Finite Group Databasebeta

← #13140394

Subgroups — #13140394

Order 1728.

532 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 85 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
1C3×#63710C211yesmaximal, characteristic
2#11721836C211yesmaximal, characteristic, complemented
3#11720869C212yesmaximal, complemented
4C9×#833C211yesmaximal, characteristic, complemented
5#11720291C212yesmaximal, complemented
6#10591513—31nomaximal
7#10588426—31nomaximal
8C2×C3×D4×C9C2211yescharacteristic
9C2×#6226C2211yescharacteristic, complemented
10C2×C3×#611—22no—
11C3×#3699—22no—
12C3×D4×D9—22no—
13C3×#3697C2212yes—
14#88551—22no—
15C4×C3⋊Dic9C2211yescharacteristic, complemented
16C9×#208C2212yescomplemented
17#88418—22no—
18#88415—22no—
19C3×#1004—31no—
20#64145—31no—
21#63831—32no—
22#63710D311yescharacteristic
23C3×#833D311yescharacteristic
24#63568—32no—
25#63498—31no—
26#63498—61no—
27#63471—32no—
28C9×#100—31no—
29#63384—32no—
30C23×C3×C9D412yes—
31C22×C3×D9—22no—
32C3×D4×C9—22no—
33C2×C3×C4×C9C2311yescharacteristic
34C2×C3⋊Dic9—21no—
35C2×C3⋊Dic9—21no—
36#6226—22no—
37C3×#611—24no—
38C3×#611—22no—
39C2×C3×Dic9—22no—
40C3×D36D412yes—
41C3×C4×D9—22no—
42C3×Dic18D412yes—
43C9×C3⋊C8—22no—
44#5242—91no—
45#4746—91no—
46C2×C32×D4C2×D311yescharacteristic
47C2×#634—31no—
48C2×C3×#67—62no—
49C3×#237—62no—
50C3×D3×D4—62no—
51C3×#235—32no—
52#3757—62no—
53C4×C3⋊Dic3—31no—
54C3×#208C2×D312yes—
55#3719—62no—
56#3716—62no—
57C2×D4×C9—21no—
58C2×D4×C9C2×D311yescharacteristic
59C2×#611—22no—
60#3699—22no—
61D4×D9—22no—
62#3697C2×D312yes—
63C2×Dic18—31no—
64C2×Dic18—61no—
65C9×M16—32no—
66#3676—64no—
67#3676—62no—
68C4×Dic9—31no—
69C4×Dic9—61no—
70#3664—62no—
71Dic36—62no—
72C22×C3×C9—22no—
73C22×C3×C9—42no—
74C22×C3×C9C2×D411yescharacteristic
75C2×C3×D9—22no—
76C2×C3×D9—82no—
77C3×C4×C9C2×D412yes—
78C3⋊Dic9—41no—
79C3⋊Dic9—41no—
80C3×Dic9—22no—
81C3×H(5,𝔽2)—91no—
82#1004—31no—
83#951—92no—
84#936—91no—
85#936—91no—
86#936—181no—
87#909—92no—
88C3×#100—31no—
89#837—92no—
90#833D911yescharacteristic, complemented
91#825—92no—
92C23×C32D1212yes—
93C22×C3×D3—62no—
94C32×D4—22no—
95C2×C32×C4C22×D311yescharacteristic
96C2×C3⋊Dic3—61no—
97C2×C3⋊Dic3—61no—
98#634—62no—
99C3×#67—64no—
100C3×#67—62no—
101C2×C3×Dic3—62no—
102C3×D12—32no—
103C3×C4×D3—62no—
104C3×Dic6—32no—
105C23×C9—22no—
106C23×C9#6712yes—
107C22×D9—22no—
108C3×C3⋊C8—22no—
109D4×C9—22no—
110D4×C9—42no—
111C2×C4×C9—21no—
112C2×C4×C9C22×D311yescharacteristic
113#611—24no—
114#611—22no—
115C2×Dic9—22no—
116C2×Dic9—62no—
117C2×Dic9—62no—
118C2×Dic9—61no—
119C2×Dic9—61no—
120D36#6712yes—
121C4×D9—22no—
122Dic18—62no—
123Dic18—122no—
124Dic18#6712yes—
125C8×C9—62no—
126#455—271noSylow
127C2×C3×C9—21no—
128C2×C3×C9—42no—
129C2×C3×C9#12011yescharacteristic
130C3×D9—42no—
131C3×#41—92no—
132C3×#41—182no—
133C2×C3×D4—21no—
134C2×C3×D4—182no—
135C2×C3×D4—182no—
136C2×C3×D4C2×D911yescharacteristic
137C2×C3×D4D3211yescharacteristic
138C2×#67—62no—
139#237—62no—
140D3×D4—62no—
141#235—32no—
142C2×Dic6—91no—
143C2×Dic6—91no—
144C2×Dic6—181no—
145C3×M16—32no—
146#217—184no—
147#217—182no—
148#217—182no—
149#216—182no—
150#214—182no—
151C4×Dic3—91no—
152C4×Dic3—91no—
153C4×Dic3—181no—
154#208C2×D912yescomplemented
155Dic12—182no—
156#204—182no—
157C22×C32—22no—
158C22×C32—42no—
159C22×C32C2×D1211yescharacteristic
160C2×C3×D3—62no—
161C2×C3×D3—242no—
162C32×C4D3×D412yes—
