Finite Group Databasebeta

← C3×#10596514 #13181033

Subgroups — C3×#10596514 #13181033

Order 1728.

976 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 976 conjugacy classes of subgroups — too many to draw. Only its 170 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×#64313C212yesmaximal, complemented
2C3×#64287C211yesmaximal, characteristic, complemented
3C3×#64237C212yesmaximal, complemented
4C3×#64236C214yesmaximal, complemented
5C2×C3×#3774C212yesmaximal, complemented
6C3×#64225C211yesmaximal, characteristic, complemented
7C3×#64224C212yesmaximal, complemented
8C3×#64223C211yesmaximal, characteristic, complemented
9#10596514C311yesmaximal, characteristic, direct product factor
10C3×#3830C2211yescharacteristic
11C32×D3×D4—22no—
12C3×D3×#67—24no—
13C2×C3×#625C2214yescomplemented
14C3×#3805C2214yescomplemented
15C3×#3802C2212yescomplemented
16C3×#3798C2212yescomplemented
17C3×#3795—22no—
18C3×#3792C2212yescomplemented
19C2×C3×#622C2212yescomplemented
20C3×#3788C2211yescharacteristic, complemented
21C3×#3787C2211yescharacteristic, complemented
22C3×#3776C2211yescharacteristic, complemented
23C3×#3775C2212yescomplemented
24C3×#3775C2214yescomplemented
25C3×#3774C2211yescharacteristic, complemented
26C3×#3774C2214yescomplemented
27C3×#3774C2212yescomplemented
28C3×#3774C2212yescomplemented
29C3×#1004—62no—
30#64313C2×C312yescomplemented
31#64287C2×C311yescharacteristic, complemented
32#64237C2×C312yescomplemented
33#64236C2×C314yescomplemented
34C2×#3774C2×C312yescomplemented
35#64225C2×C311yescharacteristic, complemented
36#64224C2×C312yescomplemented
37#64223C2×C311yescharacteristic, complemented
38C22×C32×D3—24no—
39C2×C3×D32—22no—
40C33×D4D411yescharacteristic
41C3×#638D412yes—
42C2×C3×C3⋊Dic3C2312yes—
43C3×C4×C3⋊D3C2311yescharacteristic
44C3×#634D411yescharacteristic
45C32×#67—24no—
46C32×D12—22no—
47C32×C4×D3—22no—
48C3×#627C2312yes—
49C3×#626—22no—
50C3×#625—24no—
51C3×#625C2312yes—
52C3×#625C2314yes—
53C3×D3×Dic3—24no—
54C3×#622C2311yescharacteristic
55C3×#622C2311yescharacteristic
56C3×#622C2312yes—
57C3×#576—91nomaximal
58C2×C32×D4—62no—
59#3830C22×C311yescharacteristic
60C2×C3×#67—64no—
61C2×C3×#67—64no—
62C3×#237—62no—
63C3×#237—64no—
64C3×#237—62no—
65C3×D3×D4—22no—
66C3×D3×D4—22no—
67C3×D3×D4—42no—
68C3×D3×D4—62no—
69C3×D3×D4—64no—
70C3×#235—62no—
71C3×#235—62no—
72D3×#67—24no—
73C2×#625C22×C314yescomplemented
74#3805C22×C314yescomplemented
75#3802C22×C312yescomplemented
76#3798C22×C312yescomplemented
77#3795—22no—
78#3792C22×C312yescomplemented
79C2×#622C22×C312yescomplemented
80#3788C22×C311yescharacteristic, complemented
81#3787C22×C311yescharacteristic, complemented
82#3776C22×C311yescharacteristic, complemented
83#3775C22×C312yescomplemented
84#3775C22×C314yescomplemented
85#3774C22×C311yescharacteristic, complemented
86#3774C22×C314yescomplemented
87#3774C22×C312yescomplemented
