Finite Group Databasebeta

← #13161473

Subgroups — #13161473

Order 1728.

426 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 71 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#11723654C212yesmaximal, complemented
2C3×#63628C211yesmaximal, characteristic, complemented
3C3×#63626C211yesmaximal, characteristic, complemented
4C3×C16×C3⋊D3C211yesmaximal, characteristic, complemented
5#11722068C212yesmaximal, complemented
6#10590244—31nomaximal
7#10590243—31nomaximal
8#10590241—31nomaximal
9#89041—22no—
10C3×#3746C2212yes—
11C3×#3745C2212yes—
12C3×C8×C3⋊D3C2211yescharacteristic
13#88811—22no—
14C33×C16C2211yescharacteristic
15C3×C3⋊(C3⋊C16)C2211yescharacteristic
16#63810—32no—
17#63808—32no—
18#63807—32no—
19#63628D311yescharacteristic, complemented
20#63626D311yescharacteristic, complemented
21C16×C3⋊D3D311yescharacteristic, complemented
22C3×Dic24—31no—
23C3×Dic24—31no—
24C3×Dic24—61no—
25C3×D48—31no—
26C3×D48—31no—
27C3×D48—61no—
28C3×D3×C16—31no—
29C3×D3×C16—31no—
30C3×D3×C16—61no—
31#63549—32no—
32#63548—32no—
33#63547—32no—
34C3×C4⋊(C3⋊D3)—22no—
35C3×C4×C3⋊D3D411yescharacteristic
36C3×#634—22no—
37#6295—42no—
38#6294—42no—
39C33×C8C2311yescharacteristic
40C3×C3⋊(C3⋊C8)D411yescharacteristic
41#4919—91no—
42#4912—91no—
43#4904—91no—
44#3799—62no—
45#3796—62no—
46#3795—62no—
47#3746C2×D312yes—
48#3745C2×D312yes—
49C8×C3⋊D3C2×D311yescharacteristic
50C3×Dic12—32no—
51C3×Dic12—32no—
52C3×Dic12—62no—
53C3×D24—32no—
54C3×D24—32no—
55C3×D24—62no—
56C3×D3×C8—31no—
57C3×D3×C8—31no—
58C3×D3×C8—61no—
59#3717—62no—
60#3716—62no—
61#3715—62no—
62C32×C16—21no—
63C32×C16—21no—
64C32×C16—21no—
65C32×C16—41no—
66C32×C16C2×D311yescharacteristic
67C32×C16C2×D311yescharacteristic
68C32×C16C2×D311yescharacteristic
69C3⋊(C3⋊C16)C2×D311yescharacteristic
70C3×C3⋊C16—31no—
71C3×C3⋊C16—31no—
72C3×C3⋊C16—61no—
73C2×C3×C3⋊D3—42no—
74C2×C3×C3⋊D3D811yescharacteristic
75C33×C4C2×D411yescharacteristic
76C3×C3⋊Dic3—42no—
77C3×C3⋊Dic3D811yescharacteristic
78#919—92no—
79#912—92no—
80#904—92no—
81C3×Q32—91no—
82C3×D16—91no—
83C2×C3×C16—91no—
84#828—92no—
85#827—92no—
86Dic24—31no—
87Dic24—31no—
88Dic24—61no—
89#800—92no—
90D48—31no—
91D48—31no—
92D48—61no—
93D3×C16—31no—
94D3×C16—31no—
95D3×C16—61no—
96C4⋊(C3⋊D3)—22no—
97C4×C3⋊D3D1211yescharacteristic
98#634—22no—
99C3×D12—62no—
100C3×D12—62no—
101C3×D12—122no—
102C3×C4×D3—31no—
103C3×C4×D3—31no—
104C3×C4×D3—61no—
105C3×Dic6—62no—
106C3×Dic6—62no—
107C3×Dic6—122no—
108#626—122no—
109#626—122no—
110#625—122no—
111#624—122no—
112D3×Dic3—122no—
113D3×Dic3—122no—
114C32×C8—21no—
115C32×C8—21no—
116C32×C8—21no—
117C32×C8—41no—
118C32×C8C22×D311yescharacteristic
119C32×C8C22×D311yescharacteristic
120C32×C8C22×D311yescharacteristic
121C3⋊(C3⋊C8)D1211yescharacteristic
122C3×C3⋊C8—31no—
123C3×C3⋊C8—31no—
124C3×C3⋊C8—61no—
125#508—271noSylow
126C2×C33C2×D811yescharacteristic
127C3×C3⋊D3—21no—
128C3×C3⋊D3—82no—
129#239—182no—
130#237—182no—
131#235—182no—
132C3×Q16—92no—
133C3×D8—92no—
134C2×C3×C8—91no—
135#215—182no—
136#214—182no—
137Dic12—32no—
138Dic12—32no—
139Dic12—62no—
140D24—32no—
141D24—32no—
142D24—62no—
143#204—182no—
144D3×C8—31no—
145D3×C8—31no—
146D3×C8—61no—
147C3×C16—21no—
148C3×C16—21no—
149C3×C16—21no—
150C3×C16—41no—
151C3×C16—91no—
152C3×C16D3211yescharacteristic
153C3×C16D3211yescharacteristic
154C3×C16D3211yescharacteristic
155C3⋊C16—31no—
156C3⋊C16—31no—
157C3⋊C16—61no—
158C2×C3⋊D3—42no—
159C2×C3⋊D3D2411yescharacteristic
160C2×C3×D3—31no—
161C2×C3×D3—31no—
162C2×C3×D3—61no—
163C2×C3×D3—122no—
164C2×C3×D3—122no—
165C2×C3×D3—242no—
