Finite Group Databasebeta

← #13166901

Subgroups — #13166901

Order 1728.

360 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 43 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#11723653C211yesmaximal, characteristic, complemented
2C3×#63797C211yesmaximal, characteristic, complemented
3#11723090C211yesmaximal, characteristic, complemented
4#11722767C212yesmaximal, complemented
5#11722765C212yesmaximal, complemented
6#10593277—31nomaximal
7#89041—21no—
8C3×#3802—21no—
9C3×#3744—21no—
10C3×C8×C3⋊D3C2211yescharacteristic
11C32×D24—21no—
12#88812C2211yescharacteristic, complemented
13#88810C2211yescharacteristic, complemented
14C3×#3713C2212yes—
15C3⋊#3708C2212yescomplemented
16C3×D3×D8—61no—
17#63809—31no—
18#63806—61no—
19#63797D311yescharacteristic, complemented
20#63765—31no—
21#63735—32no—
22#63734—32no—
23C2×C3×D32—41no—
24C3×C4⋊(C3⋊D3)—21no—
25C3×C4×C3⋊D3D411yescharacteristic
26C3×#634—21no—
27C32×D12—22no—
28#6295—41no—
29#6294—41no—
30C3×#625—41no—
31C33×C8D411yescharacteristic
32C3×C3⋊(C3⋊C8)C2311yescharacteristic
33#5147—91nomaximal
34C3×D3×D4—62no—
35#3802—21no—
36#3799—61no—
37#3796—61no—
38#3795—61no—
39C32×D8—61no—
40#3744—21no—
41C8×C3⋊D3C2×D311yescharacteristic
42C3×#213—62no—
43C3×D24—21no—
44C3×D24—21no—
45C3×D24—41no—
46C3×#204—61no—
47C3×#204—61no—
48C3×#204—121no—
49C3×D3×C8—61no—
50C3×D3×C8—61no—
51#3719—61no—
52#3718—31no—
53#3714—61no—
54#3713—31no—
55#3713C2×D312yes—
56#3708—32no—
57C2×C3×C3⋊D3—41no—
58C2×C3×C3⋊D3D811yescharacteristic
59C2×C32×D3—42no—
60C3×D32—82no—
61C33×C4C2×D411yescharacteristic
62C3×C3⋊Dic3—41no—
63C3×C3⋊Dic3D811yescharacteristic
64C2×C3×D8—91no—
65#916—181no—
66D3×D8—61no—
67#904—91no—
68#827—92no—
69#826—92no—
70#812—91no—
71C22×C3×D3—122no—
72C2×D32—41no—
73C32×D4—62no—
74C4⋊(C3⋊D3)—21no—
75C4×C3⋊D3#6711yescharacteristic
76#634—21no—
77C3×#67—122no—
78C3×D12—22no—
79C3×D12—22no—
80C3×D12—42no—
81C3×D12—61no—
82C3×D12—61no—
83C3×D12—121no—
84C3×C4×D3—61no—
85C3×C4×D3—61no—
86C3×Dic6—61no—
87C3×Dic6—61no—
88C3×Dic6—121no—
89#626—121no—
90#626—121no—
91#625—41no—
92#625—121no—
93#624—121no—
94D3×Dic3—121no—
95D3×Dic3—121no—
96C32×C8—21no—
97C32×C8—21no—
98C32×C8—41no—
99C32×C8—41no—
100C32×C8#6711yescharacteristic
101C3⋊(C3⋊C8)C22×D311yescharacteristic
102C3×C3⋊C8—61no—
103C3×C3⋊C8—61no—
104#509—271noSylow
105C2×C33C2×D811yescharacteristic
106C3×C3⋊D3—21no—
107C3×C3⋊D3—81no—
108C32×D3—82no—
109C2×C3×D4—181no—
110#239—181no—
111#237—181no—
112D3×D4—62no—
113#235—181no—
114C3×SD16—181no—
115C3×D8—61no—
116C3×D8—61no—
117C3×D8—92no—
118C3×D8—121no—
119C2×C3×C8—91no—
120#216—181no—
121#213—62no—
122#213—181no—
123Dic12—91no—
124D24—21no—
125D24—91no—
126#204—61no—
127#204—61no—
128#204—121no—
129D3×C8—61no—
130D3×C8—61no—
131C3⋊C16—92no—
132C22×C32—122no—
133C2×C3⋊D3—41no—
134C2×C3⋊D3#21311yescharacteristic
135C2×C3×D3—42no—
136C2×C3×D3—42no—
137C2×C3×D3—61no—
138C2×C3×D3—61no—
139C2×C3×D3—82no—
140C2×C3×D3—121no—
141C2×C3×D3—121no—
142C2×C3×D3—241no—
143C2×C3×D3—244no—
144D32—82no—
145C32×C4—21no—
146C32×C4—21no—
147C32×C4—41no—
148C32×C4—41no—
149C32×C4C2×#6711yescharacteristic
