Finite Group Databasebeta

← C32×#1604 #11768653

Subgroups — C32×#1604 #11768653

Order 1152.

846 conjugacy classes of subgroups; only the 204 normal subgroups are shown.

normal subgroups only

This group has 846 conjugacy classes of subgroups — too many to draw. Only its 204 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×#537C211yesmaximal, characteristic
2C32×#448C212yesmaximal, complemented
3C32×#395C211yesmaximal, characteristic, complemented
4C32×#327C212yesmaximal, complemented
5C32×#324C211yesmaximal, characteristic, complemented
6C3×#1604C314yesmaximal, direct product factor
7C2×C32×#41—21no—
8C22×C32×Q8C2211yescharacteristic
9C2×C32×SD16—22no—
10C32×C22⋊Q8—22no—
11C32×#121—22no—
12C32×#121—21no—
13C32×#121C2212yes—
14C2×C32×C4⋊C4—21no—
15C2×C32×C4⋊C4C2211yescharacteristic
16C32×#103—22no—
17C32×C22⋊C8C2212yes—
18C32×#95—21no—
19C32×#95C2211yescharacteristic
20C3×#537C2×C314yes—
21C3×#448C2×C318yescomplemented
22C3×#395C2×C314yescomplemented
23C3×#327C2×C318yescomplemented
24C3×#324C2×C314yescomplemented
25C32×#41—42no—
26C32×#41—42no—
27C2×C32×Q8—21no—
28C2×C32×Q8—22no—
29C2×C32×Q8—42no—
30C2×C32×D4—21no—
31C2×C32×D4—22no—
32C2×C32×D4—21no—
33C2×C32×D4D412yes—
34C22×C32×C4—22no—
35C22×C32×C4—21no—
36C22×C32×C4—21no—
37C22×C32×C4D411yescharacteristic
38C22×C32×C4D411yescharacteristic
39C22×C32×C4C2311yescharacteristic
40C32×SD16—44no—
41C2×C32×C8—22no—
42C32×C4⋊C4—21no—
43C32×C4⋊C4—22no—
44C32×C4⋊C4—42no—
45C32×C4⋊C4D412yes—
46C32×C22⋊C4—22no—
47C32×C22⋊C4—22no—
48#1604C3211yescharacteristic, direct product factor, Sylow
49C2×C3×#41—24no—
50C22×C3×Q8C22×C314yes—
51C2×C3×SD16—28no—
52C3×C22⋊Q8—28no—
53C3×#121—28no—
54C3×#121—24no—
55C3×#121C22×C318yes—
56C2×C3×C4⋊C4—24no—
57C2×C3×C4⋊C4C22×C314yes—
58C3×#103—28no—
59C3×C22⋊C8C22×C318yes—
60C3×#95—24no—
61C3×#95C22×C314yes—
62C23×C32—22no—
63C23×C32C2×D411yescharacteristic
64C32×Q8—24no—
65C32×Q8—42no—
66C32×Q8—42no—
67C32×D4—42no—
68C32×D4—42no—
69C32×D4—42no—
70C32×D4—42no—
71C32×D4—81no—
72C2×C32×C4—21no—
73C2×C32×C4—21no—
74C2×C32×C4—22no—
75C2×C32×C4—22no—
76C2×C32×C4—21no—
77C2×C32×C4—22no—
78C2×C32×C4—41no—
79C2×C32×C4—44no—
80C2×C32×C4—41no—
81C2×C32×C4—41no—
82C2×C32×C4—41no—
83C2×C32×C4—41no—
84C2×C32×C4—41no—
85C2×C32×C4SD1612yescomplemented
86C2×C32×C4C2×D412yes—
87C32×C8—42no—
88#537C2×C3211yescharacteristic
89#448C2×C3212yescomplemented
90#395C2×C3211yescharacteristic, complemented
91#327C2×C3212yescomplemented
92#324C2×C3211yescharacteristic, complemented
93C3×#41—48no—
94C3×#41—48no—
95C2×C3×Q8—24no—
96C2×C3×Q8—28no—
97C2×C3×Q8—48no—
98C2×C3×D4—24no—
99C2×C3×D4—28no—
100C2×C3×D4—24no—
101C2×C3×D4C3×D418yes—
102C22×C3×C4—28no—
103C22×C3×C4—24no—
104C22×C3×C4—24no—
105C22×C3×C4C3×D414yes—
106C22×C3×C4C3×D414yes—
107C22×C3×C4C23×C314yes—
108C3×SD16—416no—
109C2×C3×C8—28no—
110C3×C4⋊C4—24no—
111C3×C4⋊C4—28no—
112C3×C4⋊C4—48no—
113C3×C4⋊C4C3×D418yes—
114C3×C22⋊C4—28no—
115C3×C22⋊C4—28no—
116C22×C32—21no—
117C22×C32—21no—
118C22×C32—42no—
119C22×C32—42no—
120C22×C32—42no—
121C22×C32#12011yescharacteristic
122C22×C32C2×SD1611yescharacteristic
123C22×C32AGL(1,ℤ/8)11yescharacteristic
124C32×C4—22no—
125C32×C4—22no—
126C32×C4—42no—
127C32×C4—42no—
128C32×C4—42no—
129C32×C4—41no—
130C2×#41—21no—
131C22×Q8C22×C3211yescharacteristic
132C2×SD16—22no—
133C22⋊Q8—22no—
