Finite Group Databasebeta

← C2×#11722714 #13166390

Subgroups — C2×#11722714 #13166390

Order 1728.

456 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 98 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×C4⋊(C3⋊#157)C211yesmaximal, characteristic
2C2×C3×C4⋊#157C211yesmaximal, characteristic
3C2×C3⋊D3×C3⋊C8C211yesmaximal, characteristic
4#11722714C214yesmaximal, direct product factor
5C2×#63768—31nomaximal
6C22×C3⋊#157—21no—
7C22×C3×#157—21no—
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C4⋊(C3⋊#157)—21no—
10C4⋊(C3⋊#157)C2212yes—
11C3×C4⋊#157—21no—
12C3×C4⋊#157C2212yes—
13C2×C3⋊(C3⋊(C3⋊C8))C411yescharacteristic, complemented
14C2×C32×C3⋊C8C411yescharacteristic, complemented
15C3⋊D3×C3⋊C8C412yescomplemented
16C3⋊D3×C3⋊C8C2212yes—
17C2×C4⋊#157—31no—
18C2×C4⋊#157D311yescharacteristic
19C2×C8×C3⋊D3—31no—
20C2×D3×C3⋊C8—62no—
21#63768—34no—
22C22×C3×C3⋊D3D411yescharacteristic
23C2×C3⋊#157—22no—
24C2×C3⋊#157—42no—
25C2×C3×#157—22no—
26C2×C3×#157—42no—
27C2×C33×C4C2×C411yescharacteristic
28C2×C3×C3⋊Dic3Q811yescharacteristic
29C3×C4×C3⋊D3C2×C411yescharacteristic
30C3×C4×C3⋊D3D411yescharacteristic
31C3×C4×C3⋊D3Q811yescharacteristic
32C3×C4×C3⋊D3C2311yescharacteristic
33C3⋊(C3⋊(C3⋊C8))C2×C412yescomplemented
34C32×C3⋊C8C2×C412yescomplemented
35C2×#808—91nomaximal
36C22×#157—21no—
37C22×#157—61no—
38C2×C4×C3⋊D3C2×D311yescharacteristic
39C2×C3×C4×D3—62no—
40C4⋊#157—21no—
41C4⋊#157—32no—
42C4⋊#157—61no—
43C4⋊#157C2×D312yes—
44C2×C32×C8—31no—
45C2×C3⋊(C3⋊C8)—31no—
46C2×C3⋊(C3⋊C8)—62no—
47C2×C3⋊(C3⋊C8)—122no—
48C8×C3⋊D3—32no—
49C8×C3⋊D3—32no—
50C2×C3×C3⋊C8—22no—
51D3×C3⋊C8—64no—
52D3×C3⋊C8—64no—
53C22×C33C4⋊C411yescharacteristic
54C2×C3×C3⋊D3C4⋊C411yescharacteristic
55C2×C3×C3⋊D3SD1612yes—
56C2×C3×C3⋊D3SD1612yes—
57C2×C3×C3⋊D3C2×D411yescharacteristic
58C3⋊#157—42no—
59C3×#157—42no—
60C33×C4C4⋊C411yescharacteristic
61C33×C4C22×C411yescharacteristic
62C3×C3⋊Dic3C4⋊C411yescharacteristic
63C3×C3⋊Dic3C2×Q811yescharacteristic
64C2×C3×C4⋊C4—91no—
65C2×C4⋊Dic3—91no—
66C22×C3⋊C8—91no—
67C2×D3×C8—182no—
68#808—94no—
69C22×C3⋊D3#6711yescharacteristic
70C22×C3×D3—62no—
71C2×#157—22no—
72C2×#157—42no—
73C2×#157—62no—
74C2×#157—122no—
75C2×C32×C4—22no—
76C2×C32×C4—42no—
77C2×C32×C4C4×D311yescharacteristic
78C2×C3⋊Dic3Dic611yescharacteristic
