Finite Group Databasebeta

← C2×C22⋊#6188 #13147430

Subgroups — C2×C22⋊#6188 #13147430

Order 1728.

775 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 775 conjugacy classes of subgroups — too many to draw. Only its 111 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
1C23×#1084C211yesmaximal, characteristic
2C22×#6188C212yesmaximal, complemented
3C22⋊#6188C214yesmaximal, direct product factor
4C24×H(3,𝔽3)C411yescharacteristic
5C22×#1084—24no—
6C22×#1084C411yescharacteristic
7C22×#1084C412yes—
8C22×#1084C2212yes—
9C22×#1084C2211yescharacteristic
10C2×#6188—24no—
11C2×#6188C2214yescomplemented
12C23×C3×Dic3—62no—
13C23×H(3,𝔽3)—22no—
14C23×H(3,𝔽3)—22no—
15C23×H(3,𝔽3)C814yescomplemented
16C23×H(3,𝔽3)C2×C412yes—
17C23×H(3,𝔽3)C2×C411yescharacteristic
18C2×#1084—22no—
19C2×#1084—24no—
20C2×#1084—24no—
21C2×#1084C2×C412yes—
22C2×#1084C2×C411yescharacteristic
23C2×#1084D414yes—
24C2×#1084C2311yescharacteristic
25#6188—24no—
26C2×C3×C22⋊C8—91nomaximal
27C24×C32—22no—
28C22×C3×Dic3—64no—
29C22×C3×Dic3—62no—
30C22×C3×Dic3—62no—
31C22×C3×Dic3—64no—
32C22×C3×Dic3—616no—
33C22×H(3,𝔽3)—28no—
34C22×H(3,𝔽3)—22no—
35C22×H(3,𝔽3)—22no—
36C22×H(3,𝔽3)C22⋊C412yes—
37C22×H(3,𝔽3)C2×C812yes—
38C22×H(3,𝔽3)C2×C814yescomplemented
39C22×H(3,𝔽3)M1612yes—
40C22×H(3,𝔽3)C22×C411yescharacteristic
41#1084—22no—
42#1084C22⋊C412yes—
43#1084C2×D412yes—
44C23×C3×C4—91no—
45C23×Dic3—62no—
46C22×C3×C8—92no—
47C3×C22⋊C8—94no—
48C23×C32—28no—
49C23×C32—28no—
50C23×C32—28no—
51C23×C32—24no—
52C23×C32—22no—
53C2×C3×Dic3—62no—
54C2×C3×Dic3—64no—
55C2×C3×Dic3—68no—
56C2×C3×Dic3—62no—
57C2×C3×Dic3—616no—
58C2×C3×Dic3—616no—
59C2×C3×Dic3—68no—
60C2×C22⋊C8—91noSylow
61C2×H(3,𝔽3)—24no—
62C2×H(3,𝔽3)C22⋊C814yescomplemented
63C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
64C2×H(3,𝔽3)C22×C811yescharacteristic
65C2×H(3,𝔽3)C2×M1611yescharacteristic
66C24×C3—62no—
67C24×C3#15711yescharacteristic
68C22×C3×C4—91no—
69C22×C3×C4—92no—
70C22×C3×C4—92no—
71C22×C3×C4—91no—
72C22×C3×C4—184no—
73C22×Dic3—62no—
74C22×Dic3—64no—
75C22×Dic3—64no—
76C22×Dic3—62no—
77C22×Dic3—616no—
78C2×C3×C8—94no—
79C2×C3×C8—184no—
80C22×C32—28no—
81C22×C32—28no—
82C22×C32—232no—
83C22×C32—22no—
84C22×C32—24no—
85C22×C32—28no—
86C22×C32—24no—
87C22×C32—24no—
88C3×Dic3—64no—
89C3×Dic3—64no—
90C3×Dic3—68no—
91C23×C4—91no—
92C22×C8—92no—
93C22⋊C8—94no—
94H(3,𝔽3)C2×C22⋊C811yescharacteristic, complemented, Sylow
95C23×C3—22no—
96C23×C3—22no—
97C23×C3—68no—
98C23×C3—68no—
99C23×C3—68no—
100C23×C3—64no—
101C23×C3—62no—
102C23×C3#62214yes—
