Finite Group Databasebeta

← #13138849

Subgroups — #13138849

Order 1728.

505 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 505 conjugacy classes of subgroups — too many to draw. Only its 99 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×#3745⋊C3C211yesmaximal, characteristic
2#11721132C212yesmaximal, complemented
3#11720551C211yesmaximal, characteristic, complemented
4C8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
5#11720308C212yesmaximal, complemented
6#10590777—31nomaximal
7#10590763—31nomaximal
8C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
9C2×((C3⋊D3)⋊C3)⋊C4—22no—
10C2×C8×H(3,𝔽3)C2211yescharacteristic
11C4⋊#1084C2212yes—
12C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
13C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
14#3745⋊C3C414yescomplemented
15#88450—22no—
16C2×#3745D311yescharacteristic
17C2×C3×D24—31no—
18#63898—32no—
19#63894—32no—
20C8×C3⋊Dic3D311yescharacteristic, complemented
21C3×#818—31no—
22C3×#818—31no—
23C3×#818—61no—
24C3×C8×Dic3—31no—
25#63564—32no—
26#63562—32no—
27C22×(C3⋊D3)⋊C3—22no—
28C2×C4×H(3,𝔽3)C2311yescharacteristic
29C2×#1084—22no—
30C2×(C3⋊Dic3)⋊C3D411yescharacteristic
31C2×(C3⋊Dic3)⋊C3D411yescharacteristic
32C4⋊((C3⋊D3)⋊C3)—22no—
33C4⋊((C3⋊D3)⋊C3)C2×C414yescomplemented
34((C3⋊D3)⋊C3)⋊C4—44no—
35C8×H(3,𝔽3)C2×C412yescomplemented
36(C3⋊(C3⋊C8))⋊C3—21no—
37Dic3×D8—91no—
38#4872—91no—
39C4×D24—91no—
40C2×C4⋊(C3⋊D3)C2×D312yes—
41C2×C3×D12—32no—
42C2×#624—62no—
43C2×D3×Dic3—62no—
44C2×C32×C8—21no—
45C2×C32×C8C2×D311yescharacteristic
46C2×C32×C8C2×D311yescharacteristic
47C4×C3⋊Dic3C2×D311yescharacteristic
48C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
49#3745Dic314yescomplemented
50C3×C4⋊Dic3—32no—
51C3×C4⋊Dic3—32no—
52C3×C4⋊Dic3—62no—
53C3×C4×Dic3—31no—
54C2×C3×C3⋊C8—31no—
55C3×D24—34no—
56#3722—62no—
57(C2×D3)⋊Dic3—62no—
58C22×H(3,𝔽3)C2×D411yescharacteristic
59C2×(C3⋊D3)⋊C3—24no—
60C2×(C3⋊D3)⋊C3—42no—
61C2×(C3⋊D3)⋊C3—42no—
62C4×H(3,𝔽3)C22×C411yescharacteristic
63C4×H(3,𝔽3)#4111yescharacteristic
64#1084—42no—
65(C3⋊Dic3)⋊C3—21no—
66(C3⋊Dic3)⋊C3D812yescomplemented
67C2×C3×D8—91no—
68D4×Dic3—92no—
69C2×D24—31no—
70C2×D24—91no—
71#893—92no—
72C4×D12—92no—
73C3×#107—91no—
74C3×C4×C8—91no—
75#832—92no—
76#821—92no—
77#818—31no—
78#818—31no—
79#818—61no—
80C8×Dic3—31no—
81C8×Dic3—91no—
82#809—92no—
83C22×C3⋊D3—22no—
84C22×C3×D3—62no—
85C2×C32×C4—21no—
86C2×C32×C4C22×D311yescharacteristic
87C2×C32×C4C22×D311yescharacteristic
88C2×C3⋊Dic3D1211yescharacteristic
