Finite Group Databasebeta

← C32×#1665 #11768714

Subgroups — C32×#1665 #11768714

Order 1152.

564 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#492C211yesmaximal, characteristic, complemented
2C32×#474C211yesmaximal, characteristic, complemented
3C32×#459C211yesmaximal, characteristic
4C32×C8×D4C211yesmaximal, characteristic, complemented
5C32×#334C211yesmaximal, characteristic, complemented
6C32×#332C211yesmaximal, characteristic, complemented
7C32×#331C211yesmaximal, characteristic, complemented
8C3×#1665C314yesmaximal, direct product factor
9C2×C32×SD16—21no—
10C2×C32×SD16—21no—
11C2×C32×D8—21no—
12C22×C32×C8—21no—
13C32×C4⋊Q8C2211yescharacteristic
14C32×#127C2211yescharacteristic
15C32×C4×D4C2211yescharacteristic
16C32×#107—21no—
17C32×C4⋊C8C2211yescharacteristic
18C32×C4⋊C8C2211yescharacteristic
19C32×C4⋊C8C2211yescharacteristic
20C32×#102—21no—
21C32×#102—21no—
22C32×C22⋊C8—21no—
23C32×C4×C8C2211yescharacteristic
24C3×#492C2×C314yescomplemented
25C3×#474C2×C314yescomplemented
26C3×#459C2×C314yes—
27C3×C8×D4C2×C314yescomplemented
28C3×#334C2×C314yescomplemented
29C3×#332C2×C314yescomplemented
30C3×#331C2×C314yescomplemented
31C2×C32×Q8—21no—
32C2×C32×D4—21no—
33C2×C32×D4—21no—
34C2×C32×D4—21no—
35C2×C32×D4D411yescharacteristic
36C22×C32×C4—21no—
37C32×SD16—42no—
38C32×SD16—42no—
39C32×D8—42no—
40C2×C32×C8—21no—
41C2×C32×C8—21no—
42C2×C32×C8—21no—
43C2×C32×C8—42no—
44C2×C32×C8D411yescharacteristic
45C2×C32×C8D411yescharacteristic
46C32×C4⋊C4—21no—
47C32×C4⋊C4—21no—
48C32×C4⋊C4D411yescharacteristic
49C32×C22⋊C4—21no—
50C32×C42C2311yescharacteristic
51#1665C3211yescharacteristic, direct product factor, Sylow
52C2×C3×SD16—24no—
53C2×C3×SD16—24no—
54C2×C3×D8—24no—
55C22×C3×C8—24no—
56C3×C4⋊Q8C22×C314yes—
57C3×#127C22×C314yes—
58C3×C4×D4C22×C314yes—
59C3×#107—24no—
60C3×C4⋊C8C22×C314yes—
61C3×C4⋊C8C22×C314yes—
62C3×C4⋊C8C22×C314yes—
63C3×#102—24no—
64C3×#102—24no—
65C3×C22⋊C8—24no—
66C3×C4×C8C22×C314yes—
67C23×C32—21no—
68C23×C32—41no—
69C32×Q8—41no—
70C32×Q8—41no—
71C32×D4—21no—
72C32×D4—41no—
73C32×D4—41no—
74C32×D4—41no—
75C32×D4—41no—
76C32×D4—81no—
77C32×D4SD1612yescomplemented
78C2×C32×C4—21no—
79C2×C32×C4—41no—
80C2×C32×C4—41no—
81C2×C32×C4—41no—
82C2×C32×C4C2×D411yescharacteristic
83C2×C32×C4C2×D411yescharacteristic
84C2×C32×C4#4111yescharacteristic
85C32×C8—21no—
86C32×C8—41no—
87C32×C8—41no—
88C32×C8—41no—
89C32×C8D812yescomplemented
90#492C2×C3211yescharacteristic, complemented
91#474C2×C3211yescharacteristic, complemented
92#459C2×C3211yescharacteristic
93C8×D4C2×C3211yescharacteristic, complemented
94#334C2×C3211yescharacteristic, complemented
95#332C2×C3211yescharacteristic, complemented
96#331C2×C3211yescharacteristic, complemented
97C2×C3×Q8—24no—
98C2×C3×D4—24no—
99C2×C3×D4—24no—
100C2×C3×D4—24no—
101C2×C3×D4C3×D414yes—
102C22×C3×C4—24no—
103C3×SD16—48no—
104C3×SD16—48no—
105C3×D8—48no—
106C2×C3×C8—24no—
107C2×C3×C8—24no—
108C2×C3×C8—24no—
109C2×C3×C8—48no—
110C2×C3×C8C3×D414yes—
111C2×C3×C8C3×D414yes—
112C3×C4⋊C4—24no—
113C3×C4⋊C4—24no—
114C3×C4⋊C4C3×D414yes—
115C3×C22⋊C4—24no—
116C3×C42C23×C314yes—
117C22×C32—22no—
118C22×C32—41no—
119C22×C32—41no—
120C22×C32—81no—
121C22×C32—81no—
122C22×C32—81no—
123C22×C32#12111yescharacteristic
124C32×C4—21no—
125C32×C4—41no—
126C32×C4—81no—
127C32×C4C2×D811yescharacteristic
128C32×C4C2×SD1611yescharacteristic
129C32×C4#13511yescharacteristic
130C32×C4AGL(1,ℤ/8)11yescharacteristic
131C2×SD16—21no—
132C2×SD16—21no—
133C2×D8—21no—
134C22×C8—21no—
135C4⋊Q8C22×C3211yescharacteristic
136#127C22×C3211yescharacteristic
