Finite Group Databasebeta

← #10592414⋊C3 #13141537

Subgroups — #10592414⋊C3 #13141537

Order 1728.

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

normal subgroups only

This group has 474 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
1C2×#3744⋊C3C212yesmaximal
2C2×#3746⋊C3C212yesmaximal, complemented
3#64084⋊C3C211yesmaximal, characteristic
4#64078⋊C3C211yesmaximal, characteristic, complemented
5H(3,𝔽3)×#97C211yesmaximal, characteristic, complemented
6#10592414C311yesmaximal, characteristic, complemented
7C3×#4715—31nomaximal
8C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
9C2×#634⋊C3C2211yescharacteristic
10C2×#634⋊C3C2212yes—
11C2×C8×H(3,𝔽3)C2212yes—
12C42×H(3,𝔽3)C2211yescharacteristic
13#3752⋊C3—22no—
14C4⋊((C3⋊Dic3)⋊C3)—22no—
15#3744⋊C3—24no—
16#3746⋊C3—24no—
17C2×#3746C2×C312yescomplemented
18C2×#3744C2×C312yes—
19#64084C2×C311yescharacteristic
20#64078C2×C311yescharacteristic, complemented
21C2×C3×Dic12—32no—
22C2×C3×#204—32no—
23C3×#875—31no—
24C3×#869—31no—
25C32×#97—21no—
26C32×#97D311yescharacteristic, complemented
27C32×#97C2×C311yescharacteristic, complemented
28C22×(C3⋊D3)⋊C3—21no—
29C2×C4×H(3,𝔽3)D412yes—
30C2×C4×H(3,𝔽3)C2311yescharacteristic
31C2×(C3⋊Dic3)⋊C3—22no—
32C2×(C3⋊Dic3)⋊C3—21no—
33C4⋊((C3⋊D3)⋊C3)—22no—
34#634⋊C3—22no—
35#634⋊C3—22no—
36#634⋊C3—22no—
37C8×H(3,𝔽3)D414yes—
38C3×#497—91no—
39#4715—31no—
40#4715—91no—
41C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
42C2×#634C22×C312yes—
43C2×#634C22×C311yescharacteristic
44C2×C3×D12—31no—
45C2×C3×Dic6—32no—
46C2×C3×Dic6—31no—
47C2×C32×C8—22no—
48C2×C32×C8C2×D312yes—
49C2×C32×C8C22×C312yes—
50C32×C42—21no—
51C32×C42C2×D311yescharacteristic
52C32×C42C22×C311yescharacteristic
53#3752—22no—
54C4⋊(C3⋊Dic3)—22no—
55#3746—24no—
56#3744—24no—
57C3×(C2×D3)⋊C4—62no—
58C3×C4⋊Dic3—62no—
59C3×Dic12—64no—
60C3×#204—64no—
61C22×H(3,𝔽3)C2×D411yescharacteristic
62C2×(C3⋊D3)⋊C3—41no—
63C2×(C3⋊D3)⋊C3—42no—
64C4×H(3,𝔽3)—22no—
65C4×H(3,𝔽3)C2×D412yes—
66(C3⋊Dic3)⋊C3—41no—
67(C3⋊Dic3)⋊C3—42no—
68C2×C3×Q16—92no—
69C2×C3×SD16—92no—
70C3×C4⋊Q8—91no—
71C3×#124—91no—
72C2×Dic12—32no—
73C2×Dic12—92no—
74C2×#204—32no—
75C2×#204—92no—
76#875—31no—
77#875—91no—
78#869—31no—
79#869—91no—
80C3×#97—31no—
81C3×#97—31no—
82C3×#97—61no—
83C3×#97C3×D311yescharacteristic
84C22×C3⋊D3—21no—
85C22×C3×D3—61no—
86C2×C32×C4—22no—
87C2×C32×C4—21no—
88C2×C32×C4D1212yes—
89C2×C32×C4C3×D412yes—
90C2×C32×C4C22×D311yescharacteristic
91C2×C32×C4C23×C311yescharacteristic
92C2×C3⋊Dic3—22no—
93C2×C3⋊Dic3—21no—
94C4⋊(C3⋊D3)—22no—
95#634—22no—
96#634—22no—
97#634—22no—
98C2×C3×Dic3—62no—
99C2×C3×Dic3—61no—
100C3×D12—62no—
101C3×Dic6—62no—
102C3×Dic6—62no—
103C3×Dic6—62no—
104C32×C8—24no—
105C32×C8D1214yes—
106C32×C8C3×D414yes—
107#497—91noSylow
108C2×H(3,𝔽3)#12711yescharacteristic
109C2×H(3,𝔽3)#13712yes—
110(C3⋊D3)⋊C3—81no—
111C2×C3×Q8—91no—
112C2×C3×Q8—92no—
113C2×C3×D4—91no—
114C2×D12—31no—
115C2×D12—91no—
116C2×Dic6—32no—
117C2×Dic6—31no—
118C2×Dic6—92no—
119C2×Dic6—91no—
120C3×Q16—184no—
121C3×SD16—184no—
122C2×C3×C8—32no—
123C2×C3×C8—32no—
124C2×C3×C8—62no—
125C2×C3×C8C2×C3×D312yes—
126C3×C4⋊C4—182no—
127C3×C22⋊C4—182no—
128C3×C42—31no—
129C3×C42—31no—
130C3×C42—61no—
131C3×C42C2×C3×D311yescharacteristic
132(C2×D3)⋊C4—62no—
133(C2×D3)⋊C4—182no—
134C4⋊Dic3—62no—
135C4⋊Dic3—182no—
136Dic12—64no—
137Dic12—184no—
138#204—64no—
139#204—184no—
140C22×C32—21no—
141C22×C32C2×D1211yescharacteristic
