Finite Group Databasebeta

← C3×#74160 #11806041

Subgroups — C3×#74160 #11806041

Order 1152.

1197 conjugacy classes of subgroups.

The subgroup lattice has 429 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×D12C211yesmaximal, characteristic
2C32×#382C211yesmaximal, characteristic, complemented
3C3×#4988C212yesmaximal
4C3×Dic3×C4⋊C4C212yesmaximal, complemented
5C3×C4×C4⋊Dic3C211yesmaximal, characteristic, complemented
6C3×#4670C214yesmaximal, complemented
7C3×#4666C212yesmaximal, complemented
8C3×#4664C212yesmaximal, complemented
9C3×#2357—31nomaximal
10#74160C311yesmaximal, characteristic, direct product factor
11C22×C3×D12C2211yescharacteristic
12C22×C3×C4×D3C2212yes—
13C2×C32×C4⋊C4C2212yes—
14C2×C32×C42C2211yescharacteristic
15C2×C3×(C2×D3)⋊C4C2214yes—
16C2×C3×(C2×D3)⋊C4C2214yes—
17C2×C3×(C2×D3)⋊C4C2212yes—
18C2×C3×C4⋊Dic3C2212yes—
19C2×C3×C4⋊Dic3C2211yescharacteristic
20C2×C3×C4×Dic3C2212yes—
21C2×C3×C4×Dic3C2214yes—
22C3×C4×D12C418yescomplemented
23C32×#95C2212yes—
24C32×#95C2212yes—
25C3×C3⋊#95C2211yescharacteristic
26C3×C3⋊#95C2214yes—
27C3×C3⋊#95C2211yescharacteristic
28C2×C3×C4×D4—31no—
29C2×C4×D12C2×C311yescharacteristic
30C3×#390—32no—
31C3×#388—34no—
32C3×#382—21no—
33C3×#382D311yescharacteristic, complemented
34C3×#382C2×C311yescharacteristic, complemented
35C3×#381—32no—
36C3×C4×C4⋊C4—32no—
37C3×C4×C4⋊C4—31no—
38C3×C4×C22⋊C4—32no—
39#4988C2×C312yes—
40Dic3×C4⋊C4C2×C312yescomplemented
41C4×C4⋊Dic3C2×C311yescharacteristic, complemented
42#4670C2×C314yescomplemented
43#4666C2×C312yescomplemented
44#4664C2×C312yescomplemented
45C23×C3×D3C2312yes—
46C22×C32×C4C2314yes—
47C22×C32×C4C2311yescharacteristic
48C22×C32×C4C2311yescharacteristic
49C22×C32×C4C2311yescharacteristic
50C22×C3×Dic3C2312yes—
51C22×C3×Dic3C2312yes—
52C22×C3×Dic3C2312yes—
53C2×C3×D12—22no—
54C2×C3×D12—22no—
55C2×C3×D12C2×C414yes—
56C2×C3×C4×D3—44no—
57C2×C3×C4×D3C2×C418yes—
58C32×C4⋊C4—24no—
59C32×C42C2×C414yes—
60C3×(C2×D3)⋊C4—28no—
61C3×(C2×D3)⋊C4—44no—
62C3×(C2×D3)⋊C4C2×C418yes—
63C3×C4⋊Dic3—24no—
64C3×C4⋊Dic3C2×C414yes—
65C3×C4×Dic3—24no—
66C3×C4×Dic3—44no—
67#2357—31noSylow
68C22×C3×D4—31no—
69C23×C3×C4—32no—
70C22×D12C22×C311yescharacteristic
71C22×C4×D3C22×C312yes—
72C3×C4×D4—38no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—31no—
75C2×C3×C4⋊C4—32no—
76C2×C3×C4⋊C4C2×D312yes—
77C2×C3×C4⋊C4C22×C312yes—
78C2×C3×C22⋊C4—32no—
79C2×C3×C22⋊C4—34no—
80C2×C3×C22⋊C4—34no—
81C2×C3×C42—21no—
82C2×C3×C42—32no—
83C2×C3×C42—34no—
84C2×C3×C42C2×D311yescharacteristic
85C2×C3×C42C22×C311yescharacteristic
86C2×(C2×D3)⋊C4C22×C314yes—
87C2×(C2×D3)⋊C4C22×C314yes—
88C2×(C2×D3)⋊C4C22×C312yes—
89C2×C4⋊Dic3C22×C312yes—
