Finite Group Databasebeta

← C3×#73415 #11803123

Subgroups — C3×#73415 #11803123

Order 1152.

1169 conjugacy classes of subgroups.

The subgroup lattice has 431 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×D12C211yesmaximal, characteristic
2C32×#376C211yesmaximal, characteristic, complemented
3C3×#4939C212yesmaximal
4C3×C4×C4⋊Dic3C211yesmaximal, characteristic, complemented
5C3×#4928C212yesmaximal, complemented
6C3×#4669C214yesmaximal, complemented
7C3×#4667C212yesmaximal, complemented
8C3×#4664C212yesmaximal, complemented
9C3×#2319—31nomaximal
10#73415C311yesmaximal, characteristic, direct product factor
11C22×C3×D12C2211yescharacteristic
12C22×C3×C4×D3C2212yes—
13C2×C32×C42C2212yes—
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×Dic3⋊C4C2214yes—
21C2×C3×C4×Dic3C2212yes—
22C3×C4×D12C418yescomplemented
23C32×#95C2211yescharacteristic
24C32×#95C2211yescharacteristic
25C32×#95C2212yes—
26C3×C3⋊#95C2212yes—
27C3×C3⋊#95C2214yes—
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—31no—
36C3×#376—21no—
37C3×#376D311yescharacteristic, complemented
38C3×#376C2×C311yescharacteristic, complemented
39#4939C2×C312yes—
40C4×C4⋊Dic3C2×C311yescharacteristic, complemented
41#4928C2×C312yescomplemented
42#4669C2×C314yescomplemented
43#4667C2×C312yescomplemented
44#4664C2×C312yescomplemented
45C23×C3×D3C2312yes—
46C22×C32×C4C2312yes—
47C22×C32×C4C2312yes—
48C22×C32×C4C2311yescharacteristic
49C22×C32×C4C2311yescharacteristic
50C22×C32×C4C2311yescharacteristic
51C22×C3×Dic3C2314yes—
52C22×C3×Dic3C2312yes—
53C2×C3×D12—24no—
54C2×C3×D12C2×C414yes—
55C2×C3×C4×D3—44no—
56C2×C3×C4×D3C2×C418yes—
57C32×C42—24no—
58C32×C42C2×C414yes—
59C3×(C2×D3)⋊C4—44no—
60C3×(C2×D3)⋊C4—44no—
61C3×(C2×D3)⋊C4C2×C418yes—
62C3×C4⋊Dic3—24no—
63C3×C4⋊Dic3C2×C414yes—
64C3×Dic3⋊C4—44no—
65C3×C4×Dic3—42no—
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—31no—
73C2×C3×C4⋊C4—32no—
74C2×C3×C4⋊C4—34no—
75C2×C3×C22⋊C4—32no—
76C2×C3×C22⋊C4—34no—
77C2×C3×C22⋊C4—34no—
78C2×C3×C42—22no—
79C2×C3×C42—21no—
80C2×C3×C42—32no—
81C2×C3×C42C2×D312yes—
82C2×C3×C42C2×D311yescharacteristic
83C2×C3×C42C22×C312yes—
84C2×C3×C42C22×C311yescharacteristic
85C2×(C2×D3)⋊C4C22×C314yes—
86C2×(C2×D3)⋊C4C22×C314yes—
87C2×(C2×D3)⋊C4C22×C312yes—
88C2×C4⋊Dic3C22×C312yes—
89C2×C4⋊Dic3C22×C311yescharacteristic
90C2×Dic3⋊C4C22×C314yes—
91C2×C4×Dic3C22×C312yes—
92C4×D12C3×C418yescomplemented
93C3×#95—21no—
94C3×#95—21no—
95C3×#95—22no—
96C3×#95—32no—
97C3×#95—34no—
98C3×#95C2×D311yescharacteristic
99C3×#95C2×D311yescharacteristic
100C3×#95C2×D312yes—
101C3×#95C22×C311yescharacteristic
