Finite Group Databasebeta

← #13164946

Subgroups — #13164946

Order 1728.

2512 conjugacy classes of subgroups.

The subgroup lattice has 359 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×3−1+2C211yesmaximal, characteristic, complemented
2#11722642C214yesmaximal, complemented
3#11722641C214yesmaximal, complemented
4#41×AGL(1,ℤ/9)C218yesmaximal, complemented
5#11722630C212yesmaximal, complemented
6#63710⋊C3C216yesmaximal, complemented
7C2×#3697⋊C3C216yesmaximal, complemented
8C3×#5964—31nomaximal
9#10593139C311yesmaximal, characteristic, complemented
10#41×3−1+2C2214yescomplemented
11#41×3−1+2C2214yescomplemented
12C2×Q8×3−1+2C2211yescharacteristic, complemented
13C2×D4×3−1+2C2213yescomplemented
14C22×C4×3−1+2C2213yescomplemented
15C2×#611⋊C3C22112yescomplemented
16#88724C2218yescomplemented
17Q8×AGL(1,ℤ/9)C2218yescomplemented
18#3699⋊C3C22124yescomplemented
19D4×AGL(1,ℤ/9)C22124yescomplemented
20#3697⋊C3C2214yescomplemented
21#3697⋊C3C22112yescomplemented
22#3697⋊C3C22112yescomplemented
23#3697⋊C3C22112yescomplemented
24C2×C4⋊AGL(1,ℤ/9)C2216yescomplemented
25C2×C4×AGL(1,ℤ/9)C22112yescomplemented
26C2×3−1+2⋊Q8C2216yescomplemented
27C2×C32×#41D311yescharacteristic
28C3×#1010—34no—
29C3×#1009—34no—
30C3×D3×#41—38no—
31C3×#1007—32no—
32C3×#1004—36no—
33C2×C3×#235—36no—
34C2×C9×#41—21no—
35C2×C9×#41C2×C311yescharacteristic, complemented
36#63716C2×C314yescomplemented
37#63715C2×C314yescomplemented
38#41×D9C2×C318yescomplemented
39#63713C2×C312yescomplemented
40#63710C2×C316yescomplemented
41C2×#3697C2×C316yescomplemented
42C23×3−1+2C2313yes—
43C22×AGL(1,ℤ/9)C23112yes—
44Q8×3−1+2C2314yes—
45D4×3−1+2C23112yes—
46C2×C4×3−1+2C2316yes—
47C2×C4×3−1+2C2316yes—
48C2×C4×3−1+2C2311yescharacteristic
49C2×C4×3−1+2C2313yes—
50#611⋊C3C2318yes—
51#611⋊C3C23124yes—
52C2×3−1+2⋊C4C23112yes—
53C4⋊AGL(1,ℤ/9)C2314yes—
54C4⋊AGL(1,ℤ/9)C23112yes—
55C4×AGL(1,ℤ/9)C2318yes—
56C4×AGL(1,ℤ/9)C23124yes—
573−1+2⋊Q8C2314yes—
583−1+2⋊Q8C23112yes—
59C3×#585—91no—
60#5964—31no—
61C32×#41C2×D314yes—
62C32×#41C2×D314yes—
63C2×C32×Q8C2×D311yescharacteristic
64C2×C32×D4C2×D313yes—
65C22×C32×C4C2×D313yes—
66C2×C3×#67—312no—
67C3×#239—38no—
68C3×D3×Q8—38no—
69C3×#237—324no—
70C3×D3×D4—324no—
71C3×#235—34no—
72C3×#235—312no—
73C3×#235—312no—
74C3×#235—312no—
75C2×C3×D12—36no—
76C2×C3×C4×D3—312no—
77C2×C3×Dic6—36no—
78C9×#41—24no—
79C9×#41—24no—
80C9×#41C22×C314yescomplemented
81C9×#41C22×C314yescomplemented
82C2×Q8×C9—21no—
83C2×Q8×C9C22×C311yescharacteristic, complemented
84C2×D4×C9—23no—
85C2×D4×C9C22×C313yescomplemented
86C22×C4×C9—23no—
87C22×C4×C9C22×C313yescomplemented
