Finite Group Databasebeta

← #13143851

Subgroups — #13143851

Order 1728.

1045 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 1045 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C22×C4×#272C211yesmaximal, characteristic
2C2×#88527C212yesmaximal, complemented
3H(3,𝔽3)×C22⋊C8C211yesmaximal, characteristic, complemented
4#11720573C211yesmaximal, characteristic, complemented
5#11720553C212yesmaximal, complemented
6C3×#4726—34nomaximal
7C23×#272C411yescharacteristic
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C22×#1084C411yescharacteristic
10C2×C4×#272—24no—
11C2×C4×#272C411yescharacteristic
12C2×C4×#272C411yescharacteristic
13C2×C4×#272C2211yescharacteristic
14C2×C4×#272C2211yescharacteristic
15C2×C8×H(3,𝔽3)C2212yescomplemented
16C2×#6180C2212yescomplemented
17#88527—24no—
18C22×C3×C4×D3—34no—
19C2×C3×ΓL(1,𝔽25)—38no—
20C32×C22⋊C8D314yescomplemented
21C3×#830—34no—
22C3×(C2×D3)⋊C8—38no—
23C23×H(3,𝔽3)C2×C411yescharacteristic
24C22×#272—22no—
25C22×#272—22no—
26C22×#272—22no—
27C22×#272C2×C411yescharacteristic
28C22×#272C2×C411yescharacteristic
29C2×C4×H(3,𝔽3)—22no—
30C2×C4×H(3,𝔽3)C2×C411yescharacteristic
31C2×C4×H(3,𝔽3)C2311yescharacteristic
32C2×#1084—22no—
33C2×#1084C2×C411yescharacteristic
34C2×#1084C2×C411yescharacteristic
35C4×#272—22no—
36C4×#272—22no—
37C4×#272—22no—
38C4×#272D414yes—
39C8×H(3,𝔽3)—22no—
40#6180—22no—
41C3×#407—91no—
42#4726—34no—
43C23×C3×D3—34no—
44C22×C32×C4C2×D314yes—
45C22×C3×Dic3—34no—
46C2×C3×C4×D3—34no—
47C2×C3×C4×D3—34no—
48C2×C3×C4×D3—34no—
49C2×C3×C4×D3—34no—
50C2×C3×C4×D3—616no—
51C2×C32×C8C2×D318yescomplemented
52C2×C3×C3⋊C8—38no—
53C3×ΓL(1,𝔽25)—616no—
54C22×H(3,𝔽3)—21no—
55C22×H(3,𝔽3)—21no—
56C22×H(3,𝔽3)M1612yescomplemented
57C22×H(3,𝔽3)C22×C411yescharacteristic
58C2×#272—28no—
59C2×#272—21no—
60C2×#272—21no—
61C2×#272—21no—
62C2×#272—21no—
63C2×#272C22⋊C412yes—
64C2×#272M1612yescomplemented
65C4×H(3,𝔽3)—21no—
66C4×H(3,𝔽3)C2×D412yes—
67#1084—21no—
68#1084C22⋊C412yes—
69C23×C3×C4—91no—
70C22×C4×D3—34no—
71C2×C3×M16—92no—
72C2×ΓL(1,𝔽25)—38no—
73C3×C22⋊C8—34no—
74C3×C22⋊C8—91no—
75C3×C22⋊C8—92no—
76C3×C22⋊C8C3⋊D311yescharacteristic
77#830—34no—
78(C2×D3)⋊C8—38no—
79C23×C32C4×D314yes—
80C22×C3×D3—34no—
81C22×C3×D3—34no—
82C22×C3×D3—68no—
83C22×C3×D3—68no—
84C22×C3×D3—68no—
85C2×C32×C4—28no—
86C2×C32×C4C4×D314yes—
87C2×C32×C4C22×D314yes—
88C2×C3×Dic3—34no—
89C2×C3×Dic3—34no—
90C2×C3×Dic3—68no—
91C3×C4×D3—316no—
92C3×C4×D3—68no—
93C3×C4×D3—68no—
94C3×C4×D3—68no—
95C32×C8—28no—
96C3×C3⋊C8—68no—
97#407—91noSylow
98C2×H(3,𝔽3)—22no—
99C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
100C2×H(3,𝔽3)C2×M1611yescharacteristic
101C2×H(3,𝔽3)C2×M1611yescharacteristic
102#272—22no—
103#272—22no—
104C24×C3—91no—
105C23×D3—34no—
106C22×C3×C4—34no—
107C22×C3×C4—91no—
108C22×C3×C4—91no—
109C22×C3×C4—91no—
110C22×C3×C4—91no—
111C22×C3×C4—91no—
112C22×C3×C4—184no—
113C22×C3×C4C2×C3⋊D311yescharacteristic
114C22×Dic3—34no—
115C2×C4×D3—34no—
116C2×C4×D3—34no—
117C2×C4×D3—34no—
118C2×C4×D3—34no—
119C2×C4×D3—616no—
120C3×M16—184no—
121C2×C3×C8—38no—
122C2×C3×C8—92no—
123C2×C3×C8C2×C3⋊D312yes—
124C2×C3⋊C8—38no—
125ΓL(1,𝔽25)—616no—
126C22×C32—24no—
127C22×C32—24no—
128C22×C32ΓL(1,𝔽25)18yescomplemented
129C22×C32C2×C4×D314yes—
130C2×C3×D3—38no—
131C2×C3×D3—38no—
132C2×C3×D3—632no—
133C2×C3×D3—64no—
134C2×C3×D3—64no—
135C2×C3×D3—64no—
136C2×C3×D3—64no—
137C32×C4—24no—
138C32×C4D3×D418yes—
139C3×Dic3—38no—
140C3×Dic3—64no—
141C23×C4—91no—
142C2×M16—92no—
143C22⋊C8—91no—
144C22⋊C8—92no—
145C22⋊C8#27211yescharacteristic, complemented
