Finite Group Databasebeta

← C2×C32×#984 #13181362

Subgroups — C2×C32×#984 #13181362

Order 1728.

950 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 950 conjugacy classes of subgroups — too many to draw. Only its 174 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×#41⋊C3C211yesmaximal, characteristic
2C32×#984C214yesmaximal, direct product factor
3C2×C32×#226C212yesmaximal, complemented
4C2×C3×#984C314yesmaximal, direct product factor
5C2×C32×#137—31nomaximal
6C2×C32×Dic6—41nomaximal
7C32×#41⋊C3C2212yes—
8C2×C32×SL(2,𝔽3)C2211yescharacteristic
9C32×#226C2214yescomplemented
10C2×C32×#41D311yescharacteristic
11C22×C32×Q8—31no—
12C2×C3×#41⋊C3—24no—
13C2×C3×#41⋊C3C2×C314yes—
14C3×#984C2×C3116yesdirect product factor
15C2×C3×#226C2×C318yescomplemented
16C32×#137—34no—
17C32×#137—34no—
18C2×C32×Q16—32no—
19C2×C32×SD16—32no—
20C2×C32×M16—31no—
21C2×C33×C4—41no—
22C2×C32×Dic3—42no—
23C32×Dic6—44no—
24C32×SL(2,𝔽3)C2311yescharacteristic
25C2×#984C3211yescharacteristic, direct product factor
26C2×C3×#137—34no—
27C32×#41—32no—
28C32×#41C2×D312yes—
29C2×C32×Q8—32no—
30C2×C32×Q8—34no—
31C2×C32×Q8—62no—
32C2×C32×Q8—61no—
33C2×C32×Q8C2×D311yescharacteristic
34C2×C32×D4—31no—
35C22×C32×C4—31no—
36C22×C32×C4—61no—
37C2×C3×Dic6—44no—
38C3×#41⋊C3—28no—
39C3×#41⋊C3C22×C318yes—
40C2×C3×SL(2,𝔽3)—24no—
41C2×C3×SL(2,𝔽3)C22×C314yes—
42C3×#226C22×C3116yescomplemented
43C32×Q16—34no—
44C32×Q16—34no—
45C32×SD16—38no—
46C32×M16—34no—
47C2×C32×C8—32no—
48C22×C33—41no—
49C33×C4—42no—
50C32×Dic3—44no—
51C2×C3×#41C3×D314yes—
52C22×C3×Q8—34no—
53C2×#41⋊C3—24no—
54C2×#41⋊C3C2×C3211yescharacteristic
55#984C2×C3214yesdirect product factor
56C2×#226C2×C3212yescomplemented
57C3×#137—316no—
58C3×#137—316no—
59C2×C3×Q16—38no—
60C2×C3×SD16—38no—
61C2×C3×M16—34no—
62C23×C32—31no—
63C32×Q8—31no—
64C32×Q8—34no—
65C32×Q8—62no—
66C32×Q8—64no—
67C32×Q8C22×D311yescharacteristic
68C32×D4—32no—
69C32×D4—61no—
70C2×C32×C4—31no—
71C2×C32×C4—32no—
72C2×C32×C4—32no—
73C2×C32×C4—44no—
74C2×C32×C4—64no—
75C2×C32×C4—62no—
76C2×C32×C4—84no—
77C2×C32×C4S411yescharacteristic
78C2×C3×Dic3—48no—
79C3×Dic6—416no—
80C3×SL(2,𝔽3)—24no—
81C3×SL(2,𝔽3)C23×C314yes—
82C32×C8—34no—
83C2×#137—31noSylow
84C2×C33—42no—
85C2×C33—41no—
86C3×#41—38no—
87C3×#41C2×C3×D318yes—
88C2×C3×Q8—38no—
89C2×C3×Q8—316no—
90C2×C3×Q8—68no—
91C2×C3×Q8—64no—
92C2×C3×Q8C2×C3×D314yes—
93C2×C3×D4—34no—
94C22×C3×C4—34no—
95C22×C3×C4—64no—
96C2×Dic6—41no—
97#41⋊C3—28no—
98#41⋊C3C22×C3212yes—
99C2×SL(2,𝔽3)—24no—
100C2×SL(2,𝔽3)C22×C3211yescharacteristic
101#226C22×C3214yescomplemented
102C3×Q16—316no—
103C3×Q16—316no—
104C3×SD16—332no—
105C3×M16—316no—
106C2×C3×C8—38no—
107C22×C32—32no—
108C22×C32—44no—
109C22×C32—62no—
110C22×C32—84no—
111C22×C32C2×S411yescharacteristic
112C32×C4—31no—
113C32×C4—31no—
114C32×C4—48no—
115C32×C4—64no—
116C32×C4—88no—
