Finite Group Databasebeta

← #13167525

Subgroups — #13167525

Order 1728.

420 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 55 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×#3788C211yesmaximal, characteristic
2C2×#88943C211yesmaximal, characteristic, complemented
3#11723671C212yesmaximal
4#11722796C211yesmaximal, characteristic, complemented
5#11722794C212yesmaximal, complemented
6#10593449—31nomaximal
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C2×#6296C2212yes—
9C2×C3×#622C2211yescharacteristic
10C2×C3×#622C2211yescharacteristic
11C3×#3788—22no—
12#88943—22no—
13C3×#3750—22no—
14#88815C2212yes—
15#88814—22no—
16C2×#3788D311yescharacteristic
17C2×#3776—31no—
18#63865—32no—
19#63851—62no—
20#63742—31no—
21#63740—32no—
22C22×C3×C3⋊D3D411yescharacteristic
23C2×C33×C4D411yescharacteristic
24C2×C3×C3⋊Dic3—22no—
25C2×C3×C3⋊Dic3—22no—
26C2×C3×C3⋊Dic3C2311yescharacteristic
27C3×C4×C3⋊D3—21no—
28C3×C4×C3⋊D3—21no—
29#6296—22no—
30#6296—22no—
31C3×#622—21no—
32C3×#622D412yes—
33#5137—91nomaximal
34C2×C4×C3⋊D3C2×D311yescharacteristic
35C2×C3×C4×D3—62no—
36C2×#627—32no—
37C2×#627—62no—
38C2×#622C2×D311yescharacteristic
39C2×#622C2×D311yescharacteristic
40#3788—22no—
41#3776—62no—
42#3750—22no—
43C3×Dic3⋊C4—62no—
44C3×Dic3⋊C4—62no—
45C3×Dic3⋊C4—122no—
46#3724—32no—
47Dic3⋊Dic3—62no—
48#3722—62no—
49(C2×D3)⋊Dic3—62no—
50(C2×D3)⋊Dic3—62no—
51C22×C33C2×D411yescharacteristic
52C2×C3×C3⋊D3—21no—
53C2×C3×C3⋊D3—22no—
54C33×C4—21no—
55C3×C3⋊Dic3—42no—
56C3×C3⋊Dic3—42no—
57C3×C3⋊Dic3C2×D411yescharacteristic
58C3×C3⋊Dic3#4111yescharacteristic
59C2×C3×M16—91no—
60#924—92no—
61C2×Dic12—91no—
62#896—182no—
63#817—91no—
64#816—92no—
65C22×C3⋊D3D1211yescharacteristic
66C22×C3×D3—62no—
67C2×C32×C4—22no—
68C2×C32×C4—42no—
69C2×C32×C4D1211yescharacteristic
70C2×C3⋊Dic3—22no—
71C2×C3⋊Dic3—22no—
72C2×C3⋊Dic3C22×D311yescharacteristic
73C4×C3⋊D3—21no—
74C4×C3⋊D3—21no—
75C2×C3×Dic3—62no—
76C2×C3×Dic3—62no—
77C2×C3×Dic3—62no—
78C2×C3×Dic3—62no—
79C2×C3×Dic3—62no—
80C2×C3×Dic3—122no—
81C2×C3×Dic3—122no—
82C3×C4×D3—122no—
83C3×C4×D3—122no—
84#627—62no—
85#627—62no—
86#627—122no—
87#627—122no—
88#622—21no—
89#622#6712yes—
90#470—271noSylow
91C2×C33#12111yescharacteristic
92C2×C33#13712yes—
93C3×C3⋊D3—41no—
94C22×C3×C4—91no—
95C2×C4×D3—62no—
96C2×Dic6—92no—
97C2×Dic6—182no—
98C3×M16—182no—
99C2×C3×C8—91no—
100C2×C3×C8—91no—
101C3×C4⋊C4—182no—
102#217—182no—
103(C2×D3)⋊C4—182no—
104(C2×D3)⋊C4—182no—
105C4⋊Dic3—92no—
106Dic3⋊C4—62no—
107Dic3⋊C4—62no—
108Dic3⋊C4—122no—
109Dic3⋊C4—182no—
110Dic12—182no—
111C22×C32—22no—
112C22×C32—42no—
113C22×C32C2×D1211yescharacteristic
114C2×C3⋊D3—22no—
115C2×C3⋊D3—21no—
116C2×C3×D3—64no—
117C2×C3×D3—122no—
118C2×C3×D3—122no—
119C32×C4—21no—
120C32×C4—42no—
