Finite Group Databasebeta

← #13167602

Subgroups — #13167602

Order 1728.

444 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 63 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
1C22×C3×#622C211yesmaximal, characteristic
2C2×#88943C211yesmaximal, characteristic, complemented
3#11723721C212yesmaximal
4#11722797C211yesmaximal, characteristic, complemented
5#11722794C212yesmaximal, complemented
6#10593527—31nomaximal
7C22×C3×C3⋊Dic3C2211yescharacteristic
8C2×#6296C2212yes—
9C2×C3×#622—22no—
10C2×C3×#622C2211yescharacteristic
11C2×C3×#622C2211yescharacteristic
12#88943—22no—
13C3×#3757—22no—
14#88815—22no—
15#88815C2212yes—
16C22×#622D311yescharacteristic
17C2×#3776—31no—
18#63982—32no—
19#63964—62no—
20#63743—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#5116—91nomaximal
34C22×C3⋊Dic3C2×D311yescharacteristic
35C22×C3×Dic3—62no—
36C2×#627—32no—
37C2×#627—62no—
38C2×#622—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—21no—
53C22×C33Q1612yescomplemented
54C22×C33C2×D411yescharacteristic
55C3×C3⋊Dic3—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#878—182no—
64#818—91no—
65#816—92no—
66C23×C32—22no—
67C23×C32—42no—
68C23×C32D1211yescharacteristic
69C2×C3⋊Dic3—22no—
70C2×C3⋊Dic3—22no—
71C2×C3⋊Dic3—21no—
72C2×C3⋊Dic3—21no—
73C2×C3⋊Dic3D1211yescharacteristic
74C2×C3⋊Dic3C22×D311yescharacteristic
75C2×C3×Dic3—62no—
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—
83C2×C3×Dic3—122no—
84C2×C3×Dic3—122no—
85#627—62no—
86#627—62no—
87#627—122no—
88#627—122no—
89#622—21no—
90#622#6712yes—
91#467—271noSylow
92C2×C33—22no—
93C2×C33#12111yescharacteristic
94C2×C33C2×Q1611yescharacteristic
95C2×C33#13511yescharacteristic
96C22×C3×C4—91no—
97C22×Dic3—62no—
98C2×Dic6—92no—
99C2×Dic6—182no—
100C2×C3×C8—91no—
101C2×C3×C8—91no—
102C2×C3×C8—182no—
103C3×C22⋊C4—182no—
104#217—62no—
105#217—62no—
106#217—122no—
107C4⋊Dic3—92no—
108C4⋊Dic3—182no—
109Dic3⋊C4—182no—
110Dic3⋊C4—182no—
111Dic3⋊C4—182no—
112Dic12—182no—
113C22×C32—22no—
114C22×C32—24no—
115C22×C32—21no—
116C22×C32—21no—
117C22×C32—42no—
118C22×C32—44no—
119C22×C32—42no—
120C22×C32—42no—
121C22×C32—82no—
122C22×C32—82no—
123C22×C32Dic1212yescomplemented
124C22×C32C2×D1211yescharacteristic
125C3⋊Dic3—21no—
126C3⋊Dic3—42no—
127C3⋊Dic3—42no—
128C3⋊Dic3#23511yescharacteristic
129C3⋊Dic3C2×#6711yescharacteristic
130C3×Dic3—62no—
131C3×Dic3—62no—
132C3×Dic3—122no—
133C3×Dic3—122no—
134C3×Dic3—122no—
135C3×Dic3—122no—
136C3×Dic3—122no—
137C3×Dic3—242no—
138C3×Dic3—242no—
139C2×Q16—271no—
140C22×C8—91no—
141C22⋊Q8—272no—
142#107—271no—
143#103—272no—
144C33#46711yescharacteristic, complemented, Sylow
145C23×C3—22no—
146C23×C3—42no—
