Finite Group Databasebeta

← C3×#79832 #11860920

Subgroups — C3×#79832 #11860920

Order 1152.

1022 conjugacy classes of subgroups.

The subgroup lattice has 491 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#919C211yesmaximal, characteristic
2C3×#5574C212yesmaximal, complemented
3C32×#441C211yesmaximal, characteristic, complemented
4C3×#5184C211yesmaximal, characteristic, complemented
5C3×C4×#215C212yesmaximal, complemented
6C3×#4869C211yesmaximal, characteristic, complemented
7C3×D3×#106C211yesmaximal, characteristic, complemented
8C3×#4803C212yesmaximal, complemented
9C3×#4788C212yesmaximal, complemented
10C3×#4711C211yesmaximal, characteristic, complemented
11C3×#4708C211yesmaximal, characteristic, complemented
12C3×#2952—31nomaximal
13#79832C311yesmaximal, characteristic, direct product factor
14C2×C3×#239C2212yes—
15C2×C32×Q16C2211yescharacteristic
16C32×C4×Q8C2212yes—
17C3×Q8×Dic3C2212yes—
18C2×C3×#215C2212yes—
19C3×#919C418yescomplemented
20C2×C3×D24C2211yescharacteristic
21C2×C3×ΓL(1,𝔽25)C2211yescharacteristic
22C2×C3×D3×C8C2211yescharacteristic
23C3×#893—22no—
24C3×#893C2212yes—
25C3×D3×C4⋊C4—22no—
26C3×D3×C4⋊C4C2212yes—
27C3×C4×D12—22no—
28C3×C4×D12C2212yes—
29C3×#872—22no—
30C3×#872C2211yescharacteristic
31C32×#106C2211yescharacteristic
32C32×#103C2212yes—
33C32×#97C2211yescharacteristic
34C3×#835C2212yes—
35C3×#821C2212yes—
36C3×#817C2211yescharacteristic
37C3×#815C2211yescharacteristic
38C3×C8×Dic3C2211yescharacteristic
39C3×#809C2212yes—
40C3×#808C2212yes—
41C3×C4×C3⋊C8C2211yescharacteristic
42C2×C3×#135—31no—
43C3×#520—32no—
44C2×#919C2×C311yescharacteristic
45#5574C2×C312yescomplemented
46C3×#442—31no—
47C3×#442—31no—
48C3×#441—21no—
49C3×#441—31no—
50C3×#441D311yescharacteristic, complemented
51C3×#441C2×C311yescharacteristic, complemented
52C3×C4×SD16—32no—
53C3×C4×SD16—32no—
54C2×C3×#106—31no—
55C3×#417—32no—
56C3×#405—31no—
57#5184C2×C311yescharacteristic, complemented
58C4×#215C2×C312yescomplemented
59#4869C2×C311yescharacteristic, complemented
60D3×#106C2×C311yescharacteristic, complemented
61#4803C2×C312yescomplemented
62#4788C2×C312yescomplemented
63#4711C2×C311yescharacteristic, complemented
64#4708C2×C311yescharacteristic, complemented
65C2×C32×Q8C2312yes—
66C3×#239—24no—
67C3×#239C2×C418yescomplemented
68C2×C3×D12—22no—
69C2×C3×D12C2312yes—
70C2×C3×C4×D3—22no—
71C2×C3×C4×D3—22no—
72C2×C3×C4×D3—22no—
73C2×C3×C4×D3—22no—
74C2×C3×C4×D3C2311yescharacteristic
75C32×Q16C2×C414yescomplemented
76C2×C32×C8C2311yescharacteristic
