Finite Group Databasebeta

← C3×#68949 #11788659

Subgroups — C3×#68949 #11788659

Order 1152.

669 conjugacy classes of subgroups.

The subgroup lattice has 509 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#909C211yesmaximal, characteristic
2C2×C3×#901C211yesmaximal, characteristic
3C2×C3×#824C211yesmaximal, characteristic, complemented
4C3×#5133C211yesmaximal, characteristic
5C2×C3×#816C211yesmaximal, characteristic, complemented
6C32×#341C211yesmaximal, characteristic, complemented
7C3×#4554C211yesmaximal, characteristic, complemented
8C3×#1908—31nomaximal
9#68949C311yesmaximal, characteristic, direct product factor
10C2×C3×#235C2211yescharacteristic
11C22×C3×Dic6C2211yescharacteristic
12C2×C32×M16C2211yescharacteristic
13C2×C32×M16C2211yescharacteristic
14C22×C32×C8C2211yescharacteristic
15C2×C3×C4⋊Dic3C2211yescharacteristic
16C2×C3×#208C2211yescharacteristic
17C3×#909—22no—
18C3×#909C414yescomplemented
19C2×C3×Dic12—21no—
20C2×C3×#204—21no—
21C3×#901—22no—
22C3×#901—22no—
23C2×C3×ΓL(1,𝔽25)—21no—
24C2×C3×D3×C8—21no—
25C3×#824—22no—
26C3×#821—21no—
27C3×#816—21no—
28C3×#816—21no—
29C3×#816—21no—
30C2×C3×#137—31no—
31C2×C3×#131—31no—
32C2×#909C2×C311yescharacteristic
33C2×#901C2×C311yescharacteristic
34C3×#417—31no—
35C2×C3×#103—31no—
36C2×C3×#101—31no—
37C2×#824C2×C311yescharacteristic, complemented
38#5133C2×C311yescharacteristic
39C2×#816C2×C311yescharacteristic, complemented
40C3×#343—31no—
41C3×#341—21no—
42C3×#341D311yescharacteristic, complemented
43C3×#341C2×C311yescharacteristic, complemented
44#4554C2×C311yescharacteristic, complemented
45C22×C32×C4C2311yescharacteristic
46C2×C3×#67—21no—
47C22×C3×Dic3—21no—
48C22×C3×Dic3—21no—
49C3×#235—21no—
50C3×#235—21no—
51C3×#235C2×C412yes—
52C3×#235D412yes—
53C2×C3×D12D411yescharacteristic
54C2×C3×C4×D3—21no—
55C2×C3×Dic6—21no—
56C2×C3×Dic6—21no—
57C2×C3×Dic6—21no—
58C2×C3×Dic6—21no—
59C2×C3×Dic6—21no—
60C2×C3×Dic6C2×C412yes—
61C2×C3×Dic6D411yescharacteristic
62C32×M16—21no—
63C32×M16—21no—
64C32×M16—21no—
65C32×M16C2×C412yes—
66C2×C32×C8—21no—
67C2×C32×C8—21no—
68C2×C32×C8—21no—
69C2×C32×C8—21no—
70C2×C32×C8D411yescharacteristic
71C2×C32×C8D411yescharacteristic
72C3×C4⋊Dic3—21no—
73C3×C4⋊Dic3—21no—
74C3×#208—21no—
75C3×#208D412yes—
76C2×C3×C3⋊C8—21no—
77C3×Dic12—24no—
78C3×#204—24no—
79C3×ΓL(1,𝔽25)—24no—
80C3×D3×C8—24no—
81#1908—31noSylow
82C2×C3×#41—31no—
83C22×C3×Q8—31no—
84C2×#235C22×C311yescharacteristic
85C22×Dic6C22×C311yescharacteristic
86C3×#137—34no—
87C3×#137—62no—
88C2×C3×Q16—61no—
89C2×C3×SD16—61no—
90C3×#131—62no—
91C3×#131—62no—
92C2×C3×M16—21no—
93C2×C3×M16—21no—
94C2×C3×M16—31no—
95C2×C3×M16—61no—
96C2×C3×M16C2×D311yescharacteristic
97C2×C3×M16C2×D311yescharacteristic
98C2×C3×M16C22×C311yescharacteristic
99C2×C3×M16C22×C311yescharacteristic
100C22×C3×C8—21no—
101C22×C3×C8—61no—
102C22×C3×C8C2×D311yescharacteristic
103C22×C3×C8C22×C311yescharacteristic
104C2×C3×C4⋊C4—31no—
105C2×C4⋊Dic3C22×C311yescharacteristic
