Finite Group Databasebeta

← C32×#2203 #11769252

Subgroups — C32×#2203 #11769252

Order 1152.

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

normal subgroups only

This group has 648 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×#456C212yesmaximal, complemented
3C32×#454C211yesmaximal, characteristic
4C32×#356C212yesmaximal, complemented
5C32×#355C211yesmaximal, characteristic, complemented
6C3×#2203C314yesmaximal, direct product factor
7C32×2−1+4C2211yescharacteristic
8C32×#137—22no—
9C32×AGL(1,ℤ/8)—21no—
10C32×C4⋊Q8C2212yes—
11C32×#126—22no—
12C32×#125—21no—
13C32×#124C2211yescharacteristic
14C32×#123—21no—
15C32×C22⋊Q8—22no—
16C32×#104—21no—
17C32×#104—22no—
18C32×#101C2211yescharacteristic
19C32×#101C2212yes—
20C3×#563C2×C314yescomplemented
21C3×#456C2×C318yescomplemented
22C3×#454C2×C314yes—
23C3×#356C2×C318yescomplemented
24C3×#355C2×C314yescomplemented
25C32×#41—21no—
26C32×#41—22no—
27C32×#41—41no—
28C2×C32×Q8—41no—
29C2×C32×Q8D412yes—
30C2×C32×Q8C2311yescharacteristic
31C2×C32×D4D411yescharacteristic
32C22×C32×C4—21no—
33C32×Q16—42no—
34C32×SD16—42no—
35C32×SD16—41no—
36C32×D8—41no—
37C32×M16—21no—
38C32×M16—22no—
39C32×C4⋊C4—21no—
40C32×C4⋊C4—22no—
41C32×C4⋊C4—21no—
42C32×C4⋊C4—22no—
43C32×C4⋊C4—22no—
44C32×C22⋊C4—22no—
45C32×C22⋊C4—21no—
46C32×C22⋊C4—22no—
47C32×C42D411yescharacteristic
48C32×C42D412yes—
49#2203C3211yescharacteristic, direct product factor, Sylow
50C3×2−1+4C22×C314yes—
51C3×#137—28no—
52C3×AGL(1,ℤ/8)—24no—
53C3×C4⋊Q8C22×C318yes—
54C3×#126—28no—
55C3×#125—24no—
56C3×#124C22×C314yes—
57C3×#123—24no—
58C3×C22⋊Q8—28no—
59C3×#104—24no—
60C3×#104—28no—
61C3×#101C22×C314yes—
62C3×#101C22×C318yes—
63C23×C32—21no—
64C32×Q8—21no—
65C32×Q8—22no—
66C32×Q8—41no—
67C32×Q8—42no—
68C32×D4—21no—
69C32×D4—22no—
70C32×D4—41no—
71C32×D4—41no—
72C2×C32×C4—21no—
73C2×C32×C4—22no—
74C2×C32×C4—22no—
75C2×C32×C4—41no—
76C2×C32×C4—42no—
77C2×C32×C4—81no—
78C2×C32×C4C2×D412yes—
79C2×C32×C4C2×D411yescharacteristic
80C32×C8—41no—
81C32×C8—42no—
82#563C2×C3211yescharacteristic, complemented
83#456C2×C3212yescomplemented
84#454C2×C3211yescharacteristic
85#356C2×C3212yescomplemented
86#355C2×C3211yescharacteristic, complemented
87C3×#41—24no—
88C3×#41—28no—
89C3×#41—44no—
90C2×C3×Q8—44no—
91C2×C3×Q8C3×D418yes—
92C2×C3×Q8C23×C314yes—
93C2×C3×D4C3×D414yes—
94C22×C3×C4—24no—
95C3×Q16—48no—
96C3×SD16—48no—
97C3×SD16—44no—
98C3×D8—44no—
99C3×M16—24no—
100C3×M16—28no—
101C3×C4⋊C4—24no—
102C3×C4⋊C4—28no—
103C3×C4⋊C4—24no—
104C3×C4⋊C4—28no—
105C3×C4⋊C4—28no—
106C3×C22⋊C4—28no—
107C3×C22⋊C4—24no—
108C3×C22⋊C4—28no—
109C3×C42C3×D414yes—
110C3×C42C3×D418yes—
111C22×C32—42no—
112C22×C32—41no—
113C22×C32—41no—
114C22×C32#12011yescharacteristic
115C32×C4—21no—
116C32×C4—22no—
117C32×C4—41no—
