Finite Group Databasebeta

← #69510

Subgroups — #69510

Order 384.

288 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 89 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×#924C211yesmaximal, characteristic
2C3×#380C211yesmaximal, characteristic
3C2×#830C211yesmaximal, characteristic
4#4452C214yesmaximal, complemented
5#1520—31nomaximal, Sylow
6C23×C3×C4C2211yescharacteristic
7C22×Dic6C411yescharacteristic, complemented
8C2×C3×C4⋊C4—21no—
9C2×C3×C22⋊C4—21no—
10C2×#217—21no—
11C2×C4⋊Dic3C411yescharacteristic, complemented
12#924—21no—
13#924—21no—
14#924C412yescomplemented
15#924C2212yes—
16C2×Dic3⋊C4—21no—
17C22×C3⋊C8—21no—
18C3×#95C2212yes—
19#830—21no—
20#830C2212yes—
21C2×C22⋊Q8—31no—
22C2×C22⋊C8—31no—
23#380D311yescharacteristic
24#328—34no—
25C24×C3D411yescharacteristic
26C22×C3×C4—22no—
27C22×C3×C4—42no—
28C22×C3×C4C2×C411yescharacteristic
29C22×C3×C4C2×C411yescharacteristic
30C22×C3×C4D411yescharacteristic
31C22×C3×C4D411yescharacteristic
32C22×C3×C4D411yescharacteristic
33C22×C3×C4C2311yescharacteristic
34C22×Dic3—21no—
35C22×Dic3—21no—
36C2×Dic6—21no—
37C2×Dic6—42no—
38C2×Dic6C2×C412yes—
39C3×C4⋊C4—42no—
40C3×C22⋊C4—42no—
41#217—22no—
42#217—22no—
43C4⋊Dic3—21no—
44C4⋊Dic3C2×C412yes—
45Dic3⋊C4—22no—
46Dic3⋊C4—22no—
47C2×C3⋊C8—22no—
48C2×C3⋊C8—42no—
49C22×Q8—31no—
50C23×C4C2×D311yescharacteristic
51C22×C8—61no—
52C22⋊Q8—32no—
53C22⋊Q8—32no—
54C22⋊Q8—61no—
55C22⋊Q8—61no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—31no—
58C2×C4⋊C4—61no—
59C2×C22⋊C4—21no—
60C2×C22⋊C4—61no—
61C22⋊C8—32no—
62C22⋊C8—61no—
63#95C2×D312yes—
64C23×C3—41no—
65C23×C3—41no—
66C23×C3C22⋊C411yescharacteristic
67C23×C3C22⋊C411yescharacteristic
68C23×C3SD1612yes—
69C23×C3Q1612yes—
70C23×C3C2×D411yescharacteristic
71C2×C3×C4—22no—
72C2×C3×C4—22no—
73C2×C3×C4—22no—
74C2×C3×C4—41no—
75C2×C3×C4—41no—
76C2×C3×C4—41no—
77C2×C3×C4—42no—
78C2×C3×C4—42no—
79C2×C3×C4—42no—
80C2×C3×C4C22⋊C411yescharacteristic
81C2×C3×C4C22⋊C411yescharacteristic
82C2×C3×C4C22×C411yescharacteristic
83C2×C3×C4C2×D411yescharacteristic
84C2×Dic3—22no—
85C2×Dic3—22no—
86C2×Dic3—42no—
87C2×Dic3—42no—
88Dic6—42no—
89Dic6—42no—
90C3⋊C8—42no—
91C24#6711yescharacteristic
92C2×Q8—32no—
93C2×Q8—61no—
94C2×Q8—122no—
95C22×C4—22no—
96C22×C4—42no—
97C22×C4—61no—
98C22×C4—61no—
99C22×C4C4×D311yescharacteristic
100C22×C4C4×D311yescharacteristic
101C22×C4D1211yescharacteristic
102C22×C4D1211yescharacteristic
103C22×C4#6711yescharacteristic
104C22×C4C22×D311yescharacteristic
105C2×C8—62no—
106C2×C8—122no—
107C4⋊C4—32no—
108C4⋊C4—42no—
109C4⋊C4—61no—
110C4⋊C4—62no—
111C4⋊C4—62no—
112C22⋊C4—42no—
113C22⋊C4—62no—
114C22⋊C4—62no—
115C22×C3—21no—
116C22×C3—21no—
117C22×C3—21no—
118C22×C3—21no—
119C22×C3—21no—
120C22×C3—21no—
121C22×C3—42no—
122C22×C3—42no—
123C22×C3#10312yes—
124C22×C3#10312yes—
125C22×C3C2×C22⋊C411yescharacteristic
126C22×C3C2×SD1611yescharacteristic
127C22×C3C2×Q1611yescharacteristic
128C3×C4—21no—
129C3×C4—21no—
130C3×C4—21no—
131C3×C4—82no—
132C3×C4#9912yes—
133Dic3—42no—
134Dic3—42no—
135C23—41no—
136C23—41no—
137C23(C2×D3)⋊C411yescharacteristic
138C23(C2×D3)⋊C411yescharacteristic
139C23#21412yes—
140C23#21612yes—
141C23C2×#6711yescharacteristic
142Q8—122no—
143Q8—122no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4—62no—
154C2×C4—62no—
155C2×C4—122no—
156C2×C4—122no—
157C2×C4(C2×D3)⋊C411yescharacteristic
158C2×C4(C2×D3)⋊C411yescharacteristic
159C2×C4C2×C4×D311yescharacteristic
160C2×C4C2×D1211yescharacteristic
161C8—122no—
162C2×C3—21no—
163C2×C3—22no—
164C2×C3—22no—
165C2×C3—21no—
166C2×C3C2×#9911yescharacteristic
167C2×C3C2×#10311yescharacteristic
168C2×C3#42111yescharacteristic
169C22—21no—
170C22—21no—
171C22—21no—
172C22—21no—
173C22—21no—
174C22—21no—
175C22—42no—
176C22—42no—
177C22#81012yes—
178C22#81012yes—
179C22C2×(C2×D3)⋊C411yescharacteristic
180C22C2×#21411yescharacteristic
181C22C2×#21611yescharacteristic
182C4—21no—
183C4—21no—
184C4—21no—
185C4—82no—
186C4—122no—
187C4—122no—
188C4#80612yes—
189C3#152011yescharacteristic, complemented, Sylow
190C2—21no—
191C2—22no—
192C2—22no—
193C2—21no—
194C2C2×#80611yescharacteristic
195C2C2×#81011yescharacteristic
196C2#500511yescharacteristic
subgroupG/H
centerC22C2×(C2×D3)⋊C4
commutatorC2×C3×C4C22×C4
FrattiniC22×C4C22×D3
FittingC3×#380C2
socleC22×C3C2×C22⋊C4
prime radicalC24×C3D4
cosocleC22×C3×C4C23
hypercenter#380D3
hypocenterC3#1520
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
4. #4452C2semidirect
7. C22×Dic6C4semidirect
11. C2×C4⋊Dic3C4semidirect
14. #924C4semidirect
189. C3#1520semidirect

Maximal subgroups

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

subgroupindex
C2×#9242
C3×#3802
C2×#8302
#44522
#15203