Finite Group Databasebeta

← #13144737

Subgroups — #13144737

Order 1728.

660 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 111 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
2#11721863C211yesmaximal, characteristic, complemented
3#11721848C211yesmaximal, characteristic
4C2×#88527C211yesmaximal, characteristic, complemented
5#11720583C211yesmaximal, characteristic, complemented
6#11720578C211yesmaximal, characteristic, complemented
7#11720551C211yesmaximal, characteristic, complemented
8C3×#5174—34nomaximal
9C2×Q8×H(3,𝔽3)C2211yescharacteristic
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×#6235—21no—
12C2×C4×#272C2211yescharacteristic
13SD16×H(3,𝔽3)—22no—
14C2×C8×H(3,𝔽3)C2211yescharacteristic
15#88552—21no—
16#88542—21no—
17C4⋊#1084C2211yescharacteristic
18C4⋊#1084C2211yescharacteristic
19#88540—21no—
20C2×#6180C2211yescharacteristic
21#88527—22no—
22C2×C32×SD16D314yescomplemented
23C3×#946—34no—
24C3×#938—34no—
25C2×C3×ΓL(1,𝔽25)—34no—
26C3×#835—34no—
27C3×#832—34no—
28C3×#818—34no—
29C23×H(3,𝔽3)—21no—
30C22×#272D411yescharacteristic
31Q8×H(3,𝔽3)—21no—
32Q8×H(3,𝔽3)—21no—
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—
38C2×#1084—21no—
39C2×#1084—21no—
40C2×#1084D411yescharacteristic
41C4×#272—21no—
42C4×#272—21no—
43C8×H(3,𝔽3)D412yescomplemented
44#6180—21no—
45C3×#468—91no—
46#5174—34no—
47C2×C32×Q8C2×D314yes—
48C2×C32×D4C2×D314yes—
49C2×C3×#67—64no—
50C2×C3×C4×D3—34no—
51C32×SD16—28no—
52C2×C32×C8C2×D314yes—
53C3×#217—64no—
54C3×(C2×D3)⋊C4—64no—
55C3×C4⋊Dic3—34no—
56C3×C4⋊Dic3—34no—
57C3×Dic3⋊C4—64no—
58C2×C3×C3⋊C8—34no—
59C3×ΓL(1,𝔽25)—68no—
60C22×H(3,𝔽3)—41no—
61C22×H(3,𝔽3)—41no—
62C22×H(3,𝔽3)—41no—
63C22×H(3,𝔽3)C2×D411yescharacteristic
64C2×#272—21no—
65C2×#272—21no—
66C2×#272—21no—
67C4×H(3,𝔽3)—41no—
68C4×H(3,𝔽3)C2×D411yescharacteristic
69C4×H(3,𝔽3)#4111yescharacteristic
70#1084—21no—
71#1084—41no—
72#1084—41no—
73C2×C3×SD16—34no—
74C2×C3×SD16C3⋊D311yescharacteristic
75C2×C3×M16—91no—
76C3×C22⋊Q8—91no—
77C3×#121—91no—
78#946—34no—
79#938—34no—
80C2×ΓL(1,𝔽25)—34no—
81C3×#107—91no—
82C3×#103—91no—
83C3×#102—91no—
84#835—34no—
85#832—34no—
86#818—34no—
87C23×C32—24no—
88C22×C3×D3—34no—
89C32×Q8—24no—
90C32×Q8—24no—
91C32×D4—24no—
92C32×D4—24no—
93C2×C32×C4—24no—
94C2×C32×C4C22×D314yes—
95C3×#67—128no—
96C2×C3×Dic3—34no—
97C2×C3×Dic3—64no—
98C2×C3×Dic3—64no—
99C3×C4×D3—64no—
100C3×C4×D3—64no—
101C32×C8#6718yescomplemented
102C3×C3⋊C8—64no—
103#468—91noSylow
104C2×H(3,𝔽3)—81no—
105C2×H(3,𝔽3)#12111yescharacteristic
106C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
107C2×H(3,𝔽3)#13711yescharacteristic
108#272—41no—
109C2×C3×Q8—34no—
110C2×C3×Q8C2×C3⋊D311yescharacteristic
111C2×C3×D4—34no—
112C2×C3×D4—181no—
113C2×C3×D4C2×C3⋊D311yescharacteristic
114C22×C3×C4—91no—
115C2×#67—64no—
116C2×C4×D3—34no—
117C3×SD16—22no—
118C3×SD16—68no—
119C3×M16—182no—
120C2×C3×C8—34no—
121C2×C3×C8—91no—
122C2×C3×C8C2×C3⋊D311yescharacteristic
123C3×C4⋊C4—91no—
124C3×C4⋊C4—91no—
125C3×C4⋊C4—181no—
126C3×C22⋊C4—181no—
127C3×C22⋊C4—181no—
128#217—64no—
129(C2×D3)⋊C4—64no—
130C4⋊Dic3—34no—
131C4⋊Dic3—34no—
132Dic3⋊C4—64no—
133C2×C3⋊C8—34no—
134ΓL(1,𝔽25)—68no—
135C22×C32—44no—
136C22×C32—44no—
137C22×C32—44no—
138C22×C32D3×D414yes—
