Finite Group Databasebeta

← #13141049

Subgroups — #13141049

Order 1728.

356 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C9×#950C211yesmaximal, characteristic
2C3×#63510C211yesmaximal, characteristic, complemented
3#11720877C211yesmaximal, characteristic, complemented
4#11720585C211yesmaximal, characteristic, complemented
5#11720300C211yesmaximal, characteristic, complemented
6#11720280C211yesmaximal, characteristic, complemented
7#11720275C211yesmaximal, characteristic, complemented
8#10591674—31nomaximal
9#10588877—31nomaximal
10C3×C9×#41C2211yescharacteristic
11C9×#235C2211yescharacteristic
12C4×C3⋊Dic9C2211yescharacteristic
13#88532C2211yescharacteristic
14C9×#216C411yescharacteristic, complemented
15C9×#215C411yescharacteristic, complemented
16C9×#214C411yescharacteristic, complemented
17C9×#213C411yescharacteristic, complemented
18C2×C9×C3⋊C8C2211yescharacteristic
19C3×#3667—21no—
20C3×#3667C2211yescharacteristic
21C2×C3×C9⋊C8—21no—
22C2×C3×C9⋊C8C2211yescharacteristic
23#88420—21no—
24D3×C9⋊C8—21no—
25C3×#950D311yescharacteristic
26C3×#948—31no—
27#63814—31no—
28#63664—31no—
29#63577—31no—
30#63568—31no—
31#63555—31no—
32C9×#135—31no—
33#63510—31no—
34#63510D311yescharacteristic, complemented
35#63396—31no—
36#63396—31no—
37#63396—61no—
38#63375—31no—
39#63372—31no—
40C3×Q8×C9C2×C411yescharacteristic
41C3×D4×C9—21no—
42C3×D4×C9C2×C411yescharacteristic
43C2×C3×C4×C9—21no—
44C2×C3×C4×C9C2311yescharacteristic
45C2×C3⋊Dic9—21no—
46C9×#67—21no—
47C9×D12C2×C411yescharacteristic
48C4×D3×C9—21no—
49C9×Dic6C2×C411yescharacteristic
50C3⋊(C9⋊C8)—21no—
51C9×C3⋊C8C2×C411yescharacteristic
52C9×C3⋊C8C2×C411yescharacteristic
53C3×C9⋊C8—21no—
54C3×C9⋊C8D411yescharacteristic
55C3×C9⋊C8D411yescharacteristic
56#5197—91no—
57#5145—91no—
58C32×#41C2×D311yescharacteristic
59C3×#235C2×D311yescharacteristic
60C4×C3⋊Dic3—31no—
61#3748—31no—
62C3×#216Dic311yescharacteristic
63C3×#215Dic311yescharacteristic
64C3×#214Dic311yescharacteristic
65C3×#213Dic311yescharacteristic
66C3×#208—31no—
67C3×#208—61no—
68C2×C3×C3⋊C8—31no—
69C2×C3×C3⋊C8—61no—
70C2×C3×C3⋊C8C2×D311yescharacteristic
71#3711—61no—
72D3×C3⋊C8—61no—
73C9×#41—21no—
74C9×#41—31no—
75C9×#41C2×D311yescharacteristic
76C9×Q16—31no—
77C9×SD16—31no—
78C9×SD16—31no—
79C9×D8—31no—
80C2×C8×C9—31no—
81C4×Dic9—31no—
82C4×Dic9—61no—
83#3667—21no—
84#3667—31no—
85#3667—61no—
86#3667—61no—
87#3667C2×D311yescharacteristic
88C2×C9⋊C8—21no—
89C2×C9⋊C8—61no—
90C2×C9⋊C8C2×D311yescharacteristic
91C22×C3×C9—21no—
92C22×C3×C9#4111yescharacteristic
93C2×D3×C9—21no—
94C3×C4×C9—21no—
95C3×C4×C9C22×C411yescharacteristic
96C3×C4×C9C2×D411yescharacteristic
97C3⋊Dic9—41no—
98C9×Dic3—21no—
99C3×#135—31no—
100C3×#131—91no—
101#950D911yescharacteristic
102#948—31no—
103#948—91no—
104#901—91no—
105#837—91no—
106#837—91no—
107#837—91no—
108#837—181no—
109#825—91no—
110#822—91no—
111#819—91no—
112#815—91no—
113#815—91no—
