Finite Group Databasebeta

← C32×#1351 #11768400

Subgroups — C32×#1351 #11768400

Order 1152.

486 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 120 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
1C32×#393C211yesmaximal, characteristic
2C32×#342C211yesmaximal, characteristic
3C32×#323C211yesmaximal, characteristic
4C3×#1351C314yesmaximal, direct product factor
5C22×C32×Q8C411yescharacteristic, complemented
6C2×C32×C22⋊C4—21no—
7C2×C32×C22⋊C4—21no—
8C2×C32×C22⋊C4C2211yescharacteristic
9C32×C22⋊C8—21no—
10C32×#95—21no—
11C32×#95C411yescharacteristic, complemented
12C3×#393C2×C314yes—
13C3×#342C2×C314yes—
14C3×#323C2×C314yes—
15C24×C32D411yescharacteristic
16C2×C32×Q8—42no—
17C2×C32×Q8—41no—
18C22×C32×C4—21no—
19C22×C32×C4—21no—
20C22×C32×C4—41no—
21C22×C32×C4C2×C411yescharacteristic
22C22×C32×C4D411yescharacteristic
23C2×C32×C8—41no—
24C32×C22⋊C4—22no—
25C32×C22⋊C4—41no—
26C32×C22⋊C4—42no—
27C32×C22⋊C4—41no—
28#1351C3211yescharacteristic, direct product factor, Sylow
29C22×C3×Q8C3×C414yescomplemented
30C2×C3×C22⋊C4—24no—
31C2×C3×C22⋊C4—24no—
32C2×C3×C22⋊C4C22×C314yes—
33C3×C22⋊C8—24no—
34C3×#95—24no—
35C3×#95C3×C414yescomplemented
36C23×C32—21no—
37C23×C32—21no—
38C23×C32—41no—
39C23×C32—41no—
40C23×C32C22⋊C411yescharacteristic
41C23×C32SD1611yescharacteristic
42C23×C32Q1611yescharacteristic
43C32×Q8—42no—
44C32×Q8—42no—
45C2×C32×C4—22no—
46C2×C32×C4—21no—
47C2×C32×C4—21no—
48C2×C32×C4—41no—
49C2×C32×C4—41no—
50C2×C32×C4—41no—
51C2×C32×C4—41no—
52C2×C32×C4—42no—
53C2×C32×C4—41no—
54C2×C32×C4—41no—
55C2×C32×C4—41no—
56C2×C32×C4—81no—
57C2×C32×C4—81no—
58C32×C8—81no—
59#393C2×C3211yescharacteristic
60#342C2×C3211yescharacteristic
61#323C2×C3211yescharacteristic
62C24×C3C3×D414yes—
63C2×C3×Q8—48no—
64C2×C3×Q8—44no—
65C22×C3×C4—24no—
66C22×C3×C4—24no—
67C22×C3×C4—44no—
68C22×C3×C4C2×C3×C414yes—
69C22×C3×C4C3×D414yes—
70C2×C3×C8—44no—
71C3×C22⋊C4—28no—
72C3×C22⋊C4—44no—
73C3×C22⋊C4—48no—
74C3×C22⋊C4—44no—
75C22×C32—21no—
76C22×C32—21no—
77C22×C32—21no—
78C22×C32—21no—
79C22×C32—41no—
80C22×C32—41no—
81C22×C32—41no—
82C22×C32—41no—
83C22×C32—81no—
84C22×C32#9911yescharacteristic
85C22×C32#10311yescharacteristic
86C22×C32#10411yescharacteristic
87C32×C4—42no—
88C32×C4—41no—
89C32×C4—41no—
90C32×C4—81no—
91C32×C4—82no—
92C22×Q8C32×C411yescharacteristic, complemented
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4C22×C3211yescharacteristic
96C22⋊C8—21no—
97#95—21no—
98#95C32×C411yescharacteristic, complemented
99C23×C3—24no—
100C23×C3—24no—
101C23×C3—44no—
102C23×C3—44no—
103C23×C3C3×C22⋊C414yes—
104C23×C3C3×SD1614yes—
105C23×C3C3×Q1614yes—
106C3×Q8—48no—
107C3×Q8—48no—
