Finite Group Databasebeta

← C3×#73761 #11803105

Subgroups — C3×#73761 #11803105

Order 1152.

1177 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×D12C211yesmaximal, characteristic
2C32×C4×C4⋊C4C211yesmaximal, characteristic, complemented
3C3×#4988C212yesmaximal
4C3×#4972C212yesmaximal, complemented
5C3×#4940C214yesmaximal, complemented
6C3×#4932C211yesmaximal, characteristic, complemented
7C3×#4667C212yesmaximal, complemented
8C3×#4664C212yesmaximal, complemented
9C3×#2319—31nomaximal
10#73761C311yesmaximal, characteristic, direct product factor
11C22×C3×D12C2211yescharacteristic
12C22×C3×C4×D3C2212yes—
13C2×C32×C4⋊C4C2212yes—
14C2×C32×C42C2212yes—
15C2×C32×C42C2211yescharacteristic
16C2×C3×(C2×D3)⋊C4C2214yes—
17C2×C3×(C2×D3)⋊C4C2214yes—
18C2×C3×(C2×D3)⋊C4C2212yes—
19C2×C3×C4⋊Dic3C2211yescharacteristic
20C2×C3×Dic3⋊C4C2214yes—
21C2×C3×C4×Dic3C2212yes—
22C3×C4×D12C418yescomplemented
23C32×#95C2212yes—
24C3×C3⋊#95C2211yescharacteristic
25C3×C3⋊#95C2214yes—
26C3×C3⋊#95C2211yescharacteristic
27C3×C3⋊#95C2212yes—
28C2×C3×C4×D4—31no—
29C2×C4×D12C2×C311yescharacteristic
30C3×#390—32no—
31C3×#389—32no—
32C3×#388—34no—
33C3×#381—32no—
34C3×#380—32no—
35C3×C4×C4⋊C4—21no—
36C3×C4×C4⋊C4D311yescharacteristic, complemented
37C3×C4×C4⋊C4C2×C311yescharacteristic, complemented
38C3×#376—31no—
39#4988C2×C312yes—
40#4972C2×C312yescomplemented
41#4940C2×C314yescomplemented
42#4932C2×C311yescharacteristic, complemented
43#4667C2×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—24no—
54C2×C3×D12C2×C414yes—
55C2×C3×C4×D3—44no—
56C2×C3×C4×D3C2×C418yes—
57C32×C4⋊C4—24no—
58C32×C42—42no—
59C32×C42C2×C414yes—
60C3×(C2×D3)⋊C4—44no—
61C3×(C2×D3)⋊C4—44no—
62C3×(C2×D3)⋊C4C2×C418yes—
63C3×C4⋊Dic3C2×C414yes—
64C3×Dic3⋊C4—44no—
65C3×C4×Dic3—24no—
66#2319—31noSylow
67C22×C3×D4—31no—
68C23×C3×C4—32no—
69C22×D12C22×C311yescharacteristic
70C22×C4×D3C22×C312yes—
71C3×C4×D4—38no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—31no—
74C2×C3×C4⋊C4—34no—
75C2×C3×C4⋊C4C2×D312yes—
76C2×C3×C4⋊C4C22×C312yes—
77C2×C3×C22⋊C4—32no—
78C2×C3×C22⋊C4—34no—
79C2×C3×C22⋊C4—34no—
80C2×C3×C42—21no—
81C2×C3×C42—22no—
82C2×C3×C42—32no—
83C2×C3×C42C2×D312yes—
84C2×C3×C42C2×D311yescharacteristic
85C2×C3×C42C22×C312yes—
86C2×C3×C42C22×C311yescharacteristic
87C2×(C2×D3)⋊C4C22×C314yes—
88C2×(C2×D3)⋊C4C22×C314yes—
89C2×(C2×D3)⋊C4C22×C312yes—
90C2×C4⋊Dic3C22×C311yescharacteristic
91C2×Dic3⋊C4C22×C314yes—
92C2×C4×Dic3C22×C312yes—
93C4×D12C3×C418yescomplemented
94C3×#95—22no—
95C3×#95—31no—
96C3×#95—31no—
