Finite Group Databasebeta

← C3×#10596532 #13181051

Subgroups — C3×#10596532 #13181051

Order 1728.

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

normal subgroups only

This group has 960 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
1C2×C3×#3805C212yesmaximal, complemented
2C2×C3×#3792C211yesmaximal, characteristic, complemented
3C3×#64239C214yesmaximal, complemented
4C3×#64237C214yesmaximal, complemented
5C3×#64236C214yesmaximal, complemented
6#10596532C311yesmaximal, characteristic, direct product factor
7C22×C3×C3⋊Dic3C2211yescharacteristic
8C2×C32×#67—22no—
9C2×C3×#627C2212yescomplemented
10C2×C3×#625C2212yescomplemented
11C3×#3805—22no—
12C3×#3805—22no—
13C3×#3805C2214yes—
14C3×#3805C2214yescomplemented
15C2×C3×D3×Dic3—22no—
16C3×#3792C2214yescomplemented
17C2×C3×#622C2212yescomplemented
18C3×#3776C2214yescomplemented
19C3×#3775C2218yescomplemented
20C3×#3774C2214yescomplemented
21C2×C3×#237—62no—
22C2×#3805C2×C312yescomplemented
23C2×#3792C2×C311yescharacteristic, complemented
24#64239C2×C314yescomplemented
25#64237C2×C314yescomplemented
26#64236C2×C314yescomplemented
27C23×C33D411yescharacteristic
28C22×C32×D3—22no—
29C2×C3×C3⋊Dic3D412yes—
30C2×C3×C3⋊Dic3D411yescharacteristic
31C2×C3×C3⋊Dic3C2311yescharacteristic
32C2×C3×C3⋊Dic3C2312yes—
33C32×#67—24no—
34C32×#67—24no—
35C2×C32×Dic3—22no—
36C3×#627—22no—
37C3×#627C2314yes—
38C3×#625—22no—
39C3×#625C2314yes—
40C3×D3×Dic3—24no—
41C3×D3×Dic3—24no—
42C3×#622C2314yes—
43C3×#575—91nomaximal
44C2×C32×D4—62no—
45C22×C3⋊Dic3C22×C311yescharacteristic
46C2×C3×#67—22no—
47C2×C3×#67—22no—
48C2×C3×#67—42no—
49C2×C3×#67—62no—
50C22×C3×Dic3—62no—
51C22×C3×Dic3—62no—
52C3×#237—64no—
53C3×#237—64no—
54C3×#237—64no—
55C3×#237—64no—
56C2×C3×C4×D3—62no—
57C2×C3×Dic6—62no—
58C2×#627C22×C312yescomplemented
59C2×#625C22×C312yescomplemented
60#3805—22no—
61#3805—22no—
62#3805C22×C314yes—
63#3805C22×C314yescomplemented
64C2×D3×Dic3—22no—
65#3792C22×C314yescomplemented
66C2×#622C22×C312yescomplemented
67#3776C22×C314yescomplemented
68#3775C22×C318yescomplemented
69#3774C22×C314yescomplemented
70C22×C33—41no—
71C22×C33C2×D412yes—
72C22×C33C2×D411yescharacteristic
73C2×C32×D3—24no—
74C2×C32×D3—44no—
75C3×C3⋊Dic3C2×D411yescharacteristic
76C3×C3⋊Dic3C2×D412yes—
77C3×C3⋊Dic3C2411yescharacteristic
78C32×Dic3—24no—
79C2×C3×#41—92no—
80C2×#237—62no—
81C3×#137—94no—
82C3×AGL(1,ℤ/8)—94no—
83C3×#135—94no—
84C2×C3×M16—91no—
85C23×C32—22no—
86C23×C32—42no—
87C23×C32—62no—
88C23×C32C3×D411yescharacteristic
89C22×C3×D3—22no—
90C22×C3×D3—22no—
91C22×C3×D3—42no—
92C32×D4—64no—
93C32×D4—64no—
94C2×C32×C4—62no—
95C2×C3⋊Dic3C3×D412yes—
96C2×C3⋊Dic3C3×D411yescharacteristic
97C2×C3⋊Dic3C23×C311yescharacteristic
98C2×C3⋊Dic3C23×C312yes—
99C3×#67—24no—
100C3×#67—24no—
