Finite Group Databasebeta

← #13165729

Subgroups — #13165729

Order 1728.

432 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 60 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×#88943C211yesmaximal, characteristic
2C2×#88938C212yesmaximal
3C2×C3×#3776C211yesmaximal, characteristic
4#11723035C212yesmaximal
5Dic3×#622C211yesmaximal, characteristic
6#10593450—31nomaximal
7C2×#6296C2212yes—
8C2×C3×#627C2212yes—
9C2×C3⋊#622C2211yescharacteristic
10C2×C3×#622C2211yescharacteristic
11#88943—22no—
12#88938—24no—
13C3×#3776—22no—
14#88806—22no—
15Dic3×C3⋊Dic3C2211yescharacteristic
16C3×Dic3⋊Dic3—22no—
17C2×#3776—32no—
18C2×#3776—31no—
19C2×#3776D311yescharacteristic
20#63919—32no—
21#63839—62no—
22C4×#622—31no—
23C2×C3⋊(C3⋊Dic3)D411yescharacteristic
24C2×C3×C3⋊Dic3—22no—
25C2×C3×C3⋊Dic3C2311yescharacteristic
26C2×C32×Dic3—22no—
27C2×C32×Dic3D411yescharacteristic
28#6296—22no—
29#6296—22no—
30C3×#627—22no—
31C3×#627—22no—
32C3⋊#622D412yes—
33C3×#622D412yes—
34#5183—91nomaximal
35C2×C3×Dic6—62no—
36C2×#627—32no—
37C2×#627—62no—
38C2×#627C2×D312yes—
39C2×#622—31no—
40C2×#622C2×D311yescharacteristic
41#3776—22no—
42#3776—64no—
43#3776—62no—
44#3750—62no—
45C4×C3⋊Dic3—31no—
46C3×C4⋊Dic3—62no—
47C3×Dic3⋊C4—62no—
48#3724—62no—
49#3724—62no—
50#3724—122no—
51Dic3⋊Dic3—22no—
52Dic32—62no—
53C22×C33C2×D411yescharacteristic
54C3⋊(C3⋊Dic3)Q1612yescomplemented
55C3×C3⋊Dic3—42no—
56C3×C3⋊Dic3C2×D411yescharacteristic
57C3×C3⋊Dic3C2×D411yescharacteristic
58C32×Dic3—42no—
59C32×Dic3Q1612yescomplemented
60C2×C3×Q16—91no—
61#944—92no—
62C2×#216—92no—
63C2×Dic12—91no—
64#888—182no—
65C8×Dic3—91no—
66C2×C32×C4—31no—
67C2×C32×C4—62no—
68C2×C3⋊Dic3—22no—
69C2×C3⋊Dic3—31no—
70C2×C3⋊Dic3—62no—
71C2×C3⋊Dic3—122no—
72C2×C3⋊Dic3C22×D311yescharacteristic
73C2×C3×Dic3—22no—
74C2×C3×Dic3—22no—
75C2×C3×Dic3—22no—
76C2×C3×Dic3—42no—
77C2×C3×Dic3—62no—
78C2×C3×Dic3—62no—
79C2×C3×Dic3—62no—
80C2×C3×Dic3—122no—
81C3×Dic6—122no—
82C3×Dic6—122no—
83#627—22no—
84#627—22no—
85#627—62no—
86#627—62no—
87#627—122no—
88#627—122no—
89#622—32no—
90#622#6712yes—
91#494—271noSylow
92C2×C33#12711yescharacteristic
93C2×C33C2×Q1611yescharacteristic
94C2×C33C2×Q1611yescharacteristic
95C2×C3×Q8—92no—
96C2×Dic6—62no—
97C2×Dic6—92no—
98C2×Dic6—182no—
99C3×Q16—182no—
100C2×C3×C8—91no—
101C3×C4⋊C4—182no—
102#216—184no—
103C4⋊Dic3—62no—
104Dic3⋊C4—62no—
105Dic3⋊C4—182no—
106Dic3⋊C4—182no—
107Dic3⋊C4—182no—
108Dic3⋊C4—362no—
109C4×Dic3—91no—
110C4×Dic3—182no—
111C2×C3⋊C8—91no—
112Dic12—182no—
113C22×C32—22no—
114C22×C32—42no—
115C22×C32D3×D411yescharacteristic
116C32×C4—32no—
117C32×C4—122no—
118C3⋊Dic3—32no—
119C3⋊Dic3—42no—
120C3⋊Dic3—64no—
121C3⋊Dic3—124no—
122C3⋊Dic3D3×D411yescharacteristic
