Finite Group Databasebeta

← #13139006

Subgroups — #13139006

Order 1728.

438 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 87 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×#88530C211yesmaximal, characteristic
2#11721083C212yesmaximal
3#63644⋊C3C211yesmaximal, characteristic
4#63642⋊C3C211yesmaximal, characteristic
5#11720326C212yesmaximal
6#10590819—31nomaximal
7#10590812—31nomaximal
8C2×#6229C2212yes—
9C2×C8×H(3,𝔽3)C2211yescharacteristic
10#88530C414yescomplemented
11C4⋊((C3⋊Dic3)⋊C3)C2212yes—
12C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
13C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
14#88449—22no—
15#88447—22no—
16C2×C3×Dic12—31no—
17C2×C3×Dic12—31no—
18C2×C3×Dic12—61no—
19#63843—32no—
20Dic3×Dic6—32no—
21#63644D311yescharacteristic
22#63642D311yescharacteristic
23C3×#817—31no—
24C3×#815—31no—
25#63566—32no—
26#63565—32no—
27C2×C4×H(3,𝔽3)C2311yescharacteristic
28C2×#1084—22no—
29#6229—22no—
30#6229C2×C414yes—
31C2×(C3⋊Dic3)⋊C3—22no—
32C2×(C3⋊Dic3)⋊C3D412yes—
33C8×H(3,𝔽3)C2×C412yes—
34(C3⋊(C3⋊C8))⋊C3—21no—
35#5184—91no—
36#4853—91no—
37#4716—91no—
38C2×C3×Dic6—32no—
39C2×C3×Dic6—32no—
40C2×C3×Dic6—62no—
41C2×C32×C8—21no—
42C2×C32×C8C2×D311yescharacteristic
43C2×C32×C8C2×D311yescharacteristic
44C4⋊(C3⋊Dic3)C2×D312yes—
45C4×C3⋊Dic3C2×D311yescharacteristic
46C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
47C3×C4⋊Dic3—32no—
48C3×C4×Dic3—31no—
49C2×C3×C3⋊C8—31no—
50C3×Dic12—34no—
51C3×Dic12—34no—
52C3×Dic12—64no—
53Dic3⋊Dic3—62no—
54Dic3⋊Dic3—62no—
55Dic32—62no—
56Dic32—62no—
57C22×H(3,𝔽3)C2×D411yescharacteristic
58C4×H(3,𝔽3)C22×C411yescharacteristic
59C4×H(3,𝔽3)#4111yescharacteristic
60#1084—24no—
61(C3⋊Dic3)⋊C3—22no—
62(C3⋊Dic3)⋊C3—42no—
63C2×C3×Q16—91no—
64Q8×Dic3—92no—
65C2×Dic12—31no—
66C2×Dic12—31no—
67C2×Dic12—61no—
68#887—92no—
69C4×Dic6—92no—
70C3×#106—91no—
71C3×#97—91no—
72#835—92no—
73#817—31no—
74#817—91no—
75#816—92no—
76#815—31no—
77#815—91no—
78#810—92no—
79C2×C32×C4—21no—
80C2×C32×C4C22×D311yescharacteristic
81C2×C32×C4C22×D311yescharacteristic
82C2×C3⋊Dic3—22no—
83C2×C3⋊Dic3D1212yes—
84C2×C3×Dic3—32no—
85C2×C3×Dic3—62no—
86C2×C3×Dic3—62no—
87C2×C3×Dic3—62no—
88C2×C3×Dic3—122no—
89C3×Dic6—34no—
90C3×Dic6—34no—
91C3×Dic6—64no—
92C3×Dic6—62no—
93C3×Dic6—62no—
94C3×Dic6—122no—
95C32×C8—22no—
96C32×C8C4×D312yes—
97C32×C8C4×D312yes—
98C3⋊(C3⋊C8)—21no—
99C3×C3⋊C8—61no—
100#441—271noSylow
101C2×H(3,𝔽3)C4×D411yescharacteristic
102C2×H(3,𝔽3)#13712yes—
103C2×C3×Q8—92no—
104C2×Dic6—32no—
105C2×Dic6—32no—
106C2×Dic6—62no—
107C3×Q16—94no—
108C2×C3×C8—31no—
109C2×C3×C8—31no—
110C2×C3×C8—61no—
111C2×C3×C8—91no—
112C2×C3×C8D3211yescharacteristic
113C3×C4⋊C4—92no—
114C3×C42—91no—
115C4⋊Dic3—32no—
116C4⋊Dic3—92no—
117C4⋊Dic3—182no—
118Dic3⋊C4—182no—
119Dic3⋊C4—182no—
120C4×Dic3—31no—
121C4×Dic3—91no—
122C4×Dic3—182no—
123C4×Dic3—182no—
124C4×Dic3—182no—
125C2×C3⋊C8—31no—
126C2×C3⋊C8—91no—
127Dic12—34no—