77C2×C32×C8C2311yescharacteristic
78C32×C4⋊C4—22no—
79C32×C4⋊C4C2312yes—
80C32×C42—22no—
81C32×C42C2311yescharacteristic
82C3×#215C2×C418yescomplemented
83C3×(C2×D3)⋊C4—22no—
84C3×(C2×D3)⋊C4—22no—
85C3×(C2×D3)⋊C4D412yes—
86C3×C4⋊Dic3—22no—
87C3×C4⋊Dic3C2312yes—
88C3×Dic3⋊C4—22no—
89C3×Dic3⋊C4D412yes—
90C3×C4×Dic3—22no—
91C3×C4×Dic3C2311yescharacteristic
92C2×C3×C3⋊C8C2311yescharacteristic
93C2×C3×C3⋊C8C2311yescharacteristic
94C3×D24C2×C414yescomplemented
95C3×ΓL(1,𝔽25)—22no—
96C3×D3×C8C2×C414yescomplemented
97#2952—31noSylow
98C2×C3×#41—32no—
99C2×#239C22×C312yes—
100C3×#135—38no—
101C2×C3×Q16—21no—
102C2×C3×Q16C2×D311yescharacteristic
103C2×C3×Q16C22×C311yescharacteristic
104C2×C3×SD16—32no—
105C2×C3×D8—31no—
106C2×C3×M16—31no—
107C22×C3×C8—31no—
108C3×C4×Q8—22no—
109C3×C4×Q8—32no—
110C3×C4×Q8C2×D312yes—
111C3×C4×Q8C22×C312yes—
112C3×C4×D4—32no—
113C3×C4×D4—32no—
114C3×C4×D4—62no—
115C3×C4×D4—62no—
116C3×#117—31no—
117C3×#117—62no—
118C2×C3×C4⋊C4—32no—
119C2×C3×C4⋊C4—62no—
120Q8×Dic3C22×C312yes—
121C2×#215C22×C312yes—
122#919C3×C418yescomplemented
123C2×D24C22×C311yescharacteristic
124C2×ΓL(1,𝔽25)C22×C311yescharacteristic
125C2×D3×C8C22×C311yescharacteristic
126#893—22no—
127#893C22×C312yes—
128D3×C4⋊C4—22no—
129D3×C4⋊C4C22×C312yes—
130C4×D12—22no—
131C4×D12C22×C312yes—
132#872—22no—
133#872C22×C311yescharacteristic
134C3×#106—21no—
135C3×#106—31no—
136C3×#106—32no—
137C3×#106C2×D311yescharacteristic
138C3×#106C22×C311yescharacteristic
139C3×#103—22no—
140C3×#103—32no—
141C3×#103C2×D312yes—
142C3×#103C22×C312yes—
143C3×#102—32no—
144C3×#102—32no—
145C3×#97—21no—
146C3×#97—31no—
147C3×#97C2×D311yescharacteristic
148C3×#97C22×C311yescharacteristic
149C3×C4×C8—31no—
150C3×C4×C8—31no—
151#835C22×C312yes—
152#821C22×C312yes—
153#817C22×C311yescharacteristic
154#815C22×C311yescharacteristic
155C8×Dic3C22×C311yescharacteristic
156#809C22×C312yes—
157#808C22×C312yes—
158C4×C3⋊C8C22×C311yescharacteristic
159C22×C3×D3—22no—
160C22×C3×D3C2×D411yescharacteristic
161C32×Q8—22no—
162C32×Q8C22×C414yes—
163C2×C32×C4—22no—
164C2×C32×C4—22no—
165C2×C32×C4C2×D412yes—
166C2×C32×C4C2411yescharacteristic
167C2×C3×Dic3—22no—
168C2×C3×Dic3C2×D412yes—
169C2×C3×Dic3C2×D411yescharacteristic
170C3×D12—28no—
171C3×D12—22no—
172C3×D12C22×C414yes—
173C3×C4×D3—28no—
174C3×C4×D3—82no—
175C3×C4×D3—82no—
176C3×C4×D3—82no—
177C3×C4×D3C22×C412yes—
178C3×C4×D3#4112yes—
179C32×C8—21no—
180C32×C8C22×C412yes—
181C3×C3⋊C8—21no—
182C3×C3⋊C8C22×C412yes—
183C2×#135—31no—
184#520—32no—
185#442—31no—
186#442—31no—
187#441—31no—
