Finite Group Databasebeta

← #13162058

Subgroups — #13162058

Order 1728.

1274 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 1274 conjugacy classes of subgroups — too many to draw. Only its 187 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×C3×#3757C213yesmaximal
2C2×#88815C213yesmaximal, complemented
3#11722082C211yesmaximal, characteristic, complemented
4#10591715—33nomaximal
5C24×C33C2211yescharacteristic
6C22×C3×C3⋊Dic3C2213yes—
7C22×C3×C3⋊Dic3C2213yes—
8C3×#3757—26no—
9#88815—26no—
10C2×#3757D313yes—
11C2×C3×#217—36no—
12C2×C3×#217—63no—
13C2×#3724—33no—
14C2×Dic3⋊Dic3—36no—
15#63579—33no—
16C23×C33—23no—
17C23×C33—23no—
18C23×C33Q812yes—
19C23×C33C2311yescharacteristic
20C2×C3×C3⋊Dic3—26no—
21C2×C3×C3⋊Dic3—23no—
22C2×C3×C3⋊Dic3—26no—
23C2×C3×C3⋊Dic3D416yes—
24#4949—93no—
25C24×C32—23no—
26C24×C32—41no—
27C24×C32C2×D313yes—
28C22×C3⋊Dic3C2×D313yes—
29C22×C3⋊Dic3C2×D313yes—
30C22×C3×Dic3—36no—
31C22×C3×Dic3—36no—
32C22×C3×Dic3—63no—
33C22×C3×Dic3—63no—
34#3757—26no—
35C3×#217—612no—
36C3×#217—612no—
37#3724—66no—
38Dic3⋊Dic3—612no—
39C22×C33—26no—
40C22×C33—43no—
41C22×C33—41no—
42C22×C33C2×D413yes—
43C22×C33C2×Q811yescharacteristic
44C22×C33#4113yes—
45C3×C3⋊Dic3—26no—
46C3×C3⋊Dic3—43no—
47C2×C3×C22⋊C4—93no—
48C2×#217—36no—
49C2×#217—63no—
50C2×C4⋊Dic3—93no—
51C2×Dic3⋊C4—96no—
52C3⋊#95—93no—
53C23×C32—23no—
54C23×C32—23no—
55C23×C32—26no—
56C23×C32—26no—
57C23×C32—26no—
58C23×C32—26no—
59C23×C32—26no—
60C23×C32—23no—
61C23×C32—42no—
62C23×C32—41no—
63C23×C32—46no—
64C23×C32—46no—
65C23×C32—46no—
66C23×C32—46no—
67C23×C32Dic616yes—
68C23×C32C22×D313yes—
69C2×C3⋊Dic3—23no—
70C2×C3⋊Dic3—26no—
71C2×C3⋊Dic3—26no—
72C2×C3⋊Dic3#6716yes—
73C2×C3×Dic3—312no—
74C2×C3×Dic3—66no—
75C2×C3×Dic3—66no—
76C2×C3×Dic3—66no—
77C2×C3×Dic3—66no—
78C2×C3×Dic3—66no—
79C2×C3×Dic3—66no—
80C2×C3×Dic3—66no—
81C2×C3×Dic3—126no—
82C2×C3×Dic3—126no—
83#399—271noSylow
84C2×C33—42no—
85C2×C33C22⋊Q813yes—
86C2×C33#12313yes—
87C2×C33#12711yescharacteristic
88C24×C3—23no—
89C24×C3—41no—
90C24×C3D3213yes—
91C22×C3×C4—93no—
92C22×C3×C4—93no—
93C22×Dic3—36no—
94C22×Dic3—36no—
95C22×Dic3—63no—
96C22×Dic3—63no—
97C3×C22⋊C4—186no—
98#217—612no—
99#217—612no—
100C4⋊Dic3—186no—
101Dic3⋊C4—1812no—
102C22×C32—26no—
103C22×C32—212no—
104C22×C32—212no—
105C22×C32—26no—
106C22×C32—23no—
107C22×C32—23no—
108C22×C32—26no—
109C22×C32—23no—
110C22×C32—43no—
111C22×C32—41no—
112C22×C32—43no—
113C22×C32—412no—
114C22×C32—412no—
115C22×C32—412no—
116C22×C32—46no—
117C22×C32—46no—
118C22×C32—412no—
119C22×C32—44no—
120C22×C32—46no—
121C22×C32—43no—
122C22×C32—43no—
123C22×C32C2×Dic613yes—
124C22×C32#23513yes—
125C22×C32D3×D416yes—
126C22×C32#23716yes—
127C22×C32C2×#6713yes—
128C3⋊Dic3—26no—
129C3⋊Dic3—43no—
130C3×Dic3—612no—
131C3×Dic3—126no—
