Finite Group Databasebeta

← #13166334

Subgroups — #13166334

Order 1728.

380 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 72 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
1Dic6×C3⋊Dic3C211yesmaximal, characteristic
2C4×C3⋊#622C211yesmaximal, characteristic
3C3×C4⋊#622C211yesmaximal, characteristic
4#11722732C212yesmaximal
5#11722724C212yesmaximal
6#10593835—31nomaximal
7C2×C32×Dic6C411yescharacteristic
8C2×C3⋊#622C2211yescharacteristic
9C2×C3⋊#622C2211yescharacteristic
10C2×C3×#622C2212yes—
11C4⋊(C3⋊(C3⋊Dic3))C411yescharacteristic
12C3×C4×C3⋊Dic3C2211yescharacteristic
13Dic3⋊(C3⋊Dic3)C412yes—
14Dic3×C3⋊Dic3C2212yes—
15Q8×C3⋊Dic3—31no—
16Dic3×Dic6—62no—
17#63779—32no—
18#63777—32no—
19C4⋊#622D311yescharacteristic
20C4×#622—31no—
21C2×C33×C4C2×C411yescharacteristic
22C2×C3⋊(C3⋊Dic3)C2×C412yes—
23C2×C3×C3⋊Dic3C2×C411yescharacteristic
24C2×C3×C3⋊Dic3C2311yescharacteristic
25C2×C32×Dic3C2×C412yes—
26C32×Dic6—22no—
27C3⋊#622—21no—
28C3⋊#622Q812yescomplemented
29C3×#622—22no—
30#4831—91nomaximal
31C2×C32×Q8—31no—
32C2×C3×Dic6—22no—
33C2×#622—31no—
34C2×#622—31no—
35C2×#622C2×D312yes—
36C4⋊(C3⋊Dic3)—31no—
37C4⋊(C3⋊Dic3)—32no—
38C4⋊(C3⋊Dic3)—62no—
39C4⋊(C3⋊Dic3)—122no—
40C4×C3⋊Dic3—32no—
41C4×C3⋊Dic3C2×D311yescharacteristic
42C3×C4×Dic3—62no—
43Dic3⋊Dic3—64no—
44Dic32—64no—
45C22×C33C22×C411yescharacteristic
46C33×C4M1612yes—
47C3⋊(C3⋊Dic3)—22no—
48C3×C3⋊Dic3—21no—
49C3×C3⋊Dic3C2×Q811yescharacteristic
50C3×C3⋊Dic3#4111yescharacteristic
51C32×Dic3—22no—
52Q8×Dic3—182no—
53C4×Dic6—91no—
54C3×C4⋊C8—91no—
55#815—92no—
56Dic3⋊C8—92no—
57C4×C3⋊C8—91no—
58C32×Q8—62no—
59C2×C32×C4—22no—
60C2×C32×C4—32no—
61C2×C32×C4—42no—
62C2×C32×C4C4×D311yescharacteristic
63C2×C3⋊Dic3—32no—
64C2×C3⋊Dic3—64no—
65C2×C3⋊Dic3—124no—
66C2×C3⋊Dic3C4×D311yescharacteristic
67C2×C3⋊Dic3C22×D311yescharacteristic
68C2×C3×Dic3—24no—
69C2×C3×Dic3—62no—
70C2×C3×Dic3—62no—
71C3×Dic6—28no—
72#622—22no—
73#622—32no—
74#622—61no—
75#446—271noSylow
76C2×C33C4×Q811yescharacteristic
77C2×C33C2×M1611yescharacteristic
78C2×C33#13111yescharacteristic
79C2×C3×Q8—62no—
80C2×Dic6#15711yescharacteristic
81C2×C3×C8—92no—
82C3×C42—91no—
83C4⋊Dic3—91no—
84C4⋊Dic3—184no—
85C4⋊Dic3—182no—
86C4⋊Dic3—362no—
87Dic3⋊C4—92no—
88C4×Dic3—62no—
89C4×Dic3—92no—
90C4×Dic3—184no—
91C2×C3⋊C8—91no—
92C2×C3⋊C8—91no—
93C22×C32—22no—
94C22×C32—42no—
95C22×C32C2×C4×D311yescharacteristic
96C32×C4—24no—
97C32×C4—44no—
98C32×C4—62no—
99C32×C4ΓL(1,𝔽25)12yes—
100C3⋊Dic3—21no—
101C3⋊Dic3—62no—
102C3⋊Dic3—124no—
103C3⋊Dic3—244no—
104C3⋊Dic3#23711yescharacteristic
105C3⋊Dic3D3×Q811yescharacteristic
106C3×Dic3—28no—
107C3×Dic3—62no—
108C3×Dic3—62no—
109C3×Dic3—122no—
110C4×Q8—271no—
111C4⋊C8—91no—
112C4⋊C8—272no—
113#97—272no—
114C4×C8—271no—
115C33#44611yescharacteristic, complemented, Sylow
116C3×Q8—68no—
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—22no—
130C3×C8—182no—
131C3⋊C8—92no—
132C3⋊C8—181no—
133C2×C32—22no—
134C2×C32—22no—
135C2×C32—22no—
136C2×C32—42no—
137C2×C32—42no—
138C2×C32—42no—
139C2×C32#88711yescharacteristic
140C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
141C2×C32#90711yescharacteristic
142C2×Q8—31no—
143C2×C8—92no—
144C2×C8—271no—
145C2×C8—271no—
146C4⋊C4—272no—
147C4⋊C4—271no—
148C42—91no—
149C42—272no—
150C22×C3—22no—
151C22×C3—42no—
152C22×C3C22×#15711yescharacteristic
153C3×C4—24no—
154C3×C4—44no—
155C3×C4—68no—
156C3×C4—91no—
157C3×C4—91no—
158C3×C4—181no—
159C3×C4#379212yes—
160Dic3—22no—
161Dic3—62no—
162Dic3—62no—
163Dic3—122no—
164Dic3—182no—
165Dic3—364no—
166Dic3—724no—
167C32—22no—
168C32—42no—
169C32#483111yescharacteristic, complemented
170Q8—62no—
171C2×C4—32no—
172C2×C4—91no—
173C2×C4—91no—
174C2×C4—272no—
175C2×C4D3×#15711yescharacteristic
176C8—182no—
177C8—272no—
178C8—541no—
179C2×C3—22no—
180C2×C3—22no—
181C2×C3—22no—
182C2×C3—42no—
183C2×C3—42no—
184C2×C3—42no—
185C2×C3#6428911yescharacteristic
186C2×C3Q8×#15711yescharacteristic
187C2×C3C2×#379211yescharacteristic
188C22C2×D3×#15711yescharacteristic
189C4—62no—
190C4—91no—
191C4—91no—
192C4—181no—
193C4—542no—
194C4#8892512yes—
195C3—22no—
196C3—42no—
197C3#1059383511yescharacteristic, complemented
198C2#1172424511yescharacteristic
199C2#1172425311yescharacteristic
200C2C2×#8892511yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×#157
commutator, socle, prime radicalC22×C33C22×C4
FittingC2×C33×C4C2×C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×C4D3×#157
hypocenterC33#446
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
28. C3⋊#622Q8semidirect
115. C33#446semidirect
169. C32#4831semidirect
197. C3#10593835semidirect

Maximal subgroups

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

subgroupindex
Dic6×C3⋊Dic32
C4×C3⋊#6222
C3×C4⋊#6222
#117227322
#117227242
#105938353
#48319