Finite Group Databasebeta

← #13167368

Subgroups — #13167368

Order 1728.

460 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 63 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#11724344C211yesmaximal, characteristic
2#11724339C212yesmaximal, complemented
3#11724335C211yesmaximal, characteristic, complemented
4#11723646C212yesmaximal, complemented
5C3×C8×#157C211yesmaximal, characteristic, complemented
6#10593292—31nomaximal
7C2×#157⋊D3—22no—
8C4⋊#1113C2212yes—
9#89040C2212yes—
10C3×C4×#157C2211yescharacteristic
11C3×#3787C2211yescharacteristic
12#88942—24no—
13#88940—22no—
14C3×#3744—22no—
15C3×C8×C3⋊D3C2211yescharacteristic
16#88811—22no—
17#64231—31no—
18#64227—32no—
19#64222—31no—
20#63796—32no—
21D3×#204—62no—
22C8×#157D311yescharacteristic, complemented
23C2×#1113—22no—
24C2×C3×#157D412yes—
25#157⋊D3—44no—
26C3×C4⋊(C3⋊D3)—22no—
27C3×C4×C3⋊D3C2311yescharacteristic
28C3×#634—22no—
29#6296—22no—
30#6295—22no—
31#6294—22no—
32C3×#622D412yescomplemented
33C33×C8D411yescharacteristic, complemented
34C3×C3⋊(C3⋊C8)D411yescharacteristic, complemented
35#4693—91nomaximal
36C2×GO+(4,𝔽2)—62no—
37#3802—32no—
38D3×D12—62no—
39#3797—32no—
40D3×Dic6—62no—
41C4×#157C2×D311yescharacteristic
42#3787C2×D311yescharacteristic
43#3775—64no—
44#3773—62no—
45#3744—22no—
46C8×C3⋊D3C2×D311yescharacteristic
47C3×#204—62no—
48C3×#204—62no—
49C3×#204—122no—
50C3×D3×C8—62no—
51#3717—62no—
52#3716—62no—
53#3715—62no—
54C2×C3×C3⋊D3—42no—
55C2×C3×C3⋊D3C2×D411yescharacteristic
56#1113—44no—
57C3×#157SD1614yescomplemented
58C33×C4C2×D411yescharacteristic
59C3×C3⋊Dic3—42no—
60C3×C3⋊Dic3C2×D411yescharacteristic
61D3×SD16—182no—
62C2×#204—92no—
63C2×#204—92no—
64#874—91no—
65#869—91no—
66C3×C4×C8—91no—
67C2×D32—62no—
68C2×D32—122no—
69C2×#157D1212yes—
70GO+(4,𝔽2)—124no—
71C4⋊(C3⋊D3)—22no—
72C4×C3⋊D3C22×D311yescharacteristic
73#634—22no—
74C3×D12—62no—
75C3×D12—62no—
76C3×D12—122no—
77C3×C4×D3—62no—
78C3×Dic6—62no—
79C3×Dic6—62no—
80C3×Dic6—122no—
81#627—62no—
82#627—122no—
83#626—122no—
84#625—62no—
85#624—62no—
86D3×Dic3—122no—
87#622D1212yescomplemented
88C32×C8—22no—
89C32×C8—42no—
90C32×C8D1211yescharacteristic, complemented
91C3⋊(C3⋊C8)D1211yescharacteristic, complemented
92C3×C3⋊C8—62no—
93#492—271noSylow
94C2×C33#12711yescharacteristic
95C3×C3⋊D3—82no—
96C3×C3⋊D3C2×SD1612yes—
97D3×Q8—182no—
98D3×D4—182no—
99C2×D12—92no—
100C2×D12—182no—
101C2×Dic6—92no—
102C3×SD16—182no—
103C2×C3×C8—91no—
104C2×C3×C8—91no—
105C3×C42—91no—
106#215—182no—
107#214—182no—
108C4⋊Dic3—182no—
109#204—62no—
110#204—62no—
111#204—122no—
112#204—182no—
113#204—184no—
114D3×C8—62no—
115C2×C3⋊D3—42no—
116C2×C3⋊D3C2×D1211yescharacteristic
117C2×C3×D3—62no—
118C2×C3×D3—122no—
119C2×C3×D3—122no—
120C2×C3×D3—242no—
121D32—124no—
122D32—244no—
123#157#20414yescomplemented
124C32×C4—22no—
125C32×C4—42no—
126C32×C4C2×D1211yescharacteristic
