Finite Group Databasebeta

← C3×#82317 #11800907

Subgroups — C3×#82317 #11800907

Order 1152.

1773 conjugacy classes of subgroups.

The subgroup lattice has 461 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×C4×D4C211yesmaximal, characteristic
2C2×C3×D4×Dic3C212yesmaximal
3C3×C4×#217C212yesmaximal, complemented
4C2×C3×C3⋊#95C212yesmaximal, complemented
5C3×Dic3×C4⋊C4C212yesmaximal, complemented
6C3×Dic3×C22⋊C4C214yesmaximal, complemented
7C3×C4×C4⋊Dic3C211yesmaximal, characteristic, complemented
8C3×#4932C211yesmaximal, characteristic, complemented
9C3×#2275—31nomaximal
10#82317C311yesmaximal, characteristic, direct product factor
11C22×C32×D4C2211yescharacteristic
12C23×C32×C4C2212yes—
13C23×C3×Dic3C2214yes—
14C32×C4×D4C418yescomplemented
15C2×C32×C4⋊C4C2211yescharacteristic
16C2×C32×C22⋊C4C2212yes—
17C2×C32×C42C2211yescharacteristic
18C2×C3×#217C2214yes—
19C3×D4×Dic3C4116yescomplemented
20C2×C3×C4⋊Dic3C2212yes—
21C2×C3×C4×Dic3C2212yes—
22C2×C3×C4×Dic3C2214yes—
23C2×C3×C4×Dic3C2212yes—
24C3×C3⋊#95C2212yes—
25C3×C3⋊#95C2211yescharacteristic
26C3×C3⋊#95C2211yescharacteristic
27C3×C3⋊#95C2212yes—
28C3×C3⋊#95C2214yes—
29C2×C3×C4×D4—21no—
30C2×C3×C4×D4—32no—
31C2×C3×C4×D4D311yescharacteristic
32C2×C3×C4×D4C2×C311yescharacteristic
33C2×D4×Dic3C2×C312yes—
34C3×C4×C4⋊C4—32no—
35C3×C4×C4⋊C4—31no—
36C3×C4×C22⋊C4—32no—
37C3×C4×C22⋊C4—34no—
38C3×#376—31no—
39C2×C3×#95—32no—
40C4×#217C2×C312yescomplemented
41C2×C3⋊#95C2×C312yescomplemented
42Dic3×C4⋊C4C2×C312yescomplemented
43Dic3×C22⋊C4C2×C314yescomplemented
44C4×C4⋊Dic3C2×C311yescharacteristic, complemented
45#4932C2×C311yescharacteristic, complemented
46C24×C32C2312yes—
47C2×C32×D4C2×C418yes—
48C2×C32×D4C2×C414yes—
49C22×C32×C4—44no—
50C22×C32×C4C2×C418yescomplemented
51C22×C32×C4C2312yes—
52C22×C32×C4C2311yescharacteristic
53C22×C32×C4C2311yescharacteristic
54C22×C32×C4C2311yescharacteristic
55C22×C3×Dic3—44no—
56C22×C3×Dic3—44no—
57C22×C3×Dic3C2×C4116yescomplemented
58C22×C3×Dic3C2312yes—
59C22×C3×Dic3C2314yes—
60C22×C3×Dic3C2312yes—
61C32×C4⋊C4C2×C414yescomplemented
62C32×C22⋊C4C2×C418yescomplemented
63C32×C42C2×C414yescomplemented
64C3×#217C2×C4116yescomplemented
65C3×C4⋊Dic3C2×C418yescomplemented
66C3×C4×Dic3—44no—
67C3×C4×Dic3—42no—
68C3×C4×Dic3C2×C418yescomplemented
69#2275—31noSylow
70C22×C3×D4—21no—
71C22×C3×D4C2×D311yescharacteristic
72C22×C3×D4C22×C311yescharacteristic
73C23×C3×C4—22no—
74C23×C3×C4—34no—
75C23×C3×C4C2×D312yes—
76C23×C3×C4C22×C312yes—
77C23×Dic3C22×C314yes—
78C3×C4×D4—28no—
79C3×C4×D4—316no—
80C3×C4×D4Dic318yescomplemented
81C3×C4×D4C3×C418yescomplemented
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—32no—
84C2×C3×C4⋊C4C2×D311yescharacteristic
85C2×C3×C4⋊C4C22×C311yescharacteristic
86C2×C3×C22⋊C4—22no—
87C2×C3×C22⋊C4—34no—
88C2×C3×C22⋊C4C2×D312yes—
89C2×C3×C22⋊C4C22×C312yes—
