Finite Group Databasebeta

← #13167524

Subgroups — #13167524

Order 1728.

428 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 59 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×#3787C211yesmaximal, characteristic
2C2×#88943C211yesmaximal, characteristic, complemented
3#11723671C212yesmaximal
4#11722797C211yesmaximal, characteristic, complemented
5#11722794C212yesmaximal, complemented
6#10593446—31nomaximal
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C2×#6296C2212yes—
9C2×C3×#622C2211yescharacteristic
10C2×C3×#622C2211yescharacteristic
11C3×#3787—22no—
12#88943—22no—
13C3×#3750—22no—
14#88815C2212yes—
15#88814—22no—
16C2×#3787D311yescharacteristic
17C2×#3776—31no—
18#63865—32no—
19#63851—62no—
20#63743—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#5116—91nomaximal
34C2×C4×C3⋊D3C2×D311yescharacteristic
35C2×C3×C4×D3—62no—
36C2×#627—32no—
37C2×#627—62no—
38C2×#622C2×D311yescharacteristic
39C2×#622C2×D311yescharacteristic
40#3787—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—21no—
54C2×C3×C3⋊D3Q1612yescomplemented
55C33×C4—21no—
56C3×C3⋊Dic3—42no—
57C3×C3⋊Dic3—42no—
58C3×C3⋊Dic3C2×D411yescharacteristic
59C3×C3⋊Dic3#4111yescharacteristic
60C22×C3×C8—91no—
61#924—92no—
62C2×Dic12—91no—
63#896—182no—
64#818—91no—
65#816—92no—
66C22×C3⋊D3D1211yescharacteristic
67C22×C3×D3—62no—
68C2×C32×C4—22no—
69C2×C32×C4—42no—
70C2×C32×C4D1211yescharacteristic
71C2×C3⋊Dic3—22no—
72C2×C3⋊Dic3—22no—
73C2×C3⋊Dic3C22×D311yescharacteristic
74C4×C3⋊D3—21no—
75C4×C3⋊D3—21no—
76C2×C3×Dic3—62no—
77C2×C3×Dic3—62no—
78C2×C3×Dic3—62no—
79C2×C3×Dic3—62no—
80C2×C3×Dic3—62no—
81C2×C3×Dic3—122no—
82C2×C3×Dic3—122no—
83C3×C4×D3—122no—
84C3×C4×D3—122no—
85#627—62no—
86#627—62no—
87#627—122no—
88#627—122no—
89#622—21no—
90#622#6712yes—
91#467—271noSylow
92C2×C33#12111yescharacteristic
93C2×C33C2×Q1611yescharacteristic
94C2×C33#13511yescharacteristic
95C3×C3⋊D3—22no—
96C22×C3×C4—91no—
97C2×C4×D3—62no—
98C2×Dic6—92no—
99C2×Dic6—182no—
100C2×C3×C8—91no—
101C2×C3×C8—91no—
102C2×C3×C8—182no—
103C3×C4⋊C4—182no—
104#217—182no—
105(C2×D3)⋊C4—182no—
106(C2×D3)⋊C4—182no—
107C4⋊Dic3—92no—
108Dic3⋊C4—62no—
109Dic3⋊C4—62no—
110Dic3⋊C4—122no—
111Dic3⋊C4—182no—
112Dic12—182no—
113C22×C32—22no—
114C22×C32—42no—
115C22×C32C2×D1211yescharacteristic
116C2×C3⋊D3—21no—
117C2×C3⋊D3—21no—
118C2×C3⋊D3Dic1212yescomplemented
119C2×C3×D3—64no—
120C2×C3×D3—122no—
121C2×C3×D3—122no—
122C32×C4—21no—
123C32×C4—42no—
124C32×C4—82no—
125C3⋊Dic3—42no—
126C3⋊Dic3—42no—
127C3⋊Dic3#23511yescharacteristic
128C3⋊Dic3C2×#6711yescharacteristic
129C3×Dic3—62no—
130C3×Dic3—62no—
131C3×Dic3—122no—
