Finite Group Databasebeta

← C3×#10588502 #13141966

Subgroups — C3×#10588502 #13141966

Order 1728.

548 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#63715C211yesmaximal, characteristic, complemented
2C3×#63512C211yesmaximal, characteristic, complemented
3C3×#63470C211yesmaximal, characteristic, complemented
4C3×#63436C211yesmaximal, characteristic, complemented
5C3×C9×#104C211yesmaximal, characteristic, complemented
6C3×#63383C211yesmaximal, characteristic, complemented
7C3×#63364C211yesmaximal, characteristic, complemented
8C3×#4822—31nomaximal
9#10588502C311yesmaximal, characteristic, direct product factor
10C3×C9×#41C2211yescharacteristic
11C3×#3701—21no—
12C3×D4×D9—22no—
13C3×D4×D9—21no—
14C3×#3697—21no—
15C3×#3697C2211yescharacteristic
16C2×C3×D36—21no—
17C2×C3×D36—22no—
18C2×C3×D36C2211yescharacteristic
19C2×C3×D36C2211yescharacteristic, complemented
20C3×C9×M16C2211yescharacteristic, complemented
21C3×C42×C9C2211yescharacteristic, complemented
22C3×#3675—21no—
23C3×#3673—21no—
24C3×#3667C2211yescharacteristic, complemented
25C3×D72—21no—
26C3×#3664—21no—
27C3×#1009—31no—
28C3×#949—31no—
29C3×#908—31no—
30C3×#874—31no—
31#63715C2×C311yescharacteristic, complemented
32C32×#104D311yescharacteristic
33C3×#823—31no—
34C3×#805—31no—
35#63512C2×C311yescharacteristic, complemented
36#63470C2×C311yescharacteristic, complemented
37#63436C2×C311yescharacteristic, complemented
38C9×#104—21no—
39C9×#104C2×C311yescharacteristic, complemented
40#63383C2×C311yescharacteristic, complemented
41#63364C2×C311yescharacteristic, complemented
42C22×C3×D9—22no—
43C22×C3×D9—21no—
44C22×C3×D9D412yes—
45C3×Q8×C9D411yescharacteristic, complemented
46C3×D4×C9—21no—
47C3×D4×C9D411yescharacteristic, complemented
48C2×C3×C4×C9—21no—
49C2×C3×C4×C9—21no—
50C2×C3×C4×C9C2311yescharacteristic
51C3×#611—22no—
52C3×#611—21no—
53C3×D36—22no—
54C3×D36—21no—
55C3×D36—21no—
56C3×D36—21no—
57C3×D36—21no—
58C3×D36—44no—
59C3×D36D411yescharacteristic, complemented
60C3×C4×D9—22no—
61C3×C4×D9—21no—
62C3×Dic18D411yescharacteristic, complemented
63C3×C8×C9—21no—
64C3×C9⋊C8—21no—
65C3×#453—91no—
66#4822—31no—
67C32×#41C2×D311yescharacteristic
68C3×#239—61no—
69C3×D3×D4—62no—
70C3×D3×D4—61no—
71C3×#235—31no—
72C3×#235—61no—
73C2×C3×D12—31no—
74C2×C3×D12—31no—
75C2×C3×D12—61no—
76C2×C3×D12—62no—
77C32×M16C2×D311yescharacteristic
78C32×C42C2×D311yescharacteristic
79C3×#215—61no—
80C3×#213—61no—
81C3×#208—31no—
82C3×D24—61no—
83C3×#204—61no—
84C9×#41—21no—
85C9×#41C22×C311yescharacteristic
86#3701—21no—
87D4×D9—22no—
88D4×D9—21no—
89#3697—21no—
90#3697C22×C311yescharacteristic
91C2×D36—21no—
92C2×D36—22no—
93C2×D36C22×C311yescharacteristic
94C2×D36C22×C311yescharacteristic, complemented
95C9×M16—21no—
96C9×M16C22×C311yescharacteristic, complemented
97C42×C9—21no—
98C42×C9C22×C311yescharacteristic, complemented
99#3675—21no—
100#3673—21no—
101#3667C22×C311yescharacteristic, complemented
102D72—21no—
103#3664—21no—
104C22×C3×C9—21no—
105C22×C3×C9C2×D411yescharacteristic
106C2×C3×D9—22no—
107C2×C3×D9—21no—
108C2×C3×D9—42no—
109C2×C3×D9—42no—
110C2×C3×D9—41no—
111C2×C3×D9—82no—
112C3×C4×C9—22no—
113C3×C4×C9—21no—
114C3×C4×C9C2×D411yescharacteristic
115C3×C4×C9C2×D411yescharacteristic
116C3×Dic9—21no—
117C3×H(5,𝔽2)—91no—
118#1009—31no—