88#3774C22×C312yescomplemented
89C22×C33C2×D412yes—
90C2×C3×C3⋊D3C2×D411yescharacteristic
91C2×C32×D3—24no—
92C2×C32×D3—22no—
93C2×C32×D3—84no—
94C3×D32—82no—
95C33×C4C2×D411yescharacteristic
96C3×C3⋊Dic3C2×D412yes—
97C3×C3⋊Dic3C2411yescharacteristic
98C32×Dic3—22no—
99C3×H(5,𝔽2)—92no—
100#1004—62no—
101C3×AGL(1,ℤ/8)—94no—
102C3×AGL(1,ℤ/8)—92no—
103C3×#135—92no—
104C3×#135—91no—
105C2×C3×D8—92no—
106C2×C3×D8—91no—
107C3×#131—91no—
108C23×C32—64no—
109C22×C3×D3—24no—
110C22×C3×D3—24no—
111C22×C3×D3—44no—
112C22×C3×D3—64no—
113C2×D32—22no—
114C32×D4—22no—
115C32×D4—42no—
116C32×D4—62no—
117C32×D4—64no—
118C32×D4C3×D411yescharacteristic
119C2×C32×C4—62no—
120#638C3×D412yes—
121C2×C3⋊Dic3C23×C312yes—
122C4×C3⋊D3C23×C311yescharacteristic
123#634C3×D411yescharacteristic
124C3×#67—24no—
125C3×#67—24no—
126C3×#67—44no—
127C3×#67—62no—
128C3×#67—64no—
129C3×#67—64no—
130C3×#67—62no—
131C3×#67—64no—
132C3×#67—64no—
133C2×C3×Dic3—64no—
134C2×C3×Dic3—64no—
135C3×D12—22no—
136C3×D12—22no—
137C3×D12—42no—
138C3×D12—62no—
139C3×C4×D3—22no—
140C3×C4×D3—22no—
141C3×C4×D3—42no—
142C3×C4×D3—62no—
143C3×C4×D3—64no—
144C3×Dic6—62no—
145C3×Dic6—62no—
146#627C23×C312yes—
147#626—22no—
148#625—24no—
149#625C23×C312yes—
150#625C23×C314yes—
151D3×Dic3—24no—
152#622C23×C311yescharacteristic
153#622C23×C311yescharacteristic
154#622C23×C312yes—
155#576—91noSylow
156C2×C33—22no—
157C2×C33C22×D411yescharacteristic
158C3×C3⋊D3—21no—
159C32×D3—42no—
160C32×D3—44no—
161C3×#41—92no—
162C3×#41—91no—
163C3×#41—94no—
164C3×#41—182no—
165C2×C3×D4—62no—
166C2×C3×D4—62no—
167C2×C3×D4—92no—
168C2×C3×D4—94no—
169C2×C3×D4—122no—
170C2×C3×D4—184no—
171C2×#67—64no—
172C2×#67—64no—
173#237—64no—
174#237—62no—
175#237—62no—
176D3×D4—22no—
177D3×D4—62no—
178D3×D4—64no—
179#235—62no—
180#235—62no—
181C3×Q16—91no—
182C3×SD16—92no—
183C3×SD16—94no—
184C3×D8—91no—
185C3×D8—92no—
186C3×D8—94no—
187C3×D8—92no—
188C3×M16—91no—
189C3×M16—92no—
190C2×C3×C8—91no—
191C2×C3×C8—92no—
192C22×C32—24no—
193C22×C32—44no—
194C22×C32—64no—
195C22×C32—62no—
196C22×C32—244no—
197C22×C32C2×C3×D412yes—
198C2×C3⋊D3C2×C3×D411yescharacteristic
199C2×C3×D3—22no—
200C2×C3×D3—24no—
201C2×C3×D3—24no—
202C2×C3×D3—22no—
203C2×C3×D3—42no—
204C2×C3×D3—44no—
205C2×C3×D3—62no—
206C2×C3×D3—84no—
207C2×C3×D3—84no—
208C2×C3×D3—88no—
209C2×C3×D3—244no—
210D32—82no—
211C32×C4—22no—
212C32×C4—42no—
213C32×C4—62no—
214C32×C4C2×C3×D411yescharacteristic
215C3⋊Dic3C2×C3×D412yes—
216C3⋊Dic3C24×C311yescharacteristic
217C3×Dic3—22no—
218C3×Dic3—22no—
219C3×Dic3—42no—
220C3×Dic3—64no—
221C3×Dic3—62no—
222H(5,𝔽2)—92no—
223AGL(1,ℤ/8)—92no—
224AGL(1,ℤ/8)—94no—
225#135—91no—
226#135—92no—
227C2×D8—91no—
228C2×D8—92no—
229#131—91no—
230C33#57611yescharacteristic, complemented, Sylow
231C23×C3—64no—
232C23×C3—64no—
233C23×C3—124no—
234C23×C3—182no—
235C22×D3—24no—
236C22×D3—64no—
237C3×Q8—92no—
238C3×Q8—91no—