114C32×Q8C2×Dic311yescharacteristic
115C32×D4—21no—
116C32×D4C2×Dic311yescharacteristic
117C2×C32×C4—21no—
118C2×C32×C4C22×D311yescharacteristic
119C2×C3⋊Dic3—61no—
120C3×#67—21no—
121C3×D12C2×Dic311yescharacteristic
122C3×C4×D3—21no—
123C3×Dic6C2×Dic311yescharacteristic
124C3⋊(C3⋊C8)—61no—
125C3×C3⋊C8—31no—
126C3×C3⋊C8—31no—
127C3×C3⋊C8—61no—
128C3×C3⋊C8C2×Dic311yescharacteristic
129C3×C3⋊C8C2×Dic311yescharacteristic
130Q8×C9—21no—
131Q8×C9—31no—
132Q8×C9C4×D311yescharacteristic
133D4×C9—21no—
134D4×C9—21no—
135D4×C9—31no—
136D4×C9—41no—
137D4×C9—61no—
138D4×C9C4×D311yescharacteristic
139C2×C4×C9—21no—
140C2×C4×C9—21no—
141C2×C4×C9—41no—
142C2×C4×C9—61no—
143C2×C4×C9C22×D311yescharacteristic
144C2×Dic9—61no—
145C2×Dic9—121no—
146C8×C9—31no—
147C8×C9—31no—
148C9⋊C8—21no—
149C9⋊C8—61no—
150C9⋊C8—121no—
151C9⋊C8#6711yescharacteristic
152C9⋊C8#6711yescharacteristic
153#444—271noSylow
154C2×C3×C9—21no—
155C2×C3×C9—41no—
156C2×C3×C9C4×D411yescharacteristic
157D3×C9—41no—
158C3×#41—21no—
159C3×#41—31no—
160C3×#41C2×D911yescharacteristic
161C3×#41D3211yescharacteristic
162#235C2×D911yescharacteristic
163C3×Q16—31no—
164C3×SD16—31no—
165C3×SD16—31no—
166C3×D8—31no—
167C3×M16—91no—
168C3×M16—181no—
169C2×C3×C8—31no—
170C2×C3×C8—91no—
171C2×C3×C8—181no—
172#216Dic911yescharacteristic, complemented
173#215Dic911yescharacteristic, complemented
174#214Dic911yescharacteristic, complemented
175#213Dic911yescharacteristic, complemented
176C4×Dic3—91no—
177C4×Dic3—91no—
178C4×Dic3—181no—
179#208—31no—
180#208—61no—
181#208—91no—
182#208—91no—
183#208—181no—
184#208—181no—
185C2×C3⋊C8—31no—
186C2×C3⋊C8—61no—
187C2×C3⋊C8—181no—
188C2×C3⋊C8C2×D911yescharacteristic
189ΓL(1,𝔽25)—181no—
190D3×C8—181no—
191C22×C32—21no—
192C22×C32#23711yescharacteristic
193C2×C3×D3—21no—
194C32×C4—21no—
195C32×C4D3×D411yescharacteristic
196C32×C4C22×Dic311yescharacteristic
197C3⋊Dic3—121no—
198C3×Dic3—21no—
199C22×C9—21no—
200C22×C9—21no—
201C22×C9—41no—
202C22×C9—61no—
203C22×C9#23511yescharacteristic
204C4×C9—21no—
205C4×C9—21no—
206C4×C9—21no—
207C4×C9—41no—
208C4×C9—61no—
209C4×C9C2×C4×D311yescharacteristic
210C4×C9C2×#6711yescharacteristic
211Dic9—121no—
212Dic9—122no—
213#135—31no—
214#131—91no—
215#131—271no—
216#108—271no—
217#104—271no—
218#104—271no—
219#97—271no—
220C3×C9#44411yescharacteristic, complemented, Sylow
221C3×Q8—21no—
222C3×Q8—31no—
223C3×Q8C2×Dic911yescharacteristic
224C3×Q8D3×Dic311yescharacteristic
225C3×D4—21no—
226C3×D4—21no—
227C3×D4—21no—
228C3×D4—31no—
229C3×D4—41no—
230C3×D4—61no—
231C3×D4C2×Dic911yescharacteristic
232C3×D4D3×Dic311yescharacteristic
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—41no—
237C2×C3×C4—61no—
238C2×C3×C4C22×D911yescharacteristic
239C2×C3×C4C2×D3211yescharacteristic
240#67—21no—
241C2×Dic3—181no—
242C2×Dic3—181no—
243C2×Dic3—361no—
244D12C2×Dic911yescharacteristic
245C4×D3—21no—
246Dic6C2×Dic911yescharacteristic
247C3×C8—31no—
248C3×C8—31no—
