Finite Group Databasebeta

← #13167603

Subgroups — #13167603

Order 1728.

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

normal subgroups only

This group has 436 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×#3792C211yesmaximal, characteristic
2C2×#88943C211yesmaximal, characteristic, complemented
3#11723721C212yesmaximal
4#11722796C211yesmaximal, characteristic, complemented
5#11722794C212yesmaximal, complemented
6#10593508—31nomaximal
7C22×C3×C3⋊Dic3C2211yescharacteristic
8C2×#6296C2212yes—
9C3×#3792—22no—
10C2×C3×#622C2211yescharacteristic
11C2×C3×#622C2211yescharacteristic
12#88943—22no—
13C3×#3757—22no—
14#88815—22no—
15#88815C2212yes—
16C2×#3792D311yescharacteristic
17C2×#3776—31no—
18#63982—32no—
19#63964—62no—
20#63742—31no—
21#63740—32no—
22C23×C33D411yescharacteristic
23C2×C3×C3⋊Dic3—21no—
24C2×C3×C3⋊Dic3—21no—
25C2×C3×C3⋊Dic3—22no—
26C2×C3×C3⋊Dic3—22no—
27C2×C3×C3⋊Dic3D411yescharacteristic
28C2×C3×C3⋊Dic3C2311yescharacteristic
29#6296—22no—
30#6296—22no—
31C3×#622—21no—
32C3×#622D412yes—
33#5137—91nomaximal
34C22×C3⋊Dic3C2×D311yescharacteristic
35C22×C3×Dic3—62no—
36C2×#627—32no—
37C2×#627—62no—
38#3792—22no—
39C2×#622C2×D311yescharacteristic
40C2×#622C2×D311yescharacteristic
41#3776—62no—
42#3757—22no—
43C3×#217—62no—
44C3×#217—62no—
45C3×#217—122no—
46#3724—32no—
47#3724—62no—
48Dic3⋊Dic3—62no—
49Dic3⋊Dic3—62no—
50Dic3⋊Dic3—62no—
51C22×C33—21no—
52C22×C33—22no—
53C22×C33C2×D411yescharacteristic
54C3×C3⋊Dic3—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#878—182no—
63#817—91no—
64#816—92no—
65C23×C32—22no—
66C23×C32—42no—
67C23×C32D1211yescharacteristic
68C2×C3⋊Dic3—22no—
69C2×C3⋊Dic3—22no—
70C2×C3⋊Dic3—21no—
71C2×C3⋊Dic3—21no—
72C2×C3⋊Dic3D1211yescharacteristic
73C2×C3⋊Dic3C22×D311yescharacteristic
74C2×C3×Dic3—62no—
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—
82C2×C3×Dic3—122no—
83C2×C3×Dic3—122no—
84#627—62no—
85#627—62no—
86#627—122no—
87#627—122no—
88#622—21no—
89#622#6712yes—
90#470—271noSylow
91C2×C33—41no—
92C2×C33#12111yescharacteristic
93C2×C33#13712yes—
94C22×C3×C4—91no—
95C22×Dic3—62no—
96C2×Dic6—92no—
97C2×Dic6—182no—
98C3×M16—182no—
99C2×C3×C8—91no—
100C2×C3×C8—91no—
101C3×C22⋊C4—182no—
102#217—62no—
103#217—62no—
104#217—122no—
105C4⋊Dic3—92no—
106C4⋊Dic3—182no—
107Dic3⋊C4—182no—
108Dic3⋊C4—182no—
109Dic3⋊C4—182no—
110Dic12—182no—
111C22×C32—24no—
112C22×C32—22no—
113C22×C32—22no—
114C22×C32—21no—
115C22×C32—42no—
116C22×C32—42no—
117C22×C32—42no—
118C22×C32—44no—
119C22×C32—82no—
120C22×C32—82no—
121C22×C32C2×D1211yescharacteristic
122C3⋊Dic3—21no—
123C3⋊Dic3—42no—
124C3⋊Dic3—42no—
125C3⋊Dic3#23511yescharacteristic
126C3⋊Dic3C2×#6711yescharacteristic
