Finite Group Databasebeta

← #13139959

Subgroups — #13139959

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
1C22×#6180C211yesmaximal, characteristic
2#64135⋊C3C211yesmaximal, characteristic
3C2×#3748⋊C3C211yesmaximal, characteristic
4#11720312C212yesmaximal, complemented
5#11720310C212yesmaximal, complemented
6#10591059—31nomaximal
7#10591055—31nomaximal
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C2×#6180C414yescomplemented
10C2×#6180C2211yescharacteristic
11C2×#6180C2211yescharacteristic
12#3757⋊C3C412yes—
13C4⋊((C3⋊Dic3)⋊C3)C412yes—
14C4×(C3⋊Dic3)⋊C3C2212yes—
15#3748⋊C3C414yescomplemented
16C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
17#64135D311yescharacteristic
18C2×#3748D311yescharacteristic
19C3×#926—31no—
20C2×C3×#208—31no—
21C22×C3×C3⋊C8—31no—
22C22×C3×C3⋊C8—31no—
23C22×C3×C3⋊C8—61no—
24#63555—32no—
25#63554—32no—
26#63553—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×(C3⋊Dic3)⋊C3C2×C414yes—
34#6180C2×C412yes—
35#6180C2×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#3757Dic312yes—
44C4⋊(C3⋊Dic3)Dic312yes—
45C4×C3⋊Dic3C2×D312yes—
46#3748Dic314yescomplemented
47C2×C3⋊(C3⋊C8)C2×D312yes—
48C3×#217—32no—
49C3×C4⋊Dic3—32no—
50C3×C4×Dic3—32no—
51C3×#208—34no—
52C2×C3×C3⋊C8—34no—
53C2×C3×C3⋊C8—34no—
54C2×C3×C3⋊C8—31no—
55C2×C3×C3⋊C8—31no—
56C2×C3×C3⋊C8—31no—
57C2×C3×C3⋊C8—31no—
58C2×C3×C3⋊C8—32no—
59C2×C3×C3⋊C8—64no—
60C2×C3×C3⋊C8—61no—
61C2×C3×C3⋊C8—61no—
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(C3⋊Dic3)⋊C3—24no—
69C2×C3×M16—91no—
70C22×C3×C8—91no—
71C3×#117—91no—
72#926—31no—
73#926—91no—
74C2×#208—31no—
75C2×#208—91no—
76C22×C3⋊C8—31no—
77C22×C3⋊C8—31no—
78C22×C3⋊C8—61no—
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×C32C2×Dic311yescharacteristic
88C2×C32×C4—21no—
89C2×C32×C4—22no—
90C2×C32×C4—22no—
91C2×C32×C4—21no—
92C2×C32×C4C4×D311yescharacteristic
93C2×C32×C4C4×D312yes—
94C2×C32×C4C4×D312yes—
95C2×C32×C4C4×D312yes—
96C2×C32×C4C2×Dic311yescharacteristic
97C2×C32×C4C2×Dic312yes—
98C2×C32×C4C22×D311yescharacteristic
99C2×C32×C4C22×D311yescharacteristic
100C2×C3⋊Dic3C2×Dic314yes—
101C2×C3×Dic3—34no—
102C3⋊(C3⋊C8)C2×Dic314yes—
103C3×C3⋊C8—34no—
104C3×C3⋊C8—32no—
105C3×C3⋊C8—32no—
106C3×C3⋊C8—32no—
107C3×C3⋊C8—32no—
108C3×C3⋊C8—62no—
109C3×C3⋊C8—62no—
110#405—271noSylow
111C2×H(3,𝔽3)—22no—
112C2×H(3,𝔽3)C2×C4211yescharacteristic
113C2×H(3,𝔽3)#13112yes—
114C22×C3×C4—31no—
115C22×C3×C4—31no—
116C22×C3×C4—61no—
117C22×C3×C4D3211yescharacteristic
118C3×M16—94no—
119C2×C3×C8—92no—
120C2×C3×C8—94no—
121C2×C3×C8—91no—
122C2×C3×C8—91no—
123C3×C4⋊C4—92no—
124C3×C22⋊C4—92no—
125C3×C42—92no—
126#217—32no—
127#217—92no—
128C4⋊Dic3—32no—
129C4⋊Dic3—92no—
130C4×Dic3—32no—
131C4×Dic3—92no—
132#208—34no—
133#208—94no—
134C2×C3⋊C8—31no—
135C2×C3⋊C8—31no—
136C2×C3⋊C8—31no—
137C2×C3⋊C8—31no—
138C2×C3⋊C8—34no—
139C2×C3⋊C8—34no—
140C2×C3⋊C8—32no—
141C2×C3⋊C8—64no—
142C2×C3⋊C8—61no—
143C2×C3⋊C8—61no—
144C2×C3⋊C8—92no—
145C22×C32—24no—
146C22×C32—22no—
147C22×C32—22no—
148C22×C32—22no—
149C22×C32—21no—
150C22×C32C4×Dic312yes—
151C22×C32C4×Dic312yes—
152C22×C32C2×C4×D311yescharacteristic
153C22×C32C22×Dic311yescharacteristic
