Finite Group Databasebeta

← #13146362

Subgroups — #13146362

Order 1728.

264 conjugacy classes of subgroups; only the 33 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 33 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#11722391C211yesmaximal, characteristic, complemented
2C3×#63692C211yesmaximal, characteristic, complemented
3C3⋊(#131⋊C9)C211yesmaximal, characteristic, complemented
4#10591677—31nomaximal
5#10589509—31nomaximal
6C3×#41⋊C9C2211yescharacteristic
7C3⋊#3689—21no—
8C3×#3689—21no—
9#88417—41nomaximal
10#64158—31no—
11C3×#949—31no—
12C3×#948D311yescharacteristic
13C2×#3714—31no—
14#63692—31no—
15#63692—61no—
16#63692D311yescharacteristic, complemented
17#63577—31no—
18#63559—31no—
19#63558—31no—
20#131⋊C9—31no—
21C4⋊(C3⋊D9)—41no—
22C3×D36—41no—
23C3×Q8⋊C9D411yescharacteristic
24C9×C3⋊C8—41no—
25#5199—91no—
26#5142—91no—
27C32×#41C2×D311yescharacteristic
28C2×C4⋊(C3⋊D3)—31no—
29C2×C3×D12—31no—
30#3755—61no—
31#3753—61no—
32#3748—31no—
33C3×#215—61no—
34C3×#213—61no—
35C3×#208—31no—
36C3×#208—31no—
37C2×C3×C3⋊C8—31no—
38#3714—61no—
39#3714—61no—
40#41⋊C9—21no—
41#41⋊C9C2×D311yescharacteristic
42#3689—21no—
43#3689—62no—
44#3689—61no—
45D72—121no—
46C2×C3⋊D9—81no—
47C2×C3×D9—81no—
48C3×C4×C9—41no—
49C3×AGL(1,ℤ/8)—91no—
50C3×#131—31no—
51#949—31no—
52#949—91no—
53#949—91no—
54#949—181no—
55#948D911yescharacteristic, complemented
56C2×#213—91no—
57C2×D24—91no—
58#833—91no—
59#823—91no—
60#823—91no—
61#823—91no—
62#822—91no—
63#819—91no—
64C22×C3⋊D3—61no—
65C22×C3×D3—61no—
66C32×Q8D1211yescharacteristic
67C32×D4—31no—
68C2×C32×C4—31no—
69C4⋊(C3⋊D3)—61no—
70C4⋊(C3⋊D3)—61no—
71C3×D12—61no—
72C3×D12—61no—
73C3⋊(C3⋊C8)—61no—
74C3×C3⋊C8—31no—
75C3×C3⋊C8—61no—
76C3×C3⋊C8S411yescharacteristic
77D36—41no—
78D36—121no—
79D36—241no—
80Q8⋊C9—21no—
81Q8⋊C9#6711yescharacteristic
82C8×C9—121no—
83#472—271noSylow
84C2×C3×C9—41no—
85C3⋊D9—161no—
86C3×D9—161no—
87C3×#41—21no—
88C3×#41C2×D911yescharacteristic
89C3×#41D3211yescharacteristic
90C2×C3×D4—91no—
91C2×D12—31no—
92C2×D12—91no—
93C2×D12—91no—
94C2×D12—181no—
95C3×SD16—181no—
96C3×D8—181no—
97C3×M16—91no—
98C3×M16—91no—
99C2×C3×C8—91no—
100#215—61no—
101#215—182no—
102#215—181no—
103#215—181no—
104#213—61no—
105#213—181no—
106#213—181no—
107#213—181no—
108#213—182no—
109#213—181no—
110#208—31no—
111#208—31no—
112#208—91no—
113#208—91no—
114#208—181no—
115C2×C3⋊C8—31no—
116D24—181no—
117D24—181no—
118C22×C32—31no—
119C2×C3⋊D3—121no—
120C2×C3⋊D3—241no—
121C2×C3×D3—121no—
122C2×C3×D3—241no—
123C32×C4—31no—
124C32×C4C2×S411yescharacteristic
125C2×D9—81no—
126C2×D9—241no—
127C2×D9—242no—
128C4×C9—41no—
129C4×C9—81no—
130AGL(1,ℤ/8)—91no—
131AGL(1,ℤ/8)—271no—
