Finite Group Databasebeta

← #13166292

Subgroups — #13166292

Order 1728.

456 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 102 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
1#11724514C211yesmaximal, characteristic
2C3×#64283C211yesmaximal, characteristic
3C2×C3⋊D3×C3⋊C8C211yesmaximal, characteristic
4#11722705C212yesmaximal, complemented
5#11722703C212yesmaximal, complemented
6#10593765—31nomaximal
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8#88994C412yes—
9C4⋊(C3⋊#157)C412yes—
10C4×C3⋊#157C2212yes—
11C3×#3792C412yescomplemented
12C3×#3788C412yescomplemented
13C3×#3787C2212yes—
14C2×C3⋊(C3⋊(C3⋊C8))C411yescharacteristic, complemented
15C2×C32×C3⋊C8C411yescharacteristic, complemented
16C3⋊D3×C3⋊C8C411yescharacteristic, complemented
17C3⋊D3×C3⋊C8C411yescharacteristic, complemented
18C3⋊D3×C3⋊C8C2211yescharacteristic
19C3⋊D3×C3⋊C8C2211yescharacteristic
20#64286—31no—
21#64283D311yescharacteristic
22C2×C8×C3⋊D3—31no—
23C2×D3×C3⋊C8—62no—
24#63767—32no—
25C8×#157—32no—
26C22×C3×C3⋊D3C2×C411yescharacteristic
27C2×C3⋊#157C2×C414yes—
28C2×C33×C4C2×C411yescharacteristic
29C2×C3×C3⋊Dic3C2×C411yescharacteristic
30C3×C4×C3⋊D3C2×C411yescharacteristic
31C3×C4×C3⋊D3C2×C411yescharacteristic
32C3×C4×C3⋊D3C2×C411yescharacteristic
33C3×C4×C3⋊D3C2311yescharacteristic
34C3×#622C2×C414yes—
35C3⋊(C3⋊(C3⋊C8))C2×C411yescharacteristic
36C3⋊(C3⋊(C3⋊C8))C2×C411yescharacteristic
37C32×C3⋊C8C2×C411yescharacteristic
38C32×C3⋊C8C2×C411yescharacteristic
39#5122—91nomaximal
40C2×C4×C3⋊D3C2×D311yescharacteristic
41C2×C3×C4×D3—62no—
42C22⋊#157—32no—
43#3792Dic312yescomplemented
44C4⋊#157—32no—
45C4×#157—32no—
46#3788Dic312yescomplemented
47#3787C2×D312yes—
48C2×C32×C8—31no—
49C2×C3⋊(C3⋊C8)—31no—
50C2×C3⋊(C3⋊C8)—62no—
51C2×C3⋊(C3⋊C8)—122no—
52C8×C3⋊D3—31no—
53C8×C3⋊D3—31no—
54C8×C3⋊D3—31no—
55C8×C3⋊D3—31no—
56C2×C3×C3⋊C8—22no—
57D3×C3⋊C8—62no—
58D3×C3⋊C8—62no—
59D3×C3⋊C8—62no—
60D3×C3⋊C8—62no—
61C22×C33C4211yescharacteristic
62C2×C3×C3⋊D3—22no—
63C2×C3×C3⋊D3C4211yescharacteristic
64C2×C3×C3⋊D3C22×C411yescharacteristic
65C3⋊#157—24no—
66C33×C4C4211yescharacteristic
67C33×C4C22×C411yescharacteristic
68C3×C3⋊Dic3C4211yescharacteristic
69C3×C3⋊Dic3C22×C411yescharacteristic
70C2×C3×M16—91no—
71#926—91no—
72C22×C3⋊C8—91no—
73C2×D3×C8—182no—
74#815—92no—
75C8×Dic3—92no—
76C22×C3⋊D3C4×D311yescharacteristic
77C22×C3×D3—62no—
78C2×#157—34no—
79C2×C32×C4—22no—
80C2×C32×C4—42no—
81C2×C32×C4C4×D311yescharacteristic
82C2×C3⋊Dic3C2×Dic311yescharacteristic
83C4×C3⋊D3C4×D311yescharacteristic
84C4×C3⋊D3C4×D311yescharacteristic
85C4×C3⋊D3C2×Dic311yescharacteristic
86C4×C3⋊D3C22×D311yescharacteristic
87C2×C3×Dic3—62no—
88C3×C4×D3—62no—
89C3×C4×D3—62no—
90C3×C4×D3—62no—
91C3×C4×D3—62no—
92#622C2×Dic314yes—
93C32×C8—31no—
94C32×C8—31no—
95C3⋊(C3⋊C8)—31no—
96C3⋊(C3⋊C8)—31no—
97C3⋊(C3⋊C8)—62no—
98C3⋊(C3⋊C8)—62no—
99C3⋊(C3⋊C8)—122no—
100C3⋊(C3⋊C8)—122no—
101C3×C3⋊C8—22no—
102C3×C3⋊C8—22no—
103#405—271noSylow
104C2×C33—21no—
105C2×C33C2×C4211yescharacteristic
106C3×C3⋊D3—21no—
107C3×C3⋊D3#13112yes—
108C22×C3×C4—91no—
109C2×C4×D3—62no—
110C3×M16—92no—
111C3×M16—92no—
112C2×C3×C8—62no—
113C2×C3×C8—92no—
114#217—92no—
115C4⋊Dic3—92no—
116C4×Dic3—92no—
117C2×C3⋊C8—91no—
118C2×C3⋊C8—91no—
119C2×C3⋊C8—91no—
120C2×C3⋊C8—91no—
121C2×C3⋊C8—91no—
122C2×C3⋊C8—182no—
123C2×C3⋊C8—362no—
124C2×C3⋊C8#15711yescharacteristic, complemented
125D3×C8—182no—
126D3×C8—182no—
127D3×C8—182no—
128D3×C8—182no—