89C2×C3⋊Dic3D1211yescharacteristic
90C4⋊(C3⋊D3)—22no—
91C4⋊(C3⋊D3)C2×Dic314yescomplemented
92C2×C3×Dic3—31no—
93C2×C3×Dic3—31no—
94C2×C3×Dic3—62no—
95C2×C3×Dic3—62no—
96C2×C3×Dic3—122no—
97C3×D12—34no—
98C3×D12—62no—
99#624—124no—
100D3×Dic3—124no—
101C32×C8—22no—
102C32×C8C4×D312yescomplemented
103C32×C8C2×Dic312yescomplemented
104C3⋊(C3⋊C8)—21no—
105C3×C3⋊C8—61no—
106C4×D8—271noSylow
107C2×H(3,𝔽3)C4×D411yescharacteristic
108C2×H(3,𝔽3)C2×D811yescharacteristic
109C2×H(3,𝔽3)#13511yescharacteristic
110(C3⋊D3)⋊C3—44no—
111C2×C3×D4—92no—
112C22×Dic3—182no—
113C2×D12—32no—
114C2×D12—92no—
115C2×C4×D3—182no—
116C2×C4×D3—182no—
117C3×D8—94no—
118C2×C3×C8—31no—
119C2×C3×C8—31no—
120C2×C3×C8—61no—
121C2×C3×C8—91no—
122C2×C3×C8D3211yescharacteristic
123C3×C4⋊C4—92no—
124C3×C42—91no—
125#217—182no—
126(C2×D3)⋊C4—182no—
127(C2×D3)⋊C4—182no—
128C4⋊Dic3—32no—
129C4⋊Dic3—32no—
130C4⋊Dic3—62no—
131C4×Dic3—31no—
132C4×Dic3—91no—
133C2×C3⋊C8—31no—
134C2×C3⋊C8—91no—
135D24—34no—
136D24—94no—
137C22×C32—21no—
138C22×C32C2×D1211yescharacteristic
139C22×C32D3×D411yescharacteristic
140C2×C3⋊D3—24no—
141C2×C3⋊D3—42no—
142C2×C3⋊D3—42no—
143C2×C3×D3—64no—
144C2×C3×D3—122no—
145C2×C3×D3—122no—
146C32×C4—21no—
147C32×C4—21no—
148C32×C4C2×C4×D311yescharacteristic
149C32×C4#23511yescharacteristic
150C32×C4#23711yescharacteristic
151C32×C4C22×Dic311yescharacteristic
152C3⋊Dic3—21no—
153C3⋊Dic3D2412yescomplemented
154C3×Dic3—32no—
155C3×Dic3—61no—
156C3×Dic3—122no—
157C3×Dic3—122no—
158C3×Dic3—242no—
159C2×D8—91no—
160C4×D4—272no—
161#107—91no—
162#102—272no—
163C4×C8—91no—
164H(3,𝔽3)C4×D811yescharacteristic, complemented, Sylow
165C23×C3—182no—
166C22×D3—62no—
167C22×D3—182no—
168C3×D4—94no—
169C3×D4—182no—
170C2×C3×C4—31no—
171C2×C3×C4—31no—
172C2×C3×C4—61no—
173C2×C3×C4—91no—
174C2×C3×C4—91no—
175C2×C3×C4—182no—
176C2×C3×C4C2×D3211yescharacteristic
177C2×Dic3—31no—
178C2×Dic3—31no—
179C2×Dic3—62no—
180C2×Dic3—62no—
181C2×Dic3—91no—
182C2×Dic3—91no—
183C2×Dic3—122no—
184C2×Dic3—364no—
185D12—34no—
186D12—62no—
187D12—94no—
188D12—182no—
189C4×D3—364no—
190C4×D3—364no—
191C3×C8—32no—
192C3×C8—32no—
193C3×C8—62no—
194C3×C8—181no—
195C3×C8D3×Dic312yes—
196C3⋊C8—61no—
197C3⋊C8—181no—
198C2×C32—21no—
199C2×C32—21no—
200C2×C32—21no—
201C2×C32C4×D1211yescharacteristic
202C2×C32C2×D2411yescharacteristic
203C2×C32#90411yescharacteristic
204C2×C32D3×D811yescharacteristic
205C2×C32#91211yescharacteristic
206C2×C32D4×Dic311yescharacteristic
207C3⋊D3—44no—
208C3×D3—124no—
209C2×D4—92no—
210C22×C4—542no—