90C2×C4⋊Dic3C22×C311yescharacteristic
91C2×C4×Dic3C22×C312yes—
92C2×C4×Dic3C22×C314yes—
93C4×D12C3×C418yescomplemented
94C3×#95—22no—
95C3×#95—22no—
96C3×#95—31no—
97C3×#95—31no—
98C3×#95—34no—
99C3×#95C2×D312yes—
100C3×#95C2×D312yes—
101C3×#95C22×C312yes—
102C3×#95C22×C312yes—
103C3⋊#95C22×C311yescharacteristic
104C3⋊#95C22×C314yes—
105C3⋊#95C22×C311yescharacteristic
106C23×C32C2411yescharacteristic
107C22×C3×D3—24no—
108C22×C3×D3—22no—
109C22×C3×D3—22no—
110C22×C3×D3—24no—
111C22×C3×D3C22×C414yes—
112C2×C32×C4—22no—
113C2×C32×C4—22no—
114C2×C32×C4—24no—
115C2×C32×C4—24no—
116C2×C32×C4—24no—
117C2×C32×C4C22×C412yes—
118C2×C32×C4C22×C412yes—
119C2×C32×C4C22×C412yes—
120C2×C32×C4#4112yes—
121C2×C32×C4#4112yes—
122C2×C3×Dic3—22no—
123C2×C3×Dic3—22no—
124C2×C3×Dic3—24no—
125C2×C3×Dic3—22no—
126C2×C3×Dic3—22no—
127C2×C3×Dic3—22no—
128C2×C3×Dic3C22×C414yes—
129C2×C3×Dic3#4114yes—
130C3×D12—28no—
131C3×C4×D3—48no—
132C2×C4×D4—31no—
133#390—32no—
134#388—34no—
135#382C3×D311yescharacteristic, complemented
136#381—32no—
137C4×C4⋊C4—32no—
138C4×C4⋊C4—31no—
139C4×C22⋊C4—32no—
140C24×C3—32no—
141C23×D3C23×C312yes—
142C2×C3×D4—34no—
143C2×C3×D4—62no—
144C2×C3×D4—62no—
145C22×C3×C4—21no—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—24no—
149C22×C3×C4—32no—
150C22×C3×C4—32no—
151C22×C3×C4—38no—
152C22×C3×C4—32no—
153C22×C3×C4—124no—
154C22×C3×C4C22×D314yes—
155C22×C3×C4C22×D311yescharacteristic
156C22×C3×C4C22×D311yescharacteristic
157C22×C3×C4C22×D311yescharacteristic
158C22×C3×C4C23×C314yes—
159C22×C3×C4C23×C311yescharacteristic
160C22×C3×C4C23×C311yescharacteristic
161C22×C3×C4C23×C311yescharacteristic
162C22×Dic3C23×C312yes—
163C22×Dic3C23×C312yes—
164C22×Dic3C23×C312yes—
165C2×D12—22no—
166C2×D12—22no—
167C2×D12C2×C3×C414yes—
168C2×C4×D3—44no—
169C2×C4×D3C2×C3×C418yes—
170C3×C4⋊C4—24no—
171C3×C4⋊C4—24no—
172C3×C4⋊C4—34no—
173C3×C4⋊C4—44no—
174C3×C4⋊C4—64no—
175C3×C22⋊C4—38no—
176C3×C22⋊C4—68no—
177C3×C22⋊C4—124no—
178C3×C42—24no—
179C3×C42—64no—
180C3×C42—124no—
181C3×C42C4×D314yes—
182C3×C42C2×C3×C414yes—
183(C2×D3)⋊C4—28no—
184(C2×D3)⋊C4—44no—
185(C2×D3)⋊C4C2×C3×C418yes—
186C4⋊Dic3—24no—
187C4⋊Dic3C2×C3×C414yes—
188C4×Dic3—24no—
189C4×Dic3—44no—
190C22×C32C23×C411yescharacteristic
191C22×C32C2×#4111yescharacteristic
192C22×C32C2×#4111yescharacteristic
193C22×C32C2×#4112yes—
194C22×C32H(5,𝔽2)12yes—
195C2×C3×D3—24no—
196C2×C3×D3—24no—
197C2×C3×D3—24no—
198C2×C3×D3—44no—
199C2×C3×D3—44no—
200C32×C4—22no—
201C32×C4—22no—
202C32×C4—44no—
203C32×C4#11714yes—
204C3×Dic3—24no—
205C3×Dic3—24no—
206C3×Dic3—42no—
207C22×D4—31no—
208C23×C4—32no—
209C4×D4—38no—
210C2×C4⋊C4—31no—
211C2×C4⋊C4—32no—
212C2×C4⋊C4C2×C3×D312yes—
