Finite Group Databasebeta

← #13161466

Subgroups — #13161466

Order 1728.

426 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 95 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#11723651C211yesmaximal, characteristic
2C3×#63625C212yesmaximal, complemented
3C3×C16×C3⋊D3C211yesmaximal, characteristic, complemented
4#11722066C211yesmaximal, characteristic, complemented
5#11722065C212yesmaximal, complemented
6#10590229—31nomaximal
7#10590228—32nomaximal
8#89042C411yescharacteristic
9C3×C8×C3⋊D3C2212yes—
10C3×C8×C3⋊D3C2211yescharacteristic
11#88809C411yescharacteristic
12#88809C412yes—
13C33×C16C2211yescharacteristic
14C3×C3⋊(C3⋊C16)C2211yescharacteristic
15C3×C3⋊(C3⋊C16)C2212yes—
16#63803—32no—
17#63803—31no—
18#63625D312yescomplemented
19C16×C3⋊D3D311yescharacteristic, complemented
20C3×#798—32no—
21C3×#798—32no—
22C3×#798—62no—
23C3×D3×C16—32no—
24C3×D3×C16—61no—
25#63544—31no—
26#63543—32no—
27#63542—32no—
28D3×C3⋊C16—32no—
29D3×C3⋊C16—32no—
30C3×C4×C3⋊D3C2×C412yes—
31C3×C4×C3⋊D3C2×C411yescharacteristic
32#6296C811yescharacteristic
33#6295C812yes—
34#6295C811yescharacteristic
35C33×C8C2311yescharacteristic
36C3×C3⋊(C3⋊C8)C2×C412yes—
37C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
38#4907—92no—
39#4901—91no—
40#3798—32no—
41#3798—31no—
42C8×C3⋊D3C2×D312yes—
43C8×C3⋊D3C2×D311yescharacteristic
44C3×D3×C8—32no—
45C3×D3×C8—32no—
46C3×D3×C8—32no—
47C3×D3×C8—61no—
48C3×D3×C8—62no—
49#3712—32no—
50#3712—31no—
51#3711—32no—
52#3711—32no—
53#3711—32no—
54C32×C16—21no—
55C32×C16—22no—
56C32×C16—41no—
57C32×C16C2×D312yes—
58C32×C16C2×D311yescharacteristic
59C3⋊(C3⋊C16)C2×D311yescharacteristic
60C3⋊(C3⋊C16)C2×D312yes—
61C3×C3⋊C16—32no—
62C3×C3⋊C16—32no—
63C3×C3⋊C16—32no—
64C3×C3⋊C16—61no—
65C3×C3⋊C16—62no—
66C2×C3×C3⋊D3C2×C811yescharacteristic
67C2×C3×C3⋊D3C2×C812yes—
68C33×C4C22×C411yescharacteristic
69C3×C3⋊Dic3C2×C812yes—
70C3×C3⋊Dic3C2×C811yescharacteristic
71#901—91no—
72#901—92no—
73C3×M32—92no—
74C2×C3×C16—91no—
75#812—91no—
76C2×C3⋊C16—92no—
77#798—32no—
78#798—32no—
79#798—62no—
80#798—92no—
81D3×C16—32no—
82D3×C16—61no—
83D3×C16—92no—
84D3×C16—92no—
85C4×C3⋊D3C4×D312yes—
86C4×C3⋊D3C4×D311yescharacteristic
87C3×C4×D3—32no—
88C3×C4×D3—32no—
89C3×C4×D3—32no—
90C3×C4×D3—61no—
91C3×C4×D3—62no—
92#627—31no—
93#627—32no—
94#626—32no—
95#626—32no—
96#626—32no—
97#625—31no—
98#625—32no—
99C32×C8—21no—
100C32×C8—22no—
101C32×C8—41no—
102C32×C8C22×D312yes—
103C32×C8C22×D311yescharacteristic
104C3⋊(C3⋊C8)C4×D312yes—
105C3⋊(C3⋊C8)C4×D311yescharacteristic
106C3×C3⋊C8—32no—
107C3×C3⋊C8—32no—
108C3×C3⋊C8—32no—
109C3×C3⋊C8—61no—
110C3×C3⋊C8—62no—
111#504—271noSylow
112C2×C33C22×C811yescharacteristic
113C3×C3⋊D3—21no—
114C3×C3⋊D3—22no—
115#235—91no—
116#235—92no—
117C2×C3×C8—92no—
118C2×C3×C8—91no—
119#208—91no—
120#208—92no—
121ΓL(1,𝔽25)—92no—
122ΓL(1,𝔽25)—92no—
123ΓL(1,𝔽25)—92no—
124D3×C8—32no—
125D3×C8—32no—
126D3×C8—32no—
127D3×C8—61no—
128D3×C8—62no—
129C3×C16—21no—
130C3×C16—22no—
131C3×C16—41no—
132C3×C16—91no—
133C3×C16—92no—
134C3×C16D3212yes—
135C3×C16D3211yescharacteristic
136C3⋊C16—32no—
137C3⋊C16—32no—
