Finite Group Databasebeta

← #13166391

Subgroups — #13166391

Order 1728.

424 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 78 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×#64286C211yesmaximal, characteristic
3C2×C3⋊D3×C3⋊C8C211yesmaximal, characteristic
4#11722715C212yesmaximal, complemented
5#11722714C212yesmaximal, complemented
6#10593768—31nomaximal
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8#88994—21no—
9C4⋊(C3⋊#157)C2212yes—
10C4×C3⋊#157—21no—
11C3×C22⋊#157—21no—
12C3×C4⋊#157C2212yes—
13C3×C4×#157—21no—
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#64286D311yescharacteristic
22C2×C8×C3⋊D3—31no—
23C2×D3×C3⋊C8—62no—
24#63769—32no—
25#63768—32no—
26C22×C3×C3⋊D3D411yescharacteristic
27C2×C3⋊#157—22no—
28C2×C3×#157—22no—
29C2×C33×C4C2×C411yescharacteristic
30C2×C3×C3⋊Dic3Q811yescharacteristic
31C3×C4×C3⋊D3C2×C411yescharacteristic
32C3×C4×C3⋊D3D411yescharacteristic
33C3×C4×C3⋊D3Q811yescharacteristic
34C3×C4×C3⋊D3C2311yescharacteristic
35C3⋊(C3⋊(C3⋊C8))C2×C411yescharacteristic
36C3⋊(C3⋊(C3⋊C8))C2×C411yescharacteristic
37C32×C3⋊C8C2×C411yescharacteristic
38C32×C3⋊C8C2×C411yescharacteristic
39#4998—91nomaximal
40C2×C4×C3⋊D3C2×D311yescharacteristic
41C2×C3×C4×D3—62no—
42C22⋊#157—21no—
43C22⋊#157—61no—
44C4⋊#157—32no—
45C4⋊#157C2×D312yes—
46C4×#157—21no—
47C4×#157—61no—
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×C33C4⋊C411yescharacteristic
62C2×C3×C3⋊D3—22no—
63C2×C3×C3⋊D3C4⋊C411yescharacteristic
64C2×C3×C3⋊D3C2×D411yescharacteristic
65C3⋊#157—42no—
66C3×#157—42no—
67C33×C4C4⋊C411yescharacteristic
68C33×C4C22×C411yescharacteristic
69C3×C3⋊Dic3C4⋊C411yescharacteristic
70C3×C3⋊Dic3C2×Q811yescharacteristic
71C3×#117—91no—
72#926—91no—
73C22×C3⋊C8—91no—
74C2×D3×C8—182no—
75#808—92no—
76#807—92no—
77C22×C3⋊D3#6711yescharacteristic
78C22×C3×D3—62no—
79C2×#157—22no—
80C2×#157—62no—
81C2×C32×C4—22no—
82C2×C32×C4—42no—
83C2×C32×C4C4×D311yescharacteristic
84C2×C3⋊Dic3Dic611yescharacteristic
85C4×C3⋊D3Dic611yescharacteristic
86C4×C3⋊D3C4×D311yescharacteristic
87C4×C3⋊D3#6711yescharacteristic
88C4×C3⋊D3C22×D311yescharacteristic
89C2×C3×Dic3—62no—
90C3×C4×D3—62no—
91C3×C4×D3—62no—
92C3×C4×D3—62no—
93C3×C4×D3—62no—
94C32×C8—31no—
95C32×C8—31no—
96C3⋊(C3⋊C8)—31no—
97C3⋊(C3⋊C8)—31no—
98C3⋊(C3⋊C8)—62no—
99C3⋊(C3⋊C8)—62no—
100C3⋊(C3⋊C8)—122no—
101C3⋊(C3⋊C8)—122no—
102C3×C3⋊C8—22no—
103C3×C3⋊C8—22no—
104#427—271noSylow
105C2×C33—21no—
106C2×C33C2×C4⋊C411yescharacteristic
107C3×C3⋊D3—21no—
108C3×C3⋊D3#13512yes—
109C22×C3×C4—91no—
110C2×C4×D3—62no—
111C2×C3×C8—62no—
112C3×C4⋊C4—92no—
113C3×C22⋊C4—181no—
114C3×C42—181no—
115#217—181no—
116C4⋊Dic3—92no—
117C4×Dic3—181no—
118C2×C3⋊C8—91no—
119C2×C3⋊C8—91no—
120C2×C3⋊C8—91no—
121C2×C3⋊C8—91no—
122C2×C3⋊C8—91no—
123C2×C3⋊C8—182no—
124C2×C3⋊C8—362no—
125C2×C3⋊C8#15711yescharacteristic, complemented
126D3×C8—182no—
127D3×C8—182no—
128D3×C8—182no—
129D3×C8—182no—
130C22×C32—22no—
131C22×C32—42no—
132C22×C32Dic3⋊C411yescharacteristic
133C2×C3⋊D3—22no—
134C2×C3⋊D3Dic3⋊C411yescharacteristic
135C2×C3⋊D3C2×#6711yescharacteristic
136C2×C3×D3—62no—
137C2×C3×D3—62no—
138C2×C3×D3—68no—
139#157—42no—
140#157—122no—
