Finite Group Databasebeta

← #13147340

Subgroups — #13147340

Order 1728.

281 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 281 conjugacy classes of subgroups — too many to draw. Only its 55 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#11721755C211yesmaximal, characteristic
2#11720785C211yesmaximal, characteristic
3#11720785C211yesmaximal, characteristic
4#11720783C211yesmaximal, characteristic
5#11720783C211yesmaximal, characteristic
6C4⋊#6188C211yesmaximal, characteristic
7C4×#6188C211yesmaximal, characteristic
8C2×#6229C411yescharacteristic
9C2×#6188C2211yescharacteristic
10C2×#6188C2211yescharacteristic
11C2×#6188C2211yescharacteristic
12C2×#6188C2211yescharacteristic
13C4⋊C4×H(3,𝔽3)C411yescharacteristic
14#88540C411yescharacteristic
15#88540C411yescharacteristic
16C4×#1084C2211yescharacteristic
17C4×#1084C2211yescharacteristic
18C4×#1084C2211yescharacteristic
19C3×#887—62no—
20C2×C4×H(3,𝔽3)C2×C411yescharacteristic
21C2×C4×H(3,𝔽3)C2×C411yescharacteristic
22C2×C4×H(3,𝔽3)C2×C411yescharacteristic
23C2×#1084C2×C411yescharacteristic
24C2×#1084C2×C411yescharacteristic
25C2×#1084C2×C411yescharacteristic
26C2×#1084C2311yescharacteristic
27#6229—22no—
28#6188—21no—
29#6188—21no—
30#6188—21no—
31#6188Q812yescomplemented
32C3×#446—91nomaximal
33C2×C3×Dic6—62no—
34C32×C4⋊C4—22no—
35C3×Dic3⋊C4—62no—
36C3×Dic3⋊C4—62no—
37C3×C4×Dic3—62no—
38C3×C4×Dic3—62no—
39C3×C4×Dic3—62no—
40C22×H(3,𝔽3)C22×C411yescharacteristic
41C4×H(3,𝔽3)—21no—
42C4×H(3,𝔽3)—21no—
43C4×H(3,𝔽3)—21no—
44#1084—21no—
45#1084—21no—
46#1084M1612yes—
47#1084C2×Q811yescharacteristic
48#1084#4111yescharacteristic
49C3×C4×Q8—91no—
50#887—62no—
51C3×C4⋊C8—91no—
52C3×C4⋊C8—91no—
53C3×C4⋊C8—91no—
54C3×#97—91no—
55C3×#97—91no—
56C3×C4×C8—91no—
57C2×C32×C4—22no—
58C2×C32×C4—22no—
59C2×C32×C4—22no—
60C2×C3×Dic3—62no—
61C2×C3×Dic3—62no—
62C2×C3×Dic3—62no—
63C2×C3×Dic3—62no—
64C3×Dic6—68no—
65#446—91noSylow
66C2×H(3,𝔽3)C4×Q811yescharacteristic
67C2×H(3,𝔽3)C2×M1611yescharacteristic
68C2×H(3,𝔽3)#13111yescharacteristic
69C2×C3×Q8—91no—
70C2×Dic6—62no—
71C2×C3×C8—91no—
72C2×C3×C8—91no—
73C2×C3×C8—91no—
74C2×C3×C8—91no—
75C3×C4⋊C4—62no—
76C3×C4⋊C4—91no—
77C3×C4⋊C4—91no—
78C3×C4⋊C4#15711yescharacteristic
79C3×C42—91no—
80C3×C42—91no—
81C3×C42—91no—
82Dic3⋊C4—62no—
83Dic3⋊C4—62no—
84C4×Dic3—62no—
85C4×Dic3—62no—
86C4×Dic3—62no—
87C22×C32—22no—
88C32×C4—24no—
89C32×C4—42no—
90C32×C4—42no—
91C3×Dic3—64no—
92C3×Dic3—62no—
93C3×Dic3—62no—
94C3×Dic3—64no—
95C3×Dic3—122no—
96C4×Q8—91no—
97C4⋊C8—91no—
98C4⋊C8—91no—
99C4⋊C8—91no—
100#97—91no—
101#97—91no—
102C4×C8—91no—
103H(3,𝔽3)#44611yescharacteristic, complemented, Sylow
104C3×Q8—182no—
105C2×C3×C4—62no—
106C2×C3×C4—62no—
107C2×C3×C4—62no—
108C2×C3×C4—91no—
109C2×C3×C4—91no—
110C2×C3×C4—91no—
111C2×C3×C4—91no—
112C2×C3×C4C2×#15711yescharacteristic
113C2×C3×C4C2×#15711yescharacteristic
114C2×C3×C4C2×#15711yescharacteristic
115C2×Dic3—62no—
116C2×Dic3—62no—
117C2×Dic3—62no—
118C2×Dic3—62no—
119Dic6—68no—
120C3×C8—92no—
121C3×C8—181no—
122C3×C8—181no—
123C3×C8—181no—
124C2×C32—22no—
125C2×C32—22no—
126C2×C32—22no—
127C2×Q8—91no—
128C2×C8—91no—
129C2×C8—91no—
130C2×C8—91no—
131C2×C8—91no—
132C4⋊C4—91no—
133C4⋊C4—91no—
134C4⋊C4#108811yescharacteristic
135C42—91no—
136C42—91no—
137C42—91no—
138C22×C3—62no—
139C22×C3C22×#15711yescharacteristic
140C3×C4—21no—
141C3×C4—21no—
142C3×C4—21no—
143C3×C4—64no—
144C3×C4—92no—
145C3×C4—91no—
146C3×C4—91no—
147C3×C4—122no—
148C3×C4—122no—
149C3×C4—181no—
150C3×C4—181no—
151Dic3—64no—
152Dic3—62no—
153Dic3—62no—
154Dic3—64no—
155Dic3—122no—
156C32—22no—
157Q8—182no—
158C2×C4—91no—
159C2×C4—91no—
160C2×C4—91no—
161C2×C4—91no—
162C2×C4C2×#108811yescharacteristic
163C2×C4C2×#108811yescharacteristic
164C2×C4C2×#108811yescharacteristic
165C8—92no—
166C8—181no—
167C8—181no—
168C8—181no—
169C2×C3—62no—
170C2×C3—62no—
171C2×C3—62no—
172C2×C3#6428311yescharacteristic
173C2×C3#6428711yescharacteristic
174C2×C3Q8×#15711yescharacteristic
175C22C22×#108811yescharacteristic
176C4—21no—
177C4—21no—
178C4—21no—
179C4—92no—
180C4—91no—
181C4—91no—
182C4—181no—
183C4—181no—
184C3—62no—
185C3#1059375411yescharacteristic
186C2#1172244511yescharacteristic
187C2#1172244911yescharacteristic
188C2Q8×#108811yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C22×#157
commutator, prime radicalC22×H(3,𝔽3)C22×C4
FittingC4⋊C4×H(3,𝔽3)C4
cosocleC2×#1084C23
hypercenterC3×C4⋊C4#157
hypocenterH(3,𝔽3)#446
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
31. #6188Q8semidirect
103. H(3,𝔽3)#446semidirect

Maximal subgroups

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

subgroupindex
#117217552
#117207852
#117207832
C4⋊#61882
C4×#61882
C3×#4469