Finite Group Databasebeta

← #13167372

Subgroups — #13167372

Order 1728.

436 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 55 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#11724341C211yesmaximal, characteristic, complemented
2#11724336C211yesmaximal, characteristic
3#11724334C212yesmaximal, complemented
4#11723646C212yesmaximal, complemented
5C3×C8×#157C211yesmaximal, characteristic, complemented
6#10593296—31nomaximal
7C4⋊#1113C2212yes—
8#89040C2212yes—
9C3×C4×#157C2211yescharacteristic
10C3×#3787C2211yescharacteristic
11#88943—22no—
12#88941—22no—
13#88939—24no—
14C3×#3744—22no—
15C3×C8×C3⋊D3C2211yescharacteristic
16#88811—22no—
17#64228—31no—
18#64225—31no—
19#64221—32no—
20#63796—32no—
21D3×#204—62no—
22C8×#157D311yescharacteristic, complemented
23C2×#1113—22no—
24C2×C3×#157D412yes—
25C3×C4⋊(C3⋊D3)—22no—
26C3×C4×C3⋊D3C2311yescharacteristic
27C3×#634—22no—
28#6296—22no—
29#6295—22no—
30#6294—22no—
31C3×#622D412yes—
32C33×C8D411yescharacteristic
33C3×C3⋊(C3⋊C8)D411yescharacteristic
34#4696—91nomaximal
35#3802—32no—
36D3×D12—62no—
37#3797—32no—
38D3×Dic6—62no—
39C4×#157C2×D311yescharacteristic
40#3787C2×D311yescharacteristic
41#3776—62no—
42#3774—62no—
43D32⋊C4—64no—
44#3744—22no—
45C8×C3⋊D3C2×D311yescharacteristic
46C3×#204—62no—
47C3×#204—62no—
48C3×#204—122no—
49C3×D3×C8—62no—
50#3717—62no—
51#3716—62no—
52#3715—62no—
53C2×C3×C3⋊D3—42no—
54C2×C3×C3⋊D3C2×D411yescharacteristic
55#1113—44no—
56C3×#157—22no—
57C33×C4C2×D411yescharacteristic
58C3×C3⋊Dic3—42no—
59C3×C3⋊Dic3C2×D411yescharacteristic
60D3×SD16—182no—
61C2×Dic12—91no—
62C2×D24—91no—
63C2×#204—92no—
64#875—92no—
65C3×C4×C8—91no—
66C2×D32—62no—
67C2×D32—122no—
68C2×#157D1212yes—
69C4⋊(C3⋊D3)—22no—
70C4×C3⋊D3C22×D311yescharacteristic
71#634—22no—
72C3×D12—62no—
73C3×D12—62no—
74C3×D12—122no—
75C3×C4×D3—62no—
76C3×Dic6—62no—
77C3×Dic6—62no—
78C3×Dic6—122no—
79#627—62no—
80#627—122no—
81#626—122no—
82#625—62no—
83#624—62no—
84D3×Dic3—122no—
85#622D1212yes—
86C32×C8—22no—
87C32×C8—42no—
88C32×C8D1211yescharacteristic
89C3⋊(C3⋊C8)D1211yescharacteristic
90C3×C3⋊C8—62no—
91#495—271noSylow
92C2×C33#12711yescharacteristic
93C3×C3⋊D3—82no—
94C3×C3⋊D3#13512yes—
95D3×Q8—182no—
96D3×D4—182no—
97C2×D12—92no—
98C2×Dic6—92no—
99C3×SD16—182no—
100C2×C3×C8—91no—
101C2×C3×C8—91no—
102C3×C42—91no—
103#215—182no—
104#214—182no—
105(C2×D3)⋊C4—184no—
106Dic12—182no—
107D24—182no—
108#204—62no—
109#204—62no—
110#204—122no—
111#204—182no—
112D3×C8—62no—
113C2×C3⋊D3—42no—
114C2×C3⋊D3C2×D1211yescharacteristic
115C2×C3×D3—62no—
116C2×C3×D3—122no—
117C2×C3×D3—122no—
118C2×C3×D3—242no—
119D32—124no—
120D32—244no—
