Finite Group Databasebeta

← C4×C4⋊#1088 #13147320

Subgroups — C4×C4⋊#1088 #13147320

Order 1728.

507 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 507 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C2×C4⋊#1088C212yesmaximal
2C2×C4×#1088C212yesmaximal
3C42×#272C211yesmaximal, characteristic
4#11720787C212yesmaximal
5C22×#1088C2214yes—
6C2×C4×#272C2211yescharacteristic
7C2×C4×#272C2211yescharacteristic
8C2×C4×#272C2211yescharacteristic
9C4⋊#1088C418yesdirect product factor
10C4×#1088—24no—
11C42×H(3,𝔽3)C411yescharacteristic, complemented
12C4×#1084C411yescharacteristic, complemented
13C4×#1084C411yescharacteristic, complemented
14C4×#1084C411yescharacteristic, complemented
15C3×C42×D3—62no—
16C22×#272C2311yescharacteristic
17C2×#1088—24no—
18C2×#1088—24no—
19C2×#1088C2×C418yes—
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
26C4×#272—22no—
27C4×#272C2×C414yes—
28C4×#272D412yes—
29C4×#272Q812yes—
30C3×C4×C4⋊C4—91nomaximal
31C2×C3×C4×D3—62no—
32C2×C3×C4×D3—62no—
33C2×C3×C4×D3—62no—
34C32×C42—22no—
35C3×C4×Dic3—62no—
36C3×C4×Dic3—62no—
37C3×C4×Dic3—62no—
38C22×H(3,𝔽3)C22×C411yescharacteristic
39C2×#272C22×C412yes—
40C2×#272C2×D411yescharacteristic
41C2×#272C2×Q811yescharacteristic
42C2×#272#4112yes—
43#1088—28no—
44C4×H(3,𝔽3)—21no—
45C4×H(3,𝔽3)C4212yescomplemented
46C4×H(3,𝔽3)C4⋊C412yescomplemented
47#1084—21no—
48#1084C4212yescomplemented
49#1084C4⋊C412yescomplemented
50C2×C3×C4⋊C4—92no—
51C2×C3×C42—91no—
52C2×C3×C42—92no—
53C42×D3—62no—
54C3×#95—92no—
55C22×C3×D3—62no—
56C2×C32×C4—22no—
57C2×C32×C4—22no—
58C2×C32×C4—22no—
59C2×C3×Dic3—62no—
60C2×C3×Dic3—62no—
61C2×C3×Dic3—62no—
62C3×C4×D3—68no—
63C3×C4×D3—64no—
64C3×C4×D3—64no—
65C3×C4×D3—68no—
66C4×C4⋊C4—91noSylow
67C2×H(3,𝔽3)C2×C4211yescharacteristic
68C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
69C2×H(3,𝔽3)#11711yescharacteristic
70#272C4×D412yes—
71#272C4×Q812yes—
72C22×C3×C4—91no—
73C22×C3×C4—91no—
74C22×C3×C4—94no—
75C22×C3×C4—91no—
76C2×C4×D3—62no—
77C2×C4×D3—62no—
78C2×C4×D3—62no—
79C3×C4⋊C4—98no—
80C3×C42—62no—
81C3×C42—91no—
82C3×C42—91no—
83C3×C42—91no—
84C3×C42—184no—
85C3×C42#15711yescharacteristic
86C4×Dic3—62no—
87C4×Dic3—62no—
88C4×Dic3—62no—
89C22×C32—22no—
90C2×C3×D3—64no—
91C2×C3×D3—62no—
92C2×C3×D3—62no—
93C2×C3×D3—64no—
94C32×C4—24no—
95C32×C4—24no—
96C32×C4—24no—
97C3×Dic3—64no—
98C3×Dic3—64no—
99C3×Dic3—64no—
100C2×C4⋊C4—92no—
101C2×C42—91no—
102C2×C42—92no—
103#95—92no—
104H(3,𝔽3)C4×C4⋊C411yescharacteristic, complemented, Sylow
105C23×C3—91no—
106C22×D3—62no—
107C2×C3×C4—62no—
108C2×C3×C4—62no—
109C2×C3×C4—62no—
110C2×C3×C4—91no—
111C2×C3×C4—91no—
112C2×C3×C4—94no—
113C2×C3×C4—92no—
