Finite Group Databasebeta

← #13167367

Subgroups — #13167367

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#11724339C212yesmaximal, complemented
2#11724334C212yesmaximal, complemented
3#11723649C211yesmaximal, characteristic, complemented
4#11723647C211yesmaximal, characteristic
5C3×C8×#157C211yesmaximal, characteristic, complemented
6#10593290—31nomaximal
7C4⋊#1113C2212yes—
8#89040C2212yes—
9C3×C4×#157C2211yescharacteristic
10C3×#3787C2211yescharacteristic
11#88942—24no—
12#88939—24no—
13C3×#3746—21no—
14C3×#3745—21no—
15C3×C8×C3⋊D3C2211yescharacteristic
16#88812—21no—
17#88810—21no—
18#64227—32no—
19#64221—32no—
20#63802—31no—
21D3×Dic12—61no—
22#63797—31no—
23D3×D24—61no—
24C8×#157D311yescharacteristic, complemented
25C2×#1113—22no—
26C2×C3×#157D412yes—
27C3×C4⋊(C3⋊D3)—22no—
28C3×C4×C3⋊D3C2311yescharacteristic
29C3×#634—22no—
30#6296—22no—
31#6295—22no—
32#6294—22no—
33C3×#622D412yes—
34C33×C8D411yescharacteristic
35C3×C3⋊(C3⋊C8)D411yescharacteristic
36#4696—91nomaximal
37#3802—32no—
38D3×D12—62no—
39#3797—32no—
40D3×Dic6—62no—
41C4×#157C2×D311yescharacteristic
42#3787C2×D311yescharacteristic
43#3775—64no—
44D32⋊C4—64no—
45#3746—21no—
46#3745—21no—
47C8×C3⋊D3C2×D311yescharacteristic
48C3×Dic12—61no—
49C3×Dic12—61no—
50C3×Dic12—121no—
51C3×D24—61no—
52C3×D24—61no—
53C3×D24—121no—
54C3×D3×C8—61no—
55C3×D3×C8—61no—
56#3719—62no—
57#3718—61no—
58#3714—62no—
59#3713—61no—
60C2×C3×C3⋊D3—42no—
61C2×C3×C3⋊D3C2×D411yescharacteristic
62#1113—44no—
63C3×#157—22no—
64C33×C4C2×D411yescharacteristic
65C3×C3⋊Dic3—42no—
66C3×C3⋊Dic3C2×D411yescharacteristic
67D3×Q16—181no—
68D3×D8—181no—
69C2×Dic12—91no—
70C2×D24—91no—
71C2×#204—92no—
72#875—92no—
73C3×C4×C8—91no—
74C2×D32—62no—
75C2×D32—122no—
76C2×#157D1212yes—
77C4⋊(C3⋊D3)—22no—
78C4×C3⋊D3C22×D311yescharacteristic
79#634—22no—
80C3×D12—62no—
81C3×D12—62no—
82C3×D12—122no—
83C3×C4×D3—61no—
84C3×C4×D3—61no—
85C3×Dic6—62no—
86C3×Dic6—62no—
87C3×Dic6—122no—
88#627—62no—
89#627—122no—
90#626—122no—
91#625—62no—
92#624—62no—
93D3×Dic3—122no—
94#622D1212yes—
95C32×C8—21no—
96C32×C8—21no—
97C32×C8—41no—
98C32×C8—41no—
99C32×C8D1211yescharacteristic
100C3⋊(C3⋊C8)D1211yescharacteristic
101C3×C3⋊C8—61no—
102C3×C3⋊C8—61no—
103#495—271noSylow
104C2×C33#12711yescharacteristic
105C3×C3⋊D3—82no—
106C3×C3⋊D3#13512yes—
107D3×Q8—182no—
108D3×D4—182no—
109C2×D12—92no—
110C2×Dic6—92no—
111C3×Q16—181no—
112C3×D8—181no—
113C2×C3×C8—91no—
114C2×C3×C8—91no—
115C3×C42—91no—
116#216—182no—
117#213—182no—
118(C2×D3)⋊C4—184no—
119Dic12—61no—
120Dic12—61no—
121Dic12—121no—
122Dic12—181no—
123D24—61no—
124D24—61no—
125D24—121no—
126D24—181no—
127#204—184no—
128D3×C8—61no—
129D3×C8—61no—
130C2×C3⋊D3—42no—
131C2×C3⋊D3C2×D1211yescharacteristic
132C2×C3×D3—61no—
133C2×C3×D3—61no—
134C2×C3×D3—122no—
135C2×C3×D3—122no—
136C2×C3×D3—242no—
137D32—124no—
138D32—244no—
139#157—22no—
