Finite Group Databasebeta

← #13138748

Subgroups — #13138748

Order 1728.

418 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 93 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×#3746⋊C3C211yesmaximal, characteristic
2#11721084C212yesmaximal
3#11720548C211yesmaximal, characteristic
4#63644⋊C3C211yesmaximal, characteristic
5#11720313C212yesmaximal
6#10590812—31nomaximal
7#10590810—31nomaximal
8C2×#634⋊C3C2212yes—
9C2×C8×H(3,𝔽3)C2211yescharacteristic
10C4×#1084C2211yescharacteristic
11C2×#6180C2211yescharacteristic
12C4⋊((C3⋊Dic3)⋊C3)C2212yes—
13#3746⋊C3C414yescomplemented
14#88448—22no—
15#88447—22no—
16C2×#3746D311yescharacteristic
17C2×C3×Dic12—31no—
18#63844—32no—
19Dic3×Dic6—32no—
20#63644D311yescharacteristic
21C3×#817—31no—
22C3×#815—31no—
23C3×#815—31no—
24C3×#815—61no—
25#63567—32no—
26#63566—32no—
27C2×C4×H(3,𝔽3)C2311yescharacteristic
28C2×#1084D412yes—
29C2×(C3⋊Dic3)⋊C3—22no—
30C2×(C3⋊Dic3)⋊C3—22no—
31#634⋊C3—22no—
32#634⋊C3C2×C414yes—
33C8×H(3,𝔽3)C2×C412yes—
34#6180—21no—
35#5184—91no—
36#4853—91no—
37#4716—91no—
38C2×#634C2×D312yes—
39C2×C3×Dic6—32no—
40C2×C32×C8—21no—
41C2×C32×C8C2×D311yescharacteristic
42C2×C32×C8C2×D311yescharacteristic
43C4⋊(C3⋊Dic3)C2×D312yes—
44#3746Dic314yescomplemented
45C3×C4⋊Dic3—32no—
46C3×C4×Dic3—31no—
47C3×C4×Dic3—31no—
48C3×C4×Dic3—61no—
49C2×C3×C3⋊C8—31no—
50C2×C3×C3⋊C8—31no—
51C2×C3×C3⋊C8—61no—
52C3×Dic12—34no—
53#3724—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—22no—
61(C3⋊Dic3)⋊C3—24no—
62(C3⋊Dic3)⋊C3—42no—
63C2×C3×Q16—91no—
64Q8×Dic3—92no—
65C2×Dic12—31no—
66C2×Dic12—91no—
67#887—92no—
68C4×Dic6—92no—
69C3×#106—91no—
70C3×#97—91no—
71#835—92no—
72#817—31no—
73#817—91no—
74#816—92no—
75#815—31no—
76#815—31no—
77#815—61no—
78#810—92no—
79C2×C32×C4—21no—
80C2×C32×C4C22×D311yescharacteristic
81C2×C32×C4C22×D311yescharacteristic
82C2×C3⋊Dic3—22no—
83C2×C3⋊Dic3—22no—
84#634—22no—
85#634C2×Dic314yes—
86C2×C3×Dic3—32no—
87C2×C3×Dic3—32no—
88C2×C3×Dic3—62no—
89C2×C3×Dic3—62no—
90C2×C3×Dic3—62no—
91C3×Dic6—34no—
92C3×Dic6—62no—
93C32×C8—22no—
94C32×C8C4×D312yes—
95C32×C8C2×Dic312yes—
96C3×C3⋊C8—62no—
97C3×C3⋊C8—61no—
98C3×C3⋊C8—61no—
99#441—271noSylow
100C2×H(3,𝔽3)C4×D411yescharacteristic
101C2×H(3,𝔽3)#13712yes—
102C2×C3×Q8—92no—
103C2×Dic6—32no—
104C2×Dic6—92no—
105C3×Q16—94no—
106C2×C3×C8—31no—
107C2×C3×C8—31no—
108C2×C3×C8—61no—
109C2×C3×C8—91no—
110C2×C3×C8D3211yescharacteristic
111C3×C4⋊C4—92no—
112C3×C42—91no—
113C4⋊Dic3—32no—
114C4⋊Dic3—92no—
115C4⋊Dic3—182no—
116Dic3⋊C4—182no—
117Dic3⋊C4—182no—
118C4×Dic3—31no—
119C4×Dic3—31no—
120C4×Dic3—61no—
121C4×Dic3—182no—
122C4×Dic3—182no—
123C4×Dic3—182no—
124C2×C3⋊C8—31no—
125C2×C3⋊C8—31no—
126C2×C3⋊C8—61no—
127Dic12—34no—
128Dic12—94no—
129C22×C32—21no—
130C22×C32D3×D411yescharacteristic
