Finite Group Databasebeta

← C2×#11720784 #13147382

Subgroups — C2×#11720784 #13147382

Order 1728.

775 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 775 conjugacy classes of subgroups — too many to draw. Only its 95 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×C4×#272C211yesmaximal, characteristic
2C22×#6188C212yesmaximal, complemented
3#11720784C214yesmaximal, direct product factor
4C23×#272C411yescharacteristic
5C22×C4×H(3,𝔽3)C411yescharacteristic
6C22×#1084C2211yescharacteristic
7C2×C4×#272—24no—
8C2×C4×#272C412yes—
9C2×C4×#272C2212yes—
10C2×#6188—24no—
11C2×#6188C2214yescomplemented
12C22×C3×C4×D3—62no—
13C23×H(3,𝔽3)C2×C411yescharacteristic
14C22×#272—22no—
15C22×#272—22no—
16C22×#272C814yescomplemented
17C22×#272C2×C412yes—
18C2×C4×H(3,𝔽3)—22no—
19C2×C4×H(3,𝔽3)C2×C412yes—
20C2×#1084C2×C411yescharacteristic
21C2×#1084D414yes—
22C2×#1084C2311yescharacteristic
23C4×#272—24no—
24C4×#272—24no—
25#6188—24no—
26C2×C3×C22⋊C8—91nomaximal
27C23×C3×D3—62no—
28C22×C32×C4—22no—
29C22×C3×Dic3—62no—
30C2×C3×C4×D3—64no—
31C2×C3×C4×D3—64no—
32C2×C3×C4×D3—616no—
33C22×H(3,𝔽3)C22⋊C412yes—
34C22×H(3,𝔽3)C2×C812yes—
35C22×H(3,𝔽3)M1612yes—
36C22×H(3,𝔽3)C22×C411yescharacteristic
37C2×#272—28no—
38C2×#272—22no—
39C2×#272—22no—
40C2×#272C2×C814yescomplemented
41C4×H(3,𝔽3)—22no—
42#1084C22⋊C412yes—
43#1084C2×D412yes—
44C23×C3×C4—91no—
45C22×C4×D3—62no—
46C22×C3×C8—92no—
47C3×C22⋊C8—94no—
48C23×C32—22no—
49C22×C3×D3—68no—
50C22×C3×D3—64no—
51C22×C3×D3—68no—
52C22×C3×D3—68no—
53C2×C32×C4—28no—
54C2×C32×C4—24no—
55C2×C3×Dic3—62no—
56C2×C3×Dic3—68no—
57C2×C3×Dic3—62no—
58C3×C4×D3—616no—
59C3×C4×D3—616no—
60C2×C22⋊C8—91noSylow
61C2×H(3,𝔽3)C22⋊C814yescomplemented
62C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
63C2×H(3,𝔽3)C22×C811yescharacteristic
64C2×H(3,𝔽3)C2×M1611yescharacteristic
65#272—24no—
66C24×C3—91no—
67C23×D3—62no—
68C22×C3×C4—62no—
69C22×C3×C4—91no—
70C22×C3×C4—92no—
71C22×C3×C4—92no—
72C22×C3×C4—184no—
73C22×C3×C4#15711yescharacteristic
74C22×Dic3—62no—
75C2×C4×D3—64no—
76C2×C4×D3—64no—
77C2×C4×D3—616no—
78C2×C3×C8—94no—
79C2×C3×C8—184no—
80C22×C32—22no—
81C22×C32—24no—
82C22×C32—24no—
83C22×C32—24no—
84C2×C3×D3—68no—
85C2×C3×D3—68no—
86C2×C3×D3—68no—
87C2×C3×D3—632no—
88C32×C4—28no—
89C3×Dic3—64no—
90C3×Dic3—64no—
91C23×C4—91no—
92C22×C8—92no—
93C22⋊C8—94no—
94H(3,𝔽3)C2×C22⋊C811yescharacteristic, complemented, Sylow
95C23×C3—62no—
96C23×C3—92no—
97C23×C3—94no—