119C3×AGL(1,ℤ/8)—91no—
120C3×AGL(1,ℤ/8)—91no—
121C3×#127—91no—
122#949—31no—
123#908—31no—
124#874—31no—
125C3×#104—21no—
126C3×#104—91no—
127C3×#104D911yescharacteristic, complemented
128C3×#104C3×D311yescharacteristic
129C3×#100—91no—
130#823—31no—
131#805—31no—
132C22×C3×D3—32no—
133C22×C3×D3—62no—
134C22×C3×D3—61no—
135C32×Q8D1211yescharacteristic
136C32×D4—21no—
137C32×D4D1211yescharacteristic
138C2×C32×C4—21no—
139C2×C32×C4—21no—
140C2×C32×C4C22×D311yescharacteristic
141C3×#67—62no—
142C3×#67—61no—
143C3×D12—31no—
144C3×D12—62no—
145C3×D12—61no—
146C3×D12—61no—
147C3×D12—61no—
148C3×D12—61no—
149C3×D12—124no—
150C3×C4×D3—62no—
151C3×C4×D3—61no—
152C3×Dic6—31no—
153C22×D9—22no—
154C22×D9—21no—
155C22×D9C3×D412yes—
156C32×C8—21no—
157C3×C3⋊C8—61no—
158Q8×C9—21no—
159Q8×C9C3×D411yescharacteristic, complemented
160D4×C9—21no—
161D4×C9—21no—
162D4×C9—41no—
163D4×C9C3×D411yescharacteristic, complemented
164C2×C4×C9—21no—
165C2×C4×C9—21no—
166C2×C4×C9—21no—
167C2×C4×C9—41no—
168C2×C4×C9—41no—
169C2×C4×C9C23×C311yescharacteristic
170#611—22no—
171#611—21no—
172D36—22no—
173D36—21no—
174D36—21no—
175D36—21no—
176D36—21no—
177D36—44no—
178D36C3×D411yescharacteristic, complemented
179C4×D9—22no—
180C4×D9—21no—
181Dic18C3×D411yescharacteristic, complemented
182C8×C9—21no—
183C8×C9—41no—
184C9⋊C8—21no—
185#453—91noSylow
186C2×C3×C9—21no—
187C2×C3×C9—41no—
188C2×C3×C9#12011yescharacteristic
189C3×D9—42no—
190C3×D9—41no—
191C3×D9—81no—
192C3×#41—21no—
193C3×#41—91no—
194C3×#41—181no—
195C3×#41—181no—
196C3×#41C2×D911yescharacteristic
197C3×#41C2×C3×D311yescharacteristic
198C2×C3×D4—91no—
199C2×C3×D4—91no—
200C2×C3×D4—181no—
201C2×C3×D4—182no—
202C2×C3×D4—182no—
203C2×C3×D4—181no—
204#239—61no—
205D3×D4—62no—
206D3×D4—61no—
207#235—31no—
208#235—61no—
209C2×D12—31no—
210C2×D12—31no—
211C2×D12—61no—
212C2×D12—62no—
213C3×SD16—181no—
214C3×SD16—181no—
215C3×D8—181no—
216C3×D8—181no—
217C3×M16—21no—
218C3×M16—91no—
219C3×M16C2×D911yescharacteristic, complemented
220C3×M16C2×C3×D311yescharacteristic
221C3×C42—21no—
222C3×C42C2×D911yescharacteristic, complemented
223C3×C42C2×C3×D311yescharacteristic
224#215—61no—
225#213—61no—
226#208—31no—
227D24—61no—
228#204—61no—
229C22×C32—21no—
230C22×C32D3×D411yescharacteristic
231C2×C3×D3—61no—
232C2×C3×D3—62no—
233C2×C3×D3—122no—
234C2×C3×D3—122no—
235C2×C3×D3—121no—
236C2×C3×D3—242no—
237C32×C4—22no—
238C32×C4—21no—
239C32×C4C2×D1211yescharacteristic
240C32×C4D3×D411yescharacteristic
241C3×Dic3—61no—
242C22×C9—21no—
243C22×C9—21no—
244C22×C9—41no—
245C22×C9C2×C3×D411yescharacteristic
246C2×D9—22no—
247C2×D9—21no—
248C2×D9—42no—
249C2×D9—42no—
250C2×D9—41no—
251C2×D9—82no—
252C4×C9—22no—
253C4×C9—21no—
254C4×C9—21no—
255C4×C9—21no—
256C4×C9—41no—
257C4×C9—42no—
258C4×C9C2×C3×D411yescharacteristic
259C4×C9C2×C3×D411yescharacteristic
260Dic9—21no—
261H(5,𝔽2)—91no—
262AGL(1,ℤ/8)—91no—
263AGL(1,ℤ/8)—91no—
264#127—91no—
265#104—91no—
266#104C3×D911yescharacteristic, complemented
267#100—91no—
268C3×C9#45311yescharacteristic, complemented, Sylow
269C23×C3—92no—
270C23×C3—182no—
271C23×C3—181no—
272C22×D3—32no—
273C22×D3—62no—
274C22×D3—61no—
275C3×Q8—21no—
276C3×Q8—91no—
277C3×Q8D3611yescharacteristic, complemented
278C3×Q8C3×D1211yescharacteristic
279C3×D4—21no—
280C3×D4—21no—
