Finite Group Databasebeta

← #13139961

Subgroups — #13139961

Order 1728.

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

normal subgroups only

This group has 523 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
1C2×#88538C211yesmaximal, characteristic
2#64135⋊C3C211yesmaximal, characteristic
3C22×(C3⋊(C3⋊C8))⋊C3C211yesmaximal, characteristic
4#11720312C212yesmaximal, complemented
5#11720310C212yesmaximal, complemented
6#10591059—31nomaximal
7#10591057—31nomaximal
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9#88538C414yescomplemented
10C2×#6180C2212yes—
11#3757⋊C3C412yes—
12C4⋊((C3⋊Dic3)⋊C3)C412yes—
13C4×(C3⋊Dic3)⋊C3C2212yes—
14C2×(C3⋊(C3⋊C8))⋊C3C414yescomplemented
15C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
16C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
17#64135D311yescharacteristic
18C22×C3⋊(C3⋊C8)D311yescharacteristic
19C3×#926—31no—
20C2×C3×#208—31no—
21C2×C3×#208—31no—
22C2×C3×#208—61no—
23C22×C3×C3⋊C8—31no—
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×C414yes—
35(C3⋊(C3⋊C8))⋊C3C2×C412yes—
36(C3⋊(C3⋊C8))⋊C3C2×C412yes—
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—
46C2×C3⋊(C3⋊C8)Dic314yescomplemented
47C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
48C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
49C3×#217—32no—
50C3×C4⋊Dic3—32no—
51C3×C4×Dic3—32no—
52C3×#208—34no—
53C3×#208—34no—
54C3×#208—64no—
55C2×C3×C3⋊C8—34no—
56C2×C3×C3⋊C8—31no—
57C2×C3×C3⋊C8—31no—
58C2×C3×C3⋊C8—32no—
59C2×C3×C3⋊C8—32no—
60C2×C3×C3⋊C8—62no—
61C22×H(3,𝔽3)—22no—
62C22×H(3,𝔽3)C4212yes—
63C22×H(3,𝔽3)C22×C411yescharacteristic
64C4×H(3,𝔽3)C4212yes—
65C4×H(3,𝔽3)C22×C411yescharacteristic
66C4×H(3,𝔽3)C22×C411yescharacteristic
67(C3⋊Dic3)⋊C3—24no—
68C2×C3×M16—91no—
69C22×C3×C8—91no—
70C3×#117—91no—
71#926—31no—
72#926—91no—
73C2×#208—31no—
74C2×#208—31no—
75C2×#208—61no—
76C22×C3⋊C8—31no—
77C22×C3⋊C8—91no—
78#815—92no—
79#815—92no—
80C8×Dic3—92no—
81C8×Dic3—92no—
82#803—92no—
83C4×C3⋊C8—92no—
84C23×C32—21no—
85C23×C32C4×D311yescharacteristic
86C23×C32C2×Dic311yescharacteristic
87C2×C32×C4—21no—
88C2×C32×C4—22no—
89C2×C32×C4—22no—
90C2×C32×C4—21no—
91C2×C32×C4C4×D311yescharacteristic
92C2×C32×C4C4×D312yes—
93C2×C32×C4C4×D312yes—
94C2×C32×C4C4×D312yes—
95C2×C32×C4C2×Dic311yescharacteristic
96C2×C32×C4C2×Dic312yes—
97C2×C32×C4C22×D311yescharacteristic
98C2×C32×C4C22×D311yescharacteristic
99C2×C3⋊Dic3C2×Dic314yes—
100C2×C3×Dic3—34no—
101C3⋊(C3⋊C8)C2×Dic312yes—
102C3⋊(C3⋊C8)C2×Dic312yes—
103C3×C3⋊C8—32no—
104C3×C3⋊C8—32no—
105C3×C3⋊C8—34no—
106C3×C3⋊C8—34no—
107C3×C3⋊C8—64no—
108#405—271noSylow
109C2×H(3,𝔽3)—22no—
110C2×H(3,𝔽3)C2×C4211yescharacteristic
111C2×H(3,𝔽3)#13112yes—
112C22×C3×C4—31no—
113C22×C3×C4—31no—
114C22×C3×C4—61no—
115C22×C3×C4D3211yescharacteristic
116C3×M16—94no—
117C2×C3×C8—91no—
118C2×C3×C8—91no—
119C2×C3×C8—94no—
120C2×C3×C8—92no—
121C3×C4⋊C4—92no—
122C3×C22⋊C4—92no—
123C3×C42—92no—
124#217—32no—
125#217—92no—
126C4⋊Dic3—32no—
127C4⋊Dic3—92no—
128C4×Dic3—32no—
129C4×Dic3—92no—
130#208—34no—
131#208—34no—
132#208—64no—
133C2×C3⋊C8—32no—
134C2×C3⋊C8—32no—
135C2×C3⋊C8—34no—
136C2×C3⋊C8—31no—
137C2×C3⋊C8—31no—
138C2×C3⋊C8—62no—
139C2×C3⋊C8—91no—
140C2×C3⋊C8—91no—
141C2×C3⋊C8—94no—
142C22×C32—22no—
143C22×C32—22no—
144C22×C32—22no—
145C22×C32—21no—
146C22×C32—42no—
