Finite Group Databasebeta

← C32×#5395 #13172949

Subgroups — C32×#5395 #13172949

Order 1728.

526 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 90 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
1C2×C32×GL(2,𝔽3)C211yesmaximal, characteristic
2C2×C32×#226C211yesmaximal, characteristic, complemented
3C32×C4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4C3×#5395C314yesmaximal, direct product factor
5C32×#464—31nomaximal
6C2×C32×SL(2,𝔽3)C2211yescharacteristic
7C32×GL(2,𝔽3)—21no—
8C32×#226—21no—
9C32×(C2×D3)⋊C4—41nomaximal
10C2×C3×GL(2,𝔽3)C2×C314yes—
11C2×C3×#226C2×C314yescomplemented
12C2×C32×Q16—31no—
13C2×C32×SD16—31no—
14C32×#124—31no—
15C32×C4×Q8D311yescharacteristic, complemented
16C3×C4×SL(2,𝔽3)—24no—
17C3×C4×SL(2,𝔽3)C2×C314yescomplemented
18C32×C4⋊C8—31no—
19C32×#103—31no—
20C32×#102—31no—
21C22×C32×D3—41no—
22C2×C33×C4—41no—
23C2×C32×Dic3—41no—
24C32×SL(2,𝔽3)D411yescharacteristic
25#5395C3211yescharacteristic, direct product factor
26C3×#464—34no—
27C2×C32×Q8—31no—
28C2×C32×Q8C2×D311yescharacteristic
29C2×C32×D4—31no—
30C2×C3×SL(2,𝔽3)—24no—
31C2×C3×SL(2,𝔽3)C22×C314yes—
32C3×GL(2,𝔽3)—24no—
33C3×#226—24no—
34C32×Q16—61no—
35C32×Q16—61no—
36C32×SD16—61no—
37C32×SD16—61no—
38C2×C32×C8—31no—
39C2×C32×C8—31no—
40C32×C4⋊C4—31no—
41C32×C22⋊C4—61no—
42C32×C22⋊C4—61no—
43C32×C42—31no—
44C3×(C2×D3)⋊C4—44no—
45C22×C33—41no—
46C2×C32×D3—42no—
47C2×C32×D3—81no—
48C2×C32×D3—81no—
49C33×C4—81no—
50C32×Dic3—81no—
51C2×GL(2,𝔽3)C2×C3211yescharacteristic
52C2×#226C2×C3211yescharacteristic, complemented
53C2×C3×Q16—34no—
54C2×C3×SD16—34no—
55C3×#124—34no—
56C3×C4×Q8C3×D314yescomplemented
57C4×SL(2,𝔽3)—24no—
58C4×SL(2,𝔽3)C2×C3211yescharacteristic, complemented
59C3×C4⋊C8—34no—
60C3×#103—34no—
61C3×#102—34no—
62C23×C32—61no—
63C22×C3×D3—44no—
64C32×Q8—31no—
65C32×Q8—61no—
66C32×Q8—61no—
67C32×Q8D1211yescharacteristic
68C32×D4—61no—
69C32×D4—61no—
70C2×C32×C4—31no—
71C2×C32×C4—31no—
72C2×C32×C4—44no—
73C2×C32×C4—61no—
74C2×C32×C4—84no—
75C2×C32×C4S411yescharacteristic
76C2×C3×Dic3—44no—
77C3×SL(2,𝔽3)—24no—
78C3×SL(2,𝔽3)C3×D414yes—
79C32×C8—61no—
80C32×C8—61no—
81#464—31noSylow
82C2×C33—41no—
83C2×C33—41no—
84C2×C33—41no—
85C32×D3—82no—
86C2×C3×Q8—34no—
87C2×C3×Q8C2×C3×D314yes—
88C2×C3×D4—34no—
89C2×SL(2,𝔽3)—24no—
90C2×SL(2,𝔽3)C22×C3211yescharacteristic
91GL(2,𝔽3)—21no—
92#226—21no—
93C3×Q16—64no—
94C3×Q16—64no—
95C3×SD16—64no—
96C3×SD16—64no—
97C2×C3×C8—34no—
98C2×C3×C8—34no—
99C3×C4⋊C4—34no—
100C3×C22⋊C4—64no—
101C3×C22⋊C4—64no—
102C3×C42—34no—
103(C2×D3)⋊C4—41no—
104C22×C32—44no—
105C22×C32—84no—
106C22×C32—121no—
107C22×C32—121no—
108C22×C32—121no—
109C22×C32C2×S411yescharacteristic
110C2×C3×D3—48no—
111C2×C3×D3—84no—
112C2×C3×D3—84no—
113C32×C4—21no—
114C32×C4—31no—
115C32×C4—31no—
116C32×C4—61no—
117C32×C4—84no—
118C32×C4—88no—
119C32×C4—121no—
120C3×Dic3—84no—
121C2×Q16—31no—
