Finite Group Databasebeta

← #13143850

Subgroups — #13143850

Order 1728.

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

normal subgroups only

This group has 737 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
1C4×#6235C211yesmaximal, characteristic
2C22×#6180C211yesmaximal, characteristic, complemented
3C2×C8×#272C211yesmaximal, characteristic, complemented
4H(3,𝔽3)×C22⋊C8C211yesmaximal, characteristic, complemented
5#11720553C211yesmaximal, characteristic, complemented
6#11720547C211yesmaximal, characteristic, complemented
7C8×#1084C211yesmaximal, characteristic, complemented
8C3×#67⋊C8—34nomaximal
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C2×#6235C411yescharacteristic
11C2×C4×#272C2211yescharacteristic
12C2×C8×H(3,𝔽3)C2211yescharacteristic
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14#88552C411yescharacteristic
15#88542C411yescharacteristic
16#88540C411yescharacteristic
17C4×#1084C2211yescharacteristic
18C2×#6180—22no—
19C2×#6180C2211yescharacteristic
20C2×#6180C2211yescharacteristic
21C8×#272—22no—
22C3×C4×#67—34no—
23C22×C3×C3⋊C8—34no—
24C2×C3×D3×C8—34no—
25C32×C22⋊C8D314yescomplemented
26C3×(C2×D3)⋊C8—34no—
27C3×Dic3⋊C8—34no—
28C3×C8×Dic3—34no—
29C23×H(3,𝔽3)C2×C411yescharacteristic
30C22×#272C2×C411yescharacteristic
31C2×C4×H(3,𝔽3)—21no—
32C2×C4×H(3,𝔽3)—21no—
33C2×C4×H(3,𝔽3)C2×C411yescharacteristic
34C2×C4×H(3,𝔽3)C2311yescharacteristic
35#6235C814yescomplemented
36C2×#1084C2×C411yescharacteristic
37C2×#1084C2×C411yescharacteristic
38C2×#1084C2×C411yescharacteristic
39C4×#272—21no—
40C4×#272—21no—
41C8×H(3,𝔽3)—21no—
42C8×H(3,𝔽3)—21no—
43#6180—21no—
44#6180D412yescomplemented
45C3×C8×D4—91no—
46#67⋊C8—34no—
47C22×C32×C4C2×D314yes—
48C2×C3×#67—34no—
49C2×C3×C4×D3—34no—
50C2×C32×C8C2×D314yes—
51C2×C32×C8C2×D314yes—
52C3×#217—34no—
53C3×(C2×D3)⋊C4—34no—
54C3×Dic3⋊C4—34no—
55C3×C4×Dic3—34no—
56C2×C3×C3⋊C8—34no—
57C2×C3×C3⋊C8—34no—
58C2×C3×C3⋊C8—68no—
59C3×D3×C8—68no—
60C22×H(3,𝔽3)—21no—
61C22×H(3,𝔽3)—21no—
62C22×H(3,𝔽3)C2×C812yescomplemented
63C22×H(3,𝔽3)C22×C411yescharacteristic
64C2×#272—21no—
65C2×#272—21no—
66C2×#272C2×C812yescomplemented
67C4×H(3,𝔽3)—21no—
68C4×H(3,𝔽3)C2×D411yescharacteristic
69C4×H(3,𝔽3)#4111yescharacteristic
70#1084—21no—
71#1084—21no—
72#1084C2×C812yescomplemented
73C22×C3×C8—91no—
74C22×C3×C8—91no—
75C3×C4×D4—91no—
76C4×#67—34no—
77C22×C3⋊C8—34no—
78C2×D3×C8—34no—
79C3×C4⋊C8—91no—
80C3×C22⋊C8—34no—
81C3×C22⋊C8—91no—
