Finite Group Databasebeta

← #13138830

Subgroups — #13138830

Order 1728.

491 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 491 conjugacy classes of subgroups — too many to draw. Only its 123 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×C8×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2#11721129C212yesmaximal
3#11720551C211yesmaximal, characteristic, complemented
4#63645⋊C3C211yesmaximal, characteristic, complemented
5#11720306C212yesmaximal, complemented
6#10590687—31nomaximal
7D3×#818—31nomaximal
8C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
9C2×((C3⋊D3)⋊C3)⋊C4—22no—
10C2×C8×H(3,𝔽3)C2211yescharacteristic
11C4⋊#1084C2212yes—
12C4⋊((C3⋊Dic3)⋊C3)C2212yes—
13C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
14C8×(C3⋊D3)⋊C3C414yescomplemented
15#88448—22no—
16C2×C8×C3⋊D3D311yescharacteristic
17C2×C3×D3×C8—31no—
18#63900—32no—
19D3×C4⋊Dic3—32no—
20#63645D311yescharacteristic, complemented
21C3×#818—31no—
22C3×#818—31no—
23C3×#818—31no—
24C3×#818—61no—
25#63576—32no—
26#63575—32no—
27C22×(C3⋊D3)⋊C3D411yescharacteristic
28C2×C4×H(3,𝔽3)C2311yescharacteristic
29C2×#1084—22no—
30C2×(C3⋊Dic3)⋊C3—22no—
31C2×(C3⋊Dic3)⋊C3D411yescharacteristic
32C4×(C3⋊D3)⋊C3C2×C412yes—
33C4×(C3⋊D3)⋊C3Q812yes—
34((C3⋊D3)⋊C3)⋊C4—44no—
35C8×H(3,𝔽3)C2×C412yescomplemented
36(C3⋊(C3⋊C8))⋊C3C2×C412yescomplemented
37C2×#818—91no—
38D3×#107—91no—
39D3×#107—91no—
40C2×C4×C3⋊D3C2×D311yescharacteristic
41C2×C3×C4×D3—31no—
42C2×#624—62no—
43C2×D3×Dic3—62no—
44C2×C32×C8—21no—
45C2×C32×C8C2×D311yescharacteristic
46C2×C32×C8C2×D311yescharacteristic
47C4⋊(C3⋊Dic3)C2×D312yes—
48C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
49C8×C3⋊D3Dic314yescomplemented
50C3×C4⋊Dic3—32no—
51C3×C4⋊Dic3—32no—
52C3×C4⋊Dic3—32no—
53C3×C4⋊Dic3—62no—
54C2×C3×C3⋊C8—31no—
55C3×D3×C8—34no—
56#3724—62no—
57Dic3⋊Dic3—62no—
58C22×H(3,𝔽3)C2×D411yescharacteristic
59C2×(C3⋊D3)⋊C3C4⋊C412yes—
60C2×(C3⋊D3)⋊C3D812yes—
61C2×(C3⋊D3)⋊C3Q1612yes—
62C4×H(3,𝔽3)C22×C411yescharacteristic
63C4×H(3,𝔽3)C2×Q811yescharacteristic
64#1084—42no—
65(C3⋊Dic3)⋊C3—42no—
66(C3⋊Dic3)⋊C3C4⋊C412yes—
67C22×C3×C8—91no—
68C2×C4⋊Dic3—92no—
69C2×D3×C8—31no—
70C2×D3×C8—91no—
71D3×C4⋊C4—92no—
72D3×C4⋊C4—92no—
73C3×#107—91no—
74C3×#107—91no—
75#818—31no—
76#818—31no—
77#818—31no—
78#818—61no—
79#818—91no—
80#818—92no—
81#807—92no—
82#807—92no—
83C22×C3⋊D3D1211yescharacteristic
84C22×C3×D3—31no—
85C2×C32×C4—21no—
86C2×C32×C4C22×D311yescharacteristic
87C2×C32×C4C22×D311yescharacteristic
88C2×C3⋊Dic3—22no—