140C32×C4—21no—
141C32×C4—21no—
142C32×C4—41no—
143C32×C4—41no—
144C32×C4C2×D1211yescharacteristic
145C3⋊Dic3—42no—
146C3⋊Dic3C2×D1211yescharacteristic
147C3×Dic3—61no—
148C3×Dic3—61no—
149C3×Dic3—122no—
150C3×Dic3—122no—
151C3×Dic3—242no—
152C2×Q16—271no—
153C2×SD16—272no—
154C2×D8—271no—
155#124—272no—
156C4×C8—91no—
157C33#49511yescharacteristic, complemented, Sylow
158C22×D3—182no—
159C22×D3—362no—
160C3×Q8—182no—
161C3×D4—182no—
162C2×C3×C4—91no—
163C2×C3×C4—92no—
164#67—362no—
165C2×Dic3—182no—
166D12—62no—
167D12—62no—
168D12—122no—
169D12—182no—
170C4×D3—61no—
171C4×D3—61no—
172C4×D3—362no—
173Dic6—62no—
174Dic6—62no—
175Dic6—122no—
176Dic6—182no—
177Dic6—362no—
178C3×C8—21no—
179C3×C8—21no—
180C3×C8—41no—
181C3×C8—41no—
182C3×C8—92no—
183C3×C8—91no—
184C3×C8GO+(4,𝔽2)11yescharacteristic
185C3⋊C8—61no—
186C3⋊C8—61no—
187C2×C32—21no—
188C2×C32—21no—
189C2×C32—41no—
190C2×C32—41no—
191C2×C32#87411yescharacteristic
192C3⋊D3—82no—
193C3⋊D3#90412yes—
194C3×D3—62no—
195C3×D3—62no—
196C3×D3—242no—
197C3×D3—242no—
198C3×D3—482no—
199C2×Q8—272no—
200C2×D4—272no—
201Q16—181no—
202Q16—541no—
203SD16—544no—
204D8—181no—
205D8—541no—
206C2×C8—91no—
207C2×C8—91no—
208C22⋊C4—544no—
209C42—91no—
210C22×C3—91no—
211C22×C3—362no—
212C2×D3—61no—
213C2×D3—61no—
214C2×D3—122no—
215C2×D3—122no—
216C2×D3—242no—
217C2×D3—364no—
218C2×D3—724no—
219C3×C4—21no—
220C3×C4—21no—
221C3×C4—41no—
222C3×C4—41no—
223C3×C4—91no—
224C3×C4—182no—
225C3×C4—362no—
226C3×C4C2×GO+(4,𝔽2)11yescharacteristic
227Dic3—61no—
228Dic3—61no—
229Dic3—122no—
230Dic3—122no—
231Dic3—242no—
232C32—21no—
233C32—21no—
234C32—41no—
235C32—41no—
236C32#469611yescharacteristic, complemented
237C23—542no—
238Q8—182no—
239Q8—542no—
240D4—182no—
241D4—542no—
242C2×C4—91no—
243C2×C4—92no—
244C2×C4—542no—
245C8—92no—
246C8—91no—
247C8#157⋊D311yescharacteristic
248C2×C3—21no—
249C2×C3—21no—
250C2×C3—41no—
251C2×C3—41no—
252C2×C3—92no—
253C2×C3—722no—
254C2×C3#6423111yescharacteristic
255D3—62no—
256D3—62no—
257D3—242no—
258D3—242no—
259D3—482no—
260C22—91no—
261C22—362no—
262C22—1084no—
263C4—91no—
264C4—182no—
265C4—362no—
266C4C2×#157⋊D311yescharacteristic
267C3—21no—
268C3—21no—
269C3—41no—
270C3—41no—
271C3#1059329011yescharacteristic, complemented
272C2—92no—
273C2—722no—
274C2#1172434411yescharacteristic
subgroupG/H
centerC2#11724344
commutator, cosocleC3×C4×C3⋊D3C23
FrattiniC4C2×#157⋊D3
FittingC33×C8D4
socleC2×C33#127
prime radical#11723647C2
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. #11724339C2semidirect
2. #11724334C2semidirect
3. #11723649C2semidirect
5. C3×C8×#157C2semidirect
24. C8×#157D3semidirect
157. C33#495semidirect
236. C32#4696semidirect
271. C3#10593290semidirect

Maximal subgroups

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

subgroupindex
#117243392
#117243342
#117236492
#117236472
C3×C8×#1572
#105932903
#46969