150C3⋊Dic3—41no—
151C3⋊Dic3#21311yescharacteristic
152C3×Dic3—61no—
153C3×Dic3—61no—
154C3×Dic3—121no—
155C3×Dic3—121no—
156C3×Dic3—241no—
157#135—271no—
158C2×D8—91no—
159SD32—272no—
160D16—272no—
161M32—271no—
162C33#50911yescharacteristic, complemented, Sylow
163C23×C3—361no—
164C22×D3—122no—
165C3×Q8—181no—
166C3×D4—62no—
167C3×D4—62no—
168C3×D4—122no—
169C3×D4—181no—
170C3×D4—361no—
171C2×C3×C4—91no—
172#67—122no—
173#67—361no—
174#67—361no—
175C2×Dic3—361no—
176D12—22no—
177D12—61no—
178D12—61no—
179D12—121no—
180D12—361no—
181C4×D3—61no—
182C4×D3—61no—
183C4×D3—361no—
184C4×D3—361no—
185Dic6—61no—
186Dic6—61no—
187Dic6—121no—
188C3×C8—21no—
189C3×C8—21no—
190C3×C8—41no—
191C3×C8—41no—
192C3×C8—91no—
193C3×C8GO+(4,𝔽2)11yescharacteristic
194C3⋊C8—61no—
195C3⋊C8—61no—
196C2×C32—21no—
197C2×C32—21no—
198C2×C32—41no—
199C2×C32—41no—
200C2×C32—242no—
201C2×C32C2×#21311yescharacteristic
202C3⋊D3—21no—
203C3⋊D3—81no—
204C3×D3—62no—
205C3×D3—82no—
206C3×D3—82no—
207C3×D3—121no—
208C3×D3—162no—
209C3×D3—241no—
210C3×D3—241no—
211C3×D3—481no—
212#41—541no—
213C2×D4—181no—
214Q16—271no—
215SD16—181no—
216D8—61no—
217D8—92no—
218D8—271no—
219C2×C8—91no—
220C16—272no—
221C22×C3—91no—
222C22×C3—122no—
223C22×C3—122no—
224C22×C3—242no—
225C22×C3—361no—
226C22×C3—722no—
227C2×D3—42no—
228C2×D3—61no—
229C2×D3—61no—
230C2×D3—121no—
231C2×D3—121no—
232C2×D3—241no—
233C2×D3—244no—
234C3×C4—21no—
235C3×C4—21no—
236C3×C4—41no—
237C3×C4—41no—
238C3×C4—91no—
239C3×C4—361no—
240C3×C4C2×GO+(4,𝔽2)11yescharacteristic
241Dic3—61no—
242Dic3—61no—
243Dic3—121no—
244Dic3—121no—
245Dic3—241no—
246C32—21no—
247C32—21no—
248C32—41no—
249C32—41no—
250C32#514711yescharacteristic, complemented
251C23—361no—
252Q8—181no—
253D4—62no—
254D4—181no—
255D4—361no—
256D4—1081no—
257C2×C4—91no—
258C2×C4—1081no—
259C8—91no—
260C8D32⋊D311yescharacteristic
261C2×C3—21no—
262C2×C3—21no—
263C2×C3—41no—
264C2×C3—41no—
265C2×C3—181no—
266C2×C3—242no—
267C2×C3—242no—
268C2×C3—482no—
269C2×C3—721no—
270C2×C3#6422511yescharacteristic
271D3—62no—
272D3—82no—
273D3—121no—
274D3—241no—
275D3—241no—
276D3—481no—
277C22—91no—
278C22—122no—
279C22—361no—
280C22—722no—
281C4—91no—
282C4—361no—
283C4C2×D32⋊D311yescharacteristic
284C3—21no—
285C3—21no—
286C3—41no—
287C3—41no—
288C3#1059327711yescharacteristic, complemented
289C2—181no—
290C2—242no—
291C2—721no—
292C2#1172430611yescharacteristic
subgroupG/H
centerC2#11724306
commutator, cosocleC3×C3⋊(C3⋊C8)C23
FrattiniC4C2×D32⋊D3
FittingC33×C8D4
socleC2×C33C2×D8
hypercenterC8D32⋊D3
hypocenterC33#509
LCSG ▷ C3×C3⋊(C3⋊C8) ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
1. #11723653C2semidirect
2. C3×#63797C2semidirect
3. #11723090C2semidirect
4. #11722767C2semidirect
5. #11722765C2semidirect
12. #88812C22semidirect
13. #88810C22semidirect
15. C3⋊#3708C22semidirect
19. #63797D3semidirect
162. C33#509semidirect
250. C32#5147semidirect
288. C3#10593277semidirect

Maximal subgroups

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

subgroupindex
#117236532
C3×#637972
#117230902
#117227672
#117227652
#105932773
#51479