Finite Group Databasebeta

← #13167068

Subgroups — #13167068

Order 1728.

440 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 79 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×#88988C211yesmaximal, characteristic
2#11723673C211yesmaximal, characteristic
3C3×#63874C211yesmaximal, characteristic
4#11722779C212yesmaximal, complemented
5#11722778C212yesmaximal, complemented
6#10593389—31nomaximal
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C2×C3⋊#622C2211yescharacteristic
9C2×C3⋊#622C2211yescharacteristic
10#88988C414yescomplemented
11C3×C4×C3⋊Dic3—21no—
12C32×C4×Dic3—21no—
13#88815C2212yes—
14#88814—21no—
15C3×#3724C2212yes—
16C3×#3722—21no—
17C2×#3787—31no—
18C3×#872—61no—
19#63874—31no—
20#63874D311yescharacteristic
21#63855—61no—
22#63743—32no—
23#63742—32no—
24C22×C3×C3⋊D3D411yescharacteristic
25C2×C33×C4D411yescharacteristic
26C2×C3×C3⋊Dic3—22no—
27C2×C3×C3⋊Dic3C2311yescharacteristic
28C3×C4×C3⋊D3C2×C412yes—
29C3×C4×C3⋊D3Q812yes—
30C2×C32×Dic3—22no—
31C3⋊#622C2×C412yes—
32C3⋊#622C2×C412yes—
33#4998—91nomaximal
34C2×C4×C3⋊D3C2×D311yescharacteristic
35C2×C3×C4×D3—61no—
36C2×C3×C4×D3—61no—
37C2×#622—31no—
38C2×#622—31no—
39#3787—34no—
40C32×C42—61no—
41C4×C3⋊Dic3—21no—
42C3×(C2×D3)⋊C4—62no—
43C3×Dic3⋊C4—62no—
44C3×C4×Dic3—21no—
45C3×C4×Dic3—21no—
46C3×C4×Dic3—41no—
47C3×C4×Dic3—61no—
48C3×C4×Dic3—61no—
49C3×C4×Dic3—121no—
50#3724—32no—
51#3724C2×D312yes—
52Dic3⋊Dic3—62no—
53#3722—21no—
54#3722—61no—
55(C2×D3)⋊Dic3—62no—
56C22×C33C2×D411yescharacteristic
57C2×C3×C3⋊D3—22no—
58C2×C3×C3⋊D3C4⋊C412yes—
59C33×C4C4⋊C412yes—
60C3×C3⋊Dic3—42no—
61C3×C3⋊Dic3C22×C411yescharacteristic
62C3×C3⋊Dic3C2×Q811yescharacteristic
63C32×Dic3—42no—
64C3×#117—91no—
65#926—91no—
66C22×C3⋊C8—91no—
67#872—61no—
68#872—181no—
69#808—92no—
70#807—92no—
71C22×C3⋊D3#6711yescharacteristic
72C22×C3×D3—61no—
73C22×C3×D3—61no—
74C2×C32×C4—21no—
75C2×C32×C4—21no—
76C2×C32×C4—41no—
77C2×C32×C4—41no—
78C2×C32×C4—62no—
79C2×C32×C4#6711yescharacteristic
80C2×C3⋊Dic3—22no—
81C2×C3⋊Dic3C22×D311yescharacteristic
82C4×C3⋊D3Dic612yes—
83C4×C3⋊D3C4×D312yes—
84C2×C3×Dic3—22no—
85C2×C3×Dic3—22no—
86C2×C3×Dic3—42no—
87C2×C3×Dic3—61no—
88C2×C3×Dic3—61no—
89C2×C3×Dic3—62no—
90C2×C3×Dic3—62no—
91C2×C3×Dic3—122no—
92C3×C4×D3—62no—
93C3×C4×D3—62no—
94C3×C4×D3—62no—
95C3×C4×D3—62no—
96#622—32no—
97#622—32no—
98#427—271noSylow
99C2×C33C2×C4⋊C411yescharacteristic
100C2×C33#13512yes—
101C3×C3⋊D3—22no—
102C22×C3×C4—91no—
103C2×C4×D3—61no—
104C2×C4×D3—61no—
105C3×C4⋊C4—92no—
106C3×C22⋊C4—181no—
107C3×C42—61no—
108C3×C42—61no—
109C3×C42—121no—
110C3×C42—181no—
111#217—181no—
112(C2×D3)⋊C4—62no—
113(C2×D3)⋊C4—182no—
114C4⋊Dic3—92no—
115Dic3⋊C4—62no—
116Dic3⋊C4—182no—
117C4×Dic3—21no—
118C4×Dic3—61no—
119C4×Dic3—61no—
120C4×Dic3—121no—
121C2×C3⋊C8—91no—
122C2×C3⋊C8—91no—
123C2×C3⋊C8—94no—
124C22×C32—21no—
125C22×C32—21no—
126C22×C32—41no—
127C22×C32—41no—
128C22×C32C2×#6711yescharacteristic
129C2×C3⋊D3—22no—
130C2×C3⋊D3Dic3⋊C412yes—
131C2×C3×D3—62no—
132C2×C3×D3—62no—
133C2×C3×D3—122no—
134C2×C3×D3—122no—
135C32×C4—22no—
136C32×C4—22no—
137C32×C4—42no—
138C32×C4—42no—
