Finite Group Databasebeta

← C32×#1659 #11768708

Subgroups — C32×#1659 #11768708

Order 1152.

684 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 180 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×H(3,𝔽4)C211yesmaximal, characteristic
2C32×#497C211yesmaximal, characteristic, complemented
3C32×#489C212yesmaximal, complemented
4C32×#329C212yesmaximal, complemented
5C32×#329C211yesmaximal, characteristic, complemented
6C3×#1659C314yesmaximal, direct product factor
7C2×C32×Q16—21no—
8C2×C32×SD16—21no—
9C32×C4⋊Q8C2211yescharacteristic, complemented
10C32×#124—21no—
11C32×#124—22no—
12C32×#124C2212yes—
13C32×#124C2211yescharacteristic
14C32×#120—21no—
15C32×#120—22no—
16C32×#103—22no—
17C32×#102—22no—
18C32×#97C2212yescomplemented
19C32×#97C2211yescharacteristic, complemented
20C3×H(3,𝔽4)C2×C314yes—
21C3×#497C2×C314yescomplemented
22C3×#489C2×C318yescomplemented
23C3×#329C2×C314yescomplemented
24C3×#329C2×C318yescomplemented
25C24×C32—21no—
26C2×C32×Q8—21no—
27C2×C32×Q8D411yescharacteristic
28C2×C32×Q8D412yes—
29C2×C32×D4—21no—
30C2×C32×D4—22no—
31C2×C32×D4D411yescharacteristic
32C2×C32×D4D412yes—
33C32×Q16—42no—
34C32×SD16—42no—
35C2×C32×C8—21no—
36C2×C32×C8—22no—
37C32×C4⋊C4—22no—
38C32×C22⋊C4—22no—
39C32×C22⋊C4—21no—
40C32×C22⋊C4—22no—
41C32×C22⋊C4—22no—
42C32×C22⋊C4—22no—
43C32×C42—21no—
44C32×C42C2311yescharacteristic
45#1659C3211yescharacteristic, direct product factor, Sylow
46C2×C3×Q16—24no—
47C2×C3×SD16—24no—
48C3×C4⋊Q8C22×C314yescomplemented
49C3×#124—28no—
50C3×#124—24no—
51C3×#124C22×C318yes—
52C3×#124C22×C314yes—
53C3×#120—28no—
54C3×#120—24no—
55C3×#103—28no—
56C3×#102—28no—
57C3×#97C22×C318yescomplemented
58C3×#97C22×C314yescomplemented
59C23×C32—21no—
60C23×C32—22no—
61C23×C32—81no—
62C23×C32—82no—
63C32×Q8—41no—
64C32×Q8—42no—
65C32×Q8—41no—
66C32×Q8—41no—
67C32×D4—41no—
68C32×D4—42no—
69C32×D4—81no—
70C32×D4—82no—
71C2×C32×C4—21no—
72C2×C32×C4—22no—
73C2×C32×C4—41no—
74C2×C32×C4C2×D412yes—
75C2×C32×C4C2×D411yescharacteristic
76C32×C8—22no—
77C32×C8—42no—
78H(3,𝔽4)C2×C3211yescharacteristic
79#497C2×C3211yescharacteristic, complemented
80#489C2×C3212yescomplemented
81#329C2×C3211yescharacteristic, complemented
82#329C2×C3212yescomplemented
83C24×C3—24no—
84C2×C3×Q8—24no—
85C2×C3×Q8C3×D418yes—
86C2×C3×Q8C3×D414yes—
87C2×C3×D4—24no—
88C2×C3×D4—28no—
89C2×C3×D4C3×D418yes—
90C2×C3×D4C3×D414yes—
91C3×Q16—48no—
92C3×SD16—48no—
93C2×C3×C8—24no—
94C2×C3×C8—28no—
95C3×C4⋊C4—28no—
96C3×C22⋊C4—28no—
97C3×C22⋊C4—24no—
98C3×C22⋊C4—28no—
99C3×C22⋊C4—28no—
100C3×C22⋊C4—28no—
101C3×C42—24no—
102C3×C42C23×C314yes—
103C22×C32—41no—
104C22×C32—42no—
105C22×C32—42no—
106C22×C32—42no—
107C22×C32—42no—
108C22×C32—84no—
109C22×C32#12011yescharacteristic
110C32×C4—21no—
111C32×C4—22no—
112C32×C4—41no—
113C32×C4—42no—
114C32×C4—81no—
115C2×Q16—21no—
116C2×SD16—21no—
117C4⋊Q8C22×C3211yescharacteristic, complemented
118#124—22no—
119#124—21no—
