Finite Group Databasebeta

← #13178763

Subgroups — #13178763

Order 1728.

1228 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 1228 conjugacy classes of subgroups — too many to draw. Only its 183 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
1C3×#64155C213yesmaximal, complemented
2#11723700C213yesmaximal
3#11723692C216yesmaximal, complemented
4#11723671C213yesmaximal, complemented
5#10595434—33nomaximal
6C22×#1113C2211yescharacteristic
7C2×C33×Q8C2211yescharacteristic, complemented
8C2×C3×C4×C3⋊D3C2213yes—
9C2×#6296C2211yescharacteristic, complemented
10C2×#6296C2212yescomplemented
11C2×#6294C2213yes—
12C3×#3752C2216yes—
13C3×C4⋊(C3⋊Dic3)C2213yescomplemented
14C3×#3750C2216yescomplemented
15#88815C2213yescomplemented
16#88814C2216yes—
17#64155D313yescomplemented
18C3×#946—36no—
19C3×#946—63no—
20#63921—33no—
21D3×(C2×D3)⋊C4—36no—
22#63899—36no—
23#63895—36no—
24#63893—36no—
25#63865—33no—
26#63851—36no—
27C22×C3×C3⋊D3C2313yes—
28C2×#1113—41no—
29C2×#1113—41no—
30C2×#1113Q814yescomplemented
31C33×Q8—41no—
32C2×C33×C4C2313yes—
33C2×C3×C3⋊Dic3C2313yes—
34C2×C3×C3⋊Dic3C2316yes—
35C3×C4×C3⋊D3—43no—
36#6296—42no—
37#6296—41no—
38#6294—43no—
39#5634—93no—
40C22×D32—33no—
41C2×C32×Q8—23no—
42C2×C32×Q8—41no—
43C2×C32×Q8C2×D313yescomplemented
44C2×C4×C3⋊D3C2×D313yes—
45C2×C3×C4×D3—36no—
46C2×C3×C4×D3—63no—
47C2×#627—33no—
48C2×#627—36no—
49C2×#624—33no—
50C2×D3×Dic3—36no—
51#3752C2×D316yes—
52C4⋊(C3⋊Dic3)C2×D313yescomplemented
53#3750C2×D316yescomplemented
54C3×(C2×D3)⋊C4—36no—
55C3×(C2×D3)⋊C4—36no—
56C3×(C2×D3)⋊C4—66no—
57C3×C4⋊Dic3—36no—
58C3×C4⋊Dic3—63no—
59C3×Dic3⋊C4—36no—
60C3×Dic3⋊C4—36no—
61C3×Dic3⋊C4—66no—
62#3724—33no—
63Dic3⋊Dic3—36no—
64#3722—36no—
65(C2×D3)⋊Dic3—36no—
66(C2×D3)⋊Dic3—36no—
67C22×C33C2411yescharacteristic
68C2×C3×C3⋊D3—23no—
69C2×C3×C3⋊D3—23no—
70C2×C3×C3⋊D3C2×Q816yes—
71#1113—44no—
72C33×C4—23no—
73C3×C3⋊Dic3—26no—
74C3×C3⋊Dic3—23no—
75C3×C22⋊Q8—93no—
76#946—36no—
77#946—63no—
78C2×(C2×D3)⋊C4—93no—
79#924—93no—
80#897—96no—
81#896—96no—
82#896—96no—
83D3×C22⋊C4—96no—
84#878—96no—
85C22×C3⋊D3C22×D313yes—
86C22×C3×D3—36no—
87C22×C3×D3—63no—
88C2×D32—312no—
89C2×D32—123no—
90C2×D32—123no—
91C32×Q8—43no—
92C32×Q8—46no—
93C32×Q8—44no—
94C2×C32×C4—23no—
95C2×C32×C4—26no—
96C2×C32×C4—43no—
97C2×C32×C4C22×D313yes—
98C2×C32×C4C22×D316yes—
99C2×C3⋊Dic3C22×D313yes—
100C2×C3⋊Dic3C22×D316yes—
101C4×C3⋊D3—43no—
102C2×C3×Dic3—36no—
103C2×C3×Dic3—36no—
104C2×C3×Dic3—36no—
105C2×C3×Dic3—63no—
106C2×C3×Dic3—66no—
107C3×C4×D3—126no—
108C3×C4×D3—126no—
109#627—123no—
110#627—126no—
111#624—123no—
112D3×Dic3—126no—
113#543—271noSylow
114C2×C33C22×Q811yescharacteristic
115C2×C33H(5,𝔽2)11yescharacteristic
116C2×C33H(5,𝔽2)11yescharacteristic
117C3×C3⋊D3—26no—
118C23×D3—93no—
119C2×C3×Q8—23no—
120C2×C3×Q8—41no—
121C2×C3×Q8D3213yescomplemented
122C22×C3×C4—93no—
123C22×Dic3—93no—
124C2×C4×D3—36no—
125C2×C4×D3—63no—
126C2×C4×D3—96no—
127C2×Dic6—96no—
128C2×Dic6—93no—
129C3×C4⋊C4—93no—
130C3×C4⋊C4—96no—
131C3×C22⋊C4—96no—
132#217—96no—
133(C2×D3)⋊C4—36no—
134(C2×D3)⋊C4—36no—
135(C2×D3)⋊C4—66no—
136(C2×D3)⋊C4—96no—
137(C2×D3)⋊C4—96no—
138C4⋊Dic3—36no—
139C4⋊Dic3—63no—
140C4⋊Dic3—93no—
141Dic3⋊C4—36no—
142Dic3⋊C4—36no—
143Dic3⋊C4—66no—
144Dic3⋊C4—96no—
145C22×C32—23no—
146C22×C32—41no—
147C22×C32C23×D313yes—
148C2×C3⋊D3—23no—
149C2×C3⋊D3—23no—
