Finite Group Databasebeta

← C3×#70123 #11788653

Subgroups — C3×#70123 #11788653

Order 1152.

697 conjugacy classes of subgroups.

The subgroup lattice has 519 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#942C211yesmaximal, characteristic
2C2×C3×#907C211yesmaximal, characteristic
3C2×C32×#101C211yesmaximal, characteristic, complemented
4C2×C3×#835C211yesmaximal, characteristic, complemented
5C3×#5133C211yesmaximal, characteristic
6C3×#4558C211yesmaximal, characteristic, complemented
7C3×#4554C211yesmaximal, characteristic, complemented
8C3×#1908—31nomaximal
9#70123C311yesmaximal, characteristic, direct product factor
10C22×C32×Q8C2211yescharacteristic
11C2×C3×#235C2211yescharacteristic
12C2×C32×M16C2211yescharacteristic
13C2×C32×M16C2211yescharacteristic
14C2×C3×#216—21no—
15C3×#942—22no—
16C3×#942C414yescomplemented
17C2×C3×#215—21no—
18C2×C3×C4⋊Dic3C2211yescharacteristic
19C2×C3×#208C2211yescharacteristic
20C22×C3×C3⋊C8C2211yescharacteristic
21C3×#907—22no—
22C3×#907—22no—
23C2×C3×ΓL(1,𝔽25)—21no—
24C2×C3×D3×C8—21no—
25C32×#101—22no—
26C3×#835—21no—
27C3×#835—21no—
28C3×#821—21no—
29C3×#816—21no—
30C2×C3×#137—31no—
31C2×C3×#131—31no—
32C2×#942C2×C311yescharacteristic
33C2×#907C2×C311yescharacteristic
34C3×#417—31no—
35C2×C3×#103—31no—
36C2×C3×#101—21no—
37C2×C3×#101D311yescharacteristic, complemented
38C2×C3×#101C2×C311yescharacteristic, complemented
39C2×#835C2×C311yescharacteristic, complemented
40#5133C2×C311yescharacteristic
41C3×#343—31no—
42C3×#341—31no—
43#4558C2×C311yescharacteristic, complemented
44#4554C2×C311yescharacteristic, complemented
45C2×C32×Q8—21no—
46C2×C32×Q8—21no—
47C2×C32×Q8—21no—
48C2×C32×Q8—21no—
49C2×C32×Q8—21no—
50C2×C32×Q8C2×C412yes—
51C22×C32×C4—21no—
52C22×C32×C4C2311yescharacteristic
53C2×C3×#67—21no—
54C22×C3×Dic3—21no—
55C3×#235—21no—
56C3×#235—21no—
57C3×#235C2×C412yes—
58C3×#235D412yes—
59C2×C3×D12D411yescharacteristic
60C2×C3×C4×D3—21no—
61C2×C3×Dic6D411yescharacteristic
62C32×M16—21no—
63C32×M16—21no—
64C32×M16—21no—
65C32×M16D412yes—
66C2×C32×C8—21no—
67C2×C32×C8—21no—
68C3×#216—24no—
69C3×#215—24no—
70C3×C4⋊Dic3—21no—
71C3×C4⋊Dic3—21no—
72C3×#208—21no—
73C3×#208C2×C412yes—
74C2×C3×C3⋊C8—21no—
75C2×C3×C3⋊C8—21no—
76C2×C3×C3⋊C8—21no—
77C2×C3×C3⋊C8D411yescharacteristic
78C2×C3×C3⋊C8D411yescharacteristic
79C3×ΓL(1,𝔽25)—24no—
80C3×D3×C8—24no—
81#1908—31noSylow
82C2×C3×#41—31no—
83C22×C3×Q8—21no—
84C22×C3×Q8C2×D311yescharacteristic
85C22×C3×Q8C22×C311yescharacteristic
86C2×#235C22×C311yescharacteristic
87C3×#137—34no—
88C3×#137—62no—
89C2×C3×Q16—61no—
90C2×C3×SD16—61no—
91C3×#131—62no—
92C3×#131—62no—
93C2×C3×M16—21no—
94C2×C3×M16—21no—
95C2×C3×M16—31no—
96C2×C3×M16—61no—
97C2×C3×M16C2×D311yescharacteristic
98C2×C3×M16C2×D311yescharacteristic