137C4×D4C22×C3211yescharacteristic
138#107—21no—
139C4⋊C8C22×C3211yescharacteristic
140C4⋊C8C22×C3211yescharacteristic
141C4⋊C8C22×C3211yescharacteristic
142#102—21no—
143#102—21no—
144C22⋊C8—21no—
145C4×C8C22×C3211yescharacteristic
146C23×C3—24no—
147C23×C3—44no—
148C3×Q8—44no—
149C3×Q8—44no—
150C3×D4—24no—
151C3×D4—44no—
152C3×D4—44no—
153C3×D4—44no—
154C3×D4—44no—
155C3×D4—84no—
156C3×D4C3×SD1618yescomplemented
157C2×C3×C4—24no—
158C2×C3×C4—44no—
159C2×C3×C4—44no—
160C2×C3×C4—44no—
161C2×C3×C4C2×C3×D414yes—
162C2×C3×C4C2×C3×D414yes—
163C2×C3×C4C3×#4114yes—
164C3×C8—24no—
165C3×C8—44no—
166C3×C8—44no—
167C3×C8—44no—
168C3×C8C3×D818yescomplemented
169C2×C32—42no—
170C2×C32—161no—
171C2×C32#45911yescharacteristic
172C2×C32#46511yescharacteristic
173C2×C32#47111yescharacteristic
174C2×Q8—21no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4C32×D411yescharacteristic
179C22×C4—21no—
180SD16—42no—
181SD16—42no—
182D8—42no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—42no—
187C2×C8C32×D411yescharacteristic
188C2×C8C32×D411yescharacteristic
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4C32×D411yescharacteristic
192C22⋊C4—21no—
193C42C23×C3211yescharacteristic
194C22×C3—28no—
195C22×C3—44no—
196C22×C3—44no—
197C22×C3—84no—
198C22×C3—84no—
199C22×C3—84no—
200C22×C3C3×#12114yes—
201C3×C4—24no—
202C3×C4—44no—
203C3×C4—84no—
204C3×C4C2×C3×D814yes—
205C3×C4C2×C3×SD1614yes—
206C3×C4C3×#13514yes—
207C3×C4C3×AGL(1,ℤ/8)14yes—
208C32#166511yescharacteristic, direct product factor, Sylow
209C23—21no—
210C23—41no—
211Q8—41no—
212Q8—41no—
213D4—21no—
214D4—41no—
215D4—41no—
216D4—41no—
217D4—41no—
218D4—81no—
219D4C32×SD1612yescomplemented
220C2×C4—21no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4C2×C32×D411yescharacteristic
225C2×C4C2×C32×D411yescharacteristic
226C2×C4C32×#4111yescharacteristic
227C8—21no—
228C8—41no—
229C8—41no—
230C8—41no—
231C8C32×D812yescomplemented
232C2×C3—48no—
233C2×C3—164no—
234C2×C3C3×#45914yes—
235C2×C3C3×#46514yes—
236C2×C3C3×#47114yes—
237C22—22no—
238C22—41no—
239C22—41no—
240C22—81no—
241C22—81no—
242C22—81no—
243C22C32×#12111yescharacteristic
244C4—21no—
245C4—41no—
246C4—81no—
247C4C2×C32×D811yescharacteristic
248C4C2×C32×SD1611yescharacteristic
249C4C32×#13511yescharacteristic
250C4C32×AGL(1,ℤ/8)11yescharacteristic
251C3C3×#166514yesdirect product factor
252C2—42no—
253C2—161no—
254C2C32×#45911yescharacteristic
255C2C32×#46511yescharacteristic
256C2C32×#47111yescharacteristic
subgroupG/H
center, socleC22×C32#121
commutator, Frattini, cosocleC42C23×C32
prime radicalC32×#459C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C32 ◁ C32×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C32×#492C2semidirect
2. C32×#474C2semidirect
4. C32×C8×D4C2semidirect
5. C32×#334C2semidirect
6. C32×#332C2semidirect
7. C32×#331C2semidirect
8. C3×#1665C3direct (N × H)
24. C3×#492C2×C3semidirect
25. C3×#474C2×C3semidirect
27. C3×C8×D4C2×C3semidirect
28. C3×#334C2×C3semidirect
29. C3×#332C2×C3semidirect
30. C3×#331C2×C3semidirect
51. #1665C32direct (N × H)
77. C32×D4SD16semidirect
89. C32×C8D8semidirect
90. #492C2×C32semidirect
91. #474C2×C32semidirect
93. C8×D4C2×C32semidirect
94. #334C2×C32semidirect
95. #332C2×C32semidirect
96. #331C2×C32semidirect
156. C3×D4C3×SD16semidirect
168. C3×C8C3×D8semidirect
208. C32#1665direct (N × H)
219. D4C32×SD16semidirect
231. C8C32×D8semidirect
251. C3C3×#1665direct (N × H)

Maximal subgroups

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

subgroupindex
C32×#4922
C32×#4742
C32×#4592
C32×C8×D42
C32×#3342
C32×#3322
C32×#3312
C3×#16653