Finite Group Databasebeta

← D3×#63663 #13161274

Subgroups — D3×#63663 #13161274

Order 1728.

1028 conjugacy classes of subgroups; only the 248 normal subgroups are shown.

normal subgroups only

This group has 1028 conjugacy classes of subgroups — too many to draw. Only its 248 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×C32×D3×Q8C211yesmaximal, characteristic
2D3×#3748C212yesmaximal
3#11722206C211yesmaximal, characteristic, complemented
4C3×#63663C211yesmaximal, characteristic, complemented
5#11722028C212yesmaximal, complemented
6C2×#63663—31nomaximal
7D3×#836—34nomaximal
8C2×C33×Q8C2211yescharacteristic
9C32×D3×Q8—22no—
10C32×D3×Q8—22no—
11C2×C32×C4×D3C412yes—
12C2×C32×C4×D3C2211yescharacteristic
13C2×C32×Dic6C412yes—
14C2×C32×Dic6C2211yescharacteristic
15#88855C2212yes—
16C3×#3748C2212yes—
17#88786—22no—
18D3×C3⋊(C3⋊C8)—22no—
19C22×C32×Q8—31no—
20C2×C3×D3×Q8D314yes—
21C2×#3748—32no—
22D3×#208—38no—
23#63663—31no—
24#63663—32no—
25#63663—34no—
26#63663—64no—
27#63663D311yescharacteristic, direct product factor
28C3×#836—34no—
29#63560—38no—
30C22×C32×D3C2×C411yescharacteristic
31C33×Q8—22no—
32C2×C33×C4C2×C412yes—
33C2×C33×C4C2311yescharacteristic
34C2×C32×Dic3C2×C412yes—
35C2×C32×Dic3C2×C411yescharacteristic
36C32×C4×D3—22no—
37C32×C4×D3—22no—
38C32×C4×D3D412yes—
39C32×C4×D3D412yes—
40C32×Dic6—22no—
41C32×Dic6—24no—
42C3⋊(C3⋊(C3⋊C8))—22no—
43C3×C3⋊(C3⋊C8)—22no—
44C2×#836—94no—
45D3×#101—91no—
46C2×C32×Q8—24no—
47C2×C32×Q8—32no—
48C2×C32×Q8—31no—
49C2×C32×Q8—62no—
50C2×C32×Q8—62no—
51C2×C32×Q8C2×D311yescharacteristic
52C2×C32×Q8C2×D314yes—
53C22×C32×C4—31no—
54C22×C32×C4—32no—
55C3×D3×Q8—28no—
56C3×D3×Q8—28no—
57C2×C3×C4×D3Dic318yes—
58C2×C3×C4×D3C2×D314yes—
59C2×C3×Dic6Dic318yes—
60C2×C3×Dic6C2×D314yes—
61#3748—32no—
62#3748—38no—
63#3748—62no—
64#3748—68no—
65#3748C2×D312yes—
66C2×C3⋊(C3⋊C8)—62no—
67C3×#208—38no—
68#3711—68no—
69D3×C3⋊C8—68no—
70C22×C33C22×C411yescharacteristic
71C2×C32×D3—22no—
72C2×C32×D3C22⋊C411yescharacteristic
73C2×C32×D3C22⋊C411yescharacteristic
74C33×C4—22no—
75C33×C4C2×D412yes—
76C32×Dic3—22no—
77C32×Dic3C22⋊C412yes—
78C22×C3×Q8—34no—
79C2×D3×Q8C3⋊D311yescharacteristic
80C2×#208—98no—
81D3×M16—92no—
82C3×#101—91no—
83#836—34no—
84#836—98no—
85#836—94no—
86#836—91no—
87#836—184no—
88#824—92no—
89C23×C32—31no—
90C22×C3×D3C2×Dic314yes—
91C32×Q8—22no—
92C32×Q8—28no—
93C32×Q8—48no—
94C32×Q8—64no—
95C32×Q8—62no—
96C2×C32×C4—28no—
97C2×C32×C4—24no—
98C2×C32×C4—31no—
99C2×C32×C4—32no—
100C2×C32×C4—32no—
101C2×C32×C4—32no—
102C2×C32×C4—62no—
103C2×C32×C4—62no—
104C2×C32×C4C4×D312yes—
105C2×C32×C4C2×Dic318yes—
106C2×C32×C4C22×D314yes—
107C2×C32×C4C22×D311yescharacteristic
108C2×C3×Dic3C2×Dic318yes—
109C2×C3×Dic3C2×Dic314yes—
110C3×C4×D3—28no—
111C3×C4×D3—28no—
112C3×C4×D3#6718yes—
113C3×C4×D3#6718yes—
114C3×Dic6—28no—
115C3×Dic6—216no—
116C3⋊(C3⋊C8)—22no—
117C3⋊(C3⋊C8)—62no—
118C3⋊(C3⋊C8)—68no—
119C3⋊(C3⋊C8)—128no—
120C3×C3⋊C8—68no—
121C2×#101—271noSylow
122C2×C33—21no—
123C2×C33C2×C22⋊C411yescharacteristic
124C32×D3—21no—
125C32×D3#10112yescomplemented
126C2×C3×Q8—24no—
127C2×C3×Q8—38no—
128C2×C3×Q8—34no—
129C2×C3×Q8—68no—
130C2×C3×Q8—68no—
131C2×C3×Q8D3214yes—
132C2×C3×Q8C2×C3⋊D311yescharacteristic
133C22×C3×C4—34no—
134C22×C3×C4—38no—
135D3×Q8—22no—
136D3×Q8—22no—
137C2×C4×D3C3⋊Dic312yes—
138C2×C4×D3C2×C3⋊D311yescharacteristic
139C2×Dic6C3⋊Dic312yes—
140C2×Dic6C2×C3⋊D311yescharacteristic
141C3×M16—92no—
142#208—38no—
143#208—98no—
144#208—92no—
145#208—188no—
146#208—188no—
147C2×C3⋊C8—188no—
