Finite Group Databasebeta

← #13170808

Subgroups — #13170808

Order 1728.

1974 conjugacy classes of subgroups; only the 324 normal subgroups are shown.

normal subgroups only

This group has 1974 conjugacy classes of subgroups — too many to draw. Only its 324 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×#1113C211yesmaximal, characteristic
2C2×#88949C212yesmaximal
3C3⋊D3×C4⋊Dic3C213yesmaximal, complemented
4C3×#63900C213yesmaximal, complemented
5#11722910C216yesmaximal, complemented
6D32×C4⋊C4—33nomaximal
7C22×#1113C2211yescharacteristic
8C2×C3×C4×C3⋊D3C2213yes—
9C2×#6294C2213yes—
10C4×#1113C418yescomplemented
11C2×Dic3×C3⋊D3C2216yes—
12C2×C3×#624C2216yes—
13#88949—28no—
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic, complemented
15C32×C4⋊Dic3C2213yescomplemented
16#88807C2216yescomplemented
17C3×#3724C2216yescomplemented
18C2×C4×D32—33no—
19C2×C4×D32—36no—
20C4⋊C4×C3⋊D3—33no—
21C3×D3×C4⋊C4—36no—
22#63900D313yescomplemented
23D3×C4⋊Dic3—36no—
24D3×C4⋊Dic3—63no—
25#63883—36no—
26D3×Dic3⋊C4—312no—
27C22×C3×C3⋊D3C2313yes—
28C2×#1113C2×C414yes—
29C2×#1113D414yes—
30C2×#1113Q814yes—
31C2×C33×C4C2311yescharacteristic
32C2×C3⋊(C3⋊Dic3)C2312yes—
33C2×C3×C3⋊Dic3C2313yes—
34C3×C4×C3⋊D3C2×C4112yescomplemented
35C2×C32×Dic3C2316yes—
36#6294C2×C4112yescomplemented
37Dic3×C3⋊D3—212no—
38C3×#624—212no—
39C2×D3×C4⋊C4—93no—
40C22×D32—33no—
41C2×C4×C3⋊D3—36no—
42C2×C4×C3⋊D3C2×D313yes—
43C2×C3×C4×D3—36no—
44C2×C3×C4×D3—312no—
45C2×C3×C4×D3—63no—
46C2×#624—33no—
47C2×#624C2×D316yes—
48C2×D3×Dic3—36no—
49C2×D3×Dic3—312no—
50C2×D3×Dic3—66no—
51C4×D32—324no—
52C4×D32—624no—
53C32×C4⋊C4—33no—
54C4⋊(C3⋊Dic3)—33no—
55C4⋊(C3⋊Dic3)—63no—
56C4⋊(C3⋊Dic3)—121no—
57#3750—36no—
58C3×C4⋊Dic3—23no—
59C3×C4⋊Dic3C2×D316yescomplemented
60C3×Dic3⋊C4—312no—
61#3724C2×D316yescomplemented
62Dic3⋊Dic3—312no—
63Dic3⋊Dic3—66no—
64C22×C33C2411yescharacteristic
65C2×C3×C3⋊D3C22×C416yes—
66C2×C3×C3⋊D3C2×D416yes—
67C2×C3×C3⋊D3C2×Q816yes—
68#1113C4⋊C4116yescomplemented
69C33×C4C22×C412yescomplemented
70C3⋊(C3⋊Dic3)—22no—
71C3×C3⋊Dic3C22×C416yescomplemented
72C32×Dic3—26no—
73C22×C4×D3—93no—
74C22×C4×D3—96no—
75C2×C3×C4⋊C4—93no—
76C2×C4⋊Dic3—93no—
77C2×Dic3⋊C4—96no—
78D3×C4⋊C4—36no—
79D3×C4⋊C4—912no—
80D3×C4⋊C4—96no—
81D3×C4⋊C4—183no—
82C22×C3⋊D3C22×D313yes—
83C22×C3×D3—36no—
84C22×C3×D3—63no—
85C2×D32—312no—
86C2×D32—312no—
87C2×D32—312no—
88C2×C32×C4—23no—
89C2×C32×C4—36no—
90C2×C32×C4—41no—
