Finite Group Databasebeta

← #13167067

Subgroups — #13167067

Order 1728.

408 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 67 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
1#11723678C211yesmaximal, characteristic
2C3×#63873C211yesmaximal, characteristic
3C4×C3⋊#622C211yesmaximal, characteristic
4#11722779C214yesmaximal
5#10593388—31nomaximal
6C2×C3×#634—21no—
7C2×C32×Dic6—21no—
8C2×C3⋊#622C2212yes—
9C3×C4×C3⋊Dic3C2211yescharacteristic
10#88815—21no—
11#88815C2212yes—
12C3×#3724—21no—
13C3×#3724C2212yes—
14C3×#944—61no—
15#63873—31no—
16#63873D311yescharacteristic
17#63866—61no—
18C4×#622—31no—
19#63743—34no—
20C2×C33×C4D411yescharacteristic
21C2×C3×C3⋊Dic3—22no—
22C2×C3×C3⋊Dic3D411yescharacteristic
23C2×C3×C3⋊Dic3C2311yescharacteristic
24C3×#634—42no—
25C2×C32×Dic3—22no—
26C32×Dic6—42no—
27C3⋊#622Q814yescomplemented
28#5078—91nomaximal
29C2×C32×Q8—61no—
30C2×#634—21no—
31C2×C3×Dic6—21no—
32C2×C3×Dic6—21no—
33C2×C3×Dic6—41no—
34C2×C3×Dic6—61no—
35C2×C3×Dic6—61no—
36C2×C3×Dic6—121no—
37C2×#622—32no—
38C4×C3⋊Dic3C2×D311yescharacteristic
39C3×Dic3⋊C4—62no—
40C3×Dic3⋊C4—62no—
41C3×C4×Dic3—61no—
42C3×C4×Dic3—61no—
43#3724—21no—
44#3724—32no—
45#3724—61no—
46#3724C2×D312yes—
47Dic3⋊Dic3—62no—
48Dic3⋊Dic3—62no—
49C22×C33C2×D411yescharacteristic
50C33×C4D812yes—
51C3×C3⋊Dic3—42no—
52C3×C3⋊Dic3Q1612yes—
53C3×C3⋊Dic3C2×Q812yes—
54C32×Dic3—42no—
55C3×C4⋊Q8—91no—
56#944—61no—
57#944—181no—
58#869—91no—
59#807—94no—
60C4×C3⋊C8—91no—
61C32×Q8—122no—
62C2×C32×C4—21no—
63C2×C32×C4—21no—
64C2×C32×C4—41no—
65C2×C32×C4—41no—
66C2×C32×C4—62no—
67C2×C32×C4#6711yescharacteristic
68C2×C3⋊Dic3—22no—
69C2×C3⋊Dic3#6711yescharacteristic
70C2×C3⋊Dic3C22×D311yescharacteristic
71#634—42no—
72C2×C3×Dic3—22no—
73C2×C3×Dic3—22no—
74C2×C3×Dic3—42no—
75C2×C3×Dic3—61no—
76C2×C3×Dic3—61no—
77C2×C3×Dic3—61no—
78C2×C3×Dic3—61no—
79C2×C3×Dic3—62no—
80C2×C3×Dic3—62no—
81C2×C3×Dic3—122no—
82C3×Dic6—42no—
83C3×Dic6—44no—
84C3×Dic6—42no—
85C3×Dic6—122no—
86C3×Dic6—124no—
87C3×Dic6—122no—
88#622—34no—
89#500—271noSylow
90C2×C33C4⋊Q811yescharacteristic
91C2×C33C2×D811yescharacteristic
92C2×C33C2×Q1611yescharacteristic
93C2×C3×Q8—61no—
94C2×C3×Q8—61no—
95C2×C3×Q8—121no—
96C2×C3×Q8—181no—
97C2×Dic6—21no—
98C2×Dic6—61no—
99C2×Dic6—61no—
100C2×Dic6—121no—
101C3×C4⋊C4—92no—
102C3×C4⋊C4—181no—
103C3×C42—91no—
104C4⋊Dic3—92no—
105C4⋊Dic3—181no—
106Dic3⋊C4—62no—
107Dic3⋊C4—62no—
108Dic3⋊C4—182no—
109Dic3⋊C4—182no—
110C4×Dic3—61no—
111C4×Dic3—61no—
112C2×C3⋊C8—92no—
113C22×C32—21no—
114C22×C32—21no—
115C22×C32—41no—
116C22×C32—41no—
117C22×C32C2×#6711yescharacteristic
118C32×C4—22no—
119C32×C4—22no—
120C32×C4—42no—
121C32×C4—42no—
122C32×C4—122no—
123C32×C4#21312yes—
124C3⋊Dic3—42no—
125C3⋊Dic3#21612yes—
126C3⋊Dic3D3×Q812yes—
127C3×Dic3—42no—
128C3×Dic3—44no—
129C3×Dic3—42no—
130C3×Dic3—62no—
