Finite Group Databasebeta

← C32×#2201 #11769250

Subgroups — C32×#2201 #11769250

Order 1152.

678 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 678 conjugacy classes of subgroups — too many to draw. Only its 168 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×#563C211yesmaximal, characteristic, complemented
2C32×#458C212yesmaximal
3C32×#455C211yesmaximal, characteristic
4C32×#354C211yesmaximal, characteristic, complemented
5C32×#352C212yesmaximal, complemented
6C3×#2201C314yesmaximal, direct product factor
7C32×H(5,𝔽2)C2211yescharacteristic
8C32×#137—21no—
9C32×C4⋊Q8—21no—
10C32×#126—22no—
11C32×#125—21no—
12C32×#124C2211yescharacteristic
13C32×#123—22no—
14C32×#123C2212yes—
15C32×C22⋊Q8—22no—
16C32×#104—21no—
17C32×#100C2211yescharacteristic
18C32×#99—22no—
19C32×#99C2212yes—
20C3×#563C2×C314yescomplemented
21C3×#458C2×C318yes—
22C3×#455C2×C314yes—
23C3×#354C2×C314yescomplemented
24C3×#352C2×C318yescomplemented
25C32×#41—21no—
26C32×#41—41no—
27C2×C32×Q8—21no—
28C2×C32×Q8D411yescharacteristic
29C2×C32×D4—22no—
30C2×C32×D4—41no—
31C2×C32×D4C2311yescharacteristic
32C22×C32×C4D412yes—
33C32×Q16—41no—
34C32×SD16—41no—
35C32×M16—21no—
36C32×C4⋊C4—21no—
37C32×C4⋊C4—22no—
38C32×C4⋊C4—22no—
39C32×C4⋊C4—21no—
40C32×C4⋊C4—22no—
41C32×C4⋊C4—42no—
42C32×C22⋊C4—22no—
43C32×C22⋊C4—22no—
44C32×C22⋊C4—22no—
45C32×C22⋊C4—42no—
46C32×C22⋊C4D412yes—
47C32×C42—21no—
48C32×C42D411yescharacteristic
49#2201C3211yescharacteristic, direct product factor, Sylow
50C3×H(5,𝔽2)C22×C314yes—
51C3×#137—24no—
52C3×C4⋊Q8—24no—
53C3×#126—28no—
54C3×#125—24no—
55C3×#124C22×C314yes—
56C3×#123—28no—
57C3×#123C22×C318yes—
58C3×C22⋊Q8—28no—
59C3×#104—24no—
60C3×#100C22×C314yes—
61C3×#99—28no—
62C3×#99C22×C318yes—
63C23×C32—22no—
64C23×C32C2×D412yes—
65C32×Q8—21no—
66C32×Q8—41no—
67C32×Q8—81no—
68C32×D4—21no—
69C32×D4—41no—
70C32×D4—42no—
71C32×D4—41no—
72C2×C32×C4—22no—
73C2×C32×C4—21no—
74C2×C32×C4—22no—
75C2×C32×C4—21no—
76C2×C32×C4—22no—
77C2×C32×C4—41no—
78C2×C32×C4—42no—
79C2×C32×C4—42no—
80C2×C32×C4C2×D411yescharacteristic
81C32×C8—41no—
82#563C2×C3211yescharacteristic, complemented
83#458C2×C3212yes—
84#455C2×C3211yescharacteristic
85#354C2×C3211yescharacteristic, complemented
86#352C2×C3212yescomplemented
87C3×#41—24no—
88C3×#41—44no—
89C2×C3×Q8—24no—
90C2×C3×Q8C3×D414yes—
91C2×C3×D4—28no—
92C2×C3×D4—44no—
93C2×C3×D4C23×C314yes—
94C22×C3×C4C3×D418yes—
95C3×Q16—44no—
96C3×SD16—44no—
97C3×M16—24no—
98C3×C4⋊C4—24no—
99C3×C4⋊C4—28no—
100C3×C4⋊C4—28no—
101C3×C4⋊C4—24no—
102C3×C4⋊C4—28no—
103C3×C4⋊C4—48no—
104C3×C22⋊C4—28no—
105C3×C22⋊C4—28no—
106C3×C22⋊C4—28no—
107C3×C22⋊C4—48no—
108C3×C22⋊C4C3×D418yes—
109C3×C42—24no—
110C3×C42C3×D414yes—
111C22×C32—22no—
112C22×C32—42no—
113C22×C32—41no—
114C22×C32—82no—
115C22×C32#12011yescharacteristic