211D8—94no—
212C2×C8—91no—
213C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
214C4⋊C4—92no—
215C22⋊C4—542no—
216C42—91no—
217C22×C3—31no—
218C22×C3—31no—
219C22×C3—61no—
220C22×C3—184no—
221C22×C3—362no—
222C22×C3—362no—
223C22×C3D3×D1211yescharacteristic
224C2×D3—64no—
225C2×D3—122no—
226C2×D3—122no—
227C2×D3—184no—
228C2×D3—362no—
229C2×D3—362no—
230C3×C4—31no—
231C3×C4—31no—
232C3×C4—31no—
233C3×C4—31no—
234C3×C4—61no—
235C3×C4—61no—
236C3×C4—92no—
237C3×C4—181no—
238C3×C4—362no—
239C3×C4#379511yescharacteristic
240C3×C4C2×D3×Dic311yescharacteristic
241Dic3—32no—
242Dic3—61no—
243Dic3—92no—
244Dic3—122no—
245Dic3—122no—
246Dic3—181no—
247Dic3—242no—
248C32—21no—
249C32C4×D2411yescharacteristic, complemented
250C32Dic3×D811yescharacteristic, complemented
251C23—182no—
252D4—94no—
253D4—182no—
254C2×C4—91no—
255C2×C4—91no—
256C2×C4—182no—
257C2×C4—1084no—
258C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
259C8—181no—
260C8((C3⋊D3)⋊C3)⋊C412yescomplemented
261C2×C3—31no—
262C2×C3—31no—
263C2×C3—31no—
264C2×C3—31no—
265C2×C3—31no—
266C2×C3—31no—
267C2×C3—61no—
268C2×C3—61no—
269C2×C3—61no—
270C2×C3—364no—
271C2×C3D3×D2411yescharacteristic
272C2×C3#6380711yescharacteristic
273C2×C3Dic3×D1211yescharacteristic
274D3—124no—
275D3—364no—
276C22—184no—
277C22—362no—
278C22—362no—
279C22#8865411yescharacteristic
280C4—92no—
281C4—181no—
282C4—362no—
283C4#8864811yescharacteristic
284C4C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
285C3—31no—
286C3—31no—
287C3—61no—
288C3Dic3×D2411yescharacteristic
289C2—364no—
290C2#1172084711yescharacteristic
291C2#1172085511yescharacteristic
292C2#1172113211yescharacteristic
subgroupG/H
centerC22#88654
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3D3×D12
prime radicalC2×#3745⋊C3C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C4×D8
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. #11721132C2semidirect
3. #11720551C2semidirect
4. C8×(C3⋊Dic3)⋊C3C2semidirect
5. #11720308C2semidirect
14. #3745⋊C3C4semidirect
20. C8×C3⋊Dic3D3semidirect
33. C4⋊((C3⋊D3)⋊C3)C2×C4semidirect
35. C8×H(3,𝔽3)C2×C4semidirect
49. #3745Dic3semidirect
66. (C3⋊Dic3)⋊C3D8semidirect
91. C4⋊(C3⋊D3)C2×Dic3semidirect
102. C32×C8C4×D3semidirect
103. C32×C8C2×Dic3semidirect
153. C3⋊Dic3D24semidirect
164. H(3,𝔽3)C4×D8semidirect
249. C32C4×D24semidirect
250. C32Dic3×D8semidirect
260. C8((C3⋊D3)⋊C3)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#3745⋊C32
#117211322
#117205512
C8×(C3⋊Dic3)⋊C32
#117203082
#105907773
#105907633