Finite Group Databasebeta

← #13166300

Subgroups — #13166300

Order 1728.

420 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 82 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
1#11723406C211yesmaximal, characteristic
2C4⋊(C3⋊#622)C211yesmaximal, characteristic
3C3×C4×#622C211yesmaximal, characteristic
4#11722732C212yesmaximal
5Dic3×#622C212yesmaximal
6#10593750—31nomaximal
7C2×#6308C411yescharacteristic
8C2×C3⋊#622C2212yes—
9C2×C3×#622C2211yescharacteristic
10C2×C3×#622C2211yescharacteristic
11C3×C4×C3⋊Dic3C2211yescharacteristic
12C32×C4⋊Dic3C411yescharacteristic
13#88806C412yes—
14Dic3×C3⋊Dic3C2212yes—
15#64096—31no—
16#63844—62no—
17#63779—32no—
18C4⋊#622—31no—
19C4×#622—32no—
20C4×#622D311yescharacteristic
21C2×C33×C4C2×C411yescharacteristic
22C2×C3⋊(C3⋊Dic3)C2×C412yes—
23#6308C814yescomplemented
24C2×C3×C3⋊Dic3C2×C411yescharacteristic
25C2×C3×C3⋊Dic3C2311yescharacteristic
26C2×C32×Dic3C2×C412yes—
27C3⋊#622—22no—
28C3×#622—21no—
29C3×#622Q812yescomplemented
30C8×Dic6—91nomaximal
31C2×#634—31no—
32C2×#634—62no—
33C2×#634—122no—
34C2×#622—32no—
35C2×#622C2×D311yescharacteristic
36C2×#622C2×D311yescharacteristic
37C32×C4⋊C4—31no—
38#3750—32no—
39C4×C3⋊Dic3—32no—
40C4×C3⋊Dic3C2×D311yescharacteristic
41C3×C4⋊Dic3—22no—
42C3×C4×Dic3—62no—
43#3724—64no—
44Dic32—64no—
45C22×C33C22×C411yescharacteristic
46C33×C4C2×C812yes—
47C3⋊(C3⋊Dic3)C2×C814yes—
48C3×C3⋊Dic3—21no—
49C3×C3⋊Dic3C2×Q811yescharacteristic
50C3×C3⋊Dic3#4111yescharacteristic
51C32×Dic3—22no—
52#887—182no—
53C4×Dic6—91no—
54C3×C4×C8—91no—
55Dic3⋊C8—92no—
56C8×Dic3—92no—
57C4⋊(C3⋊C8)—91no—
58C2×C32×C4—22no—
59C2×C32×C4—32no—
60C2×C32×C4—42no—
61C2×C32×C4C4×D311yescharacteristic
62C2×C3⋊Dic3—32no—
63C2×C3⋊Dic3—64no—
64C2×C3⋊Dic3—124no—
65C2×C3⋊Dic3C4×D311yescharacteristic
66C2×C3⋊Dic3C22×D311yescharacteristic
67#634—34no—
68#634—68no—
69#634—128no—
70C2×C3×Dic3—24no—
71C2×C3×Dic3—62no—
72C2×C3×Dic3—62no—
73#622—21no—
74#622—62no—
75#622Dic612yescomplemented
76C8×Q8—271noSylow
77C2×C33C4×Q811yescharacteristic
78C2×C33C22×C811yescharacteristic
79C2×C33#13111yescharacteristic
80C2×Dic6—91no—
81C2×Dic6—182no—
82C2×Dic6—362no—
83C2×C3×C8—91no—
84C2×C3×C8—91no—
85C3×C4⋊C4—62no—
86C3×C42—91no—
87C4⋊Dic3#15711yescharacteristic
88Dic3⋊C4—92no—
89Dic3⋊C4—184no—
90C4×Dic3—62no—
91C4×Dic3—92no—
92C4×Dic3—184no—
93C2×C3⋊C8—92no—
94C22×C32—22no—
95C22×C32—42no—
96C22×C32C2×C4×D311yescharacteristic
97C32×C4—24no—
98C32×C4—44no—
99C32×C4—62no—
100C32×C4D3×C812yes—
101C3⋊Dic3—21no—
102C3⋊Dic3—34no—
103C3⋊Dic3—68no—