139C2×C3×D3—64no—
140C2×C3×D3—64no—
141C2×C3×D3—64no—
142C32×C4—44no—
143C32×C4#23714yes—
144C32×C4C2×#6714yes—
145C3×Dic3—64no—
146C3×Dic3—124no—
147C3×Dic3—124no—
148C2×SD16#27211yescharacteristic, complemented
149C2×M16—91no—
150C22⋊Q8—91no—
151#121—91no—
152#107—91no—
153#103—91no—
154#102—91no—
155H(3,𝔽3)#46811yescharacteristic, complemented, Sylow
156C23×C3—21no—
157C23×C3—64no—
158C23×C3—91no—
159C22×D3—34no—
160C3×Q8—21no—
161C3×Q8—21no—
162C3×Q8—64no—
163C3×Q8—64no—
164C3×D4—21no—
165C3×D4—21no—
166C3×D4—64no—
167C3×D4—64no—
168C3×D4—362no—
169C2×C3×C4—21no—
170C2×C3×C4—34no—
171C2×C3×C4—64no—
172C2×C3×C4—91no—
173C2×C3×C4—181no—
174C2×C3×C4—181no—
175C2×C3×C4—181no—
176C2×C3×C4—181no—
177C2×C3×C4C22×C3⋊D311yescharacteristic
178#67—128no—
179C2×Dic3—34no—
180C2×Dic3—64no—
181C2×Dic3—64no—
182C4×D3—64no—
183C4×D3—64no—
184C3×C8—38no—
185C3×C8—181no—
186C3×C8#63812yes—
187C3⋊C8—64no—
188C2×C32—84no—
189C2×C32#91414yes—
190C2×C32#91514yes—
191C2×C32#93814yes—
192C3×D3—124no—
193C2×Q8C2×#27211yescharacteristic
194C2×D4—181no—
195C2×D4C2×#27211yescharacteristic
196C22×C4—91no—
197SD16—22no—
198M16—182no—
199C2×C8—91no—
200C2×C8C2×#27211yescharacteristic
201C4⋊C4—91no—
202C4⋊C4—91no—
203C4⋊C4—181no—
204C22⋊C4—181no—
205C22⋊C4—181no—
206C22×C3—34no—
207C22×C3—41no—
208C22×C3—41no—
209C22×C3—41no—
210C22×C3—124no—
211C22×C3—124no—
212C22×C3—124no—
213C22×C3—181no—
214C22×C3—181no—
215C22×C3—181no—
216C22×C3D4×C3⋊D311yescharacteristic
217C2×D3—64no—
218C2×D3—64no—
219C2×D3—64no—
220C3×C4—34no—
221C3×C4—34no—
222C3×C4—41no—
223C3×C4—124no—
224C3×C4—181no—
225C3×C4—361no—
226C3×C4—361no—
227C3×C4#383011yescharacteristic
228C3×C4C2×#63811yescharacteristic
229Dic3—64no—
230Dic3—124no—
231Dic3—124no—
232C32#517414yescomplemented
233C23—21no—
234C23—91no—
235Q8—21no—
236Q8—21no—
237D4—21no—
238D4—21no—
239D4—362no—
240C2×C4—21no—
241C2×C4—91no—
242C2×C4—181no—
243C2×C4—181no—
244C2×C4—181no—
245C2×C4—181no—
246C2×C4C22×#27211yescharacteristic
247C8—181no—
248C8#623512yescomplemented
249C2×C3—34no—
250C2×C3—34no—
251C2×C3—34no—
252C2×C3—81no—
253C2×C3—244no—
254C2×C3—361no—
255C2×C3#6412311yescharacteristic
256C2×C3#6412411yescharacteristic
257C2×C3#6414711yescharacteristic
258D3—124no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—181no—
263C22—181no—
264C22—181no—
265C22D4×#27211yescharacteristic
266C4—41no—
267C4—181no—
268C4—361no—
269C4—361no—
270C4#8874411yescharacteristic
271C4C2×#623511yescharacteristic
272C3—34no—
273C3#1059287311yescharacteristic
274C2—81no—
275C2—361no—
276C2#1172180011yescharacteristic
277C2#1172180111yescharacteristic
278C2#1172184811yescharacteristic
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#11721848C2
hypercenterC2×C3×SD16C3⋊D3
hypocenterH(3,𝔽3)#468
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. #11721863C2semidirect
4. C2×#88527C2semidirect
5. #11720583C2semidirect
6. #11720578C2semidirect
7. #11720551C2semidirect
22. C2×C32×SD16D3semidirect
43. C8×H(3,𝔽3)D4semidirect
101. C32×C8#67semidirect
148. C2×SD16#272semidirect
155. H(3,𝔽3)#468semidirect
232. C32#5174semidirect
248. C8#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
#117218632
#117218482
C2×#885272
#117205832
#117205782
#117205512
C3×#51743