108C2×C3×C4—28no—
109C2×C3×C4—24no—
110C2×C3×C4—24no—
111C2×C3×C4—44no—
112C2×C3×C4—44no—
113C2×C3×C4—44no—
114C2×C3×C4—44no—
115C2×C3×C4—48no—
116C2×C3×C4—44no—
117C2×C3×C4—44no—
118C2×C3×C4—44no—
119C2×C3×C4—84no—
120C2×C3×C4—84no—
121C3×C8—84no—
122C2×C32—21no—
123C2×C32—21no—
124C2×C32—41no—
125C2×C32—41no—
126C2×C32#32811yescharacteristic
127C2×C32AΣL(1,𝔽16)11yescharacteristic
128C2×C32#35411yescharacteristic
129C24C32×D411yescharacteristic
130C2×Q8—42no—
131C2×Q8—41no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—41no—
135C22×C4C2×C32×C411yescharacteristic
136C22×C4C32×D411yescharacteristic
137C2×C8—41no—
138C22⋊C4—22no—
139C22⋊C4—41no—
140C22⋊C4—42no—
141C22⋊C4—41no—
142C22×C3—24no—
143C22×C3—24no—
144C22×C3—24no—
145C22×C3—24no—
146C22×C3—44no—
147C22×C3—44no—
148C22×C3—44no—
149C22×C3—44no—
150C22×C3—84no—
151C22×C3C3×#9914yes—
152C22×C3C3×#10314yes—
153C22×C3C3×#10414yes—
154C3×C4—48no—
155C3×C4—44no—
156C3×C4—44no—
157C3×C4—84no—
158C3×C4—88no—
159C32#135111yescharacteristic, direct product factor, Sylow
160C23—21no—
161C23—21no—
162C23—41no—
163C23—41no—
164C23C32×C22⋊C411yescharacteristic
165C23C32×SD1611yescharacteristic
166C23C32×Q1611yescharacteristic
167Q8—42no—
168Q8—42no—
169C2×C4—22no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—42no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—81no—
181C2×C4—81no—
182C8—81no—
183C2×C3—24no—
184C2×C3—24no—
185C2×C3—44no—
186C2×C3—44no—
187C2×C3C3×#32814yes—
188C2×C3C3×AΣL(1,𝔽16)14yes—
189C2×C3C3×#35414yes—
190C22—21no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—41no—
195C22—41no—
196C22—41no—
197C22—41no—
198C22—81no—
199C22C32×#9911yescharacteristic
200C22C32×#10311yescharacteristic
201C22C32×#10411yescharacteristic
202C4—42no—
203C4—41no—
204C4—41no—
205C4—81no—
206C4—82no—
207C3C3×#135114yesdirect product factor
208C2—21no—
209C2—21no—
210C2—41no—
211C2—41no—
212C2C32×#32811yescharacteristic
213C2C32×AΣL(1,𝔽16)11yescharacteristic
214C2C32×#35411yescharacteristic
subgroupG/H
center, socleC22×C32#99
commutatorC22×C4C2×C32×C4
Frattini, cosocleC2×C22⋊C4C22×C32
prime radicalC24×C32D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C32 ◁ C23×C32 ◁ C2×C32×C22⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
4. C3×#1351C3direct (N × H)
5. C22×C32×Q8C4semidirect
11. C32×#95C4semidirect
28. #1351C32direct (N × H)
29. C22×C3×Q8C3×C4semidirect
35. C3×#95C3×C4semidirect
92. C22×Q8C32×C4semidirect
98. #95C32×C4semidirect
159. C32#1351direct (N × H)
207. C3C3×#1351direct (N × H)

Maximal subgroups

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

subgroupindex
C32×#3932
C32×#3422
C32×#3232
C3×#13513