106C2×#208C22×C311yescharacteristic
107#909—22no—
108#909C3×C414yescomplemented
109C2×Dic12—21no—
110C2×#204—21no—
111#901—22no—
112#901—22no—
113C2×ΓL(1,𝔽25)—21no—
114C2×D3×C8—21no—
115C3×#103—61no—
116C3×#103—61no—
117C3×#103—61no—
118C3×#102—61no—
119C3×#101—62no—
120#824—22no—
121#821—21no—
122#816—21no—
123#816—21no—
124#816—21no—
125C23×C32#4111yescharacteristic
126C22×C3×D3—21no—
127C2×C32×C4C22×C411yescharacteristic
128C2×C32×C4C2×D411yescharacteristic
129C2×C32×C4C2×D411yescharacteristic
130C2×C32×C4C2×D411yescharacteristic
131C2×C32×C4C2×D411yescharacteristic
132C2×C32×C4#4111yescharacteristic
133C3×#67—22no—
134C3×#67—22no—
135C2×C3×Dic3—21no—
136C2×C3×Dic3—22no—
137C2×C3×Dic3—41no—
138C2×C3×Dic3—41no—
139C2×C3×Dic3—41no—
140C2×C3×Dic3—81no—
141C3×D12—21no—
142C3×D12C22⋊C412yes—
143C3×C4×D3—22no—
144C3×C4×D3—22no—
145C3×Dic6—21no—
146C3×Dic6—22no—
147C3×Dic6—22no—
148C3×Dic6—41no—
149C3×Dic6—41no—
150C3×Dic6C22⋊C412yes—
151C32×C8—22no—
152C32×C8—21no—
153C32×C8—21no—
154C32×C8—41no—
155C3×C3⋊C8—22no—
156C2×#137—31no—
157C2×#131—31no—
158#417—31no—
159C2×#103—31no—
160C2×#101—31no—
161#343—31no—
162#341C3×D311yescharacteristic, complemented
163C3×#41—32no—
164C3×#41—32no—
165C3×#41—61no—
166C3×#41—61no—
167C2×C3×Q8—32no—
168C2×C3×Q8—31no—
169C2×C3×Q8—61no—
170C2×C3×Q8—61no—
171C2×C3×Q8—61no—
172C2×C3×Q8—61no—
173C2×C3×Q8—61no—
174C2×C3×D4—31no—
175C2×C3×D4—61no—
176C22×C3×C4—21no—
177C22×C3×C4—61no—
178C22×C3×C4—61no—
179C22×C3×C4—61no—
180C22×C3×C4C22×D311yescharacteristic
181C22×C3×C4C23×C311yescharacteristic
182C2×#67—21no—
183C22×Dic3—21no—
184C22×Dic3—21no—
185#235—21no—
186#235—21no—
187#235C2×C3×C412yes—
188#235C3×D412yes—
189C2×D12C3×D411yescharacteristic
190C2×C4×D3—21no—
191C2×Dic6—21no—
192C2×Dic6—21no—
193C2×Dic6—21no—
194C2×Dic6—21no—
195C2×Dic6—21no—
196C2×Dic6C2×C3×C412yes—
197C2×Dic6C3×D411yescharacteristic
198C3×Q16—64no—
199C3×SD16—64no—
200C3×M16—21no—
201C3×M16—21no—
202C3×M16—22no—
203C3×M16—21no—
204C3×M16—21no—
205C3×M16—21no—
206C3×M16—21no—
207C3×M16—22no—
208C3×M16—32no—
209C3×M16—41no—
210C3×M16—41no—
211C3×M16—64no—
212C3×M16—61no—
213C3×M16C4×D312yes—
214C3×M16C2×C3×C412yes—
215C2×C3×C8—21no—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—41no—
226C2×C3×C8—41no—
227C2×C3×C8—41no—
228C2×C3×C8—41no—
229C2×C3×C8—64no—
230C2×C3×C8—61no—
231C2×C3×C8D1211yescharacteristic
232C2×C3×C8D1211yescharacteristic
233C2×C3×C8C3×D411yescharacteristic
234C2×C3×C8C3×D411yescharacteristic
235C3×C4⋊C4—61no—
236C3×C4⋊C4—61no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239#208—21no—
240#208C3×D412yes—
241C2×C3⋊C8—21no—
242Dic12—24no—
243#204—24no—
244ΓL(1,𝔽25)—24no—
245D3×C8—24no—
246C22×C32—21no—
247C22×C32—21no—
248C22×C32C4×D411yescharacteristic
249C22×C32#12111yescharacteristic
250C22×C32#12311yescharacteristic
251C2×C3×D3—22no—
252C2×C3×D3—41no—
253C2×C3×D3—41no—
254C32×C4C2×C22⋊C411yescharacteristic
255C32×C4C4×D411yescharacteristic
256C32×C4#12011yescharacteristic
257C32×C4#12111yescharacteristic