154C32×C4—22no—
155C32×C4—21no—
156C32×C4—21no—
157C32×C4C4×Dic312yes—
158C32×C4C4×Dic312yes—
159C32×C4C2×C4×D311yescharacteristic
160C32×C4C2×C4×D311yescharacteristic
161C32×C4C2×C4×D311yescharacteristic
162C32×C4C22×Dic311yescharacteristic
163C3⋊Dic3—24no—
164C3×Dic3—64no—
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×C3D3×Dic311yescharacteristic
175C2×C3×C4—31no—
176C2×C3×C4—32no—
177C2×C3×C4—32no—
178C2×C3×C4—31no—
179C2×C3×C4—32no—
180C2×C3×C4—32no—
181C2×C3×C4—31no—
182C2×C3×C4—31no—
183C2×C3×C4—62no—
184C2×C3×C4—61no—
185C2×C3×C4—62no—
186C2×C3×C4—61no—
187C2×C3×C4—94no—
188C2×C3×C4D3×Dic311yescharacteristic
189C2×C3×C4D3×Dic312yes—
190C2×C3×C4#62412yes—
191C2×C3×C4C2×D3211yescharacteristic
192C2×Dic3—34no—
193C2×Dic3—94no—
194C3×C8—94no—
195C3×C8—92no—
196C3×C8—92no—
197C3⋊C8—32no—
198C3⋊C8—32no—
199C3⋊C8—32no—
200C3⋊C8—32no—
201C3⋊C8—34no—
202C3⋊C8—62no—
203C3⋊C8—62no—
204C3⋊C8—94no—
205C2×C32—24no—
206C2×C32—22no—
207C2×C32—22no—
208C2×C32—22no—
209C2×C32—21no—
210C2×C32#90712yes—
211C2×C32C2×C4×Dic311yescharacteristic
212C2×C32C2×C4×Dic311yescharacteristic
213C2×C32#94812yes—
214C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
215M16—94no—
216C2×C8—94no—
217C2×C8—91no—
218C2×C8—92no—
219C2×C8—91no—
220C4⋊C4—92no—
221C22⋊C4—92no—
222C42—92no—
223C22×C3—22no—
224C22×C3—31no—
225C22×C3—31no—
226C22×C3—32no—
227C22×C3—32no—
228C22×C3—62no—
229C22×C3—62no—
230C22×C3—64no—
231C22×C3—61no—
232C22×C3—62no—
233C22×C3Dic3212yes—
234C22×C3C2×D3×Dic311yescharacteristic
235C3×C4—31no—
236C3×C4—31no—
237C3×C4—31no—
238C3×C4—31no—
239C3×C4—32no—
240C3×C4—32no—
241C3×C4—61no—
242C3×C4—61no—
243C3×C4—62no—
244C3×C4—184no—
245C3×C4Dic3212yes—
246C3×C4C2×D3×Dic311yescharacteristic
247C3×C4C2×#62411yescharacteristic
248Dic3—64no—
249Dic3—184no—
250C32—21no—
251C32#499711yescharacteristic, complemented
252C32#512211yescharacteristic, complemented
253C23((C3⋊D3)⋊C3)⋊C411yescharacteristic
254C2×C4—94no—
255C2×C4((C3⋊D3)⋊C3)⋊C412yes—
256C2×C4((C3⋊D3)⋊C3)⋊C411yescharacteristic
257C2×C4#619912yes—
258C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
259C8—94no—
260C8—92no—
261C8—92no—
262C2×C3—22no—
263C2×C3—31no—
264C2×C3—31no—
265C2×C3—32no—
266C2×C3—32no—
267C2×C3—62no—
268C2×C3—62no—
269C2×C3—64no—
270C2×C3—61no—
271C2×C3—62no—
272C2×C3#6381512yes—
273C2×C3C2×Dic3211yescharacteristic
274C22—22no—
275C22#8844712yes—
276C22C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
277C4—184no—
278C4#8844712yes—
279C4C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
280C4C2×#619911yescharacteristic
281C3—31no—
282C3—31no—
283C3—61no—
284C3#1059105711yescharacteristic
285C2—22no—
286C2#1172105512yes—
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×D3×Dic3
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. #11720312C2semidirect
5. #11720310C2semidirect
9. C2×#6180C4semidirect
15. #3748⋊C3C4semidirect
46. #3748Dic3semidirect
170. H(3,𝔽3)#405semidirect
251. C32#4997semidirect
252. C32#5122semidirect

Maximal subgroups

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

subgroupindex
C22×#61802
#64135⋊C32
C2×#3748⋊C32
#117203122
#117203102
#105910593
#105910553