127C3×Dic3—62no—
128C3×Dic3—62no—
129C3×Dic3—122no—
130C3×Dic3—122no—
131C3×Dic3—122no—
132C3×Dic3—122no—
133C3×Dic3—122no—
134C3×Dic3—242no—
135C3×Dic3—242no—
136C2×Q16—271no—
137C2×M16—91no—
138C22⋊Q8—272no—
139#106—271no—
140#103—272no—
141C33#47011yescharacteristic, complemented, Sylow
142C23×C3—22no—
143C23×C3—42no—
144C23×C3GO+(4,𝔽2)11yescharacteristic
145C2×C3×C4—91no—
146C2×C3×C4—91no—
147C2×C3×C4—181no—
148C2×C3×C4—181no—
149C2×C3×C4—182no—
150C2×C3×C4—182no—
151C2×Dic3—62no—
152C2×Dic3—62no—
153C2×Dic3—62no—
154C2×Dic3—62no—
155C2×Dic3—62no—
156C2×Dic3—62no—
157C2×Dic3—122no—
158C2×Dic3—122no—
159C2×Dic3—122no—
160C2×Dic3—122no—
161Dic6—182no—
162Dic6—182no—
163Dic6—362no—
164Dic6—362no—
165C3×C8—92no—
166C3×C8—181no—
167C2×C32—22no—
168C2×C32—22no—
169C2×C32—22no—
170C2×C32—42no—
171C2×C32—42no—
172C2×C32—42no—
173C2×C32—41no—
174C2×C32—44no—
175C2×C32—84no—
176C2×C32#90912yes—
177C2×C32#93011yescharacteristic
178C2×Q8—272no—
179C22×C4—91no—
180Q16—542no—
181M16—182no—
182C2×C8—91no—
183C2×C8—91no—
184C4⋊C4—272no—
185C4⋊C4—542no—
186C22⋊C4—182no—
187C22×C3—24no—
188C22×C3—22no—
189C22×C3—21no—
190C22×C3—22no—
191C22×C3—42no—
192C22×C3—42no—
193C22×C3—44no—
194C22×C3—42no—
195C22×C3—82no—
196C22×C3—82no—
197C22×C3C2×GO+(4,𝔽2)11yescharacteristic
198C3×C4—91no—
199C3×C4—91no—
200C3×C4—181no—
201C3×C4—362no—
202C3×C4—362no—
203Dic3—62no—
204Dic3—62no—
205Dic3—122no—
206Dic3—122no—
207Dic3—122no—
208Dic3—122no—
209Dic3—122no—
210Dic3—242no—
211Dic3—242no—
212C32—22no—
213C32—42no—
214C32#513711yescharacteristic, complemented
215C23#157⋊D311yescharacteristic
216Q8—542no—
217Q8—542no—
218C2×C4—91no—
219C2×C4—91no—
220C2×C4—181no—
221C2×C4—181no—
222C2×C4—182no—
223C2×C4—182no—
224C8—92no—
225C8—181no—
226C2×C3—22no—
227C2×C3—22no—
228C2×C3—22no—
229C2×C3—44no—
230C2×C3—41no—
231C2×C3—42no—
232C2×C3—42no—
233C2×C3—42no—
234C2×C3—84no—
235C2×C3#6423912yes—
236C2×C3#6424211yescharacteristic
237C22—22no—
238C22—21no—
239C22C2×#157⋊D311yescharacteristic
240C4—91no—
241C4—91no—
242C4—181no—
243C4—362no—
244C4—362no—
245C3—22no—
246C3—42no—
247C3#1059350811yescharacteristic, complemented
248C2—41no—
249C2#1172435212yes—
250C2#1172435511yescharacteristic
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutator, cosocleC2×C3×C3⋊Dic3C23
Fitting, prime radicalC23×C33D4
socleC22×C33C2×D4
hypercenterC23#157⋊D3
hypocenterC33#470
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×#88943C2semidirect
4. #11722796C2semidirect
5. #11722794C2semidirect
141. C33#470semidirect
214. C32#5137semidirect
247. C3#10593508semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37922
C2×#889432
#117237212
#117227962
#117227942
#105935083
#51379