Finite Group Databasebeta

← C4⋊C4×#1088 #13147343

Subgroups — C4⋊C4×#1088 #13147343

Order 1728.

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

normal subgroups only

This group has 475 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
1C2×C4⋊#1088C211yesmaximal, characteristic
2C2×C4×#1088C212yesmaximal
3C2×C4×#1088C211yesmaximal, characteristic
4C4⋊C4×#272C211yesmaximal, characteristic
5#11720787C212yesmaximal
6C22×#1088C2211yescharacteristic
7C22×#1088C2212yes—
8C22×#1088C2211yescharacteristic
9C2×C4×#272C2212yes—
10C2×C4×#272C2211yescharacteristic
11C4⋊#1088C414yescomplemented
12C4×#1088—24no—
13C4×#1088C414yescomplemented
14C4⋊C4×H(3,𝔽3)C411yescharacteristic, complemented
15C4⋊#1084C411yescharacteristic, complemented
16#88540C412yescomplemented
17C3×D3×C4⋊C4—62no—
18C22×#272C2311yescharacteristic
19C2×#1088—22no—
20C2×#1088—21no—
21C2×#1088—21no—
22C2×#1088—22no—
23C2×#1088C2×C412yes—
24C2×#1088C2×C414yes—
25C2×#1088C2×C412yes—
26C2×#1088D412yes—
27C2×#1088Q812yes—
28C2×C4×H(3,𝔽3)C2×C412yes—
29C2×C4×H(3,𝔽3)C2×C411yescharacteristic
30C2×#1084C2×C412yes—
31C2×#1084C2×C411yescharacteristic
32C4×#272—22no—
33C4×#272—22no—
34C4×#272C2×C412yes—
35C4×#272C2×C412yes—
36C3×C4×C4⋊C4—91nomaximal
37C2×C3×C4×D3—62no—
38C2×C3×C4×D3—64no—
39C32×C4⋊C4—22no—
40C3×C4⋊Dic3—62no—
41C3×Dic3⋊C4—64no—
42C22×H(3,𝔽3)C22×C411yescharacteristic
43C2×#272C22×C411yescharacteristic
44C2×#272C22×C411yescharacteristic
45C2×#272C2×D411yescharacteristic
46C2×#272C2×Q811yescharacteristic
47C2×#272#4111yescharacteristic
48C2×#272#4111yescharacteristic
49#1088—22no—
50#1088—24no—
51#1088C4⋊C414yesdirect product factor
52C4×H(3,𝔽3)—22no—
53C4×H(3,𝔽3)C4212yescomplemented
54#1084—22no—
55#1084C4212yescomplemented
56C2×C3×C4⋊C4—91no—
57C2×C3×C4⋊C4—91no—
58C2×C3×C42—91no—
59C2×C3×C42—92no—
60D3×C4⋊C4—62no—
61C3×#95—92no—
62C22×C3×D3—62no—
63C2×C32×C4—22no—
64C2×C32×C4—24no—
65C2×C3×Dic3—62no—
66C2×C3×Dic3—64no—
67C3×C4×D3—64no—
68C3×C4×D3—64no—
69C3×C4×D3—124no—
70C3×C4×D3—124no—
71C4×C4⋊C4—91noSylow
72C2×H(3,𝔽3)C2×C4211yescharacteristic
73C2×H(3,𝔽3)C4×D411yescharacteristic
74C2×H(3,𝔽3)C4×Q811yescharacteristic
75#272C2×C4⋊C411yescharacteristic
76#272#11711yescharacteristic
77#272C4×D411yescharacteristic
78#272C4×Q811yescharacteristic
79C22×C3×C4—91no—
80C22×C3×C4—91no—
81C22×C3×C4—91no—
82C22×C3×C4—92no—
83C22×C3×C4—92no—
84C2×C4×D3—62no—
85C2×C4×D3—64no—
86C3×C4⋊C4—62no—
87C3×C4⋊C4—94no—
88C3×C4⋊C4—92no—
89C3×C4⋊C4—91no—
90C3×C4⋊C4#15711yescharacteristic
91C3×C42—94no—
92C3×C42—184no—
93C4⋊Dic3—62no—
94Dic3⋊C4—64no—
95C22×C32—22no—
96C2×C3×D3—62no—
97C2×C3×D3—62no—
98C2×C3×D3—62no—
99C2×C3×D3—62no—
100C2×C3×D3—62no—
101C2×C3×D3—62no—
102C32×C4—24no—
103C32×C4—44no—
104C3×Dic3—64no—
105C3×Dic3—124no—
106C2×C4⋊C4—91no—
107C2×C4⋊C4—91no—
108C2×C42—91no—
109C2×C42—92no—
110#95—92no—
111H(3,𝔽3)C4×C4⋊C411yescharacteristic, complemented, Sylow
