Finite Group Databasebeta

← #10592393⋊C3 #13141460

Subgroups — #10592393⋊C3 #13141460

Order 1728.

457 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 457 conjugacy classes of subgroups — too many to draw. Only its 112 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
1#11721385C211yesmaximal, characteristic
2#64078⋊C3C211yesmaximal, characteristic, complemented
3C4×C8×H(3,𝔽3)C211yesmaximal, characteristic, complemented
4#63648⋊C3C214yesmaximal, complemented
5#10592393C311yesmaximal, characteristic, complemented
6C3×#4694—31nomaximal
7C2×C4⋊((C3⋊D3)⋊C3)—21no—
8C2×C4⋊((C3⋊D3)⋊C3)—21no—
9C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
10C2×#634⋊C3—21no—
11C2×C8×H(3,𝔽3)C2212yes—
12C42×H(3,𝔽3)C2211yescharacteristic
13C4⋊((C3⋊Dic3)⋊C3)—21no—
14C4⋊((C3⋊Dic3)⋊C3)C2212yes—
15#64083C2×C311yescharacteristic
16#64078C2×C311yescharacteristic, complemented
17C3×#874—31no—
18C3×#869—31no—
19C32×C4×C8—21no—
20C32×C4×C8D311yescharacteristic, complemented
21C32×C4×C8C2×C311yescharacteristic, complemented
22#63648C2×C314yescomplemented
23C3×#821—34no—
24C22×(C3⋊D3)⋊C3—22no—
25C2×C4×H(3,𝔽3)D411yescharacteristic
26C2×C4×H(3,𝔽3)D411yescharacteristic
27C2×C4×H(3,𝔽3)C2311yescharacteristic
28C2×(C3⋊Dic3)⋊C3—22no—
29C4⋊((C3⋊D3)⋊C3)—24no—
30C4⋊((C3⋊D3)⋊C3)—42no—
31C4⋊((C3⋊D3)⋊C3)—42no—
32#634⋊C3—42no—
33C8×H(3,𝔽3)—22no—
34C3×#486—91no—
35#4694—31no—
36#4694—91no—
37C2×C4⋊(C3⋊D3)—21no—
38C2×C4⋊(C3⋊D3)—21no—
39C2×C4⋊(C3⋊D3)C22×C312yes—
40C2×#634—21no—
41C2×C3×D12—32no—
42C2×C3×D12—61no—
43C2×C3×D12—61no—
44C2×C3×Dic6—61no—
45C2×C32×C8—22no—
46C2×C32×C8C2×D312yes—
47C2×C32×C8C22×C312yes—
48C32×C42—21no—
49C32×C42C2×D311yescharacteristic
50C32×C42C22×C311yescharacteristic
51C4⋊(C3⋊Dic3)—21no—
52C4⋊(C3⋊Dic3)C22×C312yes—
53C3×C4⋊Dic3—32no—
54C3×C4⋊Dic3—61no—
55C22×H(3,𝔽3)C2×D411yescharacteristic
56C2×(C3⋊D3)⋊C3—42no—
57C2×(C3⋊D3)⋊C3—42no—
58C2×(C3⋊D3)⋊C3—42no—
59C4×H(3,𝔽3)D812yes—
60C4×H(3,𝔽3)SD1612yes—
61C4×H(3,𝔽3)#4112yes—
62(C3⋊Dic3)⋊C3—42no—
63C3×C4⋊Q8—91no—
64C3×#127—91no—
65#874—31no—
66#874—91no—
67#869—31no—
68#869—91no—
69C3×#102—94no—
70C3×C4×C8—31no—
71C3×C4×C8—31no—
72C3×C4×C8—61no—
73C3×C4×C8C3×D311yescharacteristic
74#821—34no—
75#821—94no—
76C22×C3⋊D3—22no—
77C22×C3×D3—62no—
78C2×C32×C4—21no—
79C2×C32×C4—21no—
80C2×C32×C4—21no—
81C2×C32×C4D1211yescharacteristic
82C2×C32×C4D1211yescharacteristic
83C2×C32×C4C3×D411yescharacteristic
84C2×C32×C4C3×D411yescharacteristic
