Finite Group Databasebeta

← #13144725

Subgroups — #13144725

Order 1728.

704 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 113 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×SD16×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2Q8×#1084C211yesmaximal, characteristic, complemented
3C2×#88549C211yesmaximal, characteristic, complemented
4#11721850C211yesmaximal, characteristic
5#11720578C211yesmaximal, characteristic, complemented
6#11720554C211yesmaximal, characteristic, complemented
7#11720547C211yesmaximal, characteristic, complemented
8C3×#5163—34nomaximal
9C2×Q8×H(3,𝔽3)C2211yescharacteristic
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×#6235—21no—
12C2×#6235—21no—
13C2×C4⋊#272C2211yescharacteristic
14SD16×H(3,𝔽3)—22no—
15C2×C8×H(3,𝔽3)C2211yescharacteristic
16#88549—22no—
17C4⋊#1084—21no—
18C4⋊#1084C2211yescharacteristic
19C4×#1084—21no—
20C4×#1084C2211yescharacteristic
21C2×#6180C2211yescharacteristic
22C2×C32×SD16D314yescomplemented
23C3×Q8×Dic3—34no—
24C2×C3×#215—34no—
25C3×#940—34no—
26C3×#832—34no—
27C3×#821—34no—
28C3×Dic3⋊C8—34no—
29C23×H(3,𝔽3)—21no—
30C22×#272—21no—
31Q8×H(3,𝔽3)—21no—
32Q8×H(3,𝔽3)D412yescomplemented
33D4×H(3,𝔽3)—21no—
34D4×H(3,𝔽3)—21no—
35C2×C4×H(3,𝔽3)—21no—
36C2×C4×H(3,𝔽3)C2311yescharacteristic
37#6235—42no—
38#6235—42no—
39C2×#1084—21no—
40C2×#1084D411yescharacteristic
41C2×#1084D411yescharacteristic
42C4⋊#272—21no—
43C4⋊#272—21no—
44C8×H(3,𝔽3)—21no—
45#6180—21no—
46C3×#460—91no—
47#5163—34no—
48C2×C32×Q8C2×D314yes—
49C2×C32×D4C2×D314yes—
50C2×C3×#67—64no—
51C2×C3×#67—64no—
52C2×C3×D12—34no—
53C32×SD16—28no—
54C2×C32×C8C2×D314yes—
55C3×#215—68no—
56C3×C4⋊Dic3—34no—
57C3×C4⋊Dic3—64no—
58C3×C4×Dic3—34no—
59C3×C4×Dic3—64no—
60C2×C3×C3⋊C8—34no—
61C22×H(3,𝔽3)—41no—
62C22×H(3,𝔽3)—41no—
63C22×H(3,𝔽3)—41no—
64C22×H(3,𝔽3)C2×D411yescharacteristic
65C2×#272—41no—
66C2×#272—41no—
67C2×#272—41no—
68C4×H(3,𝔽3)—22no—
69C4×H(3,𝔽3)C2×D411yescharacteristic
70C4×H(3,𝔽3)#4111yescharacteristic
71#1084—21no—
72#1084—41no—
73#1084SD1612yescomplemented
74C2×C3×SD16—34no—
75C2×C3×SD16—91no—
76C2×C3×SD16C3⋊D311yescharacteristic
77C3×#127—91no—
78C3×C4×Q8—91no—
79Q8×Dic3—34no—
80C2×#215—34no—
81#940—34no—
82C3×C4⋊C8—91no—
83C3×#102—91no—
84C3×#102—91no—
85#832—34no—
86#821—34no—
87Dic3⋊C8—34no—
88C23×C32—24no—
89C22×C3×D3—64no—
90C32×Q8—24no—
91C32×Q8#6718yescomplemented
92C32×D4—24no—
93C32×D4—24no—
94C2×C32×C4—24no—
95C2×C32×C4C22×D314yes—
96C3×#67—128no—
97C3×#67—128no—
98C2×C3×Dic3—34no—
99C2×C3×Dic3—34no—
100C2×C3×Dic3—64no—
101C3×D12—64no—
102C3×D12—64no—
103C32×C8—24no—
104C3×C3⋊C8—64no—
105#460—91noSylow
106C2×H(3,𝔽3)—81no—
107C2×H(3,𝔽3)#12111yescharacteristic
108C2×H(3,𝔽3)C2×SD1611yescharacteristic
109C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
110#272—81no—
111C2×C3×Q8—34no—
112C2×C3×Q8C2×C3⋊D311yescharacteristic
113C2×C3×D4—34no—
114C2×C3×D4—91no—
115C2×C3×D4—181no—
116C2×C3×D4—181no—
117C2×C3×D4C2×C3⋊D311yescharacteristic
118C2×#67—64no—
119C2×#67—64no—
120C2×D12—34no—
121C3×SD16—22no—
122C3×SD16—68no—
123C3×SD16—182no—
124C2×C3×C8—34no—
125C2×C3×C8—91no—
126C2×C3×C8C2×C3⋊D311yescharacteristic
127C3×C4⋊C4—91no—
128C3×C4⋊C4—181no—
129C3×C42—91no—
130C3×C42—181no—
131#215—68no—
132C4⋊Dic3—34no—
133C4⋊Dic3—64no—
134C4×Dic3—34no—
135C4×Dic3—64no—
136C2×C3⋊C8—34no—
137C22×C32—44no—
138C22×C32—44no—
139C22×C32—44no—
140C22×C32D3×D414yes—
141C2×C3×D3—124no—
142C2×C3×D3—124no—
143C2×C3×D3—124no—