141C32×C4—22no—
142C32×C4—22no—
143C32×C4—42no—
144C32×C4—42no—
145C32×C4Dic3⋊C411yescharacteristic
146C32×C4C2×C4×D311yescharacteristic
147C3⋊Dic3Dic3⋊C411yescharacteristic
148C3⋊Dic3C2×Dic611yescharacteristic
149C3×Dic3—62no—
150C3×Dic3—62no—
151C22×C8—271no—
152#117—91no—
153#117—271no—
154#107—272no—
155#106—272no—
156C33#42711yescharacteristic, complemented, Sylow
157C23×C3—91no—
158C22×D3—62no—
159C2×C3×C4—22no—
160C2×C3×C4—42no—
161C2×C3×C4—91no—
162C2×C3×C4—91no—
163C2×C3×C4—91no—
164C2×C3×C4—91no—
165C2×C3×C4—91no—
166C2×C3×C4—182no—
167C2×C3×C4C2×#15711yescharacteristic
168C2×Dic3—62no—
169C2×Dic3—182no—
170C4×D3—62no—
171C4×D3—62no—
172C4×D3—62no—
173C4×D3—62no—
174C3×C8—62no—
175C3×C8—62no—
176C3⋊C8—91no—
177C3⋊C8—91no—
178C3⋊C8—182no—
179C3⋊C8—182no—
180C3⋊C8—362no—
181C3⋊C8—362no—
182C3⋊C8C2×#15711yescharacteristic
183C3⋊C8C2×#15711yescharacteristic
184C2×C32—21no—
185C2×C32—24no—
186C2×C32—22no—
187C2×C32—44no—
188C2×C32—42no—
189C2×C32C2×Dic3⋊C411yescharacteristic
190C3⋊D3—21no—
191C3⋊D3#95012yes—
192C3×D3—64no—
193C3×D3—64no—
194C22×C4—91no—
195C2×C8—31no—
196C2×C8—271no—
197C2×C8—271no—
198C2×C8—271no—
199C2×C8—271no—
200C2×C8—271no—
201C4⋊C4—92no—
202C4⋊C4—272no—
203C22⋊C4—181no—
204C22⋊C4—541no—
205C42—181no—
206C42—541no—
207C22×C3—22no—
208C22×C3—42no—
209C22×C3—91no—
210C22×C3—91no—
211C22×C3—182no—
212C22×C3C4⋊#15711yescharacteristic
213C2×D3—62no—
214C2×D3—62no—
215C2×D3—68no—
216C3×C4—22no—
217C3×C4—22no—
218C3×C4—42no—
219C3×C4—42no—
220C3×C4—91no—
221C3×C4—91no—
222C3×C4—362no—
223C3×C4C4⋊#15711yescharacteristic
224C3×C4C22×#15711yescharacteristic
225Dic3—62no—
226Dic3—62no—
227Dic3—362no—
228C32—22no—
229C32—42no—
230C32#499811yescharacteristic, complemented
231C23—91no—
232C2×C4—91no—
233C2×C4—91no—
234C2×C4—91no—
235C2×C4—91no—
236C2×C4—91no—
237C2×C4—182no—
238C2×C4—542no—
239C2×C4D3×#15711yescharacteristic
240C8—31no—
241C8—31no—
242C8—271no—
243C8—271no—
244C2×C3—24no—
245C2×C3—21no—
246C2×C3—22no—
247C2×C3—44no—
248C2×C3—42no—
249C2×C3—92no—
250C2×C3—181no—
251C2×C3C2×C4⋊#15711yescharacteristic
252D3—64no—
253D3—64no—
254C22—91no—
255C22—91no—
256C22—182no—
257C22Dic3⋊#15711yescharacteristic
258C4—91no—
259C4—91no—
260C4—362no—
261C4—1082no—
262C4Dic3⋊#15711yescharacteristic
263C4C2×D3×#15711yescharacteristic
264C3—22no—
265C3—42no—
266C3#1059376811yescharacteristic, complemented
267C2—21no—
268C2—92no—
269C2—181no—
270C2C2×Dic3⋊#15711yescharacteristic
subgroupG/H
centerC4Dic3⋊#157
commutatorC33×C4C22×C4
FrattiniC4C2×D3×#157
FittingC2×C33×C4C2×C4
socleC2×C33C2×C4⋊C4
prime radicalC22×C3×C3⋊D3D4
cosocleC3×C4×C3⋊D3C23
hypercenterC2×C4D3×#157
hypocenterC33#427
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. #11722715C2semidirect
5. #11722714C2semidirect
14. C2×C3⋊(C3⋊(C3⋊C8))C4semidirect
15. C2×C32×C3⋊C8C4semidirect
16. C3⋊D3×C3⋊C8C4semidirect
17. C3⋊D3×C3⋊C8C4semidirect
125. C2×C3⋊C8#157semidirect
156. C33#427semidirect
230. C32#4998semidirect
266. C3#10593768semidirect

Maximal subgroups

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

subgroupindex
#117245142
C3×#642862
C2×C3⋊D3×C3⋊C82
#117227152
#117227142
#105937683
#49989