Finite Group Databasebeta

← #13152176

Subgroups — #13152176

Order 1728.

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

normal subgroups only

This group has 416 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
1C22×#6204C211yesmaximal, characteristic
2C4×#6204C212yesmaximal, complemented
3C32×#519C311yesmaximal, characteristic
4#519⋊C9C313yesmaximal, complemented
5C2×#6204C412yescomplemented
6C2×#6204C2211yescharacteristic
7C22⋊C4×3−1+2—41nomaximal
8C22×C32×Q8C2×C311yescharacteristic
9C32×C4×Q8—32no—
10C32×C4×Q8C2×C312yes—
11C32×#117—31no—
12C32×#117—31no—
13C22×Q8⋊C9C2×C313yes—
14C4×Q8⋊C9C2×C316yescomplemented
15C23×3−1+2—41no—
16C2×C4×3−1+2—42no—
17#6204C2×C411yescharacteristic
18C3×#519—31no—
19C3×#519C3211yescharacteristic
20C2×C32×Q8—32no—
21C2×C32×Q8—31no—
22C2×C32×Q8C3×C412yes—
23C2×C32×Q8C22×C311yescharacteristic
24C22×C32×C4—31no—
25C32×C4⋊C4—32no—
26C32×C4⋊C4—32no—
27C32×C22⋊C4—31no—
28C32×C22⋊C4A411yescharacteristic
29C32×C42—32no—
30C32×C42—32no—
31C2×Q8⋊C9C3×C416yescomplemented
32C2×Q8⋊C9C22×C313yes—
33C9×C22⋊C4—43no—
34C22×3−1+2—41no—
35C22×3−1+2—42no—
36C22×3−1+2—82no—
37C4×3−1+2—82no—
38C22×C3×Q8—31no—
39C22×C3×Q8C2×C3211yescharacteristic
40C3×C4×Q8—36no—
41C3×C4×Q8—32no—
42C3×C4×Q8—32no—
43C3×C4×Q8C2×C3212yes—
44C3×#117—33no—
45C3×#117—33no—
46C3×#117—31no—
47C3×#117—31no—
48C23×C32C2×A411yescharacteristic
49C32×Q8—32no—
50C32×Q8—31no—
51C32×Q8—32no—
52C32×Q8C2×C3×C411yescharacteristic
53C2×C32×C4—32no—
54C2×C32×C4—32no—
55C2×C32×C4—31no—
56C2×C32×C4—31no—
57C2×C32×C4—32no—
58C2×C32×C4C2×A412yes—
59C23×C9—43no—
60C2×C4×C9—46no—
61Q8⋊C9C2×C3×C413yes—
62#5193−1+211yescharacteristic, complemented, Sylow
63C2×3−1+2—41no—
64C2×3−1+2—42no—
65C2×3−1+2—82no—
66C2×C3×Q8—33no—
67C2×C3×Q8—36no—
68C2×C3×Q8—31no—
69C2×C3×Q8—32no—
70C2×C3×Q8—32no—
71C2×C3×Q8—31no—
72C2×C3×Q8C32×C412yes—
73C2×C3×Q8C22×C3211yescharacteristic
74C22×C3×C4—33no—
75C22×C3×C4—31no—
76C3×C4⋊C4—32no—
77C3×C4⋊C4—36no—
78C3×C4⋊C4—36no—
79C3×C4⋊C4—32no—
80C3×C22⋊C4—33no—
81C3×C22⋊C4—31no—
82C3×C22⋊C4—31no—
83C3×C22⋊C4C3×A411yescharacteristic
84C3×C42—36no—
85C3×C42—32no—
86C3×C42—36no—
87C3×C42—32no—
88C22×C32—22no—
89C22×C32C4×A412yes—
90C22×C32C22×A411yescharacteristic
91C32×C4—22no—
92C32×C4—31no—
93C32×C4—32no—
94C32×C4—31no—
95C32×C4—62no—
96C22×C9—43no—
97C22×C9—46no—
98C22×C9—86no—
99C4×C9—86no—
100C22×Q8C2×3−1+211yescharacteristic