131C22×C32D3×D411yescharacteristic
132C32×C4—21no—
133C32×C4—21no—
134C32×C4C2×C4×D311yescharacteristic
135C32×C4#23711yescharacteristic
136C32×C4#23911yescharacteristic
137C32×C4C22×Dic311yescharacteristic
138C3⋊Dic3—24no—
139C3⋊Dic3—42no—
140C3×Dic3—64no—
141C3×Dic3—62no—
142C3×Dic3—62no—
143C3×Dic3—122no—
144C3×Dic3—122no—
145C2×Q16—91no—
146C4×Q8—272no—
147#106—91no—
148#103—272no—
149#97—91no—
150H(3,𝔽3)#44111yescharacteristic, complemented, Sylow
151C3×Q8—94no—
152C3×Q8—182no—
153C2×C3×C4—31no—
154C2×C3×C4—31no—
155C2×C3×C4—61no—
156C2×C3×C4—92no—
157C2×C3×C4—182no—
158C2×C3×C4—182no—
159C2×C3×C4C2×D3211yescharacteristic
160C2×Dic3—32no—
161C2×Dic3—32no—
162C2×Dic3—62no—
163C2×Dic3—62no—
164C2×Dic3—62no—
165C2×Dic3—182no—
166C2×Dic3—182no—
167Dic6—34no—
168Dic6—62no—
169Dic6—94no—
170Dic6—182no—
171C3×C8—32no—
172C3×C8—32no—
173C3×C8—62no—
174C3×C8—181no—
175C3×C8D3×Dic312yes—
176C3⋊C8—61no—
177C3⋊C8—61no—
178C3⋊C8—62no—
179C2×C32—22no—
180C2×C32—21no—
181C2×C32#89311yescharacteristic
182C2×C32#91512yes—
183C2×C32#91812yes—
184C2×C32D4×Dic311yescharacteristic
185C2×Q8—92no—
186Q16—94no—
187C2×C8—91no—
188C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
189C4⋊C4—92no—
190C4⋊C4—542no—
191C42—91no—
192C42—542no—
193C22×C3—31no—
194C22×C3—31no—
195C22×C3—61no—
196C22×C3#380211yescharacteristic
197C3×C4—31no—
198C3×C4—31no—
199C3×C4—31no—
200C3×C4—31no—
201C3×C4—61no—
202C3×C4—61no—
203C3×C4—184no—
204C3×C4—182no—
205C3×C4—362no—
206C3×C4#379611yescharacteristic
207C3×C4C2×D3×Dic311yescharacteristic
208Dic3—62no—
209Dic3—62no—
210Dic3—64no—
211Dic3—122no—
212Dic3—122no—
213Dic3—184no—
214Dic3—362no—
215C32—21no—
216C32#485311yescharacteristic, complemented
217C32#518411yescharacteristic, complemented
218Q8—94no—
219Q8—182no—
220C2×C4—92no—
221C2×C4—182no—
222C2×C4—182no—
223C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
224C8—181no—
225C8((C3⋊D3)⋊C3)⋊C412yes—
226C2×C3—32no—
227C2×C3—32no—
228C2×C3—31no—
229C2×C3—31no—
230C2×C3—62no—
231C2×C3—61no—
232C2×C3#6381112yes—
233C2×C3#6389811yescharacteristic
234C22#8865511yescharacteristic
235C4—182no—
236C4—184no—
237C4—362no—
238C4#8864911yescharacteristic
239C4C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
240C3—31no—
241C3—31no—
242C3—61no—
243C3#1059081911yescharacteristic
244C2#1172084512yes—
245C2#1172113611yescharacteristic
subgroupG/H
centerC22#88655
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3#3802
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
13. #3746⋊C3C4semidirect
44. #3746Dic3semidirect
150. H(3,𝔽3)#441semidirect
216. C32#4853semidirect
217. C32#5184semidirect

Maximal subgroups

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

subgroupindex
C2×#3746⋊C32
#117210842
#117205482
#63644⋊C32
#117203132
#105908123
#105908103