97C3×#95—32no—
98C3×#95—34no—
99C3×#95C2×D312yes—
100C3×#95C22×C312yes—
101C3⋊#95C22×C311yescharacteristic
102C3⋊#95C22×C314yes—
103C3⋊#95C22×C311yescharacteristic
104C3⋊#95C22×C312yes—
105C23×C32C2411yescharacteristic
106C22×C3×D3—24no—
107C22×C3×D3—22no—
108C22×C3×D3—24no—
109C22×C3×D3—22no—
110C22×C3×D3C22×C414yes—
111C2×C32×C4—21no—
112C2×C32×C4—21no—
113C2×C32×C4—24no—
114C2×C32×C4—22no—
115C2×C32×C4—24no—
116C2×C32×C4—24no—
117C2×C32×C4C22×C412yes—
118C2×C32×C4C22×C412yes—
119C2×C32×C4C22×C412yes—
120C2×C32×C4#4114yes—
121C2×C3×Dic3—24no—
122C2×C3×Dic3—22no—
123C2×C3×Dic3—22no—
124C2×C3×Dic3—22no—
125C2×C3×Dic3—22no—
126C2×C3×Dic3—22no—
127C2×C3×Dic3—22no—
128C2×C3×Dic3C22×C414yes—
129C3×D12—28no—
130C3×C4×D3—48no—
131C2×C4×D4—31no—
132#390—32no—
133#389—32no—
134#388—34no—
135#381—32no—
136#380—32no—
137C4×C4⋊C4C3×D311yescharacteristic, complemented
138#376—31no—
139C24×C3—32no—
140C23×D3C23×C312yes—
141C2×C3×D4—34no—
142C2×C3×D4—64no—
143C22×C3×C4—21no—
144C22×C3×C4—21no—
145C22×C3×C4—21no—
146C22×C3×C4—24no—
147C22×C3×C4—32no—
148C22×C3×C4—38no—
149C22×C3×C4—32no—
150C22×C3×C4—32no—
151C22×C3×C4—124no—
152C22×C3×C4C22×D314yes—
153C22×C3×C4C22×D311yescharacteristic
154C22×C3×C4C22×D311yescharacteristic
155C22×C3×C4C22×D311yescharacteristic
156C22×C3×C4C23×C314yes—
157C22×C3×C4C23×C311yescharacteristic
158C22×C3×C4C23×C311yescharacteristic
159C22×C3×C4C23×C311yescharacteristic
160C22×Dic3C23×C312yes—
161C22×Dic3C23×C312yes—
162C22×Dic3C23×C312yes—
163C2×D12—24no—
164C2×D12C2×C3×C414yes—
165C2×C4×D3—44no—
166C2×C4×D3C2×C3×C418yes—
167C3×C4⋊C4—28no—
168C3×C4⋊C4—24no—
169C3×C4⋊C4—24no—
170C3×C4⋊C4—34no—
171C3×C4⋊C4—124no—
172C3×C22⋊C4—38no—
173C3×C22⋊C4—124no—
174C3×C22⋊C4—124no—
175C3×C42—24no—
176C3×C42—42no—
177C3×C42—42no—
178C3×C42—44no—
179C3×C42—64no—
180C3×C42C4×D314yes—
181C3×C42C2×C3×C414yes—
182(C2×D3)⋊C4—44no—
183(C2×D3)⋊C4—44no—
184(C2×D3)⋊C4C2×C3×C418yes—
185C4⋊Dic3C2×C3×C414yes—
186Dic3⋊C4—44no—
187C4×Dic3—24no—
188C22×C32C23×C411yescharacteristic
189C22×C32C2×#4112yes—
190C22×C32H(5,𝔽2)11yescharacteristic
191C22×C32H(5,𝔽2)12yes—
192C22×C322−1+411yescharacteristic
193C2×C3×D3—24no—
194C2×C3×D3—24no—
195C2×C3×D3—24no—
196C2×C3×D3—44no—
197C2×C3×D3—44no—
198C32×C4—22no—
199C32×C4—22no—
200C32×C4—44no—
201C32×C4#11714yes—
202C3×Dic3—24no—
203C3×Dic3—42no—
204C3×Dic3—42no—
205C22×D4—31no—
206C23×C4—32no—
207C4×D4—38no—
208C2×C4⋊C4—31no—
209C2×C4⋊C4—34no—
210C2×C4⋊C4C2×C3×D312yes—
211C2×C22⋊C4—32no—
212C2×C22⋊C4—34no—
213C2×C22⋊C4—34no—
214C2×C42—32no—