239C3×D4—22no—
240C3×D4—42no—
241C3×D4—62no—
242C3×D4—64no—
243C3×D4—64no—
244C3×D4—62no—
245C3×D4—92no—
246C3×D4—94no—
247C3×D4—92no—
248C3×D4—122no—
249C3×D4—124no—
250C3×D4—182no—
251C3×D4—184no—
252C3×D4GO+(4,𝔽2)11yescharacteristic
253C2×C3×C4—62no—
254C2×C3×C4—62no—
255C2×C3×C4—92no—
256C2×C3×C4—91no—
257C2×C3×C4—122no—
258C2×C3×C4—184no—
259#67—24no—
260#67—62no—
261#67—64no—
262#67—64no—
263#67—62no—
264#67—64no—
265#67—64no—
266C2×Dic3—64no—
267C2×Dic3—64no—
268D12—22no—
269D12—62no—
270C4×D3—22no—
271C4×D3—62no—
272C4×D3—64no—
273Dic6—62no—
274Dic6—62no—
275C3×C8—91no—
276C3×C8—91no—
277C3×C8—92no—
278C2×C32—22no—
279C2×C32—22no—
280C2×C32—42no—
281C2×C32—44no—
282C2×C32—84no—
283C2×C32—122no—
284C2×C32—124no—
285C2×C32C22×C3×D411yescharacteristic
286C3⋊D3—21no—
287C3×D3—42no—
288C3×D3—44no—
289C3×D3—44no—
290C3×D3—44no—
291C3×D3—42no—
292C3×D3—84no—
293C3×D3—122no—
294#41—92no—
295#41—91no—
296#41—94no—
297#41—182no—
298C2×D4—62no—
299C2×D4—92no—
300C2×D4—94no—
301C2×D4—184no—
302Q16—91no—
303SD16—92no—
304SD16—94no—
305D8—92no—
306D8—91no—
307D8—92no—
308D8—94no—
309M16—91no—
310M16—92no—
311C2×C8—91no—
312C2×C8—92no—
313C22×C3—24no—
314C22×C3—44no—
315C22×C3—64no—
316C22×C3—62no—
317C22×C3—62no—
318C22×C3—64no—
319C22×C3—91no—
320C22×C3—122no—
321C22×C3—124no—
322C22×C3—244no—
323C22×C3—244no—
324C22×C3—248no—
325C22×C3—722no—
326C22×C3C2×GO+(4,𝔽2)12yes—
327C2×D3—22no—
328C2×D3—24no—
329C2×D3—62no—
330C2×D3—84no—
331C2×D3—244no—
332C3×C4—22no—
333C3×C4—42no—
334C3×C4—62no—
335C3×C4—62no—
336C3×C4—91no—
337C3×C4—92no—
338C3×C4—122no—
339C3×C4C2×GO+(4,𝔽2)11yescharacteristic
340Dic3—22no—
341Dic3—64no—
342Dic3—62no—
343C32—22no—
344C32—42no—
345C32C3×#57611yescharacteristic, complemented
346C23—64no—
347C23—182no—
348Q8—92no—
349Q8—91no—
350D4—64no—
351D4—62no—
352D4—92no—
353D4—94no—
354D4—92no—
355D4—182no—
356D4—184no—
357D4C3×GO+(4,𝔽2)11yescharacteristic
358C2×C4—62no—
359C2×C4—92no—
360C2×C4—91no—
361C2×C4—184no—
362C8—91no—
363C8—91no—
364C8—92no—
365C2×C3—22no—
366C2×C3—22no—
367C2×C3—44no—
368C2×C3—42no—
369C2×C3—84no—
370C2×C3—124no—
371C2×C3—122no—
372C2×C3—124no—
373C2×C3—122no—
374C2×C3—124no—
375C2×C3—181no—
376C2×C3—244no—
377C2×C3C22×GO+(4,𝔽2)11yescharacteristic
378D3—42no—
379D3—44no—
380D3—122no—
381C22—62no—
382C22—64no—
383C22—91no—
384C22—244no—
385C22—722no—
386C22C2×C3×GO+(4,𝔽2)12yes—
387C4—62no—
388C4—91no—
389C4—92no—
390C4C2×C3×GO+(4,𝔽2)11yescharacteristic
391C3—22no—
392C3—42no—
393C3#1059651411yescharacteristic, direct product factor
394C2—22no—
395C2—122no—
396C2—124no—
397C2—181no—
398C2C22×C3×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
centerC2×C3C22×GO+(4,𝔽2)
commutator, cosocleC3⋊Dic3C24×C3
FrattiniC2C22×C3×GO+(4,𝔽2)
FittingC33×D4D4
socleC2×C33C22×D4
hypercenterC3×D4GO+(4,𝔽2)
hypocenterC32C3×#576
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C3×D4

