Finite Group Databasebeta

← #13161268

Subgroups — #13161268

Order 1728.

1052 conjugacy classes of subgroups; only the 172 normal subgroups are shown.

normal subgroups only

This group has 1052 conjugacy classes of subgroups — too many to draw. Only its 172 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×C3×#3832C211yesmaximal, characteristic
2C3⋊D3×#208C212yesmaximal, complemented
3#11722206C211yesmaximal, characteristic, complemented
4C32×#836C211yesmaximal, characteristic, complemented
5#11722026C212yesmaximal, complemented
6#10592450—31nomaximal
7#10591637—34nomaximal
8C2×C33×Q8C2211yescharacteristic
9C3×#3832—22no—
10C3×#3832—22no—
11C2×C3×C4⋊(C3⋊D3)C412yes—
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×C4×C3⋊D3C412yes—
14C2×C3×C4×C3⋊D3C2211yescharacteristic
15#88855C2212yes—
16C32×#208C2212yes—
17#88787—22no—
18C3⋊D3×C3⋊C8—22no—
19C2×#3832D311yescharacteristic
20C2×C3×#239—34no—
21M16×C3⋊D3—32no—
22D3×#208—38no—
23C32×#101—31no—
24#63663—31no—
25#63663—34no—
26#63663—64no—
27#63650—32no—
28C3×#836D314yescomplemented
29#63558—38no—
30C22×C3×C3⋊D3C2×C412yes—
31C22×C3×C3⋊D3C2×C411yescharacteristic
32C33×Q8—22no—
33C2×C33×C4C2×C412yes—
34C2×C33×C4C2311yescharacteristic
35C2×C3×C3⋊Dic3C2×C411yescharacteristic
36C3×C4⋊(C3⋊D3)—24no—
37C3×C4⋊(C3⋊D3)—22no—
38C3×C4×C3⋊D3—22no—
39C3×C4×C3⋊D3—22no—
40C3×C4×C3⋊D3D412yes—
41C3×C4×C3⋊D3D412yes—
42C3⋊(C3⋊(C3⋊C8))—22no—
43C32×C3⋊C8—22no—
44#5237—91no—
45#4751—94no—
46C2×C32×Q8—24no—
47C2×C32×Q8C2×D314yes—
48C2×C32×Q8C2×D311yescharacteristic
49#3832—22no—
50#3832—22no—
51C2×C4⋊(C3⋊D3)Dic312yes—
52C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
53C2×C4×C3⋊D3Dic312yes—
54C2×C4×C3⋊D3C2×D311yescharacteristic
55C3×#239—68no—
56C3×#239—68no—
57C2×C3×D12—38no—
58C2×C3×D12—34no—
59C2×C3×C4×D3—34no—
60C2×C3×C4×D3—38no—
61C32×M16—32no—
62#3748—32no—
63#3748—38no—
64#3748—68no—
65#3743—62no—
66C8×C3⋊D3—62no—
67C3×#208C2×D318yes—
68#3711—68no—
69D3×C3⋊C8—68no—
70C22×C33C22×C411yescharacteristic
71C2×C3×C3⋊D3—22no—
72C2×C3×C3⋊D3—41no—
73C2×C3×C3⋊D3—42no—
74C2×C3×C3⋊D3C22⋊C412yes—
75C33×C4—22no—
76C33×C4C2×D412yes—
77C3×C3⋊Dic3C22⋊C411yescharacteristic
78C3×C3⋊Dic3C22⋊C411yescharacteristic
79C2×C3×#41—91no—
80C2×#239—34no—
81C2×#208—92no—
82D3×M16—98no—
83C3×#101—34no—
84#836—94no—
85#836—91no—
86#836—184no—
87#836C3⋊D311yescharacteristic, complemented
88#833—92no—
89#823—98no—
90C22×C3⋊D3C2×Dic312yes—
91C22×C3⋊D3C2×Dic311yescharacteristic
92C22×C3×D3—34no—
93C22×C3×D3—38no—
94C32×Q8—22no—
95C32×Q8—28no—
96C32×Q8—48no—
97C2×C32×C4—28no—
98C2×C32×C4—24no—
99C2×C32×C4C4×D318yes—
100C2×C32×C4C2×Dic312yes—
101C2×C32×C4C22×D314yes—
102C2×C32×C4C22×D311yescharacteristic
103C2×C3⋊Dic3C2×Dic311yescharacteristic
104C4⋊(C3⋊D3)—24no—
105C4⋊(C3⋊D3)—22no—
106C4×C3⋊D3—22no—
107C4×C3⋊D3—22no—
108C4×C3⋊D3#6712yes—
109C4×C3⋊D3#6712yes—
110C2×C3×Dic3—34no—
111C3×D12—616no—
112C3×D12—68no—
113C3×C4×D3—38no—
114C3×C4×D3—38no—
115C3×C4×D3—68no—
116C3×C4×D3—68no—
117C32×C8—62no—
118C3⋊(C3⋊C8)—62no—
119C3⋊(C3⋊C8)—68no—
120C3⋊(C3⋊C8)—128no—
121C3×C3⋊C8—28no—
122#413—271noSylow
123C2×C33—21no—
124C2×C33C2×C22⋊C411yescharacteristic
125C3×C3⋊D3—22no—
126C3×C3⋊D3—42no—
127C3×#41—182no—
128C3×#41—182no—
129C2×C3×Q8—24no—
130C2×C3×Q8D3214yes—
131C2×C3×Q8C2×C3⋊D311yescharacteristic
132C2×C3×D4—92no—
133C2×C3×D4—91no—
134C22×C3×C4—91no—
135C22×C3×C4—92no—
136#239—68no—
137#239—68no—
138C2×D12—38no—
139C2×D12—34no—
140C2×C4×D3—34no—
141C2×C4×D3—38no—
142C3×M16—38no—
143#208—98no—