215C2×C42C2×C3×D312yes—
216C2×C42C2×C3×D311yescharacteristic
217#95—31no—
218#95—31no—
219#95—32no—
220#95—34no—
221#95C2×C3×D312yes—
222C23×C3—21no—
223C23×C3—34no—
224C23×C3—64no—
225C23×C3—62no—
226C23×C3—62no—
227C23×C3—64no—
228C23×C3C23×D311yescharacteristic
229C23×C3C24×C311yescharacteristic
230C22×D3—24no—
231C22×D3—22no—
232C22×D3—24no—
233C22×D3—22no—
234C22×D3C22×C3×C414yes—
235C3×D4—68no—
236C2×C3×C4—22no—
237C2×C3×C4—22no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—22no—
241C2×C3×C4—21no—
242C2×C3×C4—22no—
243C2×C3×C4—21no—
244C2×C3×C4—28no—
245C2×C3×C4—24no—
246C2×C3×C4—24no—
247C2×C3×C4—24no—
248C2×C3×C4—24no—
249C2×C3×C4—24no—
250C2×C3×C4—24no—
251C2×C3×C4—22no—
252C2×C3×C4—22no—
253C2×C3×C4—22no—
254C2×C3×C4—24no—
255C2×C3×C4—34no—
256C2×C3×C4—44no—
257C2×C3×C4—42no—
258C2×C3×C4—44no—
259C2×C3×C4—64no—
260C2×C3×C4—62no—
261C2×C3×C4—62no—
262C2×C3×C4—62no—
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#23514yes—
271C2×C3×C4C22×C3×C412yes—
272C2×C3×C4C22×C3×C412yes—
273C2×C3×C4C22×C3×C412yes—
274C2×C3×C4C3×#4114yes—
275C2×Dic3—24no—
276C2×Dic3—22no—
277C2×Dic3—22no—
278C2×Dic3—22no—
279C2×Dic3—22no—
280C2×Dic3—22no—
281C2×Dic3—22no—
282C2×Dic3C22×C3×C414yes—
283D12—28no—
284C4×D3—48no—
285C2×C32C2×#11711yescharacteristic
286C2×C32#51811yescharacteristic
287C2×C32#52011yescharacteristic
288C2×C32#54012yes—
289C2×C32#54212yes—
290C3×D3—44no—
291C24—32no—
292C2×D4—34no—
293C2×D4—64no—
294C22×C4—32no—
295C22×C4—38no—
296C22×C4—32no—
297C22×C4—32no—
298C22×C4—124no—
299C22×C4C22×C3×D314yes—
300C22×C4C22×C3×D311yescharacteristic
301C22×C4C22×C3×D311yescharacteristic
302C22×C4C22×C3×D311yescharacteristic
303C4⋊C4—24no—
304C4⋊C4—34no—
305C4⋊C4—124no—
306C22⋊C4—38no—
307C22⋊C4—124no—
308C22⋊C4—124no—
309C42—42no—
310C42—64no—
311C42C3×C4×D314yes—
312C22×C3—21no—
313C22×C3—22no—
314C22×C3—21no—
315C22×C3—21no—
316C22×C3—22no—
317C22×C3—64no—
318C22×C3—64no—
319C22×C3—64no—
320C22×C3—124no—
321C22×C3—124no—
322C22×C3C22×C4×D311yescharacteristic
323C22×C3C2×#23512yes—
324C22×C3#100411yescharacteristic
325C22×C3#100711yescharacteristic
326C22×C3#100912yes—
327C22×C3C23×C3×C411yescharacteristic
328C22×C3C2×C3×#4112yes—
329C22×C3C3×H(5,𝔽2)11yescharacteristic
330C22×C3C3×H(5,𝔽2)12yes—
331C22×C3C3×2−1+411yescharacteristic
332C2×D3—24no—
333C2×D3—24no—
334C2×D3—24no—
335C2×D3—44no—
336C2×D3—44no—
337C3×C4—24no—
338C3×C4—22no—
339C3×C4—22no—
340C3×C4—22no—
341C3×C4—22no—
342C3×C4—24no—
343C3×C4—42no—
344C3×C4—48no—
345C3×C4—44no—
346C3×C4—44no—
347C3×C4—64no—
