Finite Group Databasebeta

← #13181521

Subgroups — #13181521

Order 1728.

920 conjugacy classes of subgroups; only the 225 normal subgroups are shown.

normal subgroups only

This group has 920 conjugacy classes of subgroups — too many to draw. Only its 225 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×C3×Q8×C3⋊D3C211yesmaximal, characteristic
2Q8×C3⋊#157C218yesmaximal, complemented
3#11724514C216yesmaximal
4#10596646—31nomaximal
5C2×C33×Q8C411yescharacteristic, complemented
6C3×Q8×C3⋊D3C414yescomplemented
7C3×Q8×C3⋊D3C2214yes—
8C2×C3×C4×C3⋊D3C2213yes—
9C2×C3×#634C413yescomplemented
10#88994C2214yes—
11C4⋊(C3⋊#157)C22112yes—
12C4×C3⋊#157C22112yes—
13C2×Q8×C3⋊D3D311yescharacteristic
14C2×C3×D3×Q8—62no—
15Q8×#157—38no—
16#64286—36no—
17C22×C3×C3⋊D3C2311yescharacteristic
18C2×C3⋊#157C2318yes—
19C33×Q8C2×C414yes—
20C2×C33×C4C2×C413yes—
21C2×C3×C3⋊Dic3C2×C413yes—
22C3×C4×C3⋊D3C2×C416yes—
23C3×C4×C3⋊D3C2316yes—
24C3×#634C2×C4112yes—
25#5812—91nomaximal
26C2×C32×Q8—22no—
27C2×C32×Q8—42no—
28C2×C32×Q8Dic311yescharacteristic, complemented
29Q8×C3⋊D3Dic314yescomplemented
30Q8×C3⋊D3C2×D314yes—
31C2×C4×C3⋊D3C2×D313yes—
32C2×#634Dic313yescomplemented
33C3×D3×Q8—68no—
34C3×D3×Q8—68no—
35C2×C3×C4×D3—66no—
36C2×C3×Dic6—66no—
37C22⋊#157—34no—
38C4⋊#157—312no—
39C4×#157—312no—
40C22×C33C22×C411yescharacteristic
41C2×C3×C3⋊D3—22no—
42C2×C3×C3⋊D3C22×C411yescharacteristic
43C2×C3×C3⋊D3C2411yescharacteristic
44C3⋊#157—28no—
45C33×C4C22×C416yes—
46C3×C3⋊Dic3C22×C416yes—
47C22×C3×Q8—91no—
48C2×D3×Q8—62no—
49Q8×Dic3—98no—
50#926—96no—
51C22×C3⋊D3C22×D311yescharacteristic
52C22×C3×D3—62no—
53C2×#157—38no—
54C32×Q8—28no—
55C32×Q8—48no—
56C32×Q8C2×Dic314yes—
57C2×C32×C4—26no—
58C2×C32×C4—46no—
59C2×C32×C4C2×Dic313yes—
60C2×C3⋊Dic3C2×Dic313yes—
61C4×C3⋊D3C2×Dic316yes—
62C4×C3⋊D3C22×D316yes—
63#634C2×Dic3112yes—
64C2×C3×Dic3—66no—
65C3×C4×D3—612no—
66C3×C4×D3—612no—
67C3×Dic6—624no—
68#519—271noSylow
69C2×C33—21no—
70C2×C33C23×C411yescharacteristic
71C3×C3⋊D3—21no—
72C3×C3⋊D32−1+412yes—
73C2×C3×Q8—22no—
74C2×C3×Q8—42no—
75C2×C3×Q8—94no—
76C2×C3×Q8—94no—
77C2×C3×Q8—93no—
78C2×C3×Q8#15711yescharacteristic, complemented
79C22×C3×C4—93no—
80D3×Q8—68no—
81D3×Q8—68no—
82C2×C4×D3—66no—
83C2×Dic6—66no—
84#217—94no—
85C4⋊Dic3—912no—
86C4×Dic3—912no—
87C22×C32—22no—
88C22×C32—42no—
89C22×C32C22×Dic311yescharacteristic
90C2×C3⋊D3—22no—
91C2×C3⋊D3C22×Dic311yescharacteristic
92C2×C3⋊D3C23×D311yescharacteristic
93C2×C3×D3—62no—
94C2×C3×D3—62no—