88C2×#611C22×C3112yescomplemented
89#3701C22×C318yescomplemented
90Q8×D9C22×C318yescomplemented
91#3699C22×C3124yescomplemented
92D4×D9C22×C3124yescomplemented
93#3697C22×C314yescomplemented
94#3697C22×C3112yescomplemented
95#3697C22×C3112yescomplemented
96#3697C22×C3112yescomplemented
97C2×D36C22×C316yescomplemented
98C2×C4×D9C22×C3112yescomplemented
99C2×Dic18C22×C316yescomplemented
100C22×3−1+2—43no—
101C22×3−1+2C2416yes—
102C22×3−1+2C2411yescharacteristic
103C2×AGL(1,ℤ/9)—412no—
104C2×AGL(1,ℤ/9)C2418yes—
105C4×3−1+2C2411yescharacteristic
106C4×3−1+2C2416yes—
107C4×3−1+2C2411yescharacteristic
1083−1+2⋊C4C2418yes—
109C3×2−1+4—92no—
110C3×2−1+4—94no—
111C3×H(5,𝔽2)—96no—
112C3×H(5,𝔽2)—94no—
113C2×C3×#41—31no—
114C2×C3×#41—96no—
115C2×C3×#41—98no—
116C2×C3×#41C3×D311yescharacteristic
117#1010—34no—
118#1009—34no—
119D3×#41—38no—
120#1007—32no—
121#1004—36no—
122C2×#235—36no—
123C23×C32C22×D313yes—
124C22×C3×D3—312no—
125C32×Q8C22×D314yes—
126C32×D4C22×D3112yes—
127C2×C32×C4C22×D316yes—
128C2×C32×C4C22×D316yes—
129C2×C32×C4C22×D311yescharacteristic
130C2×C32×C4C22×D313yes—
131C3×#67—38no—
132C3×#67—324no—
133C2×C3×Dic3—312no—
134C3×D12—34no—
135C3×D12—312no—
136C3×C4×D3—38no—
137C3×C4×D3—324no—
138C3×Dic6—34no—
139C3×Dic6—312no—
140C23×C9—23no—
141C23×C9C23×C313yes—
142C22×D9C23×C3112yes—
143Q8×C9—24no—
144Q8×C9C23×C314yes—
145D4×C9—212no—
146D4×C9C23×C3112yes—
147C2×C4×C9—21no—
148C2×C4×C9—23no—
149C2×C4×C9—26no—
150C2×C4×C9—26no—
151C2×C4×C9C23×C316yes—
152C2×C4×C9C23×C316yes—
153C2×C4×C9C23×C311yescharacteristic
154C2×C4×C9C23×C313yes—
155#611C23×C318yes—
156#611C23×C3124yes—
157C2×Dic9C23×C3112yes—
158D36C23×C314yes—
159D36C23×C3112yes—
160C4×D9C23×C318yes—
161C4×D9C23×C3124yes—
162Dic18C23×C314yes—
163Dic18C23×C3112yes—
164#585—91noSylow
165C2×3−1+2—21no—
166C2×3−1+2—26no—
167C2×3−1+2C2511yescharacteristic
168AGL(1,ℤ/9)—28no—
169C3×#41—34no—
170C3×#41—34no—
171C3×#41—94no—
172C3×#41—912no—
173C3×#41—912no—
174C3×#41—912no—
175C3×#41—924no—
176C3×#41—98no—
177C3×#41C2×C3×D314yes—
178C3×#41C2×C3×D314yes—
179C2×C3×Q8—31no—
180C2×C3×Q8—96no—
181C2×C3×Q8—98no—
182C2×C3×Q8C2×C3×D311yescharacteristic
183C2×C3×D4—33no—
184C2×C3×D4—96no—
185C2×C3×D4—924no—
186C2×C3×D4—912no—
187C2×C3×D4C2×C3×D313yes—
188C22×C3×C4—33no—
189C22×C3×C4—912no—
190C22×C3×C4C2×C3×D313yes—
191C2×#67—312no—
192#239—38no—
193D3×Q8—38no—
194#237—324no—
195D3×D4—324no—
196#235—34no—
197#235—312no—
198#235—312no—
199#235—312no—
200C2×D12—36no—
201C2×C4×D3—312no—
202C2×Dic6—36no—
203C22×C32—43no—
204C22×C32C23×D316yes—
205C22×C32C23×D311yescharacteristic
206C2×C3×D3—38no—
207C2×C3×D3—1212no—