348C3×C4—122no—
349C3×C4—122no—
350C3×C4#87214yes—
351C3×C4C3×#11714yes—
352Dic3—24no—
353Dic3—42no—
354Dic3—42no—
355C32#231911yescharacteristic, complemented, Sylow
356C23—34no—
357C23—64no—
358C23—62no—
359C23—62no—
360C23—64no—
361C23C23×C3×D311yescharacteristic
362D4—68no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—34no—
370C2×C4—64no—
371C2×C4—62no—
372C2×C4—62no—
373C2×C4—62no—
374C2×C4—62no—
375C2×C4—62no—
376C2×C4—62no—
377C2×C4—128no—
378C2×C4C2×C3×C4×D312yes—
379C2×C4C2×C3×C4×D312yes—
380C2×C4C2×C3×C4×D312yes—
381C2×C4C3×#23514yes—
382C2×C3—21no—
383C2×C3—21no—
384C2×C3—22no—
385C2×C3—22no—
386C2×C3—21no—
387C2×C3—124no—
388C2×C3C2×#87211yescharacteristic
389C2×C3#550411yescharacteristic
390C2×C3#552111yescharacteristic
391C2×C3#556212yes—
392C2×C3#558212yes—
393C2×C3C2×C3×#11711yescharacteristic
394C2×C3C3×#51811yescharacteristic
395C2×C3C3×#52011yescharacteristic
396C2×C3C3×#54012yes—
397C2×C3C3×#54212yes—
398D3—44no—
399C22—64no—
400C22—64no—
401C22—64no—
402C22—124no—
403C22—124no—
404C22C22×C3×C4×D311yescharacteristic
405C22C2×C3×#23512yes—
406C22C3×#100411yescharacteristic
407C22C3×#100711yescharacteristic
408C22C3×#100912yes—
409C4—22no—
410C4—22no—
411C4—44no—
412C4—64no—
413C4—122no—
414C4—122no—
415C4C3×#87214yes—
416C3—21no—
417C3#7376111yescharacteristic, direct product factor
418C3C3×#231911yescharacteristic, complemented
419C2—124no—
420C2C2×C3×#87211yescharacteristic
421C2C3×#550411yescharacteristic
422C2C3×#552111yescharacteristic
423C2C3×#556212yes—
424C2C3×#558212yes—
subgroupG/H
centerC23×C3C23×D3
commutatorC22×C3C23×C3×C4
FrattiniC23C23×C3×D3
FittingC32×C4×C4⋊C4C2
socleC23×C32C24
prime radicalC22×C3×D12C22
cosocleC23×C3C24×C3
hypercenterC3×C4×C4⋊C4D3
hypocenterC3C3×#2319
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C3×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C32×C4×C4⋊C4C2semidirect
4. C3×#4972C2semidirect
5. C3×#4940C2semidirect
6. C3×#4932C2semidirect
7. C3×#4667C2semidirect
8. C3×#4664C2semidirect
10. #73761C3direct (N × H)
22. C3×C4×D12C4semidirect
36. C3×C4×C4⋊C4D3semidirect
37. C3×C4×C4⋊C4C2×C3semidirect
40. #4972C2×C3semidirect
41. #4940C2×C3semidirect
42. #4932C2×C3semidirect
43. #4667C2×C3semidirect
44. #4664C2×C3semidirect
93. C4×D12C3×C4semidirect
137. C4×C4⋊C4C3×D3semidirect
355. C32#2319semidirect
417. C3#73761direct (N × H)
418. C3C3×#2319semidirect

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×C4×C4⋊C42
C3×#49882
C3×#49722
C3×#49402
C3×#49322
C3×#46672
C3×#46642
C3×#23193
#737613