129C22×C32—22no—
130C22×C32—42no—
131C22×C32C4×Dic311yescharacteristic
132C2×C3⋊D3—22no—
133C2×C3⋊D3C4×Dic311yescharacteristic
134C2×C3⋊D3C2×C4×D311yescharacteristic
135C2×C3×D3—62no—
136C2×C3×D3—62no—
137C2×C3×D3—68no—
138#157—64no—
139C32×C4—22no—
140C32×C4—22no—
141C32×C4—42no—
142C32×C4—42no—
143C32×C4C4×Dic311yescharacteristic
144C32×C4C2×C4×D311yescharacteristic
145C3⋊Dic3C4×Dic311yescharacteristic
146C3⋊Dic3C22×Dic311yescharacteristic
147C3×Dic3—62no—
148C3×Dic3—62no—
149C2×M16—91no—
150C22×C8—271no—
151#117—271no—
152#97—272no—
153C4×C8—272no—
154C33#40511yescharacteristic, complemented, Sylow
155C23×C3—91no—
156C22×D3—62no—
157C2×C3×C4—22no—
158C2×C3×C4—42no—
159C2×C3×C4—91no—
160C2×C3×C4—91no—
161C2×C3×C4—91no—
162C2×C3×C4—91no—
163C2×C3×C4—91no—
164C2×C3×C4C2×#15711yescharacteristic
165C2×Dic3—62no—
166C2×Dic3—94no—
167C4×D3—62no—
168C4×D3—62no—
169C4×D3—62no—
170C4×D3—62no—
171C3×C8—62no—
172C3×C8—62no—
173C3×C8—94no—
174C3⋊C8—91no—
175C3⋊C8—91no—
176C3⋊C8—182no—
177C3⋊C8—182no—
178C3⋊C8—362no—
179C3⋊C8—362no—
180C3⋊C8C2×#15711yescharacteristic
181C3⋊C8C2×#15711yescharacteristic
182C2×C32—21no—
183C2×C32—24no—
184C2×C32—22no—
185C2×C32—44no—
186C2×C32—42no—
187C2×C32C2×C4×Dic311yescharacteristic
188C3⋊D3—21no—
189C3⋊D3#90712yes—
190C3×D3—64no—
191C3×D3—64no—
192C22×C4—91no—
193M16—92no—
194M16—92no—
195C2×C8—31no—
196C2×C8—92no—
197C2×C8—271no—
198C2×C8—271no—
199C2×C8—271no—
200C2×C8—271no—
201C2×C8—271no—
202C4⋊C4—272no—
203C22⋊C4—272no—
204C42—272no—
205C22×C3—22no—
206C22×C3—42no—
207C22×C3—91no—
208C22×C3—91no—
209C22×C3—182no—
210C22×C3C4×#15711yescharacteristic
211C2×D3—62no—
212C2×D3—62no—
213C2×D3—68no—
214C3×C4—22no—
215C3×C4—22no—
216C3×C4—42no—
217C3×C4—42no—
218C3×C4—91no—
219C3×C4—91no—
220C3×C4C4×#15711yescharacteristic
221C3×C4C22×#15711yescharacteristic
222Dic3—62no—
223Dic3—62no—
224Dic3—184no—
225C32—22no—
226C32—42no—
227C32#512211yescharacteristic, complemented
228C23—91no—
229C2×C4—91no—
230C2×C4—91no—
231C2×C4—91no—
232C2×C4—91no—
233C2×C4—91no—
234C2×C4—274no—
235C2×C4D3×#15711yescharacteristic
236C8—31no—
237C8—31no—
238C8—94no—
239C8—271no—
240C8—271no—
241C2×C3—24no—
242C2×C3—21no—
243C2×C3—22no—
244C2×C3—44no—
245C2×C3—42no—
246C2×C3—92no—
247C2×C3—181no—
248C2×C3C2×C4×#15711yescharacteristic
249D3—64no—
250D3—64no—
251C22—91no—
252C22—91no—
253C22—182no—
254C22Dic3×#15711yescharacteristic
255C4—91no—
256C4—91no—
257C4—544no—
258C4Dic3×#15711yescharacteristic
259C4C2×D3×#15711yescharacteristic
260C3—22no—
261C3—42no—
262C3#1059376511yescharacteristic, complemented
263C2—21no—
264C2—92no—
265C2—181no—
266C2C2×Dic3×#15711yescharacteristic
subgroupG/H
center, FrattiniC4C2×D3×#157
commutator, socleC2×C33C2×C42
FittingC2×C33×C4C2×C4
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC3×C4×C3⋊D3C23
hypercenterC2×C4D3×#157
hypocenterC33#405
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. #11722705C2semidirect
5. #11722703C2semidirect
11. C3×#3792C4semidirect
12. C3×#3788C4semidirect
14. C2×C3⋊(C3⋊(C3⋊C8))C4semidirect
15. C2×C32×C3⋊C8C4semidirect
16. C3⋊D3×C3⋊C8C4semidirect
17. C3⋊D3×C3⋊C8C4semidirect
43. #3792Dic3semidirect
46. #3788Dic3semidirect
124. C2×C3⋊C8#157semidirect
154. C33#405semidirect
227. C32#5122semidirect
262. C3#10593765semidirect

Maximal subgroups

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

subgroupindex
#117245142
C3×#642832
C2×C3⋊D3×C3⋊C82
#117227052
#117227032
#105937653
#51229