102C3×#95C22×C311yescharacteristic
103C3×#95C22×C312yes—
104C3⋊#95C22×C312yes—
105C3⋊#95C22×C314yes—
106C23×C32C2411yescharacteristic
107C22×C3×D3—24no—
108C22×C3×D3—22no—
109C22×C3×D3—24no—
110C22×C3×D3—22no—
111C22×C3×D3C22×C414yes—
112C2×C32×C4—22no—
113C2×C32×C4—22no—
114C2×C32×C4—21no—
115C2×C32×C4—22no—
116C2×C32×C4—21no—
117C2×C32×C4—22no—
118C2×C32×C4—22no—
119C2×C32×C4—22no—
120C2×C32×C4—22no—
121C2×C32×C4C22×C412yes—
122C2×C32×C4C22×C412yes—
123C2×C32×C4C22×C412yes—
124C2×C32×C4#4114yes—
125C2×C3×Dic3—24no—
126C2×C3×Dic3—24no—
127C2×C3×Dic3—24no—
128C2×C3×Dic3—24no—
129C2×C3×Dic3C22×C414yes—
130C3×D12—28no—
131C3×C4×D3—48no—
132C2×C4×D4—31no—
133#390—32no—
134#389—32no—
135#388—34no—
136#381—32no—
137#380—32no—
138C4×C4⋊C4—31no—
139#376C3×D311yescharacteristic, complemented
140C24×C3—32no—
141C23×D3C23×C312yes—
142C2×C3×D4—34no—
143C2×C3×D4—64no—
144C22×C3×C4—21no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—21no—
148C22×C3×C4—21no—
149C22×C3×C4—32no—
150C22×C3×C4—38no—
151C22×C3×C4—34no—
152C22×C3×C4—124no—
153C22×C3×C4C22×D312yes—
154C22×C3×C4C22×D312yes—
155C22×C3×C4C22×D311yescharacteristic
156C22×C3×C4C22×D311yescharacteristic
157C22×C3×C4C22×D311yescharacteristic
158C22×C3×C4C23×C312yes—
159C22×C3×C4C23×C312yes—
160C22×C3×C4C23×C311yescharacteristic
161C22×C3×C4C23×C311yescharacteristic
162C22×C3×C4C23×C311yescharacteristic
163C22×Dic3C23×C314yes—
164C22×Dic3C23×C312yes—
165C2×D12—24no—
166C2×D12C2×C3×C414yes—
167C2×C4×D3—44no—
168C2×C4×D3C2×C3×C418yes—
169C3×C4⋊C4—34no—
170C3×C4⋊C4—64no—
171C3×C4⋊C4—124no—
172C3×C22⋊C4—38no—
173C3×C22⋊C4—124no—
174C3×C22⋊C4—124no—
175C3×C42—28no—
176C3×C42—24no—
177C3×C42—24no—
178C3×C42—24no—
179C3×C42—122no—
180C3×C42C4×D314yes—
181C3×C42C2×C3×C414yes—
182(C2×D3)⋊C4—44no—
183(C2×D3)⋊C4—44no—
184(C2×D3)⋊C4C2×C3×C418yes—
185C4⋊Dic3—24no—
186C4⋊Dic3C2×C3×C414yes—
187Dic3⋊C4—44no—
188C4×Dic3—42no—
189C22×C32C23×C411yescharacteristic
190C22×C32C2×#4112yes—
191C22×C32H(5,𝔽2)12yes—
192C22×C32H(5,𝔽2)11yescharacteristic
193C22×C322−1+411yescharacteristic
194C2×C3×D3—24no—
195C2×C3×D3—24no—
196C2×C3×D3—24no—
197C2×C3×D3—44no—
198C2×C3×D3—44no—
199C32×C4—22no—
200C32×C4—22no—
201C32×C4—42no—
202C32×C4—42no—
203C32×C4#11714yes—
204C3×Dic3—24no—
205C3×Dic3—44no—
206C22×D4—31no—
207C23×C4—32no—
208C4×D4—38no—
209C2×C4⋊C4—31no—
210C2×C4⋊C4—32no—
211C2×C4⋊C4—34no—
212C2×C22⋊C4—32no—
213C2×C22⋊C4—34no—
214C2×C22⋊C4—34no—
215C2×C42—32no—
216C2×C42C2×C3×D312yes—
217C2×C42C2×C3×D311yescharacteristic
218#95—32no—
219#95—34no—
220#95C2×C3×D311yescharacteristic
