Finite Group Databasebeta

← #13159557

Subgroups — #13159557

Order 1728.

1084 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 1084 conjugacy classes of subgroups — too many to draw. Only its 188 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×C8×C3⋊(C3⋊D3)C211yesmaximal, characteristic
2#11723503C212yesmaximal
3C3×#63645C211yesmaximal, characteristic, complemented
4C32×#818C211yesmaximal, characteristic, complemented
5#11722056C212yesmaximal, complemented
6C3⋊D3×#107—31nomaximal
7#10590687—34nomaximal
8C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
9C2×#6289—22no—
10C2×C33×C8C2211yescharacteristic
11C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
12C8×C3⋊(C3⋊D3)C414yescomplemented
13C3×C4⋊(C3⋊Dic3)C2212yes—
14C32×C4⋊Dic3C2212yes—
15#88806—22no—
16C2×C8×C3⋊D3—31no—
17C2×C8×C3⋊D3—34no—
18C2×C8×C3⋊D3—64no—
19C4⋊C4×C3⋊D3—32no—
20#63900—38no—
21C32×#107—31no—
22#63645D311yescharacteristic, complemented
23#63634—32no—
24C3×#818—34no—
25C3×#818D314yescomplemented
26#63576—38no—
27C22×C3⋊(C3⋊D3)D411yescharacteristic
28C2×C33×C4C2311yescharacteristic
29C2×C3⋊(C3⋊Dic3)D411yescharacteristic
30C4×C3⋊(C3⋊D3)C2×C412yes—
31C4×C3⋊(C3⋊D3)Q812yes—
32C2×C3×C3⋊Dic3—22no—
33C2×C32×Dic3—22no—
34#6289—44no—
35C33×C8C2×C412yescomplemented
36C3⋊(C3⋊(C3⋊C8))C2×C412yescomplemented
37D3×#107—94no—
38D3×#107—91no—
39C2×C4×C3⋊D3—31no—
40C2×C4×C3⋊D3—34no—
41C2×C4×C3⋊D3—64no—
42C2×C4×C3⋊D3—62no—
43C2×#624—68no—
44C2×C32×C8—24no—
45C2×C32×C8C2×D314yes—
46C2×C32×C8C2×D311yescharacteristic
47C32×C4⋊C4—32no—
48C4⋊(C3⋊Dic3)C2×D312yes—
49#3750—62no—
50C2×C3⋊(C3⋊C8)—31no—
51C2×C3⋊(C3⋊C8)—34no—
52C2×C3⋊(C3⋊C8)—64no—
53C8×C3⋊D3—34no—
54C8×C3⋊D3—316no—
55C8×C3⋊D3—616no—
56C3×C4⋊Dic3—38no—
57C3×C4⋊Dic3C2×D318yes—
58#3724—68no—
59C22×C33C2×D411yescharacteristic
60C2×C3⋊(C3⋊D3)C4⋊C412yes—
61C2×C3⋊(C3⋊D3)D812yes—
62C2×C3⋊(C3⋊D3)Q1612yes—
63C33×C4C22×C411yescharacteristic
64C33×C4C2×Q811yescharacteristic
65C3⋊(C3⋊Dic3)C4⋊C412yes—
66C3×C3⋊Dic3—42no—
67C32×Dic3—42no—
68C2×D3×C8—94no—
69C2×D3×C8—91no—
70C2×D3×C8—184no—
71D3×C4⋊C4—98no—
72D3×C4⋊C4—92no—
73C3×#107—34no—
74C3×#107—91no—
75#818—34no—
76#818C3⋊D311yescharacteristic, complemented
77#807—98no—
78#807—92no—
79C22×C3⋊D3—31no—
80C22×C3⋊D3—34no—
81C22×C3⋊D3—64no—
82C2×C32×C4—24no—
83C2×C32×C4—62no—
84C2×C32×C4C22×D314yes—
85C2×C32×C4C22×D311yescharacteristic
86C2×C3⋊Dic3—22no—
87C2×C3⋊Dic3—31no—