79C4×C3⋊D3Dic611yescharacteristic
80C4×C3⋊D3C4×D311yescharacteristic
81C4×C3⋊D3#6711yescharacteristic
82C4×C3⋊D3C22×D311yescharacteristic
83C2×C3×Dic3—62no—
84C3×C4×D3—62no—
85C3×C4×D3—62no—
86C3×C4×D3—62no—
87C3×C4×D3—62no—
88C32×C8—32no—
89C3⋊(C3⋊C8)—32no—
90C3⋊(C3⋊C8)—64no—
91C3⋊(C3⋊C8)—124no—
92C3×C3⋊C8—24no—
93C2×#106—271noSylow
94C2×C33#10612yescomplemented
95C2×C33C2×C4⋊C411yescharacteristic
96C3×C3⋊D3#10612yescomplemented
97C3×C3⋊D3C2×SD1611yescharacteristic
98C3×C3⋊D3C2×SD1611yescharacteristic
99C22×C3×C4—91no—
100C22×C3×C4—181no—
101C22×Dic3—181no—
102C2×C4×D3—62no—
103C2×C3×C8—62no—
104C3×C4⋊C4—92no—
105C3×C4⋊C4—181no—
106C4⋊Dic3—92no—
107C4⋊Dic3—181no—
108C2×C3⋊C8—91no—
109C2×C3⋊C8—92no—
110C2×C3⋊C8—92no—
111C2×C3⋊C8—182no—
112C2×C3⋊C8—362no—
113C2×C3⋊C8#15711yescharacteristic, complemented
114D3×C8—184no—
115D3×C8—184no—
116C22×C32—22no—
117C22×C32—42no—
118C22×C32Dic3⋊C411yescharacteristic
119C2×C3⋊D3Dic3⋊C411yescharacteristic
120C2×C3⋊D3#21412yes—
121C2×C3⋊D3#21512yes—
122C2×C3⋊D3C2×#6711yescharacteristic
123C2×C3×D3—62no—
124C2×C3×D3—64no—
125C2×C3×D3—64no—
126C2×C3×D3—62no—
127#157—42no—
128#157—122no—
129C32×C4—22no—
130C32×C4—22no—
131C32×C4—42no—
132C32×C4—42no—
133C32×C4Dic3⋊C411yescharacteristic
134C32×C4C2×C4×D311yescharacteristic
135C3⋊Dic3Dic3⋊C411yescharacteristic
136C3⋊Dic3C2×Dic611yescharacteristic
137C3×Dic3—62no—
138C3×Dic3—62no—
139C22×C8—271no—
140C2×C4⋊C4—91no—
141C2×C4⋊C4—271no—
142#106—274no—
143C33C2×#10611yescharacteristic, complemented, Sylow
144C23×C3—91no—
145C22×D3—62no—
146C2×C3×C4—22no—
147C2×C3×C4—42no—
148C2×C3×C4—91no—
149C2×C3×C4—91no—
150C2×C3×C4—91no—
151C2×C3×C4—91no—
152C2×C3×C4—91no—
153C2×C3×C4—182no—
154C2×C3×C4—362no—
155C2×C3×C4C2×#15711yescharacteristic
156C2×Dic3—62no—
157C2×Dic3—182no—
158C2×Dic3—362no—
159C4×D3—62no—
160C4×D3—62no—
161C4×D3—62no—
162C4×D3—62no—
163C3×C8—64no—
164C3⋊C8—92no—
165C3⋊C8—184no—
166C3⋊C8—364no—
167C3⋊C8C2×#15712yescomplemented
168C2×C32—24no—
169C2×C32—22no—
170C2×C32—44no—
171C2×C32—42no—
172C2×C32#80812yescomplemented
173C2×C32C2×Dic3⋊C411yescharacteristic
174C3⋊D3#80812yescomplemented
175C3⋊D3C2×#21411yescharacteristic
176C3⋊D3C2×#21511yescharacteristic
177C3×D3—64no—
178C3×D3—62no—
179C3×D3—62no—
180C22×C4—91no—
181C22×C4—181no—
182C22×C4—541no—
183C2×C8—31no—