82C3×C22⋊C8C3⋊D311yescharacteristic
83C3×C4×C8—91no—
84(C2×D3)⋊C8—34no—
85Dic3⋊C8—34no—
86C8×Dic3—34no—
87C23×C32C4×D314yes—
88C22×C3×D3—34no—
89C2×C32×C4—24no—
90C2×C32×C4—24no—
91C2×C32×C4C4×D314yes—
92C2×C32×C4C22×D314yes—
93C3×#67—316no—
94C2×C3×Dic3—34no—
95C2×C3×Dic3—34no—
96C2×C3×Dic3—34no—
97C3×C4×D3—64no—
98C3×C4×D3—64no—
99C32×C8—24no—
100C32×C8—24no—
101C3×C3⋊C8—38no—
102C3×C3⋊C8—64no—
103C8×D4—91noSylow
104C2×H(3,𝔽3)—22no—
105C2×H(3,𝔽3)C4×D411yescharacteristic
106C2×H(3,𝔽3)C22×C811yescharacteristic
107C2×H(3,𝔽3)#13111yescharacteristic
108#272—22no—
109C2×C3×D4—91no—
110C22×C3×C4—34no—
111C22×C3×C4—91no—
112C22×C3×C4C2×C3⋊D311yescharacteristic
113C2×#67—34no—
114C2×C4×D3—34no—
115C2×C3×C8—34no—
116C2×C3×C8—34no—
117C2×C3×C8—91no—
118C2×C3×C8—91no—
119C2×C3×C8—182no—
120C2×C3×C8—182no—
121C2×C3×C8C2×C3⋊D311yescharacteristic
122C2×C3×C8C2×C3⋊D311yescharacteristic
123C3×C4⋊C4—91no—
124C3×C22⋊C4—91no—
125C3×C22⋊C4—91no—
126C3×C42—91no—
127#217—34no—
128(C2×D3)⋊C4—34no—
129Dic3⋊C4—34no—
130C4×Dic3—34no—
131C2×C3⋊C8—34no—
132C2×C3⋊C8—34no—
133C2×C3⋊C8—68no—
134D3×C8—68no—
135C22×C32—24no—
136C22×C32—24no—
137C22×C32D3×C818yescomplemented
138C22×C32C2×C4×D314yes—
139C2×C3×D3—38no—
140C2×C3×D3—64no—
141C2×C3×D3—64no—
142C32×C4—24no—
143C32×C4D3×D414yes—
144C32×C4#23714yes—
145C3×Dic3—38no—
146C3×Dic3—64no—
147C3×Dic3—64no—
148C22×C8—91no—
149C22×C8—91no—
150C4×D4—91no—
151C4⋊C8—91no—
152C22⋊C8—91no—
153C22⋊C8#27211yescharacteristic, complemented
154C4×C8—91no—
155H(3,𝔽3)C8×D411yescharacteristic, complemented, Sylow
156C23×C3—34no—
157C23×C3—91no—
158C23×C3C4×C3⋊D311yescharacteristic
159C22×D3—34no—
160C3×D4—94no—
161C2×C3×C4—21no—
162C2×C3×C4—21no—
163C2×C3×C4—34no—
164C2×C3×C4—34no—
165C2×C3×C4—64no—
166C2×C3×C4—64no—
167C2×C3×C4—91no—
168C2×C3×C4—91no—
169C2×C3×C4—91no—
170C2×C3×C4—181no—
171C2×C3×C4—181no—
172C2×C3×C4C4×C3⋊D311yescharacteristic
173C2×C3×C4C22×C3⋊D311yescharacteristic
174#67—316no—
175C2×Dic3—34no—
176C2×Dic3—34no—
177C2×Dic3—34no—
178C4×D3—64no—
179C4×D3—64no—
180C3×C8—21no—
181C3×C8—21no—
182C3×C8—64no—
183C3×C8—64no—
184C3×C8—92no—
185C3×C8—181no—
186C3⋊C8—38no—
187C3⋊C8—64no—
188C2×C32—28no—
189C2×C32#67⋊C414yes—
190C2×C32C2×D3×C814yes—
191C2×C32#90714yes—
192C3×D3—68no—
193C2×D4—91no—
194C22×C4—91no—
195C22×C4C2×#27211yescharacteristic
196C2×C8—91no—
197C2×C8—91no—
198C2×C8—182no—
199C2×C8—182no—
200C2×C8C2×#27211yescharacteristic
201C2×C8C2×#27211yescharacteristic
