Finite Group Databasebeta

← #13143810

Subgroups — #13143810

Order 1728.

682 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 682 conjugacy classes of subgroups — too many to draw. Only its 133 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×#88527C211yesmaximal, characteristic, complemented
2C2×C8×#272C211yesmaximal, characteristic, complemented
3C4×C4⋊#272C211yesmaximal, characteristic
4H(3,𝔽3)×#97C211yesmaximal, characteristic, complemented
5#11720553C211yesmaximal, characteristic, complemented
6#11720553C211yesmaximal, characteristic, complemented
7C4⋊#6180C211yesmaximal, characteristic, complemented
8C3×#4706—34nomaximal
9C2×C4⋊#272C411yescharacteristic
10C2×C4×#272C2211yescharacteristic
11C2×C4×#272C2211yescharacteristic
12C2×C8×H(3,𝔽3)C2211yescharacteristic
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14C42×H(3,𝔽3)C2211yescharacteristic
15#88542C411yescharacteristic
16#88542C411yescharacteristic
17C4⋊#1084C411yescharacteristic
18C2×#6180C2211yescharacteristic
19C2×#6180C2211yescharacteristic
20#88527—22no—
21C8×#272—22no—
22C2×C3×ΓL(1,𝔽25)—34no—
23C2×C3×D3×C8—34no—
24C3×C4×D12—34no—
25C32×#97D314yescomplemented
26C3×(C2×D3)⋊C8—34no—
27C3×(C2×D3)⋊C8—34no—
28C3×C4⋊(C3⋊C8)—34no—
29C22×#272C2×C411yescharacteristic
30C22×#272C2×C411yescharacteristic
31C2×C4×H(3,𝔽3)C2×C411yescharacteristic
32C2×C4×H(3,𝔽3)C2×C411yescharacteristic
33C2×C4×H(3,𝔽3)C2311yescharacteristic
34C2×#1084C2×C411yescharacteristic
35C2×#1084C2×C411yescharacteristic
36C4⋊#272—22no—
37C4×#272—21no—
38C4×#272—21no—
39C4×#272—21no—
40C4×#272—21no—
41C8×H(3,𝔽3)—21no—
42C8×H(3,𝔽3)D412yescomplemented
43#6180—21no—
44#6180—21no—
45C3×#435—91no—
46#4706—34no—
47C2×C3×D12—34no—
48C2×C3×C4×D3—34no—
49C2×C3×C4×D3—34no—
50C2×C32×C8C2×D314yes—
51C2×C32×C8C2×D314yes—
52C32×C42C2×D314yes—
53C3×(C2×D3)⋊C4—34no—
54C3×(C2×D3)⋊C4—34no—
55C3×C4⋊Dic3—34no—
56C2×C3×C3⋊C8—34no—
57C2×C3×C3⋊C8—34no—
58C3×ΓL(1,𝔽25)—68no—
59C3×D3×C8—68no—
60C22×H(3,𝔽3)C22×C411yescharacteristic
61C2×#272—21no—
62C2×#272—21no—
63C2×#272—21no—
64C2×#272—21no—
65C2×#272—21no—
66C2×#272M1612yescomplemented
67C4×H(3,𝔽3)—21no—
68C4×H(3,𝔽3)—21no—
69C4×H(3,𝔽3)C2×D411yescharacteristic
70C4×H(3,𝔽3)#4111yescharacteristic
71#1084—21no—
72#1084—21no—
73C2×C3×M16—91no—
74C22×C3×C8—91no—
75C3×C4×D4—91no—
76C2×ΓL(1,𝔽25)—34no—
77C2×D3×C8—34no—
78C4×D12—34no—
79C3×C4⋊C8—91no—
80C3×C22⋊C8—91no—
81C3×C22⋊C8—91no—
82C3×#97—34no—
83C3×#97C3⋊D311yescharacteristic
84(C2×D3)⋊C8—34no—
85(C2×D3)⋊C8—34no—
86C4⋊(C3⋊C8)—34no—
87C22×C3×D3—34no—
88C22×C3×D3—34no—
89C2×C32×C4C4×D314yes—
90C2×C32×C4C4×D314yes—
91C2×C32×C4C22×D314yes—
92C2×C3×Dic3—34no—
93C2×C3×Dic3—34no—
94C3×D12—68no—
95C3×C4×D3—64no—
96C3×C4×D3—64no—
97C3×C4×D3—64no—
98C3×C4×D3—64no—
99C32×C8—24no—
100C32×C8D1218yescomplemented
101C3×C3⋊C8—64no—
102C3×C3⋊C8—64no—
103#435—91noSylow
104C2×H(3,𝔽3)C4×D411yescharacteristic
105C2×H(3,𝔽3)C2×M1611yescharacteristic
106C2×H(3,𝔽3)#13111yescharacteristic
107#272—22no—
108#272—41no—
109C2×C3×D4—91no—
110C22×C3×C4—91no—
111C22×C3×C4—91no—
112C2×D12—34no—
113C2×C4×D3—34no—
114C2×C4×D3—34no—
115C3×M16—182no—
116C2×C3×C8—34no—
117C2×C3×C8—34no—
118C2×C3×C8—91no—
119C2×C3×C8—91no—
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—34no—
127C3×C42C2×C3⋊D311yescharacteristic
128(C2×D3)⋊C4—34no—
129(C2×D3)⋊C4—34no—
130C4⋊Dic3—34no—
131C2×C3⋊C8—34no—
132C2×C3⋊C8—34no—
133ΓL(1,𝔽25)—68no—
134D3×C8—68no—
135C22×C32C2×C4×D314yes—
136C2×C3×D3—38no—
137C2×C3×D3—64no—
138C2×C3×D3—64no—
139C2×C3×D3—64no—
140C2×C3×D3—64no—
141C2×C3×D3—64no—
142C32×C4—24no—
143C32×C4—24no—
144C32×C4C2×D1214yes—