127C3⋊Dic3—42no—
128C3⋊Dic3C2×D1211yescharacteristic
129C3×Dic3—62no—
130C3×Dic3—122no—
131C3×Dic3—122no—
132C3×Dic3—242no—
133C2×SD16—272no—
134C2×SD16—272no—
135C4⋊Q8—271no—
136#127—271no—
137C4×C8—91no—
138C33#49211yescharacteristic, complemented, Sylow
139C22×D3—182no—
140C22×D3—362no—
141C3×Q8—182no—
142C3×D4—182no—
143C2×C3×C4—92no—
144C2×C3×C4—91no—
145#67—362no—
146C2×Dic3—182no—
147D12—62no—
148D12—62no—
149D12—122no—
150D12—182no—
151D12—364no—
152C4×D3—62no—
153C4×D3—362no—
154Dic6—62no—
155Dic6—62no—
156Dic6—122no—
157Dic6—182no—
158Dic6—362no—
159C3×C8—22no—
160C3×C8—42no—
161C3×C8—92no—
162C3×C8—91no—
163C3×C8GO+(4,𝔽2)11yescharacteristic, complemented
164C3⋊C8—62no—
165C2×C32—22no—
166C2×C32—42no—
167C2×C32#87411yescharacteristic
168C3⋊D3—82no—
169C3⋊D3C2×#20412yes—
170C3×D3—64no—
171C3×D3—242no—
172C3×D3—242no—
173C3×D3—482no—
174C2×Q8—272no—
175C2×D4—272no—
176C2×D4—542no—
177SD16—182no—
178SD16—542no—
179SD16—544no—
180C2×C8—91no—
181C2×C8—91no—
182C4⋊C4—542no—
183C42—91no—
184C22×C3—91no—
185C22×C3—362no—
186C2×D3—62no—
187C2×D3—122no—
188C2×D3—122no—
189C2×D3—242no—
190C2×D3—364no—
191C2×D3—724no—
192C3×C4—22no—
193C3×C4—42no—
194C3×C4—94no—
195C3×C4—91no—
196C3×C4—362no—
197C3×C4C2×GO+(4,𝔽2)11yescharacteristic
198Dic3—62no—
199Dic3—122no—
200Dic3—122no—
201Dic3—242no—
202C32—22no—
203C32—42no—
204C32#469311yescharacteristic, complemented
205C23—542no—
206Q8—182no—
207Q8—542no—
208D4—182no—
209D4—542no—
210D4—1084no—
211C2×C4—92no—
212C2×C4—91no—
213C2×C4—542no—
214C8—92no—
215C8—91no—
216C8#157⋊D311yescharacteristic, complemented
217C2×C3—22no—
218C2×C3—42no—
219C2×C3—92no—
220C2×C3—722no—
221C2×C3#6423111yescharacteristic
222D3—64no—
223D3—242no—
224D3—242no—
225D3—482no—
226C22—91no—
227C22—362no—
228C22—1084no—
229C4—94no—
230C4—91no—
231C4—362no—
232C4C2×#157⋊D311yescharacteristic
233C3—22no—
234C3—42no—
235C3#1059329211yescharacteristic, complemented
236C2—92no—
237C2—722no—
238C2#1172434411yescharacteristic
subgroupG/H
centerC2#11724344
commutator, cosocleC3×C4×C3⋊D3C23
FrattiniC4C2×#157⋊D3
FittingC33×C8D4
socleC2×C33#127
prime radical#11724344C2
hypercenterC8#157⋊D3
hypocenterC33#492
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
2. #11724339C2semidirect
3. #11724335C2semidirect
4. #11723646C2semidirect
5. C3×C8×#157C2semidirect
22. C8×#157D3semidirect
32. C3×#622D4semidirect
33. C33×C8D4semidirect
34. C3×C3⋊(C3⋊C8)D4semidirect
57. C3×#157SD16semidirect
87. #622D12semidirect
90. C32×C8D12semidirect
91. C3⋊(C3⋊C8)D12semidirect
123. #157#204semidirect
138. C33#492semidirect
163. C3×C8GO+(4,𝔽2)semidirect
204. C32#4693semidirect
216. C8#157⋊D3semidirect
235. C3#10593292semidirect

Maximal subgroups

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

subgroupindex
#117243442
#117243392
#117243352
#117236462
C3×C8×#1572
#105932923
#46939