112C23×C3—91no—
113C22×D3—62no—
114C2×C3×C4—62no—
115C2×C3×C4—64no—
116C2×C3×C4—92no—
117C2×C3×C4—92no—
118C2×C3×C4—92no—
119C2×C3×C4—91no—
120C2×C3×C4—92no—
121C2×C3×C4—92no—
122C2×C3×C4—92no—
123C2×C3×C4—94no—
124C2×C3×C4—92no—
125C2×C3×C4—182no—
126C2×C3×C4—182no—
127C2×C3×C4—182no—
128C2×C3×C4—181no—
129C2×C3×C4—181no—
130C2×C3×C4—182no—
131C2×C3×C4C2×#15712yes—
132C2×C3×C4C2×#15711yescharacteristic
133C2×Dic3—62no—
134C2×Dic3—64no—
135C4×D3—64no—
136C4×D3—64no—
137C4×D3—124no—
138C4×D3—124no—
139C2×C32—22no—
140C2×C32—22no—
141C2×C32—22no—
142C3×D3—62no—
143C3×D3—62no—
144C3×D3—62no—
145C3×D3—62no—
146C22×C4—91no—
147C22×C4—91no—
148C22×C4—91no—
149C22×C4—92no—
150C22×C4—92no—
151C4⋊C4—94no—
152C4⋊C4—92no—
153C4⋊C4—91no—
154C4⋊C4#108811yescharacteristic, direct product factor
155C42—94no—
156C42—184no—
157C22×C3—62no—
158C22×C3—91no—
159C22×C3—91no—
160C22×C3—91no—
161C22×C3—91no—
162C22×C3—91no—
163C22×C3—91no—
164C22×C3C22×#15711yescharacteristic
165C2×D3—62no—
166C2×D3—62no—
167C2×D3—62no—
168C2×D3—62no—
169C2×D3—62no—
170C2×D3—62no—
171C3×C4—22no—
172C3×C4—64no—
173C3×C4—94no—
174C3×C4—92no—
175C3×C4—124no—
176C3×C4—184no—
177C3×C4—182no—
178C3×C4—182no—
179C3×C4C4×#15712yes—
180Dic3—64no—
181Dic3—124no—
182C32—22no—
183C23—91no—
184C2×C4—92no—
185C2×C4—92no—
186C2×C4—92no—
187C2×C4—91no—
188C2×C4—92no—
189C2×C4—92no—
190C2×C4—92no—
191C2×C4—94no—
192C2×C4—92no—
193C2×C4—182no—
194C2×C4—182no—
195C2×C4—182no—
196C2×C4—181no—
197C2×C4—181no—
198C2×C4—182no—
199C2×C4C2×#108812yes—
200C2×C4C2×#108811yescharacteristic
201C2×C3—62no—
202C2×C3—62no—
203C2×C3—62no—
204C2×C3—91no—
205C2×C3—91no—
206C2×C3—91no—
207C2×C3—91no—
208C2×C3C2×C4×#15711yescharacteristic
209C2×C3D4×#15711yescharacteristic
210C2×C3Q8×#15711yescharacteristic
211D3—62no—
212D3—62no—
213D3—62no—
214D3—62no—
215C22—91no—
216C22—91no—
217C22—91no—
218C22—91no—
219C22—91no—
220C22—91no—
221C22C22×#108811yescharacteristic
222C4—22no—
223C4—94no—
224C4—92no—
225C4—184no—
226C4—182no—
227C4—182no—
228C4C4×#108812yescomplemented
229C3—62no—
230C3C4⋊C4×#15711yescharacteristic
231C2—91no—
232C2—91no—
233C2—91no—
234C2—91no—
235C2C2×C4×#108811yescharacteristic
236C2D4×#108811yescharacteristic
237C2Q8×#108811yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C22×#157
commutatorC2×H(3,𝔽3)C2×C42
FittingC4⋊C4×H(3,𝔽3)C4
prime radical, cosocleC22×#272C23
hypercenterC3×C4⋊C4#157
hypocenterH(3,𝔽3)C4×C4⋊C4
LCSG ▷ C2×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
11. C4⋊#1088C4semidirect
13. C4×#1088C4semidirect
14. C4⋊C4×H(3,𝔽3)C4semidirect
15. C4⋊#1084C4semidirect
16. #88540C4semidirect
51. #1088C4⋊C4direct (N × H)
53. C4×H(3,𝔽3)C42semidirect
55. #1084C42semidirect
111. H(3,𝔽3)C4×C4⋊C4semidirect
154. C4⋊C4#1088direct (N × H)
228. C4C4×#1088semidirect

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
C4⋊C4×#2722
#117207872
C3×C4×C4⋊C49