Finite Group Databasebeta

← #13139962

Subgroups — #13139962

Order 1728.

509 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 509 conjugacy classes of subgroups — too many to draw. Only its 139 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
2C2×C4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic
3C2×#3748⋊C3C211yesmaximal, characteristic
4#11720311C214yesmaximal, complemented
5#10591120—31nomaximal
6#10591115—31nomaximal
7C22×C4×H(3,𝔽3)C2211yescharacteristic
8C22×(C3⋊Dic3)⋊C3C412yes—
9#88538—22no—
10C2×#6180C2212yes—
11C4×(C3⋊Dic3)⋊C3C411yescharacteristic
12C4×(C3⋊Dic3)⋊C3C411yescharacteristic
13C4×(C3⋊Dic3)⋊C3C2212yes—
14#3748⋊C3—22no—
15C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
16C2×C4×C3⋊Dic3D311yescharacteristic
17C2×#3748D311yescharacteristic
18C2×C3×C4×Dic3—31no—
19C2×C3×#208—31no—
20C2×C3×#208—31no—
21C2×C3×#208—31no—
22C2×C3×#208—61no—
23#63572—34no—
24Dic3⋊(C3⋊C8)—34no—
25C23×H(3,𝔽3)C2×C411yescharacteristic
26C2×C4×H(3,𝔽3)C2×C411yescharacteristic
27C2×C4×H(3,𝔽3)D412yes—
28C2×C4×H(3,𝔽3)Q812yes—
29C2×C4×H(3,𝔽3)C2311yescharacteristic
30C2×(C3⋊Dic3)⋊C3—24no—
31C2×(C3⋊Dic3)⋊C3C2×C412yes—
32C2×(C3⋊Dic3)⋊C3C2×C412yes—
33#6180—22no—
34(C3⋊(C3⋊C8))⋊C3—22no—
35#5118—91no—
36#5118—91no—
37#4924—91no—
38C22×C32×C4—21no—
39C22×C32×C4C2×D311yescharacteristic
40C22×C32×C4C2×D311yescharacteristic
41C22×C3⋊Dic3Dic312yes—
42C22×C3×Dic3—32no—
43C4×C3⋊Dic3Dic311yescharacteristic
44C4×C3⋊Dic3Dic311yescharacteristic
45C4×C3⋊Dic3C2×D312yes—
46#3748—22no—
47C2×C3⋊(C3⋊C8)C2×D312yes—
48C3×C4×Dic3—31no—
49C3×C4×Dic3—31no—
50C3×C4×Dic3—32no—
51C3×#208—64no—
52C3×#208—62no—
53C3×#208—62no—
54C3×#208—62no—
55C2×C3×C3⋊C8—32no—
56C2×C3×C3⋊C8—32no—
57C2×C3×C3⋊C8—32no—
58C2×C3×C3⋊C8—62no—
59C22×H(3,𝔽3)—22no—
60C22×H(3,𝔽3)C4⋊C412yes—
61C22×H(3,𝔽3)C22×C411yescharacteristic
62C4×H(3,𝔽3)C4⋊C412yes—
63C4×H(3,𝔽3)C2×D411yescharacteristic
64C4×H(3,𝔽3)C2×Q811yescharacteristic
65(C3⋊Dic3)⋊C3—22no—
66(C3⋊Dic3)⋊C3M1614yescomplemented
67C2×C3×M16—91no—
68C2×C3×M16—91no—
69C2×C3×C42—91no—
70C2×C4×Dic3—31no—
71C2×C4×Dic3—91no—
72C2×#208—31no—
73C2×#208—31no—
74C2×#208—31no—
75C2×#208—61no—
76C2×#208—91no—
77Dic3⋊C8—94no—
78Dic3⋊C8—94no—
79C4⋊(C3⋊C8)—94no—
80C23×C32—21no—
81C23×C32C4×D311yescharacteristic
82C23×C32C2×Dic311yescharacteristic
83C2×C32×C4—21no—
84C2×C32×C4—22no—
85C2×C32×C4—22no—
86C2×C32×C4—21no—
87C2×C32×C4Dic612yes—