88C2×C3⋊Dic3—34no—
89C2×C3⋊Dic3—64no—
90C4×C3⋊D3—32no—
91C4×C3⋊D3—32no—
92C4×C3⋊D3—38no—
93C4×C3⋊D3—38no—
94C4×C3⋊D3—68no—
95C4×C3⋊D3—68no—
96C4×C3⋊D3—124no—
97C2×C3×Dic3—28no—
98C2×C3×Dic3—68no—
99#624—1216no—
100C32×C8—28no—
101C32×C8C4×D318yescomplemented
102C32×C8C4×D312yescomplemented
103C3⋊(C3⋊C8)—32no—
104C3⋊(C3⋊C8)—38no—
105C3⋊(C3⋊C8)—68no—
106C2×#107—271noSylow
107C2×C33C2×C4⋊C411yescharacteristic
108C2×C33C2×D811yescharacteristic
109C2×C33C2×Q1611yescharacteristic
110C3⋊(C3⋊D3)#10714yescomplemented
111C2×C4×D3—94no—
112C2×C4×D3—91no—
113C2×C4×D3—184no—
114C2×C4×D3—182no—
115C2×C4×D3—188no—
116C2×C3×C8—24no—
117C2×C3×C8D3214yes—
118C2×C3×C8C2×C3⋊D311yescharacteristic
119C3×C4⋊C4—38no—
120C3×C4⋊C4—92no—
121C4⋊Dic3—38no—
122C4⋊Dic3C2×C3⋊D312yes—
123Dic3⋊C4—182no—
124Dic3⋊C4—188no—
125C2×C3⋊C8—94no—
126C2×C3⋊C8—91no—
127C2×C3⋊C8—184no—
128D3×C8—916no—
129D3×C8—94no—
130D3×C8—1816no—
131C22×C32—24no—
132C22×C32D3×D414yes—
133C22×C32D3×D411yescharacteristic
134C2×C3⋊D3—32no—
135C2×C3⋊D3—32no—
136C2×C3⋊D3—32no—
137C2×C3⋊D3—38no—
138C2×C3⋊D3—38no—
139C2×C3⋊D3—38no—
140C2×C3⋊D3—68no—
141C2×C3⋊D3—68no—
142C2×C3⋊D3—68no—
143C32×C4—24no—
144C32×C4—24no—
145C32×C4—122no—
146C32×C4C2×C4×D314yes—
147C32×C4C2×C4×D311yescharacteristic
148C32×C4D3×Q814yes—
149C32×C4D3×Q811yescharacteristic
150C3⋊Dic3—32no—
151C3⋊Dic3—38no—
152C3⋊Dic3—42no—
153C3⋊Dic3—68no—
154C3×Dic3—48no—
155C3×Dic3—128no—
156C22×C8—271no—
157C2×C4⋊C4—272no—
158#107—31no—
159#107—91no—
160#107—272no—
161C33C2×#10711yescharacteristic, complemented, Sylow
162C22×D3—94no—
163C22×D3—91no—
164C22×D3—184no—
165C2×C3×C4—24no—
166C2×C3×C4—68no—
167C2×C3×C4—182no—
168C2×C3×C4C2×D3214yes—
169C2×C3×C4C22×C3⋊D311yescharacteristic
170C2×Dic3—22no—
171C2×Dic3—68no—
172C2×Dic3—94no—
173C2×Dic3—91no—
174C2×Dic3—184no—
175C4×D3—98no—
176C4×D3—98no—
177C4×D3—92no—
178C4×D3—92no—
179C4×D3—188no—
180C4×D3—188no—
181C4×D3—364no—
182C4×D3—3616no—
183C3×C8—28no—
184C3×C8#62418yescomplemented
185C3×C8C4×C3⋊D312yescomplemented
186C3⋊C8—98no—
187C3⋊C8—92no—
188C3⋊C8—188no—
189C2×C32—24no—
190C2×C32—24no—
191C2×C32—24no—
192C2×C32D3×C4⋊C414yes—
193C2×C32D3×C4⋊C411yescharacteristic
194C2×C32D3×D814yes—
195C2×C32D3×D811yescharacteristic
196C2×C32D3×Q1614yes—
197C2×C32D3×Q1611yescharacteristic
198C3⋊D3—34no—
199C3⋊D3—316no—
200C3⋊D3—616no—
201C22×C4—271no—
202C22×C4—542no—
203C2×C8—271no—
204C2×C8—274no—
