Finite Group Databasebeta

← #13139921

Subgroups — #13139921

Order 1728.

531 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 531 conjugacy classes of subgroups — too many to draw. Only its 155 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#11721826C211yesmaximal, characteristic
2C2×#3748⋊C3C211yesmaximal, characteristic
3C22×(C3⋊(C3⋊C8))⋊C3C211yesmaximal, characteristic
4#11720303C212yesmaximal, complemented
5#11720302C212yesmaximal, complemented
6#10591057—31nomaximal
7#10591055—31nomaximal
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9#88552C412yes—
10C4⋊#1084C412yes—
11C4×#1084C2212yes—
12#3748⋊C3C414yescomplemented
13C2×(C3⋊(C3⋊C8))⋊C3C414yescomplemented
14C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
15C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
16C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
17C2×#3748D311yescharacteristic
18C22×C3⋊(C3⋊C8)D311yescharacteristic
19C3×#926—31no—
20C3×#926—31no—
21C3×#926—61no—
22C2×C3×#208—31no—
23C22×C3×C3⋊C8—31no—
24#63554—32no—
25#63554—32no—
26Dic3×C3⋊C8—32no—
27Dic3×C3⋊C8—32no—
28C23×H(3,𝔽3)C2×C411yescharacteristic
29C2×C4×H(3,𝔽3)C2×C411yescharacteristic
30C2×C4×H(3,𝔽3)C2×C412yes—
31C2×C4×H(3,𝔽3)C2×C412yes—
32C2×C4×H(3,𝔽3)C2311yescharacteristic
33C2×#1084C2×C414yes—
34(C3⋊(C3⋊C8))⋊C3C2×C412yes—
35(C3⋊(C3⋊C8))⋊C3C2×C412yes—
36(C3⋊(C3⋊C8))⋊C3C2×C414yes—
37#5122—91no—
38#5101—91no—
39#4997—91no—
40C22×C32×C4—21no—
41C22×C32×C4C2×D311yescharacteristic
42C22×C32×C4C2×D311yescharacteristic
43#3748Dic314yescomplemented
44C2×C3⋊(C3⋊C8)Dic314yescomplemented
45C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
46C2×C3⋊(C3⋊C8)C2×D312yes—
47C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
48C3×#217—32no—
49C3×#217—32no—
50C3×#217—62no—
51C3×C4⋊Dic3—32no—
52C3×C4⋊Dic3—32no—
53C3×C4⋊Dic3—62no—
54C3×C4×Dic3—32no—
55C3×C4×Dic3—32no—
56C3×C4×Dic3—62no—
57C3×#208—34no—
58C2×C3×C3⋊C8—34no—
59C2×C3×C3⋊C8—31no—
60C2×C3×C3⋊C8—31no—
61C2×C3×C3⋊C8—32no—
62C22×H(3,𝔽3)—22no—
63C22×H(3,𝔽3)C4212yes—
64C22×H(3,𝔽3)C22×C411yescharacteristic
65C4×H(3,𝔽3)C4212yes—
66C4×H(3,𝔽3)C22×C411yescharacteristic
67C4×H(3,𝔽3)C22×C411yescharacteristic
68#1084—24no—
69C2×C3×M16—91no—
70C22×C3×C8—91no—
71C3×#117—91no—
72#926—31no—
73#926—31no—
74#926—61no—
75C2×#208—31no—
76C2×#208—91no—
77C22×C3⋊C8—31no—
78C22×C3⋊C8—91no—
79#815—92no—
80#815—92no—
81C8×Dic3—92no—
82C8×Dic3—92no—
83#803—92no—
84C4×C3⋊C8—92no—
85C23×C32—21no—
86C23×C32C4×D311yescharacteristic
87C23×C32C4×D311yescharacteristic
88C2×C32×C4—22no—
89C2×C32×C4—22no—
90C2×C32×C4—21no—
91C2×C32×C4—21no—
92C2×C32×C4C4×D312yes—
93C2×C32×C4C4×D312yes—
94C2×C32×C4C4×D311yescharacteristic
95C2×C32×C4C4×D311yescharacteristic
96C2×C32×C4C2×Dic312yes—
97C2×C32×C4C2×Dic312yes—
98C2×C32×C4C22×D311yescharacteristic
99C2×C32×C4C22×D311yescharacteristic
100C2×C3×Dic3—34no—
101C2×C3×Dic3—34no—
102C2×C3×Dic3—64no—
103C3⋊(C3⋊C8)C2×Dic312yes—
104C3⋊(C3⋊C8)C2×Dic312yes—
105C3⋊(C3⋊C8)C2×Dic314yes—
106C3×C3⋊C8—32no—
107C3×C3⋊C8—32no—
108C3×C3⋊C8—34no—
109#405—271noSylow
110C2×H(3,𝔽3)—22no—
111C2×H(3,𝔽3)C2×C4211yescharacteristic
112C2×H(3,𝔽3)#13112yes—
113C22×C3×C4—31no—
114C22×C3×C4—31no—
115C22×C3×C4—61no—
116C22×C3×C4D3211yescharacteristic
117C3×M16—94no—
118C2×C3×C8—91no—
119C2×C3×C8—92no—
120C2×C3×C8—91no—
121C2×C3×C8—94no—
122C3×C4⋊C4—92no—
123C3×C22⋊C4—92no—
124C3×C42—92no—
125#217—32no—
126#217—32no—
127#217—62no—
128C4⋊Dic3—32no—
129C4⋊Dic3—32no—
130C4⋊Dic3—62no—
131C4×Dic3—32no—
132C4×Dic3—32no—
133C4×Dic3—62no—
134#208—34no—
135#208—94no—
136C2×C3⋊C8—34no—
137C2×C3⋊C8—31no—
138C2×C3⋊C8—32no—
139C2×C3⋊C8—31no—
140C2×C3⋊C8—91no—
141C2×C3⋊C8—92no—
142C2×C3⋊C8—91no—
143C2×C3⋊C8—94no—
144C22×C32—22no—
145C22×C32—22no—
