#17933