145C32×C4#23514yes—
146C3×Dic3—64no—
147C3×Dic3—64no—
148C2×M16—91no—
149C22×C8—91no—
150C4×D4—91no—
151C4⋊C8—91no—
152C22⋊C8—91no—
153C22⋊C8—91no—
154#97#27211yescharacteristic, complemented
155H(3,𝔽3)#43511yescharacteristic, complemented, Sylow
156C23×C3—91no—
157C23×C3—91no—
158C22×D3—34no—
159C22×D3—34no—
160C3×D4—182no—
161C2×C3×C4—34no—
162C2×C3×C4—34no—
163C2×C3×C4—34no—
164C2×C3×C4—91no—
165C2×C3×C4—91no—
166C2×C3×C4—181no—
167C2×C3×C4—181no—
168C2×C3×C4—181no—
169C2×C3×C4—181no—
170C2×C3×C4C4×C3⋊D311yescharacteristic
171C2×C3×C4C4×C3⋊D311yescharacteristic
172C2×C3×C4C22×C3⋊D311yescharacteristic
173C2×Dic3—34no—
174C2×Dic3—34no—
175D12—68no—
176C4×D3—64no—
177C4×D3—64no—
178C4×D3—64no—
179C4×D3—64no—
180C3×C8—21no—
181C3×C8—38no—
182C3×C8—64no—
183C3×C8—181no—
184C3×C8—181no—
185C3×C8C4⋊(C3⋊D3)12yes—
186C3⋊C8—64no—
187C3⋊C8—64no—
188C2×C32C4×D1214yes—
189C2×C32D3×M1614yes—
190C2×C32#90714yes—
191C3×D3—68no—
192C3×D3—124no—
193C2×D4—91no—
194C22×C4—91no—
195C22×C4—91no—
196M16—182no—
197C2×C8—91no—
198C2×C8—91no—
199C2×C8—182no—
200C2×C8C2×#27211yescharacteristic
201C2×C8C2×#27211yescharacteristic
202C4⋊C4—91no—
203C22⋊C4—91no—
204C22⋊C4—91no—
205C42C2×#27211yescharacteristic
206C22×C3—34no—
207C22×C3—92no—
208C22×C3—181no—
209C22×C3—181no—
210C22×C3—181no—
211C22×C3—181no—
212C22×C3—181no—
213C22×C3C2×C4×C3⋊D311yescharacteristic
214C2×D3—38no—
215C2×D3—64no—
216C2×D3—64no—
217C2×D3—64no—
218C2×D3—64no—
219C2×D3—64no—
220C3×C4—21no—
221C3×C4—21no—
222C3×C4—34no—
223C3×C4—34no—
224C3×C4—64no—
225C3×C4—64no—
226C3×C4—181no—
227C3×C4—181no—
228C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
229C3×C4#382811yescharacteristic
230Dic3—64no—
231Dic3—64no—
232C32#470614yescomplemented
233C23—91no—
234C23—91no—
235D4—182no—
236C2×C4—91no—
237C2×C4—91no—
238C2×C4—181no—
239C2×C4—181no—
240C2×C4—181no—
241C2×C4—181no—
242C2×C4C4×#27211yescharacteristic
243C2×C4C4×#27211yescharacteristic
244C2×C4C22×#27211yescharacteristic
245C8—21no—
246C8—181no—
247C8—181no—
248C8C4⋊#27212yescomplemented
249C2×C3—34no—
250C2×C3—34no—
251C2×C3—34no—
252C2×C3—182no—
253C2×C3—361no—
254C2×C3C4×C4⋊(C3⋊D3)11yescharacteristic
255C2×C3M16×C3⋊D311yescharacteristic
256C2×C3#6411611yescharacteristic
257D3—68no—
258D3—124no—
259C22—92no—
260C22—181no—
261C22—181no—
262C22—181no—
263C22—181no—
264C22—181no—
265C22C2×C4×#27211yescharacteristic
266C4—21no—
267C4—21no—
268C4—181no—
269C4—181no—
270C4C2×C4⋊#27211yescharacteristic
271C4#8874011yescharacteristic
272C3—34no—
273C3#1059240511yescharacteristic
274C2—182no—
275C2—361no—
276C2C4×C4⋊#27211yescharacteristic
277C2M16×#27211yescharacteristic
278C2#1172178611yescharacteristic
subgroupG/H
center, FrattiniC2×C3×C4C22×C3⋊D3
commutatorC22×H(3,𝔽3)C22×C4
FittingH(3,𝔽3)×#97C2
socleC22×C3C2×C4×C3⋊D3
prime radicalC2×C4⋊#272C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC3×#97C3⋊D3
hypocenterH(3,𝔽3)#435
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3×C4 ◁ C3×#97

Complements

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

NH ≅ G/Ntype
1. C2×#88527C2semidirect
2. C2×C8×#272C2semidirect
4. H(3,𝔽3)×#97C2semidirect
5. #11720553C2semidirect
6. #11720553C2semidirect
7. C4⋊#6180C2semidirect
25. C32×#97D3semidirect
42. C8×H(3,𝔽3)D4semidirect
66. C2×#272M16semidirect
100. C32×C8D12semidirect
154. #97#272semidirect
155. H(3,𝔽3)#435semidirect
232. C32#4706semidirect
248. C8C4⋊#272semidirect

Maximal subgroups

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

subgroupindex
C2×#885272
C2×C8×#2722
C4×C4⋊#2722
H(3,𝔽3)×#972
#117205532
C4⋊#61802
C3×#47063