Finite Group Databasebeta

← #13173350

Subgroups — #13173350

Order 1728.

396 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C3×#64096C211yesmaximal, characteristic
2#11723104C212yesmaximal
3C4⋊(C3⋊#622)C211yesmaximal, characteristic
4C4×C3⋊#622C211yesmaximal, characteristic
5C4×C3⋊#622C212yesmaximal
6#10593750—31nomaximal
7C2×C3×#634C411yescharacteristic
8C2×C3⋊#622C2212yes—
9C2×C3⋊#622C2211yescharacteristic
10C2×C3⋊#622C2211yescharacteristic
11C33×C4⋊C4C411yescharacteristic
12C3×#3750C412yes—
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C3×C4×C3⋊Dic3C2212yes—
15#64096D311yescharacteristic
16C3×#887—62no—
17#63779—32no—
18C4⋊#622—31no—
19C4×#622—31no—
20C4×#622—32no—
21C2×C33×C4C2×C412yes—
22C2×C33×C4C2×C411yescharacteristic
23C2×C3×C3⋊Dic3C2×C411yescharacteristic
24C2×C3×C3⋊Dic3C2×C412yes—
25C2×C3×C3⋊Dic3C2311yescharacteristic
26C3×#634C814yescomplemented
27C3⋊#622—22no—
28C3⋊#622—21no—
29C3⋊#622Q812yescomplemented
30Q8×C3⋊C8—91nomaximal
31C2×#634Dic311yescharacteristic
32C2×C3×Dic6—62no—
33C2×#622—31no—
34C2×#622—32no—
35C2×#622—31no—
36C32×C4⋊C4—22no—
37C32×C4⋊C4—42no—
38C32×C4⋊C4Dic311yescharacteristic
39#3750Dic312yes—
40C4×C3⋊Dic3C2×D311yescharacteristic
41C4×C3⋊Dic3C2×D312yes—
42C3×Dic3⋊C4—64no—
43C3×C4×Dic3—64no—
44C3×C4×Dic3—62no—
45C22×C33C22×C411yescharacteristic
46C33×C4—22no—
47C33×C4C2×C812yes—
48C3×C3⋊Dic3—21no—
49C3×C3⋊Dic3C2×C814yes—
50C3×C3⋊Dic3C2×Q811yescharacteristic
51C3×C3⋊Dic3#4111yescharacteristic
52C3×C4×Q8—91no—
53#887—62no—
54C4⋊(C3⋊C8)—91no—
55C4⋊(C3⋊C8)—92no—
56C4×C3⋊C8—92no—
57C4×C3⋊C8—91no—
58C2×C32×C4—22no—
59C2×C32×C4—24no—
60C2×C32×C4—42no—
61C2×C32×C4—44no—
62C2×C32×C4C2×Dic312yes—
63C2×C32×C4C2×Dic311yescharacteristic
64C2×C3⋊Dic3C2×Dic311yescharacteristic
65C2×C3⋊Dic3C2×Dic312yes—
66C2×C3⋊Dic3C22×D311yescharacteristic
67#634C3⋊C814yescomplemented
68C2×C3×Dic3—62no—
69C2×C3×Dic3—64no—
70C2×C3×Dic3—62no—
71C3×Dic6—68no—
72#622—32no—
73#622—62no—
74#622—61no—
75C8×Q8—271noSylow
76C2×C33C4×Q811yescharacteristic
77C2×C33C22×C811yescharacteristic
78C2×C33#13111yescharacteristic
79C2×C3×Q8—91no—
80C2×Dic6—62no—
81C3×C4⋊C4—22no—
82C3×C4⋊C4—42no—
83C3×C4⋊C4—92no—
84C3×C4⋊C4#15711yescharacteristic
85C3×C42—92no—
86C3×C42—91no—
87Dic3⋊C4—64no—
88C4×Dic3—64no—
89C4×Dic3—62no—
90C2×C3⋊C8—91no—
91C2×C3⋊C8—92no—
92C2×C3⋊C8—91no—
93C22×C32—22no—
94C22×C32—42no—
95C22×C32C22×Dic311yescharacteristic
96C32×C4—24no—
97C32×C4—22no—
98C32×C4—44no—
99C32×C4—44no—
100C32×C4—84no—
101C32×C4C2×C3⋊C812yes—
102C3⋊Dic3—21no—
