Finite Group Databasebeta

← C3×#10596500 #13181019

Subgroups — C3×#10596500 #13181019

Order 1728.

940 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 940 conjugacy classes of subgroups — too many to draw. Only its 174 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×#64306C212yesmaximal, complemented
2C2×C3×#3787C211yesmaximal, characteristic, complemented
3C3×#64237C214yesmaximal, complemented
4C3×#64228C211yesmaximal, characteristic, complemented
5C3×#64227C212yesmaximal, complemented
6C3×#64225C211yesmaximal, characteristic, complemented
7C3×#64223C214yesmaximal, complemented
8#10596500C311yesmaximal, characteristic, direct product factor
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C32×#235—22no—
11C3×#3811—22no—
12C3×#3805C2214yescomplemented
13C3×#3802C2212yes—
14C3×C4×D32—22no—
15C3×#3798C2214yescomplemented
16C3×#3797C2212yes—
17C3×#3796—22no—
18C2×C3×#622C2212yescomplemented
19C3×#3787C2211yescharacteristic
20C3×#3787C2211yescharacteristic
21C3×#3787C2212yescomplemented
22C3×#3776C2214yescomplemented
23C3×#3775C2218yescomplemented
24C3×#3774C2214yescomplemented
25C3×D3×#41—62no—
26#64306C2×C312yescomplemented
27C2×#3787C2×C311yescharacteristic, complemented
28#64237C2×C314yescomplemented
29#64228C2×C311yescharacteristic, complemented
30#64227C2×C312yescomplemented
31#64225C2×C311yescharacteristic, complemented
32#64223C2×C314yescomplemented
33C22×C3×C3⋊D3D411yescharacteristic
34C2×C3×D32—22no—
35C2×C33×C4D411yescharacteristic
36C2×C3×C3⋊Dic3C2311yescharacteristic
37C3×C4×C3⋊D3D411yescharacteristic
38C3×C4×C3⋊D3D411yescharacteristic
39C3×C4×C3⋊D3C2311yescharacteristic
40C3×C4×C3⋊D3C2311yescharacteristic
41C32×#67—24no—
42C32×D12—22no—
43C32×C4×D3—24no—
44C32×Dic6—22no—
45C3×#627C2314yes—
46C3×#626—24no—
47C3×#625C2314yes—
48C3×#624—22no—
49C3×D3×Dic3—24no—
50C3×#622C2312yes—
51C3×#622C2312yes—
52C2×C3×#135—91nomaximal
53C32×#41—62no—
54C2×C4×C3⋊D3C22×C311yescharacteristic
55C3×#239—62no—
56C3×D3×Q8—62no—
57C3×#237—62no—
58C3×#237—64no—
59C3×D3×D4—62no—
60C3×D3×D4—64no—
61C3×#235—22no—
62C3×#235—22no—
63C3×#235—42no—
64C3×#235—64no—
65C2×C3×C4×D3—64no—
66C2×C3×C4×D3—62no—
67#3811—22no—
68#3805C22×C314yescomplemented
69#3802C22×C312yes—
70C4×D32—22no—
71#3798C22×C314yescomplemented
72#3797C22×C312yes—
73#3796—22no—
74C2×#622C22×C312yescomplemented
75#3787C22×C311yescharacteristic
76#3787C22×C311yescharacteristic
77#3787C22×C312yescomplemented
78#3776C22×C314yescomplemented
79#3775C22×C318yescomplemented
80#3774C22×C314yescomplemented
81C22×C33C2×D411yescharacteristic
82C2×C3×C3⋊D3—22no—
83C2×C3×C3⋊D3C2×D411yescharacteristic
84C2×C3×C3⋊D3C2×D411yescharacteristic
85C2×C32×D3—24no—
86C3×D32—44no—
87C33×C4C2×D411yescharacteristic
88C33×C4C2×D411yescharacteristic
89C3×C3⋊Dic3C2×D411yescharacteristic
90C3×C3⋊Dic3C2411yescharacteristic
91C32×Dic3—24no—
92C2×C3×#41—92no—
93D3×#41—62no—
94C3×#135—94no—
95C3×#135—94no—
96C2×C3×Q16—91no—
97C2×C3×SD16—92no—
98C2×C3×D8—91no—
99C22×C3×C8—91no—
100C22×C3⋊D3C3×D411yescharacteristic
101C22×C3×D3—62no—
102C22×C3×D3—64no—
103C2×D32—22no—
104C32×Q8—62no—
105C32×D4—62no—
106C32×D4—64no—
107C2×C32×C4—22no—
108C2×C32×C4—42no—
109C2×C32×C4—64no—