163C3⋊Dic3—121no—
164C3⋊Dic3—121no—
165C3×Dic3—62no—
166C22×C9—22no—
167C22×C9—21no—
168C22×C9—42no—
169C22×C9—42no—
170C22×C9—44no—
171C22×C9C2×#6711yescharacteristic
172C2×D9—22no—
173C2×D9—82no—
174C4×C9—22no—
175C4×C9C2×#6712yes—
176Dic9—22no—
177Dic9—122no—
178Dic9—122no—
179Dic9—121no—
180Dic9—121no—
181H(5,𝔽2)—91no—
182#137—272no—
183#124—271no—
184#104—272no—
185#100—31no—
186C3×C9#45511yescharacteristic, complemented, Sylow
187C23×C3—22no—
188C23×C3—182no—
189C23×C3D3612yes—
190C23×C3#62612yes—
191C22×D3—62no—
192C3×Q8—92no—
193C3×D4—22no—
194C3×D4—22no—
195C3×D4—42no—
196C3×D4—92no—
197C3×D4—184no—
198C3×D4—182no—
199C2×C3×C4—21no—
200C2×C3×C4—182no—
201C2×C3×C4—182no—
202C2×C3×C4C22×D911yescharacteristic
203C2×C3×C4C2×D3211yescharacteristic
204#67—64no—
205#67—62no—
206C2×Dic3—62no—
207C2×Dic3—182no—
208C2×Dic3—182no—
209C2×Dic3—181no—
210C2×Dic3—181no—
211C2×Dic3—181no—
212C2×Dic3—181no—
213D12—32no—
214C4×D3—62no—
215Dic6—32no—
216Dic6—182no—
217Dic6—182no—
218Dic6—362no—
219C3×C8—62no—
220C3⋊C8—22no—
221C2×C32—21no—
222C2×C32—42no—
223C2×C32#88211yescharacteristic
224C3×D3—122no—
225C2×C9—22no—
226C2×C9—21no—
227C2×C9—21no—
228C2×C9—42no—
229C2×C9—82no—
230C2×C9#95311yescharacteristic
231D9—42no—
232#41—92no—
233#41—182no—
234C2×Q8—271no—
235C2×D4—182no—
236C2×D4—182no—
237C2×D4D3×D911yescharacteristic
238Q16—542no—
239SD16—542no—
240M16—32no—
241C22⋊C4—542no—
242C42—271no—
243C22×C3—22no—
244C22×C3—22no—
245C22×C3—21no—
246C22×C3—42no—
247C22×C3—42no—
248C22×C3—42no—
249C22×C3—44no—
250C22×C3—182no—
251C22×C3—722no—
252C22×C3C2×D3611yescharacteristic
253C22×C3C2×#62611yescharacteristic
254C2×D3—62no—
255C2×D3—242no—
256C3×C4—22no—
257C3×C4—182no—
258C3×C4D4×D912yes—
259C3×C4D3×#6712yes—
260Dic3—62no—
261Dic3—362no—
262Dic3—361no—
263Dic3—362no—
264Dic3—361no—
265Dic3—361no—
266Dic3—361no—
267C32#474611yescharacteristic
268C9—21no—
269C9#524211yescharacteristic, complemented
270C23—182no—
271C23#619212yes—
272Q8—92no—
273Q8—542no—
274D4—22no—
275D4—92no—
276D4—184no—
277D4—182no—
278C2×C4—182no—
279C2×C4—182no—
280C2×C4—541no—
281C2×C4—541no—
282C2×C4C2×D3×D911yescharacteristic
283C8—62no—
284C2×C3—22no—
285C2×C3—21no—
286C2×C3—21no—
287C2×C3—21no—
288C2×C3—42no—
289C2×C3—42no—
290C2×C3—82no—
291C2×C3—362no—
292C2×C3#6344411yescharacteristic
293C2×C3#6397711yescharacteristic
294D3—122no—
295C22—22no—
296C22—42no—
297C22—182no—
298C22—722no—
299C22C2×#619211yescharacteristic
300C4—182no—
301C4—1081no—
302C4—1081no—
303C4D9×#6712yes—
304C3—21no—
305C3#1058842611yescharacteristic, complemented
306C3#1059151311yescharacteristic
307C2—21no—
308C2—42no—
309C2—362no—
310C2#1172127611yescharacteristic
subgroupG/H
centerC2#11721276
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×D4×C9C22
socleC2×C32#882
prime radicalC3×#63710C2
hypercenterC2×D4D3×D9
hypocenterC3×C9#455
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #11721836C2semidirect
3. #11720869C2semidirect
4. C9×#833C2semidirect
5. #11720291C2semidirect
9. C2×#6226C22semidirect
15. C4×C3⋊Dic9C22semidirect
16. C9×#208C22semidirect
90. #833D9semidirect
154. #208C2×D9semidirect
186. C3×C9#455semidirect
269. C9#5242semidirect
305. C3#10588426semidirect

Maximal subgroups

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

subgroupindex
C3×#637102
#117218362
#117208692
C9×#8332
#117202912
#105915133
#105884263