208C32×C4C23×D311yescharacteristic
209C32×C4C23×D316yes—
210C32×C4C23×D311yescharacteristic
211C3×Dic3—38no—
212C22×C9—21no—
213C22×C9—26no—
214C22×C9—43no—
215C22×C9—46no—
216C22×C9C24×C316yes—
217C22×C9C24×C311yescharacteristic
218C2×D9—412no—
219C2×D9C24×C318yes—
220C4×C9—21no—
221C4×C9—21no—
222C4×C9—26no—
223C4×C9C24×C311yescharacteristic
224C4×C9C24×C316yes—
225C4×C9C24×C311yescharacteristic
226Dic9C24×C318yes—
2272−1+4—94no—
2282−1+4—92no—
229H(5,𝔽2)—94no—
230H(5,𝔽2)—96no—
231C2×#41—98no—
232C2×#41—96no—
233C2×#41AGL(1,ℤ/9)11yescharacteristic, complemented
2343−1+2#58511yescharacteristic, complemented, Sylow
235C23×C3—33no—
236C23×C3—912no—
237C23×C3C22×C3×D313yes—
238C22×D3—312no—
239C3×Q8—34no—
240C3×Q8—94no—
241C3×Q8—912no—
242C3×Q8C22×C3×D314yes—
243C3×D4—312no—
244C3×D4—94no—
245C3×D4—912no—
246C3×D4—98no—
247C3×D4—924no—
248C3×D4C22×C3×D3112yes—
249C2×C3×C4—31no—
250C2×C3×C4—36no—
251C2×C3×C4—33no—
252C2×C3×C4—36no—
253C2×C3×C4—98no—
254C2×C3×C4—924no—
255C2×C3×C4—912no—
256C2×C3×C4C22×C3×D316yes—
257C2×C3×C4C22×C3×D316yes—
258C2×C3×C4C22×C3×D311yescharacteristic
259C2×C3×C4C22×C3×D313yes—
260#67—38no—
261#67—324no—
262C2×Dic3—312no—
263D12—34no—
264D12—312no—
265C4×D3—38no—
266C4×D3—324no—
267Dic6—34no—
268Dic6—312no—
269C2×C32—26no—
270C2×C32—21no—
271C2×C32C24×D311yescharacteristic
272C3×D3—68no—
273C2×C9—22no—
274C2×C9—26no—
275C2×C9—21no—
276C2×C9—21no—
277C2×C9—46no—
278C2×C9C25×C311yescharacteristic
279D9—28no—
280#41—98no—
281#41—924no—
282#41—912no—
283#41—94no—
284#41—912no—
285#41—912no—
286#41C2×AGL(1,ℤ/9)14yescomplemented
287#41C2×AGL(1,ℤ/9)14yescomplemented
288C2×Q8—98no—
289C2×Q8—96no—
290C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic, complemented
291C2×D4—912no—
292C2×D4—924no—
293C2×D4—96no—
294C2×D4C2×AGL(1,ℤ/9)13yescomplemented
295C22×C4—912no—
296C22×C4C2×AGL(1,ℤ/9)13yescomplemented
297C22×C3—31no—
298C22×C3—36no—
299C22×C3—43no—
300C22×C3—98no—
301C22×C3—123no—
302C22×C3—3612no—
303C22×C3C23×C3×D316yes—
304C22×C3C23×C3×D311yescharacteristic
305C2×D3—38no—
306C2×D3—1212no—
307C3×C4—31no—
308C3×C4—36no—
309C3×C4—31no—
310C3×C4—98no—
311C3×C4C23×C3×D311yescharacteristic
312C3×C4C23×C3×D316yes—
313C3×C4C23×C3×D311yescharacteristic
314Dic3—38no—
315C32#596411yescharacteristic
316C9—21no—
317C9C3×#58511yescharacteristic, complemented
318C23—912no—
319C23C22×AGL(1,ℤ/9)13yes—
320Q8—912no—
321Q8—94no—
322Q8C22×AGL(1,ℤ/9)14yes—
323D4—924no—
324D4—98no—
325D4—912no—
326D4—94no—
327D4C22×AGL(1,ℤ/9)112yes—
328C2×C4—912no—
329C2×C4—924no—
330C2×C4—98no—
331C2×C4C22×AGL(1,ℤ/9)16yes—
332C2×C4C22×AGL(1,ℤ/9)13yes—
