Finite Group Databasebeta

← #13173408

Subgroups — #13173408

Order 1728.

1016 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 1016 conjugacy classes of subgroups — too many to draw. Only its 109 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
1C22×C3×C4×C3⋊D3C211yesmaximal, characteristic
2C2×#88994C212yesmaximal
3#11723106C214yesmaximal, complemented
4#10593811—31nomaximal
5C23×C3×C3⋊D3C2211yescharacteristic
6C22×C3⋊#157C2214yes—
7C22×C33×C4C411yescharacteristic, complemented
8C22×C3×C3⋊Dic3C411yescharacteristic, complemented
9C2×C3×C4×C3⋊D3—24no—
10C2×C3×C4×C3⋊D3C412yescomplemented
11C2×C3×C4×C3⋊D3C2212yes—
12#88994—24no—
13C22×C4×C3⋊D3D311yescharacteristic
14C22×C3×C4×D3—62no—
15C2×C22⋊#157—32no—
16#63781—34no—
17C23×C33C2×C411yescharacteristic
18C22×C3×C3⋊D3—24no—
19C22×C3×C3⋊D3C2×C411yescharacteristic
20C22×C3×C3⋊D3D414yes—
21C22×C3×C3⋊D3C2311yescharacteristic
22C2×C3⋊#157—24no—
23C2×C3⋊#157—28no—
24C2×C33×C4—22no—
25C2×C33×C4C2×C412yes—
26C2×C3×C3⋊Dic3—22no—
27C2×C3×C3⋊Dic3C2×C412yes—
28C3×C4×C3⋊D3—24no—
29C3×C4×C3⋊D3—24no—
30#5211—91nomaximal
31C23×C3⋊D3C2×D311yescharacteristic
32C23×C3×D3—62no—
33C22×#157—34no—
34C22×C32×C4—22no—
35C22×C32×C4—42no—
36C22×C32×C4Dic311yescharacteristic, complemented
37C22×C3⋊Dic3Dic311yescharacteristic, complemented
38C2×C4×C3⋊D3—24no—
39C2×C4×C3⋊D3Dic312yescomplemented
40C2×C4×C3⋊D3C2×D312yes—
41C22×C3×Dic3—62no—
42C2×C3×C4×D3—64no—
43C2×C3×C4×D3—64no—
44C2×C3×C4×D3—616no—
45C22⋊#157—64no—
46C22×C33—22no—
47C22×C33C22⋊C412yes—
48C22×C33C22×C411yescharacteristic
49C2×C3×C3⋊D3—22no—
50C2×C3×C3⋊D3—28no—
51C2×C3×C3⋊D3C22⋊C412yes—
52C2×C3×C3⋊D3C2×D412yes—
53C2×C3×C3⋊D3#4114yes—
54C3⋊#157—44no—
55C33×C4—22no—
56C3×C3⋊Dic3—22no—
57C23×C3×C4—91no—
58C22×C4×D3—62no—
59C2×#217—92no—
60C3⋊#95—94no—
61C23×C32—22no—
62C23×C32—42no—
63C23×C32C2×Dic311yescharacteristic
64C22×C3⋊D3—24no—
65C22×C3⋊D3C2×Dic311yescharacteristic
66C22×C3⋊D3#6714yes—
67C22×C3⋊D3C22×D311yescharacteristic
68C22×C3×D3—62no—
69C22×C3×D3—68no—
70C22×C3×D3—62no—
71C22×C3×D3—616no—
72C2×#157—68no—
73C2×#157—64no—
74C2×C32×C4—22no—
75C2×C32×C4—28no—
76C2×C32×C4—24no—
77C2×C32×C4—48no—
78C2×C32×C4—44no—
79C2×C32×C4C2×Dic312yes—
80C2×C3⋊Dic3—22no—
81C2×C3⋊Dic3C2×Dic312yes—
82C4×C3⋊D3—24no—
83C4×C3⋊D3—24no—
84C2×C3×Dic3—64no—
85C2×C3×Dic3—68no—
86C3×C4×D3—616no—
87C3×C4×D3—616no—
88#381—271noSylow
89C2×C33—22no—
90C2×C33C2×C22⋊C411yescharacteristic
91C2×C33#11712yes—
92C3×C3⋊D3—22no—
93C3×C3⋊D3#12314yes—
94C24×C3—91no—
95C23×D3—62no—
96C22×C3×C4—22no—
97C22×C3×C4—42no—
98C22×C3×C4—91no—
99C22×C3×C4—92no—
100C22×C3×C4—92no—
101C22×C3×C4—184no—
102C22×C3×C4#15711yescharacteristic, complemented
103C22×Dic3—62no—
104C22×Dic3—94no—
105C2×C4×D3—64no—
106C2×C4×D3—64no—
107C2×C4×D3—616no—
108#217—184no—
109C22×C32—22no—
110C22×C32—28no—
111C22×C32—22no—
112C22×C32—24no—
113C22×C32—48no—
114C22×C32—44no—
115C22×C32—42no—
116C22×C32#21712yes—
117C22×C32C22×Dic311yescharacteristic
118C2×C3⋊D3—22no—
119C2×C3⋊D3—28no—
120C2×C3⋊D3#21712yes—
121C2×C3⋊D3#23514yes—
122C2×C3⋊D3C2×#6712yes—
123C2×C3×D3—64no—
124C2×C3×D3—64no—
