Finite Group Databasebeta

← #13154216

Subgroups — #13154216

Order 1728.

960 conjugacy classes of subgroups; only the 271 normal subgroups are shown.

normal subgroups only

This group has 960 conjugacy classes of subgroups — too many to draw. Only its 271 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
1C3×C9×#125C211yesmaximal, characteristic, complemented
2#11721754C214yesmaximal
3#11721753C212yesmaximal
4#11721746C214yesmaximal, complemented
5#11721743C212yesmaximal, complemented
6#11721709C211yesmaximal, characteristic
7#11721706C211yesmaximal, characteristic, complemented
8#10596169—31nomaximal
9#10589800—33nomaximal
10C2×C4×C3⋊D9C2211yescharacteristic
11C2×#6226C2212yes—
12C3×C9×C4⋊C4C2212yes—
13C3×C9×C4⋊C4C2214yes—
14C3×C42×C9C2211yescharacteristic
15#88536C2214yes—
16#88536C2212yes—
17C4⋊(C3⋊Dic9)C2214yes—
18C4⋊(C3⋊Dic9)C2214yes—
19#88534C2214yes—
20#88534C2212yes—
21C4×C3⋊Dic9C2211yescharacteristic
22C4×C3⋊Dic9C2214yes—
23C32×#125D311yescharacteristic
24#64107—34no—
25#64106—32no—
26#64099—34no—
27#64096—32no—
28#64081—31no—
29#64078—31no—
30C9×#125D313yescomplemented
31#63460—312no—
32#63459—36no—
33#63452—312no—
34#63449—36no—
35#63434—33no—
36#63431—33no—
37C22×C3⋊D9C2311yescharacteristic
38C2×C3×C4×C9C2314yes—
39C2×C3×C4×C9C2312yes—
40C2×C3×C4×C9C2311yescharacteristic
41C2×C3⋊Dic9C2314yes—
42C2×C3⋊Dic9C2311yescharacteristic
43C2×C3⋊Dic9C2312yes—
44C4×C3⋊D9—22no—
45#6226—24no—
46#5698—93no—
47#5698—91no—
48C2×C4×C3⋊D3—31no—
49C2×#634—32no—
50C32×C4⋊C4C2×D312yes—
51C32×C4⋊C4C2×D314yes—
52C32×C42C2×D311yescharacteristic
53#3752—32no—
54#3752—34no—
55C4⋊(C3⋊Dic3)—34no—
56C4⋊(C3⋊Dic3)—34no—
57#3750—32no—
58#3750—34no—
59C4×C3⋊Dic3—31no—
60C4×C3⋊Dic3—34no—
61C2×C4×D9—33no—
62C2×Dic18—36no—
63C9×C4⋊C4C2×D316yes—
64C9×C4⋊C4C2×D3112yes—
65C42×C9C2×D313yes—
66(C2×D9)⋊C4—36no—
67(C2×D9)⋊C4—312no—
68C4⋊Dic9—312no—
69C4⋊Dic9—312no—
70Dic9⋊C4—36no—
71Dic9⋊C4—312no—
72C4×Dic9—33no—
73C4×Dic9—312no—
74C22×C3×C9C2411yescharacteristic
75C2×C3⋊D9—21no—
76C2×C3⋊D9—22no—
77C3×C4×C9—24no—
78C3×C4×C9—22no—
79C3×C4×C9#4112yes—
80C3⋊Dic9—24no—
81C3⋊Dic9—21no—
82C3⋊Dic9—22no—
83C3×#125D913yescomplemented
84C3×#125C3⋊D311yescharacteristic
85#898—912no—
86#898—94no—
87#897—96no—
88#897—92no—
89#890—912no—
90#890—94no—
91#887—96no—
92#887—92no—
93#872—93no—
94#872—91no—
95#869—93no—
96#869—91no—
97C22×C3⋊D3—31no—
98C2×C32×C4C22×D314yes—
99C2×C32×C4C22×D312yes—
100C2×C32×C4C22×D311yescharacteristic
101C2×C3⋊Dic3—32no—
102C2×C3⋊Dic3—34no—
103C2×C3⋊Dic3—31no—
104C4×C3⋊D3—62no—
105#634—64no—
106C22×D9—33no—
107C2×C4×C9C22×D3112yes—
108C2×C4×C9C22×D316yes—
109C2×C4×C9C22×D313yes—
110C2×Dic9—36no—
111C2×Dic9—312no—
112C2×Dic9—33no—
113C4×D9—66no—
114Dic18—612no—
115#541—271noSylow
116C2×C3×C9C2×#4111yescharacteristic
117C2×C3×C92−1+412yes—
118C3⋊D9—41no—
119C2×C4×D3—93no—
120C2×C4×D3—91no—
121C2×Dic6—96no—
122C2×Dic6—92no—
123C3×C4⋊C4C2×D916yes—
124C3×C4⋊C4C2×D9112yes—
125C3×C4⋊C4C2×C3⋊D312yes—
126C3×C4⋊C4C2×C3⋊D314yes—
127C3×C42C2×D913yes—
128C3×C42C2×C3⋊D311yescharacteristic
129(C2×D3)⋊C4—96no—
130(C2×D3)⋊C4—912no—
131(C2×D3)⋊C4—92no—
132(C2×D3)⋊C4—94no—
133C4⋊Dic3—912no—
134C4⋊Dic3—912no—