281C3×D4—21no—
282C3×D4—41no—
283C3×D4—91no—
284C3×D4—182no—
285C3×D4—182no—
286C3×D4—181no—
287C3×D4—181no—
288C3×D4—181no—
289C3×D4—181no—
290C3×D4—181no—
291C3×D4—364no—
292C3×D4D3611yescharacteristic, complemented
293C3×D4C3×D1211yescharacteristic
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—182no—
302C2×C3×C4—181no—
303C2×C3×C4C22×D911yescharacteristic
304C2×C3×C4C22×C3×D311yescharacteristic
305#67—62no—
306#67—61no—
307D12—31no—
308D12—62no—
309D12—61no—
310D12—61no—
311D12—61no—
312D12—61no—
313D12—124no—
314C4×D3—62no—
315C4×D3—61no—
316Dic6—31no—
317C3×C8—21no—
318C3×C8—21no—
319C3×C8—41no—
320C3×C8—181no—
321C3⋊C8—61no—
322C2×C32—21no—
323C2×C32—41no—
324C2×C32#88211yescharacteristic
325C3×D3—122no—
326C3×D3—121no—
327C3×D3—241no—
328C2×C9—21no—
329C2×C9—21no—
330C2×C9—41no—
331C2×C9—41no—
332C2×C9—81no—
333C2×C9C3×#12011yescharacteristic
334D9—42no—
335D9—41no—
336D9—81no—
337#41—91no—
338#41—181no—
339#41—181no—
340#41C2×C3×D911yescharacteristic
341C2×D4—91no—
342C2×D4—91no—
343C2×D4—181no—
344C2×D4—182no—
345C2×D4—182no—
346C2×D4—181no—
347SD16—181no—
348SD16—181no—
349D8—181no—
350D8—181no—
351M16—91no—
352M16C2×C3×D911yescharacteristic, complemented
353C42C2×C3×D911yescharacteristic, complemented
354C22×C3—21no—
355C22×C3—21no—
356C22×C3—21no—
357C22×C3—41no—
358C22×C3—181no—
359C22×C3—182no—
360C22×C3—362no—
361C22×C3—362no—
362C22×C3—361no—
363C22×C3—722no—
364C22×C3D4×D911yescharacteristic
365C22×C3C3×D3×D411yescharacteristic
366C2×D3—61no—
367C2×D3—62no—
368C2×D3—122no—
369C2×D3—122no—
370C2×D3—121no—
371C2×D3—242no—
372C3×C4—22no—
373C3×C4—22no—
374C3×C4—21no—
375C3×C4—21no—
376C3×C4—21no—
377C3×C4—21no—
378C3×C4—41no—
379C3×C4—42no—
380C3×C4—181no—
381C3×C4C2×D3611yescharacteristic
382C3×C4D4×D911yescharacteristic
383C3×C4C2×C3×D1211yescharacteristic
384C3×C4C3×D3×D411yescharacteristic
385Dic3—61no—
386C32#482211yescharacteristic
387C9—21no—
388C9C3×#45311yescharacteristic, complemented
389C23—92no—
390C23—182no—
391C23—181no—
392Q8—91no—
393Q8C3×D3611yescharacteristic, complemented
394D4—21no—
395D4—91no—
396D4—182no—
397D4—182no—
398D4—181no—
399D4—181no—
400D4—181no—
401D4—181no—
402D4—181no—
403D4—364no—
404D4C3×D3611yescharacteristic, complemented
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—182no—
408C2×C4—181no—
409C2×C4C22×C3×D911yescharacteristic
410C8—21no—
411C8—181no—
412C2×C3—21no—
413C2×C3—21no—
414C2×C3—21no—
415C2×C3—41no—
416C2×C3—41no—
417C2×C3—41no—
418C2×C3—81no—
419C2×C3—362no—
420C2×C3—361no—
421C2×C3—721no—
422C2×C3#6344411yescharacteristic
423C2×C3C3×#88211yescharacteristic
424D3—122no—
425D3—121no—
426D3—241no—
427C22—21no—
428C22—181no—
429C22—182no—
430C22—362no—
431C22—362no—
432C22—361no—
433C22—722no—
434C22C3×D4×D911yescharacteristic
435C4—22no—
436C4—21no—
437C4—181no—
438C4C2×C3×D3611yescharacteristic
439C4C3×D4×D911yescharacteristic
440C3—21no—
441C3#1058850211yescharacteristic, direct product factor
442C3C3×#482211yescharacteristic
443C2—21no—
444C2—41no—
445C2—362no—
446C2—361no—
447C2—721no—
448C2C3×#6344411yescharacteristic
subgroupG/H
centerC2×C3#63444
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×#104C2
socleC2×C32#882
hypercenterC3×#104D9
hypocenterC9C3×#453
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#104