150C2×C3⋊D3D3×Q816yes—
151C2×C3×D3—312no—
152C2×C3×D3—66no—
153C2×C3×D3—66no—
154C2×C3×D3—66no—
155C2×C3×D3—123no—
156C2×C3×D3—123no—
157D32—1212no—
158C32×C4—23no—
159C32×C4—26no—
160C32×C4—26no—
161C32×C4—46no—
162C32×C4—46no—
163C3⋊Dic3—23no—
164C3⋊Dic3—26no—
165C3×Dic3—66no—
166C3×Dic3—66no—
167C3×Dic3—66no—
168C3×Dic3—123no—
169C3×Dic3—126no—
170C22⋊Q8—93no—
171C22⋊Q8—276no—
172C22⋊Q8—273no—
173C2×C22⋊C4—273no—
174C33#54311yescharacteristic, complemented, Sylow
175C23×C3—93no—
176C22×D3—36no—
177C22×D3—63no—
178C22×D3—912no—
179C22×D3—363no—
180C22×D3—363no—
181C3×Q8—43no—
182C3×Q8—46no—
183C3×Q8—44no—
184C2×C3×C4—23no—
185C2×C3×C4—26no—
186C2×C3×C4—43no—
187C2×C3×C4—96no—
188C2×C3×C4—93no—
189C2×C3×C4—363no—
190C2×C3×C4C2×D3213yes—
191C2×C3×C4C2×D3216yes—
192C2×Dic3—36no—
193C2×Dic3—36no—
194C2×Dic3—36no—
195C2×Dic3—63no—
196C2×Dic3—66no—
197C2×Dic3—363no—
198C4×D3—126no—
199C4×D3—126no—
200C4×D3—366no—
201Dic6—366no—
202Dic6—363no—
203C2×C32—23no—
204C2×C32—23no—
205C2×C32—23no—
206C2×C32—41no—
207C2×C32—41no—
208C2×C32—41no—
209C2×C32#100413yes—
210C2×C32C2×D3×Q813yes—
211C2×C32#100913yes—
212C3⋊D3—26no—
213C3×D3—612no—
214C3×D3—126no—
215C24—271no—
216C2×Q8—272no—
217C2×Q8—271no—
218C2×Q8#111311yescharacteristic, complemented
219C22×C4—93no—
220C22×C4—273no—
221C4⋊C4—96no—
222C4⋊C4—93no—
223C4⋊C4—273no—
224C22⋊C4—96no—
225C22⋊C4—276no—
226C22×C3—23no—
227C22×C3—41no—
228C22×C3—96no—
229C22×C3—183no—
230C22×C3—183no—
231C22×C3C22×D3213yes—
232C2×D3—312no—
233C2×D3—66no—
234C2×D3—66no—
235C2×D3—66no—
236C2×D3—123no—
237C2×D3—123no—
238C2×D3—3612no—
239C3×C4—26no—
240C3×C4—26no—
241C3×C4—23no—
242C3×C4—46no—
243C3×C4—46no—
244C3×C4—186no—
245C3×C4—183no—
246Dic3—66no—
247Dic3—66no—
248Dic3—66no—
249Dic3—123no—
250Dic3—126no—
251C32—23no—
252C32—41no—
253C32#563413yescomplemented
254C23—93no—
255C23—274no—
256C23—1081no—
257C23—1081no—
258Q8—41no—
259Q8—1082no—
260Q8—1081no—
261C2×C4—96no—
262C2×C4—93no—
263C2×C4—363no—
264C2×C4—1083no—
265C2×C4C2×#111313yes—
266C2×C3—23no—
267C2×C3—23no—
268C2×C3—23no—
269C2×C3—41no—
270C2×C3—41no—
271C2×C3—41no—
272C2×C3—186no—
273C2×C3D32×Q813yes—
274C2×C3#6432013yes—
275C2×C3#6433013yes—
276D3—612no—
277D3—126no—
278C22—96no—
279C22—183no—
280C22—183no—
281C22—1084no—
282C22C22×#111311yescharacteristic
283C4—23no—
284C4—186no—
285C4—183no—
286C3—23no—
287C3—41no—
288C3#1059543413yescomplemented
289C2—186no—
290C2Q8×#111311yescharacteristic
291C2#1172462711yescharacteristic
292C2#1172463311yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC2×C33×Q8C22
prime radicalC22×#1113C22
hypercenterC2×Q8#1113
hypocenterC33#543
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C3×#64155C2semidirect
3. #11723692C2semidirect
4. #11723671C2semidirect
7. C2×C33×Q8C22semidirect
9. C2×#6296C22semidirect
10. C2×#6296C22semidirect
13. C3×C4⋊(C3⋊Dic3)C22semidirect
14. C3×#3750C22semidirect
15. #88815C22semidirect
17. #64155D3semidirect
30. C2×#1113Q8semidirect
43. C2×C32×Q8C2×D3semidirect
52. C4⋊(C3⋊Dic3)C2×D3semidirect
53. #3750C2×D3semidirect
121. C2×C3×Q8D32semidirect
174. C33#543semidirect
218. C2×Q8#1113semidirect
253. C32#5634semidirect
288. C3#10595434semidirect

Maximal subgroups

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

subgroupindex
C3×#641552
#117237002
#117236922
#117236712
#105954343