Finite Group Databasebeta

← #13138810

Subgroups — #13138810

Order 1728.

414 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 95 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#11721082C212yesmaximal
3#11720550C211yesmaximal, characteristic
4#63642⋊C3C211yesmaximal, characteristic
5#11720309C212yesmaximal
6#10590819—31nomaximal
7#10590810—31nomaximal
8C2×#634⋊C3C2212yes—
9C2×C8×H(3,𝔽3)C2211yescharacteristic
10C4⋊#1084C2212yes—
11C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
12C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
13#3746⋊C3C414yescomplemented
14#88448—22no—
15#88447—22no—
16C2×#3746D311yescharacteristic
17C2×C3×Dic12—31no—
18#63844—32no—
19#63843—32no—
20#63642D311yescharacteristic
21C3×#817—31no—
22C3×#817—31no—
23C3×#817—61no—
24C3×#815—31no—
25#63567—32no—
26#63565—32no—
27C2×C4×H(3,𝔽3)C2311yescharacteristic
28C2×#1084—22no—
29C2×(C3⋊Dic3)⋊C3—22no—
30C2×(C3⋊Dic3)⋊C3D412yes—
31#634⋊C3—22no—
32#634⋊C3C2×C414yes—
33C8×H(3,𝔽3)C2×C412yes—
34(C3⋊(C3⋊C8))⋊C3—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⋊Dic3C2×D311yescharacteristic
44C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
45#3746Dic314yescomplemented
46C3×C4⋊Dic3—32no—
47C3×C4⋊Dic3—32no—
48C3×C4⋊Dic3—62no—
49C3×C4×Dic3—31no—
50C2×C3×C3⋊C8—31no—
51C3×Dic12—34no—
52#3724—62no—
53Dic3⋊Dic3—62no—
54Dic32—62no—
55Dic32—62no—
56C22×H(3,𝔽3)C2×D411yescharacteristic
57C4×H(3,𝔽3)C22×C411yescharacteristic
58C4×H(3,𝔽3)#4111yescharacteristic
59#1084—42no—
60(C3⋊Dic3)⋊C3—24no—
61(C3⋊Dic3)⋊C3—22no—
62C2×C3×Q16—91no—
63Q8×Dic3—92no—
64C2×Dic12—31no—
65C2×Dic12—91no—
66#887—92no—
67C4×Dic6—92no—
68C3×#106—91no—
69C3×#97—91no—
70#835—92no—
71#817—31no—
72#817—31no—
73#817—61no—
74#816—92no—
75#815—31no—
76#815—91no—
77#810—92no—
78C2×C32×C4—21no—
79C2×C32×C4C22×D311yescharacteristic
80C2×C32×C4C22×D311yescharacteristic
81C2×C3⋊Dic3—22no—
82C2×C3⋊Dic3D1212yes—
83#634—22no—
84#634C2×Dic314yes—
85C2×C3×Dic3—32no—
86C2×C3×Dic3—62no—
87C2×C3×Dic3—62no—
88C2×C3×Dic3—62no—
89C2×C3×Dic3—122no—
90C3×Dic6—34no—
91C3×Dic6—62no—
92C32×C8—22no—
93C32×C8C4×D312yes—
94C32×C8C2×Dic312yes—
95C3⋊(C3⋊C8)—21no—
96C3×C3⋊C8—61no—
97#441—271noSylow
98C2×H(3,𝔽3)C4×D411yescharacteristic
99C2×H(3,𝔽3)#13712yes—
100C2×C3×Q8—92no—
101C2×Dic6—32no—
102C2×Dic6—92no—
103C3×Q16—94no—
104C2×C3×C8—31no—
105C2×C3×C8—31no—
106C2×C3×C8—61no—
107C2×C3×C8—91no—
108C2×C3×C8D3211yescharacteristic
109C3×C4⋊C4—92no—
110C3×C42—91no—
111C4⋊Dic3—32no—
112C4⋊Dic3—32no—
113C4⋊Dic3—62no—
114C4⋊Dic3—182no—
115Dic3⋊C4—182no—
116Dic3⋊C4—182no—
117C4×Dic3—31no—
118C4×Dic3—91no—
119C4×Dic3—182no—
120C4×Dic3—182no—
121C4×Dic3—182no—
122C2×C3⋊C8—31no—
123C2×C3⋊C8—91no—
124Dic12—34no—