188#441C3×D311yescharacteristic, complemented
189C4×SD16—32no—
190C4×SD16—32no—
191C2×#106—31no—
192#417—32no—
193#405—31no—
194C3×#41—38no—
195C3×#41—64no—
196C2×C3×Q8—22no—
197C2×C3×Q8C22×D312yes—
198C2×C3×Q8C23×C312yes—
199C2×C3×D4—32no—
200C2×C3×D4—62no—
201C22×C3×C4—31no—
202C22×C3×C4—62no—
203C22×C3×C4—62no—
204C22×C3×C4—62no—
205C22×C3×C4—62no—
206#239—24no—
207#239C2×C3×C418yescomplemented
208C2×D12—22no—
209C2×D12C23×C312yes—
210C2×C4×D3—22no—
211C2×C4×D3—22no—
212C2×C4×D3—22no—
213C2×C4×D3—22no—
214C2×C4×D3C23×C311yescharacteristic
215C3×Q16—24no—
216C3×Q16C4×D314yescomplemented
217C3×Q16C2×C3×C414yescomplemented
218C3×SD16—38no—
219C3×D8—34no—
220C3×M16—62no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—31no—
224C2×C3×C8—34no—
225C2×C3×C8—31no—
226C2×C3×C8C22×D311yescharacteristic
227C2×C3×C8C22×D311yescharacteristic
228C2×C3×C8C23×C311yescharacteristic
229C2×C3×C8C23×C311yescharacteristic
230C3×C4⋊C4—22no—
231C3×C4⋊C4—22no—
232C3×C4⋊C4—22no—
233C3×C4⋊C4—32no—
234C3×C4⋊C4—32no—
235C3×C4⋊C4—42no—
236C3×C4⋊C4—62no—
237C3×C4⋊C4—62no—
238C3×C4⋊C4C22×D312yes—
239C3×C4⋊C4C23×C312yes—
240C3×C22⋊C4—32no—
241C3×C22⋊C4—62no—
242C3×C22⋊C4—62no—
243C3×C42—22no—
244C3×C42—22no—
245C3×C42—21no—
246C3×C42—31no—
247C3×C42—42no—
248C3×C42—62no—
249C3×C42C22×D311yescharacteristic
250C3×C42C23×C311yescharacteristic
251#215C2×C3×C418yescomplemented
252(C2×D3)⋊C4—22no—
253(C2×D3)⋊C4—22no—
254(C2×D3)⋊C4C3×D412yes—
255C4⋊Dic3—22no—
256C4⋊Dic3C23×C312yes—
257Dic3⋊C4—22no—
258Dic3⋊C4C3×D412yes—
259C4×Dic3—22no—
260C4×Dic3C23×C311yescharacteristic
261C2×C3⋊C8C23×C311yescharacteristic
262C2×C3⋊C8C23×C311yescharacteristic
263D24C2×C3×C414yescomplemented
264ΓL(1,𝔽25)—22no—
265D3×C8C2×C3×C414yescomplemented
266C22×C32C22×D411yescharacteristic
267C2×C3×D3—22no—
268C2×C3×D3—24no—
269C2×C3×D3—42no—
270C2×C3×D3—42no—
271C2×C3×D3C4×D412yes—
272C32×C4—24no—
273C32×C4—22no—
274C32×C4—42no—
275C32×C4C23×C411yescharacteristic
276C32×C4C2×#4111yescharacteristic
277C3×Dic3—22no—
278C3×Dic3—42no—
279C3×Dic3C4×D412yes—
280C2×#41—32no—
281#135—38no—
282C2×Q16C2×C3×D311yescharacteristic
283C2×SD16—32no—
284C2×D8—31no—
285C2×M16—31no—
286C22×C8—31no—
287C4×Q8—32no—
288C4×Q8C2×C3×D312yes—
289C4×D4—32no—
290C4×D4—32no—
291C4×D4—62no—
292C4×D4—62no—
293#117—31no—
294#117—62no—
295C2×C4⋊C4—32no—
296C2×C4⋊C4—62no—
297#106—31no—
298#106—32no—
299#106C2×C3×D311yescharacteristic
300#103—32no—
301#103C2×C3×D312yes—
302#102—32no—
303#102—32no—
304#97—31no—
305#97C2×C3×D311yescharacteristic
306C4×C8—31no—