101C3×#67—24no—
102C3×#67—24no—
103C3×#67—44no—
104C3×#67—44no—
105C3×#67—64no—
106C3×#67—64no—
107C2×C3×Dic3—22no—
108C2×C3×Dic3—22no—
109C2×C3×Dic3—42no—
110C2×C3×Dic3—62no—
111C2×C3×Dic3—62no—
112C2×C3×Dic3—62no—
113C2×C3×Dic3—62no—
114C2×C3×Dic3—62no—
115C2×C3×Dic3—62no—
116C2×C3×Dic3—64no—
117C2×C3×Dic3—64no—
118C3×C4×D3—64no—
119C3×C4×D3—64no—
120C3×Dic6—64no—
121C3×Dic6—64no—
122#627—22no—
123#627C23×C314yes—
124#625—22no—
125#625C23×C314yes—
126D3×Dic3—24no—
127D3×Dic3—24no—
128#622C23×C314yes—
129#575—91noSylow
130C2×C33—22no—
131C2×C33—21no—
132C2×C33C22×D411yescharacteristic
133C32×D3—44no—
134C3×#41—94no—
135C3×#41—94no—
136C3×#41—182no—
137C3×#41—182no—
138C2×C3×Q8—92no—
139C2×C3×D4—62no—
140C2×C3×D4—62no—
141C2×C3×D4—92no—
142C2×C3×D4—122no—
143C22×C3×C4—91no—
144C22×C3×C4—182no—
145C2×#67—22no—
146C2×#67—62no—
147C22×Dic3—62no—
148C22×Dic3—62no—
149#237—64no—
150#237—64no—
151#237—64no—
152#237—64no—
153C2×C4×D3—62no—
154C2×Dic6—62no—
155C3×Q16—94no—
156C3×SD16—98no—
157C3×D8—94no—
158C3×M16—94no—
159C2×C3×C8—92no—
160C22×C32—22no—
161C22×C32—22no—
162C22×C32—22no—
163C22×C32—42no—
164C22×C32—42no—
165C22×C32—42no—
166C22×C32—41no—
167C22×C32—44no—
168C22×C32—64no—
169C22×C32—84no—
170C22×C32—124no—
171C22×C32C2×C3×D412yes—
172C22×C32C2×C3×D411yescharacteristic
173C2×C3×D3—24no—
174C2×C3×D3—24no—
175C2×C3×D3—44no—
176C2×C3×D3—44no—
177C2×C3×D3—44no—
178C2×C3×D3—84no—
179C32×C4—64no—
180C3⋊Dic3C2×C3×D411yescharacteristic
181C3⋊Dic3C2×C3×D412yes—
182C3⋊Dic3C24×C311yescharacteristic
183C3×Dic3—24no—
184C3×Dic3—24no—
185C3×Dic3—44no—
186C3×Dic3—62no—
187C3×Dic3—62no—
188C3×Dic3—62no—
189C3×Dic3—62no—
190C2×#41—92no—
191#137—94no—
192AGL(1,ℤ/8)—94no—
193#135—94no—
194C2×M16—91no—
195C33#57511yescharacteristic, complemented, Sylow
196C23×C3—22no—
197C23×C3—42no—
198C23×C3—62no—
199C23×C3—62no—
200C23×C3—122no—
201C23×C3GO+(4,𝔽2)11yescharacteristic
202C22×D3—22no—
203C3×Q8—94no—
204C3×Q8—182no—
205C3×D4—64no—
206C3×D4—64no—
207C3×D4—64no—
208C3×D4—64no—
209C3×D4—94no—
210C3×D4—124no—
211C3×D4—124no—
212C3×D4—182no—
213C2×C3×C4—62no—
214C2×C3×C4—62no—
215C2×C3×C4—91no—
216C2×C3×C4—92no—
217C2×C3×C4—92no—
218C2×C3×C4—91no—
219C2×C3×C4—122no—
220C2×C3×C4—184no—
221C2×C3×C4—184no—
222#67—24no—
223#67—24no—
224#67—64no—
225#67—64no—
226C2×Dic3—22no—
227C2×Dic3—62no—
228C2×Dic3—62no—
229C2×Dic3—62no—
230C2×Dic3—62no—
231C2×Dic3—62no—
232C2×Dic3—62no—
233C2×Dic3—64no—
234C2×Dic3—64no—
235C4×D3—64no—
236C4×D3—64no—
237Dic6—64no—
238Dic6—64no—
239C3×C8—94no—
240C2×C32—24no—
241C2×C32—22no—
242C2×C32—22no—
243C2×C32—21no—
244C2×C32—42no—
245C2×C32—42no—
246C2×C32—42no—
247C2×C32—44no—
248C2×C32—82no—
249C2×C32—82no—
250C2×C32—124no—
251C2×C32C22×C3×D411yescharacteristic
252C3×D3—44no—
253C3×D3—44no—