99C2×C3×M16C22×C311yescharacteristic
100C2×C3×M16C22×C311yescharacteristic
101C22×C3×C8—31no—
102C22×C3×C8—61no—
103C2×C3×C4⋊C4—31no—
104C2×#216—21no—
105#942—22no—
106#942C3×C414yescomplemented
107C2×#215—21no—
108C2×C4⋊Dic3C22×C311yescharacteristic
109C2×#208C22×C311yescharacteristic
110C22×C3⋊C8C22×C311yescharacteristic
111#907—22no—
112#907—22no—
113C2×ΓL(1,𝔽25)—21no—
114C2×D3×C8—21no—
115C3×#103—61no—
116C3×#103—61no—
117C3×#103—61no—
118C3×#102—61no—
119C3×#101—24no—
120C3×#101—22no—
121C3×#101—22no—
122#835—21no—
123#835—21no—
124#821—21no—
125#816—21no—
126C23×C32#4111yescharacteristic
127C22×C3×D3—21no—
128C32×Q8—22no—
129C32×Q8—22no—
130C32×Q8—41no—
131C32×Q8—41no—
132C2×C32×C4—22no—
133C2×C32×C4—41no—
134C2×C32×C4—41no—
135C2×C32×C4C22×C411yescharacteristic
136C2×C32×C4C2×D411yescharacteristic
137C2×C32×C4C2×D411yescharacteristic
138C2×C32×C4C2×D411yescharacteristic
139C2×C32×C4C2×D411yescharacteristic
140C2×C32×C4#4111yescharacteristic
141C3×#67—22no—
142C3×#67—22no—
143C2×C3×Dic3—21no—
144C2×C3×Dic3—41no—
145C2×C3×Dic3—81no—
146C3×D12—21no—
147C3×D12C22⋊C412yes—
148C3×C4×D3—22no—
149C3×C4×D3—22no—
150C3×Dic6—21no—
151C3×Dic6C22⋊C412yes—
152C32×C8—22no—
153C32×C8—41no—
154C3×C3⋊C8—22no—
155C3×C3⋊C8—21no—
156C3×C3⋊C8—21no—
157C2×#137—31no—
158C2×#131—31no—
159#417—31no—
160C2×#103—31no—
161C2×#101C3×D311yescharacteristic, complemented
162#343—31no—
163#341—31no—
164C3×#41—32no—
165C3×#41—32no—
166C3×#41—61no—
167C3×#41—61no—
168C2×C3×Q8—22no—
169C2×C3×Q8—21no—
170C2×C3×Q8—21no—
171C2×C3×Q8—21no—
172C2×C3×Q8—21no—
173C2×C3×Q8—21no—
174C2×C3×Q8—21no—
175C2×C3×Q8—21no—
176C2×C3×Q8—21no—
177C2×C3×Q8—21no—
178C2×C3×Q8—21no—
179C2×C3×Q8—22no—
180C2×C3×Q8—31no—
181C2×C3×Q8—41no—
182C2×C3×Q8—41no—
183C2×C3×Q8—41no—
184C2×C3×Q8—41no—
185C2×C3×Q8C4×D312yes—
186C2×C3×Q8C2×C3×C412yes—
187C2×C3×D4—31no—
188C2×C3×D4—61no—
189C22×C3×C4—22no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—61no—
194C22×C3×C4—61no—
195C22×C3×C4C22×D311yescharacteristic
196C22×C3×C4C23×C311yescharacteristic
197C2×#67—21no—
198C22×Dic3—21no—
199#235—21no—
200#235—21no—
201#235C2×C3×C412yes—
202#235C3×D412yes—
203C2×D12C3×D411yescharacteristic
204C2×C4×D3—21no—
205C2×Dic6C3×D411yescharacteristic
206C3×Q16—64no—
207C3×SD16—64no—
208C3×M16—21no—
209C3×M16—21no—
210C3×M16—22no—
211C3×M16—21no—
212C3×M16—21no—
213C3×M16—21no—
214C3×M16—21no—
215C3×M16—22no—
216C3×M16—32no—
217C3×M16—41no—
218C3×M16—41no—
219C3×M16—64no—
220C3×M16—61no—
221C3×M16#6712yes—
222C3×M16C3×D412yes—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8—21no—
227C2×C3×C8—31no—
228C2×C3×C8—31no—
229C2×C3×C8—41no—
230C2×C3×C8—41no—
231C2×C3×C8—64no—
232C2×C3×C8—61no—
233C2×C3×C8—61no—
234C2×C3×C8—61no—
235C3×C4⋊C4—61no—
236C3×C4⋊C4—61no—
237#216—24no—
