Finite Group Databasebeta

← C32×#2204 #11769253

Subgroups — C32×#2204 #11769253

Order 1152.

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

normal subgroups only

This group has 708 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×#555C211yesmaximal, characteristic, complemented
2C32×#456C211yesmaximal, characteristic, complemented
3C32×#454C212yesmaximal, complemented
4C32×#356C211yesmaximal, characteristic, complemented
5C32×#355C212yesmaximal, complemented
6C3×#2204C314yesmaximal, direct product factor
7C32×2−1+4C2211yescharacteristic
8C32×#137—21no—
9C32×AGL(1,ℤ/8)—22no—
10C32×C4⋊Q8C2211yescharacteristic
11C32×#124—21no—
12C32×#124C2212yes—
13C32×#121—22no—
14C32×C4×D4—21no—
15C32×#117—22no—
16C32×#104—22no—
17C32×#104—21no—
18C32×#101C2212yes—
19C32×#101C2211yescharacteristic
20C3×#555C2×C314yescomplemented
21C3×#456C2×C314yescomplemented
22C3×#454C2×C318yescomplemented
23C3×#356C2×C314yescomplemented
24C3×#355C2×C318yescomplemented
25C32×#41—22no—
26C32×#41—21no—
27C32×#41—41no—
28C2×C32×Q8—41no—
29C2×C32×Q8D411yescharacteristic
30C2×C32×Q8C2311yescharacteristic
31C2×C32×D4—21no—
32C2×C32×D4—21no—
33C2×C32×D4D412yes—
34C22×C32×C4—22no—
35C32×Q16—41no—
36C32×SD16—41no—
37C32×SD16—42no—
38C32×D8—42no—
39C32×M16—22no—
40C32×M16—21no—
41C32×C4⋊C4—21no—
42C32×C4⋊C4—22no—
43C32×C22⋊C4—22no—
44C32×C22⋊C4—22no—
45C32×C22⋊C4—22no—
46C32×C42—21no—
47C32×C42D412yes—
48C32×C42D411yescharacteristic
49#2204C3211yescharacteristic, direct product factor, Sylow
50C3×2−1+4C22×C314yes—
51C3×#137—24no—
52C3×AGL(1,ℤ/8)—28no—
53C3×C4⋊Q8C22×C314yes—
54C3×#124—24no—
55C3×#124C22×C318yes—
56C3×#121—28no—
57C3×C4×D4—24no—
58C3×#117—28no—
59C3×#104—28no—
60C3×#104—24no—
61C3×#101C22×C318yes—
62C3×#101C22×C314yes—
63C23×C32—22no—
64C32×Q8—22no—
65C32×Q8—21no—
66C32×Q8—41no—
67C32×Q8—41no—
68C32×D4—22no—
69C32×D4—21no—
70C32×D4—42no—
71C32×D4—42no—
72C32×D4—41no—
73C32×D4—81no—
74C2×C32×C4—21no—
75C2×C32×C4—22no—
76C2×C32×C4—21no—
77C2×C32×C4—44no—
78C2×C32×C4—42no—
79C2×C32×C4—41no—
80C2×C32×C4C2×D411yescharacteristic
81C2×C32×C4C2×D412yes—
82C32×C8—42no—
83C32×C8—41no—
84#555C2×C3211yescharacteristic, complemented
85#456C2×C3211yescharacteristic, complemented
86#454C2×C3212yescomplemented
87#356C2×C3211yescharacteristic, complemented
88#355C2×C3212yescomplemented
89C3×#41—28no—
90C3×#41—24no—
91C3×#41—44no—
92C2×C3×Q8—44no—
93C2×C3×Q8C3×D414yes—
94C2×C3×Q8C23×C314yes—
95C2×C3×D4—24no—
96C2×C3×D4—24no—
97C2×C3×D4C3×D418yes—
98C22×C3×C4—28no—
99C3×Q16—44no—
100C3×SD16—44no—
101C3×SD16—48no—
102C3×D8—48no—
103C3×M16—28no—
104C3×M16—24no—
105C3×C4⋊C4—24no—
106C3×C4⋊C4—28no—
107C3×C22⋊C4—28no—
108C3×C22⋊C4—28no—
109C3×C22⋊C4—28no—
110C3×C42—24no—
111C3×C42C3×D418yes—
112C3×C42C3×D414yes—
113C22×C32—42no—
114C22×C32—42no—
115C22×C32—41no—
116C22×C32—42no—
117C22×C32#12011yescharacteristic
118C32×C4—22no—