90C2×C3×C42—21no—
91C2×C3×C42—32no—
92C2×C3×C42—34no—
93C2×C3×C42—32no—
94C2×C3×C42C2×D311yescharacteristic
95C2×C3×C42C22×C311yescharacteristic
96C2×#217C22×C314yes—
97D4×Dic3C3×C4116yescomplemented
98C2×C4⋊Dic3C22×C312yes—
99C2×C4×Dic3C22×C312yes—
100C2×C4×Dic3C22×C314yes—
101C2×C4×Dic3C22×C312yes—
102C3×#95—31no—
103C3×#95—34no—
104C3×#95—32no—
105C3×#95—32no—
106C3×#95—31no—
107C3⋊#95C22×C312yes—
108C3⋊#95C22×C311yescharacteristic
109C3⋊#95C22×C311yescharacteristic
110C3⋊#95C22×C312yes—
111C3⋊#95C22×C314yes—
112C23×C32—22no—
113C23×C32—24no—
114C23×C32—22no—
115C23×C32C22×C418yes—
116C23×C32C22×C414yes—
117C23×C32C2411yescharacteristic
118C32×D4C42116yescomplemented
119C2×C32×C4—22no—
120C2×C32×C4—21no—
121C2×C32×C4—24no—
122C2×C32×C4—21no—
123C2×C32×C4—48no—
124C2×C32×C4C22×C414yes—
125C2×C32×C4C22×C412yes—
126C2×C32×C4C22×C412yes—
127C2×C32×C4C22×C414yes—
128C2×C32×C4C22×C412yes—
129C2×C3×Dic3—22no—
130C2×C3×Dic3—24no—
131C2×C3×Dic3—24no—
132C2×C3×Dic3—22no—
133C2×C3×Dic3—22no—
134C2×C3×Dic3—22no—
135C2×C3×Dic3—416no—
136C2×C3×Dic3C22×C414yes—
137C2×C3×Dic3C22×C414yes—
138C2×C3×Dic3C22×C418yes—
139C2×C4×D4—32no—
140C2×C4×D4C3×D311yescharacteristic
141C4×C4⋊C4—32no—
142C4×C4⋊C4—31no—
143C4×C22⋊C4—32no—
144C4×C22⋊C4—34no—
145#376—31no—
146C2×#95—32no—
147C24×C3—22no—
148C24×C3C22×D312yes—
149C24×C3C23×C312yes—
150C2×C3×D4—28no—
151C2×C3×D4—24no—
152C2×C3×D4C4×D318yes—
153C2×C3×D4C2×Dic314yes—
154C2×C3×D4C2×C3×C418yes—
155C2×C3×D4C2×C3×C414yes—
156C22×C3×C4—28no—
157C22×C3×C4—21no—
158C22×C3×C4—21no—
159C22×C3×C4—21no—
160C22×C3×C4—22no—
161C22×C3×C4—32no—
162C22×C3×C4—32no—
163C22×C3×C4—34no—
164C22×C3×C4—316no—
165C22×C3×C4—44no—
166C22×C3×C4—44no—
167C22×C3×C4—48no—
168C22×C3×C4—124no—
169C22×C3×C4—124no—
170C22×C3×C4C2×Dic318yescomplemented
171C22×C3×C4C2×C3×C418yescomplemented
172C22×C3×C4C22×D311yescharacteristic
173C22×C3×C4C22×D311yescharacteristic
174C22×C3×C4C22×D312yes—
175C22×C3×C4C22×D311yescharacteristic
176C22×C3×C4C23×C311yescharacteristic
177C22×C3×C4C23×C311yescharacteristic
178C22×C3×C4C23×C312yes—
179C22×C3×C4C23×C311yescharacteristic
180C22×Dic3—44no—
181C22×Dic3—44no—
182C22×Dic3C2×C3×C4116yescomplemented
183C22×Dic3C23×C312yes—
184C22×Dic3C23×C312yes—
185C22×Dic3C23×C314yes—
186C3×C4⋊C4—24no—
187C3×C4⋊C4—38no—
188C3×C4⋊C4C2×Dic314yescomplemented
189C3×C4⋊C4C2×C3×C414yescomplemented
190C3×C22⋊C4—28no—
191C3×C22⋊C4—316no—
192C3×C22⋊C4C2×Dic318yescomplemented
193C3×C22⋊C4C2×C3×C418yescomplemented
194C3×C42—24no—
195C3×C42—38no—
196C3×C42—124no—
197C3×C42—122no—
198C3×C42C2×Dic314yescomplemented
199C3×C42C2×C3×C414yescomplemented
200#217C2×C3×C4116yescomplemented
201C4⋊Dic3C2×C3×C418yescomplemented
202C4×Dic3—44no—
203C4×Dic3—42no—
204C4×Dic3C2×C3×C418yescomplemented
205C22×C32—28no—
206C22×C32—24no—
207C22×C32—28no—
208C22×C32—24no—
