Finite Group Databasebeta

← C2×C3×#131⋊C9 #13152217

Subgroups — C2×C3×#131⋊C9 #13152217

Order 1728.

418 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 146 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×C3×#41⋊C9C211yesmaximal, characteristic
2C3×#131⋊C9C212yesmaximal, direct product factor
3C2×C32×#131C311yesmaximal, characteristic
4C2×#131⋊C9C313yesmaximal, direct product factor
5C3×#41⋊C9C411yescharacteristic
6C3×#41⋊C9C2211yescharacteristic
7C2×C3×Q8⋊C9C411yescharacteristic
8C2×C3×C8×C9—41nomaximal
9C2×C32×#41C2×C311yescharacteristic
10C32×#131—32no—
11C32×#131C2×C312yes—
12C2×C32×M16—31no—
13C22×C32×C8—31no—
14C2×#41⋊C9C2×C313yes—
15#131⋊C9C2×C316yesdirect product factor
16C2×C3×C4×C9—41no—
17C3×Q8⋊C9C2×C411yescharacteristic
18C3×C8×C9—42no—
19C2×C3×#131C913yescomplemented
20C2×C3×#131C3211yescharacteristic
21C32×#41—31no—
22C32×#41—31no—
23C32×#41C3×C411yescharacteristic
24C32×#41C22×C311yescharacteristic
25C2×C32×Q8C3×C411yescharacteristic
26C2×C32×D4—31no—
27C22×C32×C4—31no—
28C32×M16—32no—
29C32×M16—32no—
30C2×C32×C8—31no—
31C2×C32×C8—32no—
32C2×C32×C8—32no—
33C2×C32×C8A411yescharacteristic
34#41⋊C9C3×C413yes—
35#41⋊C9C22×C313yes—
36C2×Q8⋊C9C3×C413yes—
37C2×C8×C9—43no—
38C22×C3×C9—41no—
39C3×C4×C9—41no—
40C3×C4×C9—41no—
41C2×C3×#41C2×C913yes—
42C2×C3×#41C2×C3211yescharacteristic
43C3×#131—36no—
44C3×#131—32no—
45C3×#131C2×C916yescomplemented
46C3×#131C2×C3212yes—
47C2×C3×M16—33no—
48C2×C3×M16—31no—
49C22×C3×C8—33no—
50C22×C3×C8—31no—
51C23×C32—31no—
52C32×Q8—31no—
53C32×Q8C2×C3×C411yescharacteristic
54C32×D4—31no—
55C32×D4—31no—
56C32×D4—32no—
57C2×C32×C4—31no—
58C2×C32×C4—31no—
59C2×C32×C4—31no—
60C2×C32×C4—31no—
61C2×C32×C4—31no—
62C2×C32×C4C2×A411yescharacteristic
63C32×C8—32no—
64C32×C8C2×A412yes—
65C2×C4×C9—43no—
66Q8⋊C9C2×C3×C413yes—
67C8×C9—46no—
68C2×#131C3×C911yescharacteristic, complemented, Sylow
69C2×C3×C9—42no—
70C2×C3×C9—41no—
71C3×#41—33no—
72C3×#41—31no—
73C3×#41—33no—
74C3×#41—31no—
75C3×#41C4×C913yes—
76C3×#41C22×C913yes—
77C3×#41C32×C411yescharacteristic
78C3×#41C22×C3211yescharacteristic
79C2×C3×Q8C4×C913yes—
80C2×C3×Q8C32×C411yescharacteristic
81C2×C3×D4—33no—
82C2×C3×D4—31no—
83C22×C3×C4—33no—
84C22×C3×C4—31no—
85C3×M16—32no—
86C3×M16—36no—
87C3×M16—36no—
88C3×M16—32no—
89C2×C3×C8—33no—
90C2×C3×C8—31no—
91C2×C3×C8—36no—
92C2×C3×C8—36no—
93C2×C3×C8—32no—
94C2×C3×C8—32no—
95C2×C3×C8C22⋊C913yes—
96C2×C3×C8C3×A411yescharacteristic
97C22×C32—31no—
98C22×C32—31no—
99C22×C32—62no—
100C22×C32C4×A411yescharacteristic
101C32×C4—31no—
102C32×C4—31no—
103C32×C4C4×A411yescharacteristic
104C32×C4C22×A411yescharacteristic
105C22×C9—43no—
106C4×C9—43no—
107C4×C9—43no—
108C2×#41C2×C3×C911yescharacteristic
109#131—32no—
110#131C2×C3×C912yescomplemented
111C2×M16—31no—
112C22×C8—31no—
113C3×C9—41noSylow
114C23×C3—33no—