131C3×Dic3—62no—
132C3×Dic3—62no—
133C3×Dic3—62no—
134C3×Dic3—122no—
135C3×Dic3—124no—
136C3×Dic3—122no—
137C4⋊Q8—91no—
138C4⋊Q8—271no—
139#107—274no—
140C4×C8—271no—
141C33#50011yescharacteristic, complemented, Sylow
142C3×Q8—122no—
143C3×Q8—124no—
144C3×Q8—122no—
145C3×Q8—362no—
146C2×C3×C4—21no—
147C2×C3×C4—21no—
148C2×C3×C4—41no—
149C2×C3×C4—41no—
150C2×C3×C4—62no—
151C2×C3×C4—62no—
152C2×C3×C4—91no—
153C2×C3×C4—91no—
154C2×C3×C4—122no—
155C2×C3×C4—182no—
156C2×C3×C4GO+(4,𝔽2)11yescharacteristic
157C2×Dic3—22no—
158C2×Dic3—61no—
159C2×Dic3—61no—
160C2×Dic3—61no—
161C2×Dic3—61no—
162C2×Dic3—62no—
163C2×Dic3—62no—
164C2×Dic3—122no—
165Dic6—42no—
166Dic6—122no—
167Dic6—124no—
168Dic6—122no—
169C3⋊C8—94no—
170C2×C32—21no—
171C2×C32—21no—
172C2×C32—21no—
173C2×C32—21no—
174C2×C32—21no—
175C2×C32—21no—
176C2×C32—41no—
177C2×C32—41no—
178C2×C32—41no—
179C2×C32—41no—
180C2×C32—41no—
181C2×C32—41no—
182C2×C32C2×#21311yescharacteristic
183C2×C32C2×#21611yescharacteristic
184C2×C32#94411yescharacteristic
185C2×Q8—61no—
186C2×Q8—181no—
187C2×C8—272no—
188C4⋊C4—92no—
189C4⋊C4—181no—
190C4⋊C4—272no—
191C4⋊C4—541no—
192C42—91no—
193C22×C3—21no—
194C22×C3—21no—
195C22×C3—41no—
196C22×C3—41no—
197C22×C3C2×GO+(4,𝔽2)11yescharacteristic
198C3×C4—22no—
199C3×C4—22no—
200C3×C4—42no—
201C3×C4—42no—
202C3×C4—92no—
203C3×C4—92no—
204C3×C4—122no—
205C3×C4—124no—
206C3×C4—122no—
207C3×C4—362no—
208C3×C4#377412yes—
209Dic3—42no—
210Dic3—62no—
211Dic3—62no—
212Dic3—62no—
213Dic3—62no—
214Dic3—122no—
215Dic3—124no—
216Dic3—122no—
217C32—21no—
218C32—21no—
219C32—41no—
220C32—41no—
221C32#507811yescharacteristic, complemented
222Q8—122no—
223Q8—362no—
224C2×C4—62no—
225C2×C4—91no—
226C2×C4—91no—
227C2×C4—182no—
228C2×C4D32⋊D311yescharacteristic
229C8—274no—
230C2×C3—21no—
231C2×C3—21no—
232C2×C3—21no—
233C2×C3—21no—
234C2×C3—21no—
235C2×C3—21no—
236C2×C3—41no—
237C2×C3—41no—
238C2×C3—41no—
239C2×C3—41no—
240C2×C3—41no—
241C2×C3—41no—
242C2×C3#6422211yescharacteristic
243C2×C3#6422811yescharacteristic
244C2×C3C2×#377411yescharacteristic
245C22C2×D32⋊D311yescharacteristic
246C4—92no—
247C4—92no—
248C4—122no—
249C4—362no—
250C4#8893512yes—
251C3—21no—
252C3—21no—
253C3—41no—
254C3—41no—
255C3#1059338811yescharacteristic, complemented
256C2#1172430211yescharacteristic
257C2#1172431111yescharacteristic
258C2C2×#8893511yescharacteristic
subgroupG/H
center, FrattiniC22C2×D32⋊D3
commutator, cosocleC2×C3×C3⋊Dic3C23
FittingC2×C33×C4D4
socle, prime radicalC22×C33C2×D4
hypercenterC2×C4D32⋊D3
hypocenterC33#500
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
27. C3⋊#622Q8semidirect
141. C33#500semidirect
221. C32#5078semidirect
255. C3#10593388semidirect

Maximal subgroups

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

subgroupindex
#117236782
C3×#638732
C4×C3⋊#6222
#117227792
#105933883
#50789