Finite Group Databasebeta

← #13139282

Subgroups — #13139282

Order 1728.

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

normal subgroups only

This group has 349 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
2#11720833C211yesmaximal, characteristic, complemented
3#63656⋊C3C212yesmaximal, complemented
4#11720254C212yesmaximal, complemented
5#11720252C211yesmaximal, characteristic, complemented
6#10591042—32nomaximal
7#88748—21no—
8#88651—21no—
9Q16×H(3,𝔽3)C2211yescharacteristic
10#88549—21no—
11#88529C2211yescharacteristic
12C8×#272C2211yescharacteristic
13#3746⋊C3C2212yes—
14#88433—21no—
15(C3⋊(C3⋊C16))⋊C3C2212yes—
16C3×#919—32no—
17C3×#919—61no—
18D3×Dic12—32no—
19#63656D312yescomplemented
20C3×#829—32no—
21#63548—32no—
22#63547—32no—
23#63543—32no—
24Q8×H(3,𝔽3)—21no—
25C4⋊#272—21no—
26C4⋊#272—41no—
27C4×#272—41no—
28C4×#272D411yescharacteristic
29#634⋊C3—22no—
30#6199—41no—
31#6196—41no—
32C8×H(3,𝔽3)C2311yescharacteristic
33#6180D411yescharacteristic
34#5195—91no—
35#4918—92no—
36C3×#239—62no—
37C3×#239—62no—
38D3×Dic6—62no—
39C32×Q16—21no—
40C32×Q16C2×D312yes—
41#3746C2×D312yes—
42C3×#215—62no—
43C3×#215—62no—
44C3×Dic12—32no—
45C3×D24—32no—
46C3×D24—61no—
47C3×D3×C8—32no—
48C3×D3×C8—61no—
49#3719—62no—
50C3⋊(C3⋊C16)C2×D312yes—
51C3×C3⋊C16—32no—
52C2×#272—41no—
53C2×#272D811yescharacteristic
54C4×H(3,𝔽3)—41no—
55C4×H(3,𝔽3)C2×D411yescharacteristic
56#1084D811yescharacteristic
57(C3⋊Dic3)⋊C3—42no—
58C3×#135—91no—
59#919—32no—
60#919—61no—
61D3×Q16—92no—
62C2×Dic12—91no—
63C3×Q32—92no—
64#829—32no—
65#829—92no—
66#828—92no—
67#827—92no—
68#812—91no—
69#800—92no—
70#798—92no—
71C32×Q8—22no—
72C32×Q8—22no—
73#634—22no—
74C3×D12—62no—
75C3×D12—62no—
76C3×D12—122no—
77C3×D12—122no—
78C3×C4×D3—32no—
79C3×C4×D3—61no—
80C3×C4×D3—122no—
81C3×C4×D3—122no—
82C3×Dic6—62no—
83#627—122no—
84D3×Dic3—122no—
85C32×C8—21no—
86C32×C8C22×D312yes—
87C3×C3⋊C8—32no—
88C3×C3⋊C8—61no—
89#510—271noSylow
90C2×H(3,𝔽3)C2×D811yescharacteristic
91#272—21no—
92#272—81no—
93C3×#41—181no—
94#239—62no—
95#239—62no—
96D3×Q8—182no—
97C2×Dic6—181no—
98C3×Q16—32no—
99C3×Q16—61no—
100C3×Q16—92no—
101C3×Q16D3211yescharacteristic
102C3×SD16—181no—
103C3×D8—91no—
104C2×C3×C8—91no—
105#216—182no—
106#215—62no—
107#215—62no—
108Dic12—32no—
109Dic12—92no—
110Dic12—181no—
111D24—32no—
112D24—61no—
113D3×C8—32no—
114D3×C8—61no—
115C3×C16—92no—
116C3⋊C16—32no—
117C3⋊C16—92no—
118C2×C3×D3—32no—
119C2×C3×D3—61no—
120C2×C3×D3—122no—
121C2×C3×D3—122no—
122C32×C4—21no—
123C32×C4—42no—
124C32×C4—42no—
125C32×C4D3×D412yes—
126C3⋊Dic3—42no—
127C3×Dic3—32no—
128C3×Dic3—61no—
129C3×Dic3—122no—
130#135—91no—
131C2×Q16—271no—
132Q32—92no—
133SD32—272no—
134M32—271no—
135H(3,𝔽3)#51011yescharacteristic, complemented, Sylow
136C3×Q8—21no—
137C3×Q8—62no—
138C3×Q8—62no—
139C3×Q8—182no—
140C3×D4—181no—
141C3×D4—361no—
142C2×C3×C4—91no—
143C2×C3×C4—361no—
144C2×Dic3—361no—
145D12—62no—
146D12—62no—
147D12—122no—
148D12—122no—
149C4×D3—32no—
150C4×D3—61no—
151C4×D3—122no—
152C4×D3—122no—
153C4×D3—362no—
154Dic6—62no—
155Dic6—182no—
156Dic6—362no—
157Dic6—361no—
158C3×C8—32no—
159C3×C8—61no—
160C3×C8—91no—
161C3×C8C2×D3211yescharacteristic
162C3⋊C8—32no—
163C3⋊C8—61no—
164C2×C32—21no—
165C2×C32D3×D812yes—
166C3×D3—62no—
167C3×D3—121no—
168C3×D3—242no—
169C3×D3—242no—
170#41—181no—
171C2×Q8—541no—
172Q16—92no—
173Q16—541no—
174Q16((C3⋊D3)⋊C3)⋊C211yescharacteristic
175SD16—181no—
176D8—91no—
177C2×C8—91no—
178C16—92no—
179C22×C3—91no—
180C22×C3—361no—
181C2×D3—32no—
182C2×D3—61no—
183C2×D3—122no—
184C2×D3—122no—
185C3×C4—32no—
186C3×C4—41no—
187C3×C4—61no—
188C3×C4—91no—
189C3×C4—122no—
190C3×C4—122no—
191C3×C4—362no—
192C3×C4#381111yescharacteristic
193Dic3—32no—
194Dic3—61no—
195Dic3—122no—
196Dic3—362no—
197C32—21no—
198C32#491812yescomplemented
199Q8—21no—
200Q8—182no—
201Q8—1081no—
202D4—181no—
203D4—361no—
204C2×C4—91no—
205C2×C4—361no—
206C2×C4—1081no—
207C8—91no—
208C8C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
209C2×C3—32no—
210C2×C3—61no—
211C2×C3—181no—
212C2×C3—721no—
213C2×C3#6392611yescharacteristic
214D3—62no—
215D3—121no—
216D3—242no—
217D3—242no—
218C22—91no—
219C22—361no—
220C4—41no—
221C4—91no—
222C4—362no—
223C4#8867711yescharacteristic
224C3—32no—
225C3—61no—
226C3#1059104411yescharacteristic
227C2—181no—
228C2—721no—
229C2#1172119011yescharacteristic
subgroupG/H
centerC2#11721190
commutator, cosocleC8×H(3,𝔽3)C23
FrattiniC3×C8C2×D32
FittingQ16×H(3,𝔽3)C22
socleC2×C3#63926
prime radical#11721808C2
hypercenterQ16((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#510
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
2. #11720833C2semidirect
3. #63656⋊C3C2semidirect
4. #11720254C2semidirect
5. #11720252C2semidirect
19. #63656D3semidirect
135. H(3,𝔽3)#510semidirect
198. C32#4918semidirect

Maximal subgroups

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

subgroupindex
#117218082
#117208332
#63656⋊C32
#117202542
#117202522
#105910423