128Dic12—34no—
129Dic12—64no—
130C22×C32—21no—
131C22×C32C2×D1211yescharacteristic
132C22×C32D3×D411yescharacteristic
133C32×C4—21no—
134C32×C4—21no—
135C32×C4C2×C4×D311yescharacteristic
136C32×C4C2×C4×D311yescharacteristic
137C32×C4#23511yescharacteristic
138C32×C4#23911yescharacteristic
139C3⋊Dic3—22no—
140C3⋊Dic3—42no—
141C3×Dic3—64no—
142C3×Dic3—64no—
143C3×Dic3—62no—
144C3×Dic3—124no—
145C3×Dic3—122no—
146C2×Q16—91no—
147C4×Q8—272no—
148#106—91no—
149#103—272no—
150#97—91no—
151H(3,𝔽3)#44111yescharacteristic, complemented, Sylow
152C3×Q8—94no—
153C3×Q8—182no—
154C2×C3×C4—31no—
155C2×C3×C4—31no—
156C2×C3×C4—61no—
157C2×C3×C4—92no—
158C2×C3×C4—182no—
159C2×C3×C4—182no—
160C2×C3×C4C2×D3211yescharacteristic
161C2×Dic3—32no—
162C2×Dic3—62no—
163C2×Dic3—62no—
164C2×Dic3—62no—
165C2×Dic3—92no—
166C2×Dic3—122no—
167C2×Dic3—182no—
168Dic6—34no—
169Dic6—34no—
170Dic6—64no—
171Dic6—62no—
172Dic6—62no—
173Dic6—122no—
174C3×C8—32no—
175C3×C8—32no—
176C3×C8—62no—
177C3×C8—181no—
178C3×C8#62412yes—
179C3⋊C8—61no—
180C3⋊C8—181no—
181C2×C32—22no—
182C2×C32—21no—
183C2×C32C4×D1211yescharacteristic
184C2×C32#89311yescharacteristic
185C2×C32#90912yes—
186C2×C32#91512yes—
187C2×Q8—92no—
188Q16—94no—
189C2×C8—91no—
190C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
191C4⋊C4—92no—
192C4⋊C4—542no—
193C42—91no—
194C42—542no—
195C22×C3—31no—
196C22×C3—31no—
197C22×C3—61no—
198C22×C3D3×D1211yescharacteristic
199C3×C4—31no—
200C3×C4—31no—
201C3×C4—31no—
202C3×C4—31no—
203C3×C4—61no—
204C3×C4—61no—
205C3×C4—182no—
206C3×C4—184no—
207C3×C4—362no—
208C3×C4#379911yescharacteristic
209C3×C4C2×#62411yescharacteristic
210Dic3—64no—
211Dic3—64no—
212Dic3—62no—
213Dic3—124no—
214Dic3—122no—
215Dic3—182no—
216Dic3—362no—
217C32—21no—
218C32#471611yescharacteristic, complemented
219C32#485311yescharacteristic, complemented
220Q8—94no—
221Q8—182no—
222C2×C4—92no—
223C2×C4—182no—
224C2×C4—182no—
225C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
226C8—181no—
227C8#619912yes—
228C2×C3—32no—
229C2×C3—32no—
230C2×C3—31no—
231C2×C3—31no—
232C2×C3—62no—
233C2×C3—61no—
234C2×C3#6380012yes—
235C2×C3#6389411yescharacteristic
236C22#8865411yescharacteristic
237C4—182no—
238C4—184no—
239C4—362no—
240C4#8865211yescharacteristic
241C4C2×#619911yescharacteristic
242C3—31no—
243C3—31no—
244C3—61no—
245C3#1059081011yescharacteristic
246C2#1172085212yes—
247C2#1172114911yescharacteristic
subgroupG/H
centerC22#88654
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3D3×D12
prime radicalC22×H(3,𝔽3)C2×D4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#441
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
10. #88530C4semidirect
151. H(3,𝔽3)#441semidirect
218. C32#4716semidirect
219. C32#4853semidirect

Maximal subgroups

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

subgroupindex
C2×#885302
#117210832
#63644⋊C32
#63642⋊C32
#117203262
#105908193
#105908123