146C22×C32—22no—
147C22×C32—21no—
148C22×C32—42no—
149C22×C32C4×Dic312yes—
150C22×C32C4×Dic312yes—
151C22×C32C2×C4×D311yescharacteristic
152C22×C32C2×C4×D311yescharacteristic
153C32×C4—22no—
154C32×C4—21no—
155C32×C4—21no—
156C32×C4C4×Dic312yes—
157C32×C4C4×Dic312yes—
158C32×C4C2×C4×D311yescharacteristic
159C32×C4C2×C4×D311yescharacteristic
160C32×C4C22×Dic311yescharacteristic
161C32×C4C22×Dic311yescharacteristic
162C3×Dic3—64no—
163C3×Dic3—64no—
164C3×Dic3—124no—
165C2×M16—91no—
166C22×C8—91no—
167#117—91no—
168#97—272no—
169C4×C8—272no—
170H(3,𝔽3)#40511yescharacteristic, complemented, Sylow
171C23×C3—31no—
172C23×C3—31no—
173C23×C3—61no—
174C23×C3#62411yescharacteristic
175C2×C3×C4—32no—
176C2×C3×C4—32no—
177C2×C3×C4—32no—
178C2×C3×C4—32no—
179C2×C3×C4—31no—
180C2×C3×C4—31no—
181C2×C3×C4—31no—
182C2×C3×C4—31no—
183C2×C3×C4—62no—
184C2×C3×C4—62no—
185C2×C3×C4—61no—
186C2×C3×C4—61no—
187C2×C3×C4—94no—
188C2×C3×C4D3×Dic312yes—
189C2×C3×C4D3×Dic312yes—
190C2×C3×C4#62411yescharacteristic
191C2×C3×C4C2×D3211yescharacteristic
192C2×Dic3—34no—
193C2×Dic3—34no—
194C2×Dic3—64no—
195C3×C8—92no—
196C3×C8—92no—
197C3×C8—94no—
198C3⋊C8—32no—
199C3⋊C8—32no—
200C3⋊C8—34no—
201C3⋊C8—92no—
202C3⋊C8—92no—
203C3⋊C8—94no—
204C2×C32—22no—
205C2×C32—22no—
206C2×C32—22no—
207C2×C32—21no—
208C2×C32—42no—
209C2×C32#90112yes—
210C2×C32#90712yes—
211C2×C32C2×C4×Dic311yescharacteristic
212C2×C32C2×C4×Dic311yescharacteristic
213C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
214M16—94no—
215C2×C8—94no—
216C2×C8—91no—
217C2×C8—92no—
218C2×C8—91no—
219C4⋊C4—92no—
220C22⋊C4—92no—
221C42—92no—
222C22×C3—22no—
223C22×C3—31no—
224C22×C3—31no—
225C22×C3—32no—
226C22×C3—32no—
227C22×C3—62no—
228C22×C3—62no—
229C22×C3—61no—
230C22×C3—62no—
231C22×C3—122no—
232C22×C3Dic3212yes—
233C22×C3C2×#62411yescharacteristic
234C3×C4—31no—
235C3×C4—31no—
236C3×C4—31no—
237C3×C4—31no—
238C3×C4—32no—
239C3×C4—32no—
240C3×C4—61no—
241C3×C4—61no—
242C3×C4—62no—
243C3×C4—184no—
244C3×C4Dic3212yes—
245C3×C4C2×D3×Dic311yescharacteristic
246C3×C4C2×D3×Dic311yescharacteristic
247Dic3—64no—
248Dic3—64no—
249Dic3—124no—
250C32—21no—
251C32#510111yescharacteristic, complemented
252C32#512211yescharacteristic, complemented
253C23#619911yescharacteristic
254C2×C4—94no—
255C2×C4((C3⋊D3)⋊C3)⋊C412yes—
256C2×C4((C3⋊D3)⋊C3)⋊C412yes—
257C2×C4#619911yescharacteristic
258C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
259C8—92no—
260C8—92no—
261C8—94no—
262C2×C3—22no—
263C2×C3—31no—
264C2×C3—31no—
265C2×C3—32no—
266C2×C3—32no—
267C2×C3—62no—
268C2×C3—62no—
269C2×C3—61no—
270C2×C3—62no—
271C2×C3—122no—
272C2×C3#6381712yes—
273C2×C3C2×Dic3211yescharacteristic
274C22—22no—
275C22#8844712yes—
276C22C2×#619911yescharacteristic
277C4—184no—
278C4#8844712yes—
279C4C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
280C4C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
281C3—31no—
282C3—31no—
283C3—61no—
284C3#1059105911yescharacteristic
285C2—22no—
286C2#1172106412yes—
287C2C2×#8844711yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C2×C42
FrattiniC2×C3×C4C2×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×#624
prime radicalC23×H(3,𝔽3)C2×C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#405
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
4. #11720303C2semidirect
5. #11720302C2semidirect
12. #3748⋊C3C4semidirect
13. C2×(C3⋊(C3⋊C8))⋊C3C4semidirect
43. #3748Dic3semidirect
44. C2×C3⋊(C3⋊C8)Dic3semidirect
170. H(3,𝔽3)#405semidirect
251. C32#5101semidirect
252. C32#5122semidirect

Maximal subgroups

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

subgroupindex
#117218262
C2×#3748⋊C32
C22×(C3⋊(C3⋊C8))⋊C32
#117203032
#117203022
#105910573
#105910553