Finite Group Databasebeta

← #13144729

Subgroups — #13144729

Order 1728.

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

normal subgroups only

This group has 792 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
2#11721864C211yesmaximal, characteristic, complemented
3C2×#88549C211yesmaximal, characteristic, complemented
4#11721850C211yesmaximal, characteristic, complemented
5C2×#88545C211yesmaximal, characteristic, complemented
6C2×#88529C211yesmaximal, characteristic, complemented
7#11720548C211yesmaximal, characteristic, complemented
8C3×#5176—34nomaximal
9C2×Q8×H(3,𝔽3)C2211yescharacteristic
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×#6235—22no—
12C2×C4⋊#272C2211yescharacteristic
13C2×C4⋊#272C2211yescharacteristic
14SD16×H(3,𝔽3)—22no—
15C2×C8×H(3,𝔽3)C2211yescharacteristic
16#88549—22no—
17#88545—22no—
18#88542—22no—
19C4×#1084C2211yescharacteristic
20C2×#6180C2211yescharacteristic
21#88529—22no—
22C2×C32×SD16D314yescomplemented
23C3×#947—34no—
24C2×C3×#215—34no—
25C3×#940—34no—
26C2×C3×#213—34no—
27C2×C3×D24—34no—
28C3×#815—34no—
29C23×H(3,𝔽3)—21no—
30C22×#272—21no—
31C22×#272—21no—
32Q8×H(3,𝔽3)—21no—
33Q8×H(3,𝔽3)—21no—
34D4×H(3,𝔽3)—21no—
35D4×H(3,𝔽3)—21no—
36C2×C4×H(3,𝔽3)—21no—
37C2×C4×H(3,𝔽3)C2311yescharacteristic
38#6235—44no—
39C2×#1084D412yes—
40C4⋊#272—21no—
41C4⋊#272—21no—
42C4⋊#272—21no—
43C4⋊#272—21no—
44C8×H(3,𝔽3)D412yescomplemented
45#6180D412yescomplemented
46C3×#496—91no—
47#5176—34no—
48C2×C32×Q8C2×D314yes—
49C2×C32×D4C2×D314yes—
50C2×C3×#67—68no—
51C2×C3×D12—34no—
52C2×C3×D12—34no—
53C32×SD16—28no—
54C2×C32×C8C2×D314yes—
55C3×#215—68no—
56C3×#213—68no—
57C3×(C2×D3)⋊C4—68no—
58C3×C4×Dic3—34no—
59C2×C3×C3⋊C8—34no—
60C3×D24—68no—
61C22×H(3,𝔽3)—41no—
62C22×H(3,𝔽3)—42no—
63C22×H(3,𝔽3)C2×D411yescharacteristic
64C2×#272—41no—
65C2×#272—41no—
66C2×#272—42no—
67C2×#272—42no—
68C4×H(3,𝔽3)—41no—
69C4×H(3,𝔽3)C2×D411yescharacteristic
70C4×H(3,𝔽3)C2×D411yescharacteristic
71#1084—22no—
72C2×C3×SD16—34no—
73C2×C3×SD16—91no—
74C2×C3×SD16C3⋊D311yescharacteristic
75C2×C3×D8—91no—
76C2×C3×D8—91no—
77C3×#127—91no—
78C3×#124—91no—
79#947—34no—
80C2×#215—34no—
81#940—34no—
82C2×#213—34no—
83C2×D24—34no—
84C3×#97—91no—
85#815—34no—
86C23×C32—24no—
87C22×C3×D3—64no—
88C22×C3×D3—64no—
89C32×Q8—24no—
90C32×Q8—24no—
91C32×D4—24no—
92C32×D4—24no—
93C2×C32×C4—24no—
94C2×C32×C4C22×D314yes—
95C3×#67—1216no—
96C2×C3×Dic3—38no—
97C3×D12—64no—
98C3×D12—64no—
99C3×D12—64no—
100C3×D12—64no—
101C32×C8#6718yescomplemented
102C3×C3⋊C8—38no—
103#496—91noSylow
104C2×H(3,𝔽3)—81no—
105C2×H(3,𝔽3)#12711yescharacteristic
106C2×H(3,𝔽3)AGL(1,ℤ/8)12yes—
107#272—81no—
108#272—81no—
109C2×C3×Q8—34no—
110C2×C3×Q8C2×C3⋊D311yescharacteristic
111C2×C3×D4—34no—
112C2×C3×D4—91no—
113C2×C3×D4—91no—
114C2×C3×D4—182no—
115C2×C3×D4C2×C3⋊D311yescharacteristic
116C2×#67—68no—
117C2×D12—34no—
118C2×D12—34no—
119C3×SD16—22no—
120C3×SD16—68no—
121C3×SD16—182no—
122C3×D8—182no—
123C3×D8—182no—
124C2×C3×C8—34no—
125C2×C3×C8—91no—
126C2×C3×C8C2×C3⋊D311yescharacteristic
127C3×C22⋊C4—182no—
128C3×C42—91no—
129#215—68no—
130#213—68no—
131(C2×D3)⋊C4—68no—
132C4×Dic3—34no—
