Finite Group Databasebeta

← C3×#10596511 #13181030

Subgroups — C3×#10596511 #13181030

Order 1728.

1028 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 1028 conjugacy classes of subgroups — too many to draw. Only its 182 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×C4×#157C211yesmaximal, characteristic, complemented
3C3×#64242C214yesmaximal, complemented
4C3×#64231C211yesmaximal, characteristic, complemented
5C3×C4×GO+(4,𝔽2)C214yesmaximal, complemented
6C3×#64222C211yesmaximal, characteristic, complemented
7C3×#64221C212yesmaximal, complemented
8#10596511C311yesmaximal, characteristic, direct product factor
9C22×C3×#157C2212yes—
10C2×C3×GO+(4,𝔽2)C2214yes—
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C32×#235—22no—
13C3×#3811C2214yes—
14C3×#3805—22no—
15C3×#3802C2212yes—
16C3×C4×D32C2214yes—
17C3×#3798—22no—
18C3×#3797C2212yes—
19C3×#3796—22no—
20C3×C4×#157C2211yescharacteristic
21C3×C4×#157C2211yescharacteristic
22C3×C4×#157C2212yes—
23C3×#3773C2214yes—
24C3×D32⋊C4C2218yes—
25C3×D3×#41—62no—
26#64306C2×C312yescomplemented
27C2×C4×#157C2×C311yescharacteristic, complemented
28#64242C2×C314yescomplemented
29#64231C2×C311yescharacteristic, complemented
30C4×GO+(4,𝔽2)C2×C314yescomplemented
31#64222C2×C311yescharacteristic, complemented
32#64221C2×C312yescomplemented
33C22×C3×C3⋊D3C2311yescharacteristic
34C2×C3×D32C2314yes—
35C2×C3×#157—24no—
36C2×C3×#157C2312yes—
37C2×C3×#157C2312yes—
38C3×GO+(4,𝔽2)—28no—
39C2×C33×C4D411yescharacteristic, complemented
40C2×C3×C3⋊Dic3D411yescharacteristic, complemented
41C3×C4×C3⋊D3D411yescharacteristic, complemented
42C3×C4×C3⋊D3D411yescharacteristic, complemented
43C3×C4×C3⋊D3C2311yescharacteristic
44C3×C4×C3⋊D3C2311yescharacteristic
45C32×#67—24no—
46C32×D12—22no—
47C32×C4×D3—24no—
48C32×Dic6—22no—
49C3×#627—22no—
50C3×#626—24no—
51C3×#625—22no—
52C3×#624C2314yes—
53C3×D3×Dic3—24no—
54C3×#532—91nomaximal
55C22×#157C22×C312yes—
56C2×GO+(4,𝔽2)C22×C314yes—
57C32×#41—62no—
58C2×C4×C3⋊D3C22×C311yescharacteristic
59C3×#239—62no—
60C3×D3×Q8—62no—
61C3×#237—62no—
62C3×#237—64no—
63C3×D3×D4—62no—
64C3×D3×D4—64no—
65C3×#235—22no—
66C3×#235—22no—
67C3×#235—42no—
68C3×#235—64no—
69C2×C3×C4×D3—64no—
70C2×C3×C4×D3—62no—
71#3811C22×C314yes—
72#3805—22no—
73#3802C22×C312yes—
74C4×D32C22×C314yes—
75#3798—22no—
76#3797C22×C312yes—
77#3796—22no—
78C4×#157C22×C311yescharacteristic
79C4×#157C22×C311yescharacteristic