125C2×C3×D3—68no—
126C2×C3×D3—632no—
127C2×C3×D3—68no—
128#157—124no—
129C32×C4—22no—
130C32×C4—28no—
131C32×C4—48no—
132C3⋊Dic3—22no—
133C3×Dic3—68no—
134C23×C4—91no—
135C2×C22⋊C4—272no—
136#95—274no—
137C33#38111yescharacteristic, complemented, Sylow
138C23×C3—22no—
139C23×C3—42no—
140C23×C3—91no—
141C23×C3—91no—
142C23×C3—94no—
143C23×C3—184no—
144C23×C3C2×#15711yescharacteristic
145C22×D3—62no—
146C22×D3—68no—
147C22×D3—62no—
148C22×D3—616no—
149C2×C3×C4—28no—
150C2×C3×C4—22no—
151C2×C3×C4—24no—
152C2×C3×C4—48no—
153C2×C3×C4—44no—
154C2×C3×C4—92no—
155C2×C3×C4—184no—
156C2×C3×C4—184no—
157C2×C3×C4—182no—
158C2×C3×C4C2×#15712yes—
159C2×Dic3—64no—
160C2×Dic3—68no—
161C2×Dic3—184no—
162C2×Dic3—188no—
163C4×D3—616no—
164C4×D3—616no—
165C2×C32—22no—
166C2×C32—28no—
167C2×C32—24no—
168C2×C32—22no—
169C2×C32—48no—
170C2×C32—42no—
171C2×C32—44no—
172C2×C32#92612yes—
173C2×C32C2×#21711yescharacteristic
174C3⋊D3—22no—
175C3⋊D3#92914yes—
176C3×D3—68no—
177C3×D3—68no—
178C24—91no—
179C22×C4—91no—
180C22×C4—92no—
181C22×C4—92no—
182C22×C4—184no—
183C22×C4—274no—
184C22×C4C3⋊#15711yescharacteristic, complemented
185C22⋊C4—544no—
186C22×C3—28no—
187C22×C3—22no—
188C22×C3—22no—
189C22×C3—24no—
190C22×C3—48no—
191C22×C3—42no—
192C22×C3—44no—
193C22×C3—94no—
194C22×C3—92no—
195C22×C3—92no—
196C22×C3—188no—
197C22×C3—182no—
198C22×C3C22⋊#15712yes—
199C22×C3C22×#15711yescharacteristic
200C2×D3—64no—
201C2×D3—64no—
202C2×D3—68no—
203C2×D3—632no—
204C2×D3—68no—
205C3×C4—28no—
206C3×C4—22no—
207C3×C4—48no—
208C3×C4—182no—
209Dic3—68no—
210Dic3—364no—
211C32—22no—
212C32—42no—
213C32#521111yescharacteristic, complemented
214C23—91no—
215C23—91no—
216C23—94no—
217C23—184no—
218C23C2×C3⋊#15711yescharacteristic
219C2×C4—22no—
220C2×C4—92no—
221C2×C4—184no—
222C2×C4—184no—
223C2×C4—182no—
224C2×C4—544no—
225C2×C4—548no—
226C2×C4C2×C3⋊#15712yes—
227C2×C3—28no—
228C2×C3—22no—
229C2×C3—24no—
230C2×C3—22no—
231C2×C3—48no—
232C2×C3—44no—
233C2×C3—42no—
234C2×C3—94no—
235C2×C3—182no—
236C2×C3#6428612yes—
237C2×C3C2×C22⋊#15711yescharacteristic
238D3—68no—
239D3—68no—
240C22—22no—
241C22—94no—
242C22—92no—
243C22—92no—
244C22—188no—
245C22—182no—
246C22#8899412yes—
247C22C22×C3⋊#15711yescharacteristic
248C4—22no—
249C4—182no—
250C4—1084no—
251C3—22no—
252C3—42no—
253C3#1059381111yescharacteristic, complemented
254C2—22no—
255C2—94no—
256C2—182no—
257C2#1172451412yes—
258C2C2×#8899411yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊#157
commutator, socleC22×C33C22×C4
FittingC22×C33×C4C4
prime radicalC23×C3×C3⋊D3C22
cosocleC22×C3×C3⋊D3C23
hypercenterC22×C4C3⋊#157
hypocenterC33#381
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. #11723106C2semidirect
7. C22×C33×C4C4semidirect
8. C22×C3×C3⋊Dic3C4semidirect
10. C2×C3×C4×C3⋊D3C4semidirect
36. C22×C32×C4Dic3semidirect
37. C22×C3⋊Dic3Dic3semidirect
39. C2×C4×C3⋊D3Dic3semidirect
102. C22×C3×C4#157semidirect
137. C33#381semidirect
184. C22×C4C3⋊#157semidirect
213. C32#5211semidirect
253. C3#10593811semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4×C3⋊D32
C2×#889942
#117231062
#105938113
#52119