Finite Group Databasebeta

← #13173432

Subgroups — #13173432

Order 1728.

404 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 83 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×Q8×C3⋊Dic3C211yesmaximal, characteristic
2#11723104C212yesmaximal
3#11723102C212yesmaximal
4C4⋊(C3⋊#622)C211yesmaximal, characteristic
5C4×C3⋊#622C211yesmaximal, characteristic
6#10593835—31nomaximal
7C2×C33×Q8C411yescharacteristic
8C2×C3⋊#622C2211yescharacteristic
9C2×C3⋊#622C2212yes—
10C2×C3⋊#622C2211yescharacteristic
11C3×C4⋊(C3⋊Dic3)C412yes—
12C3×C4⋊(C3⋊Dic3)C411yescharacteristic
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C3×C4×C3⋊Dic3C2212yes—
15Q8×C3⋊Dic3D311yescharacteristic
16C3×Q8×Dic3—62no—
17#63779—32no—
18#63777—32no—
19C4⋊#622—31no—
20C4×#622—31no—
21C33×Q8—22no—
22C2×C33×C4C2×C412yes—
23C2×C33×C4C2×C411yescharacteristic
24C2×C3×C3⋊Dic3C2×C411yescharacteristic
25C2×C3×C3⋊Dic3C2×C412yes—
26C2×C3×C3⋊Dic3C2311yescharacteristic
27C3⋊#622—21no—
28C3⋊#622—22no—
29C3⋊#622Q812yescomplemented
30#5024—91nomaximal
31C2×C32×Q8—22no—
32C2×C32×Q8—42no—
33C2×C32×Q8Dic311yescharacteristic
34C2×#622—31no—
35C2×#622—31no—
36C2×#622—32no—
37C4⋊(C3⋊Dic3)Dic312yes—
38C4⋊(C3⋊Dic3)Dic311yescharacteristic
39C4×C3⋊Dic3C2×D311yescharacteristic
40C4×C3⋊Dic3C2×D312yes—
41C3×C4⋊Dic3—64no—
42C3×C4⋊Dic3—62no—
43C3×C4×Dic3—62no—
44C3×C4×Dic3—64no—
45C22×C33C22×C411yescharacteristic
46C33×C4—22no—
47C33×C4M1612yes—
48C3×C3⋊Dic3—21no—
49C3×C3⋊Dic3—22no—
50C3×C3⋊Dic3C2×Q811yescharacteristic
51C3×C3⋊Dic3#4111yescharacteristic
52C3×C4×Q8—91no—
53Q8×Dic3—62no—
54C4⋊(C3⋊C8)—91no—
55C4⋊(C3⋊C8)—92no—
56#803—92no—
57C4×C3⋊C8—91no—
58C32×Q8—22no—
59C32×Q8—28no—
60C32×Q8—48no—
61C2×C32×C4—24no—
62C2×C32×C4—22no—
63C2×C32×C4—44no—
64C2×C32×C4—42no—
65C2×C32×C4C2×Dic312yes—
66C2×C32×C4C2×Dic311yescharacteristic
67C2×C3⋊Dic3C2×Dic311yescharacteristic
68C2×C3⋊Dic3C2×Dic312yes—
69C2×C3⋊Dic3C22×D311yescharacteristic
70C2×C3×Dic3—62no—
71C2×C3×Dic3—62no—
72C2×C3×Dic3—64no—
73#622—32no—
74#622—61no—
75#622—62no—
76#446—271noSylow
77C2×C33C4×Q811yescharacteristic
78C2×C33C2×M1611yescharacteristic
79C2×C33#13111yescharacteristic
80C2×C3×Q8—22no—
81C2×C3×Q8—42no—
82C2×C3×Q8#15711yescharacteristic
83C3×C4⋊C4—92no—
84C3×C4⋊C4—91no—
85C3×C42—91no—
86C3×C42—92no—
87C4⋊Dic3—64no—
88C4⋊Dic3—62no—
89C4×Dic3—62no—
90C4×Dic3—64no—
91C2×C3⋊C8—91no—
92C2×C3⋊C8—91no—
93C2×C3⋊C8—92no—
94C22×C32—22no—
95C22×C32—42no—
96C22×C32C22×Dic311yescharacteristic