122C2×SD16—31no—
123#124—31no—
124C4×Q8C32×D311yescharacteristic, complemented
125C4⋊C8—31no—
126#103—31no—
127#102—31no—
128C33—41noSylow
129C23×C3—64no—
130C22×D3—41no—
131C3×Q8—34no—
132C3×Q8—64no—
133C3×Q8—64no—
134C3×Q8C3×D1214yes—
135C3×D4—64no—
136C3×D4—64no—
137C2×C3×C4—34no—
138C2×C3×C4—34no—
139C2×C3×C4—41no—
140C2×C3×C4—64no—
141C2×C3×C4—84no—
142C2×C3×C4C3×S414yes—
143C2×Dic3—41no—
144SL(2,𝔽3)—24no—
145SL(2,𝔽3)C32×D411yescharacteristic
146C3×C8—64no—
147C3×C8—64no—
148C2×C32—44no—
149C2×C32—44no—
150C2×C32—44no—
151C2×C32—84no—
152C2×C32—84no—
153C2×C32—84no—
154C2×C32—241no—
155C2×C32C4⋊S411yescharacteristic
156C2×C32#98311yescharacteristic
157C2×C32#98511yescharacteristic
158C3×D3—88no—
159C2×Q8—31no—
160C2×Q8C2×C32×D311yescharacteristic
161C2×D4—31no—
162Q16—61no—
163Q16—61no—
164SD16—61no—
165SD16—61no—
166C2×C8—31no—
167C2×C8—31no—
168C4⋊C4—31no—
169C22⋊C4—61no—
170C22⋊C4—61no—
171C42—31no—
172C22×C3—41no—
173C22×C3—84no—
174C22×C3—124no—
175C22×C3—124no—
176C22×C3—124no—
177C22×C3C2×C3×S414yes—
178C2×D3—42no—
179C2×D3—81no—
180C2×D3—81no—
181C3×C4—24no—
182C3×C4—34no—
183C3×C4—34no—
184C3×C4—64no—
185C3×C4—81no—
186C3×C4—88no—
187C3×C4—124no—
188Dic3—81no—
189C32—44no—
190C32—84no—
191C32#539511yescharacteristic, direct product factor
192C23—61no—
193Q8—31no—
194Q8—61no—
195Q8—61no—
196Q8C32×D1211yescharacteristic
197D4—61no—
198D4—61no—
199C2×C4—31no—
200C2×C4—31no—
201C2×C4—61no—
202C2×C4C32×S411yescharacteristic
203C8—61no—
204C8—61no—
205C2×C3—41no—
206C2×C3—41no—
207C2×C3—41no—
208C2×C3—84no—
209C2×C3—84no—
210C2×C3—84no—
211C2×C3—244no—
212C2×C3C3×C4⋊S414yes—
213C2×C3C3×#98314yes—
214C2×C3C3×#98514yes—
215D3—82no—
216C22—121no—
217C22—121no—
218C22—121no—
219C22C2×C32×S411yescharacteristic
220C4—21no—
221C4—31no—
222C4—31no—
223C4—61no—
224C4—121no—
225C3—41no—
226C3—84no—
227C3C3×#539514yesdirect product factor
228C2—241no—
229C2C32×C4⋊S411yescharacteristic
230C2C32×#98311yescharacteristic
231C2C32×#98511yescharacteristic
subgroupG/H
center, socleC22×C32C2×S4
commutator, cosocleC2×SL(2,𝔽3)C22×C32
FrattiniC22C2×C32×S4
FittingC32×C4×Q8D3
prime radicalC2×C32×GL(2,𝔽3)C2
hypercenterC2×C32×C4S4
hypocenterSL(2,𝔽3)C32×D4
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C32 ◁ C2×C32×C4

Complements

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

NH ≅ G/Ntype
2. C2×C32×#226C2semidirect
3. C32×C4×SL(2,𝔽3)C2semidirect
4. C3×#5395C3direct (N × H)
11. C2×C3×#226C2×C3semidirect
15. C32×C4×Q8D3semidirect
17. C3×C4×SL(2,𝔽3)C2×C3semidirect
25. #5395C32direct (N × H)
52. C2×#226C2×C32semidirect
56. C3×C4×Q8C3×D3semidirect
58. C4×SL(2,𝔽3)C2×C32semidirect
124. C4×Q8C32×D3semidirect
191. C32#5395direct (N × H)
227. C3C3×#5395direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×GL(2,𝔽3)2
C2×C32×#2262
C32×C4×SL(2,𝔽3)2
C3×#53953
C32×#4643
C32×(C2×D3)⋊C44