132C2×D8—271no—
133#131—31no—
134#108—271no—
135#100—271no—
136#100—271no—
137C3×C9—41noSylow
138C23×C3—181no—
139C22×D3—61no—
140C22×D3—181no—
141C22×D3—181no—
142C22×D3—361no—
143C3×Q8—21no—
144C3×Q8D3611yescharacteristic
145C3×Q8#62611yescharacteristic
146C3×D4—31no—
147C3×D4—31no—
148C3×D4—61no—
149C3×D4—181no—
150C3×D4—181no—
151C2×C3×C4—31no—
152C2×C3×C4—31no—
153C2×C3×C4—61no—
154D12—61no—
155D12—61no—
156D12—181no—
157D12—181no—
158D12—182no—
159D12—181no—
160D12—181no—
161D12—361no—
162C3×C8—31no—
163C3×C8—91no—
164C3×C8—181no—
165C3⋊C8—31no—
166C3⋊C8—61no—
167C3⋊C8—181no—
168C3⋊C8—181no—
169C3⋊C8—182no—
170C3⋊C8(C22⋊C9)⋊C211yescharacteristic
171C2×C32—61no—
172C2×C32C4⋊S411yescharacteristic
173C3⋊D3—241no—
174C3×D3—241no—
175C2×C9—41no—
176C2×C9—81no—
177D9—161no—
178D9—481no—
179D9—482no—
180#41D3×D911yescharacteristic
181C2×D4—91no—
182C2×D4—271no—
183SD16—181no—
184SD16—541no—
185D8—181no—
186D8—541no—
187D8—541no—
188D8—541no—
189M16—91no—
190M16—91no—
191M16—271no—
192C2×C8—91no—
193C22×C3—31no—
194C22×C3—31no—
195C22×C3—61no—
196C22×C3—361no—
197C22×C3—721no—
198C2×D3—121no—
199C2×D3—241no—
200C2×D3—362no—
201C2×D3—361no—
202C2×D3—361no—
203C2×D3—722no—
204C2×D3—721no—
205C2×D3—721no—
206C3×C4—21no—
207C3×C4—31no—
208C3×C4—31no—
209C3×C4—61no—
210C3×C4C2×(C22⋊C9)⋊C211yescharacteristic
211C3×C4D3×S411yescharacteristic
212C32#540711yescharacteristic
213C9—41no—
214C9—81no—
215C23—181no—
216C23—541no—
217Q8#619211yescharacteristic
218D4—31no—
219D4—181no—
220D4—181no—
221D4—541no—
222D4—541no—
223C2×C4—31no—
224C8—31no—
225C8—91no—
226C8—181no—
227C8—541no—
228C2×C3—21no—
229C2×C3—61no—
230C2×C3—61no—
231C2×C3—121no—
232C2×C3—721no—
233C2×C3C4⋊((C22⋊C9)⋊C2)11yescharacteristic
234C2×C3#6420711yescharacteristic
235D3—241no—
236D3—722no—
237D3—721no—
238D3—721no—
239C22—31no—
240C22—361no—
241C22—721no—
242C22—1081no—
243C22—2161no—
244C4—31no—
245C4D3×(C22⋊C9)⋊C211yescharacteristic
246C3—21no—
247C3#1058950911yescharacteristic, complemented
248C3#1059322511yescharacteristic
249C2—61no—
250C2—721no—
251C2—2161no—
252C2#1172226411yescharacteristic
subgroupG/H
centerC2#11722264
commutator, cosocleC3×#41⋊C9C22
FrattiniC3×C4D3×S4
FittingC32×#41C2×D3
socleC2×C32C4⋊S4
hypercenterC4D3×(C22⋊C9)⋊C2
hypocenterC3×Q8⋊C9D4
LCSG ▷ C3×#41⋊C9 ▷ C3×Q8⋊C9
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. #11722391C2semidirect
2. C3×#63692C2semidirect
3. C3⋊(#131⋊C9)C2semidirect
16. #63692D3semidirect
55. #948D9semidirect
247. C3#10589509semidirect

Maximal subgroups

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

subgroupindex
#117223912
C3×#636922
C3⋊(#131⋊C9)2
#105916773
#105895093
#884174