144C32×C4—28no—
145C32×C4#23714yes—
146C32×C4C2×#6714yes—
147C3×Dic3—38no—
148C3×Dic3—64no—
149C3×Dic3—124no—
150C2×SD16—91no—
151C2×SD16#27211yescharacteristic, complemented
152#127—91no—
153C4×Q8—91no—
154C4⋊C8—91no—
155#102—91no—
156#102—91no—
157H(3,𝔽3)#46011yescharacteristic, complemented, Sylow
158C23×C3—21no—
159C23×C3—64no—
160C23×C3—181no—
161C22×D3—64no—
162C3×Q8—21no—
163C3×Q8—38no—
164C3×Q8—64no—
165C3×Q8#63812yes—
166C3×D4—21no—
167C3×D4—21no—
168C3×D4—64no—
169C3×D4—64no—
170C3×D4—181no—
171C3×D4—181no—
172C3×D4—362no—
173C3×D4—362no—
174C2×C3×C4—21no—
175C2×C3×C4—34no—
176C2×C3×C4—64no—
177C2×C3×C4—91no—
178C2×C3×C4—91no—
179C2×C3×C4—181no—
180C2×C3×C4C22×C3⋊D311yescharacteristic
181#67—128no—
182#67—128no—
183C2×Dic3—34no—
184C2×Dic3—34no—
185C2×Dic3—64no—
186D12—64no—
187D12—64no—
188C3×C8—21no—
189C3×C8—64no—
190C3×C8—181no—
191C3⋊C8—64no—
192C2×C32—84no—
193C2×C32D3×SD1614yes—
194C2×C32#91414yes—
195C2×C32#93914yes—
196C3×D3—244no—
197C2×Q8C2×#27211yescharacteristic
198C2×D4—91no—
199C2×D4—181no—
200C2×D4—181no—
201C2×D4C2×#27211yescharacteristic
202SD16—22no—
203SD16—182no—
204C2×C8—91no—
205C2×C8C2×#27211yescharacteristic
206C4⋊C4—91no—
207C4⋊C4—181no—
208C42—91no—
209C42—181no—
210C22×C3—34no—
211C22×C3—41no—
212C22×C3—41no—
213C22×C3—41no—
214C22×C3—124no—
215C22×C3—124no—
216C22×C3—124no—
217C22×C3—361no—
218C22×C3—361no—
219C22×C3—361no—
220C22×C3D4×C3⋊D311yescharacteristic
221C2×D3—124no—
222C2×D3—124no—
223C2×D3—124no—
224C3×C4—22no—
225C3×C4—34no—
226C3×C4—34no—
227C3×C4—68no—
228C3×C4—92no—
229C3×C4—181no—
230C3×C4—361no—
231C3×C4#383011yescharacteristic
232C3×C4C2×#63811yescharacteristic
233Dic3—38no—
234Dic3—64no—
235Dic3—124no—
236C32#516314yescomplemented
237C23—21no—
238C23—181no—
239Q8—21no—
240Q8#623512yescomplemented
241D4—21no—
242D4—21no—
243D4—181no—
244D4—181no—
245D4—362no—
246D4—362no—
247C2×C4—21no—
248C2×C4—91no—
249C2×C4—91no—
250C2×C4—181no—
251C2×C4C22×#27211yescharacteristic
252C8—21no—
253C8—181no—
254C2×C3—34no—
255C2×C3—34no—
256C2×C3—34no—
257C2×C3—81no—
258C2×C3—244no—
259C2×C3—721no—
260C2×C3SD16×C3⋊D311yescharacteristic
261C2×C3#6412311yescharacteristic
262C2×C3#6414811yescharacteristic
263D3—244no—
264C22—41no—
265C22—41no—
266C22—41no—
267C22—361no—
268C22—361no—
269C22—361no—
270C22D4×#27211yescharacteristic
271C4—22no—
272C4—92no—
273C4—181no—
274C4—361no—
275C4#8874411yescharacteristic
276C4C2×#623511yescharacteristic
277C3—34no—
278C3#1059286211yescharacteristic
279C2—81no—
280C2—721no—
281C2SD16×#27211yescharacteristic
282C2#1172180011yescharacteristic
283C2#1172184911yescharacteristic
subgroupG/H
center, socleC22×C3D4×C3⋊D3
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC2×SD16×H(3,𝔽3)C2
prime radical#11721850C2
hypercenterC2×C3×SD16C3⋊D3
hypocenterH(3,𝔽3)#460
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×SD16

Complements

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

NH ≅ G/Ntype
1. C2×SD16×H(3,𝔽3)C2semidirect
2. Q8×#1084C2semidirect
3. C2×#88549C2semidirect
5. #11720578C2semidirect
6. #11720554C2semidirect
7. #11720547C2semidirect
22. C2×C32×SD16D3semidirect
32. Q8×H(3,𝔽3)D4semidirect
73. #1084SD16semidirect
91. C32×Q8#67semidirect
151. C2×SD16#272semidirect
157. H(3,𝔽3)#460semidirect
236. C32#5163semidirect
240. Q8#6235semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×H(3,𝔽3)2
Q8×#10842
C2×#885492
#117218502
#117205782
#117205542
#117205472
C3×#51633