114C2×C3×C4—92no—
115C2×C3×C4—98no—
116C2×C3×C4—91no—
117C2×C3×C4—182no—
118C2×C3×C4—184no—
119C2×C3×C4—184no—
120C2×C3×C4C2×#15711yescharacteristic
121C2×C3×C4C2×#15711yescharacteristic
122C2×C3×C4C2×#15711yescharacteristic
123C2×Dic3—62no—
124C2×Dic3—62no—
125C2×Dic3—62no—
126C4×D3—68no—
127C4×D3—64no—
128C4×D3—64no—
129C4×D3—68no—
130C2×C32—22no—
131C2×C32—22no—
132C2×C32—22no—
133C3×D3—64no—
134C3×D3—64no—
135C22×C4—91no—
136C22×C4—91no—
137C22×C4—94no—
138C22×C4—91no—
139C4⋊C4—98no—
140C42—91no—
141C42—91no—
142C42—91no—
143C42—184no—
144C42#108811yescharacteristic, complemented
145C22×C3—62no—
146C22×C3—92no—
147C22×C3—91no—
148C22×C3—91no—
149C22×C3—92no—
150C22×C3C22×#15711yescharacteristic
151C2×D3—64no—
152C2×D3—62no—
153C2×D3—62no—
154C2×D3—64no—
155C3×C4—21no—
156C3×C4—64no—
157C3×C4—64no—
158C3×C4—64no—
159C3×C4—92no—
160C3×C4—92no—
161C3×C4—188no—
162C3×C4—181no—
163C3×C4C4×#15712yes—
164C3×C4C4⋊#15712yes—
165Dic3—64no—
166Dic3—64no—
167Dic3—64no—
168C32—22no—
169C23—91no—
170C2×C4—91no—
171C2×C4—91no—
172C2×C4—94no—
173C2×C4—92no—
174C2×C4—92no—
175C2×C4—98no—
176C2×C4—91no—
177C2×C4—182no—
178C2×C4—184no—
179C2×C4—184no—
180C2×C4C2×#108811yescharacteristic
181C2×C4C2×#108811yescharacteristic
182C2×C4C2×#108811yescharacteristic
183C2×C3—62no—
184C2×C3—62no—
185C2×C3—62no—
186C2×C3—92no—
187C2×C3—92no—
188C2×C3C2×C4×#15711yescharacteristic
189C2×C3C2×C4⋊#15711yescharacteristic
190C2×C3#6428611yescharacteristic
191D3—64no—
192D3—64no—
193C22—92no—
194C22—91no—
195C22—91no—
196C22—92no—
197C22C22×#108811yescharacteristic
198C4—21no—
199C4—92no—
200C4—92no—
201C4—188no—
202C4—181no—
203C4C4×#108812yescomplemented
204C4C4⋊#108812yesdirect product factor
205C3—62no—
206C3C4×C4⋊#15711yescharacteristic
207C2—92no—
208C2—92no—
209C2C2×C4×#108811yescharacteristic
210C2C2×C4⋊#108811yescharacteristic
211C2#1172244811yescharacteristic
subgroupG/H
centerC2×C3×C4C2×#157
commutatorC2×H(3,𝔽3)C2×C42
Frattini, socleC22×C3C22×#157
FittingC42×H(3,𝔽3)C4
prime radical, cosocleC22×#272C23
hypercenterC3×C42#157
hypocenterH(3,𝔽3)C4×C4⋊C4
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3×C4 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
9. C4⋊#1088C4direct (N × H)
11. C42×H(3,𝔽3)C4semidirect
12. C4×#1084C4semidirect
13. C4×#1084C4semidirect
14. C4×#1084C4semidirect
45. C4×H(3,𝔽3)C42semidirect
46. C4×H(3,𝔽3)C4⋊C4semidirect
48. #1084C42semidirect
49. #1084C4⋊C4semidirect
104. H(3,𝔽3)C4×C4⋊C4semidirect
144. C42#1088semidirect
203. C4C4×#1088semidirect
204. C4C4⋊#1088direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4⋊#10882
C2×C4×#10882
C42×#2722
#117207872
C3×C4×C4⋊C49