205C2×C8D3×C3⋊D311yescharacteristic
206C4⋊C4—32no—
207C4⋊C4—92no—
208C4⋊C4—542no—
209C22×C3—24no—
210C22×C3#380214yes—
211C22×C3D4×C3⋊D311yescharacteristic
212C2×D3—98no—
213C2×D3—98no—
214C2×D3—92no—
215C2×D3—92no—
216C2×D3—92no—
217C2×D3—98no—
218C2×D3—188no—
219C2×D3—188no—
220C2×D3—188no—
221C3×C4—24no—
222C3×C4—24no—
223C3×C4—128no—
224C3×C4—362no—
225C3×C4#379714yes—
226C3×C4C2×#62414yes—
227C3×C4C2×C4×C3⋊D311yescharacteristic
228C3×C4Q8×C3⋊D311yescharacteristic
229Dic3—42no—
230Dic3—92no—
231Dic3—98no—
232Dic3—128no—
233Dic3—188no—
234C32—24no—
235C32D3×#10714yescomplemented
236C32D3×#10711yescharacteristic, complemented
237C23—271no—
238C2×C4—62no—
239C2×C4—182no—
240C2×C4—271no—
241C2×C4—272no—
242C2×C4—272no—
243C2×C4—1084no—
244C2×C4C2×D3×C3⋊D311yescharacteristic
245C8—272no—
246C8#628912yescomplemented
247C2×C3—24no—
248C2×C3—24no—
249C2×C3—24no—
250C2×C3#6379714yes—
251C2×C3#6380214yes—
252C2×C3#6390014yes—
253C2×C3C4⋊C4×C3⋊D311yescharacteristic
254C2×C3D8×C3⋊D311yescharacteristic
255C2×C3Q16×C3⋊D311yescharacteristic
256D3—94no—
257D3—916no—
258D3—1816no—
259C22—272no—
260C22—272no—
261C22—272no—
262C22#8902611yescharacteristic
263C4—122no—
264C4—272no—
265C4—362no—
266C4#8901911yescharacteristic
267C4C2×#628911yescharacteristic
268C3—24no—
269C3#1059068714yescomplemented
270C3C3⋊D3×#10711yescharacteristic, complemented
271C2—274no—
272C2#1172332211yescharacteristic
273C2#1172333011yescharacteristic
274C2#1172350311yescharacteristic
subgroupG/H
centerC22#89026
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC22×C3⋊(C3⋊D3)D4
cosocleC2×C33×C4C23
hypercenterC2×C8D3×C3⋊D3
hypocenterC33C2×#107
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
3. C3×#63645C2semidirect
4. C32×#818C2semidirect
5. #11722056C2semidirect
12. C8×C3⋊(C3⋊D3)C4semidirect
22. #63645D3semidirect
25. C3×#818D3semidirect
35. C33×C8C2×C4semidirect
36. C3⋊(C3⋊(C3⋊C8))C2×C4semidirect
76. #818C3⋊D3semidirect
101. C32×C8C4×D3semidirect
102. C32×C8C4×D3semidirect
110. C3⋊(C3⋊D3)#107semidirect
161. C33C2×#107semidirect
184. C3×C8#624semidirect
185. C3×C8C4×C3⋊D3semidirect
235. C32D3×#107semidirect
236. C32D3×#107semidirect
246. C8#6289semidirect
269. C3#10590687semidirect
270. C3C3⋊D3×#107semidirect

Maximal subgroups

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

subgroupindex
C2×C8×C3⋊(C3⋊D3)2
#117235032
C3×#636452
C32×#8182
#117220562
C3⋊D3×#1073
#105906873