89C2×C3⋊Dic3D1211yescharacteristic
90C4×C3⋊D3Dic612yes—
91C4×C3⋊D3C2×Dic312yes—
92C2×C3×Dic3—31no—
93C2×C3×Dic3—62no—
94C2×C3×Dic3—62no—
95C2×C3×Dic3—62no—
96C2×C3×Dic3—122no—
97C3×C4×D3—32no—
98C3×C4×D3—32no—
99#624—124no—
100D3×Dic3—124no—
101C32×C8—22no—
102C32×C8C4×D312yescomplemented
103C32×C8C2×Dic312yescomplemented
104C3⋊(C3⋊C8)C2×Dic312yescomplemented
105C3×C3⋊C8—32no—
106C2×#107—271noSylow
107C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
108C2×H(3,𝔽3)C2×D811yescharacteristic
109C2×H(3,𝔽3)C2×Q1611yescharacteristic
110(C3⋊D3)⋊C3#10714yescomplemented
111C22×C3×C4—91no—
112C22×Dic3—182no—
113C2×C4×D3—31no—
114C2×C4×D3—91no—
115C2×C4×D3—182no—
116C2×C4×D3—182no—
117C2×C3×C8—31no—
118C2×C3×C8—31no—
119C2×C3×C8—61no—
120C2×C3×C8—94no—
121C2×C3×C8—91no—
122C2×C3×C8D3211yescharacteristic
123C3×C4⋊C4—92no—
124C3×C4⋊C4—92no—
125C4⋊Dic3—32no—
126C4⋊Dic3—32no—
127C4⋊Dic3—32no—
128C4⋊Dic3—62no—
129C4⋊Dic3—92no—
130C4⋊Dic3—182no—
131Dic3⋊C4—182no—
132Dic3⋊C4—182no—
133C2×C3⋊C8—31no—
134C2×C3⋊C8—91no—
135D3×C8—34no—
136D3×C8—94no—
137C22×C32—21no—
138C22×C32C2×D1211yescharacteristic
139C22×C32D3×D411yescharacteristic
140C2×C3⋊D3D2412yes—
141C2×C3⋊D3Dic1212yes—
142C2×C3⋊D3C4⋊Dic312yes—
143C2×C3×D3—32no—
144C2×C3×D3—32no—
145C2×C3×D3—32no—
146C32×C4—21no—
147C32×C4—21no—
148C32×C4C2×Dic611yescharacteristic
149C32×C4C2×C4×D311yescharacteristic
150C32×C4D3×Q811yescharacteristic
151C32×C4C22×Dic311yescharacteristic
152C3⋊Dic3—42no—
153C3⋊Dic3C4⋊Dic312yes—
154C3×Dic3—32no—
155C3×Dic3—122no—
156C3×Dic3—122no—
157C3×Dic3—122no—
158C3×Dic3—242no—
159C22×C8—91no—
160C2×C4⋊C4—272no—
161#107—91no—
162#107—91no—
163#107—272no—
164H(3,𝔽3)C2×#10711yescharacteristic, complemented, Sylow
165C23×C3—91no—
166C22×D3—31no—
167C22×D3—91no—
168C2×C3×C4—31no—
169C2×C3×C4—31no—
170C2×C3×C4—61no—
171C2×C3×C4—92no—
172C2×C3×C4—92no—
173C2×C3×C4—91no—
174C2×C3×C4—182no—
175C2×C3×C4—182no—
176C2×C3×C4C2×D3211yescharacteristic
177C2×Dic3—31no—
178C2×Dic3—62no—
179C2×Dic3—62no—
180C2×Dic3—62no—
181C2×Dic3—91no—
182C2×Dic3—122no—
183C2×Dic3—182no—
184C2×Dic3—364no—
185C4×D3—32no—
186C4×D3—32no—
187C4×D3—92no—
188C4×D3—92no—
189C4×D3—364no—
190C4×D3—364no—
191C3×C8—32no—
192C3×C8—32no—
193C3×C8—62no—
194C3×C8—92no—
195C3×C8D3×Dic312yes—
196C3⋊C8—32no—
197C3⋊C8—92no—
198C2×C32—21no—
199C2×C32—21no—
200C2×C32—21no—
201C2×C32D3×C4⋊C411yescharacteristic
202C2×C32C2×D2411yescharacteristic
203C2×C32C2×Dic1211yescharacteristic
204C2×C32D3×D811yescharacteristic