333C2×C4C22×AGL(1,ℤ/9)16yes—
334C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
335C2×C3—26no—
336C2×C3—21no—
337C2×C3—31no—
338C2×C3—61no—
339C2×C3—66no—
340C2×C3—188no—
341C2×C3C24×C3×D311yescharacteristic
342D3—68no—
343C22—43no—
344C22—98no—
345C22—3612no—
346C22C23×AGL(1,ℤ/9)16yes—
347C22C23×AGL(1,ℤ/9)11yescharacteristic
348C4—98no—
349C4C23×AGL(1,ℤ/9)16yes—
350C4C23×AGL(1,ℤ/9)11yescharacteristic
351C4C23×AGL(1,ℤ/9)11yescharacteristic
352C3—31no—
353C3C3×#596411yescharacteristic
354C2—21no—
355C2—26no—
356C2—188no—
357C2C24×AGL(1,ℤ/9)11yescharacteristic
subgroupG/H
centerC4C23×AGL(1,ℤ/9)
commutator, cosocleC2×C9C25×C3
Frattini, socleC2×C3C24×C3×D3
FittingC2×#41×3−1+2C2
hypercenterC2×#41AGL(1,ℤ/9)
hypocenterC9C3×#585
LCSG ▷ C2×C9 ▷ C9
UCSC1 ◁ C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. C2×#41×3−1+2C2semidirect
2. #11722642C2semidirect
3. #11722641C2semidirect
4. #41×AGL(1,ℤ/9)C2semidirect
5. #11722630C2semidirect
6. #63710⋊C3C2semidirect
7. C2×#3697⋊C3C2semidirect
9. #10593139C3semidirect
10. #41×3−1+2C22semidirect
11. #41×3−1+2C22semidirect
12. C2×Q8×3−1+2C22semidirect
13. C2×D4×3−1+2C22semidirect
14. C22×C4×3−1+2C22semidirect
15. C2×#611⋊C3C22semidirect
16. #88724C22semidirect
17. Q8×AGL(1,ℤ/9)C22semidirect
18. #3699⋊C3C22semidirect
19. D4×AGL(1,ℤ/9)C22semidirect
20. #3697⋊C3C22semidirect
21. #3697⋊C3C22semidirect
22. #3697⋊C3C22semidirect
23. #3697⋊C3C22semidirect
24. C2×C4⋊AGL(1,ℤ/9)C22semidirect
25. C2×C4×AGL(1,ℤ/9)C22semidirect
26. C2×3−1+2⋊Q8C22semidirect
35. C2×C9×#41C2×C3semidirect
36. #63716C2×C3semidirect
37. #63715C2×C3semidirect
38. #41×D9C2×C3semidirect
39. #63713C2×C3semidirect
40. #63710C2×C3semidirect
41. C2×#3697C2×C3semidirect
80. C9×#41C22×C3semidirect
81. C9×#41C22×C3semidirect
83. C2×Q8×C9C22×C3semidirect
85. C2×D4×C9C22×C3semidirect
87. C22×C4×C9C22×C3semidirect
88. C2×#611C22×C3semidirect
89. #3701C22×C3semidirect
90. Q8×D9C22×C3semidirect
91. #3699C22×C3semidirect
92. D4×D9C22×C3semidirect
93. #3697C22×C3semidirect
94. #3697C22×C3semidirect
95. #3697C22×C3semidirect
96. #3697C22×C3semidirect
97. C2×D36C22×C3semidirect
98. C2×C4×D9C22×C3semidirect
99. C2×Dic18C22×C3semidirect
233. C2×#41AGL(1,ℤ/9)semidirect
234. 3−1+2#585semidirect
286. #41C2×AGL(1,ℤ/9)semidirect
287. #41C2×AGL(1,ℤ/9)semidirect
290. C2×Q8C2×AGL(1,ℤ/9)semidirect
294. C2×D4C2×AGL(1,ℤ/9)semidirect
296. C22×C4C2×AGL(1,ℤ/9)semidirect
317. C9C3×#585semidirect

Maximal subgroups

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

subgroupindex
C2×#41×3−1+22
#117226422
#117226412
#41×AGL(1,ℤ/9)2
#117226302
#63710⋊C32
C2×#3697⋊C32
C3×#59643
#105931393