120#124C22×C3212yes—
121#124C22×C3211yescharacteristic
122#120—22no—
123#120—21no—
124#103—22no—
125#102—22no—
126#97C22×C3212yescomplemented
127#97C22×C3211yescharacteristic, complemented
128C23×C3—24no—
129C23×C3—28no—
130C23×C3—84no—
131C23×C3—88no—
132C3×Q8—44no—
133C3×Q8—48no—
134C3×Q8—44no—
135C3×Q8—44no—
136C3×D4—44no—
137C3×D4—48no—
138C3×D4—84no—
139C3×D4—88no—
140C2×C3×C4—24no—
141C2×C3×C4—28no—
142C2×C3×C4—44no—
143C2×C3×C4C2×C3×D418yes—
144C2×C3×C4C2×C3×D414yes—
145C3×C8—28no—
146C3×C8—48no—
147C2×C32—81no—
148C2×C32—82no—
149C2×C32#45311yescharacteristic
150C2×C32#45512yes—
151C24—21no—
152C2×Q8—21no—
153C2×Q8C32×D412yes—
154C2×Q8C32×D411yescharacteristic
155C2×D4—21no—
156C2×D4—22no—
157C2×D4C32×D412yes—
158C2×D4C32×D411yescharacteristic
159Q16—42no—
160SD16—42no—
161C2×C8—21no—
162C2×C8—22no—
163C4⋊C4—22no—
164C22⋊C4—22no—
165C22⋊C4—21no—
166C22⋊C4—22no—
167C22⋊C4—22no—
168C22⋊C4—22no—
169C42—21no—
170C42C23×C3211yescharacteristic
171C22×C3—44no—
172C22×C3—48no—
173C22×C3—48no—
174C22×C3—48no—
175C22×C3—48no—
176C22×C3—816no—
177C22×C3C3×#12014yes—
178C3×C4—24no—
179C3×C4—28no—
180C3×C4—44no—
181C3×C4—48no—
182C3×C4—84no—
183C32#165911yescharacteristic, direct product factor, Sylow
184C23—21no—
185C23—22no—
186C23—81no—
187C23—82no—
188Q8—41no—
189Q8—42no—
190Q8—41no—
191Q8—41no—
192D4—41no—
193D4—42no—
194D4—81no—
195D4—82no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—41no—
199C2×C4C2×C32×D412yes—
200C2×C4C2×C32×D411yescharacteristic
201C8—22no—
202C8—42no—
203C2×C3—84no—
204C2×C3—88no—
205C2×C3C3×#45314yes—
206C2×C3C3×#45518yes—
207C22—41no—
208C22—42no—
209C22—42no—
210C22—42no—
211C22—42no—
212C22—84no—
213C22C32×#12011yescharacteristic
214C4—21no—
215C4—22no—
216C4—41no—
217C4—42no—
218C4—81no—
219C3C3×#165914yesdirect product factor
220C2—81no—
221C2—82no—
222C2C32×#45311yescharacteristic
223C2C32×#45512yes—
subgroupG/H
center, socleC22×C32#120
commutator, Frattini, cosocleC42C23×C32
prime radicalC32×H(3,𝔽4)C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C32 ◁ C32×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C32×#497C2semidirect
3. C32×#489C2semidirect
4. C32×#329C2semidirect
5. C32×#329C2semidirect
6. C3×#1659C3direct (N × H)
9. C32×C4⋊Q8C22semidirect
18. C32×#97C22semidirect
19. C32×#97C22semidirect
21. C3×#497C2×C3semidirect
22. C3×#489C2×C3semidirect
23. C3×#329C2×C3semidirect
24. C3×#329C2×C3semidirect
45. #1659C32direct (N × H)
48. C3×C4⋊Q8C22×C3semidirect
57. C3×#97C22×C3semidirect
58. C3×#97C22×C3semidirect
79. #497C2×C32semidirect
80. #489C2×C32semidirect
81. #329C2×C32semidirect
82. #329C2×C32semidirect
117. C4⋊Q8C22×C32semidirect
126. #97C22×C32semidirect
127. #97C22×C32semidirect
183. C32#1659direct (N × H)
219. C3C3×#1659direct (N × H)

Maximal subgroups

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

subgroupindex
C32×H(3,𝔽4)2
C32×#4972
C32×#4892
C32×#3292
C3×#16593