221#95C2×C3×D311yescharacteristic
222#95C2×C3×D312yes—
223C23×C3—21no—
224C23×C3—34no—
225C23×C3—64no—
226C23×C3—62no—
227C23×C3—64no—
228C23×C3—62no—
229C23×C3C23×D311yescharacteristic
230C23×C3C24×C311yescharacteristic
231C22×D3—24no—
232C22×D3—22no—
233C22×D3—24no—
234C22×D3—22no—
235C22×D3C22×C3×C414yes—
236C3×D4—68no—
237C2×C3×C4—21no—
238C2×C3×C4—22no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—22no—
242C2×C3×C4—21no—
243C2×C3×C4—24no—
244C2×C3×C4—24no—
245C2×C3×C4—22no—
246C2×C3×C4—22no—
247C2×C3×C4—22no—
248C2×C3×C4—22no—
249C2×C3×C4—22no—
250C2×C3×C4—22no—
251C2×C3×C4—22no—
252C2×C3×C4—22no—
253C2×C3×C4—22no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—22no—
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—22no—
260C2×C3×C4—24no—
261C2×C3×C4—34no—
262C2×C3×C4—41no—
263C2×C3×C4—42no—
264C2×C3×C4—41no—
265C2×C3×C4—42no—
266C2×C3×C4—42no—
267C2×C3×C4—42no—
268C2×C3×C4—42no—
269C2×C3×C4—64no—
270C2×C3×C4—64no—
271C2×C3×C4—64no—
272C2×C3×C4—64no—
273C2×C3×C4—128no—
274C2×C3×C4C2×C4×D312yes—
275C2×C3×C4C2×C4×D312yes—
276C2×C3×C4C2×C4×D312yes—
277C2×C3×C4#23514yes—
278C2×C3×C4C22×C3×C412yes—
279C2×C3×C4C22×C3×C412yes—
280C2×C3×C4C22×C3×C412yes—
281C2×C3×C4C3×#4114yes—
282C2×Dic3—24no—
283C2×Dic3—24no—
284C2×Dic3—24no—
285C2×Dic3—24no—
286C2×Dic3C22×C3×C414yes—
287D12—28no—
288C4×D3—48no—
289C2×C32C2×#11711yescharacteristic
290C2×C32#51811yescharacteristic
291C2×C32#52011yescharacteristic
292C2×C32#54012yes—
293C2×C32#54212yes—
294C3×D3—44no—
295C24—32no—
296C2×D4—34no—
297C2×D4—64no—
298C22×C4—32no—
299C22×C4—38no—
300C22×C4—34no—
301C22×C4—124no—
302C22×C4C22×C3×D312yes—
303C22×C4C22×C3×D312yes—
304C22×C4C22×C3×D311yescharacteristic
305C22×C4C22×C3×D311yescharacteristic
306C22×C4C22×C3×D311yescharacteristic
307C4⋊C4—34no—
308C4⋊C4—64no—
309C4⋊C4—124no—
310C22⋊C4—38no—
311C22⋊C4—124no—
312C22⋊C4—124no—
313C42—24no—
314C42—122no—
315C42C3×C4×D314yes—
316C22×C3—21no—
317C22×C3—22no—
318C22×C3—22no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—64no—
322C22×C3—64no—
323C22×C3—64no—
324C22×C3—124no—
325C22×C3—124no—
326C22×C3C22×C4×D311yescharacteristic
327C22×C3C2×#23512yes—
328C22×C3#100912yes—
329C22×C3#100911yescharacteristic
330C22×C3#101011yescharacteristic
331C22×C3C23×C3×C411yescharacteristic
332C22×C3C2×C3×#4112yes—
333C22×C3C3×H(5,𝔽2)12yes—
334C22×C3C3×H(5,𝔽2)11yescharacteristic
335C22×C3C3×2−1+411yescharacteristic
336C2×D3—24no—
337C2×D3—24no—
338C2×D3—24no—
339C2×D3—44no—
340C2×D3—44no—
341C3×C4—22no—
342C3×C4—22no—
343C3×C4—22no—
344C3×C4—22no—
345C3×C4—24no—
346C3×C4—42no—
347C3×C4—42no—