213C2×C22⋊C4—32no—
214C2×C22⋊C4—34no—
215C2×C22⋊C4—34no—
216C2×C42—32no—
217C2×C42—34no—
218C2×C42C2×C3×D311yescharacteristic
219#95—31no—
220#95—31no—
221#95—34no—
222#95C2×C3×D312yes—
223#95C2×C3×D312yes—
224C23×C3—21no—
225C23×C3—34no—
226C23×C3—62no—
227C23×C3—62no—
228C23×C3—64no—
229C23×C3—64no—
230C23×C3C23×D311yescharacteristic
231C23×C3C24×C311yescharacteristic
232C22×D3—24no—
233C22×D3—22no—
234C22×D3—22no—
235C22×D3—24no—
236C22×D3C22×C3×C414yes—
237C3×D4—68no—
238C2×C3×C4—22no—
239C2×C3×C4—22no—
240C2×C3×C4—22no—
241C2×C3×C4—22no—
242C2×C3×C4—24no—
243C2×C3×C4—24no—
244C2×C3×C4—24no—
245C2×C3×C4—24no—
246C2×C3×C4—24no—
247C2×C3×C4—24no—
248C2×C3×C4—22no—
249C2×C3×C4—22no—
250C2×C3×C4—22no—
251C2×C3×C4—22no—
252C2×C3×C4—22no—
253C2×C3×C4—34no—
254C2×C3×C4—34no—
255C2×C3×C4—42no—
256C2×C3×C4—42no—
257C2×C3×C4—44no—
258C2×C3×C4—44no—
259C2×C3×C4—44no—
260C2×C3×C4—62no—
261C2×C3×C4—62no—
262C2×C3×C4—64no—
263C2×C3×C4—62no—
264C2×C3×C4—62no—
265C2×C3×C4—62no—
266C2×C3×C4—128no—
267C2×C3×C4C2×C4×D312yes—
268C2×C3×C4C2×C4×D312yes—
269C2×C3×C4C2×C4×D312yes—
270C2×C3×C4#23712yes—
271C2×C3×C4#23912yes—
272C2×C3×C4C22×C3×C412yes—
273C2×C3×C4C22×C3×C412yes—
274C2×C3×C4C22×C3×C412yes—
275C2×C3×C4C3×#4112yes—
276C2×C3×C4C3×#4112yes—
277C2×Dic3—22no—
278C2×Dic3—22no—
279C2×Dic3—24no—
280C2×Dic3—22no—
281C2×Dic3—22no—
282C2×Dic3—22no—
283C2×Dic3C22×C3×C414yes—
284C2×Dic3C3×#4114yes—
285D12—28no—
286C4×D3—48no—
287C2×C32C2×#11711yescharacteristic
288C2×C32C4×#4111yescharacteristic
289C2×C32#51811yescharacteristic
290C2×C32#55312yes—
291C2×C32#55912yes—
292C3×D3—44no—
293C24—32no—
294C2×D4—34no—
295C2×D4—62no—
296C2×D4—62no—
297C22×C4—32no—
298C22×C4—32no—
299C22×C4—38no—
300C22×C4—32no—
301C22×C4—124no—
302C22×C4C22×C3×D314yes—
303C22×C4C22×C3×D311yescharacteristic
304C22×C4C22×C3×D311yescharacteristic
305C22×C4C22×C3×D311yescharacteristic
306C4⋊C4—24no—
307C4⋊C4—34no—
308C4⋊C4—64no—
309C22⋊C4—38no—
310C22⋊C4—68no—
311C22⋊C4—124no—
312C42—64no—
313C42—124no—
314C42C3×C4×D314yes—
315C22×C3—21no—
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—22no—
319C22×C3—22no—
320C22×C3—64no—
321C22×C3—64no—
322C22×C3—64no—
323C22×C3—124no—
324C22×C3—124no—
325C22×C3C22×C4×D311yescharacteristic
326C22×C3C2×#23711yescharacteristic
327C22×C3C2×#23911yescharacteristic
328C22×C3D3×#4112yes—
329C22×C3#100912yes—
330C22×C3C23×C3×C411yescharacteristic
331C22×C3C2×C3×#4111yescharacteristic
332C22×C3C2×C3×#4111yescharacteristic
333C22×C3C2×C3×#4112yes—
334C22×C3C3×H(5,𝔽2)12yes—
335C2×D3—24no—
336C2×D3—24no—
337C2×D3—24no—
338C2×D3—44no—
339C2×D3—44no—
340C3×C4—22no—
341C3×C4—22no—
342C3×C4—22no—
343C3×C4—22no—
344C3×C4—24no—