91C2×C32×C4C22×D313yes—
92C2×C3⋊Dic3—36no—
93C2×C3⋊Dic3—66no—
94C2×C3⋊Dic3—122no—
95C2×C3⋊Dic3C22×D313yes—
96C4×C3⋊D3—612no—
97C4×C3⋊D3C4×D3112yescomplemented
98C2×C3×Dic3—26no—
99C2×C3×Dic3—36no—
100C2×C3×Dic3—63no—
101C2×C3×Dic3C22×D3112yes—
102C3×C4×D3—324no—
103C3×C4×D3—612no—
104C3×C4×D3—624no—
105#624—212no—
106#624—312no—
107D3×Dic3—324no—
108D3×Dic3—624no—
109D3×Dic3—1212no—
110C22×C4⋊C4—271noSylow
111C2×C33C23×C411yescharacteristic
112C2×C33C22×D411yescharacteristic
113C2×C33C22×Q811yescharacteristic
114C3×C3⋊D3C2×C4⋊C4112yescomplemented
115C23×D3—93no—
116C22×C3×C4—96no—
117C22×C3×C4—93no—
118C22×Dic3—93no—
119C22×Dic3—96no—
120C2×C4×D3—36no—
121C2×C4×D3—312no—
122C2×C4×D3—63no—
123C2×C4×D3—924no—
124C2×C4×D3—96no—
125C2×C4×D3—912no—
126C2×C4×D3—186no—
127C2×C4×D3—1824no—
128C3×C4⋊C4—36no—
129C3×C4⋊C4—63no—
130C3×C4⋊C4—96no—
131C4⋊Dic3—93no—
132C4⋊Dic3—96no—
133C4⋊Dic3—183no—
134C4⋊Dic3—361no—
135C4⋊Dic3D3213yescomplemented
136Dic3⋊C4—312no—
137Dic3⋊C4—912no—
138Dic3⋊C4—186no—
139C22×C32—23no—
140C22×C32—41no—
141C22×C32C23×D313yes—
142C2×C3⋊D3C2×C4×D316yes—
143C2×C3⋊D3D3×D416yes—
144C2×C3⋊D3D3×Q816yes—
145C2×C3×D3—312no—
146C2×C3×D3—312no—
147C2×C3×D3—312no—
148C2×C3×D3—66no—
149C2×C3×D3—66no—
150C2×C3×D3—66no—
151D32—348no—
152C32×C4—26no—
153C32×C4—42no—
154C32×C4—66no—
155C32×C4C2×C4×D316yescomplemented
156C3⋊Dic3—66no—
157C3⋊Dic3—126no—
158C3⋊Dic3—242no—
159C3⋊Dic3C2×C4×D316yescomplemented
160C3×Dic3—212no—
161C3×Dic3—312no—
162C3×Dic3—46no—
163C3×Dic3—66no—
164C23×C4—271no—
165C23×C4—272no—
166C2×C4⋊C4—93no—
167C2×C4⋊C4—273no—
168C2×C4⋊C4—276no—
169C33C22×C4⋊C411yescharacteristic, complemented, Sylow
170C23×C3—93no—
171C22×D3—36no—
172C22×D3—63no—
173C22×D3—912no—
174C22×D3—912no—
175C22×D3—912no—
176C2×C3×C4—23no—
177C2×C3×C4—312no—
178C2×C3×C4—41no—
179C2×C3×C4—66no—
180C2×C3×C4—93no—
181C2×C3×C4—912no—
182C2×C3×C4—1812no—
183C2×C3×C4C2×D3213yes—
184C2×Dic3—36no—
185C2×Dic3—63no—
186C2×Dic3—912no—
187C2×Dic3—96no—
188C2×Dic3—186no—
189C2×Dic3—1812no—
190C2×Dic3—362no—
191C2×Dic3C2×D3216yes—
192C4×D3—324no—
193C4×D3—612no—
194C4×D3—624no—
195C4×D3—924no—
196C4×D3—1824no—
197C4×D3—3612no—
198C2×C32—23no—
199C2×C32—23no—
200C2×C32—23no—
201C2×C32—41no—
202C2×C32—41no—
203C2×C32—41no—
204C2×C32C22×C4×D313yes—
205C2×C32C2×D3×D413yes—
206C2×C32C2×D3×Q813yes—
207C3⋊D3D3×C4⋊C4112yescomplemented
208C3×D3—324no—
209C3×D3—612no—
210C24—271no—
211C22×C4—96no—