209C22×C32C2×C4218yes—
210C22×C32C23×C412yes—
211C22×C32C23×C411yescharacteristic
212C22×C32H(5,𝔽2)12yes—
213C22×C32H(5,𝔽2)11yescharacteristic
214C22×C322−1+411yescharacteristic
215C32×C4—22no—
216C32×C4—24no—
217C32×C4—22no—
218C32×C4C2×C4214yes—
219C3×Dic3—24no—
220C3×Dic3—28no—
221C3×Dic3—24no—
222C22×D4C2×C3×D311yescharacteristic
223C23×C4—34no—
224C23×C4C2×C3×D312yes—
225C4×D4—316no—
226C4×D4C3×Dic318yescomplemented
227C2×C4⋊C4—32no—
228C2×C4⋊C4C2×C3×D311yescharacteristic
229C2×C22⋊C4—34no—
230C2×C22⋊C4C2×C3×D312yes—
231C2×C42—32no—
232C2×C42—34no—
233C2×C42—32no—
234C2×C42C2×C3×D311yescharacteristic
235#95—31no—
236#95—34no—
237#95—32no—
238#95—32no—
239#95—31no—
240C23×C3—24no—
241C23×C3—22no—
242C23×C3—22no—
243C23×C3—22no—
244C23×C3—24no—
245C23×C3—22no—
246C23×C3—21no—
247C23×C3—28no—
248C23×C3—24no—
249C23×C3—44no—
250C23×C3—42no—
251C23×C3—42no—
252C23×C3C2×C4×D318yes—
253C23×C3C22×Dic314yes—
254C23×C3C22×C3×C418yes—
255C23×C3C22×C3×C414yes—
256C23×C3C23×D311yescharacteristic
257C23×C3C24×C311yescharacteristic
258C3×D4—216no—
259C3×D4C4×Dic3116yescomplemented
260C3×D4C3×C42116yescomplemented
261C2×C3×C4—24no—
262C2×C3×C4—24no—
263C2×C3×C4—22no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—24no—
268C2×C3×C4—22no—
269C2×C3×C4—21no—
270C2×C3×C4—24no—
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4—24no—
275C2×C3×C4—34no—
276C2×C3×C4—34no—
277C2×C3×C4—38no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4—44no—
281C2×C3×C4—48no—
282C2×C3×C4—48no—
283C2×C3×C4—416no—
284C2×C3×C4—62no—
285C2×C3×C4—64no—
286C2×C3×C4—62no—
287C2×C3×C4—64no—
288C2×C3×C4—62no—
289C2×C3×C4—62no—
290C2×C3×C4—1216no—
291C2×C3×C4C2×C4×D314yes—
292C2×C3×C4C22×Dic312yes—
293C2×C3×C4C22×Dic312yes—
294C2×C3×C4C22×Dic314yes—
295C2×C3×C4C22×Dic312yes—
296C2×C3×C4C22×C3×C414yes—
297C2×C3×C4C22×C3×C412yes—
298C2×C3×C4C22×C3×C412yes—
299C2×C3×C4C22×C3×C414yes—
300C2×C3×C4C22×C3×C412yes—
301C2×Dic3—22no—
302C2×Dic3—24no—
303C2×Dic3—24no—
304C2×Dic3—22no—
305C2×Dic3—22no—
306C2×Dic3—22no—
307C2×Dic3—416no—
308C2×Dic3C22×C3×C414yes—
309C2×Dic3C22×C3×C414yes—
310C2×Dic3C22×C3×C418yes—
311C2×C32—28no—
312C2×C32C22×C4211yescharacteristic
313C2×C32#51811yescharacteristic
314C2×C32#51812yes—
315C2×C32#52012yes—
316C2×C32#52011yescharacteristic
317C24C22×C3×D312yes—
318C2×D4C3×C4×D318yes—
319C2×D4C2×C3×Dic314yes—
320C22×C4—32no—
321C22×C4—32no—
322C22×C4—34no—
323C22×C4—316no—
324C22×C4—44no—
325C22×C4—124no—
326C22×C4—124no—
327C22×C4C2×C3×Dic318yescomplemented
328C22×C4C22×C3×D311yescharacteristic
329C22×C4C22×C3×D311yescharacteristic
330C22×C4C22×C3×D312yes—
331C22×C4C22×C3×D311yescharacteristic
332C4⋊C4—38no—
333C4⋊C4C2×C3×Dic314yescomplemented
334C22⋊C4—316no—
335C22⋊C4C2×C3×Dic318yescomplemented
336C42—38no—
337C42—124no—
338C42—122no—
339C42C2×C3×Dic314yescomplemented