254C3×D3—84no—
255#41—94no—
256#41—94no—
257#41—182no—
258#41—182no—
259C2×Q8—92no—
260C2×D4—62no—
261C2×D4—92no—
262C22×C4—91no—
263C22×C4—182no—
264Q16—94no—
265SD16—98no—
266D8—94no—
267M16—94no—
268C2×C8—92no—
269C22×C3—22no—
270C22×C3—22no—
271C22×C3—22no—
272C22×C3—41no—
273C22×C3—44no—
274C22×C3—42no—
275C22×C3—42no—
276C22×C3—42no—
277C22×C3—64no—
278C22×C3—64no—
279C22×C3—84no—
280C22×C3—124no—
281C22×C3—124no—
282C22×C3—124no—
283C22×C3—244no—
284C22×C3C2×GO+(4,𝔽2)12yes—
285C22×C3C2×GO+(4,𝔽2)11yescharacteristic
286C2×D3—24no—
287C2×D3—44no—
288C3×C4—64no—
289C3×C4—64no—
290C3×C4—91no—
291C3×C4—91no—
292C3×C4—92no—
293C3×C4—124no—
294Dic3—24no—
295Dic3—62no—
296Dic3—62no—
297Dic3—62no—
298Dic3—62no—
299C32—22no—
300C32—42no—
301C32C3×#57511yescharacteristic, complemented
302C23—62no—
303C23C3×GO+(4,𝔽2)11yescharacteristic
304Q8—94no—
305Q8—182no—
306D4—64no—
307D4—64no—
308D4—94no—
309D4—182no—
310C2×C4—62no—
311C2×C4—91no—
312C2×C4—92no—
313C2×C4—92no—
314C2×C4—91no—
315C2×C4—184no—
316C2×C4—184no—
317C8—94no—
318C2×C3—24no—
319C2×C3—22no—
320C2×C3—21no—
321C2×C3—22no—
322C2×C3—42no—
323C2×C3—42no—
324C2×C3—44no—
325C2×C3—42no—
326C2×C3—82no—
327C2×C3—82no—
328C2×C3—124no—
329C2×C3—124no—
330C2×C3—244no—
331C2×C3C22×GO+(4,𝔽2)11yescharacteristic
332D3—44no—
333C22—41no—
334C22—64no—
335C22—124no—
336C22C2×C3×GO+(4,𝔽2)12yes—
337C22C2×C3×GO+(4,𝔽2)11yescharacteristic
338C4—64no—
339C4—91no—
340C4—91no—
341C4—92no—
342C3—22no—
343C3—42no—
344C3#1059653211yescharacteristic, direct product factor
345C2—22no—
346C2—21no—
347C2—124no—
348C2C22×C3×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
centerC2×C3C22×GO+(4,𝔽2)
commutator, cosocleC3⋊Dic3C24×C3
FrattiniC2C22×C3×GO+(4,𝔽2)
FittingC23×C33D4
socleC2×C33C22×D4
hypercenterC23×C3GO+(4,𝔽2)
hypocenterC32C3×#575
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3805C2semidirect
2. C2×C3×#3792C2semidirect
3. C3×#64239C2semidirect
4. C3×#64237C2semidirect
5. C3×#64236C2semidirect
6. #10596532C3direct (N × H)
9. C2×C3×#627C22semidirect
10. C2×C3×#625C22semidirect
14. C3×#3805C22semidirect
16. C3×#3792C22semidirect
17. C2×C3×#622C22semidirect
18. C3×#3776C22semidirect
19. C3×#3775C22semidirect
20. C3×#3774C22semidirect
22. C2×#3805C2×C3semidirect
23. C2×#3792C2×C3semidirect
24. #64239C2×C3semidirect
25. #64237C2×C3semidirect
26. #64236C2×C3semidirect
58. C2×#627C22×C3semidirect
59. C2×#625C22×C3semidirect
63. #3805C22×C3semidirect
65. #3792C22×C3semidirect
66. C2×#622C22×C3semidirect
67. #3776C22×C3semidirect
68. #3775C22×C3semidirect
69. #3774C22×C3semidirect
195. C33#575semidirect
301. C32C3×#575semidirect
344. C3#10596532direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38052
C2×C3×#37922
C3×#642392
C3×#642372
C3×#642362
#105965323
C3×#5759