103C3⋊Dic3C2×C3⋊C814yes—
104C3⋊Dic3D3×Q811yescharacteristic
105C3⋊Dic3#23911yescharacteristic
106C3×Dic3—68no—
107C3×Dic3—62no—
108C3×Dic3—62no—
109C3×Dic3—122no—
110C4×Q8—91no—
111C4⋊C8—271no—
112C4⋊C8—272no—
113C4×C8—272no—
114C4×C8—271no—
115C33C8×Q811yescharacteristic, complemented, Sylow
116C3×Q8—94no—
117C2×C3×C4—22no—
118C2×C3×C4—24no—
119C2×C3×C4—42no—
120C2×C3×C4—44no—
121C2×C3×C4—91no—
122C2×C3×C4—92no—
123C2×C3×C4—91no—
124C2×C3×C4C2×#15712yes—
125C2×C3×C4C2×#15711yescharacteristic
126C2×Dic3—64no—
127C2×Dic3—62no—
128C2×Dic3—62no—
129Dic6—68no—
130C3⋊C8—92no—
131C3⋊C8—182no—
132C3⋊C8—181no—
133C2×C32—22no—
134C2×C32—22no—
135C2×C32—22no—
136C2×C32—42no—
137C2×C32—42no—
138C2×C32—42no—
139C2×C32C22×C3⋊C811yescharacteristic
140C2×C32Q8×Dic311yescharacteristic
141C2×C32#94811yescharacteristic
142C2×Q8—91no—
143C2×C8—271no—
144C2×C8—272no—
145C2×C8—271no—
146C4⋊C4—92no—
147C4⋊C4C3⋊#15711yescharacteristic
148C42—92no—
149C42—91no—
150C22×C3—22no—
151C22×C3—42no—
152C22×C3C22×#15711yescharacteristic
153C3×C4—22no—
154C3×C4—24no—
155C3×C4—44no—
156C3×C4—44no—
157C3×C4—84no—
158C3×C4—94no—
159C3×C4—91no—
160C3×C4—91no—
161C3×C4—181no—
162C3×C4#378712yes—
163Dic3—68no—
164Dic3—62no—
165Dic3—62no—
166Dic3—122no—
167C32—22no—
168C32—42no—
169C32Q8×C3⋊C811yescharacteristic, complemented
170Q8—94no—
171C2×C4—91no—
172C2×C4—92no—
173C2×C4—91no—
174C2×C4C2×C3⋊#15712yes—
175C2×C4C2×C3⋊#15711yescharacteristic
176C8—272no—
177C8—542no—
178C8—541no—
179C2×C3—22no—
180C2×C3—22no—
181C2×C3—22no—
182C2×C3—42no—
183C2×C3—42no—
184C2×C3—42no—
185C2×C3C2×#378711yescharacteristic
186C2×C3#6428711yescharacteristic
187C2×C3Q8×#15711yescharacteristic
188C22C22×C3⋊#15711yescharacteristic
189C4—22no—
190C4—94no—
191C4—91no—
192C4—91no—
193C4—181no—
194C4#8898812yes—
195C3—22no—
196C3—42no—
197C3#1059375011yescharacteristic, complemented
198C2C2×#8898811yescharacteristic
199C2#1172451511yescharacteristic
200C2Q8×C3⋊#15711yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊#157
commutatorC2×C33C22×C8
FittingC33×C4⋊C4C4
socle, prime radicalC22×C33C22×C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC4⋊C4C3⋊#157
hypocenterC33C8×Q8
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
26. C3×#634C8semidirect
29. C3⋊#622Q8semidirect
67. #634C3⋊C8semidirect
115. C33C8×Q8semidirect
169. C32Q8×C3⋊C8semidirect
197. C3#10593750semidirect

Maximal subgroups

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

subgroupindex
C3×#640962
#117231042
C4⋊(C3⋊#622)2
C4×C3⋊#6222
#105937503
Q8×C3⋊C89