147C22×C32C4×Dic312yes—
148C22×C32C4×Dic312yes—
149C22×C32C2×C4×D311yescharacteristic
150C22×C32C22×Dic311yescharacteristic
151C32×C4—22no—
152C32×C4—21no—
153C32×C4—21no—
154C32×C4C4×Dic312yes—
155C32×C4C4×Dic312yes—
156C32×C4C2×C4×D311yescharacteristic
157C32×C4C2×C4×D311yescharacteristic
158C32×C4C2×C4×D311yescharacteristic
159C32×C4C22×Dic311yescharacteristic
160C3⋊Dic3—24no—
161C3×Dic3—64no—
162C2×M16—91no—
163C22×C8—91no—
164#117—91no—
165#97—272no—
166C4×C8—272no—
167H(3,𝔽3)#40511yescharacteristic, complemented, Sylow
168C23×C3—31no—
169C23×C3—31no—
170C23×C3—61no—
171C23×C3D3×Dic311yescharacteristic
172C2×C3×C4—31no—
173C2×C3×C4—32no—
174C2×C3×C4—32no—
175C2×C3×C4—31no—
176C2×C3×C4—32no—
177C2×C3×C4—32no—
178C2×C3×C4—31no—
179C2×C3×C4—31no—
180C2×C3×C4—62no—
181C2×C3×C4—61no—
182C2×C3×C4—62no—
183C2×C3×C4—61no—
184C2×C3×C4—94no—
185C2×C3×C4D3×Dic311yescharacteristic
186C2×C3×C4D3×Dic312yes—
187C2×C3×C4#62412yes—
188C2×C3×C4C2×D3211yescharacteristic
189C2×Dic3—34no—
190C2×Dic3—94no—
191C3×C8—92no—
192C3×C8—92no—
193C3×C8—94no—
194C3⋊C8—34no—
195C3⋊C8—34no—
196C3⋊C8—32no—
197C3⋊C8—32no—
198C3⋊C8—64no—
199C3⋊C8—92no—
200C3⋊C8—92no—
201C2×C32—22no—
202C2×C32—22no—
203C2×C32—22no—
204C2×C32—21no—
205C2×C32—42no—
206C2×C32#90112yes—
207C2×C32C2×C4×Dic311yescharacteristic
208C2×C32C2×C4×Dic311yescharacteristic
209C2×C32#94812yes—
210C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
211M16—94no—
212C2×C8—94no—
213C2×C8—91no—
214C2×C8—92no—
215C2×C8—91no—
216C4⋊C4—92no—
217C22⋊C4—92no—
218C42—92no—
219C22×C3—22no—
220C22×C3—31no—
221C22×C3—31no—
222C22×C3—32no—
223C22×C3—32no—
224C22×C3—62no—
225C22×C3—62no—
226C22×C3—61no—
227C22×C3—62no—
228C22×C3—122no—
229C22×C3Dic3212yes—
230C22×C3C2×D3×Dic311yescharacteristic
231C3×C4—31no—
232C3×C4—31no—
233C3×C4—31no—
234C3×C4—31no—
235C3×C4—32no—
236C3×C4—32no—
237C3×C4—61no—
238C3×C4—61no—
239C3×C4—62no—
240C3×C4—184no—
241C3×C4Dic3212yes—
242C3×C4C2×D3×Dic311yescharacteristic
243C3×C4C2×#62411yescharacteristic
244Dic3—64no—
245Dic3—184no—
246C32—21no—
247C32#499711yescharacteristic, complemented
248C32#510111yescharacteristic, complemented
249C23((C3⋊D3)⋊C3)⋊C411yescharacteristic
250C2×C4—94no—
251C2×C4((C3⋊D3)⋊C3)⋊C412yes—
252C2×C4((C3⋊D3)⋊C3)⋊C411yescharacteristic
253C2×C4#619912yes—
254C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
255C8—92no—
256C8—92no—
257C8—94no—
258C2×C3—22no—
259C2×C3—31no—
260C2×C3—31no—
261C2×C3—32no—
262C2×C3—32no—
263C2×C3—62no—
264C2×C3—62no—
265C2×C3—61no—
266C2×C3—62no—
267C2×C3—122no—
268C2×C3#6381412yes—
269C2×C3C2×Dic3211yescharacteristic
270C22—22no—
271C22#8844712yes—
272C22C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
273C4—184no—
274C4#8844712yes—
275C4C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
276C4C2×#619911yescharacteristic
277C3—31no—
278C3—31no—
279C3—61no—
280C3#1059105511yescharacteristic
281C2—22no—
282C2#1172105412yes—
283C2C2×#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. #88538C4semidirect
14. C2×(C3⋊(C3⋊C8))⋊C3C4semidirect
46. C2×C3⋊(C3⋊C8)Dic3semidirect
167. H(3,𝔽3)#405semidirect
247. C32#4997semidirect
248. C32#5101semidirect

Maximal subgroups

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

subgroupindex
C2×#885382
#64135⋊C32
C22×(C3⋊(C3⋊C8))⋊C32
#117203122
#117203102
#105910593
#105910573