97C32×C4—22no—
98C32×C4—28no—
99C32×C4—24no—
100C32×C4—48no—
101C32×C4—44no—
102C32×C4#20812yes—
103C3⋊Dic3—21no—
104C3⋊Dic3—22no—
105C3⋊Dic3D3×Q811yescharacteristic
106C3⋊Dic3#23911yescharacteristic
107C3×Dic3—62no—
108C3×Dic3—62no—
109C3×Dic3—122no—
110C3×Dic3—124no—
111C4×Q8—91no—
112C4⋊C8—271no—
113C4⋊C8—272no—
114#97—272no—
115C4×C8—271no—
116C33#44611yescharacteristic, complemented, Sylow
117C3×Q8—28no—
118C3×Q8—22no—
119C3×Q8—48no—
120C2×C3×C4—24no—
121C2×C3×C4—22no—
122C2×C3×C4—44no—
123C2×C3×C4—42no—
124C2×C3×C4—91no—
125C2×C3×C4—91no—
126C2×C3×C4—92no—
127C2×C3×C4C2×#15712yes—
128C2×C3×C4C2×#15711yescharacteristic
129C2×Dic3—62no—
130C2×Dic3—62no—
131C2×Dic3—64no—
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×C32C2×#20811yescharacteristic
142C2×C32Q8×Dic311yescharacteristic
143C2×C32#94811yescharacteristic
144C2×Q8C3⋊#15711yescharacteristic
145C2×C8—271no—
146C2×C8—271no—
147C2×C8—272no—
148C4⋊C4—92no—
149C4⋊C4—91no—
150C42—91no—
151C42—92no—
152C22×C3—22no—
153C22×C3—42no—
154C22×C3C22×#15711yescharacteristic
155C3×C4—28no—
156C3×C4—22no—
157C3×C4—24no—
158C3×C4—48no—
159C3×C4—44no—
160C3×C4—91no—
161C3×C4—91no—
162C3×C4—181no—
163C3×C4—182no—
164C3×C4#379212yes—
165Dic3—62no—
166Dic3—62no—
167Dic3—122no—
168Dic3—124no—
169C32—22no—
170C32—42no—
171C32#502411yescharacteristic, complemented
172Q8—22no—
173C2×C4—91no—
174C2×C4—91no—
175C2×C4—92no—
176C2×C4C2×C3⋊#15712yes—
177C2×C4C2×C3⋊#15711yescharacteristic
178C8—272no—
179C8—541no—
180C8—542no—
181C2×C3—22no—
182C2×C3—22no—
183C2×C3—22no—
184C2×C3—42no—
185C2×C3—42no—
186C2×C3—42no—
187C2×C3#6428911yescharacteristic
188C2×C3Q8×#15711yescharacteristic
189C2×C3C2×#379211yescharacteristic
190C22C22×C3⋊#15711yescharacteristic
191C4—22no—
192C4—91no—
193C4—91no—
194C4—181no—
195C4—182no—
196C4#8899312yes—
197C3—22no—
198C3—42no—
199C3#1059383511yescharacteristic, complemented
200C2#1172451711yescharacteristic
201C2Q8×C3⋊#15711yescharacteristic
202C2C2×#8899311yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊#157
commutator, socle, prime radicalC22×C33C22×C4
FittingC2×C33×Q8C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×Q8C3⋊#157
hypocenterC33#446
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
29. C3⋊#622Q8semidirect
116. C33#446semidirect
171. C32#5024semidirect
199. C3#10593835semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×C3⋊Dic32
#117231042
#117231022
C4⋊(C3⋊#622)2
C4×C3⋊#6222
#105938353
#50249