117C32×C4C2×S412yes—
118C3×Dic3—416no—
119C2×#41C32×D311yescharacteristic
120C22×Q8—31no—
121#137—34no—
122#137—34no—
123C2×Q16—32no—
124C2×SD16—32no—
125C2×M16—31no—
126C33—41noSylow
127C23×C3—34no—
128C3×Q8—34no—
129C3×Q8—316no—
130C3×Q8—68no—
131C3×Q8—616no—
132C3×Q8C22×C3×D314yes—
133C3×D4—38no—
134C3×D4—64no—
135C2×C3×C4—34no—
136C2×C3×C4—38no—
137C2×C3×C4—38no—
138C2×C3×C4—41no—
139C2×C3×C4—616no—
140C2×C3×C4—68no—
141C2×C3×C4—84no—
142C2×C3×C4C3×S414yes—
143C2×Dic3—42no—
144Dic6—44no—
145SL(2,𝔽3)—24no—
146SL(2,𝔽3)C23×C3211yescharacteristic
147C3×C8—316no—
148C2×C32—44no—
149C2×C32—48no—
150C2×C32—62no—
151C2×C32—84no—
152C2×C32—88no—
153C2×C32#98412yesdirect product factor
154C2×C32C22×S411yescharacteristic
155#41—32no—
156#41C2×C32×D312yes—
157C2×Q8—32no—
158C2×Q8—34no—
159C2×Q8—62no—
160C2×Q8—61no—
161C2×Q8C2×C32×D311yescharacteristic
162C2×D4—31no—
163C22×C4—31no—
164C22×C4—61no—
165Q16—34no—
166Q16—34no—
167SD16—38no—
168M16—34no—
169C2×C8—32no—
170C22×C3—38no—
171C22×C3—41no—
172C22×C3—68no—
173C22×C3—84no—
174C22×C3C2×C3×S414yes—
175C3×C4—34no—
176C3×C4—34no—
177C3×C4—42no—
178C3×C4—616no—
179C3×C4—88no—
180C3×C4C2×C3×S418yes—
181Dic3—44no—
182C32—44no—
183C32—84no—
184C32C2×#98411yescharacteristic, direct product factor
185C23—31no—
186Q8—31no—
187Q8—34no—
188Q8—62no—
189Q8—64no—
190Q8C22×C32×D311yescharacteristic
191D4—32no—
192D4—61no—
193C2×C4—31no—
194C2×C4—32no—
195C2×C4—32no—
196C2×C4—64no—
197C2×C4—62no—
198C2×C4C32×S411yescharacteristic
199C8—34no—
200C2×C3—42no—
201C2×C3—41no—
202C2×C3—68no—
203C2×C3—88no—
204C2×C3—84no—
205C2×C3C3×#98418yesdirect product factor
206C2×C3C22×C3×S414yes—
207C22—32no—
208C22—62no—
209C22C2×C32×S411yescharacteristic
210C4—31no—
211C4—31no—
212C4—64no—
213C4C2×C32×S412yes—
214C3—41no—
215C3—84no—
216C3C2×C3×#98414yesdirect product factor
217C2—62no—
218C2C32×#98412yesdirect product factor
219C2C22×C32×S411yescharacteristic
subgroupG/H
center, socleC22×C32C2×S4
commutator, cosocle, hypocenterSL(2,𝔽3)C23×C32
FrattiniC2C22×C32×S4
FittingC2×C32×#41D3
prime radicalC2×C32×#41⋊C3C2
hypercenterC2×C32×C4S4
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C32 ◁ C2×C32×C4

Complements

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

NH ≅ G/Ntype
2. C32×#984C2direct (N × H)
3. C2×C32×#226C2semidirect
4. C2×C3×#984C3direct (N × H)
9. C32×#226C22semidirect
14. C3×#984C2×C3direct (N × H)
15. C2×C3×#226C2×C3semidirect
25. C2×#984C32direct (N × H)
42. C3×#226C22×C3semidirect
55. #984C2×C32direct (N × H)
56. C2×#226C2×C32semidirect
101. #226C22×C32semidirect
153. C2×C32#984direct (N × H)
184. C32C2×#984direct (N × H)
205. C2×C3C3×#984direct (N × H)
216. C3C2×C3×#984direct (N × H)
218. C2C32×#984direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×#41⋊C32
C32×#9842
C2×C32×#2262
C2×C3×#9843
C2×C32×#1373
C2×C32×Dic64