205C2×C32D3×Q1611yescharacteristic
206C2×C32C2×C4⋊Dic311yescharacteristic
207C3⋊D3#81814yescomplemented
208C3×D3—34no—
209C22×C4—91no—
210C22×C4—542no—
211C2×C8—91no—
212C2×C8—94no—
213C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
214C4⋊C4—92no—
215C4⋊C4—92no—
216C4⋊C4—542no—
217C22×C3—31no—
218C22×C3—31no—
219C22×C3—61no—
220C22×C3—92no—
221C22×C3—92no—
222C22×C3—92no—
223C22×C3D3×D1211yescharacteristic
224C2×D3—32no—
225C2×D3—32no—
226C2×D3—32no—
227C2×D3—92no—
228C2×D3—92no—
229C2×D3—92no—
230C3×C4—31no—
231C3×C4—31no—
232C3×C4—31no—
233C3×C4—31no—
234C3×C4—61no—
235C3×C4—61no—
236C3×C4—92no—
237C3×C4—362no—
238C3×C4—362no—
239C3×C4D3×Dic611yescharacteristic
240C3×C4C2×D3×Dic311yescharacteristic
241Dic3—32no—
242Dic3—92no—
243Dic3—122no—
244Dic3—122no—
245Dic3—122no—
246Dic3—242no—
247Dic3—362no—
248C32—21no—
249C32D3×#10711yescharacteristic, complemented
250C32C2×#81811yescharacteristic, complemented
251C23—91no—
252C2×C4—91no—
253C2×C4—92no—
254C2×C4—92no—
255C2×C4—182no—
256C2×C4—182no—
257C2×C4—1084no—
258C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
259C8—92no—
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—94no—
271C2×C3D3×D2411yescharacteristic
272C2×C3D3×Dic1211yescharacteristic
273C2×C3D3×C4⋊Dic311yescharacteristic
274D3—34no—
275D3—94no—
276C22—92no—
277C22—92no—
278C22—92no—
279C22#8865411yescharacteristic
280C4—92no—
281C4—362no—
282C4—362no—
283C4((C3⋊D3)⋊C3)⋊Q811yescharacteristic
284C4C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
285C3—31no—
286C3—31no—
287C3—61no—
288C3D3×#81811yescharacteristic
289C2—94no—
290C2#1172084711yescharacteristic
291C2#1172085111yescharacteristic
292C2#1172112911yescharacteristic
subgroupG/H
centerC22#88654
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3D3×D12
prime radicalC22×(C3⋊D3)⋊C3D4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#107
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
3. #11720551C2semidirect
4. #63645⋊C3C2semidirect
5. #11720306C2semidirect
14. C8×(C3⋊D3)⋊C3C4semidirect
20. #63645D3semidirect
35. C8×H(3,𝔽3)C2×C4semidirect
36. (C3⋊(C3⋊C8))⋊C3C2×C4semidirect
49. C8×C3⋊D3Dic3semidirect
102. C32×C8C4×D3semidirect
103. C32×C8C2×Dic3semidirect
104. C3⋊(C3⋊C8)C2×Dic3semidirect
110. (C3⋊D3)⋊C3#107semidirect
164. H(3,𝔽3)C2×#107semidirect
207. C3⋊D3#818semidirect
249. C32D3×#107semidirect
250. C32C2×#818semidirect
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×C8×(C3⋊D3)⋊C32
#117211292
#117205512
#63645⋊C32
#117203062
#105906873
D3×#8183