88C2×C32×C4Dic612yes—
89C2×C32×C4C4×D311yescharacteristic
90C2×C32×C4D1212yes—
91C2×C32×C4C2×Dic311yescharacteristic
92C2×C32×C4#6712yes—
93C2×C32×C4C22×D311yescharacteristic
94C2×C32×C4C22×D311yescharacteristic
95C2×C3⋊Dic3—24no—
96C2×C3⋊Dic3C2×Dic312yes—
97C2×C3⋊Dic3C2×Dic312yes—
98C2×C3×Dic3—32no—
99C2×C3×Dic3—32no—
100C2×C3×Dic3—64no—
101C3⋊(C3⋊C8)—22no—
102C3×C3⋊C8—64no—
103C3×C3⋊C8—62no—
104C3×C3⋊C8—62no—
105C3×C3⋊C8—62no—
106#423—271noSylow
107C2×H(3,𝔽3)—22no—
108C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
109C2×H(3,𝔽3)C2×M1612yes—
110C22×C3×C4—31no—
111C22×C3×C4—31no—
112C22×C3×C4—61no—
113C22×C3×C4—92no—
114C22×C3×C4D3211yescharacteristic
115C22×Dic3—32no—
116C22×Dic3—92no—
117C3×M16—182no—
118C3×M16—182no—
119C2×C3×C8—92no—
120C2×C3×C8—92no—
121C3×C42—92no—
122C3×C42—91no—
123C3×C42—91no—
124C4×Dic3—31no—
125C4×Dic3—31no—
126C4×Dic3—32no—
127C4×Dic3—92no—
128C4×Dic3—91no—
129C4×Dic3—91no—
130#208—62no—
131#208—62no—
132#208—64no—
133#208—62no—
134#208—182no—
135C2×C3⋊C8—32no—
136C2×C3⋊C8—32no—
137C2×C3⋊C8—32no—
138C2×C3⋊C8—62no—
139C2×C3⋊C8—92no—
140C22×C32—22no—
141C22×C32—22no—
142C22×C32—22no—
143C22×C32—21no—
144C22×C32—42no—
145C22×C32Dic3⋊C412yes—
146C22×C32C4⋊Dic312yes—
147C22×C32C2×C4×D311yescharacteristic
148C22×C32C22×Dic311yescharacteristic
149C32×C4—22no—
150C32×C4—21no—
151C32×C4—21no—
152C32×C4Dic3⋊C412yes—
153C32×C4C4⋊Dic312yes—
154C32×C4C2×Dic611yescharacteristic
155C32×C4C2×Dic611yescharacteristic
156C32×C4C2×D1211yescharacteristic
157C32×C4C2×#6711yescharacteristic
158C3⋊Dic3—22no—
159C3⋊Dic3#20814yescomplemented
160C3×Dic3—34no—
161C3×Dic3—62no—
162C2×M16—91no—
163C2×M16—91no—
164C2×C42—91no—
165C4⋊C8—274no—
166H(3,𝔽3)#42311yescharacteristic, complemented, Sylow
167C23×C3—31no—
168C23×C3—31no—
169C23×C3—61no—
170C23×C3D3×Dic311yescharacteristic
171C2×C3×C4—32no—
172C2×C3×C4—32no—
173C2×C3×C4—31no—
174C2×C3×C4—31no—
175C2×C3×C4—32no—
176C2×C3×C4—32no—
177C2×C3×C4—31no—
178C2×C3×C4—31no—
179C2×C3×C4—62no—
180C2×C3×C4—61no—
181C2×C3×C4—62no—
182C2×C3×C4—61no—
183C2×C3×C4—92no—
184C2×C3×C4—92no—
185C2×C3×C4—184no—
186C2×C3×C4D3×Dic311yescharacteristic
187C2×C3×C4#62612yes—
188C2×C3×C4#62712yes—
189C2×C3×C4C2×D3211yescharacteristic
190C2×Dic3—32no—
191C2×Dic3—32no—
192C2×Dic3—64no—
193C2×Dic3—92no—
194C2×Dic3—92no—
195C2×Dic3—184no—
196C3×C8—182no—
197C3×C8—182no—
198C3⋊C8—62no—