135C4⋊Dic3—94no—
136C4⋊Dic3—94no—
137Dic3⋊C4—96no—
138Dic3⋊C4—912no—
139Dic3⋊C4—92no—
140Dic3⋊C4—94no—
141C4×Dic3—93no—
142C4×Dic3—912no—
143C4×Dic3—91no—
144C4×Dic3—94no—
145C22×C32C23×D311yescharacteristic
146C2×C3⋊D3—61no—
147C2×C3⋊D3—62no—
148C32×C4—24no—
149C32×C4—22no—
150C32×C4#23912yes—
151C3⋊Dic3—64no—
152C3⋊Dic3—61no—
153C3⋊Dic3—62no—
154C22×C9C23×D313yes—
155C2×D9—63no—
156C2×D9—66no—
157C4×C9—212no—
158C4×C9—26no—
159C4×C9#23916yes—
160Dic9—612no—
161Dic9—63no—
162Dic9—66no—
163C4⋊Q8—271no—
164#126—274no—
165#125—274no—
166#125C3⋊D911yescharacteristic, complemented
167C22⋊Q8—272no—
168C4×Q8—272no—
169#117—271no—
170C3×C9#54111yescharacteristic, complemented, Sylow
171C22×D3—91no—
172C22×D3—93no—
173C2×C3×C4C22×D9112yes—
174C2×C3×C4C22×D916yes—
175C2×C3×C4C22×D913yes—
176C2×C3×C4C22×C3⋊D314yes—
177C2×C3×C4C22×C3⋊D312yes—
178C2×C3×C4C22×C3⋊D311yescharacteristic
179C2×Dic3—92no—
180C2×Dic3—94no—
181C2×Dic3—91no—
182C2×Dic3—96no—
183C2×Dic3—912no—
184C2×Dic3—93no—
185C4×D3—186no—
186C4×D3—182no—
187Dic6—1812no—
188Dic6—184no—
189C2×C32C2×#23911yescharacteristic
190C2×C32#101012yes—
191C3⋊D3—121no—
192C2×C9C2×#23913yes—
193C2×C9#101016yes—
194D9—123no—
195C2×Q8—272no—
196C22×C4—271no—
197C4⋊C4—272no—
198C4⋊C4—274no—
199C4⋊C4—274no—
200C4⋊C4—274no—
201C4⋊C4C2×C3⋊D912yes—
202C4⋊C4C2×C3⋊D914yes—
203C22⋊C4—272no—
204C22⋊C4—274no—
205C42—271no—
206C42—274no—
207C42C2×C3⋊D911yescharacteristic
208C22×C3C23×D913yes—
209C22×C3C23×C3⋊D311yescharacteristic
210C2×D3—181no—
211C2×D3—182no—
212C2×D3—183no—
213C2×D3—186no—
214C3×C4—24no—
215C3×C4—22no—
216C3×C4—212no—
217C3×C4—26no—
218C3×C4#370116yes—
219C3×C4#383212yes—
220Dic3—184no—
221Dic3—181no—
222Dic3—182no—
223Dic3—1812no—
224Dic3—183no—
225Dic3—186no—
226C32#569811yescharacteristic
227C9#569813yescomplemented
228C23—271no—
229Q8—544no—
230C2×C4—272no—
231C2×C4—274no—
232C2×C4—271no—
233C2×C4—542no—
234C2×C4C22×C3⋊D914yes—
235C2×C4C22×C3⋊D912yes—
236C2×C4C22×C3⋊D911yescharacteristic
237C2×C3C2×#370113yes—
238C2×C3#6371616yes—
239C2×C3C2×#383211yescharacteristic
240C2×C3#6436712yes—
241D3—361no—
242D3—363no—
243C22—541no—
244C22—542no—
245C22C23×C3⋊D911yescharacteristic
246C4—24no—
247C4—22no—
248C4—541no—
249C4—544no—
250C4—542no—
251C4#8874612yes—
252C3#1058980013yescomplemented
253C3#1059616911yescharacteristic
254C2—1081no—
255C2C2×#8874611yescharacteristic
256C2#1172267312yes—
subgroupG/H
centerC22C23×C3⋊D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C23×C3⋊D3
FittingC3×C9×#125C2
socleC22×C32C23×D3
prime radicalC22×C3⋊D9C23
hypercenter#125C3⋊D9
hypocenterC3×C9#541
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ #125

Complements

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

NH ≅ G/Ntype
1. C3×C9×#125C2semidirect
4. #11721746C2semidirect
5. #11721743C2semidirect
7. #11721706C2semidirect
30. C9×#125D3semidirect
83. C3×#125D9semidirect
166. #125C3⋊D9semidirect
170. C3×C9#541semidirect
227. C9#5698semidirect
252. C3#10589800semidirect

Maximal subgroups

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

subgroupindex
C3×C9×#1252
#117217542
#117217532
#117217462
#117217432
#117217092
#117217062
#105961693
#105898003