139C32×C4—122no—
140C32×C4Dic3⋊C412yes—
141C3⋊Dic3—42no—
142C3⋊Dic3C2×Dic611yescharacteristic
143C3⋊Dic3C2×C4×D311yescharacteristic
144C3×Dic3—42no—
145C3×Dic3—44no—
146C3×Dic3—42no—
147C3×Dic3—61no—
148C3×Dic3—61no—
149C3×Dic3—61no—
150C3×Dic3—61no—
151C3×Dic3—122no—
152C3×Dic3—124no—
153C3×Dic3—122no—
154C22×C8—271no—
155#117—91no—
156#117—271no—
157#107—272no—
158#106—272no—
159C33#42711yescharacteristic, complemented, Sylow
160C23×C3—91no—
161C22×D3—61no—
162C22×D3—61no—
163C2×C3×C4—21no—
164C2×C3×C4—21no—
165C2×C3×C4—41no—
166C2×C3×C4—41no—
167C2×C3×C4—62no—
168C2×C3×C4—62no—
169C2×C3×C4—91no—
170C2×C3×C4—92no—
171C2×C3×C4—92no—
172C2×C3×C4—122no—
173C2×C3×C4—182no—
174C2×C3×C4GO+(4,𝔽2)11yescharacteristic
175C2×Dic3—22no—
176C2×Dic3—61no—
177C2×Dic3—61no—
178C2×Dic3—62no—
179C2×Dic3—62no—
180C2×Dic3—122no—
181C4×D3—62no—
182C4×D3—62no—
183C4×D3—62no—
184C4×D3—62no—
185C3⋊C8—92no—
186C3⋊C8—92no—
187C2×C32—22no—
188C2×C32—22no—
189C2×C32—21no—
190C2×C32—21no—
191C2×C32—41no—
192C2×C32—41no—
193C2×C32—42no—
194C2×C32—42no—
195C2×C32C2×Dic3⋊C411yescharacteristic
196C2×C32#95012yes—
197C3⋊D3—22no—
198C3×D3—122no—
199C3×D3—122no—
200C22×C4—91no—
201C2×C8—271no—
202C2×C8—271no—
203C2×C8—274no—
204C4⋊C4—92no—
205C4⋊C4—272no—
206C22⋊C4—181no—
207C22⋊C4—541no—
208C42—61no—
209C42—181no—
210C22×C3—21no—
211C22×C3—21no—
212C22×C3—41no—
213C22×C3—41no—
214C22×C3—92no—
215C22×C3—182no—
216C22×C3C2×GO+(4,𝔽2)11yescharacteristic
217C2×D3—62no—
218C2×D3—62no—
219C2×D3—122no—
220C2×D3—122no—
221C3×C4—22no—
222C3×C4—22no—
223C3×C4—42no—
224C3×C4—42no—
225C3×C4—91no—
226C3×C4—91no—
227C3×C4—122no—
228C3×C4—124no—
229C3×C4—122no—
230C3×C4—362no—
231C3×C4#377312yes—
232Dic3—42no—
233Dic3—61no—
234Dic3—61no—
235Dic3—61no—
236Dic3—61no—
237Dic3—122no—
238Dic3—124no—
239Dic3—122no—
240C32—21no—
241C32—21no—
242C32—41no—
243C32—41no—
244C32#499811yescharacteristic, complemented
245C23—91no—
246C2×C4—62no—
247C2×C4—91no—
248C2×C4—92no—
249C2×C4—92no—
250C2×C4—182no—
251C2×C4D32⋊D311yescharacteristic
252C8—272no—
253C8—272no—
254C2×C3—22no—
255C2×C3—22no—
256C2×C3—21no—
257C2×C3—21no—
258C2×C3—42no—
259C2×C3—42no—
260C2×C3—41no—
261C2×C3—41no—
262C2×C3—182no—
263C2×C3#6422312yes—
264C2×C3C2×#377311yescharacteristic
265D3—122no—
266D3—122no—
267C22—92no—
268C22—182no—
269C22C2×D32⋊D311yescharacteristic
270C4—91no—
271C4—91no—
272C4—122no—
273C4—362no—
274C4#8893412yes—
275C3—21no—
276C3—21no—
277C3—41no—
278C3—41no—
279C3#1059338911yescharacteristic, complemented
280C2—182no—
281C2#1172430312yes—
282C2C2×#8893411yescharacteristic
subgroupG/H
center, hypercenterC2×C4D32⋊D3
commutatorC3×C3⋊Dic3C22×C4
FrattiniC22C2×D32⋊D3
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
cosocleC2×C3×C3⋊Dic3C23
hypocenterC33#427
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. #11722779C2semidirect
5. #11722778C2semidirect
10. #88988C4semidirect
159. C33#427semidirect
244. C32#4998semidirect
279. C3#10593389semidirect

Maximal subgroups

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

subgroupindex
C2×#889882
#117236732
C3×#638742
#117227792
#117227782
#105933893
#49989