146H(3,𝔽3)#40711yescharacteristic, complemented, Sylow
147C23×C3—34no—
148C23×C3—91no—
149C23×C3—91no—
150C23×C3—182no—
151C23×C3—182no—
152C23×C3—182no—
153C23×C3C4×C3⋊D311yescharacteristic
154C22×D3—34no—
155C22×D3—34no—
156C22×D3—68no—
157C22×D3—68no—
158C22×D3—68no—
159C2×C3×C4—22no—
160C2×C3×C4—34no—
161C2×C3×C4—34no—
162C2×C3×C4—68no—
163C2×C3×C4—91no—
164C2×C3×C4—94no—
165C2×C3×C4—91no—
166C2×C3×C4—182no—
167C2×C3×C4—182no—
168C2×C3×C4—182no—
169C2×C3×C4—182no—
170C2×C3×C4C4×C3⋊D311yescharacteristic
171C2×C3×C4C22×C3⋊D311yescharacteristic
172C2×Dic3—34no—
173C2×Dic3—34no—
174C2×Dic3—68no—
175C4×D3—316no—
176C4×D3—68no—
177C4×D3—68no—
178C4×D3—68no—
179C3×C8—22no—
180C3×C8—68no—
181C3×C8—182no—
182C3⋊C8—68no—
183C2×C32—28no—
184C2×C32D3×C22⋊C414yes—
185C2×C32C2×ΓL(1,𝔽25)14yes—
186C2×C32D3×M1614yes—
187C3×D3—68no—
188C3×D3—68no—
189C24—91no—
190C22×C4—91no—
191C22×C4—91no—
192C22×C4—91no—
193C22×C4—91no—
194C22×C4—91no—
195C22×C4—184no—
196C22×C4C2×#27211yescharacteristic
197M16—184no—
198C2×C8—92no—
199C2×C8C2×#27212yescomplemented
200C22×C3—21no—
201C22×C3—21no—
202C22×C3—34no—
203C22×C3—38no—
204C22×C3—64no—
205C22×C3—64no—
206C22×C3—92no—
207C22×C3—92no—
208C22×C3—188no—
209C22×C3—181no—
210C22×C3—181no—
211C22×C3—181no—
212C22×C3—181no—
213C22×C3#374312yes—
214C22×C3C2×C4×C3⋊D311yescharacteristic
215C2×D3—38no—
216C2×D3—38no—
217C2×D3—64no—
218C2×D3—64no—
219C2×D3—64no—
220C2×D3—64no—
221C2×D3—632no—
222C3×C4—21no—
223C3×C4—38no—
224C3×C4—64no—
225C3×C4—92no—
226C3×C4—181no—
227C3×C4D4×C3⋊D312yes—
228Dic3—38no—
229Dic3—64no—
230C32#472614yescomplemented
231C23—91no—
232C23—91no—
233C23—182no—
234C23—182no—
235C23—182no—
236C23C4×#27211yescharacteristic
237C2×C4—22no—
238C2×C4—91no—
239C2×C4—94no—
240C2×C4—91no—
241C2×C4—182no—
242C2×C4—182no—
243C2×C4—182no—
244C2×C4—182no—
245C2×C4C4×#27211yescharacteristic
246C2×C4C22×#27211yescharacteristic
247C8—22no—
248C8—182no—
249C2×C3—22no—
250C2×C3—34no—
251C2×C3—34no—
252C2×C3—34no—
253C2×C3—68no—
254C2×C3—182no—
255C2×C3—182no—
256C2×C3C22⋊C4×C3⋊D311yescharacteristic
257C2×C3C2×#374311yescharacteristic
258C2×C3M16×C3⋊D311yescharacteristic
259D3—68no—
260D3—68no—
261C22—21no—
262C22—21no—
263C22—92no—
264C22—92no—
265C22—188no—
266C22—181no—
267C22—181no—
268C22—181no—
269C22—181no—
270C22#8852712yescomplemented
271C22C2×C4×#27211yescharacteristic
272C4—21no—
273C4—92no—
274C4—181no—
275C4D4×#27212yes—
276C3—34no—
277C3#1059242511yescharacteristic
278C2—22no—
279C2—182no—
280C2—182no—
281C2C22⋊C4×#27211yescharacteristic
282C2C2×#8852711yescharacteristic
283C2M16×#27211yescharacteristic
subgroupG/H
center, FrattiniC2×C3×C4C22×C3⋊D3
commutatorC22×H(3,𝔽3)C22×C4
FittingH(3,𝔽3)×C22⋊C8C2
socleC22×C3C2×C4×C3⋊D3
prime radicalC23×#272C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC3×C22⋊C8C3⋊D3
hypocenterH(3,𝔽3)#407
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3×C4 ◁ C3×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#88527C2semidirect
3. H(3,𝔽3)×C22⋊C8C2semidirect
4. #11720573C2semidirect
5. #11720553C2semidirect
15. C2×C8×H(3,𝔽3)C22semidirect
16. C2×#6180C22semidirect
20. C32×C22⋊C8D3semidirect
51. C2×C32×C8C2×D3semidirect
56. C22×H(3,𝔽3)M16semidirect
64. C2×#272M16semidirect
128. C22×C32ΓL(1,𝔽25)semidirect
145. C22⋊C8#272semidirect
146. H(3,𝔽3)#407semidirect
199. C2×C8C2×#272semidirect
230. C32#4726semidirect
270. C22#88527semidirect

Maximal subgroups

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

subgroupindex
C22×C4×#2722
C2×#885272
H(3,𝔽3)×C22⋊C82
#117205732
#117205532
C3×#47263