348C3×C4—44no—
349C3×C4—44no—
350C3×C4—42no—
351C3×C4—42no—
352C3×C4—42no—
353C3×C4—42no—
354C3×C4—64no—
355C3×C4—124no—
356C3×C4#87214yes—
357C3×C4C3×#11714yes—
358Dic3—24no—
359Dic3—44no—
360C32#231911yescharacteristic, complemented, Sylow
361C23—34no—
362C23—64no—
363C23—62no—
364C23—64no—
365C23—62no—
366C23C23×C3×D311yescharacteristic
367D4—68no—
368C2×C4—21no—
369C2×C4—22no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—34no—
378C2×C4—64no—
379C2×C4—64no—
380C2×C4—64no—
381C2×C4—64no—
382C2×C4—128no—
383C2×C4C2×C3×C4×D312yes—
384C2×C4C2×C3×C4×D312yes—
385C2×C4C2×C3×C4×D312yes—
386C2×C4C3×#23514yes—
387C2×C3—21no—
388C2×C3—21no—
389C2×C3—22no—
390C2×C3—22no—
391C2×C3—21no—
392C2×C3—124no—
393C2×C3C2×#87211yescharacteristic
394C2×C3#552111yescharacteristic
395C2×C3#552311yescharacteristic
396C2×C3#553412yes—
397C2×C3#553512yes—
398C2×C3C2×C3×#11711yescharacteristic
399C2×C3C3×#51811yescharacteristic
400C2×C3C3×#52011yescharacteristic
401C2×C3C3×#54012yes—
402C2×C3C3×#54212yes—
403D3—44no—
404C22—64no—
405C22—64no—
406C22—64no—
407C22—124no—
408C22—124no—
409C22C22×C3×C4×D311yescharacteristic
410C22C2×C3×#23512yes—
411C22C3×#100912yes—
412C22C3×#100911yescharacteristic
413C22C3×#101011yescharacteristic
414C4—22no—
415C4—22no—
416C4—42no—
417C4—42no—
418C4—64no—
419C4—124no—
420C4C3×#87214yes—
421C3—21no—
422C3#7341511yescharacteristic, direct product factor
423C3C3×#231911yescharacteristic, complemented
424C2—124no—
425C2C2×C3×#87211yescharacteristic
426C2C3×#552111yescharacteristic
427C2C3×#552311yescharacteristic
428C2C3×#553412yes—
429C2C3×#553512yes—
subgroupG/H
centerC23×C3C23×D3
commutatorC22×C3C23×C3×C4
FrattiniC23C23×C3×D3
FittingC32×#376C2
socleC23×C32C24
prime radicalC22×C3×D12C22
cosocleC23×C3C24×C3
hypercenterC3×#376D3
hypocenterC3C3×#2319
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C3×#376

Complements

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

NH ≅ G/Ntype
2. C32×#376C2semidirect
4. C3×C4×C4⋊Dic3C2semidirect
5. C3×#4928C2semidirect
6. C3×#4669C2semidirect
7. C3×#4667C2semidirect
8. C3×#4664C2semidirect
10. #73415C3direct (N × H)
22. C3×C4×D12C4semidirect
37. C3×#376D3semidirect
38. C3×#376C2×C3semidirect
40. C4×C4⋊Dic3C2×C3semidirect
41. #4928C2×C3semidirect
42. #4669C2×C3semidirect
43. #4667C2×C3semidirect
44. #4664C2×C3semidirect
92. C4×D12C3×C4semidirect
139. #376C3×D3semidirect
360. C32#2319semidirect
422. C3#73415direct (N × H)
423. 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×#3762
C3×#49392
C3×C4×C4⋊Dic32
C3×#49282
C3×#46692
C3×#46672
C3×#46642
C3×#23193
#734153