85C2×C32×C4C22×D311yescharacteristic
86C2×C32×C4C23×C311yescharacteristic
87C2×C3⋊Dic3—22no—
88C4⋊(C3⋊D3)—24no—
89C4⋊(C3⋊D3)—42no—
90C4⋊(C3⋊D3)—42no—
91#634—42no—
92C2×C3×Dic3—62no—
93C3×D12—64no—
94C3×D12—122no—
95C3×D12—122no—
96C3×Dic6—122no—
97C32×C8—24no—
98C32×C8—22no—
99C32×C8—22no—
100#486—91noSylow
101C2×H(3,𝔽3)#12411yescharacteristic
102C2×H(3,𝔽3)C2×D811yescharacteristic
103C2×H(3,𝔽3)C2×SD1611yescharacteristic
104(C3⋊D3)⋊C3—82no—
105C2×C3×Q8—181no—
106C2×C3×D4—92no—
107C2×C3×D4—181no—
108C2×C3×D4—181no—
109C2×D12—32no—
110C2×D12—61no—
111C2×D12—61no—
112C2×D12—92no—
113C2×D12—181no—
114C2×D12—181no—
115C2×Dic6—61no—
116C2×Dic6—181no—
117C2×C3×C8—32no—
118C2×C3×C8—32no—
119C2×C3×C8—62no—
120C2×C3×C8C2×C3×D312yes—
121C3×C4⋊C4—92no—
122C3×C4⋊C4—181no—
123C3×C42—31no—
124C3×C42—31no—
125C3×C42—61no—
126C3×C42C2×C3×D311yescharacteristic
127C4⋊Dic3—32no—
128C4⋊Dic3—61no—
129C4⋊Dic3—92no—
130C4⋊Dic3—181no—
131C22×C32—21no—
132C22×C32C2×D1211yescharacteristic
133C22×C32C2×C3×D411yescharacteristic
134C2×C3⋊D3—42no—
135C2×C3⋊D3—42no—
136C2×C3⋊D3—42no—
137C2×C3×D3—122no—
138C2×C3×D3—122no—
139C2×C3×D3—122no—
140C32×C4—22no—
141C32×C4—22no—
142C32×C4—22no—
143C32×C4#20412yes—
144C32×C4D2412yes—
145C32×C4C3×D812yes—
146C32×C4C3×SD1612yes—
147C32×C4#23512yes—
148C32×C4C3×#4112yes—
149C3⋊Dic3—42no—
150C3×Dic3—122no—
151C4⋊Q8—91no—
152#127—91no—
153#102—94no—
154C4×C8(C3⋊D3)⋊C311yescharacteristic, complemented
155H(3,𝔽3)#48611yescharacteristic, complemented, Sylow
156C23×C3—182no—
157C22×D3—62no—
158C22×D3—182no—
159C3×Q8—362no—
160C3×D4—184no—
161C3×D4—362no—
162C3×D4—362no—
163C2×C3×C4—31no—
164C2×C3×C4—31no—
165C2×C3×C4—31no—
166C2×C3×C4—31no—
167C2×C3×C4—31no—
168C2×C3×C4—31no—
169C2×C3×C4—61no—
170C2×C3×C4—61no—
171C2×C3×C4—61no—
172C2×C3×C4—182no—
173C2×C3×C4C3×D1211yescharacteristic
174C2×C3×C4C3×D1211yescharacteristic
175C2×C3×C4C22×C3×D311yescharacteristic
176C2×Dic3—62no—
177C2×Dic3—182no—
178D12—64no—
179D12—122no—
180D12—122no—
181D12—184no—
182D12—362no—
183D12—362no—
184Dic6—122no—
185Dic6—362no—
186C3×C8—22no—
187C3×C8—62no—
188C3×C8—62no—
189C3×C8—64no—
190C2×C32—21no—
191C2×C32—21no—
192C2×C32—21no—
193C2×C32#87511yescharacteristic
194C2×C32C2×#20411yescharacteristic
195C2×C32C2×D2411yescharacteristic
196C2×C32C3×#12411yescharacteristic
197C2×C32C2×C3×D811yescharacteristic
198C2×C32C2×C3×SD1611yescharacteristic
199C3⋊D3—82no—
200C3×D3—242no—
201C2×Q8—181no—
202C2×D4—92no—
203C2×D4—181no—