121C32×C4—82no—
122C3⋊Dic3—42no—
123C3⋊Dic3—42no—
124C3⋊Dic3#23511yescharacteristic
125C3⋊Dic3C2×#6711yescharacteristic
126C3×Dic3—62no—
127C3×Dic3—62no—
128C3×Dic3—122no—
129C3×Dic3—122no—
130C3×Dic3—124no—
131C3×Dic3—122no—
132C3×Dic3—122no—
133C3×Dic3—242no—
134C2×Q16—271no—
135C2×M16—91no—
136C22⋊Q8—272no—
137#106—271no—
138#103—272no—
139C33#47011yescharacteristic, complemented, Sylow
140C23×C3—91no—
141C22×D3—62no—
142C2×C3×C4—22no—
143C2×C3×C4—42no—
144C2×C3×C4—91no—
145C2×C3×C4—181no—
146C2×C3×C4—181no—
147C2×C3×C4—182no—
148C2×C3×C4—182no—
149C2×C3×C4GO+(4,𝔽2)11yescharacteristic
150C2×Dic3—62no—
151C2×Dic3—62no—
152C2×Dic3—62no—
153C2×Dic3—62no—
154C2×Dic3—62no—
155C2×Dic3—122no—
156C2×Dic3—122no—
157C4×D3—122no—
158C4×D3—122no—
159Dic6—182no—
160Dic6—182no—
161Dic6—362no—
162Dic6—362no—
163C3×C8—92no—
164C3×C8—181no—
165C2×C32—22no—
166C2×C32—22no—
167C2×C32—22no—
168C2×C32—42no—
169C2×C32—42no—
170C2×C32—42no—
171C2×C32#90912yes—
172C2×C32#93011yescharacteristic
173C3⋊D3—41no—
174C3×D3—124no—
175C2×Q8—272no—
176C22×C4—91no—
177Q16—542no—
178M16—182no—
179C2×C8—91no—
180C2×C8—91no—
181C4⋊C4—182no—
182C4⋊C4—272no—
183C22⋊C4—542no—
184C22×C3—22no—
185C22×C3—42no—
186C22×C3—182no—
187C22×C3—181no—
188C22×C3C2×GO+(4,𝔽2)11yescharacteristic
189C2×D3—64no—
190C2×D3—122no—
191C2×D3—122no—
192C3×C4—21no—
193C3×C4—42no—
194C3×C4—82no—
195C3×C4—91no—
196C3×C4—91no—
197C3×C4—362no—
198C3×C4—362no—
199Dic3—62no—
200Dic3—62no—
201Dic3—122no—
202Dic3—122no—
203Dic3—124no—
204Dic3—122no—
205Dic3—122no—
206Dic3—242no—
207C32—22no—
208C32—42no—
209C32#513711yescharacteristic, complemented
210C23—91no—
211Q8—542no—
212Q8—542no—
213C2×C4—91no—
214C2×C4—181no—
215C2×C4—181no—
216C2×C4—182no—
217C2×C4—182no—
218C2×C4#157⋊D311yescharacteristic
219C8—92no—
220C8—181no—
221C2×C3—22no—
222C2×C3—22no—
223C2×C3—22no—
224C2×C3—42no—
225C2×C3—42no—
226C2×C3—42no—
227C2×C3—361no—
228C2×C3#6422612yes—
229C2×C3#6424211yescharacteristic
230D3—124no—
231C22—182no—
232C22—181no—
233C22C2×#157⋊D311yescharacteristic
234C4—21no—
235C4—91no—
236C4—91no—
237C4—362no—
238C4—362no—
239C3—22no—
240C3—42no—
241C3#1059344911yescharacteristic, complemented
242C2—361no—
243C2#1172433812yes—
244C2#1172435511yescharacteristic
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutator, cosocleC2×C3×C3⋊Dic3C23
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
hypercenterC2×C4#157⋊D3
hypocenterC33#470
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×#88943C2semidirect
4. #11722796C2semidirect
5. #11722794C2semidirect
139. C33#470semidirect
209. C32#5137semidirect
241. C3#10593449semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37882
C2×#889432
#117236712
#117227962
#117227942
#105934493
#51379