147C23×C3GO+(4,𝔽2)11yescharacteristic
148C2×C3×C4—91no—
149C2×C3×C4—91no—
150C2×C3×C4—181no—
151C2×C3×C4—181no—
152C2×C3×C4—182no—
153C2×C3×C4—182no—
154C2×Dic3—62no—
155C2×Dic3—62no—
156C2×Dic3—62no—
157C2×Dic3—62no—
158C2×Dic3—62no—
159C2×Dic3—62no—
160C2×Dic3—122no—
161C2×Dic3—122no—
162C2×Dic3—122no—
163C2×Dic3—122no—
164Dic6—182no—
165Dic6—182no—
166Dic6—362no—
167Dic6—362no—
168C3×C8—92no—
169C3×C8—181no—
170C2×C32—22no—
171C2×C32—22no—
172C2×C32—22no—
173C2×C32—22no—
174C2×C32—42no—
175C2×C32—42no—
176C2×C32—42no—
177C2×C32—44no—
178C2×C32—84no—
179C2×C32#90411yescharacteristic
180C2×C32C2×Dic1211yescharacteristic
181C2×C32#93011yescharacteristic
182C2×Q8—272no—
183C22×C4—91no—
184Q16—542no—
185C2×C8—91no—
186C2×C8—91no—
187C2×C8—182no—
188C4⋊C4—272no—
189C4⋊C4—542no—
190C22⋊C4—182no—
191C22×C3—21no—
192C22×C3—21no—
193C22×C3—22no—
194C22×C3—24no—
195C22×C3—42no—
196C22×C3—42no—
197C22×C3—42no—
198C22×C3—44no—
199C22×C3—82no—
200C22×C3—82no—
201C22×C3#377612yescomplemented
202C22×C3C2×GO+(4,𝔽2)11yescharacteristic
203C3×C4—91no—
204C3×C4—91no—
205C3×C4—181no—
206C3×C4—362no—
207C3×C4—362no—
208Dic3—62no—
209Dic3—62no—
210Dic3—122no—
211Dic3—122no—
212Dic3—122no—
213Dic3—122no—
214Dic3—122no—
215Dic3—242no—
216Dic3—242no—
217C32—22no—
218C32—42no—
219C32#511611yescharacteristic, complemented
220C23#157⋊D311yescharacteristic
221Q8—542no—
222Q8—542no—
223C2×C4—91no—
224C2×C4—91no—
225C2×C4—181no—
226C2×C4—181no—
227C2×C4—182no—
228C2×C4—182no—
229C8—92no—
230C8—181no—
231C2×C3—22no—
232C2×C3—22no—
233C2×C3—22no—
234C2×C3—22no—
235C2×C3—44no—
236C2×C3—42no—
237C2×C3—42no—
238C2×C3—42no—
239C2×C3—84no—
240C2×C3#6423711yescharacteristic
241C2×C3C2×#377611yescharacteristic
242C2×C3#6424211yescharacteristic
243C22—21no—
244C22—21no—
245C22#8894312yescomplemented
246C22C2×#157⋊D311yescharacteristic
247C4—91no—
248C4—91no—
249C4—181no—
250C4—362no—
251C4—362no—
252C3—22no—
253C3—42no—
254C3#1059352711yescharacteristic, complemented
255C2—22no—
256C2#1172435111yescharacteristic
257C2C2×#8894311yescharacteristic
258C2#1172435511yescharacteristic
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutator, cosocleC2×C3×C3⋊Dic3C23
Fitting, prime radicalC23×C33D4
socleC22×C33C2×D4
hypercenterC23#157⋊D3
hypocenterC33#467
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. #11722797C2semidirect
5. #11722794C2semidirect
53. C22×C33Q16semidirect
123. C22×C32Dic12semidirect
144. C33#467semidirect
201. C22×C3#3776semidirect
219. C32#5116semidirect
245. C22#88943semidirect
254. C3#10593527semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#6222
C2×#889432
#117237212
#117227972
#117227942
#105935273
#51169