Finite Group Databasebeta

← C3×#71589 #11785740

Subgroups — C3×#71589 #11785740

Order 1152.

709 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 709 conjugacy classes of subgroups — too many to draw. Only its 186 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×#117C211yesmaximal, characteristic
2C3×#5207C212yesmaximal
3C3×#4530C214yesmaximal, complemented
4C3×#1822—31nomaximal
5#71589C311yesmaximal, characteristic, direct product factor
6C23×C32×C4C2211yescharacteristic
7C32×#117—24no—
8C2×C32×C4⋊C4C2212yes—
9C2×C32×C22⋊C4C412yes—
10C2×C32×C42C412yes—
11C2×C3×#208C2214yes—
12C3×#830—24no—
13C2×C3×#117—21no—
14C2×C3×#117D311yescharacteristic
15C2×C3×#117C2×C311yescharacteristic
16C3×#407—32no—
17#5207C2×C312yes—
18C3×#343—34no—
19#4530C2×C314yescomplemented
20C24×C32C2×C411yescharacteristic
21C22×C32×C4—24no—
22C22×C32×C4C2×C411yescharacteristic
23C22×C32×C4C2×C414yes—
24C22×C32×C4D414yes—
25C22×C32×C4C2311yescharacteristic
26C32×C4⋊C4—24no—
27C32×C22⋊C4—24no—
28C32×C42—24no—
29C3×#208—28no—
30C2×C3×C3⋊C8—24no—
31#1822—31noSylow
32C23×C3×C4—21no—
33C23×C3×C4C2×D311yescharacteristic
34C23×C3×C4C22×C311yescharacteristic
35C2×C3×M16—34no—
36C3×#117—28no—
37C3×#117—24no—
38C3×#117—24no—
39C2×C3×C4⋊C4—22no—
40C2×C3×C4⋊C4C2×D312yes—
41C2×C3×C4⋊C4C22×C312yes—
42C2×C3×C22⋊C4—22no—
43C2×C3×C22⋊C4Dic312yes—
44C2×C3×C22⋊C4C3×C412yes—
45C2×C3×C42—22no—
46C2×C3×C42Dic312yes—
47C2×C3×C42C3×C412yes—
48C2×#208C22×C314yes—
49C3×C22⋊C8—64no—
50#830—24no—
51C23×C32—22no—
52C23×C32—42no—
53C23×C32C22⋊C412yes—
54C23×C32C22×C411yescharacteristic
55C2×C32×C4—22no—
56C2×C32×C4—28no—
57C2×C32×C4—24no—
58C2×C32×C4—44no—
59C2×C32×C4C22⋊C412yes—
60C2×C32×C4C2×D412yes—
61C2×C32×C4#4114yes—
62C3×C3⋊C8—44no—
63C2×#117C3×D311yescharacteristic
64#407—32no—
65#343—34no—
66C24×C3—21no—
67C24×C3C2×Dic311yescharacteristic
68C24×C3C2×C3×C411yescharacteristic
69C22×C3×C4—28no—
70C22×C3×C4—24no—
71C22×C3×C4—24no—
72C22×C3×C4—21no—
73C22×C3×C4—24no—
74C22×C3×C4—24no—
75C22×C3×C4—21no—
76C22×C3×C4C2×Dic311yescharacteristic
77C22×C3×C4C2×Dic314yes—
78C22×C3×C4#6714yes—
79C22×C3×C4C2×C3×C411yescharacteristic
80C22×C3×C4C2×C3×C414yes—
81C22×C3×C4C3×D414yes—
82C22×C3×C4C22×D311yescharacteristic
83C22×C3×C4C23×C311yescharacteristic
84C3×M16—68no—
85C2×C3×C8—64no—
86C3×C4⋊C4—28no—
87C3×C4⋊C4—24no—
88C3×C4⋊C4—24no—
89C3×C22⋊C4—28no—
90C3×C22⋊C4—24no—
91C3×C22⋊C4—24no—
92C3×C42—28no—
93C3×C42—24no—
94C3×C42—24no—
95#208—28no—
96C2×C3⋊C8—24no—
97C22×C32—22no—
98C22×C32—22no—
99C22×C32—24no—
100C22×C32—44no—
101C22×C32C2×C22⋊C411yescharacteristic
102C22×C32#11712yes—
103C32×C4—22no—
104C32×C4—44no—
105C32×C4#12314yes—
106C23×C4C2×C3×D311yescharacteristic
107C2×M16—34no—
108#117—24no—
109C2×C4⋊C4C2×C3×D312yes—
110C2×C22⋊C4C3×Dic312yes—
111C2×C42C3×Dic312yes—
112C22⋊C8—64no—
113C23×C3—24no—
114C23×C3—22no—
115C23×C3—22no—
116C23×C3—22no—
117C23×C3—21no—
118C23×C3—42no—