121#157—22no—
122C32×C4—22no—
123C32×C4—42no—
124C32×C4C2×D1211yescharacteristic
125C3⋊Dic3—42no—
126C3⋊Dic3C2×D1211yescharacteristic
127C3×Dic3—62no—
128C3×Dic3—122no—
129C3×Dic3—122no—
130C3×Dic3—242no—
131C2×Q16—271no—
132C2×SD16—272no—
133C2×D8—271no—
134#124—272no—
135C4×C8—91no—
136C33#49511yescharacteristic, complemented, Sylow
137C22×D3—182no—
138C22×D3—362no—
139C3×Q8—182no—
140C3×D4—182no—
141C2×C3×C4—91no—
142C2×C3×C4—92no—
143#67—362no—
144C2×Dic3—182no—
145D12—62no—
146D12—62no—
147D12—122no—
148D12—182no—
149C4×D3—62no—
150C4×D3—362no—
151Dic6—62no—
152Dic6—62no—
153Dic6—122no—
154Dic6—182no—
155Dic6—362no—
156C3×C8—22no—
157C3×C8—42no—
158C3×C8—92no—
159C3×C8—91no—
160C3×C8GO+(4,𝔽2)11yescharacteristic
161C3⋊C8—62no—
162C2×C32—22no—
163C2×C32—42no—
164C2×C32#87411yescharacteristic
165C3⋊D3—82no—
166C3⋊D3#90412yes—
167C3×D3—64no—
168C3×D3—242no—
169C3×D3—242no—
170C3×D3—482no—
171C2×Q8—272no—
172C2×D4—272no—
173Q16—542no—
174SD16—182no—
175SD16—542no—
176D8—542no—
177C2×C8—91no—
178C2×C8—91no—
179C22⋊C4—544no—
180C42—91no—
181C22×C3—91no—
182C22×C3—362no—
183C2×D3—62no—
184C2×D3—122no—
185C2×D3—122no—
186C2×D3—242no—
187C2×D3—364no—
188C2×D3—724no—
189C3×C4—22no—
190C3×C4—42no—
191C3×C4—91no—
192C3×C4—182no—
193C3×C4—362no—
194C3×C4C2×GO+(4,𝔽2)11yescharacteristic
195Dic3—62no—
196Dic3—122no—
197Dic3—122no—
198Dic3—242no—
199C32—22no—
200C32—42no—
201C32#469611yescharacteristic, complemented
202C23—542no—
203Q8—182no—
204Q8—542no—
205D4—182no—
206D4—542no—
207C2×C4—91no—
208C2×C4—92no—
209C2×C4—542no—
210C8—92no—
211C8—91no—
212C8#157⋊D311yescharacteristic
213C2×C3—22no—
214C2×C3—42no—
215C2×C3—92no—
216C2×C3—722no—
217C2×C3#6423111yescharacteristic
218D3—64no—
219D3—242no—
220D3—242no—
221D3—482no—
222C22—91no—
223C22—362no—
224C22—1084no—
225C4—91no—
226C4—182no—
227C4—362no—
228C4C2×#157⋊D311yescharacteristic
229C3—22no—
230C3—42no—
231C3#1059329611yescharacteristic, complemented
232C2—92no—
233C2—722no—
234C2#1172434411yescharacteristic
subgroupG/H
centerC2#11724344
commutator, cosocleC3×C4×C3⋊D3C23
FrattiniC4C2×#157⋊D3
FittingC33×C8D4
socleC2×C33#127
prime radical#11724336C2
hypercenterC8#157⋊D3
hypocenterC33#495
LCSG ▷ C3×C4×C3⋊D3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
1. #11724341C2semidirect
3. #11724334C2semidirect
4. #11723646C2semidirect
5. C3×C8×#157C2semidirect
22. C8×#157D3semidirect
136. C33#495semidirect
201. C32#4696semidirect
231. C3#10593296semidirect

Maximal subgroups

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

subgroupindex
#117243412
#117243362
#117243342
#117236462
C3×C8×#1572
#105932963
#46969