Finite Group Databasebeta

← #41×C3⋊#157 #13181524

Subgroups — #41×C3⋊#157 #13181524

Order 1728.

1028 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 1028 conjugacy classes of subgroups — too many to draw. Only its 229 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×#41×C3⋊D3C211yesmaximal, characteristic
2Q8×C3⋊#157C212yesmaximal, complemented
3D4×C3⋊#157C216yesmaximal, complemented
4#11724514C213yesmaximal, complemented
5C2×C4×C3⋊#157C213yesmaximal, complemented
6#41×#157—31nomaximal
7C22×C3⋊#157C2216yes—
8C33×#41C411yescharacteristic, complemented
9C3×#3832C411yescharacteristic, complemented
10C3×Q8×C3⋊D3C2211yescharacteristic
11C3×#3830C413yescomplemented
12C3×D4×C3⋊D3C2213yes—
13C3×#3828C413yescomplemented
14C2×C3×C4×C3⋊D3C2213yes—
15#88994C2216yes—
16C4⋊(C3⋊#157)C2216yes—
17C4×C3⋊#157C2211yescharacteristic
18C4×C3⋊#157C2213yes—
19C4×C3⋊#157C2216yes—
20#41×C3⋊D3D311yescharacteristic
21C3×D3×#41—62no—
22Q8×#157—32no—
23D4×#157—36no—
24#64286—33no—
25C2×C4×#157—33no—
26C22×C3×C3⋊D3C2313yes—
27C2×C3⋊#157—212no—
28C2×C3⋊#157C2316yes—
29C2×C3⋊#157C2312yes—
30C33×Q8C2×C411yescharacteristic, complemented
31C33×D4C2×C413yescomplemented
32C2×C33×C4C2×C413yescomplemented
33C3×#638C2×C416yescomplemented
34C2×C3×C3⋊Dic3C2×C413yescomplemented
35C3×C4⋊(C3⋊D3)C2×C413yescomplemented
36C3×C4×C3⋊D3C2×C413yescomplemented
37C3×C4×C3⋊D3C2×C413yescomplemented
38C3×C4×C3⋊D3C2311yescharacteristic
39C3×C4×C3⋊D3C2313yes—
40C3×#634C2×C413yescomplemented
41Dic3×#41—91nomaximal
42C22×#157—36no—
43C32×#41—22no—
44C32×#41—42no—
45C32×#41Dic311yescharacteristic, complemented
46#3832Dic311yescharacteristic, complemented
47Q8×C3⋊D3C2×D311yescharacteristic
48#3830Dic313yescomplemented
49D4×C3⋊D3C2×D313yes—
50#3828Dic313yescomplemented
51C2×C4×C3⋊D3C2×D313yes—
52C3×#239—62no—
53C3×D3×Q8—62no—
54C3×#237—66no—
55C3×D3×D4—66no—
56C3×#235—66no—
57C2×C3×C4×D3—66no—
58C22⋊#157—36no—
59C4⋊#157—36no—
60C4×#157—31no—
61C4×#157—33no—
62C4×#157—36no—
63C22×C33C22×C413yes—
64C2×C3×C3⋊D3—26no—
65C2×C3×C3⋊D3C22×C413yes—
66C2×C3×C3⋊D3C2411yescharacteristic
67C3⋊#157—26no—
68C3⋊#157#4114yesdirect product factor
69C33×C4C22×C413yes—
70C33×C4C22×C411yescharacteristic
71C3×C3⋊Dic3C22×C413yes—
72C3×C3⋊Dic3C22×C411yescharacteristic
73C2×C3×#41—91no—
74D3×#41—62no—
75Q8×Dic3—92no—
76D4×Dic3—96no—
77#926—93no—
78C2×C4×Dic3—93no—
79C22×C3⋊D3C22×D313yes—
80C22×C3×D3—66no—
81C2×#157—32no—
82C2×#157—36no—
83C2×#157—612no—
84C32×Q8—22no—
85C32×Q8—42no—
86C32×Q8C2×Dic311yescharacteristic, complemented
87C32×D4—26no—
88C32×D4—46no—
89C32×D4C2×Dic313yescomplemented
90C2×C32×C4—26no—
91C2×C32×C4—46no—
92C2×C32×C4C2×Dic313yescomplemented
93#638C2×Dic316yescomplemented
94C2×C3⋊Dic3C2×Dic313yescomplemented
95C4⋊(C3⋊D3)C2×Dic313yescomplemented
96C4×C3⋊D3C2×Dic313yescomplemented
97C4×C3⋊D3C2×Dic313yescomplemented
98C4×C3⋊D3C22×D311yescharacteristic
99C4×C3⋊D3C22×D313yes—
100#634C2×Dic313yescomplemented
101C3×#67—612no—
102C2×C3×Dic3—66no—
103C3×D12—66no—
104C3×C4×D3—66no—
105C3×C4×D3—66no—
106C3×C4×D3—62no—
107C3×C4×D3—66no—
108C3×Dic6—66no—
109C4×#41—271noSylow
110C2×C33—23no—
111C2×C33C23×C411yescharacteristic
112C3×C3⋊D3—23no—
113C3×C3⋊D3C2×#4112yes—
114C3×#41—22no—
115C3×#41—42no—
116C3×#41—91no—
117C3×#41—93no—
118C3×#41—93no—
119C3×#41#15711yescharacteristic, complemented
