Finite Group Databasebeta

← #13139240

Subgroups — #13139240

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#11721794C211yesmaximal, characteristic, complemented
2#11720838C211yesmaximal, characteristic, complemented
3#63653⋊C3C212yesmaximal, complemented
4#11720255C212yesmaximal, complemented
5#11720252C211yesmaximal, characteristic, complemented
6#10591024—32nomaximal
7#88744—21no—
8#88655—21no—
9D8×H(3,𝔽3)C2211yescharacteristic, complemented
10#88546—21no—
11#88530C2211yescharacteristic, complemented
12C8×#272C2211yescharacteristic
13#3745⋊C3C2212yes—
14#88432—21no—
15(C3⋊(C3⋊C16))⋊C3C2212yescomplemented
16C3×#912—32no—
17C3×#912—61no—
18D3×D24—32no—
19#63653D312yescomplemented
20C3×#826—32no—
21#63549—32no—
22#63547—32no—
23#63543—32no—
24C2×((C3⋊D3)⋊C3)⋊C2—41no—
25D4×H(3,𝔽3)—21no—
26#6235—41no—
27C2×#1084—41no—
28C4×#272D411yescharacteristic
29#6229—21no—
30C4⋊((C3⋊D3)⋊C3)—22no—
31#6198—41no—
32C8×H(3,𝔽3)C2311yescharacteristic
33#6180D411yescharacteristic
34#5193—91no—
35#4911—92no—
36C3×#237—62no—
37C3×#237—62no—
38D3×D12—62no—
39C32×D8—21no—
40C32×D8C2×D312yescomplemented
41#3745C2×D312yes—
42C3×#214—62no—
43C3×#214—62no—
44C3×Dic12—32no—
45C3×Dic12—61no—
46C3×D24—32no—
47C3×D3×C8—32no—
48C3×D3×C8—61no—
49#3714—62no—
50C3⋊(C3⋊C16)C2×D312yescomplemented
51C3×C3⋊C16—32no—
52C22×H(3,𝔽3)—41no—
53C2×#272D811yescharacteristic
54C2×(C3⋊D3)⋊C3—42no—
55((C3⋊D3)⋊C3)⋊C2—82no—
56C4×H(3,𝔽3)C2×D411yescharacteristic
57#1084—41no—
58#1084D811yescharacteristic
59C3×#135—91no—
60#912—32no—
61#912—61no—
62D3×D8—92no—
63C2×D24—91no—
64C3×D16—92no—
65#828—92no—
66#827—92no—
67#826—32no—
68#826—92no—
69#812—91no—
70#800—92no—
71#798—92no—
72C2×D32—122no—
73C32×D4—22no—
74C32×D4—22no—
75C4⋊(C3⋊D3)—22no—
76C3×#67—122no—
77C3×#67—122no—
78C2×C3×Dic3—122no—
79C2×C3×Dic3—122no—
80C3×D12—62no—
81C3×C4×D3—32no—
82C3×C4×D3—61no—
83C3×Dic6—62no—
84C3×Dic6—62no—
85#626—122no—
86C32×C8—21no—
87C32×C8C22×D312yes—
88C3×C3⋊C8—32no—
89C3×C3⋊C8—61no—
90#509—271noSylow
91C2×H(3,𝔽3)—81no—
92C2×H(3,𝔽3)C2×D811yescharacteristic
93#272—21no—
94(C3⋊D3)⋊C3—82no—
95C3×#41—181no—
96#237—62no—
97#237—62no—
98D3×D4—182no—
99C2×D12—181no—
100C3×Q16—91no—
101C3×SD16—181no—
102C3×D8—32no—
103C3×D8—61no—
104C3×D8—92no—
105C3×D8D3211yescharacteristic
106C2×C3×C8—91no—
107#214—62no—
108#214—62no—
109#213—182no—
110Dic12—32no—
111Dic12—61no—
112D24—32no—
113D24—92no—
114D24—181no—
115D3×C8—32no—
116D3×C8—61no—
117C3×C16—92no—
118C3⋊C16—32no—
119C3⋊C16—92no—
120C22×C32—42no—
121C22×C32—42no—
122C2×C3⋊D3—42no—
123C2×C3×D3—32no—
124C2×C3×D3—61no—
125C2×C3×D3—122no—
126D32—244no—
127C32×C4—21no—
128C32×C4D3×D412yes—