249C3×C8—91no—
250C3×C8—91no—
251C3×C8—181no—
252C3⋊C8—31no—
253C3⋊C8—31no—
254C3⋊C8—61no—
255C3⋊C8—181no—
256C3⋊C8—181no—
257C3⋊C8—361no—
258C3⋊C8C2×Dic911yescharacteristic
259C3⋊C8C2×Dic911yescharacteristic
260C2×C32—21no—
261C2×C32—41no—
262C2×C32D4×Dic311yescharacteristic
263C3×D3—41no—
264C2×C9—21no—
265C2×C9—21no—
266C2×C9—41no—
267C2×C9—41no—
268C2×C9—81no—
269C2×C9—121no—
270C2×C9C4×#6711yescharacteristic
271#41—31no—
272#41D3×D911yescharacteristic
273Q16—31no—
274SD16—31no—
275SD16—31no—
276D8—31no—
277M16—91no—
278M16—181no—
279M16—271no—
280M16—541no—
281C2×C8—31no—
282C2×C8—91no—
283C2×C8—181no—
284C2×C8—541no—
285C42—271no—
286C22×C3—21no—
287C22×C3—21no—
288C22×C3—21no—
289C22×C3—41no—
290C22×C3—61no—
291C22×C3#369911yescharacteristic
292C22×C3#380411yescharacteristic
293C2×D3—21no—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—41no—
299C3×C4—61no—
300C3×C4D4×D911yescharacteristic
301C3×C4C22×Dic911yescharacteristic
302C3×C4C2×D3×Dic311yescharacteristic
303C3×C4D3×#6711yescharacteristic
304Dic3—21no—
305Dic3—361no—
306Dic3—361no—
307Dic3—362no—
308C32#519711yescharacteristic
309C9—21no—
310C9#514511yescharacteristic, complemented
311Q8—31no—
312Q8D3×Dic911yescharacteristic
313D4—21no—
314D4—31no—
315D4—61no—
316D4D3×Dic911yescharacteristic
317C2×C4—21no—
318C2×C4—61no—
319C2×C4—541no—
320C2×C4C2×D3×D911yescharacteristic
321C8—31no—
322C8—31no—
323C8—91no—
324C8—91no—
325C8—181no—
326C8—541no—
327C2×C3—21no—
328C2×C3—21no—
329C2×C3—21no—
330C2×C3—41no—
331C2×C3—41no—
332C2×C3—41no—
333C2×C3—81no—
334C2×C3—121no—
335C2×C3D4×Dic911yescharacteristic
336C2×C3Dic3×#6711yescharacteristic
337D3—41no—
338C22—21no—
339C22—61no—
340C22#8866211yescharacteristic
341C4—21no—
342C4—61no—
343C4—1081no—
344C4C2×D3×Dic911yescharacteristic
345C4D9×#6711yescharacteristic
346C3—21no—
347C3#1058887711yescharacteristic, complemented
348C3#1059167411yescharacteristic
349C2—21no—
350C2—41no—
351C2—121no—
352C2#67×Dic911yescharacteristic
subgroupG/H
centerC4D9×#67
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC3×C9×#41C22
socleC2×C32D4×Dic3
prime radicalC3×#950D3
cosocleC2×C3×C4×C9C23
hypercenter#41D3×D9
hypocenterC3×C9#444
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C4 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
2. C3×#63510C2semidirect
3. #11720877C2semidirect
4. #11720585C2semidirect
5. #11720300C2semidirect
6. #11720280C2semidirect
7. #11720275C2semidirect
14. C9×#216C4semidirect
15. C9×#215C4semidirect
16. C9×#214C4semidirect
17. C9×#213C4semidirect
34. #63510D3semidirect
172. #216Dic9semidirect
173. #215Dic9semidirect
174. #214Dic9semidirect
175. #213Dic9semidirect
220. C3×C9#444semidirect
310. C9#5145semidirect
347. C3#10588877semidirect

Maximal subgroups

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

subgroupindex
C9×#9502
C3×#635102
#117208772
#117205852
#117203002
#117202802
#117202752
#105916743
#105888773