142C22×C32C2×C3×D411yescharacteristic
143C2×C3⋊D3—41no—
144C2×C3⋊D3—42no—
145C2×C3×D3—121no—
146C2×C3×D3—122no—
147C32×C4—22no—
148C32×C4—22no—
149C32×C4—22no—
150C32×C4—42no—
151C32×C4C2×D1212yes—
152C32×C4C2×C3×D412yes—
153C3⋊Dic3—42no—
154C3⋊Dic3—41no—
155C3×Dic3—122no—
156C3×Dic3—121no—
157C2×Q16—92no—
158C2×SD16—92no—
159C4⋊Q8—91no—
160#124—91no—
161#97(C3⋊D3)⋊C311yescharacteristic, complemented
162H(3,𝔽3)#49711yescharacteristic, complemented, Sylow
163C23×C3—181no—
164C22×D3—61no—
165C22×D3—181no—
166C3×Q8—182no—
167C3×Q8—182no—
168C3×Q8—182no—
169C3×D4—182no—
170C2×C3×C4—32no—
171C2×C3×C4—32no—
172C2×C3×C4—31no—
173C2×C3×C4—31no—
174C2×C3×C4—62no—
175C2×C3×C4—61no—
176C2×C3×C4—181no—
177C2×C3×C4—182no—
178C2×C3×C4C3×D1212yes—
179C2×C3×C4C22×C3×D311yescharacteristic
180C2×Dic3—62no—
181C2×Dic3—61no—
182C2×Dic3—182no—
183C2×Dic3—181no—
184D12—62no—
185D12—182no—
186Dic6—62no—
187Dic6—62no—
188Dic6—62no—
189Dic6—182no—
190Dic6—182no—
191Dic6—182no—
192C3×C8—34no—
193C3×C8—34no—
194C3×C8—64no—
195C3×C8C3×D1214yes—
196C2×C32—22no—
197C2×C32—21no—
198C2×C32#87411yescharacteristic
199C2×C32#90912yes—
200C2×C32C3×#12711yescharacteristic
201C2×C32C3×#13712yes—
202C3⋊D3—81no—
203C3×D3—241no—
204C2×Q8—91no—
205C2×Q8—92no—
206C2×D4—91no—
207Q16—184no—
208SD16—184no—
209C2×C8C2×(C3⋊D3)⋊C312yes—
210C4⋊C4—182no—
211C22⋊C4—182no—
212C42C2×(C3⋊D3)⋊C311yescharacteristic
213C22×C3—31no—
214C22×C3—31no—
215C22×C3—61no—
216C22×C3—361no—
217C22×C3—362no—
218C22×C3C2×C3×D1211yescharacteristic
219C2×D3—122no—
220C2×D3—121no—
221C2×D3—361no—
222C2×D3—362no—
223C3×C4—22no—
224C3×C4—32no—
225C3×C4—32no—
226C3×C4—62no—
227C3×C4—62no—
228C3×C4—62no—
229C3×C4—122no—
230C3×C4—361no—
231C3×C4—362no—
232C3×C4C2×C3×D1212yes—
233Dic3—122no—
234Dic3—121no—
235Dic3—362no—
236Dic3—361no—
237C32—21no—
238C32#471511yescharacteristic, complemented
239C32C3×#49711yescharacteristic, complemented
240C23—181no—
241Q8—182no—
242Q8—182no—
243Q8—182no—
244D4—182no—
245C2×C4—181no—
246C2×C4—182no—
247C2×C4C4⋊((C3⋊D3)⋊C3)12yes—
248C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
249C8C4⋊((C3⋊D3)⋊C3)14yes—
250C2×C3—32no—
251C2×C3—32no—
252C2×C3—31no—
253C2×C3—31no—
254C2×C3—62no—
255C2×C3—61no—
256C2×C3—721no—
257C2×C3C3×#87411yescharacteristic
258C2×C3C3×#90912yes—
259D3—241no—
260D3—721no—
261C22—362no—
262C22—361no—
263C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
264C4—22no—
265C4—361no—
266C4—362no—
267C4C2×C4⋊((C3⋊D3)⋊C3)12yes—
268C3—31no—
269C3—31no—
270C3—61no—
271C3C3×#471511yescharacteristic
272C2—721no—
273C2#1172138511yescharacteristic
274C2#64118⋊C312yes—
subgroupG/H
centerC22C2×C4⋊((C3⋊D3)⋊C3)
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#97C2
socleC22×C3C2×C3×D12
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
hypercenter#97(C3⋊D3)⋊C3
hypocenterC32C3×#497
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #97

Complements

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

NH ≅ G/Ntype
2. C2×#3746⋊C3C2semidirect
4. #64078⋊C3C2semidirect
5. H(3,𝔽3)×#97C2semidirect
6. #10592414C3semidirect
17. C2×#3746C2×C3semidirect
20. #64078C2×C3semidirect
26. C32×#97D3semidirect
27. C32×#97C2×C3semidirect
161. #97(C3⋊D3)⋊C3semidirect
162. H(3,𝔽3)#497semidirect
238. C32#4715semidirect
239. C32C3×#497semidirect

Maximal subgroups

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

subgroupindex
C2×#3744⋊C32
C2×#3746⋊C32
#64084⋊C32
#64078⋊C32
H(3,𝔽3)×#972
#105924143
C3×#47153