95C2×C3×D3—68no—
96#157—68no—
97C32×C4—212no—
98C32×C4—412no—
99C32×C4C22×Dic316yes—
100C3⋊Dic3C22×Dic316yes—
101C3×Dic3—612no—
102C22×Q8—91no—
103C4×Q8—278no—
104#117—276no—
105C33#51911yescharacteristic, complemented, Sylow
106C23×C3—91no—
107C22×D3—62no—
108C3×Q8—28no—
109C3×Q8—48no—
110C3×Q8—912no—
111C3×Q8C2×#15714yes—
112C2×C3×C4—26no—
113C2×C3×C4—46no—
114C2×C3×C4—93no—
115C2×C3×C4—96no—
116C2×C3×C4—96no—
117C2×C3×C4C2×#15713yes—
118C2×Dic3—66no—
119C2×Dic3—98no—
120C4×D3—612no—
121C4×D3—612no—
122Dic6—624no—
123C2×C32—21no—
124C2×C32—24no—
125C2×C32—22no—
126C2×C32—44no—
127C2×C32—42no—
128C2×C32C23×Dic311yescharacteristic
129C3⋊D3—21no—
130C3⋊D3#100712yes—
131C3×D3—64no—
132C3×D3—64no—
133C2×Q8—94no—
134C2×Q8—94no—
135C2×Q8—93no—
136C2×Q8C3⋊#15711yescharacteristic, complemented
137C22×C4—93no—
138C4⋊C4—2712no—
139C22⋊C4—274no—
140C42—2712no—
141C22×C3—22no—
142C22×C3—42no—
143C22×C3—91no—
144C22×C3—91no—
145C22×C3—182no—
146C22×C3C22×#15711yescharacteristic
147C2×D3—62no—
148C2×D3—62no—
149C2×D3—68no—
150C3×C4—212no—
151C3×C4—412no—
152C3×C4—96no—
153C3×C4C22×#15716yes—
154Dic3—612no—
155Dic3—188no—
156C32—22no—
157C32—42no—
158C32#581211yescharacteristic, complemented
159C23—91no—
160Q8—912no—
161Q8C2×C3⋊#15714yes—
162C2×C4—93no—
163C2×C4—96no—
164C2×C4—96no—
165C2×C4—278no—
166C2×C4C2×C3⋊#15713yes—
167C2×C3—24no—
168C2×C3—21no—
169C2×C3—22no—
170C2×C3—44no—
171C2×C3—42no—
172C2×C3—92no—
173C2×C3—181no—
174C2×C3C23×#15711yescharacteristic
175D3—64no—
176D3—64no—
177C22—91no—
178C22—91no—
179C22—182no—
180C22C22×C3⋊#15711yescharacteristic
181C4—96no—
182C4—548no—
183C4C22×C3⋊#15716yes—
184C3—22no—
185C3—42no—
186C3#1059664611yescharacteristic, complemented
187C2—21no—
188C2—92no—
189C2—181no—
190C2C23×C3⋊#15711yescharacteristic
subgroupG/H
center, FrattiniC2C23×C3⋊#157
commutator, socleC2×C33C23×C4
FittingC2×C33×Q8C4
prime radicalC22×C3×C3⋊D3C23
cosocleC2×C3×C3⋊D3C24
hypercenterC2×Q8C3⋊#157
hypocenterC33#519
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. Q8×C3⋊#157C2semidirect
5. C2×C33×Q8C4semidirect
6. C3×Q8×C3⋊D3C4semidirect
9. C2×C3×#634C4semidirect
28. C2×C32×Q8Dic3semidirect
29. Q8×C3⋊D3Dic3semidirect
32. C2×#634Dic3semidirect
78. C2×C3×Q8#157semidirect
105. C33#519semidirect
136. C2×Q8C3⋊#157semidirect
158. C32#5812semidirect
186. C3#10596646semidirect

Maximal subgroups

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

subgroupindex
C2×C3×Q8×C3⋊D32
Q8×C3⋊#1572
#117245142
#105966463
#58129