148ΓL(1,𝔽25)—182no—
149D3×C8—182no—
150C22×C32—24no—
151C22×C32—31no—
152C22×C32—31no—
153C22×C32—62no—
154C22×C32C2×C4×D311yescharacteristic
155C22×C32C22×Dic314yes—
156C2×C3×D3—28no—
157C2×C3×D3#21714yes—
158C2×C3×D3#21714yes—
159C32×C4—22no—
160C32×C4—28no—
161C32×C4—28no—
162C32×C4—32no—
163C32×C4—48no—
164C32×C4—62no—
165C32×C4D3×D412yes—
166C32×C4C2×#6718yes—
167C3×Dic3—28no—
168C3×Dic3#21718yes—
169C22×Q8—31no—
170C2×M16—272no—
171#101—91no—
172#101—272no—
173#101—271no—
174C33C2×#10111yescharacteristic, complemented, Sylow
175C23×C3—34no—
176C22×D3C2×C3⋊Dic311yescharacteristic
177C3×Q8—28no—
178C3×Q8—22no—
179C3×Q8—48no—
180C3×Q8—616no—
181C3×Q8—68no—
182C2×C3×C4—28no—
183C2×C3×C4—24no—
184C2×C3×C4—34no—
185C2×C3×C4—38no—
186C2×C3×C4—38no—
187C2×C3×C4—38no—
188C2×C3×C4—68no—
189C2×C3×C4—68no—
190C2×C3×C4D3×Dic318yes—
191C2×C3×C4C2×C3⋊Dic312yes—
192C2×C3×C4C2×D3214yes—
193C2×C3×C4C22×C3⋊D311yescharacteristic
194C2×Dic3C2×C3⋊Dic312yes—
195C2×Dic3C2×C3⋊Dic311yescharacteristic
196C4×D3—22no—
197C4×D3—22no—
198C4×D3#63812yes—
199C4×D3#63812yes—
200Dic6—22no—
201Dic6—24no—
202C3×C8—182no—
203C3⋊C8—68no—
204C3⋊C8—188no—
205C3⋊C8—182no—
206C3⋊C8—368no—
207C2×C32—21no—
208C2×C32—24no—
209C2×C32—24no—
210C2×C32—32no—
211C2×C32—44no—
212C2×C32—61no—
213C2×C32D3×C22⋊C411yescharacteristic
214C2×C32C2×#21714yes—
215C3×D3—24no—
216C3×D3#83618yescomplemented
217C2×Q8—32no—
218C2×Q8—31no—
219C2×Q8—62no—
220C2×Q8—62no—
221C2×Q8D3×C3⋊D311yescharacteristic
222C22×C4—31no—
223C22×C4—32no—
224M16—92no—
225M16—272no—
226M16—542no—
227C2×C8—542no—
228C22×C3—24no—
229C22×C3—34no—
230C22×C3—34no—
231C22×C3—68no—
232C22×C3C2×D3×Dic314yes—
233C22×C3C22×C3⋊Dic311yescharacteristic
234C2×D3—22no—
235C2×D3#375711yescharacteristic
236C2×D3#375711yescharacteristic
237C3×C4—22no—
238C3×C4—28no—
239C3×C4—28no—
240C3×C4—38no—
241C3×C4—48no—
242C3×C4—68no—
243C3×C4D3×#6718yes—
244C3×C4C2×#63812yes—
245Dic3—22no—
246Dic3#375712yes—
247C32—24no—
248C32D3×#10111yescharacteristic, complemented
249C32C2×#83614yescomplemented
250C23—31no—
251Q8—22no—
252Q8—64no—
253Q8—62no—
254C2×C4—31no—
255C2×C4—32no—
256C2×C4—32no—
257C2×C4—32no—
258C2×C4—62no—
259C2×C4—62no—
260C2×C4D3×C3⋊Dic312yes—
261C2×C4C2×D3×C3⋊D311yescharacteristic
262C8—182no—
263C8—542no—
264C2×C3—24no—
265C2×C3—21no—
266C2×C3—24no—
267C2×C3—38no—
268C2×C3—44no—
269C2×C3—64no—
270C2×C3D3×#21714yes—
271C2×C3C2×#375711yescharacteristic
272D3—21no—
273D3#6366312yesdirect product factor
274C22—31no—
275C22—31no—
276C22—62no—
277C22C2×D3×C3⋊Dic311yescharacteristic
278C4—22no—
279C4—32no—
280C4—62no—
281C4D3×#63812yes—
282C3—24no—
283C3D3×#83614yescomplemented
284C3C2×#6366311yescharacteristic, complemented
285C2—21no—
286C2—32no—
287C2—61no—
288C2D3×#375711yescharacteristic
subgroupG/H
centerC2D3×#3757
commutatorC22×C33C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC2×C33C2×C22⋊C4
prime radicalC22×C32×D3C2×C4
cosocleC2×C33×C4C23
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33C2×#101
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
3. #11722206C2semidirect
4. C3×#63663C2semidirect
5. #11722028C2semidirect
27. #63663D3direct (N × H)
125. C32×D3#101semidirect
174. C33C2×#101semidirect
216. C3×D3#836semidirect
248. C32D3×#101semidirect
249. C32C2×#836semidirect
273. D3#63663direct (N × H)
283. C3D3×#836semidirect
284. C3C2×#63663semidirect

Maximal subgroups

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

subgroupindex
C2×C32×D3×Q82
D3×#37482
#117222062
C3×#636632
#117220282
C2×#636633
D3×#8363