118C32×C4—42no—
119C32×C4—42no—
120C32×C4—41no—
1212−1+4C22×C3211yescharacteristic
122#137—22no—
123AGL(1,ℤ/8)—21no—
124C4⋊Q8C22×C3212yes—
125#126—22no—
126#125—21no—
127#124C22×C3211yescharacteristic
128#123—21no—
129C22⋊Q8—22no—
130#104—21no—
131#104—22no—
132#101C22×C3212yes—
133#101C22×C3211yescharacteristic
134C23×C3—24no—
135C3×Q8—24no—
136C3×Q8—28no—
137C3×Q8—44no—
138C3×Q8—48no—
139C3×D4—24no—
140C3×D4—28no—
141C3×D4—44no—
142C3×D4—44no—
143C2×C3×C4—24no—
144C2×C3×C4—28no—
145C2×C3×C4—28no—
146C2×C3×C4—44no—
147C2×C3×C4—48no—
148C2×C3×C4—84no—
149C2×C3×C4C2×C3×D418yes—
150C2×C3×C4C2×C3×D414yes—
151C3×C8—44no—
152C3×C8—48no—
153C2×C32—21no—
154C2×C32—81no—
155C2×C32—81no—
156C2×C32B(4,𝔽2)11yescharacteristic
157#41—21no—
158#41—22no—
159#41—41no—
160C2×Q8—41no—
161C2×Q8C32×D412yes—
162C2×Q8C23×C3211yescharacteristic
163C2×D4C32×D411yescharacteristic
164C22×C4—21no—
165Q16—42no—
166SD16—42no—
167SD16—41no—
168D8—41no—
169M16—21no—
170M16—22no—
171C4⋊C4—21no—
172C4⋊C4—22no—
173C4⋊C4—21no—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C22⋊C4—22no—
177C22⋊C4—21no—
178C22⋊C4—22no—
179C42C32×D411yescharacteristic
180C42C32×D412yes—
181C22×C3—48no—
182C22×C3—44no—
183C22×C3—44no—
184C22×C3C3×#12014yes—
185C3×C4—24no—
186C3×C4—28no—
187C3×C4—44no—
188C3×C4—48no—
189C3×C4—48no—
190C3×C4—44no—
191C32#220311yescharacteristic, direct product factor, Sylow
192C23—21no—
193Q8—21no—
194Q8—22no—
195Q8—41no—
196Q8—42no—
197D4—21no—
198D4—22no—
199D4—41no—
200D4—41no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—41no—
205C2×C4—42no—
206C2×C4—81no—
207C2×C4C2×C32×D412yes—
208C2×C4C2×C32×D411yescharacteristic
209C8—41no—
210C8—42no—
211C2×C3—24no—
212C2×C3—84no—
213C2×C3—84no—
214C2×C3C3×B(4,𝔽2)14yes—
215C22—42no—
216C22—41no—
217C22—41no—
218C22C32×#12011yescharacteristic
219C4—21no—
220C4—22no—
221C4—41no—
222C4—42no—
223C4—42no—
224C4—41no—
225C3C3×#220314yesdirect product factor
226C2—21no—
227C2—81no—
228C2—81no—
229C2C32×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC2×C32B(4,𝔽2)
commutator, Frattini, cosocleC2×Q8C23×C32
prime radicalC32×#454C2
LCSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C32 ◁ C22×C32 ◁ C2×C32×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
1. C32×#563C2semidirect
2. C32×#456C2semidirect
4. C32×#356C2semidirect
5. C32×#355C2semidirect
6. C3×#2203C3direct (N × H)
20. C3×#563C2×C3semidirect
21. C3×#456C2×C3semidirect
23. C3×#356C2×C3semidirect
24. C3×#355C2×C3semidirect
49. #2203C32direct (N × H)
82. #563C2×C32semidirect
83. #456C2×C32semidirect
85. #356C2×C32semidirect
86. #355C2×C32semidirect
191. C32#2203direct (N × H)
225. C3C3×#2203direct (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×#4562
C32×#4542
C32×#3562
C32×#3552
C3×#22033