115C23×C3—31no—
116C3×Q8—33no—
117C3×Q8—31no—
118C3×Q8C2×C4×C913yes—
119C3×Q8C2×C32×C411yescharacteristic
120C3×D4—33no—
121C3×D4—33no—
122C3×D4—31no—
123C3×D4—31no—
124C3×D4—32no—
125C3×D4—36no—
126C2×C3×C4—33no—
127C2×C3×C4—31no—
128C2×C3×C4—31no—
129C2×C3×C4—31no—
130C2×C3×C4—33no—
131C2×C3×C4—33no—
132C2×C3×C4—33no—
133C2×C3×C4—33no—
134C2×C3×C4—31no—
135C2×C3×C4—31no—
136C2×C3×C4C2×C22⋊C913yes—
137C2×C3×C4C2×C3×A411yescharacteristic
138C3×C8—32no—
139C3×C8—36no—
140C3×C8C2×C22⋊C916yes—
141C3×C8C2×C3×A412yes—
142C2×C32—61no—
143C2×C32—61no—
144C2×C32#131⋊C312yes—
145C2×C32C2×C4×A411yescharacteristic
146C2×C9—43no—
147C2×C9—46no—
148#41—31no—
149#41—31no—
150#41C3×C4×C911yescharacteristic
151#41C22×C3×C911yescharacteristic
152C2×Q8C3×C4×C911yescharacteristic
153C2×D4—31no—
154C22×C4—31no—
155M16—32no—
156M16—32no—
157C2×C8—31no—
158C2×C8—32no—
159C2×C8—32no—
160C2×C8C3×C22⋊C911yescharacteristic
161C22×C3—31no—
162C22×C3—31no—
163C22×C3—33no—
164C22×C3—33no—
165C22×C3—66no—
166C22×C3—62no—
167C22×C3C4×C22⋊C913yes—
168C22×C3C3×C4×A411yescharacteristic
169C3×C4—31no—
170C3×C4—31no—
171C3×C4—33no—
172C3×C4—33no—
173C3×C4C4×C22⋊C913yes—
174C3×C4C22×C22⋊C913yes—
175C3×C4C3×C4×A411yescharacteristic
176C3×C4C22×C3×A411yescharacteristic
177C32C2×#131⋊C311yescharacteristic
178C9—43no—
179C23—31no—
180Q8—31no—
181Q8C2×C3×C4×C911yescharacteristic
182D4—31no—
183D4—31no—
184D4—32no—
185C2×C4—31no—
186C2×C4—31no—
187C2×C4—31no—
188C2×C4—31no—
189C2×C4—31no—
190C2×C4C2×C3×C22⋊C911yescharacteristic
191C8—32no—
192C8C2×C3×C22⋊C912yes—
193C2×C3—63no—
194C2×C3—63no—
195C2×C3—61no—
196C2×C3—61no—
197C2×C3#131⋊C916yesdirect product factor
198C2×C3C2×C4×C22⋊C913yes—
199C2×C3C3×#131⋊C312yes—
200C2×C3C2×C3×C4×A411yescharacteristic
201C22—31no—
202C22—31no—
203C22—62no—
204C22C3×C4×C22⋊C911yescharacteristic
205C4—31no—
206C4—31no—
207C4C3×C4×C22⋊C911yescharacteristic
208C4C22×C3×C22⋊C911yescharacteristic
209C3C2×#131⋊C913yesdirect product factor
210C3C2×C3×#131⋊C311yescharacteristic
211C2—61no—
212C2—61no—
213C2C3×#131⋊C912yesdirect product factor
214C2C2×C3×C4×C22⋊C911yescharacteristic
subgroupG/H
center, hypercenterC2×C32×C8A4
commutator, hypocenterQ8C2×C3×C4×C9
FrattiniC3×C4C22×C3×A4
FittingC2×C32×#131C3
socleC22×C32C4×A4
prime radicalC2×C32×#41C2×C3
cosocleC3×#41C22×C32
LCSG ▷ Q8
UCSC1 ◁ C2×C32×C8

Complements

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

NH ≅ G/Ntype
2. C3×#131⋊C9C2direct (N × H)
4. C2×#131⋊C9C3direct (N × H)
15. #131⋊C9C2×C3direct (N × H)
19. C2×C3×#131C9semidirect
45. C3×#131C2×C9semidirect
68. C2×#131C3×C9semidirect
110. #131C2×C3×C9semidirect
197. C2×C3#131⋊C9direct (N × H)
209. C3C2×#131⋊C9direct (N × H)
213. C2C3×#131⋊C9direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#41⋊C92
C3×#131⋊C92
C2×C32×#1313
C2×#131⋊C93
C2×C3×C8×C94