166C32×C4—21no—
167C32×C4—21no—
168C32×C4—21no—
169C32×C4—41no—
170C32×C4C2×D1211yescharacteristic
171C32×C4D3×D411yescharacteristic
172C32×C4D3×D411yescharacteristic
173C3⋊Dic3—42no—
174C3⋊Dic3D2411yescharacteristic
175C3×Dic3—31no—
176C3×Dic3—31no—
177C3×Dic3—61no—
178C3×Dic3—122no—
179C3×Dic3—122no—
180C3×Dic3—242no—
181#135—272no—
182Q32—91no—
183SD32—272no—
184D16—91no—
185C2×C16—91no—
186C33#50811yescharacteristic, complemented, Sylow
187C3×Q8—182no—
188C3×D4—182no—
189C2×C3×C4—91no—
190#67—362no—
191#67—362no—
192C2×Dic3—362no—
193D12—62no—
194D12—62no—
195D12—122no—
196D12—362no—
197C4×D3—31no—
198C4×D3—31no—
199C4×D3—61no—
200C4×D3—362no—
201C4×D3—362no—
202Dic6—62no—
203Dic6—62no—
204Dic6—122no—
205C3×C8—21no—
206C3×C8—21no—
207C3×C8—21no—
208C3×C8—41no—
209C3×C8—91no—
210C3×C8C2×D3211yescharacteristic
211C3×C8C2×D3211yescharacteristic
212C3×C8C2×D3211yescharacteristic
213C3⋊C8—31no—
214C3⋊C8—31no—
215C3⋊C8—61no—
216C2×C32—21no—
217C2×C32—21no—
218C2×C32—21no—
219C2×C32—41no—
220C2×C32C2×D2411yescharacteristic
221C2×C32D3×D811yescharacteristic
222C2×C32D3×D811yescharacteristic
223C3⋊D3—21no—
224C3⋊D3—82no—
225C3×D3—62no—
226C3×D3—61no—
227C3×D3—61no—
228C3×D3—242no—
229C3×D3—242no—
230C3×D3—482no—
231#41—542no—
232Q16—92no—
233SD16—542no—
234D8—92no—
235C2×C8—91no—
236C16—91no—
237C16#111311yescharacteristic
238C22×C3—91no—
239C22×C3—362no—
240C2×D3—31no—
241C2×D3—31no—
242C2×D3—61no—
243C2×D3—122no—
244C2×D3—122no—
245C2×D3—242no—
246C3×C4—21no—
247C3×C4—21no—
248C3×C4—21no—
249C3×C4—41no—
250C3×C4—91no—
251C3×C4—362no—
252C3×C4D3×D1211yescharacteristic
253C3×C4D3×D1211yescharacteristic
254C3×C4#380211yescharacteristic
255Dic3—31no—
256Dic3—31no—
257Dic3—61no—
258Dic3—122no—
259Dic3—122no—
260Dic3—242no—
261C32—21no—
262C32—21no—
263C32—21no—
264C32—41no—
265C32#490411yescharacteristic, complemented
266C32#491211yescharacteristic, complemented
267C32#491911yescharacteristic, complemented
268Q8—182no—
269D4—182no—
270D4—1082no—
271C2×C4—91no—
272C2×C4—1082no—
273C8—91no—
274C8C2×#111311yescharacteristic
275C2×C3—21no—
276C2×C3—21no—
277C2×C3—21no—
278C2×C3—41no—
279C2×C3—181no—
280C2×C3—722no—
281C2×C3D3×D2411yescharacteristic
282C2×C3D3×D2411yescharacteristic
283C2×C3#6379711yescharacteristic
284D3—62no—
285D3—61no—
286D3—61no—
287D3—242no—
288D3—242no—
289D3—482no—
290C22—91no—
291C22—362no—
292C4—91no—
293C4—362no—
294C4C4⋊#111311yescharacteristic
295C3—21no—
296C3—21no—
297C3—21no—
298C3—41no—
299C3#1059024111yescharacteristic, complemented
300C3#1059024311yescharacteristic, complemented
301C3#1059024411yescharacteristic, complemented
302C2—181no—
303C2—722no—
304C2#1172364711yescharacteristic
subgroupG/H
centerC2#11723647
commutator, cosocleC33×C8C23
FrattiniC8C2×#1113
FittingC33×C16C22
socleC2×C33C2×D8
hypercenterC16#1113
hypocenterC33#508
LCSG ▷ C33×C8 ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ C8 ◁ C16

Complements

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

NH ≅ G/Ntype
1. #11723654C2semidirect
2. C3×#63628C2semidirect
3. C3×#63626C2semidirect
4. C3×C16×C3⋊D3C2semidirect
5. #11722068C2semidirect
19. #63628D3semidirect
20. #63626D3semidirect
21. C16×C3⋊D3D3semidirect
186. C33#508semidirect
265. C32#4904semidirect
266. C32#4912semidirect
267. C32#4919semidirect
299. C3#10590241semidirect
300. C3#10590243semidirect
301. C3#10590244semidirect

Maximal subgroups

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

subgroupindex
#117236542
C3×#636282
C3×#636262
C3×C16×C3⋊D32
#117220682
#105902443
#105902433
#105902413