132C3×Dic3—126no—
133C3×Dic3—126no—
134C2×C4⋊C4—273no—
135C2×C22⋊C4—93no—
136#95—271no—
137C33#39911yescharacteristic, complemented, Sylow
138C23×C3—26no—
139C23×C3—26no—
140C23×C3—26no—
141C23×C3—23no—
142C23×C3—26no—
143C23×C3—23no—
144C23×C3—26no—
145C23×C3—23no—
146C23×C3—46no—
147C23×C3—46no—
148C23×C3—46no—
149C23×C3—46no—
150C23×C3—42no—
151C23×C3—41no—
152C23×C3#62716yes—
153C23×C3C2×D3213yes—
154C2×C3×C4—96no—
155C2×C3×C4—186no—
156C2×C3×C4—186no—
157C2×C3×C4—183no—
158C2×Dic3—312no—
159C2×Dic3—66no—
160C2×Dic3—66no—
161C2×Dic3—66no—
162C2×Dic3—66no—
163C2×Dic3—66no—
164C2×Dic3—66no—
165C2×Dic3—66no—
166C2×Dic3—126no—
167C2×Dic3—126no—
168C2×C32—26no—
169C2×C32—26no—
170C2×C32—23no—
171C2×C32—23no—
172C2×C32—23no—
173C2×C32—43no—
174C2×C32—43no—
175C2×C32—41no—
176C2×C32—412no—
177C2×C32—48no—
178C2×C32—46no—
179C2×C32#87816yes—
180C2×C32#88316yes—
181C2×C32#92413yes—
182C2×C32#93513yes—
183C2×C32#94013yes—
184C24#111311yescharacteristic
185C22×C4—93no—
186C22×C4—93no—
187C4⋊C4—546no—
188C22⋊C4—186no—
189C22×C3—212no—
190C22×C3—212no—
191C22×C3—26no—
192C22×C3—26no—
193C22×C3—23no—
194C22×C3—23no—
195C22×C3—23no—
196C22×C3—26no—
197C22×C3—412no—
198C22×C3—412no—
199C22×C3—46no—
200C22×C3—412no—
201C22×C3—46no—
202C22×C3—412no—
203C22×C3—44no—
204C22×C3—46no—
205C22×C3—43no—
206C22×C3—43no—
207C22×C3—43no—
208C22×C3—41no—
209C22×C3—43no—
210C22×C3#380416yes—
211C22×C3#380513yes—
212C22×C3C2×#62713yes—
213C22×C3D3×#6716yes—
214C22×C3#381113yes—
215C3×C4—186no—
216C3×C4—363no—
217Dic3—612no—
218Dic3—126no—
219Dic3—126no—
220Dic3—126no—
221C32—23no—
222C32—41no—
223C32#494913yescomplemented
224C23—23no—
225C23—23no—
226C23#629612yes—
227C23C2×#111311yescharacteristic
228C2×C4—96no—
229C2×C4—186no—
230C2×C4—183no—
231C2×C4—186no—
232C2×C3—23no—
233C2×C3—26no—
234C2×C3—23no—
235C2×C3—26no—
236C2×C3—23no—
237C2×C3—412no—
238C2×C3—48no—
239C2×C3—46no—
240C2×C3—43no—
241C2×C3—43no—
242C2×C3—41no—
243C2×C3#6396416yes—
244C2×C3#6396916yes—
245C2×C3#6397513yes—
246C2×C3#6397913yes—
247C2×C3#6398213yes—
248C22—26no—
249C22—43no—
250C22—41no—
251C22#8904613yes—
252C22C2×#629611yescharacteristic
253C22#8904913yes—
254C4—186no—
255C4—363no—
256C3—23no—
257C3—41no—
258C3#1059171513yescomplemented
259C2—42no—
260C2#1172372113yes—
261C2#1172372413yes—
262C2#1172372811yescharacteristic
subgroupG/H
center, FrattiniC23C2×#1113
commutator, socle, cosocleC23×C33C23
Fitting, prime radicalC24×C33C22
hypercenterC24#1113
hypocenterC33#399
LCSG ▷ C23×C33 ▷ C33
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C2×#88815C2semidirect
3. #11722082C2semidirect
137. C33#399semidirect
223. C32#4949semidirect
258. C3#10591715semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37572
C2×#888152
#117220822
#105917153