134#121—22no—
135#121—21no—
136#121C22×C3212yes—
137C2×C4⋊C4—21no—
138C2×C4⋊C4C22×C3211yescharacteristic
139#103—22no—
140C22⋊C8C22×C3212yes—
141#95—21no—
142#95C22×C3211yescharacteristic
143C23×C3—28no—
144C23×C3C2×C3×D414yes—
145C3×Q8—216no—
146C3×Q8—48no—
147C3×Q8—48no—
148C3×D4—48no—
149C3×D4—48no—
150C3×D4—48no—
151C3×D4—48no—
152C3×D4—84no—
153C2×C3×C4—24no—
154C2×C3×C4—24no—
155C2×C3×C4—28no—
156C2×C3×C4—28no—
157C2×C3×C4—24no—
158C2×C3×C4—28no—
159C2×C3×C4—44no—
160C2×C3×C4—416no—
161C2×C3×C4—44no—
162C2×C3×C4—44no—
163C2×C3×C4—44no—
164C2×C3×C4—44no—
165C2×C3×C4—44no—
166C2×C3×C4C3×SD1618yescomplemented
167C2×C3×C4C2×C3×D418yes—
168C3×C8—48no—
169C2×C32—21no—
170C2×C32—21no—
171C2×C32—82no—
172C2×C32#45011yescharacteristic
173C2×C32#45611yescharacteristic
174C2×C32B(4,𝔽2)11yescharacteristic
175#41—42no—
176#41—42no—
177C2×Q8—21no—
178C2×Q8—22no—
179C2×Q8—42no—
180C2×D4—21no—
181C2×D4—22no—
182C2×D4—21no—
183C2×D4C32×D412yes—
184C22×C4—22no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4C32×D411yescharacteristic
188C22×C4C32×D411yescharacteristic
189C22×C4C23×C3211yescharacteristic
190SD16—44no—
191C2×C8—22no—
192C4⋊C4—21no—
193C4⋊C4—22no—
194C4⋊C4—42no—
195C4⋊C4C32×D412yes—
196C22⋊C4—22no—
197C22⋊C4—22no—
198C22×C3—24no—
199C22×C3—24no—
200C22×C3—48no—
201C22×C3—48no—
202C22×C3—48no—
203C22×C3C3×#12014yes—
204C22×C3C2×C3×SD1614yes—
205C22×C3C3×AGL(1,ℤ/8)14yes—
206C3×C4—28no—
207C3×C4—28no—
208C3×C4—48no—
209C3×C4—48no—
210C3×C4—48no—
211C3×C4—44no—
212C32#160411yescharacteristic, direct product factor, Sylow
213C23—22no—
214C23C2×C32×D411yescharacteristic
215Q8—24no—
216Q8—42no—
217Q8—42no—
218D4—42no—
219D4—42no—
220D4—42no—
221D4—42no—
222D4—81no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—41no—
230C2×C4—44no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4C32×SD1612yescomplemented
237C2×C4C2×C32×D412yes—
238C8—42no—
239C2×C3—24no—
240C2×C3—24no—
241C2×C3—88no—
242C2×C3C3×#45014yes—
243C2×C3C3×#45614yes—
244C2×C3C3×B(4,𝔽2)14yes—
245C22—21no—
246C22—21no—
247C22—42no—
248C22—42no—
249C22—42no—
250C22C32×#12011yescharacteristic
251C22C2×C32×SD1611yescharacteristic
252C22C32×AGL(1,ℤ/8)11yescharacteristic
253C4—22no—
254C4—22no—
255C4—42no—
256C4—42no—
257C4—41no—
258C4—42no—
259C3C3×#160414yesdirect product factor
260C2—21no—
261C2—21no—
262C2—82no—
263C2C32×#45011yescharacteristic
264C2C32×#45611yescharacteristic
265C2C32×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC22×C32#120
commutator, Frattini, cosocleC22×C4C23×C32
prime radicalC32×#537C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C32 ◁ C22×C32×C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C32×#448C2semidirect
3. C32×#395C2semidirect
4. C32×#327C2semidirect
5. C32×#324C2semidirect
6. C3×#1604C3direct (N × H)
21. C3×#448C2×C3semidirect
22. C3×#395C2×C3semidirect
23. C3×#327C2×C3semidirect
24. C3×#324C2×C3semidirect
48. #1604C32direct (N × H)
85. C2×C32×C4SD16semidirect
89. #448C2×C32semidirect
90. #395C2×C32semidirect
91. #327C2×C32semidirect
92. #324C2×C32semidirect
166. C2×C3×C4C3×SD16semidirect
212. C32#1604direct (N × H)
236. C2×C4C32×SD16semidirect
259. C3C3×#1604direct (N × H)

Maximal subgroups

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

subgroupindex
C32×#5372
C32×#4482
C32×#3952
C32×#3272
C32×#3242
C3×#16043