123C3⋊Dic3C2×#6711yescharacteristic
124C3×Dic3—24no—
125C3×Dic3—42no—
126C3×Dic3—42no—
127C3×Dic3—62no—
128C3×Dic3—62no—
129C3×Dic3—82no—
130C3×Dic3—122no—
131C3×Dic3—122no—
132C3×Dic3—242no—
133C2×Q16—91no—
134C2×Q16—271no—
135C2×Q16—272no—
136C4⋊Q8—272no—
137C4×C8—271no—
138C33#49411yescharacteristic, complemented, Sylow
139C3×Q8—182no—
140C3×Q8—182no—
141C2×C3×C4—62no—
142C2×C3×C4—62no—
143C2×C3×C4—62no—
144C2×C3×C4—91no—
145C2×C3×C4—122no—
146C2×C3×C4—182no—
147C2×Dic3—22no—
148C2×Dic3—62no—
149C2×Dic3—62no—
150C2×Dic3—62no—
151C2×Dic3—91no—
152C2×Dic3—122no—
153C2×Dic3—182no—
154C2×Dic3—362no—
155C2×Dic3GO+(4,𝔽2)11yescharacteristic
156Dic6—122no—
157Dic6—122no—
158Dic6—182no—
159Dic6—182no—
160Dic6—362no—
161Dic6—362no—
162C3×C8—92no—
163C3⋊C8—92no—
164C2×C32—22no—
165C2×C32—22no—
166C2×C32—22no—
167C2×C32—42no—
168C2×C32—42no—
169C2×C32—42no—
170C2×C32D3×Q1611yescharacteristic
171C2×C32D3×Q1611yescharacteristic
172C2×C32#94011yescharacteristic
173C2×Q8—92no—
174C2×Q8—272no—
175Q16—182no—
176Q16—542no—
177Q16—544no—
178C2×C8—91no—
179C2×C8—271no—
180C4⋊C4—182no—
181C4⋊C4—542no—
182C42—271no—
183C22×C3—22no—
184C22×C3—42no—
185C22×C3C2×GO+(4,𝔽2)11yescharacteristic
186C3×C4—64no—
187C3×C4—91no—
188C3×C4—91no—
189C3×C4—122no—
190C3×C4—122no—
191C3×C4—242no—
192C3×C4—362no—
193Dic3—42no—
194Dic3—62no—
195Dic3—62no—
196Dic3—92no—
197Dic3—122no—
198Dic3—122no—
199Dic3—184no—
200Dic3—242no—
201Dic3—364no—
202Dic3#377612yescomplemented
203C32—22no—
204C32—42no—
205C32#518311yescharacteristic, complemented
206Q8—182no—
207Q8—182no—
208Q8—542no—
209Q8—542no—
210C2×C4—31no—
211C2×C4—62no—
212C2×C4—91no—
213C2×C4—182no—
214C2×C4—271no—
215C8—92no—
216C8—272no—
217C2×C3—22no—
218C2×C3—22no—
219C2×C3—22no—
220C2×C3—42no—
221C2×C3—42no—
222C2×C3—42no—
223C2×C3#6422811yescharacteristic
224C2×C3#6423111yescharacteristic
225C2×C3C2×#377611yescharacteristic
226C22D3×GO+(4,𝔽2)11yescharacteristic
227C4—32no—
228C4—91no—
229C4—91no—
230C4—122no—
231C4—272no—
232C4—362no—
233C3—22no—
234C3—42no—
235C3#1059345011yescharacteristic, complemented
236C2#1172412411yescharacteristic
237C2D3×#377611yescharacteristic
238C2#1172413111yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutator, cosocleC2×C3×C3⋊Dic3C23
Fitting, socle, prime radicalC22×C33C2×D4
hypocenterC33#494
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
54. C3⋊(C3⋊Dic3)Q16semidirect
59. C32×Dic3Q16semidirect
138. C33#494semidirect
202. Dic3#3776semidirect
205. C32#5183semidirect
235. C3#10593450semidirect

Maximal subgroups

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

subgroupindex
C2×#889432
C2×#889382
C2×C3×#37762
#117230352
Dic3×#6222
#105934503
#51839