80C4×#157C22×C312yes—
81#3773C22×C314yes—
82D32⋊C4C22×C318yes—
83C22×C33C2×D411yescharacteristic
84C2×C3×C3⋊D3—22no—
85C2×C3×C3⋊D3C2×D411yescharacteristic
86C2×C3×C3⋊D3C2411yescharacteristic
87C2×C32×D3—24no—
88C3×D32—28no—
89C3×#157—22no—
90C3×#157#4114yescomplemented
91C33×C4C2×D411yescharacteristic
92C33×C4C2×D411yescharacteristic
93C3×C3⋊Dic3C2×D411yescharacteristic
94C3×C3⋊Dic3C2×D411yescharacteristic
95C32×Dic3—24no—
96C2×C3×#41—92no—
97D3×#41—62no—
98C3×C4⋊Q8—91no—
99C3×#127—91no—
100C3×#124—92no—
101C3×#121—94no—
102C3×C4×D4—94no—
103C2×C3×C42—91no—
104C22×C3⋊D3C23×C311yescharacteristic
105C22×C3×D3—64no—
106C22×C3×D3—62no—
107C2×D32C23×C314yes—
108C2×#157—24no—
109C2×#157C23×C312yes—
110C2×#157C23×C312yes—
111GO+(4,𝔽2)—28no—
112C32×Q8—62no—
113C32×D4—62no—
114C32×D4—64no—
115C2×C32×C4—22no—
116C2×C32×C4—42no—
117C2×C32×C4—64no—
118C2×C32×C4C3×D411yescharacteristic, complemented
119C2×C3⋊Dic3C3×D411yescharacteristic, complemented
120C4×C3⋊D3C3×D411yescharacteristic, complemented
121C4×C3⋊D3C3×D411yescharacteristic, complemented
122C4×C3⋊D3C23×C311yescharacteristic
123C4×C3⋊D3C23×C311yescharacteristic
124C3×#67—24no—
125C3×#67—24no—
126C3×#67—44no—
127C3×#67—64no—
128C3×#67—64no—
129C2×C3×Dic3—62no—
130C2×C3×Dic3—64no—
131C3×D12—22no—
132C3×D12—22no—
133C3×D12—42no—
134C3×D12—64no—
135C3×C4×D3—24no—
136C3×C4×D3—24no—
137C3×C4×D3—44no—
138C3×C4×D3—64no—
139C3×C4×D3—62no—
140C3×C4×D3—62no—
141C3×C4×D3—62no—
142C3×C4×D3—62no—
143C3×C4×D3—64no—
144C3×Dic6—22no—
145C3×Dic6—22no—
146C3×Dic6—42no—
147C3×Dic6—64no—
148#627—22no—
149#626—24no—
150#625—22no—
151#624C23×C314yes—
152D3×Dic3—24no—
153#532—91noSylow
154C2×C33—21no—
155C2×C33C22×D411yescharacteristic
156C3×C3⋊D3—21no—
157C3×C3⋊D3C2×#4112yes—
158C32×D3—44no—
159C3×#41—62no—
160C3×#41—62no—
161C3×#41—122no—
162C3×#41—182no—
163C3×#41—182no—
164C3×#41—182no—
165C2×C3×Q8—92no—
166C2×C3×D4—94no—
167C2×C3×D4—92no—
168C2×C3×D4—94no—
169C22×C3×C4—94no—
170C22×C3×C4—91no—
171C22×C3×C4—92no—
172#239—62no—
173D3×Q8—62no—
174#237—62no—
175#237—64no—
176D3×D4—62no—
177D3×D4—64no—
178#235—22no—
179#235—64no—
180C2×C4×D3—64no—
181C2×C4×D3—62no—
182C3×C4⋊C4—94no—
183C3×C22⋊C4—98no—
184C3×C42—92no—
185C3×C42—91no—
186C3×C42—91no—
187C22×C32—22no—
188C22×C32—42no—
189C22×C32—64no—
190C22×C32C2×C3×D411yescharacteristic
191C2×C3⋊D3—22no—
192C2×C3⋊D3C2×C3×D411yescharacteristic
193C2×C3⋊D3C24×C311yescharacteristic
194C2×C3×D3—24no—
