Finite Group Databasebeta

← #13143885

Subgroups — #13143885

Order 1728.

750 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 750 conjugacy classes of subgroups — too many to draw. Only its 147 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×#88746C211yesmaximal, characteristic
2M16×C3⋊D9C212yesmaximal, complemented
3C3×C9×#101C211yesmaximal, characteristic, complemented
4#11720581C211yesmaximal, characteristic, complemented
5#11720557C212yesmaximal, complemented
6#10592450—31nomaximal
7#10588431—33nomaximal
8C2×C3×Q8×C9C2211yescharacteristic
9#88746—22no—
10#88746—22no—
11C2×C4⋊(C3⋊D9)C412yes—
12C2×C4⋊(C3⋊D9)C2211yescharacteristic
13C2×C4×C3⋊D9C412yes—
14C2×C4×C3⋊D9C2211yescharacteristic
15C3×C9×M16C2212yes—
16#88532C2212yes—
17#88523—22no—
18C8×C3⋊D9—22no—
19C2×#3832—31no—
20M16×C3⋊D3—32no—
21C2×#3701—33no—
22C32×#101D311yescharacteristic
23#63663—31no—
24#63650—32no—
25M16×D9—36no—
26C9×#101D313yescomplemented
27#63394—33no—
28#63383—36no—
29C22×C3⋊D9C2×C412yes—
30C22×C3⋊D9C2×C411yescharacteristic
31C3×Q8×C9—22no—
32C2×C3×C4×C9C2×C412yes—
33C2×C3×C4×C9C2311yescharacteristic
34C2×C3⋊Dic9C2×C411yescharacteristic
35C4⋊(C3⋊D9)—24no—
36C4⋊(C3⋊D9)—22no—
37C4×C3⋊D9—22no—
38C4×C3⋊D9—22no—
39C4×C3⋊D9D412yes—
40C4×C3⋊D9D412yes—
41C3×C8×C9—22no—
42C3⋊(C9⋊C8)—22no—
43#4751—93no—
44#4751—91no—
45C2×C32×Q8C2×D311yescharacteristic
46#3832—62no—
47#3832—62no—
48C2×C4⋊(C3⋊D3)—32no—
49C2×C4⋊(C3⋊D3)—31no—
50C2×C4×C3⋊D3—31no—
51C2×C4×C3⋊D3—32no—
52C32×M16C2×D312yes—
53#3748—32no—
54#3743—62no—
55C8×C3⋊D3—62no—
56C2×Q8×C9C2×D313yes—
57#3701—66no—
58#3701—66no—
59C2×D36—36no—
60C2×D36—33no—
61C2×C4×D9—33no—
62C2×C4×D9—36no—
63C9×M16C2×D316yes—
64#3667—36no—
65#3663—66no—
66C8×D9—66no—
67C22×C3×C9C22×C411yescharacteristic
68C2×C3⋊D9—22no—
69C2×C3⋊D9—41no—
70C2×C3⋊D9—42no—
71C2×C3⋊D9C22⋊C412yes—
72C3×C4×C9—22no—
73C3×C4×C9C2×D412yes—
74C3⋊Dic9C22⋊C411yescharacteristic
75C3⋊Dic9C22⋊C411yescharacteristic
76C2×#239—93no—
77C2×#239—91no—
78D3×M16—96no—
79D3×M16—92no—
80C3×#101D913yescomplemented
81C3×#101C3⋊D311yescharacteristic
82#836—93no—
83#836—91no—
84#823—96no—
85#823—92no—
86C22×C3⋊D3—31no—
87C22×C3⋊D3—32no—
88C32×Q8—22no—
89C2×C32×C4C4×D312yes—
90C2×C32×C4C22×D311yescharacteristic
91C2×C3⋊Dic3—31no—
92C4⋊(C3⋊D3)—64no—
93C4⋊(C3⋊D3)—62no—
94C4×C3⋊D3—32no—
95C4×C3⋊D3—32no—
96C4×C3⋊D3—62no—
97C4×C3⋊D3—62no—
98C22×D9—33no—
99C22×D9—36no—
100C32×C8—22no—
101C3⋊(C3⋊C8)—62no—
102Q8×C9—26no—
103C2×C4×C9C4×D316yes—
104C2×C4×C9C22×D313yes—
105C2×Dic9—33no—
106D36—612no—
107D36—66no—
108C4×D9—36no—
109C4×D9—36no—
110C4×D9—66no—
111C4×D9—66no—
112C8×C9—26no—
113C9⋊C8—66no—
114#413—271noSylow
115C2×C3×C9—21no—
116C2×C3×C9C2×C22⋊C411yescharacteristic
117C3⋊D9—22no—
118C3⋊D9—42no—
119C2×C3×Q8C2×D913yes—
120C2×C3×Q8C2×C3⋊D311yescharacteristic
121#239—182no—
122#239—182no—
123#239—186no—
124#239—186no—
125C2×D12—96no—
126C2×D12—93no—
127C2×D12—92no—
128C2×D12—91no—
129C2×C4×D3—93no—
130C2×C4×D3—91no—
131C2×C4×D3—96no—
132C2×C4×D3—92no—
133C3×M16C2×D916yes—
134C3×M16C2×C3⋊D312yes—
135#208—96no—
136#208—92no—
137ΓL(1,𝔽25)—182no—
138ΓL(1,𝔽25)—186no—
139D3×C8—182no—
140D3×C8—186no—
141C22×C32C2×C4×D311yescharacteristic
