Finite Group Databasebeta

← #13139278

Subgroups — #13139278

Order 1728.

377 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 377 conjugacy classes of subgroups — too many to draw. Only its 49 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
1#11721808C211yesmaximal, characteristic, complemented
2#11720838C211yesmaximal, characteristic, complemented
3#63655⋊C3C212yesmaximal, complemented
4#11720253C212yesmaximal, complemented
5#11720252C211yesmaximal, characteristic, complemented
6#10591038—32nomaximal
7#88748—21no—
8#88655—21no—
9Q16×H(3,𝔽3)C2211yescharacteristic, complemented
10#88549—21no—
11#88529C2211yescharacteristic, complemented
12C8×#272C2211yescharacteristic
13#3745⋊C3C2212yes—
14#88432—21no—
15(C3⋊(C3⋊C16))⋊C3C2212yescomplemented
16C3×#919—32no—
17C3×#919—61no—
18D3×D24—32no—
19#63655D312yescomplemented
20C3×#828—32no—
21#63546—32no—
22#63545—32no—
23#63543—32no—
24C2×((C3⋊D3)⋊C3)⋊C2—41no—
25Q8×H(3,𝔽3)—21no—
26C4⋊#272—21no—
27C4⋊#272—41no—
28C4×#272—41no—
29C4×#272D411yescharacteristic
30C4⋊((C3⋊D3)⋊C3)—22no—
31#6198—41no—
32C8×H(3,𝔽3)C2311yescharacteristic
33#6180D411yescharacteristic
34#5193—91no—
35#4914—92no—
36C3×#239—62no—
37C3×#239—62no—
38D3×D12—62no—
39C32×Q16—21no—
40C32×Q16C2×D312yescomplemented
41#3745C2×D312yes—
42C3×#215—62no—
43C3×#215—62no—
44C3×D24—32no—
45C3×D24—32no—
46C3×D24—61no—
47C3×D3×C8—32no—
48C3×D3×C8—61no—
49#3714—62no—
50C3⋊(C3⋊C16)C2×D312yescomplemented
51C3×C3⋊C16—32no—
52C2×#272—41no—
53C2×#272D811yescharacteristic
54C2×(C3⋊D3)⋊C3—42no—
55((C3⋊D3)⋊C3)⋊C2—82no—
56C4×H(3,𝔽3)—41no—
57C4×H(3,𝔽3)C2×D411yescharacteristic
58#1084D811yescharacteristic
59C3×#135—91no—
60#919—32no—
61#919—61no—
62D3×D8—92no—
63C2×D24—91no—
64C3×SD32—92no—
65#828—32no—
66#828—92no—
67#826—92no—
68#826—92no—
69#812—91no—
70D48—92no—
71#798—92no—
72C2×D32—122no—
73C32×Q8—22no—
74C32×Q8—22no—
75C4⋊(C3⋊D3)—22no—
76C3×D12—62no—
77C3×D12—62no—
78C3×D12—62no—
79C3×D12—122no—
80C3×D12—122no—
81C3×C4×D3—32no—
82C3×C4×D3—61no—
83C3×C4×D3—122no—
84C3×C4×D3—122no—
85#626—122no—
86C32×C8—21no—
87C32×C8C22×D312yes—
88C3×C3⋊C8—32no—
89C3×C3⋊C8—61no—
90#509—271noSylow
91C2×H(3,𝔽3)C2×D811yescharacteristic
92#272—21no—
93#272—81no—
94(C3⋊D3)⋊C3—82no—
95C3×#41—181no—
96#239—62no—
97#239—62no—
98D3×D4—182no—
99C2×D12—181no—
100C3×Q16—32no—
101C3×Q16—61no—
102C3×Q16D3211yescharacteristic
103C3×SD16—181no—
104C3×D8—92no—
105C3×D8—91no—
106C2×C3×C8—91no—
107#215—62no—
108#215—62no—
109#213—182no—
110D24—32no—
111D24—32no—
112D24—61no—
113D24—92no—
114D24—181no—
115D3×C8—32no—
116D3×C8—61no—
117C3×C16—92no—
118C3⋊C16—32no—
119C3⋊C16—92no—
120C2×C3⋊D3—42no—
121C2×C3×D3—32no—
122C2×C3×D3—61no—
123C2×C3×D3—122no—
124C2×C3×D3—122no—
125C2×C3×D3—122no—
126D32—244no—
127C32×C4—21no—
128C32×C4—42no—
129C32×C4—42no—
130C32×C4D3×D412yes—