125Dic12—94no—
126C22×C32—21no—
127C22×C32C2×D1211yescharacteristic
128C22×C32D3×D411yescharacteristic
129C32×C4—21no—
130C32×C4—21no—
131C32×C4C2×C4×D311yescharacteristic
132C32×C4#23511yescharacteristic
133C32×C4#23711yescharacteristic
134C32×C4C22×Dic311yescharacteristic
135C3⋊Dic3—22no—
136C3⋊Dic3—24no—
137C3×Dic3—64no—
138C3×Dic3—62no—
139C3×Dic3—122no—
140C3×Dic3—122no—
141C3×Dic3—242no—
142C2×Q16—91no—
143C4×Q8—272no—
144#106—91no—
145#103—272no—
146#97—91no—
147H(3,𝔽3)#44111yescharacteristic, complemented, Sylow
148C3×Q8—94no—
149C3×Q8—182no—
150C2×C3×C4—31no—
151C2×C3×C4—31no—
152C2×C3×C4—61no—
153C2×C3×C4—92no—
154C2×C3×C4—182no—
155C2×C3×C4—182no—
156C2×C3×C4C2×D3211yescharacteristic
157C2×Dic3—32no—
158C2×Dic3—62no—
159C2×Dic3—62no—
160C2×Dic3—62no—
161C2×Dic3—92no—
162C2×Dic3—122no—
163C2×Dic3—182no—
164Dic6—34no—
165Dic6—62no—
166Dic6—94no—
167Dic6—182no—
168C3×C8—32no—
169C3×C8—32no—
170C3×C8—62no—
171C3×C8—181no—
172C3×C8D3×Dic312yes—
173C3⋊C8—61no—
174C3⋊C8—181no—
175C2×C32—22no—
176C2×C32—21no—
177C2×C32C4×D1211yescharacteristic
178C2×C32#90912yes—
179C2×C32#91812yes—
180C2×C32D4×Dic311yescharacteristic
181C2×Q8—92no—
182Q16—94no—
183C2×C8—91no—
184C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
185C4⋊C4—92no—
186C4⋊C4—542no—
187C42—91no—
188C42—542no—
189C22×C3—31no—
190C22×C3—31no—
191C22×C3—61no—
192C22×C3D3×D1211yescharacteristic
193C3×C4—31no—
194C3×C4—31no—
195C3×C4—31no—
196C3×C4—31no—
197C3×C4—61no—
198C3×C4—61no—
199C3×C4—182no—
200C3×C4—184no—
201C3×C4—362no—
202C3×C4#379511yescharacteristic
203C3×C4C2×D3×Dic311yescharacteristic
204Dic3—62no—
205Dic3—64no—
206Dic3—122no—
207Dic3—122no—
208Dic3—184no—
209Dic3—182no—
210Dic3—242no—
211C32—21no—
212C32#471611yescharacteristic, complemented
213C32#518411yescharacteristic, complemented
214Q8—94no—
215Q8—182no—
216C2×C4—92no—
217C2×C4—182no—
218C2×C4—182no—
219C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
220C8—181no—
221C8((C3⋊D3)⋊C3)⋊C412yes—
222C2×C3—32no—
223C2×C3—32no—
224C2×C3—31no—
225C2×C3—31no—
226C2×C3—62no—
227C2×C3—61no—
228C2×C3#6380112yes—
229C2×C3Dic3×D1211yescharacteristic
230C22#8865411yescharacteristic
231C4—182no—
232C4—184no—
233C4—362no—
234C4#8864811yescharacteristic
235C4C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
236C3—31no—
237C3—31no—
238C3—61no—
239C3#1059081211yescharacteristic
240C2#1172085312yes—
241C2#1172113211yescharacteristic
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
13. #3746⋊C3C4semidirect
45. #3746Dic3semidirect
147. H(3,𝔽3)#441semidirect
212. C32#4716semidirect
213. 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
#117210822
#117205502
#63642⋊C32
#117203092
#105908193
#105908103