258C3×Dic3—22no—
259C3×Dic3—42no—
260C3×Dic3—81no—
261C2×#41—31no—
262C22×Q8—31no—
263#137—34no—
264#137—62no—
265C2×Q16—61no—
266C2×SD16—61no—
267#131—62no—
268#131—62no—
269C2×M16—31no—
270C2×M16—61no—
271C2×M16C2×C3×D311yescharacteristic
272C2×M16C2×C3×D311yescharacteristic
273C22×C8—61no—
274C22×C8C2×C3×D311yescharacteristic
275C2×C4⋊C4—31no—
276#103—61no—
277#103—61no—
278#103—61no—
279#102—61no—
280#101—62no—
281C23×C3—21no—
282C23×C3—61no—
283C23×C3#23911yescharacteristic
284C23×C3C3×#4111yescharacteristic
285C22×D3—21no—
286C3×Q8—32no—
287C3×Q8—61no—
288C3×Q8—62no—
289C3×Q8—62no—
290C3×Q8—121no—
291C3×Q8—121no—
292C3×D4—32no—
293C3×D4—62no—
294C3×D4—62no—
295C3×D4—61no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—61no—
303C2×C3×C4—62no—
304C2×C3×C4—62no—
305C2×C3×C4—62no—
306C2×C3×C4—121no—
307C2×C3×C4—121no—
308C2×C3×C4—121no—
309C2×C3×C4—241no—
310C2×C3×C4C2×C4×D311yescharacteristic
311C2×C3×C4C2×D1211yescharacteristic
312C2×C3×C4#23511yescharacteristic
313C2×C3×C4D3×D411yescharacteristic
314C2×C3×C4D3×D411yescharacteristic
315C2×C3×C4D3×D411yescharacteristic
316C2×C3×C4C22×C3×C411yescharacteristic
317C2×C3×C4C2×C3×D411yescharacteristic
318C2×C3×C4C2×C3×D411yescharacteristic
319C2×C3×C4C2×C3×D411yescharacteristic
320C2×C3×C4C2×C3×D411yescharacteristic
321C2×C3×C4C3×#4111yescharacteristic
322#67—22no—
323#67—22no—
324C2×Dic3—21no—
325C2×Dic3—22no—
326C2×Dic3—41no—
327C2×Dic3—41no—
328C2×Dic3—41no—
329C2×Dic3—81no—
330D12—21no—
331D12C3×C22⋊C412yes—
332C4×D3—22no—
333C4×D3—22no—
334Dic6—21no—
335Dic6—22no—
336Dic6—22no—
337Dic6—41no—
338Dic6—41no—
339Dic6C3×C22⋊C412yes—
340C3×C8—22no—
341C3×C8—21no—
342C3×C8—21no—
343C3×C8—22no—
344C3×C8—21no—
345C3×C8—21no—
346C3×C8—41no—
347C3×C8—41no—
348C3×C8—42no—
349C3×C8—42no—
350C3×C8—41no—
351C3×C8—41no—
352C3×C8—62no—
353C3⋊C8—22no—
354C2×C32—21no—
355C2×C32—21no—
356C2×C32#38611yescharacteristic
357C2×C32#47111yescharacteristic
358C2×C32#47311yescharacteristic
359C3×D3—42no—
360#41—32no—
361#41—32no—
362#41—61no—
363#41—61no—
364C2×Q8—32no—
365C2×Q8—31no—
366C2×Q8—61no—
367C2×Q8—61no—
368C2×Q8—61no—
369C2×Q8—61no—
370C2×Q8—61no—
371C2×D4—31no—
372C2×D4—61no—
373C22×C4—61no—
374C22×C4—61no—
375C22×C4—61no—
376C22×C4C22×C3×D311yescharacteristic
377Q16—64no—
378SD16—64no—
379M16—21no—
380M16—21no—
381M16—21no—
382M16—32no—
383M16—64no—
384M16—61no—
385M16C3×C4×D312yes—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—64no—
391C2×C8—61no—
392C2×C8C3×D1211yescharacteristic
393C2×C8C3×D1211yescharacteristic
394C4⋊C4—61no—
395C4⋊C4—61no—
396C22×C3—21no—
397C22×C3—21no—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—21no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—41no—
404C22×C3—41no—
405C22×C3—62no—
406C22×C3—121no—
407C22×C3—121no—
408C22×C3#89311yescharacteristic
409C22×C3#89411yescharacteristic
410C22×C3#89511yescharacteristic
411C22×C3C3×C4×D411yescharacteristic
412C22×C3C3×#12111yescharacteristic