307C4×C8—31no—
308C23×C3—31no—
309C23×C3—62no—
310C22×D3—22no—
311C22×D3C2×C3×D411yescharacteristic
312C3×Q8—22no—
313C3×Q8—22no—
314C3×Q8—24no—
315C3×Q8—42no—
316C3×Q8C2×C4×D314yes—
317C3×Q8C22×C3×C414yes—
318C3×D4—34no—
319C3×D4—68no—
320C3×D4—62no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—21no—
326C2×C3×C4—22no—
327C2×C3×C4—31no—
328C2×C3×C4—32no—
329C2×C3×C4—32no—
330C2×C3×C4—32no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—68no—
334C2×C3×C4—62no—
335C2×C3×C4—242no—
336C2×C3×C4—242no—
337C2×C3×C4—242no—
338C2×C3×C4D3×D412yes—
339C2×C3×C4C2×C3×D412yes—
340C2×C3×C4C23×D311yescharacteristic
341C2×C3×C4C24×C311yescharacteristic
342C2×Dic3—22no—
343C2×Dic3C2×C3×D412yes—
344C2×Dic3C2×C3×D411yescharacteristic
345D12—28no—
346D12—22no—
347D12C22×C3×C414yes—
348C4×D3—28no—
349C4×D3—82no—
350C4×D3—82no—
351C4×D3—82no—
352C4×D3C22×C3×C412yes—
353C4×D3C3×#4112yes—
354C3×C8—21no—
355C3×C8—21no—
356C3×C8—22no—
357C3×C8—32no—
358C3×C8—41no—
359C3×C8—61no—
360C3×C8C2×C4×D312yes—
361C3×C8C22×C3×C412yes—
362C3⋊C8—21no—
363C3⋊C8C22×C3×C412yes—
364C2×C32C2×C4×D411yescharacteristic
365C2×C32#57712yes—
366C3×D3—22no—
367C3×D3—44no—
368#41—38no—
369#41—64no—
370C2×Q8C22×C3×D312yes—
371C2×D4—32no—
372C2×D4—62no—
373C22×C4—31no—
374C22×C4—62no—
375C22×C4—62no—
376C22×C4—62no—
377C22×C4—62no—
378Q16C3×C4×D314yescomplemented
379SD16—38no—
380D8—34no—
381M16—62no—
382C2×C8—31no—
383C2×C8—34no—
384C2×C8—31no—
385C2×C8C22×C3×D311yescharacteristic
386C2×C8C22×C3×D311yescharacteristic
387C4⋊C4—22no—
388C4⋊C4—32no—
389C4⋊C4—32no—
390C4⋊C4—62no—
391C4⋊C4—62no—
392C4⋊C4C22×C3×D312yes—
393C22⋊C4—32no—
394C22⋊C4—62no—
395C22⋊C4—62no—
396C42—22no—
397C42—31no—
398C42—62no—
399C42C22×C3×D311yescharacteristic
400C22×C3—21no—
401C22×C3—32no—
402C22×C3—62no—
403C22×C3—64no—
404C22×C3—122no—
405C22×C3—122no—
406C22×C3C2×D3×D411yescharacteristic
407C22×C3C22×C3×D411yescharacteristic
408C2×D3—22no—
409C2×D3—24no—
410C2×D3—42no—
411C2×D3—42no—
412C2×D3C3×C4×D412yes—
413C3×C4—22no—
414C3×C4—22no—
415C3×C4—24no—
416C3×C4—24no—
417C3×C4—21no—
418C3×C4—21no—
419C3×C4—32no—
420C3×C4—42no—
421C3×C4—44no—
422C3×C4—42no—
423C3×C4—42no—
424C3×C4—62no—
425C3×C4—82no—
426C3×C4—122no—
427C3×C4C22×C4×D311yescharacteristic
428C3×C4D3×#4111yescharacteristic
429C3×C4C23×C3×C411yescharacteristic
430C3×C4C2×C3×#4111yescharacteristic
431Dic3—22no—
432Dic3—42no—
433Dic3C3×C4×D412yes—
434C32#295211yescharacteristic, complemented, Sylow
435C23—31no—
436C23—62no—
437Q8—22no—
438Q8C2×C3×C4×D314yes—
439D4—34no—
440D4—68no—