212C22×C4—93no—
213C22×C4—276no—
214C22×C4—278no—
215C22×C4—273no—
216C22×C4—548no—
217C4⋊C4—33no—
218C4⋊C4—96no—
219C4⋊C4—276no—
220C4⋊C4—271no—
221C22×C3—23no—
222C22×C3—41no—
223C22×C3—96no—
224C22×C3—96no—
225C22×C3—96no—
226C22×C3C22×D3213yes—
227C2×D3—312no—
228C2×D3—312no—
229C2×D3—312no—
230C2×D3—66no—
231C2×D3—66no—
232C2×D3—66no—
233C2×D3—948no—
234C3×C4—26no—
235C3×C4—42no—
236C3×C4—612no—
237C3×C4—96no—
238C3×C4—126no—
239C3×C4C4×D3216yescomplemented
240Dic3—26no—
241Dic3—312no—
242Dic3—66no—
243Dic3—186no—
244Dic3—366no—
245Dic3—722no—
246C32—23no—
247C32—41no—
248C32C2×D3×C4⋊C413yescomplemented
249C23—93no—
250C23—274no—
251C23—274no—
252C23—274no—
253C2×C4—36no—
254C2×C4—93no—
255C2×C4—912no—
256C2×C4—1812no—
257C2×C4—2712no—
258C2×C4—272no—
259C2×C4—5412no—
260C2×C4D3311yescharacteristic
261C2×C3—23no—
262C2×C3—23no—
263C2×C3—23no—
264C2×C3—41no—
265C2×C3—41no—
266C2×C3—41no—
267C2×C3—912no—
268C2×C3C2×C4×D3213yes—
269C2×C3D32×D413yes—
270C2×C3D32×Q813yes—
271D3—324no—
272D3—612no—
273C22—96no—
274C22—96no—
275C22—96no—
276C22—2716no—
277C22C2×D3311yescharacteristic
278C4—66no—
279C4—96no—
280C4—542no—
281C4#8894912yescomplemented
282C3—23no—
283C3—41no—
284C3D32×C4⋊C413yescomplemented
285C2—912no—
286C2#1172437811yescharacteristic
287C2#1172440311yescharacteristic
288C2C2×#8894911yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutatorC2×C33C23×C4
FittingC2×C33×C4C23
socle, cosocleC22×C33C24
prime radicalC22×#1113C22
hypercenterC2×C4D33
hypocenterC33C22×C4⋊C4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. C3⋊D3×C4⋊Dic3C2semidirect
4. C3×#63900C2semidirect
5. #11722910C2semidirect
10. C4×#1113C4semidirect
14. C4⋊(C3⋊(C3⋊Dic3))C22semidirect
15. C32×C4⋊Dic3C22semidirect
16. #88807C22semidirect
17. C3×#3724C22semidirect
22. #63900D3semidirect
34. C3×C4×C3⋊D3C2×C4semidirect
36. #6294C2×C4semidirect
59. C3×C4⋊Dic3C2×D3semidirect
61. #3724C2×D3semidirect
68. #1113C4⋊C4semidirect
69. C33×C4C22×C4semidirect
71. C3×C3⋊Dic3C22×C4semidirect
97. C4×C3⋊D3C4×D3semidirect
114. C3×C3⋊D3C2×C4⋊C4semidirect
135. C4⋊Dic3D32semidirect
155. C32×C4C2×C4×D3semidirect
159. C3⋊Dic3C2×C4×D3semidirect
169. C33C22×C4⋊C4semidirect
207. C3⋊D3D3×C4⋊C4semidirect
239. C3×C4C4×D32semidirect
248. C32C2×D3×C4⋊C4semidirect
281. C4#88949semidirect
284. C3D32×C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#11132
C2×#889492
C3⋊D3×C4⋊Dic32
C3×#639002
#117229102
D32×C4⋊C43