413C22×C3C3×#12311yescharacteristic
414C2×D3—22no—
415C2×D3—41no—
416C2×D3—41no—
417C3×C4—21no—
418C3×C4—21no—
419C3×C4—21no—
420C3×C4—21no—
421C3×C4—62no—
422C3×C4—122no—
423C3×C4—241no—
424C3×C4C4×D1211yescharacteristic
425C3×C4D3×C22⋊C411yescharacteristic
426C3×C4#88211yescharacteristic
427C3×C4#88411yescharacteristic
428C3×C4C2×C3×C22⋊C411yescharacteristic
429C3×C4C3×C4×D411yescharacteristic
430C3×C4C3×#12011yescharacteristic
431C3×C4C3×#12111yescharacteristic
432Dic3—22no—
433Dic3—42no—
434Dic3—81no—
435C32#190811yescharacteristic, complemented, Sylow
436C23—61no—
437C23C3×#23911yescharacteristic
438Q8—32no—
439Q8—61no—
440Q8—62no—
441Q8—62no—
442Q8—121no—
443Q8—121no—
444D4—32no—
445D4—62no—
446D4—62no—
447D4—61no—
448C2×C4—61no—
449C2×C4—62no—
450C2×C4—62no—
451C2×C4—62no—
452C2×C4—121no—
453C2×C4—121no—
454C2×C4—121no—
455C2×C4—241no—
456C2×C4C2×C3×C4×D311yescharacteristic
457C2×C4C2×C3×D1211yescharacteristic
458C2×C4C3×#23511yescharacteristic
459C2×C4C3×D3×D411yescharacteristic
460C2×C4C3×D3×D411yescharacteristic
461C2×C4C3×D3×D411yescharacteristic
462C8—22no—
463C8—21no—
464C8—21no—
465C8—41no—
466C8—62no—
467C2×C3—21no—
468C2×C3—21no—
469C2×C3—21no—
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—21no—
474C2×C3—41no—
475C2×C3—41no—
476C2×C3—122no—
477C2×C3#466511yescharacteristic
478C2×C3#489611yescharacteristic
479C2×C3#489811yescharacteristic
480C2×C3C3×#38611yescharacteristic
481C2×C3C3×#47111yescharacteristic
482C2×C3C3×#47311yescharacteristic
483D3—42no—
484C22—21no—
485C22—21no—
486C22—62no—
487C22—121no—
488C22—121no—
489C22C3×#89311yescharacteristic
490C22C3×#89411yescharacteristic
491C22C3×#89511yescharacteristic
492C4—62no—
493C4—122no—
494C4—241no—
495C4C3×C4×D1211yescharacteristic
496C4C3×D3×C22⋊C411yescharacteristic
497C4C3×#88211yescharacteristic
498C4C3×#88411yescharacteristic
499C3—21no—
500C3#6894911yescharacteristic, direct product factor
501C3C3×#190811yescharacteristic, complemented
502C2—21no—
503C2—21no—
504C2—122no—
505C2C3×#466511yescharacteristic
506C2C3×#489611yescharacteristic
507C2C3×#489811yescharacteristic
subgroupG/H
centerC22×C3#895
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC32×#341C2
socleC22×C32#121
prime radicalC2×C3×#235C22
cosocleC22×C3×C4C23×C3
hypercenterC3×#341D3
hypocenterC3C3×#1908
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#341

Complements

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

NH ≅ G/Ntype
3. C2×C3×#824C2semidirect
5. C2×C3×#816C2semidirect
6. C32×#341C2semidirect
7. C3×#4554C2semidirect
9. #68949C3direct (N × H)
18. C3×#909C4semidirect
37. C2×#824C2×C3semidirect
39. C2×#816C2×C3semidirect
42. C3×#341D3semidirect
43. C3×#341C2×C3semidirect
44. #4554C2×C3semidirect
108. #909C3×C4semidirect
162. #341C3×D3semidirect
435. C32#1908semidirect
500. C3#68949direct (N × H)
501. C3C3×#1908semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9092
C2×C3×#9012
C2×C3×#8242
C3×#51332
C2×C3×#8162
C32×#3412
C3×#45542
C3×#19083
#689493