103C23×C3C2×#15712yes—
104C23×C3C2×#15711yescharacteristic
105C2×C3×C4—91no—
106C2×C3×C4—91no—
107C2×C3×C4—94no—
108C2×C3×C4—92no—
109C2×C3×C4—184no—
110C2×C3×C4—184no—
111C2×C3×C4—182no—
112C2×Dic3—68no—
113C2×Dic3—62no—
114C2×Dic3—62no—
115C2×Dic3—64no—
116C2×Dic3—616no—
117C2×Dic3—616no—
118C2×Dic3—68no—
119C3×C8—184no—
120C2×C32—216no—
121C2×C32—22no—
122C2×C32—22no—
123C2×C32—22no—
124C2×C32—28no—
125C24#108811yescharacteristic
126C22×C4—91no—
127C22×C4—92no—
128C22×C4—92no—
129C22×C4—91no—
130C22×C4—184no—
131C2×C8—94no—
132C2×C8—184no—
133C22×C3—22no—
134C22×C3—22no—
135C22×C3—28no—
136C22×C3—68no—
137C22×C3—632no—
138C22×C3—68no—
139C22×C3—64no—
140C22×C3—68no—
141C22×C3—64no—
142C22×C3—64no—
143C22×C3—62no—
144C22×C3C2×#62212yes—
145C22×C3C2×#62214yes—
146C22×C3#379212yes—
147C22×C3C22⋊#15712yes—
148C22×C3C22×#15711yescharacteristic
149C3×C4—92no—
150C3×C4—92no—
151C3×C4—182no—
152Dic3—64no—
153Dic3—64no—
154Dic3—68no—
155C32—22no—
156C23—22no—
157C23—22no—
158C23#618814yescomplemented
159C23C2×#108812yes—
160C23C2×#108811yescharacteristic
161C2×C4—94no—
162C2×C4—91no—
163C2×C4—91no—
164C2×C4—92no—
165C2×C4—184no—
166C2×C4—184no—
167C2×C4—182no—
168C8—184no—
169C2×C3—24no—
170C2×C3—616no—
171C2×C3—68no—
172C2×C3—62no—
173C2×C3—62no—
174C2×C3—62no—
175C2×C3C22⋊#62214yes—
176C2×C3C22×#62211yescharacteristic
177C2×C3C2×#379211yescharacteristic
178C2×C3C2×C22⋊#15711yescharacteristic
179C22—28no—
180C22—22no—
181C22—22no—
182C22C2×#618812yes—
183C22C2×#618814yescomplemented
184C22#8863112yes—
185C22C22⋊#108812yes—
186C22C22×#108811yescharacteristic
187C4—92no—
188C4—92no—
189C4—182no—
190C3—62no—
191C3C2×C22⋊#62211yescharacteristic
192C2—24no—
193C2C22⋊#618814yesdirect product factor
194C2C22×#618811yescharacteristic
195C2C2×#8863111yescharacteristic
196C2C2×C22⋊#108811yescharacteristic
subgroupG/H
center, socleC23×C3C2×#157
commutatorC2×H(3,𝔽3)C22×C8
FrattiniC22×C3C22×#157
Fitting, prime radicalC24×H(3,𝔽3)C4
cosocleC2×#1084C23
hypercenterC24×C3#157
hypocenterH(3,𝔽3)C2×C22⋊C8
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23×C3 ◁ C24×C3

Complements

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

NH ≅ G/Ntype
2. C22×#6188C2semidirect
3. C22⋊#6188C2direct (N × H)
11. C2×#6188C22semidirect
15. C23×H(3,𝔽3)C8semidirect
38. C22×H(3,𝔽3)C2×C8semidirect
62. C2×H(3,𝔽3)C22⋊C8semidirect
94. H(3,𝔽3)C2×C22⋊C8semidirect
158. C23#6188semidirect
183. C22C2×#6188semidirect
193. C2C22⋊#6188direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#10842
C22×#61882
C22⋊#61882
C2×C3×C22⋊C89