Finite Group Databasebeta

← C2×#11720786 #13147393

Subgroups — C2×#11720786 #13147393

Order 1728.

531 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 531 conjugacy classes of subgroups — too many to draw. Only its 63 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
1C22×#6229C211yesmaximal, characteristic
2C2×#88631C212yesmaximal
3#11720786C214yesmaximal, direct product factor
4C22×C4×H(3,𝔽3)C411yescharacteristic
5C22×#1084C411yescharacteristic
6C22×#1084C2211yescharacteristic
7C2×#6229—22no—
8C2×#6229—22no—
9C2×#6229C412yes—
10C2×#6229C2212yes—
11#88631—22no—
12#88631C2214yes—
13C2×#6188—22no—
14C22×C3×Dic6—62no—
15C23×H(3,𝔽3)C2×C411yescharacteristic
16C2×C4×H(3,𝔽3)—21no—
17C2×C4×H(3,𝔽3)—21no—
18C2×C4×H(3,𝔽3)C2×C412yes—
19C2×#1084—21no—
20C2×#1084—21no—
21C2×#1084C2×C412yes—
22C2×#1084C2×C411yescharacteristic
23C2×#1084D412yes—
24C2×#1084D412yes—
25C2×#1084C2311yescharacteristic
26#6229—24no—
27#6229—24no—
28#6188—24no—
29C2×C3×#101—91nomaximal
30C22×C32×C4—22no—
31C22×C3×Dic3—62no—
32C22×C3×Dic3—62no—
33C2×C3×Dic6—64no—
34C2×C3×Dic6—64no—
35C2×C3×Dic6—68no—
36C2×C3×Dic6—68no—
37C22×H(3,𝔽3)—22no—
38C22×H(3,𝔽3)C22⋊C411yescharacteristic
39C22×H(3,𝔽3)C22⋊C411yescharacteristic
40C22×H(3,𝔽3)C22×C411yescharacteristic
41C4×H(3,𝔽3)—22no—
42#1084—22no—
43#1084C22⋊C412yes—
44#1084C2×D412yes—
45C22×C3×Q8—91no—
46C22×Dic6—62no—
47C2×C3×M16—92no—
48C3×#101—94no—
49C23×C32—22no—
50C2×C32×C4—24no—
51C2×C32×C4—24no—
52C2×C32×C4—24no—
53C2×C3×Dic3—62no—
54C2×C3×Dic3—64no—
55C2×C3×Dic3—64no—
56C2×C3×Dic3—64no—
57C2×C3×Dic3—62no—
58C2×C3×Dic3—64no—
59C2×C3×Dic3—64no—
60C3×Dic6—616no—
61C3×Dic6—616no—
62C2×#101—91noSylow
63C2×H(3,𝔽3)—21no—
64C2×H(3,𝔽3)—21no—
65C2×H(3,𝔽3)#10112yescomplemented
66C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
67C2×C3×Q8—92no—
68C2×C3×Q8—92no—
69C2×C3×Q8—182no—
70C2×C3×Q8—182no—
71C22×C3×C4—62no—
72C22×C3×C4—91no—
73C22×C3×C4—91no—
74C22×C3×C4#15711yescharacteristic
75C22×Dic3—62no—
76C22×Dic3—62no—
77C2×Dic6—64no—
78C2×Dic6—64no—
79C2×Dic6—68no—
80C2×Dic6—68no—
81C3×M16—94no—
82C3×M16—182no—
83C2×C3×C8—182no—
84C22×C32—28no—
85C22×C32—22no—
86C22×C32—22no—
87C22×C32—22no—
88C32×C4—28no—
89C3×Dic3—64no—
90C3×Dic3—64no—
91C3×Dic3—68no—
92C22×Q8—91no—
93C2×M16—92no—
94#101—94no—
95H(3,𝔽3)C2×#10111yescharacteristic, complemented, Sylow
96C23×C3—62no—
97C23×C3C2×#15711yescharacteristic
98C3×Q8—184no—
99C3×Q8—184no—
100C2×C3×C4—21no—
101C2×C3×C4—21no—
102C2×C3×C4—64no—
103C2×C3×C4—64no—
104C2×C3×C4—64no—
105C2×C3×C4—91no—
106C2×C3×C4—91no—
107C2×C3×C4—92no—
108C2×C3×C4—92no—
109C2×C3×C4—92no—
110C2×C3×C4—181no—
111C2×C3×C4—181no—
112C2×C3×C4C2×#15712yes—
113C2×Dic3—64no—
114C2×Dic3—64no—
115C2×Dic3—62no—
116C2×Dic3—62no—
117C2×Dic3—64no—
118C2×Dic3—64no—
119C2×Dic3—64no—
120Dic6—616no—
121Dic6—616no—
122C3×C8—184no—
123C2×C32—24no—
124C2×C32—24no—
125C2×C32—24no—
126C2×C32—22no—
127C2×Q8—92no—
128C2×Q8—92no—
129C2×Q8—182no—
130C2×Q8—182no—
131C22×C4—91no—
132C22×C4—91no—
133C22×C4#108811yescharacteristic
134M16—94no—
135M16—182no—
136C2×C8—182no—
137C22×C3—22no—
138C22×C3—68no—
139C22×C3—62no—
140C22×C3—62no—
141C22×C3—62no—
142C22×C3C22⋊#15711yescharacteristic
143C22×C3C22⋊#15711yescharacteristic
144C22×C3C22×#15711yescharacteristic
145C3×C4—22no—
146C3×C4—68no—
147C3×C4—92no—
148C3×C4—92no—
149C3×C4—182no—
150Dic3—64no—
151Dic3—64no—
152Dic3—68no—
153C32—22no—
154C23C2×#108811yescharacteristic
155Q8—184no—
156Q8—184no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—92no—
160C2×C4—92no—
161C2×C4—91no—
162C2×C4—91no—
163C2×C4—92no—
164C2×C4—181no—
165C2×C4—181no—
166C2×C4C2×#108812yes—
167C8—184no—
168C2×C3—21no—
169C2×C3—21no—
170C2×C3—64no—
171C2×C3—64no—
172C2×C3—64no—
173C2×C3—62no—
174C2×C3#6378012yes—
175C2×C3C2×C22⋊#15711yescharacteristic
176C22—22no—
177C22C22⋊#108811yescharacteristic
178C22C22⋊#108811yescharacteristic
179C22C22×#108811yescharacteristic
180C4—22no—
181C4—92no—
182C4—92no—
183C4—182no—
184C3—62no—
185C3C2×#6378011yescharacteristic
186C2—21no—
187C2—21no—
188C2#1172078612yesdirect product factor
189C2C2×C22⋊#108811yescharacteristic
subgroupG/H
center, socleC22×C3C22⋊#157
commutatorC22×H(3,𝔽3)C22×C4
FrattiniC22×C3C22×#157
FittingC22×C4×H(3,𝔽3)C4
prime radicalC23×H(3,𝔽3)C2×C4
cosocleC2×#1084C23
hypercenterC22×C3×C4#157
hypocenterH(3,𝔽3)C2×#101
LCSG ▷ C22×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
3. #11720786C2direct (N × H)
65. C2×H(3,𝔽3)#101semidirect
95. H(3,𝔽3)C2×#101semidirect
188. C2#11720786direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#62292
C2×#886312
#117207862
C2×C3×#1019