116C32×C4—21no—
117C32×C4—42no—
118C32×C4—41no—
119C32×C4—41no—
120C32×C4—42no—
121C32×C4—41no—
122C32×C4—82no—
123H(5,𝔽2)C22×C3211yescharacteristic
124#137—21no—
125C4⋊Q8—21no—
126#126—22no—
127#125—21no—
128#124C22×C3211yescharacteristic
129#123—22no—
130#123C22×C3212yes—
131C22⋊Q8—22no—
132#104—21no—
133#100C22×C3211yescharacteristic
134#99—22no—
135#99C22×C3212yes—
136C23×C3—28no—
137C23×C3C2×C3×D418yes—
138C3×Q8—24no—
139C3×Q8—44no—
140C3×Q8—84no—
141C3×D4—24no—
142C3×D4—44no—
143C3×D4—48no—
144C3×D4—44no—
145C2×C3×C4—28no—
146C2×C3×C4—24no—
147C2×C3×C4—28no—
148C2×C3×C4—24no—
149C2×C3×C4—28no—
150C2×C3×C4—44no—
151C2×C3×C4—48no—
152C2×C3×C4—48no—
153C2×C3×C4C2×C3×D414yes—
154C3×C8—44no—
155C2×C32—21no—
156C2×C32—42no—
157C2×C32—81no—
158C2×C32B(4,𝔽2)11yescharacteristic
159#41—21no—
160#41—41no—
161C2×Q8—21no—
162C2×Q8C32×D411yescharacteristic
163C2×D4—22no—
164C2×D4—41no—
165C2×D4C23×C3211yescharacteristic
166C22×C4C32×D412yes—
167Q16—41no—
168SD16—41no—
169M16—21no—
170C4⋊C4—21no—
171C4⋊C4—22no—
172C4⋊C4—22no—
173C4⋊C4—21no—
174C4⋊C4—22no—
175C4⋊C4—42no—
176C22⋊C4—22no—
177C22⋊C4—22no—
178C22⋊C4—22no—
179C22⋊C4—42no—
180C22⋊C4C32×D412yes—
181C42—21no—
182C42C32×D411yescharacteristic
183C22×C3—28no—
184C22×C3—48no—
185C22×C3—44no—
186C22×C3—88no—
187C22×C3C3×#12014yes—
188C3×C4—24no—
189C3×C4—48no—
190C3×C4—44no—
191C3×C4—44no—
192C3×C4—48no—
193C3×C4—44no—
194C3×C4—88no—
195C32#220111yescharacteristic, direct product factor, Sylow
196C23—22no—
197C23C2×C32×D412yes—
198Q8—21no—
199Q8—41no—
200Q8—81no—
201D4—21no—
202D4—41no—
203D4—42no—
204D4—41no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4C2×C32×D411yescharacteristic
214C8—41no—
215C2×C3—24no—
216C2×C3—48no—
217C2×C3—84no—
218C2×C3C3×B(4,𝔽2)14yes—
219C22—22no—
220C22—42no—
221C22—41no—
222C22—82no—
223C22C32×#12011yescharacteristic
224C4—21no—
225C4—42no—
226C4—41no—
227C4—41no—
228C4—42no—
229C4—41no—
230C4—82no—
231C3C3×#220114yesdirect product factor
232C2—21no—
233C2—42no—
234C2—81no—
235C2C32×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC2×C32B(4,𝔽2)
commutator, Frattini, cosocleC2×D4C23×C32
prime radicalC32×H(5,𝔽2)C22
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C32 ◁ C22×C32 ◁ C2×C32×D4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C32×#563C2semidirect
4. C32×#354C2semidirect
5. C32×#352C2semidirect
6. C3×#2201C3direct (N × H)
20. C3×#563C2×C3semidirect
23. C3×#354C2×C3semidirect
24. C3×#352C2×C3semidirect
49. #2201C32direct (N × H)
82. #563C2×C32semidirect
85. #354C2×C32semidirect
86. #352C2×C32semidirect
195. C32#2201direct (N × H)
231. C3C3×#2201direct (N × H)

Maximal subgroups

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

subgroupindex
C32×#5632
C32×#4582
C32×#4552
C32×#3542
C32×#3522
C3×#22013