340C22×C3—24no—
341C22×C3—28no—
342C22×C3—28no—
343C22×C3—24no—
344C22×C3—28no—
345C22×C3—24no—
346C22×C3—28no—
347C22×C3—24no—
348C22×C3—28no—
349C22×C3—22no—
350C22×C3—22no—
351C22×C3—21no—
352C22×C3—21no—
353C22×C3—21no—
354C22×C3—44no—
355C22×C3—48no—
356C22×C3—48no—
357C22×C3—44no—
358C22×C3C2×C4×Dic318yes—
359C22×C3C2×C3×C4218yes—
360C22×C3C22×C4×D312yes—
361C22×C3#100412yes—
362C22×C3#100911yescharacteristic
363C22×C3#101011yescharacteristic
364C22×C3C23×Dic311yescharacteristic
365C22×C3C23×C3×C412yes—
366C22×C3C23×C3×C411yescharacteristic
367C22×C3C3×H(5,𝔽2)12yes—
368C22×C3C3×H(5,𝔽2)11yescharacteristic
369C22×C3C3×2−1+411yescharacteristic
370C3×C4—24no—
371C3×C4—22no—
372C3×C4—24no—
373C3×C4—22no—
374C3×C4—22no—
375C3×C4—24no—
376C3×C4—22no—
377C3×C4—24no—
378C3×C4—44no—
379C3×C4—42no—
380C3×C4—64no—
381C3×C4—68no—
382C3×C4—64no—
383C3×C4C2×C4×Dic314yes—
384C3×C4C2×C3×C4214yes—
385Dic3—24no—
386Dic3—28no—
387Dic3—24no—
388C32#227511yescharacteristic, complemented, Sylow
389C23—24no—
390C23—22no—
391C23—22no—
392C23C2×C3×C4×D318yes—
393C23C22×C3×Dic314yes—
394C23C23×C3×D311yescharacteristic
395D4C3×C4×Dic3116yescomplemented
396C2×C4—24no—
397C2×C4—22no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—34no—
401C2×C4—34no—
402C2×C4—38no—
403C2×C4—48no—
404C2×C4—62no—
405C2×C4—64no—
406C2×C4—62no—
407C2×C4—64no—
408C2×C4—62no—
409C2×C4—62no—
410C2×C4—1216no—
411C2×C4C2×C3×C4×D314yes—
412C2×C4C22×C3×Dic312yes—
413C2×C4C22×C3×Dic312yes—
414C2×C4C22×C3×Dic314yes—
415C2×C4C22×C3×Dic312yes—
416C2×C3—28no—
417C2×C3—28no—
418C2×C3—21no—
419C2×C3—21no—
420C2×C3—22no—
421C2×C3—22no—
422C2×C3—21no—
423C2×C3—48no—
424C2×C3#554512yes—
425C2×C3#554612yes—
426C2×C3C22×C4×Dic311yescharacteristic
427C2×C3#579811yescharacteristic
428C2×C3#582711yescharacteristic
429C2×C3C22×C3×C4211yescharacteristic
430C2×C3C3×#51811yescharacteristic
431C2×C3C3×#51812yes—
432C2×C3C3×#52012yes—
433C2×C3C3×#52011yescharacteristic
434C22—24no—
435C22—28no—
436C22—28no—
437C22—24no—
438C22C2×C3×C4×Dic318yes—
439C22C22×C3×C4×D312yes—
440C22C3×#100412yes—
441C22C3×#100911yescharacteristic
442C22C3×#101011yescharacteristic
443C22C23×C3×Dic311yescharacteristic
444C4—22no—
445C4—24no—
446C4—22no—
447C4—64no—
448C4—68no—
449C4—64no—
450C4C2×C3×C4×Dic314yes—
451C3—21no—
452C3#8231711yescharacteristic, direct product factor
453C3C3×#227511yescharacteristic, complemented
454C2—28no—
455C2C3×#554512yes—
456C2C3×#554612yes—
457C2C22×C3×C4×Dic311yescharacteristic
458C2C3×#579811yescharacteristic
459C2C3×#582711yescharacteristic
subgroupG/H
centerC23×C3C23×D3
commutatorC2×C3C22×C3×C42
FrattiniC23C23×C3×D3
FittingC2×C32×C4×D4C2
socleC23×C32C24
prime radicalC22×C32×D4C22
cosocleC23×C3C24×C3
hypercenterC2×C3×C4×D4D3
hypocenterC3C3×#2275
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×D4