101C4×Q8—32no—
102C4×Q8C2×3−1+212yescomplemented
103#117—31no—
104#117—31no—
1053−1+2—41noSylow
106C23×C3—31no—
107C23×C3C2×C3×A411yescharacteristic
108C3×Q8—32no—
109C3×Q8—31no—
110C3×Q8—36no—
111C3×Q8—33no—
112C3×Q8—36no—
113C3×Q8—32no—
114C3×Q8—31no—
115C3×Q8C2×C32×C411yescharacteristic
116C2×C3×C4—33no—
117C2×C3×C4—33no—
118C2×C3×C4—36no—
119C2×C3×C4—36no—
120C2×C3×C4—31no—
121C2×C3×C4—31no—
122C2×C3×C4—32no—
123C2×C3×C4—32no—
124C2×C3×C4—36no—
125C2×C3×C4—32no—
126C2×C3×C4—32no—
127C2×C3×C4C2×C3×A412yes—
128C2×C32—22no—
129C2×C32C2×C4×A411yescharacteristic
130C2×C322−1+4⋊C312yes—
131C2×C9—43no—
132C2×C9—46no—
133C2×C9—86no—
134C2×Q8—31no—
135C2×Q8—32no—
136C2×Q8C4×3−1+212yescomplemented
137C2×Q8C22×3−1+211yescharacteristic
138C22×C4—31no—
139C4⋊C4—32no—
140C4⋊C4—32no—
141C22⋊C4—31no—
142C22⋊C4#109411yescharacteristic
143C42—32no—
144C42—32no—
145C22×C3—22no—
146C22×C3—31no—
147C22×C3—32no—
148C22×C3—62no—
149C22×C3C3×C4×A412yes—
150C22×C3C22×C3×A411yescharacteristic
151C3×C4—22no—
152C3×C4—33no—
153C3×C4—33no—
154C3×C4—36no—
155C3×C4—31no—
156C3×C4—32no—
157C3×C4—31no—
158C3×C4—62no—
159C3×C4—66no—
160C3×C4—62no—
161C32#519⋊C311yescharacteristic
162C9—43no—
163C23C2×#109411yescharacteristic
164Q8—32no—
165Q8—31no—
166Q8—32no—
167Q8C2×C4×3−1+211yescharacteristic
168C2×C4—31no—
169C2×C4—31no—
170C2×C4—32no—
171C2×C4—32no—
172C2×C4—32no—
173C2×C4C2×#109412yes—
174C2×C3—22no—
175C2×C3—31no—
176C2×C3—32no—
177C2×C3—62no—
178C2×C3C2×C3×C4×A411yescharacteristic
179C2×C3C3×2−1+4⋊C312yes—
180C22—22no—
181C22C4×#109412yes—
182C22C22×#109411yescharacteristic
183C4—22no—
184C4—31no—
185C4—31no—
186C4—32no—
187C4—62no—
188C3—31no—
189C3C3×#519⋊C311yescharacteristic
190C2—22no—
191C2C2×C4×#109411yescharacteristic
192C2#1172257912yes—
subgroupG/H
center, Frattini, socleC22×C3C22×C3×A4
commutatorC3×Q8C2×C32×C4
FittingC32×#519C3
prime radicalC23×C32C2×A4
cosocleC2×C3×Q8C22×C32
hypercenterC32×C22⋊C4A4
hypocenterQ8C2×C4×3−1+2
LCSG ▷ C3×Q8 ▷ Q8
UCSC1 ◁ C22×C3 ◁ C32×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C4×#6204C2semidirect
4. #519⋊C9C3semidirect
5. C2×#6204C4semidirect
14. C4×Q8⋊C9C2×C3semidirect
31. C2×Q8⋊C9C3×C4semidirect
62. #5193−1+2semidirect
102. C4×Q8C2×3−1+2semidirect
136. C2×Q8C4×3−1+2semidirect

Maximal subgroups

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

subgroupindex
C22×#62042
C4×#62042
C32×#5193
#519⋊C93
C22⋊C4×3−1+24