119C32×C4—22no—
120C32×C4—21no—
121C32×C4—42no—
122C32×C4—41no—
123C32×C4—41no—
1242−1+4C22×C3211yescharacteristic
125#137—21no—
126AGL(1,ℤ/8)—22no—
127C4⋊Q8C22×C3211yescharacteristic
128#124—21no—
129#124C22×C3212yes—
130#121—22no—
131C4×D4—21no—
132#117—22no—
133#104—22no—
134#104—21no—
135#101C22×C3211yescharacteristic
136#101C22×C3212yes—
137C23×C3—28no—
138C3×Q8—28no—
139C3×Q8—24no—
140C3×Q8—44no—
141C3×Q8—44no—
142C3×D4—28no—
143C3×D4—24no—
144C3×D4—48no—
145C3×D4—48no—
146C3×D4—44no—
147C3×D4—84no—
148C2×C3×C4—24no—
149C2×C3×C4—28no—
150C2×C3×C4—24no—
151C2×C3×C4—416no—
152C2×C3×C4—48no—
153C2×C3×C4—44no—
154C2×C3×C4C2×C3×D414yes—
155C2×C3×C4C2×C3×D418yes—
156C3×C8—48no—
157C3×C8—44no—
158C2×C32—21no—
159C2×C32—81no—
160C2×C32—82no—
161C2×C32B(4,𝔽2)11yescharacteristic
162#41—22no—
163#41—21no—
164#41—41no—
165C2×Q8—41no—
166C2×Q8C32×D411yescharacteristic
167C2×Q8C23×C3211yescharacteristic
168C2×D4—21no—
169C2×D4—21no—
170C2×D4C32×D412yes—
171C22×C4—22no—
172Q16—41no—
173SD16—41no—
174SD16—42no—
175D8—42no—
176M16—22no—
177M16—21no—
178C4⋊C4—21no—
179C4⋊C4—22no—
180C22⋊C4—22no—
181C22⋊C4—22no—
182C22⋊C4—22no—
183C42—21no—
184C42C32×D412yes—
185C42C32×D411yescharacteristic
186C22×C3—48no—
187C22×C3—48no—
188C22×C3—44no—
189C22×C3—48no—
190C22×C3C3×#12014yes—
191C3×C4—28no—
192C3×C4—28no—
193C3×C4—24no—
194C3×C4—48no—
195C3×C4—44no—
196C3×C4—44no—
197C32#220411yescharacteristic, direct product factor, Sylow
198C23—22no—
199Q8—22no—
200Q8—21no—
201Q8—41no—
202Q8—41no—
203D4—22no—
204D4—21no—
205D4—42no—
206D4—42no—
207D4—41no—
208D4—81no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—44no—
213C2×C4—42no—
214C2×C4—41no—
215C2×C4C2×C32×D411yescharacteristic
216C2×C4C2×C32×D412yes—
217C8—42no—
218C8—41no—
219C2×C3—24no—
220C2×C3—88no—
221C2×C3—84no—
222C2×C3C3×B(4,𝔽2)14yes—
223C22—42no—
224C22—42no—
225C22—41no—
226C22—42no—
227C22C32×#12011yescharacteristic
228C4—22no—
229C4—22no—
230C4—21no—
231C4—42no—
232C4—41no—
233C4—41no—
234C3C3×#220414yesdirect product factor
235C2—21no—
236C2—82no—
237C2—81no—
238C2C32×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC2×C32B(4,𝔽2)
commutator, Frattini, cosocleC2×Q8C23×C32
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×#555C2semidirect
2. C32×#456C2semidirect
3. C32×#454C2semidirect
4. C32×#356C2semidirect
5. C32×#355C2semidirect
6. C3×#2204C3direct (N × H)
20. C3×#555C2×C3semidirect
21. C3×#456C2×C3semidirect
22. C3×#454C2×C3semidirect
23. C3×#356C2×C3semidirect
24. C3×#355C2×C3semidirect
49. #2204C32direct (N × H)
84. #555C2×C32semidirect
85. #456C2×C32semidirect
86. #454C2×C32semidirect
87. #356C2×C32semidirect
88. #355C2×C32semidirect
197. C32#2204direct (N × H)
234. C3C3×#2204direct (N × H)

Maximal subgroups

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

subgroupindex
C32×#5552
C32×#4562
C32×#4542
C32×#3562
C32×#3552
C3×#22043