104C3⋊Dic3—128no—
105C3⋊Dic3C2×Dic611yescharacteristic
106C3⋊Dic3#23511yescharacteristic
107C3×Dic3—44no—
108C3×Dic3—62no—
109C3×Dic3—62no—
110C3×Dic3—122no—
111C4×Q8—271no—
112C4⋊C8—271no—
113C4⋊C8—272no—
114C4×C8—91no—
115C4×C8—272no—
116C33C8×Q811yescharacteristic, complemented, Sylow
117C2×C3×C4—22no—
118C2×C3×C4—42no—
119C2×C3×C4—64no—
120C2×C3×C4—91no—
121C2×C3×C4—91no—
122C2×C3×C4C2×#15711yescharacteristic
123C2×Dic3—62no—
124C2×Dic3—62no—
125C2×Dic3—92no—
126C2×Dic3—184no—
127C2×Dic3—364no—
128C2×Dic3C2×#15712yes—
129Dic6—94no—
130Dic6—188no—
131Dic6—368no—
132C3×C8—92no—
133C3×C8—181no—
134C3⋊C8—182no—
135C2×C32—22no—
136C2×C32—22no—
137C2×C32—22no—
138C2×C32—42no—
139C2×C32—42no—
140C2×C32—42no—
141C2×C32C4×Dic611yescharacteristic
142C2×C32C2×D3×C811yescharacteristic
143C2×C32#90111yescharacteristic
144C2×Q8—271no—
145C2×C8—91no—
146C2×C8—91no—
147C2×C8—272no—
148C4⋊C4—31no—
149C4⋊C4—272no—
150C42—91no—
151C42—272no—
152C22×C3—22no—
153C22×C3—42no—
154C22×C3C22×#15711yescharacteristic
155C3×C4—24no—
156C3×C4—44no—
157C3×C4—91no—
158C3×C4—91no—
159C3×C4—124no—
160C3×C4—181no—
161C3×C4#378712yes—
162Dic3—22no—
163Dic3—62no—
164Dic3—62no—
165Dic3—94no—
166Dic3—122no—
167Dic3—188no—
168Dic3—368no—
169C32—22no—
170C32—42no—
171C32C8×Dic611yescharacteristic, complemented
172Q8—274no—
173C2×C4—32no—
174C2×C4—91no—
175C2×C4—91no—
176C2×C4—272no—
177C2×C4D3×#15711yescharacteristic
178C8—92no—
179C8—181no—
180C8—542no—
181C2×C3—22no—
182C2×C3—22no—
183C2×C3—22no—
184C2×C3—42no—
185C2×C3—42no—
186C2×C3—42no—
187C2×C3C2×#378711yescharacteristic
188C2×C3#6428711yescharacteristic
189C2×C3Q8×#15711yescharacteristic
190C22C2×D3×#15711yescharacteristic
191C4—62no—
192C4—91no—
193C4—91no—
194C4—181no—
195C4—274no—
196C4#8892412yes—
197C3—22no—
198C3—42no—
199C3#1059375011yescharacteristic, complemented
200C2Dic6×#15711yescharacteristic
201C2#1172425011yescharacteristic
202C2C2×#8892411yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×#157
commutatorC2×C33C22×C8
FittingC2×C33×C4C2×C4
socle, prime radicalC22×C33C22×C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×C4D3×#157
hypocenterC33C8×Q8
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
23. #6308C8semidirect
29. C3×#622Q8semidirect
75. #622Dic6semidirect
116. C33C8×Q8semidirect
171. C32C8×Dic6semidirect
199. C3#10593750semidirect

Maximal subgroups

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

subgroupindex
#117234062
C4⋊(C3⋊#622)2
C3×C4×#6222
#117227322
Dic3×#6222
#105937503
C8×Dic69