142C2×C3⋊D3—32no—
143C2×C3⋊D3—62no—
144C2×C3⋊D3—121no—
145C2×C3⋊D3—122no—
146C32×C4—22no—
147C32×C4D3×D412yes—
148C3⋊Dic3—31no—
149C3⋊Dic3—31no—
150C22×C9C2×C4×D313yes—
151C2×D9—36no—
152C2×D9—66no—
153C2×D9—123no—
154C2×D9—126no—
155C4×C9—26no—
156C4×C9D3×D416yes—
157Dic9—33no—
158Dic9—33no—
159C2×#41—271no—
160C2×M16—272no—
161#101—271no—
162#101C3⋊D911yescharacteristic, complemented
163#100—272no—
164C3×C9#41311yescharacteristic, complemented, Sylow
165C22×D3—93no—
166C22×D3—91no—
167C22×D3—96no—
168C22×D3—92no—
169C3×Q8—26no—
170C3×Q8—22no—
171C2×C3×C4C4×D916yes—
172C2×C3×C4C22×D913yes—
173C2×C3×C4C4×C3⋊D312yes—
174C2×C3×C4C22×C3⋊D311yescharacteristic
175C2×Dic3—93no—
176C2×Dic3—91no—
177D12—184no—
178D12—1812no—
179D12—186no—
180D12—182no—
181C4×D3—96no—
182C4×D3—96no—
183C4×D3—92no—
184C4×D3—92no—
185C4×D3—182no—
186C4×D3—182no—
187C4×D3—186no—
188C4×D3—186no—
189C3×C8—26no—
190C3×C8—22no—
191C3⋊C8—186no—
192C3⋊C8—182no—
193C2×C32—21no—
194C2×C32D3×C22⋊C411yescharacteristic
195C3⋊D3—62no—
196C3⋊D3—122no—
197C2×C9—23no—
198C2×C9D3×C22⋊C413yes—
199D9—66no—
200D9—126no—
201#41—542no—
202#41—542no—
203C2×Q8C2×C3⋊D911yescharacteristic
204C2×D4—272no—
205C2×D4—271no—
206C22×C4—271no—
207C22×C4—272no—
208M16—272no—
209M16—542no—
210M16C2×C3⋊D912yes—
211C2×C8—542no—
212C22×C3C2×C4×D913yes—
213C22×C3C2×C4×C3⋊D311yescharacteristic
214C2×D3—92no—
215C2×D3—96no—
216C2×D3—182no—
217C2×D3—186no—
218C2×D3—361no—
219C2×D3—363no—
220C2×D3—362no—
221C2×D3—366no—
222C3×C4—22no—
223C3×C4—26no—
224C3×C4D4×D916yes—
225C3×C4D4×C3⋊D312yes—
226Dic3—91no—
227Dic3—91no—
228Dic3—93no—
229Dic3—93no—
230C32#475111yescharacteristic
231C9#475113yescomplemented
232C23—271no—
233C23—272no—
234Q8—22no—
235D4—544no—
236D4—542no—
237C2×C4—271no—
238C2×C4—272no—
239C2×C4—272no—
240C2×C4—542no—
241C2×C4—542no—
242C2×C4C4×C3⋊D912yes—
243C2×C4C22×C3⋊D911yescharacteristic
244C8—22no—
245C8—542no—
246C2×C3—23no—
247C2×C3—21no—
248C2×C3C22⋊C4×D913yes—
249C2×C3C22⋊C4×C3⋊D311yescharacteristic
250D3—186no—
251D3—182no—
252D3—366no—
253D3—362no—
254C22—272no—
255C22—542no—
256C22—1081no—
257C22—1082no—
258C22C2×C4×C3⋊D911yescharacteristic
259C4—22no—
260C4—271no—
261C4—271no—
262C4D4×C3⋊D912yes—
263C3#1058843113yescomplemented
264C3#1059245011yescharacteristic
265C2—21no—
266C2—542no—
267C2—1082no—
268C2C22⋊C4×C3⋊D911yescharacteristic
subgroupG/H
centerC2C22⋊C4×C3⋊D9
commutatorC22×C3×C9C22×C4
FrattiniC2×C3×C4C22×C3⋊D3
FittingC3×C9×#101C2
socleC2×C32D3×C22⋊C4
prime radicalC2×#88746C2
cosocleC2×C3×C4×C9C23
hypercenter#101C3⋊D9
hypocenterC3×C9#413
LCSG ▷ C22×C3×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ #101

Complements

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

NH ≅ G/Ntype
2. M16×C3⋊D9C2semidirect
3. C3×C9×#101C2semidirect
4. #11720581C2semidirect
5. #11720557C2semidirect
26. C9×#101D3semidirect
80. C3×#101D9semidirect
162. #101C3⋊D9semidirect
164. C3×C9#413semidirect
231. C9#4751semidirect
263. C3#10588431semidirect

Maximal subgroups

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

subgroupindex
C2×#887462
M16×C3⋊D92
C3×C9×#1012
#117205812
#117205572
#105924503
#105884313