144#208—92no—
145#208—188no—
146#208—182no—
147#208C2×C3⋊D312yes—
148C2×C3⋊C8—182no—
149ΓL(1,𝔽25)—188no—
150D3×C8—188no—
151C22×C32—24no—
152C22×C32C2×C4×D314yes—
153C22×C32C22×Dic311yescharacteristic
154C2×C3⋊D3—22no—
155C2×C3⋊D3—41no—
156C2×C3⋊D3—42no—
157C2×C3⋊D3#21712yes—
158C2×C3×D3—38no—
159C2×C3×D3—68no—
160C2×C3×D3—124no—
161C2×C3×D3—128no—
162C32×C4—22no—
163C32×C4—28no—
164C32×C4—28no—
165C32×C4—48no—
166C32×C4D3×D418yes—
167C32×C4C2×#6712yes—
168C3⋊Dic3#21711yescharacteristic
169C3⋊Dic3#21711yescharacteristic
170C3×Dic3—34no—
171C3×Dic3—34no—
172C2×#41—91no—
173C2×M16—272no—
174#101—31no—
175#101—271no—
176#100—272no—
177C33#41311yescharacteristic, complemented, Sylow
178C23×C3—91no—
179C23×C3—92no—
180C22×D3—34no—
181C22×D3—38no—
182C3×Q8—28no—
183C3×Q8—22no—
184C3×Q8—48no—
185C3×D4—184no—
186C3×D4—182no—
187C2×C3×C4—28no—
188C2×C3×C4—24no—
189C2×C3×C4—91no—
190C2×C3×C4—92no—
191C2×C3×C4—92no—
192C2×C3×C4—182no—
193C2×C3×C4—182no—
194C2×C3×C4D3×Dic318yes—
195C2×C3×C4C4×C3⋊D312yes—
196C2×C3×C4C2×D3214yes—
197C2×C3×C4C22×C3⋊D311yescharacteristic
198C2×Dic3—34no—
199D12—616no—
200D12—68no—
201C4×D3—38no—
202C4×D3—38no—
203C4×D3—68no—
204C4×D3—68no—
205C3×C8—68no—
206C3⋊C8—22no—
207C3⋊C8—188no—
208C3⋊C8—182no—
209C3⋊C8—368no—
210C2×C32—21no—
211C2×C32—24no—
212C2×C32—24no—
213C2×C32—44no—
214C2×C32D3×C22⋊C414yes—
215C2×C32C2×#21711yescharacteristic
216C3⋊D3—22no—
217C3⋊D3—42no—
218C3×D3—68no—
219C3×D3—128no—
220#41—182no—
221#41—182no—
222C2×Q8D3×C3⋊D311yescharacteristic
223C2×D4—92no—
224C2×D4—91no—
225C22×C4—91no—
226C22×C4—92no—
227M16—32no—
228M16—272no—
229M16—542no—
230C2×C8—542no—
231C22×C3—24no—
232C22×C3—92no—
233C22×C3—182no—
234C22×C3—361no—
235C22×C3—362no—
236C22×C3C2×D3×Dic314yes—
237C22×C3C2×C4×C3⋊D311yescharacteristic
238C2×D3—38no—
239C2×D3—68no—
240C2×D3—124no—
241C2×D3—128no—
242C3×C4—22no—
243C3×C4—28no—
244C3×C4—28no—
245C3×C4—48no—
246C3×C4—91no—
247C3×C4—91no—
248C3×C4D3×#6718yes—
249C3×C4D4×C3⋊D312yes—
250Dic3—34no—
251Dic3—34no—
252C32—24no—
253C32#475114yescomplemented
254C32#523711yescharacteristic, complemented
255C23—91no—
256C23—92no—
257Q8—22no—
258D4—184no—
259D4—182no—
260C2×C4—91no—
261C2×C4—92no—
262C2×C4—92no—
263C2×C4—182no—
264C2×C4—182no—
265C2×C4Dic3×C3⋊D312yes—
266C2×C4C2×D3×C3⋊D311yescharacteristic
267C8—62no—
268C8—542no—
269C2×C3—24no—
270C2×C3—21no—
271C2×C3—24no—
272C2×C3—44no—
273C2×C3—182no—
274C2×C3—362no—
275C2×C3D3×#21714yes—
276C2×C3C22⋊C4×C3⋊D311yescharacteristic
277D3—68no—
278D3—128no—
279C22—92no—
280C22—182no—
281C22—361no—
282C22—362no—
283C22C2×Dic3×C3⋊D311yescharacteristic
284C4—22no—
285C4—91no—
286C4—91no—
287C4C3⋊D3×#6712yes—
288C3—24no—
289C3#1059163714yescomplemented
290C3#1059245011yescharacteristic, complemented
291C2—21no—
292C2—182no—
293C2—362no—
294C2C3⋊D3×#21711yescharacteristic
subgroupG/H
centerC2C3⋊D3×#217
commutatorC22×C33C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC2×C33C2×C22⋊C4
prime radicalC2×C3×#3832C2
cosocleC2×C33×C4C23
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#413
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. C3⋊D3×#208C2semidirect
3. #11722206C2semidirect
4. C32×#836C2semidirect
5. #11722026C2semidirect
28. C3×#836D3semidirect
87. #836C3⋊D3semidirect
177. C33#413semidirect
253. C32#4751semidirect
254. C32#5237semidirect
289. C3#10591637semidirect
290. C3#10592450semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38322
C3⋊D3×#2082
#117222062
C32×#8362
#117220262
#105924503
#105916373