133C2×C3⋊C8—34no—
134D24—68no—
135C22×C32—48no—
136C22×C32—44no—
137C22×C32D3×D414yes—
138C2×C3×D3—128no—
139C2×C3×D3—128no—
140C2×C3×D3—124no—
141C2×C3×D3—124no—
142C32×C4—44no—
143C32×C4D3×D414yes—
144C32×C4C2×#6714yes—
145C3×Dic3—68no—
146C2×SD16—91no—
147C2×SD16#27211yescharacteristic, complemented
148C2×D8—91no—
149C2×D8—91no—
150#127—91no—
151#124—91no—
152#97—91no—
153H(3,𝔽3)#49611yescharacteristic, complemented, Sylow
154C23×C3—21no—
155C23×C3—64no—
156C23×C3—181no—
157C23×C3—181no—
158C22×D3—64no—
159C22×D3—64no—
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—181no—
169C3×D4—181no—
170C3×D4—181no—
171C3×D4—181no—
172C3×D4—364no—
173C2×C3×C4—21no—
174C2×C3×C4—34no—
175C2×C3×C4—64no—
176C2×C3×C4—92no—
177C2×C3×C4C22×C3⋊D311yescharacteristic
178#67—1216no—
179C2×Dic3—38no—
180D12—64no—
181D12—64no—
182D12—64no—
183D12—64no—
184C3×C8—38no—
185C3×C8—92no—
186C3×C8#63812yes—
187C3⋊C8—38no—
188C2×C32—84no—
189C2×C32#91418yes—
190C2×C32#94014yes—
191C3×D3—244no—
192C3×D3—244no—
193C2×Q8C2×#27211yescharacteristic
194C2×D4—91no—
195C2×D4—91no—
196C2×D4—182no—
197C2×D4C2×#27211yescharacteristic
198SD16—22no—
199SD16—182no—
200D8—182no—
201D8—182no—
202C2×C8—91no—
203C2×C8C2×#27211yescharacteristic
204C22⋊C4—182no—
205C42—91no—
206C22×C3—34no—
207C22×C3—42no—
208C22×C3—41no—
209C22×C3—128no—
210C22×C3—124no—
211C22×C3—362no—
212C22×C3—362no—
213C22×C3—361no—
214C22×C3—361no—
215C22×C3D4×C3⋊D311yescharacteristic
216C2×D3—124no—
217C2×D3—124no—
218C2×D3—128no—
219C2×D3—128no—
220C3×C4—34no—
221C3×C4—34no—
222C3×C4—41no—
223C3×C4—124no—
224C3×C4—182no—
225C3×C4D4×C3⋊D311yescharacteristic
226C3×C4C2×#63811yescharacteristic
227Dic3—68no—
228C32#517614yescomplemented
229C23—21no—
230C23—181no—
231C23—181no—
232Q8—21no—
233Q8—21no—
234D4—21no—
235D4—21no—
236D4—181no—
237D4—181no—
238D4—181no—
239D4—181no—
240D4—364no—
241C2×C4—21no—
242C2×C4—92no—
243C2×C4C22×#27211yescharacteristic
244C8—92no—
245C8#623512yescomplemented
246C2×C3—38no—
247C2×C3—34no—
248C2×C3—81no—
249C2×C3—244no—
250C2×C3—721no—
251C2×C3—721no—
252C2×C3#6412312yes—
253C2×C3#6414911yescharacteristic
254D3—244no—
255D3—244no—
256C22—42no—
257C22—41no—
258C22—362no—
259C22—362no—
260C22—361no—
261C22—361no—
262C22D4×#27211yescharacteristic
263C4—41no—
264C4—182no—
265C4D4×#27211yescharacteristic
266C4C2×#623511yescharacteristic
267C3—34no—
268C3#1059287511yescharacteristic
269C2—81no—
270C2—721no—
271C2—721no—
272C2#1172180012yes—
273C2#1172185011yescharacteristic
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
hypercenterC2×C3×SD16C3⋊D3
hypocenterH(3,𝔽3)#496
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. #11721864C2semidirect
3. C2×#88549C2semidirect
4. #11721850C2semidirect
5. C2×#88545C2semidirect
6. C2×#88529C2semidirect
7. #11720548C2semidirect
22. C2×C32×SD16D3semidirect
44. C8×H(3,𝔽3)D4semidirect
45. #6180D4semidirect
101. C32×C8#67semidirect
147. C2×SD16#272semidirect
153. H(3,𝔽3)#496semidirect
228. C32#5176semidirect
245. 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
#117218642
C2×#885492
#117218502
C2×#885452
C2×#885292
#117205482
C3×#51763