345C3×C4—42no—
346C3×C4—42no—
347C3×C4—44no—
348C3×C4—44no—
349C3×C4—64no—
350C3×C4—64no—
351C3×C4—84no—
352C3×C4—122no—
353C3×C4#89214yes—
354C3×C4C3×#11714yes—
355Dic3—24no—
356Dic3—24no—
357Dic3—42no—
358C32#235711yescharacteristic, complemented, Sylow
359C23—34no—
360C23—62no—
361C23—62no—
362C23—64no—
363C23—64no—
364C23C23×C3×D311yescharacteristic
365D4—68no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—34no—
372C2×C4—34no—
373C2×C4—62no—
374C2×C4—62no—
375C2×C4—64no—
376C2×C4—62no—
377C2×C4—62no—
378C2×C4—62no—
379C2×C4—128no—
380C2×C4C2×C3×C4×D312yes—
381C2×C4C2×C3×C4×D312yes—
382C2×C4C2×C3×C4×D312yes—
383C2×C4C3×#23712yes—
384C2×C4C3×#23912yes—
385C2×C3—21no—
386C2×C3—21no—
387C2×C3—21no—
388C2×C3—22no—
389C2×C3—22no—
390C2×C3—124no—
391C2×C3C2×#89211yescharacteristic
392C2×C3#552111yescharacteristic
393C2×C3#552211yescharacteristic
394C2×C3#568112yes—
395C2×C3#569512yes—
396C2×C3C2×C3×#11711yescharacteristic
397C2×C3C3×C4×#4111yescharacteristic
398C2×C3C3×#51811yescharacteristic
399C2×C3C3×#55312yes—
400C2×C3C3×#55912yes—
401D3—44no—
402C22—64no—
403C22—64no—
404C22—64no—
405C22—124no—
406C22—124no—
407C22C22×C3×C4×D311yescharacteristic
408C22C2×C3×#23711yescharacteristic
409C22C2×C3×#23911yescharacteristic
410C22C3×D3×#4112yes—
411C22C3×#100912yes—
412C4—22no—
413C4—22no—
414C4—44no—
415C4—64no—
416C4—64no—
417C4—122no—
418C4C3×#89214yes—
419C3—21no—
420C3#7416011yescharacteristic, direct product factor
421C3C3×#235711yescharacteristic, complemented
422C2—124no—
423C2C2×C3×#89211yescharacteristic
424C2C3×#552111yescharacteristic
425C2C3×#552211yescharacteristic
426C2C3×#568112yes—
427C2C3×#569512yes—
subgroupG/H
centerC23×C3C23×D3
commutatorC22×C3C23×C3×C4
FrattiniC23C23×C3×D3
FittingC32×#382C2
socleC23×C32C24
prime radicalC22×C3×D12C22
cosocleC23×C3C24×C3
hypercenterC3×#382D3
hypocenterC3C3×#2357
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C3×#382

Complements

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

NH ≅ G/Ntype
2. C32×#382C2semidirect
4. C3×Dic3×C4⋊C4C2semidirect
5. C3×C4×C4⋊Dic3C2semidirect
6. C3×#4670C2semidirect
7. C3×#4666C2semidirect
8. C3×#4664C2semidirect
10. #74160C3direct (N × H)
22. C3×C4×D12C4semidirect
33. C3×#382D3semidirect
34. C3×#382C2×C3semidirect
40. Dic3×C4⋊C4C2×C3semidirect
41. C4×C4⋊Dic3C2×C3semidirect
42. #4670C2×C3semidirect
43. #4666C2×C3semidirect
44. #4664C2×C3semidirect
93. C4×D12C3×C4semidirect
135. #382C3×D3semidirect
358. C32#2357semidirect
420. C3#74160direct (N × H)
421. C3C3×#2357semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×D122
C32×#3822
C3×#49882
C3×Dic3×C4⋊C42
C3×C4×C4⋊Dic32
C3×#46702
C3×#46662
C3×#46642
C3×#23573
#741603