Complements

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

NH ≅ G/Ntype
1. C3×#64313C2semidirect
2. C3×#64287C2semidirect
3. C3×#64237C2semidirect
4. C3×#64236C2semidirect
5. C2×C3×#3774C2semidirect
6. C3×#64225C2semidirect
7. C3×#64224C2semidirect
8. C3×#64223C2semidirect
9. #10596514C3direct (N × H)
13. C2×C3×#625C22semidirect
14. C3×#3805C22semidirect
15. C3×#3802C22semidirect
16. C3×#3798C22semidirect
18. C3×#3792C22semidirect
19. C2×C3×#622C22semidirect
20. C3×#3788C22semidirect
21. C3×#3787C22semidirect
22. C3×#3776C22semidirect
23. C3×#3775C22semidirect
24. C3×#3775C22semidirect
25. C3×#3774C22semidirect
26. C3×#3774C22semidirect
27. C3×#3774C22semidirect
28. C3×#3774C22semidirect
30. #64313C2×C3semidirect
31. #64287C2×C3semidirect
32. #64237C2×C3semidirect
33. #64236C2×C3semidirect
34. C2×#3774C2×C3semidirect
35. #64225C2×C3semidirect
36. #64224C2×C3semidirect
37. #64223C2×C3semidirect
73. C2×#625C22×C3semidirect
74. #3805C22×C3semidirect
75. #3802C22×C3semidirect
76. #3798C22×C3semidirect
78. #3792C22×C3semidirect
79. C2×#622C22×C3semidirect
80. #3788C22×C3semidirect
81. #3787C22×C3semidirect
82. #3776C22×C3semidirect
83. #3775C22×C3semidirect
84. #3775C22×C3semidirect
85. #3774C22×C3semidirect
86. #3774C22×C3semidirect
87. #3774C22×C3semidirect
88. #3774C22×C3semidirect
230. C33#576semidirect
345. C32C3×#576semidirect
393. C3#10596514direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#643132
C3×#642872
C3×#642372
C3×#642362
C2×C3×#37742
C3×#642252
C3×#642242
C3×#642232
#105965143
C3×#5769