119C23×C3—42no—
120C23×C3—44no—
121C23×C3#21712yes—
122C23×C3C3×C22⋊C412yes—
123C23×C3C22×Dic311yescharacteristic
124C23×C3C22×C3×C411yescharacteristic
125C2×C3×C4—24no—
126C2×C3×C4—216no—
127C2×C3×C4—22no—
128C2×C3×C4—22no—
129C2×C3×C4—28no—
130C2×C3×C4—28no—
131C2×C3×C4—28no—
132C2×C3×C4—24no—
133C2×C3×C4—24no—
134C2×C3×C4—22no—
135C2×C3×C4—24no—
136C2×C3×C4—22no—
137C2×C3×C4—48no—
138C2×C3×C4—44no—
139C2×C3×C4—44no—
140C2×C3×C4#21712yes—
141C2×C3×C4C3×C22⋊C412yes—
142C2×C3×C4#23514yes—
143C2×C3×C4C2×#6712yes—
144C2×C3×C4C2×C3×D412yes—
145C2×C3×C4C3×#4114yes—
146C3×C8—124no—
147C3⋊C8—44no—
148C2×C32—22no—
149C2×C32—42no—
150C2×C32#38111yescharacteristic
151C2×C32#41312yes—
152C24C2×C3×Dic311yescharacteristic
153C22×C4—24no—
154C22×C4C2×C3×Dic311yescharacteristic
155C22×C4C2×C3×Dic314yes—
156C22×C4C3×#6714yes—
157C22×C4C22×C3×D311yescharacteristic
158M16—68no—
159C2×C8—64no—
160C4⋊C4—24no—
161C22⋊C4—24no—
162C42—24no—
163C22×C3—24no—
164C22×C3—22no—
165C22×C3—22no—
166C22×C3—24no—
167C22×C3—22no—
168C22×C3—24no—
169C22×C3—22no—
170C22×C3—24no—
171C22×C3—22no—
172C22×C3—21no—
173C22×C3—44no—
174C22×C3—48no—
175C22×C3—44no—
176C22×C3—44no—
177C22×C3#92612yes—
178C22×C3C2×#21711yescharacteristic
179C22×C3C2×C3×C22⋊C411yescharacteristic
180C22×C3C3×#11712yes—
181C3×C4—24no—
182C3×C4—22no—
183C3×C4—22no—
184C3×C4—24no—
185C3×C4—48no—
186C3×C4—44no—
187C3×C4—44no—
188C3×C4#92914yes—
189C3×C4C3×#12314yes—
190C32#182211yescharacteristic, complemented, Sylow
191C23—22no—
192C23—42no—
193C23C3×#21712yes—
194C23C22×C3×Dic311yescharacteristic
195C2×C4—22no—
196C2×C4—28no—
197C2×C4—24no—
198C2×C4—44no—
199C2×C4C3×#21712yes—
200C2×C4C3×#23514yes—
201C2×C4C2×C3×#6712yes—
202C8—124no—
203C2×C3—24no—
204C2×C3—22no—
205C2×C3—22no—
206C2×C3—21no—
207C2×C3—22no—
208C2×C3—44no—
209C2×C3—42no—
210C2×C3—42no—
211C2×C3#521111yescharacteristic
212C2×C3#523712yes—
213C2×C3C3×#38111yescharacteristic
214C2×C3C3×#41312yes—
215C22—22no—
216C22—22no—
217C22—24no—
218C22—44no—
219C22C3×#92612yes—
220C22C2×C3×#21711yescharacteristic
221C4—22no—
222C4—44no—
223C4C3×#92914yes—
224C3—21no—
225C3#7158911yescharacteristic, direct product factor
226C3C3×#182211yescharacteristic, complemented
227C2—22no—
228C2—42no—
229C2C3×#521111yescharacteristic
230C2C3×#523712yes—
subgroupG/H
centerC22×C3C2×#217
commutatorC23×C3C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC2×C32×#117C2
socleC22×C32C2×C22⋊C4
prime radicalC24×C32C2×C4
cosocleC22×C3×C4C23×C3
hypercenterC2×C3×#117D3
hypocenterC3C3×#1822
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ C2×C3×#117

Complements

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

NH ≅ G/Ntype
3. C3×#4530C2semidirect
5. #71589C3direct (N × H)
19. #4530C2×C3semidirect
190. C32#1822semidirect
225. C3#71589direct (N × H)
226. C3C3×#1822semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#1172
C3×#52072
C3×#45302
C3×#18223
#715893