132C3×Dic3—122no—
133C3×Dic3—124no—
134C3×Dic3—122no—
135C3×Dic3—122no—
136C3×Dic3—242no—
137C2×Q16—271no—
138C22×C8—91no—
139C22⋊Q8—272no—
140#107—271no—
141#103—272no—
142C33#46711yescharacteristic, complemented, Sylow
143C23×C3—91no—
144C22×D3—62no—
145C2×C3×C4—22no—
146C2×C3×C4—42no—
147C2×C3×C4—91no—
148C2×C3×C4—181no—
149C2×C3×C4—181no—
150C2×C3×C4—182no—
151C2×C3×C4—182no—
152C2×C3×C4GO+(4,𝔽2)11yescharacteristic
153C2×Dic3—62no—
154C2×Dic3—62no—
155C2×Dic3—62no—
156C2×Dic3—62no—
157C2×Dic3—62no—
158C2×Dic3—122no—
159C2×Dic3—122no—
160C4×D3—122no—
161C4×D3—122no—
162Dic6—182no—
163Dic6—182no—
164Dic6—362no—
165Dic6—362no—
166C3×C8—92no—
167C3×C8—181no—
168C2×C32—22no—
169C2×C32—22no—
170C2×C32—22no—
171C2×C32—42no—
172C2×C32—42no—
173C2×C32—42no—
174C2×C32#90411yescharacteristic
175C2×C32C2×Dic1211yescharacteristic
176C2×C32#93011yescharacteristic
177C3⋊D3—22no—
178C3×D3—124no—
179C2×Q8—272no—
180C22×C4—91no—
181Q16—542no—
182C2×C8—91no—
183C2×C8—91no—
184C2×C8—182no—
185C4⋊C4—182no—
186C4⋊C4—272no—
187C22⋊C4—542no—
188C22×C3—22no—
189C22×C3—42no—
190C22×C3—92no—
191C22×C3—181no—
192C22×C3—181no—
193C22×C3C2×GO+(4,𝔽2)11yescharacteristic
194C2×D3—64no—
195C2×D3—122no—
196C2×D3—122no—
197C3×C4—21no—
198C3×C4—42no—
199C3×C4—82no—
200C3×C4—91no—
201C3×C4—91no—
202C3×C4—362no—
203C3×C4—362no—
204Dic3—62no—
205Dic3—62no—
206Dic3—122no—
207Dic3—122no—
208Dic3—124no—
209Dic3—122no—
210Dic3—122no—
211Dic3—242no—
212C32—22no—
213C32—42no—
214C32#511611yescharacteristic, complemented
215C23—91no—
216Q8—542no—
217Q8—542no—
218C2×C4—91no—
219C2×C4—181no—
220C2×C4—181no—
221C2×C4—182no—
222C2×C4—182no—
223C2×C4#157⋊D311yescharacteristic
224C8—92no—
225C8—181no—
226C2×C3—22no—
227C2×C3—22no—
228C2×C3—22no—
229C2×C3—42no—
230C2×C3—42no—
231C2×C3—42no—
232C2×C3—182no—
233C2×C3#6422311yescharacteristic
234C2×C3#6422811yescharacteristic
235C2×C3#6424211yescharacteristic
236D3—124no—
237C22—92no—
238C22—181no—
239C22—181no—
240C22C2×#157⋊D311yescharacteristic
241C4—21no—
242C4—91no—
243C4—91no—
244C4—362no—
245C4—362no—
246C3—22no—
247C3—42no—
248C3#1059344611yescharacteristic, complemented
249C2—182no—
250C2#1172433711yescharacteristic
251C2#1172434111yescharacteristic
252C2#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#467
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. #11722797C2semidirect
5. #11722794C2semidirect
54. C2×C3×C3⋊D3Q16semidirect
118. C2×C3⋊D3Dic12semidirect
142. C33#467semidirect
214. C32#5116semidirect
248. C3#10593446semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37872
C2×#889432
#117236712
#117227972
#117227942
#105934463
#51169