110C2×C32×C4C3×D411yescharacteristic
111C2×C3⋊Dic3C23×C311yescharacteristic
112C4×C3⋊D3C3×D411yescharacteristic
113C4×C3⋊D3C3×D411yescharacteristic
114C4×C3⋊D3C23×C311yescharacteristic
115C4×C3⋊D3C23×C311yescharacteristic
116C3×#67—24no—
117C3×#67—24no—
118C3×#67—44no—
119C3×#67—64no—
120C3×#67—64no—
121C2×C3×Dic3—62no—
122C2×C3×Dic3—64no—
123C3×D12—22no—
124C3×D12—22no—
125C3×D12—42no—
126C3×D12—64no—
127C3×C4×D3—24no—
128C3×C4×D3—24no—
129C3×C4×D3—44no—
130C3×C4×D3—62no—
131C3×C4×D3—62no—
132C3×C4×D3—62no—
133C3×C4×D3—62no—
134C3×C4×D3—64no—
135C3×C4×D3—64no—
136C3×Dic6—22no—
137C3×Dic6—22no—
138C3×Dic6—42no—
139C3×Dic6—64no—
140#627C23×C314yes—
141#626—24no—
142#625C23×C314yes—
143#624—22no—
144D3×Dic3—24no—
145#622C23×C312yes—
146#622C23×C312yes—
147C2×#135—91noSylow
148C2×C33—21no—
149C2×C33C22×D411yescharacteristic
150C3×C3⋊D3—21no—
151C3×C3⋊D3#13512yescomplemented
152C32×D3—44no—
153C3×#41—62no—
154C3×#41—62no—
155C3×#41—94no—
156C3×#41—94no—
157C3×#41—122no—
158C3×#41—182no—
159C2×C3×Q8—92no—
160C2×C3×D4—92no—
161C2×C3×D4—182no—
162C22×C3×C4—91no—
163C22×C3×C4—182no—
164#239—62no—
165D3×Q8—62no—
166#237—64no—
167#237—62no—
168D3×D4—62no—
169D3×D4—64no—
170#235—22no—
171#235—64no—
172C2×C4×D3—62no—
173C2×C4×D3—64no—
174C3×Q16—94no—
175C3×SD16—98no—
176C3×D8—94no—
177C2×C3×C8—91no—
178C2×C3×C8—91no—
179C2×C3×C8—92no—
180C2×C3×C8—92no—
181C22×C32—22no—
182C22×C32—42no—
183C22×C32—64no—
184C22×C32C2×C3×D411yescharacteristic
185C2×C3⋊D3—22no—
186C2×C3⋊D3C2×C3×D411yescharacteristic
187C2×C3⋊D3C2×C3×D411yescharacteristic
188C2×C3×D3—24no—
189C2×C3×D3—24no—
190C2×C3×D3—44no—
191C2×C3×D3—62no—
192C2×C3×D3—62no—
193C2×C3×D3—124no—
194C2×C3×D3—128no—
195D32—44no—
196C32×C4—22no—
197C32×C4—22no—
198C32×C4—42no—
199C32×C4—42no—
200C32×C4—64no—
201C32×C4C2×C3×D411yescharacteristic
202C32×C4C2×C3×D411yescharacteristic
203C3⋊Dic3C2×C3×D411yescharacteristic
204C3⋊Dic3C24×C311yescharacteristic
205C3×Dic3—24no—
206C3×Dic3—24no—
207C3×Dic3—44no—
208C3×Dic3—62no—
209C3×Dic3—62no—
210C2×#41—92no—
211#135—94no—
212#135—94no—
213C2×Q16—91no—
214C2×SD16—92no—
215C2×D8—91no—
216C22×C8—91no—
217C33C2×#13511yescharacteristic, complemented, Sylow
218C23×C3—91no—
219C23×C3—182no—
220C22×D3—62no—
221C22×D3—64no—
222C3×Q8—62no—
223C3×Q8—62no—
224C3×Q8—94no—
225C3×Q8—122no—
226C3×D4—62no—
227C3×D4—64no—
228C3×D4—64no—
229C3×D4—62no—
230C3×D4—94no—
231C3×D4—122no—
232C3×D4—124no—
233C3×D4—184no—
234C2×C3×C4—22no—
235C2×C3×C4—42no—
236C2×C3×C4—64no—
237C2×C3×C4—64no—
238C2×C3×C4—91no—
239C2×C3×C4—91no—
240C2×C3×C4—91no—
241C2×C3×C4—91no—
242C2×C3×C4—91no—
243C2×C3×C4—124no—
244C2×C3×C4—182no—
245C2×C3×C4—184no—
246C2×C3×C4GO+(4,𝔽2)11yescharacteristic
247#67—24no—
248#67—64no—
249#67—64no—
250C2×Dic3—62no—
251C2×Dic3—64no—
252D12—22no—
253D12—64no—
254C4×D3—24no—
255C4×D3—62no—
256C4×D3—62no—
257C4×D3—62no—
258C4×D3—62no—
259C4×D3—64no—
260C4×D3—64no—
261Dic6—22no—
262Dic6—64no—
263C3×C8—92no—
264C3×C8—92no—
265C2×C32—22no—
266C2×C32—21no—
267C2×C32—42no—
268C2×C32—42no—
269C2×C32—82no—
270C2×C32—124no—
271C2×C32C22×C3×D411yescharacteristic
272C3⋊D3—21no—
273C3⋊D3C3×#13512yescomplemented
274C3×D3—44no—
275C3×D3—44no—
276C3×D3—64no—