195C2×C3×D3—24no—
196C2×C3×D3—44no—
197C2×C3×D3—62no—
198C2×C3×D3—62no—
199C2×C3×D3—128no—
200C2×C3×D3—124no—
201D32—28no—
202#157—22no—
203#157C3×#4114yescomplemented
204C32×C4—22no—
205C32×C4—22no—
206C32×C4—42no—
207C32×C4—42no—
208C32×C4—64no—
209C32×C4C2×C3×D411yescharacteristic
210C32×C4C2×C3×D411yescharacteristic
211C3⋊Dic3C2×C3×D411yescharacteristic
212C3⋊Dic3C2×C3×D411yescharacteristic
213C3×Dic3—24no—
214C3×Dic3—24no—
215C3×Dic3—44no—
216C3×Dic3—62no—
217C3×Dic3—62no—
218C2×#41—92no—
219C4⋊Q8—91no—
220#127—91no—
221#124—92no—
222#121—94no—
223C4×D4—94no—
224C2×C42—91no—
225C33#53211yescharacteristic, complemented, Sylow
226C23×C3—94no—
227C23×C3—91no—
228C22×D3—64no—
229C22×D3—62no—
230C3×Q8—62no—
231C3×Q8—62no—
232C3×Q8—122no—
233C3×Q8—182no—
234C3×D4—62no—
235C3×D4—64no—
236C3×D4—64no—
237C3×D4—62no—
238C3×D4—122no—
239C3×D4—124no—
240C3×D4—184no—
241C3×D4—182no—
242C3×D4—188no—
243C2×C3×C4—22no—
244C2×C3×C4—42no—
245C2×C3×C4—64no—
246C2×C3×C4—64no—
247C2×C3×C4—94no—
248C2×C3×C4—92no—
249C2×C3×C4—91no—
250C2×C3×C4—91no—
251C2×C3×C4—91no—
252C2×C3×C4—91no—
253C2×C3×C4—91no—
254C2×C3×C4—92no—
255C2×C3×C4—124no—
256C2×C3×C4—184no—
257C2×C3×C4—184no—
258C2×C3×C4GO+(4,𝔽2)11yescharacteristic, complemented
259#67—24no—
260#67—64no—
261#67—64no—
262C2×Dic3—62no—
263C2×Dic3—64no—
264D12—22no—
265D12—64no—
266C4×D3—24no—
267C4×D3—64no—
268C4×D3—62no—
269C4×D3—62no—
270C4×D3—62no—
271C4×D3—62no—
272C4×D3—64no—
273Dic6—22no—
274Dic6—64no—
275C2×C32—22no—
276C2×C32—21no—
277C2×C32—42no—
278C2×C32—42no—
279C2×C32—82no—
280C2×C32—124no—
281C2×C32C22×C3×D411yescharacteristic
282C3⋊D3—21no—
283C3⋊D3C2×C3×#4112yes—
284C3×D3—44no—
285C3×D3—44no—
286C3×D3—64no—
287C3×D3—84no—
288C3×D3—122no—
289#41—62no—
290#41—182no—
291#41—182no—
292#41—182no—
293C2×Q8—92no—
294C2×D4—94no—
295C2×D4—92no—
296C2×D4—94no—
297C22×C4—94no—
298C22×C4—91no—
299C22×C4—92no—
300C4⋊C4—94no—
301C22⋊C4—98no—
302C42—92no—
303C42—91no—
304C42—91no—
305C22×C3—22no—
306C22×C3—42no—
307C22×C3—64no—
308C22×C3—64no—
309C22×C3—91no—
310C22×C3—91no—
311C22×C3—124no—
312C22×C3—182no—
313C22×C3—188no—
314C22×C3C2×GO+(4,𝔽2)11yescharacteristic
315C2×D3—24no—
316C2×D3—62no—
317C2×D3—62no—
318C2×D3—128no—
319C2×D3—124no—
320C3×C4—22no—
321C3×C4—22no—
322C3×C4—42no—
323C3×C4—42no—
324C3×C4—64no—
325C3×C4—64no—
326C3×C4—94no—
327C3×C4—91no—
328C3×C4—91no—
329C3×C4—124no—
330C3×C4—182no—
331C3×C4C2×GO+(4,𝔽2)11yescharacteristic