138C3⋊C16—32no—
139C3⋊C16—61no—
140C3⋊C16—62no—
141C2×C3⋊D3D3×C811yescharacteristic
142C2×C3⋊D3D3×C812yes—
143C2×C3×D3—32no—
144C2×C3×D3—32no—
145C2×C3×D3—32no—
146C2×C3×D3—61no—
147C2×C3×D3—62no—
148C32×C4—22no—
149C32×C4—21no—
150C32×C4—41no—
151C32×C4C2×C4×D311yescharacteristic
152C32×C4C2×C4×D312yes—
153C3⋊Dic3D3×C811yescharacteristic
154C3⋊Dic3D3×C812yes—
155C3×Dic3—32no—
156C3×Dic3—32no—
157C3×Dic3—32no—
158C3×Dic3—62no—
159C3×Dic3—61no—
160#131—271no—
161M32—92no—
162M32—271no—
163C2×C16—91no—
164C2×C16—272no—
165C33#50411yescharacteristic, complemented, Sylow
166C2×C3×C4—92no—
167C2×C3×C4—91no—
168#67—92no—
169#67—92no—
170#67—92no—
171D12—92no—
172D12—91no—
173C4×D3—32no—
174C4×D3—32no—
175C4×D3—32no—
176C4×D3—61no—
177C4×D3—62no—
178Dic6—92no—
179Dic6—91no—
180C3×C8—21no—
181C3×C8—22no—
182C3×C8—41no—
183C3×C8—92no—
184C3×C8—91no—
185C3×C8C2×D3212yes—
186C3×C8C2×D3211yescharacteristic
187C3⋊C8—32no—
188C3⋊C8—32no—
189C3⋊C8—32no—
190C3⋊C8—61no—
191C3⋊C8—62no—
192C2×C32—22no—
193C2×C32—21no—
194C2×C32—41no—
195C2×C32C2×D3×C811yescharacteristic
196C2×C32C2×D3×C812yes—
197C3⋊D3—21no—
198C3⋊D3—22no—
199C3×D3—62no—
200C3×D3—62no—
201C3×D3—62no—
202C3×D3—62no—
203C3×D3—122no—
204#41—271no—
205M16—272no—
206M16—271no—
207C2×C8—91no—
208C2×C8—92no—
209C16—91no—
210C16—92no—
211C16#111311yescharacteristic
212C22×C3—92no—
213C22×C3—91no—
214C2×D3—32no—
215C2×D3—32no—
216C2×D3—32no—
217C2×D3—61no—
218C2×D3—62no—
219C3×C4—21no—
220C3×C4—22no—
221C3×C4—41no—
222C3×C4—92no—
223C3×C4—91no—
224C3×C4C4×D3212yes—
225C3×C4C4×D3211yescharacteristic
226Dic3—32no—
227Dic3—32no—
228Dic3—32no—
229Dic3—61no—
230Dic3—62no—
231C32—22no—
232C32—21no—
233C32—41no—
234C32#490111yescharacteristic, complemented
235C32#490712yescomplemented
236Q8—271no—
237D4—272no—
238D4—271no—
239C2×C4—91no—
240C2×C4—92no—
241C8—91no—
242C8—92no—
243C8C2×#111311yescharacteristic
244C2×C3—21no—
245C2×C3—22no—
246C2×C3—41no—
247C2×C3—182no—
248C2×C3—181no—
249C2×C3D32×C812yes—
250C2×C3D32×C811yescharacteristic
251D3—62no—
252D3—62no—
253D3—62no—
254D3—62no—
255D3—122no—
256C22—91no—
257C22—92no—
258C4—91no—
259C4—92no—
260C4C4×#111311yescharacteristic
261C3—21no—
262C3—22no—
263C3—41no—
264C3#1059022812yescomplemented
265C3#1059022911yescharacteristic, complemented
266C2—181no—
267C2—182no—
268C2C8×#111311yescharacteristic
subgroupG/H
center, FrattiniC8C2×#1113
commutator, socleC2×C33C22×C8
FittingC33×C16C22
prime radical#89042C4
cosocleC33×C8C23
hypercenterC16#1113
hypocenterC33#504
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C8 ◁ C16

Complements

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

NH ≅ G/Ntype
2. C3×#63625C2semidirect
3. C3×C16×C3⋊D3C2semidirect
4. #11722066C2semidirect
5. #11722065C2semidirect
18. #63625D3semidirect
19. C16×C3⋊D3D3semidirect
165. C33#504semidirect
234. C32#4901semidirect
235. C32#4907semidirect
264. C3#10590228semidirect
265. C3#10590229semidirect

Maximal subgroups

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

subgroupindex
#117236512
C3×#636252
C3×C16×C3⋊D32
#117220662
#117220652
#105902293
#105902283