277C3×D3—84no—
278C3×D3—122no—
279#41—62no—
280#41—94no—
281#41—94no—
282#41—182no—
283C2×Q8—92no—
284C2×D4—92no—
285C2×D4—182no—
286C22×C4—91no—
287C22×C4—182no—
288Q16—94no—
289SD16—98no—
290D8—94no—
291C2×C8—91no—
292C2×C8—91no—
293C2×C8—92no—
294C2×C8—92no—
295C22×C3—22no—
296C22×C3—42no—
297C22×C3—64no—
298C22×C3—64no—
299C22×C3—91no—
300C22×C3—91no—
301C22×C3—124no—
302C22×C3—182no—
303C22×C3—364no—
304C22×C3C2×GO+(4,𝔽2)11yescharacteristic
305C2×D3—24no—
306C2×D3—62no—
307C2×D3—62no—
308C2×D3—124no—
309C2×D3—128no—
310C3×C4—22no—
311C3×C4—22no—
312C3×C4—42no—
313C3×C4—42no—
314C3×C4—64no—
315C3×C4—64no—
316C3×C4—91no—
317C3×C4—91no—
318C3×C4—124no—
319C3×C4C2×GO+(4,𝔽2)11yescharacteristic
320C3×C4C2×GO+(4,𝔽2)11yescharacteristic
321Dic3—24no—
322Dic3—62no—
323Dic3—62no—
324C32—22no—
325C32—42no—
326C32C2×C3×#13511yescharacteristic, complemented
327C23—91no—
328C23—182no—
329Q8—62no—
330Q8—94no—
331D4—64no—
332D4—62no—
333D4—94no—
334D4—184no—
335C2×C4—64no—
336C2×C4—91no—
337C2×C4—91no—
338C2×C4—91no—
339C2×C4—91no—
340C2×C4—91no—
341C2×C4—182no—
342C2×C4—184no—
343C2×C4C3×GO+(4,𝔽2)11yescharacteristic
344C8—92no—
345C8—92no—
346C2×C3—21no—
347C2×C3—22no—
348C2×C3—42no—
349C2×C3—42no—
350C2×C3—82no—
351C2×C3—92no—
352C2×C3—124no—
353C2×C3—124no—
354C2×C3—181no—
355C2×C3—244no—
356C2×C3C22×GO+(4,𝔽2)11yescharacteristic
357D3—44no—
358D3—64no—
359D3—122no—
360C22—64no—
361C22—91no—
362C22—91no—
363C22—182no—
364C22—364no—
365C22C2×C3×GO+(4,𝔽2)11yescharacteristic
366C4—64no—
367C4—91no—
368C4—91no—
369C4C2×C3×GO+(4,𝔽2)11yescharacteristic
370C4C2×C3×GO+(4,𝔽2)11yescharacteristic
371C3—22no—
372C3—42no—
373C3#1059650011yescharacteristic, direct product factor
374C2—21no—
375C2—92no—
376C2—124no—
377C2—181no—
378C2C22×C3×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
centerC3×C4C2×GO+(4,𝔽2)
commutator, cosocleC3⋊Dic3C24×C3
FrattiniC2C22×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC2×C33C22×D4
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C2×C3×#135
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C3×C4 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C3×#64306C2semidirect
2. C2×C3×#3787C2semidirect
3. C3×#64237C2semidirect
4. C3×#64228C2semidirect
5. C3×#64227C2semidirect
6. C3×#64225C2semidirect
7. C3×#64223C2semidirect
8. #10596500C3direct (N × H)
12. C3×#3805C22semidirect
15. C3×#3798C22semidirect
18. C2×C3×#622C22semidirect
21. C3×#3787C22semidirect
22. C3×#3776C22semidirect
23. C3×#3775C22semidirect
24. C3×#3774C22semidirect
26. #64306C2×C3semidirect
27. C2×#3787C2×C3semidirect
28. #64237C2×C3semidirect
29. #64228C2×C3semidirect
30. #64227C2×C3semidirect
31. #64225C2×C3semidirect
32. #64223C2×C3semidirect
68. #3805C22×C3semidirect
71. #3798C22×C3semidirect
74. C2×#622C22×C3semidirect
77. #3787C22×C3semidirect
78. #3776C22×C3semidirect
79. #3775C22×C3semidirect
80. #3774C22×C3semidirect
151. C3×C3⋊D3#135semidirect
217. C33C2×#135semidirect
273. C3⋊D3C3×#135semidirect
326. C32C2×C3×#135semidirect
373. C3#10596500direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#643062
C2×C3×#37872
C3×#642372
C3×#642282
C3×#642272
C3×#642252
C3×#642232
#105965003
C2×C3×#1359