120C2×C3×Q8—91no—
121C2×C3×D4—93no—
122C22×C3×C4—93no—
123C22×Dic3—96no—
124#239—62no—
125D3×Q8—62no—
126#237—66no—
127D3×D4—66no—
128#235—66no—
129C2×C4×D3—66no—
130#217—96no—
131C4⋊Dic3—96no—
132C4×Dic3—91no—
133C4×Dic3—93no—
134C4×Dic3—96no—
135C22×C32—26no—
136C22×C32—46no—
137C22×C32C22×Dic313yes—
138C2×C3⋊D3—26no—
139C2×C3⋊D3C22×Dic313yes—
140C2×C3⋊D3C23×D311yescharacteristic
141C2×C3×D3—62no—
142C2×C3×D3—66no—
143C2×C3×D3—1212no—
144#157—34no—
145#157—66no—
146C32×C4—26no—
147C32×C4—22no—
148C32×C4—42no—
149C32×C4—46no—
150C32×C4C22×Dic313yes—
151C32×C4C22×Dic311yescharacteristic
152C3⋊Dic3C22×Dic313yes—
153C3⋊Dic3C22×Dic311yescharacteristic
154C3×Dic3—66no—
155C3×Dic3—62no—
156C2×#41—91no—
157C4×Q8—272no—
158C4×D4—276no—
159#117—273no—
160C2×C42—273no—
161C33C4×#4111yescharacteristic, complemented, Sylow
162C23×C3—93no—
163C22×D3—66no—
164C3×Q8—22no—
165C3×Q8—42no—
166C3×Q8—93no—
167C3×Q8C2×#15711yescharacteristic, complemented
168C3×D4—26no—
169C3×D4—46no—
170C3×D4—96no—
171C3×D4—93no—
172C3×D4C2×#15713yescomplemented
173C2×C3×C4—26no—
174C2×C3×C4—46no—
175C2×C3×C4—93no—
176C2×C3×C4—91no—
177C2×C3×C4—93no—
178C2×C3×C4—93no—
179C2×C3×C4—93no—
180C2×C3×C4C2×#15713yescomplemented
181#67—612no—
182C2×Dic3—66no—
183C2×Dic3—92no—
184C2×Dic3—96no—
185C2×Dic3—1812no—
186D12—66no—
187C4×D3—66no—
188C4×D3—66no—
189C4×D3—62no—
190C4×D3—66no—
191Dic6—66no—
192C2×C32—22no—
193C2×C32—23no—
194C2×C32—42no—
195C2×C32—46no—
196C2×C32—86no—
197C2×C32C23×Dic311yescharacteristic
198C3⋊D3—23no—
199C3⋊D3D3×#4112yes—
200C3×D3—64no—
201C3×D3—126no—
202#41—91no—
203#41—93no—
204#41—93no—
205#41C3⋊#15711yescharacteristic, direct product factor
206C2×Q8—91no—
207C2×D4—93no—
208C22×C4—93no—
209C22×C4—276no—
210C4⋊C4—276no—
211C22⋊C4—276no—
212C42—271no—
213C42—273no—
214C42—276no—
215C22×C3—26no—
216C22×C3—46no—
217C22×C3—91no—
218C22×C3—93no—
219C22×C3—186no—
220C22×C3C22×#15713yes—
221C2×D3—62no—
222C2×D3—66no—
223C2×D3—1212no—
224C3×C4—26no—
225C3×C4—22no—
226C3×C4—46no—
227C3×C4—42no—
228C3×C4—91no—
229C3×C4—93no—
230C3×C4C22×#15713yes—
231C3×C4C22×#15711yescharacteristic
232Dic3—66no—
233Dic3—62no—
234Dic3—94no—
235Dic3—186no—
236C32—22no—
237C32—42no—
238C32Dic3×#4111yescharacteristic, complemented
239C23—93no—
240Q8—93no—
241Q8C2×C3⋊#15711yescharacteristic, complemented
242D4—96no—
243D4—93no—
244D4C2×C3⋊#15713yescomplemented
245C2×C4—93no—
246C2×C4—93no—
247C2×C4—93no—
248C2×C4—91no—
249C2×C4—93no—
250C2×C4—272no—
251C2×C4—276no—
252C2×C4—5412no—
253C2×C4C2×C3⋊#15713yescomplemented
254C2×C3—23no—
255C2×C3—22no—
256C2×C3—46no—
257C2×C3—42no—
258C2×C3—86no—
259C2×C3—92no—
260C2×C3—183no—
261C2×C3C23×#15711yescharacteristic
262D3—64no—
263D3—126no—
264C22—91no—
265C22—93no—
266C22—186no—
267C22C22×C3⋊#15713yes—
268C4—91no—
269C4—93no—
270C4—274no—
271C4—546no—
272C4C22×C3⋊#15713yes—
273C4C22×C3⋊#15711yescharacteristic
274C3—22no—
275C3—42no—
276C3#41×#15711yescharacteristic, complemented
277C2—23no—
278C2—92no—
279C2—183no—
280C2C23×C3⋊#15711yescharacteristic
subgroupG/H
centerC4C22×C3⋊#157
commutator, socleC2×C33C23×C4
FrattiniC2C23×C3⋊#157
FittingC33×#41C4
prime radicalC3×#41×C3⋊D3C2
cosocleC2×C3×C3⋊D3C24
hypercenter#41C3⋊#157
hypocenterC33C4×#41
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ #41