204C2×D4—181no—
205C2×C8C2×(C3⋊D3)⋊C312yes—
206C4⋊C4—92no—
207C4⋊C4—181no—
208C42C2×(C3⋊D3)⋊C311yescharacteristic
209C22×C3—31no—
210C22×C3—31no—
211C22×C3—61no—
212C22×C3—362no—
213C22×C3—362no—
214C22×C3—362no—
215C22×C3C2×C3×D1211yescharacteristic
216C2×D3—122no—
217C2×D3—122no—
218C2×D3—122no—
219C2×D3—362no—
220C2×D3—362no—
221C2×D3—362no—
222C3×C4—32no—
223C3×C4—32no—
224C3×C4—32no—
225C3×C4—32no—
226C3×C4—32no—
227C3×C4—32no—
228C3×C4—62no—
229C3×C4—62no—
230C3×C4—62no—
231C3×C4—362no—
232C3×C4C3×#20412yes—
233C3×C4C3×D2412yes—
234C3×C4C3×#23512yes—
235Dic3—122no—
236Dic3—362no—
237C32—21no—
238C32#469411yescharacteristic, complemented
239C32C3×#48611yescharacteristic, complemented
240C23—182no—
241Q8—362no—
242D4—184no—
243D4—362no—
244D4—362no—
245C2×C4—182no—
246C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
247C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
248C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
249C8—22no—
250C2×C3—31no—
251C2×C3—31no—
252C2×C3—31no—
253C2×C3—31no—
254C2×C3—31no—
255C2×C3—31no—
256C2×C3—61no—
257C2×C3—61no—
258C2×C3—61no—
259C2×C3—722no—
260C2×C3C3×#87511yescharacteristic
261C2×C3C2×C3×#20411yescharacteristic
262C2×C3C2×C3×D2411yescharacteristic
263D3—242no—
264D3—722no—
265C22—362no—
266C22—362no—
267C22—362no—
268C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
269C4—362no—
270C4#3744⋊C312yes—
271C4#3745⋊C312yes—
272C4#3828⋊C312yes—
273C3—31no—
274C3—31no—
275C3—61no—
276C3C3×#469411yescharacteristic
277C2—722no—
278C2#64084⋊C311yescharacteristic
279C2C2×#3744⋊C311yescharacteristic
280C2C2×#3745⋊C311yescharacteristic
subgroupG/H
centerC22C2×C4⋊((C3⋊D3)⋊C3)
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC4×C8×H(3,𝔽3)C2
socleC22×C3C2×C3×D12
prime radical#11721385C2
hypercenterC4×C8(C3⋊D3)⋊C3
hypocenterC32C3×#486
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8

Complements

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

NH ≅ G/Ntype
2. #64078⋊C3C2semidirect
3. C4×C8×H(3,𝔽3)C2semidirect
4. #63648⋊C3C2semidirect
5. #10592393C3semidirect
16. #64078C2×C3semidirect
20. C32×C4×C8D3semidirect
21. C32×C4×C8C2×C3semidirect
22. #63648C2×C3semidirect
154. C4×C8(C3⋊D3)⋊C3semidirect
155. H(3,𝔽3)#486semidirect
238. C32#4694semidirect
239. C32C3×#486semidirect

Maximal subgroups

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

subgroupindex
#117213852
#64078⋊C32
C4×C8×H(3,𝔽3)2
#63648⋊C32
#105923933
C3×#46943