98C23×C3—182no—
99C23×C3—182no—
100C23×C3C2×#15711yescharacteristic
101C22×D3—64no—
102C22×D3—68no—
103C22×D3—68no—
104C22×D3—68no—
105C2×C3×C4—22no—
106C2×C3×C4—68no—
107C2×C3×C4—64no—
108C2×C3×C4—91no—
109C2×C3×C4—91no—
110C2×C3×C4—94no—
111C2×C3×C4—184no—
112C2×C3×C4—184no—
113C2×C3×C4C2×#15712yes—
114C2×Dic3—68no—
115C2×Dic3—62no—
116C2×Dic3—62no—
117C4×D3—616no—
118C4×D3—616no—
119C3×C8—184no—
120C2×C32—22no—
121C2×C32—22no—
122C2×C32—28no—
123C2×C32—22no—
124C3×D3—616no—
125C24—91no—
126C22×C4—91no—
127C22×C4—92no—
128C22×C4—92no—
129C22×C4—184no—
130C22×C4#108811yescharacteristic
131C2×C8—94no—
132C2×C8—184no—
133C22×C3—64no—
134C22×C3—64no—
135C22×C3—64no—
136C22×C3—62no—
137C22×C3—94no—
138C22×C3—182no—
139C22×C3—188no—
140C22×C3—182no—
141C22×C3#378712yes—
142C22×C3#378812yes—
143C22×C3C22⋊#15712yes—
144C22×C3C22×#15711yescharacteristic
145C2×D3—68no—
146C2×D3—68no—
147C2×D3—632no—
148C2×D3—68no—
149C3×C4—22no—
150C3×C4—68no—
151C3×C4—92no—
152C3×C4—92no—
153Dic3—64no—
154Dic3—64no—
155C32—22no—
156C23—94no—
157C23—92no—
158C23—182no—
159C23—182no—
160C23C2×#108811yescharacteristic
161C2×C4—22no—
162C2×C4—94no—
163C2×C4—91no—
164C2×C4—91no—
165C2×C4—184no—
166C2×C4—184no—
167C2×C4C2×#108812yes—
168C8—184no—
169C2×C3—68no—
170C2×C3—62no—
171C2×C3—62no—
172C2×C3—62no—
173C2×C3—184no—
174C2×C3#6377814yes—
175C2×C3C2×#378711yescharacteristic
176C2×C3C2×#378811yescharacteristic
177C2×C3C2×C22⋊#15711yescharacteristic
178D3—616no—
179C22—94no—
180C22—188no—
181C22—182no—
182C22—182no—
183C22#8862612yes—
184C22#8862712yes—
185C22C22⋊#108812yes—
186C22C22×#108811yescharacteristic
187C4—22no—
188C4—92no—
189C4—92no—
190C3—62no—
191C3C2×#6377811yescharacteristic
192C2—184no—
193C2#1172078414yesdirect product factor
194C2C2×#8862611yescharacteristic
195C2C2×#8862711yescharacteristic
196C2C2×C22⋊#108811yescharacteristic
subgroupG/H
center, socleC23×C3C2×#157
commutatorC2×H(3,𝔽3)C22×C8
FrattiniC22×C3C22×#157
FittingC22×C4×H(3,𝔽3)C4
prime radicalC23×#272C4
cosocleC2×#1084C23
hypercenterC22×C3×C4#157
hypocenterH(3,𝔽3)C2×C22⋊C8
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
2. C22×#6188C2semidirect
3. #11720784C2direct (N × H)
11. C2×#6188C22semidirect
16. C22×#272C8semidirect
40. C2×#272C2×C8semidirect
61. C2×H(3,𝔽3)C22⋊C8semidirect
94. H(3,𝔽3)C2×C22⋊C8semidirect
193. C2#11720784direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×#2722
C22×#61882
#117207842
C2×C3×C22⋊C89