Complements

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

NH ≅ G/Ntype
1. C3×#63715C2semidirect
2. C3×#63512C2semidirect
3. C3×#63470C2semidirect
4. C3×#63436C2semidirect
5. C3×C9×#104C2semidirect
6. C3×#63383C2semidirect
7. C3×#63364C2semidirect
9. #10588502C3direct (N × H)
19. C2×C3×D36C22semidirect
20. C3×C9×M16C22semidirect
21. C3×C42×C9C22semidirect
24. C3×#3667C22semidirect
31. #63715C2×C3semidirect
35. #63512C2×C3semidirect
36. #63470C2×C3semidirect
37. #63436C2×C3semidirect
39. C9×#104C2×C3semidirect
40. #63383C2×C3semidirect
41. #63364C2×C3semidirect
45. C3×Q8×C9D4semidirect
47. C3×D4×C9D4semidirect
59. C3×D36D4semidirect
62. C3×Dic18D4semidirect
94. C2×D36C22×C3semidirect
96. C9×M16C22×C3semidirect
98. C42×C9C22×C3semidirect
101. #3667C22×C3semidirect
127. C3×#104D9semidirect
159. Q8×C9C3×D4semidirect
163. D4×C9C3×D4semidirect
178. D36C3×D4semidirect
181. Dic18C3×D4semidirect
219. C3×M16C2×D9semidirect
222. C3×C42C2×D9semidirect
266. #104C3×D9semidirect
268. C3×C9#453semidirect
277. C3×Q8D36semidirect
292. C3×D4D36semidirect
352. M16C2×C3×D9semidirect
353. C42C2×C3×D9semidirect
388. C9C3×#453semidirect
393. Q8C3×D36semidirect
404. D4C3×D36semidirect
441. C3#10588502direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#637152
C3×#635122
C3×#634702
C3×#634362
C3×C9×#1042
C3×#633832
C3×#633642
C3×#48223
#105885023