332C3×C4C2×GO+(4,𝔽2)11yescharacteristic
333Dic3—24no—
334Dic3—62no—
335Dic3—62no—
336C32—22no—
337C32—42no—
338C32C3×#53211yescharacteristic, complemented
339C23—94no—
340C23—91no—
341Q8—62no—
342Q8—182no—
343D4—64no—
344D4—62no—
345D4—184no—
346D4—182no—
347D4—188no—
348C2×C4—64no—
349C2×C4—94no—
350C2×C4—92no—
351C2×C4—91no—
352C2×C4—91no—
353C2×C4—91no—
354C2×C4—91no—
355C2×C4—91no—
356C2×C4—92no—
357C2×C4—184no—
358C2×C4—184no—
359C2×C4C3×GO+(4,𝔽2)11yescharacteristic, complemented
360C2×C3—21no—
361C2×C3—22no—
362C2×C3—42no—
363C2×C3—42no—
364C2×C3—82no—
365C2×C3—92no—
366C2×C3—124no—
367C2×C3—124no—
368C2×C3—181no—
369C2×C3—244no—
370C2×C3C22×GO+(4,𝔽2)11yescharacteristic
371D3—44no—
372D3—64no—
373D3—122no—
374C22—64no—
375C22—91no—
376C22—91no—
377C22—182no—
378C22—188no—
379C22C2×C3×GO+(4,𝔽2)11yescharacteristic
380C4—64no—
381C4—94no—
382C4—91no—
383C4—91no—
384C4—182no—
385C4C2×C3×GO+(4,𝔽2)11yescharacteristic
386C4C2×C3×GO+(4,𝔽2)11yescharacteristic
387C3—22no—
388C3—42no—
389C3#1059651111yescharacteristic, direct product factor
390C2—21no—
391C2—92no—
392C2—124no—
393C2—181no—
394C2C22×C3×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
centerC3×C4C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊D3C24×C3
FrattiniC2C22×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC2×C33C22×D4
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#532
LCSG ▷ C2×C3⋊D3 ▷ 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×C4×#157C2semidirect
3. C3×#64242C2semidirect
4. C3×#64231C2semidirect
5. C3×C4×GO+(4,𝔽2)C2semidirect
6. C3×#64222C2semidirect
7. C3×#64221C2semidirect
8. #10596511C3direct (N × H)
26. #64306C2×C3semidirect
27. C2×C4×#157C2×C3semidirect
28. #64242C2×C3semidirect
29. #64231C2×C3semidirect
30. C4×GO+(4,𝔽2)C2×C3semidirect
31. #64222C2×C3semidirect
32. #64221C2×C3semidirect
39. C2×C33×C4D4semidirect
40. C2×C3×C3⋊Dic3D4semidirect
41. C3×C4×C3⋊D3D4semidirect
42. C3×C4×C3⋊D3D4semidirect
90. C3×#157#41semidirect
118. C2×C32×C4C3×D4semidirect
119. C2×C3⋊Dic3C3×D4semidirect
120. C4×C3⋊D3C3×D4semidirect
121. C4×C3⋊D3C3×D4semidirect
203. #157C3×#41semidirect
225. C33#532semidirect
258. C2×C3×C4GO+(4,𝔽2)semidirect
338. C32C3×#532semidirect
359. C2×C4C3×GO+(4,𝔽2)semidirect
389. C3#10596511direct (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×C4×#1572
C3×#642422
C3×#642312
C3×C4×GO+(4,𝔽2)2
C3×#642222
C3×#642212
#105965113
C3×#5329