Finite Group Databasebeta

← #13181525

Subgroups — #13181525

Order 1728.

996 conjugacy classes of subgroups; only the 225 normal subgroups are shown.

normal subgroups only

This group has 996 conjugacy classes of subgroups — too many to draw. Only its 225 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⋊#157)C213yesmaximal, complemented
6#10596650—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)C2213yes—
17C4⋊(C3⋊#157)C2216yes—
18C4⋊(C3⋊#157)C2211yescharacteristic
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—46no—
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⋊D3C2313yes—
39C3×C4×C3⋊D3C2311yescharacteristic
40C3×#634C2×C413yescomplemented
41#5827—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—33no—
60C4⋊#157—36no—
61C4⋊#157—31no—
62C4×#157—36no—
63C22×C33C22×C413yes—
64C2×C3×C3⋊D3—23no—
65C2×C3×C3⋊D3—23no—
66C2×C3×C3⋊D3C22×C413yes—
67C2×C3×C3⋊D3C2411yescharacteristic
68C3⋊#157—26no—
69C3⋊#157—22no—
70C33×C4C22×C413yes—
71C33×C4C22×C411yescharacteristic
72C3×C3⋊Dic3C22×C413yes—
73C3×C3⋊Dic3C22×C411yescharacteristic
74C2×C3×#41—91no—
75D3×#41—62no—
76Q8×Dic3—92no—
77D4×Dic3—96no—
78#926—93no—
79C2×C4⋊Dic3—93no—
80C22×C3⋊D3C22×D313yes—
81C22×C3×D3—66no—
82C2×#157—36no—
83C2×#157—32no—
84C2×#157—126no—
85C32×Q8—22no—
86C32×Q8—42no—
87C32×Q8C2×Dic311yescharacteristic, complemented
88C32×D4—26no—
89C32×D4—46no—
90C32×D4C2×Dic313yescomplemented
91C2×C32×C4—26no—
92C2×C32×C4—46no—
93C2×C32×C4C2×Dic313yescomplemented
94#638C2×Dic316yescomplemented
95C2×C3⋊Dic3C2×Dic313yescomplemented
96C4⋊(C3⋊D3)C2×Dic313yescomplemented
97C4×C3⋊D3C2×Dic313yescomplemented
98C4×C3⋊D3C2×Dic313yescomplemented
99C4×C3⋊D3C22×D313yes—
100C4×C3⋊D3C22×D311yescharacteristic
101#634C2×Dic313yescomplemented
102C3×#67—612no—
103C2×C3×Dic3—66no—
104C3×D12—66no—
105C3×C4×D3—66no—
106C3×C4×D3—66no—
107C3×C4×D3—62no—
108C3×C4×D3—66no—
109C3×Dic6—66no—
110#520—271noSylow
111C2×C33—23no—
112C2×C33C23×C411yescharacteristic
113C3×C3⋊D3—23no—
114C3×C3⋊D3H(5,𝔽2)11yescharacteristic
115C3×C3⋊D32−1+411yescharacteristic
116C3×#41—22no—
117C3×#41—42no—
118C3×#41—91no—
119C3×#41—93no—
120C3×#41—93no—
121C3×#41#15711yescharacteristic, complemented
122C2×C3×Q8—91no—
123C2×C3×D4—93no—
124C22×C3×C4—93no—
125C22×Dic3—96no—
126#239—62no—
127D3×Q8—62no—
128#237—66no—
129D3×D4—66no—
130#235—66no—
131C2×C4×D3—66no—
132#217—96no—
133C4⋊Dic3—93no—
134C4⋊Dic3—96no—
135C4⋊Dic3—91no—
136C4×Dic3—96no—
137C22×C32—26no—
138C22×C32—46no—
139C22×C32C22×Dic313yes—
140C2×C3⋊D3—23no—
141C2×C3⋊D3—23no—
142C2×C3⋊D3C22×Dic313yes—
143C2×C3⋊D3C23×D311yescharacteristic
144C2×C3×D3—62no—
145C2×C3×D3—66no—
146C2×C3×D3—126no—
147C2×C3×D3—126no—
148#157—66no—
149#157—62no—
150C32×C4—26no—
151C32×C4—22no—
152C32×C4—42no—
153C32×C4—46no—
154C32×C4C22×Dic313yes—
155C32×C4C22×Dic311yescharacteristic
156C3⋊Dic3C22×Dic313yes—
157C3⋊Dic3C22×Dic311yescharacteristic
158C3×Dic3—66no—
159C3×Dic3—62no—
160C2×#41—91no—
161C4×Q8—272no—
162C4×D4—276no—
163#117—273no—
164C2×C4⋊C4—273no—
165C33#52011yescharacteristic, complemented, Sylow
166C23×C3—93no—
167C22×D3—66no—
168C3×Q8—22no—
169C3×Q8—42no—
170C3×Q8—93no—
171C3×Q8C2×#15711yescharacteristic, complemented
172C3×D4—26no—