202C4⋊C4—91no—
203C22⋊C4—91no—
204C22⋊C4—91no—
205C42—91no—
206C22×C3—21no—
207C22×C3—21no—
208C22×C3—34no—
209C22×C3—38no—
210C22×C3—64no—
211C22×C3—64no—
212C22×C3—92no—
213C22×C3—181no—
214C22×C3—181no—
215C22×C3C8×C3⋊D312yes—
216C22×C3C2×C4×C3⋊D311yescharacteristic
217C2×D3—38no—
218C2×D3—64no—
219C2×D3—64no—
220C3×C4—21no—
221C3×C4—34no—
222C3×C4—34no—
223C3×C4—64no—
224C3×C4—92no—
225C3×C4—181no—
226C3×C4—181no—
227C3×C4D4×C3⋊D311yescharacteristic
228C3×C4#383011yescharacteristic
229Dic3—38no—
230Dic3—64no—
231Dic3—64no—
232C32#67⋊C814yescomplemented
233C23—91no—
234C23C4×#27211yescharacteristic
235D4—94no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—91no—
239C2×C4—91no—
240C2×C4—91no—
241C2×C4—181no—
242C2×C4—181no—
243C2×C4C4×#27211yescharacteristic
244C2×C4C22×#27211yescharacteristic
245C8—21no—
246C8—21no—
247C8—92no—
248C8—181no—
249C2×C3—22no—
250C2×C3—34no—
251C2×C3—34no—
252C2×C3—34no—
253C2×C3—68no—
254C2×C3—182no—
255C2×C3#6409011yescharacteristic
256C2×C3C2×C8×C3⋊D311yescharacteristic
257C2×C3#6411611yescharacteristic
258D3—68no—
259C22—21no—
260C22—21no—
261C22—92no—
262C22—181no—
263C22—181no—
264C22C8×#27212yescomplemented
265C22C2×C4×#27211yescharacteristic
266C4—21no—
267C4—92no—
268C4—181no—
269C4—181no—
270C4D4×#27211yescharacteristic
271C4#8874411yescharacteristic
272C3—34no—
273C3#1059242411yescharacteristic
274C2—22no—
275C2—182no—
276C2#1172173711yescharacteristic
277C2C2×C8×#27211yescharacteristic
278C2#1172178611yescharacteristic
subgroupG/H
center, FrattiniC2×C3×C4C22×C3⋊D3
commutatorC2×H(3,𝔽3)C22×C8
FittingH(3,𝔽3)×C22⋊C8C2
socleC22×C3C2×C4×C3⋊D3
prime radicalC2×#6235C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC3×C22⋊C8C3⋊D3
hypocenterH(3,𝔽3)C8×D4
LCSG ▷ C2×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. C22×#6180C2semidirect
3. C2×C8×#272C2semidirect
4. H(3,𝔽3)×C22⋊C8C2semidirect
5. #11720553C2semidirect
6. #11720547C2semidirect
7. C8×#1084C2semidirect
25. C32×C22⋊C8D3semidirect
35. #6235C8semidirect
44. #6180D4semidirect
62. C22×H(3,𝔽3)C2×C8semidirect
66. C2×#272C2×C8semidirect
72. #1084C2×C8semidirect
137. C22×C32D3×C8semidirect
153. C22⋊C8#272semidirect
155. H(3,𝔽3)C8×D4semidirect
232. C32#67⋊C8semidirect
264. C22C8×#272semidirect

Maximal subgroups

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

subgroupindex
C4×#62352
C22×#61802
C2×C8×#2722
H(3,𝔽3)×C22⋊C82
#117205532
#117205472
C8×#10842
C3×#67⋊C83