441D4—62no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—31no—
445C2×C4—32no—
446C2×C4—32no—
447C2×C4—32no—
448C2×C4—68no—
449C2×C4—62no—
450C2×C4—242no—
451C2×C4—242no—
452C2×C4—242no—
453C2×C4C3×D3×D412yes—
454C2×C4C23×C3×D311yescharacteristic
455C8—21no—
456C8—32no—
457C8—61no—
458C8C2×C3×C4×D312yes—
459C2×C3—22no—
460C2×C3—21no—
461C2×C3—62no—
462C2×C3—124no—
463C2×C3C4×D3×D411yescharacteristic
464C2×C3#577812yes—
465C2×C3C2×C3×C4×D411yescharacteristic
466C2×C3C3×#57712yes—
467D3—22no—
468D3—44no—
469C22—32no—
470C22—62no—
471C22—64no—
472C22—122no—
473C22—122no—
474C22C2×C3×D3×D411yescharacteristic
475C4—22no—
476C4—24no—
477C4—32no—
478C4—42no—
479C4—62no—
480C4—122no—
481C4C22×C3×C4×D311yescharacteristic
482C4C3×D3×#4111yescharacteristic
483C3—21no—
484C3#7983211yescharacteristic, direct product factor
485C3C3×#295211yescharacteristic, complemented
486C2—62no—
487C2—124no—
488C2C3×C4×D3×D411yescharacteristic
489C2C3×#577812yes—
subgroupG/H
centerC22×C3C2×D3×D4
commutatorC3×C4C23×C3×C4
FrattiniC2×C4C23×C3×D3
FittingC32×#441C2
socleC22×C32C22×D4
prime radicalC2×C3×#919C2
cosocleC2×C3×C4C24×C3
hypercenterC3×#441D3
hypocenterC3C3×#2952
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#441

Complements

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

NH ≅ G/Ntype
2. C3×#5574C2semidirect
3. C32×#441C2semidirect
4. C3×#5184C2semidirect
5. C3×C4×#215C2semidirect
6. C3×#4869C2semidirect
7. C3×D3×#106C2semidirect
8. C3×#4803C2semidirect
9. C3×#4788C2semidirect
10. C3×#4711C2semidirect
11. C3×#4708C2semidirect
13. #79832C3direct (N × H)
19. C3×#919C4semidirect
45. #5574C2×C3semidirect
50. C3×#441D3semidirect
51. C3×#441C2×C3semidirect
57. #5184C2×C3semidirect
58. C4×#215C2×C3semidirect
59. #4869C2×C3semidirect
60. D3×#106C2×C3semidirect
61. #4803C2×C3semidirect
62. #4788C2×C3semidirect
63. #4711C2×C3semidirect
64. #4708C2×C3semidirect
67. C3×#239C2×C4semidirect
75. C32×Q16C2×C4semidirect
82. C3×#215C2×C4semidirect
94. C3×D24C2×C4semidirect
96. C3×D3×C8C2×C4semidirect
122. #919C3×C4semidirect
188. #441C3×D3semidirect
207. #239C2×C3×C4semidirect
216. C3×Q16C4×D3semidirect
217. C3×Q16C2×C3×C4semidirect
251. #215C2×C3×C4semidirect
263. D24C2×C3×C4semidirect
265. D3×C8C2×C3×C4semidirect
378. Q16C3×C4×D3semidirect
434. C32#2952semidirect
484. C3#79832direct (N × H)
485. C3C3×#2952semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9192
C3×#55742
C32×#4412
C3×#51842
C3×C4×#2152
C3×#48692
C3×D3×#1062
C3×#48032
C3×#47882
C3×#47112
C3×#47082
C3×#29523
#798323