129C3×Dic3—32no—
130C3×Dic3—61no—
131C3×Dic3—122no—
132C3×Dic3—122no—
133#135—91no—
134C2×D8—271no—
135SD32—272no—
136D16—92no—
137M32—271no—
138H(3,𝔽3)#50911yescharacteristic, complemented, Sylow
139C22×D3—362no—
140C22×D3—361no—
141C3×Q8—181no—
142C3×D4—21no—
143C3×D4—62no—
144C3×D4—62no—
145C3×D4—182no—
146C3×D4—361no—
147C2×C3×C4—91no—
148C2×C3×C4—361no—
149#67—122no—
150#67—122no—
151#67—362no—
152C2×Dic3—122no—
153C2×Dic3—122no—
154D12—62no—
155D12—182no—
156D12—361no—
157C4×D3—32no—
158C4×D3—61no—
159Dic6—62no—
160Dic6—62no—
161C3×C8—32no—
162C3×C8—61no—
163C3×C8—91no—
164C3×C8C2×D3211yescharacteristic
165C3⋊C8—32no—
166C3⋊C8—61no—
167C2×C32—21no—
168C2×C32—82no—
169C2×C32—82no—
170C2×C32D3×D812yes—
171C3⋊D3—82no—
172C3×D3—62no—
173C3×D3—121no—
174C3×D3—242no—
175#41—181no—
176C2×D4—541no—
177Q16—91no—
178SD16—181no—
179D8—92no—
180D8—541no—
181D8((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
182C2×C8—91no—
183C16—92no—
184C22×C3—41no—
185C22×C3—91no—
186C22×C3—122no—
187C22×C3—122no—
188C22×C3—362no—
189C2×D3—32no—
190C2×D3—61no—
191C2×D3—122no—
192C2×D3—362no—
193C2×D3—724no—
194C2×D3—722no—
195C3×C4—32no—
196C3×C4—61no—
197C3×C4—91no—
198C3×C4—361no—
199C3×C4#381111yescharacteristic
200Dic3—32no—
201Dic3—61no—
202Dic3—122no—
203Dic3—122no—
204C32—21no—
205C32#491112yescomplemented
206C23—1081no—
207Q8—181no—
208D4—21no—
209D4—182no—
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—81no—
219C2×C3—181no—
220C2×C3—242no—
221C2×C3—242no—
222C2×C3—722no—
223C2×C3#6392611yescharacteristic
224D3—62no—
225D3—121no—
226D3—242no—
227D3—722no—
228C22—41no—
229C22—91no—
230C22—362no—
231C22—2162no—
232C4—91no—
233C4—361no—
234C4#8867711yescharacteristic
235C3—32no—
236C3—61no—
237C3#1059102611yescharacteristic
238C2—81no—
239C2—181no—
240C2—722no—
241C2#1172119011yescharacteristic
subgroupG/H
centerC2#11721190
commutator, cosocleC8×H(3,𝔽3)C23
FrattiniC3×C8C2×D32
FittingD8×H(3,𝔽3)C22
socleC2×C3#63926
hypercenterD8((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 ◁ D8

Complements

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

NH ≅ G/Ntype
1. #11721794C2semidirect
2. #11720838C2semidirect
3. #63653⋊C3C2semidirect
4. #11720255C2semidirect
5. #11720252C2semidirect
9. D8×H(3,𝔽3)C22semidirect
11. #88530C22semidirect
15. (C3⋊(C3⋊C16))⋊C3C22semidirect
19. #63653D3semidirect
40. C32×D8C2×D3semidirect
50. C3⋊(C3⋊C16)C2×D3semidirect
138. H(3,𝔽3)#509semidirect
181. D8((C3⋊D3)⋊C3)⋊C2semidirect
205. C32#4911semidirect

Maximal subgroups

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

subgroupindex
#117217942
#117208382
#63653⋊C32
#117202552
#117202522
#105910243