199C3⋊C8—62no—
200C3⋊C8—64no—
201C3⋊C8—62no—
202C3⋊C8—182no—
203C2×C32—22no—
204C2×C32—22no—
205C2×C32—22no—
206C2×C32—21no—
207C2×C32—42no—
208C2×C32D3×M1612yes—
209C2×C32C2×#20812yes—
210C2×C32C2×Dic3⋊C411yescharacteristic
211C2×C32C2×C4⋊Dic311yescharacteristic
212C22×C4—92no—
213C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
214M16—182no—
215M16—182no—
216C2×C8—92no—
217C2×C8—92no—
218C42—91no—
219C42—91no—
220C42—92no—
221C22×C3—22no—
222C22×C3—31no—
223C22×C3—31no—
224C22×C3—32no—
225C22×C3—32no—
226C22×C3—62no—
227C22×C3—62no—
228C22×C3—61no—
229C22×C3—62no—
230C22×C3—122no—
231C22×C3Dic3⋊Dic312yes—
232C22×C3C2×D3×Dic311yescharacteristic
233C3×C4—31no—
234C3×C4—31no—
235C3×C4—31no—
236C3×C4—31no—
237C3×C4—32no—
238C3×C4—32no—
239C3×C4—61no—
240C3×C4—61no—
241C3×C4—62no—
242C3×C4—94no—
243C3×C4—182no—
244C3×C4Dic3⋊Dic312yes—
245C3×C4C2×#62611yescharacteristic
246C3×C4C2×#62711yescharacteristic
247Dic3—34no—
248Dic3—62no—
249Dic3—94no—
250Dic3—182no—
251C32—21no—
252C32#492411yescharacteristic, complemented
253C32#511811yescharacteristic, complemented
254C23((C3⋊D3)⋊C3)⋊C411yescharacteristic
255C2×C4—92no—
256C2×C4—92no—
257C2×C4—184no—
258C2×C4#619612yes—
259C2×C4((C3⋊D3)⋊C3)⋊C411yescharacteristic
260C2×C4#620012yes—
261C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
262C8—182no—
263C8—182no—
264C2×C3—22no—
265C2×C3—31no—
266C2×C3—31no—
267C2×C3—32no—
268C2×C3—32no—
269C2×C3—62no—
270C2×C3—62no—
271C2×C3—61no—
272C2×C3—62no—
273C2×C3—122no—
274C2×C3D3×#20812yes—
275C2×C3C2×Dic3⋊Dic311yescharacteristic
276C22—22no—
277C22#8844812yes—
278C22C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
279C4—94no—
280C4—182no—
281C4#8844812yes—
282C4C2×#619611yescharacteristic
283C4C2×#620011yescharacteristic
284C3—31no—
285C3—31no—
286C3—61no—
287C3#1059111511yescharacteristic
288C2—22no—
289C2#1172105312yes—
290C2C2×#8844811yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
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)#423
LCSG ▷ C22×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. #11720311C2semidirect
66. (C3⋊Dic3)⋊C3M16semidirect
159. C3⋊Dic3#208semidirect
166. H(3,𝔽3)#423semidirect
252. C32#4924semidirect
253. C32#5118semidirect

Maximal subgroups

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

subgroupindex
C2×#885382
C2×C4×(C3⋊Dic3)⋊C32
C2×#3748⋊C32
#117203112
#105911203
#105911153