184C2×C8—272no—
185C2×C8—272no—
186C2×C8—271no—
187C4⋊C4—92no—
188C4⋊C4—181no—
189C4⋊C4—272no—
190C4⋊C4—541no—
191C22×C3—22no—
192C22×C3—42no—
193C22×C3—92no—
194C22×C3—92no—
195C22×C3—91no—
196C22×C3—91no—
197C22×C3C4⋊#15711yescharacteristic
198C2×D3—62no—
199C2×D3—62no—
200C2×D3—64no—
201C2×D3—64no—
202C3×C4—22no—
203C3×C4—22no—
204C3×C4—42no—
205C3×C4—42no—
206C3×C4—91no—
207C3×C4—91no—
208C3×C4—362no—
209C3×C4C4⋊#15711yescharacteristic
210C3×C4C22×#15711yescharacteristic
211Dic3—62no—
212Dic3—62no—
213Dic3—362no—
214C32—22no—
215C32—42no—
216C32C2×#80811yescharacteristic, complemented
217C23—91no—
218C2×C4—91no—
219C2×C4—91no—
220C2×C4—91no—
221C2×C4—91no—
222C2×C4—91no—
223C2×C4—182no—
224C2×C4—362no—
225C2×C4—542no—
226C2×C4—1082no—
227C2×C4D3×#15711yescharacteristic
228C8—32no—
229C8—272no—
230C2×C3—24no—
231C2×C3—22no—
232C2×C3—44no—
233C2×C3—42no—
234C2×C3—91no—
235C2×C3—91no—
236C2×C3—92no—
237C2×C3#6376812yescomplemented
238C2×C3C2×C4⋊#15711yescharacteristic
239D3—64no—
240D3—62no—
241D3—62no—
242C22—92no—
243C22—92no—
244C22—91no—
245C22—91no—
246C22Dic3⋊#15711yescharacteristic
247C4—91no—
248C4—91no—
249C4—362no—
250C4—1082no—
251C4Dic3⋊#15711yescharacteristic
252C4C2×D3×#15711yescharacteristic
253C3—22no—
254C3—42no—
255C3C2×#6376811yescharacteristic, complemented
256C2—91no—
257C2—91no—
258C2—92no—
259C2#1172271412yesdirect product factor
260C2C2×Dic3⋊#15711yescharacteristic
subgroupG/H
centerC22Dic3⋊#157
commutatorC33×C4C22×C4
FrattiniC4C2×D3×#157
FittingC2×C33×C4C2×C4
socleC22×C33C4⋊C4
prime radicalC22×C3×C3⋊D3D4
cosocleC3×C4×C3⋊D3C23
hypercenterC2×C4D3×#157
hypocenterC33C2×#106
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. #11722714C2direct (N × H)
13. C2×C3⋊(C3⋊(C3⋊C8))C4semidirect
14. C2×C32×C3⋊C8C4semidirect
15. C3⋊D3×C3⋊C8C4semidirect
33. C3⋊(C3⋊(C3⋊C8))C2×C4semidirect
34. C32×C3⋊C8C2×C4semidirect
94. C2×C33#106semidirect
96. C3×C3⋊D3#106semidirect
113. C2×C3⋊C8#157semidirect
143. C33C2×#106semidirect
167. C3⋊C8C2×#157semidirect
172. C2×C32#808semidirect
174. C3⋊D3#808semidirect
216. C32C2×#808semidirect
237. C2×C3#63768semidirect
255. C3C2×#63768semidirect
259. C2#11722714direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4⋊(C3⋊#157)2
C2×C3×C4⋊#1572
C2×C3⋊D3×C3⋊C82
#117227142
C2×#637683
C2×#8089