Complements

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

NH ≅ G/Ntype
3. C3×C4×#217C2semidirect
4. C2×C3×C3⋊#95C2semidirect
5. C3×Dic3×C4⋊C4C2semidirect
6. C3×Dic3×C22⋊C4C2semidirect
7. C3×C4×C4⋊Dic3C2semidirect
8. C3×#4932C2semidirect
10. #82317C3direct (N × H)
14. C32×C4×D4C4semidirect
19. C3×D4×Dic3C4semidirect
40. C4×#217C2×C3semidirect
41. C2×C3⋊#95C2×C3semidirect
42. Dic3×C4⋊C4C2×C3semidirect
43. Dic3×C22⋊C4C2×C3semidirect
44. C4×C4⋊Dic3C2×C3semidirect
45. #4932C2×C3semidirect
50. C22×C32×C4C2×C4semidirect
57. C22×C3×Dic3C2×C4semidirect
61. C32×C4⋊C4C2×C4semidirect
62. C32×C22⋊C4C2×C4semidirect
63. C32×C42C2×C4semidirect
64. C3×#217C2×C4semidirect
65. C3×C4⋊Dic3C2×C4semidirect
68. C3×C4×Dic3C2×C4semidirect
80. C3×C4×D4Dic3semidirect
81. C3×C4×D4C3×C4semidirect
97. D4×Dic3C3×C4semidirect
118. C32×D4C42semidirect
170. C22×C3×C4C2×Dic3semidirect
171. C22×C3×C4C2×C3×C4semidirect
182. C22×Dic3C2×C3×C4semidirect
188. C3×C4⋊C4C2×Dic3semidirect
189. C3×C4⋊C4C2×C3×C4semidirect
192. C3×C22⋊C4C2×Dic3semidirect
193. C3×C22⋊C4C2×C3×C4semidirect
198. C3×C42C2×Dic3semidirect
199. C3×C42C2×C3×C4semidirect
200. #217C2×C3×C4semidirect
201. C4⋊Dic3C2×C3×C4semidirect
204. C4×Dic3C2×C3×C4semidirect
226. C4×D4C3×Dic3semidirect
259. C3×D4C4×Dic3semidirect
260. C3×D4C3×C42semidirect
327. C22×C4C2×C3×Dic3semidirect
333. C4⋊C4C2×C3×Dic3semidirect
335. C22⋊C4C2×C3×Dic3semidirect
339. C42C2×C3×Dic3semidirect
388. C32#2275semidirect
395. D4C3×C4×Dic3semidirect
452. C3#82317direct (N × H)
453. C3C3×#2275semidirect

Maximal subgroups

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

subgroupindex
C2×C32×C4×D42
C2×C3×D4×Dic32
C3×C4×#2172
C2×C3×C3⋊#952
C3×Dic3×C4⋊C42
C3×Dic3×C22⋊C42
C3×C4×C4⋊Dic32
C3×#49322
C3×#22753
#823173