173C3×D4—46no—
174C3×D4—96no—
175C3×D4—93no—
176C3×D4C2×#15713yescomplemented
177C2×C3×C4—26no—
178C2×C3×C4—46no—
179C2×C3×C4—93no—
180C2×C3×C4—93no—
181C2×C3×C4—91no—
182C2×C3×C4—93no—
183C2×C3×C4—93no—
184C2×C3×C4C2×#15713yescomplemented
185#67—612no—
186C2×Dic3—66no—
187C2×Dic3—92no—
188C2×Dic3—96no—
189C2×Dic3—366no—
190D12—66no—
191C4×D3—66no—
192C4×D3—66no—
193C4×D3—62no—
194C4×D3—66no—
195Dic6—66no—
196C2×C32—22no—
197C2×C32—23no—
198C2×C32—42no—
199C2×C32—46no—
200C2×C32—86no—
201C2×C32C23×Dic311yescharacteristic
202C3⋊D3—23no—
203C3⋊D3#100911yescharacteristic
204C3⋊D3#101011yescharacteristic
205C3×D3—62no—
206C3×D3—62no—
207C3×D3—126no—
208#41—91no—
209#41—93no—
210#41—93no—
211#41C3⋊#15711yescharacteristic, complemented
212C2×Q8—91no—
213C2×D4—93no—
214C22×C4—93no—
215C22×C4—276no—
216C4⋊C4—273no—
217C4⋊C4—276no—
218C4⋊C4—271no—
219C22⋊C4—276no—
220C42—276no—
221C22×C3—26no—
222C22×C3—46no—
223C22×C3—91no—
224C22×C3—93no—
225C22×C3—183no—
226C22×C3—183no—
227C22×C3C22×#15713yes—
228C2×D3—62no—
229C2×D3—66no—
230C2×D3—126no—
231C2×D3—126no—
232C3×C4—26no—
233C3×C4—22no—
234C3×C4—42no—
235C3×C4—46no—
236C3×C4—91no—
237C3×C4—93no—
238C3×C4C22×#15711yescharacteristic
239C3×C4C22×#15713yes—
240Dic3—66no—
241Dic3—62no—
242Dic3—186no—
243Dic3—182no—
244C32—22no—
245C32—42no—
246C32#582711yescharacteristic, complemented
247C23—93no—
248Q8—93no—
249Q8C2×C3⋊#15711yescharacteristic, complemented
250D4—96no—
251D4—93no—
252D4C2×C3⋊#15713yescomplemented
253C2×C4—93no—
254C2×C4—93no—
255C2×C4—91no—
256C2×C4—93no—
257C2×C4—93no—
258C2×C4—272no—
259C2×C4—276no—
260C2×C4—1086no—
261C2×C4C2×C3⋊#15713yescomplemented
262C2×C3—23no—
263C2×C3—22no—
264C2×C3—46no—
265C2×C3—42no—
266C2×C3—86no—
267C2×C3—91no—
268C2×C3—91no—
269C2×C3—183no—
270C2×C3C23×#15711yescharacteristic
271D3—62no—
272D3—62no—
273D3—126no—
274C22—91no—
275C22—93no—
276C22—183no—
277C22—183no—
278C22C22×C3⋊#15713yes—
279C4—91no—
280C4—93no—
281C4—546no—
282C4—542no—
283C4C22×C3⋊#15711yescharacteristic
284C4C22×C3⋊#15713yes—
285C3—22no—
286C3—42no—
287C3#1059665011yescharacteristic, complemented
288C2—23no—
289C2—91no—
290C2—91no—
291C2—183no—
292C2C23×C3⋊#15711yescharacteristic
subgroupG/H
center, FrattiniC2C23×C3⋊#157
commutator, socleC2×C33C23×C4
FittingC33×#41C4
prime radicalC3×#41×C3⋊D3C2
cosocleC2×C3×C3⋊D3C24
hypercenter#41C3⋊#157
hypocenterC33#520
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ #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⋊#157)C2semidirect
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
87. C32×Q8C2×Dic3semidirect
90. C32×D4C2×Dic3semidirect
93. C2×C32×C4C2×Dic3semidirect
94. #638C2×Dic3semidirect
95. C2×C3⋊Dic3C2×Dic3semidirect
96. C4⋊(C3⋊D3)C2×Dic3semidirect
97. C4×C3⋊D3C2×Dic3semidirect
98. C4×C3⋊D3C2×Dic3semidirect
101. #634C2×Dic3semidirect
121. C3×#41#157semidirect
165. C33#520semidirect
171. C3×Q8C2×#157semidirect
176. C3×D4C2×#157semidirect
184. C2×C3×C4C2×#157semidirect
211. #41C3⋊#157semidirect
246. C32#5827semidirect
249. Q8C2×C3⋊#157semidirect
252. D4C2×C3⋊#157semidirect
261. C2×C4C2×C3⋊#157semidirect
287. C3#10596650semidirect

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⋊#157)2
#105966503
#58279