131C3×Dic3—32no—
132C3×Dic3—61no—
133#135—91no—
134C2×D8—271no—
135SD32—92no—
136D16—272no—
137M32—271no—
138H(3,𝔽3)#50911yescharacteristic, complemented, Sylow
139C22×D3—362no—
140C22×D3—361no—
141C3×Q8—21no—
142C3×Q8—62no—
143C3×Q8—62no—
144C3×D4—182no—
145C3×D4—181no—
146C3×D4—361no—
147C2×C3×C4—91no—
148C2×C3×C4—361no—
149#67—362no—
150D12—62no—
151D12—62no—
152D12—62no—
153D12—122no—
154D12—122no—
155D12—182no—
156D12—361no—
157C4×D3—32no—
158C4×D3—61no—
159C4×D3—122no—
160C4×D3—122no—
161C3×C8—32no—
162C3×C8—61no—
163C3×C8—91no—
164C3×C8C2×D3211yescharacteristic
165C3⋊C8—32no—
166C3⋊C8—61no—
167C2×C32—21no—
168C2×C32D3×D812yes—
169C3⋊D3—82no—
170C3×D3—62no—
171C3×D3—121no—
172C3×D3—242no—
173C3×D3—242no—
174C3×D3—242no—
175#41—181no—
176C2×D4—541no—
177Q16((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
178SD16—181no—
179D8—91no—
180D8—92no—
181D8—541no—
182C2×C8—91no—
183C16—92no—
184C22×C3—91no—
185C22×C3—362no—
186C22×C3—361no—
187C2×D3—32no—
188C2×D3—61no—
189C2×D3—122no—
190C2×D3—122no—
191C2×D3—122no—
192C2×D3—362no—
193C2×D3—724no—
194C2×D3—722no—
195C3×C4—32no—
196C3×C4—41no—
197C3×C4—61no—
198C3×C4—91no—
199C3×C4—122no—
200C3×C4—122no—
201C3×C4#381111yescharacteristic
202Dic3—32no—
203Dic3—61no—
204C32—21no—
205C32#491412yescomplemented
206C23—1081no—
207Q8—21no—
208D4—182no—
209D4—181no—
210D4—361no—
211D4—1081no—
212C2×C4—91no—
213C2×C4—361no—
214C8—91no—
215C8C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
216C2×C3—32no—
217C2×C3—61no—
218C2×C3—181no—
219C2×C3—722no—
220C2×C3—721no—
221C2×C3#6392611yescharacteristic
222D3—62no—
223D3—121no—
224D3—242no—
225D3—242no—
226D3—242no—
227D3—722no—
228C22—91no—
229C22—361no—
230C22—362no—
231C22—2162no—
232C4—41no—
233C4—91no—
234C4#8867711yescharacteristic
235C3—32no—
236C3—61no—
237C3#1059104011yescharacteristic
238C2—181no—
239C2—722no—
240C2—721no—
241C2#1172119011yescharacteristic
subgroupG/H
centerC2#11721190
commutator, cosocleC8×H(3,𝔽3)C23
FrattiniC3×C8C2×D32
FittingQ16×H(3,𝔽3)C22
socleC2×C3#63926
hypercenterQ16((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#509
LCSG ▷ C8×H(3,𝔽3) ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C4 ◁ C8 ◁ Q16

Complements

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

NH ≅ G/Ntype
1. #11721808C2semidirect
2. #11720838C2semidirect
3. #63655⋊C3C2semidirect
4. #11720253C2semidirect
5. #11720252C2semidirect
9. Q16×H(3,𝔽3)C22semidirect
11. #88529C22semidirect
15. (C3⋊(C3⋊C16))⋊C3C22semidirect
19. #63655D3semidirect
40. C32×Q16C2×D3semidirect
50. C3⋊(C3⋊C16)C2×D3semidirect
138. H(3,𝔽3)#509semidirect
177. Q16((C3⋊D3)⋊C3)⋊C2semidirect
205. C32#4914semidirect

Maximal subgroups

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

subgroupindex
#117218082
#117208382
#63655⋊C32
#117202532
#117202522
#105910383