Complements

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

NH ≅ G/Ntype
2. Q8×C3⋊#157C2semidirect
3. D4×C3⋊#157C2semidirect
4. #11724514C2semidirect
5. C2×C4×C3⋊#157C2semidirect
8. C33×#41C4semidirect
9. C3×#3832C4semidirect
11. C3×#3830C4semidirect
13. C3×#3828C4semidirect
30. C33×Q8C2×C4semidirect
31. C33×D4C2×C4semidirect
32. C2×C33×C4C2×C4semidirect
33. C3×#638C2×C4semidirect
34. C2×C3×C3⋊Dic3C2×C4semidirect
35. C3×C4⋊(C3⋊D3)C2×C4semidirect
36. C3×C4×C3⋊D3C2×C4semidirect
37. C3×C4×C3⋊D3C2×C4semidirect
40. C3×#634C2×C4semidirect
45. C32×#41Dic3semidirect
46. #3832Dic3semidirect
48. #3830Dic3semidirect
50. #3828Dic3semidirect
68. C3⋊#157#41direct (N × H)
86. C32×Q8C2×Dic3semidirect
89. C32×D4C2×Dic3semidirect
92. C2×C32×C4C2×Dic3semidirect
93. #638C2×Dic3semidirect
94. C2×C3⋊Dic3C2×Dic3semidirect
95. C4⋊(C3⋊D3)C2×Dic3semidirect
96. C4×C3⋊D3C2×Dic3semidirect
97. C4×C3⋊D3C2×Dic3semidirect
100. #634C2×Dic3semidirect
119. C3×#41#157semidirect
161. C33C4×#41semidirect
167. C3×Q8C2×#157semidirect
172. C3×D4C2×#157semidirect
180. C2×C3×C4C2×#157semidirect
205. #41C3⋊#157direct (N × H)
238. C32Dic3×#41semidirect
241. Q8C2×C3⋊#157semidirect
244. D4C2×C3⋊#157semidirect
253. C2×C4C2×C3⋊#157semidirect
276. C3#41×#157semidirect

Maximal subgroups

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

subgroupindex
C3×#41×C3⋊D32
Q8×C3⋊#1572
D4×C3⋊#1572
#117245142
C2×C4×C3⋊#1572
#41×#1573
Dic3×#419