238#215—24no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241#208—21no—
242#208C2×C3×C412yes—
243C2×C3⋊C8—21no—
244C2×C3⋊C8—21no—
245C2×C3⋊C8—21no—
246C2×C3⋊C8C3×D411yescharacteristic
247C2×C3⋊C8C3×D411yescharacteristic
248ΓL(1,𝔽25)—24no—
249D3×C8—24no—
250C22×C32—21no—
251C22×C32—21no—
252C22×C32C4×D411yescharacteristic
253C22×C32#12111yescharacteristic
254C22×C32#12311yescharacteristic
255C2×C3×D3—22no—
256C2×C3×D3—41no—
257C2×C3×D3—41no—
258C32×C4—42no—
259C32×C4C2×C22⋊C411yescharacteristic
260C32×C4C4×D411yescharacteristic
261C32×C4#12011yescharacteristic
262C32×C4#12111yescharacteristic
263C3×Dic3—22no—
264C3×Dic3—81no—
265C2×#41—31no—
266C22×Q8C2×C3×D311yescharacteristic
267#137—34no—
268#137—62no—
269C2×Q16—61no—
270C2×SD16—61no—
271#131—62no—
272#131—62no—
273C2×M16—31no—
274C2×M16—61no—
275C2×M16C2×C3×D311yescharacteristic
276C2×M16C2×C3×D311yescharacteristic
277C22×C8—31no—
278C22×C8—61no—
279C2×C4⋊C4—31no—
280#103—61no—
281#103—61no—
282#103—61no—
283#102—61no—
284#101—22no—
285C23×C3—21no—
286C23×C3—61no—
287C23×C3#23511yescharacteristic
288C23×C3C3×#4111yescharacteristic
289C22×D3—21no—
290C3×Q8—22no—
291C3×Q8—22no—
292C3×Q8—22no—
293C3×Q8—22no—
294C3×Q8—32no—
295C3×Q8—42no—
296C3×Q8—42no—
297C3×Q8—42no—
298C3×Q8—42no—
299C3×Q8—41no—
300C3×Q8—41no—
301C3×Q8—41no—
302C3×Q8—41no—
303C3×Q8—61no—
304C3×D4—32no—
305C3×D4—62no—
306C3×D4—62no—
307C3×D4—61no—
308C2×C3×C4—24no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—41no—
320C2×C3×C4—41no—
321C2×C3×C4—41no—
322C2×C3×C4—41no—
323C2×C3×C4—61no—
324C2×C3×C4—62no—
325C2×C3×C4—62no—
326C2×C3×C4—121no—
327C2×C3×C4—241no—
328C2×C3×C4C2×C4×D311yescharacteristic
329C2×C3×C4D3×D411yescharacteristic
330C2×C3×C4D3×D411yescharacteristic
331C2×C3×C4D3×D411yescharacteristic
332C2×C3×C4#23711yescharacteristic
333C2×C3×C4C2×#6711yescharacteristic
334C2×C3×C4C22×C3×C411yescharacteristic
335C2×C3×C4C2×C3×D411yescharacteristic
336C2×C3×C4C2×C3×D411yescharacteristic
337C2×C3×C4C2×C3×D411yescharacteristic
338C2×C3×C4C2×C3×D411yescharacteristic
339C2×C3×C4C3×#4111yescharacteristic
340#67—22no—
341#67—22no—
342C2×Dic3—21no—
343C2×Dic3—41no—
344C2×Dic3—81no—
345D12—21no—
346D12C3×C22⋊C412yes—
347C4×D3—22no—
348C4×D3—22no—
349Dic6—21no—
350Dic6C3×C22⋊C412yes—
351C3×C8—22no—
352C3×C8—22no—
353C3×C8—42no—
354C3×C8—42no—
355C3×C8—41no—
356C3×C8—41no—
357C3×C8—62no—
358C3×C8—61no—
359C3×C8—61no—
360C3⋊C8—22no—
361C3⋊C8—21no—
362C3⋊C8—21no—
363C2×C32—21no—
364C2×C32—21no—
365C2×C32#38611yescharacteristic
366C2×C32#47111yescharacteristic
367C2×C32#47311yescharacteristic
368C3×D3—42no—
369#41—32no—
370#41—32no—
371#41—61no—
372#41—61no—
373C2×Q8—21no—
374C2×Q8—21no—
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8—21no—
378C2×Q8—31no—
379C2×Q8C3×C4×D312yes—
380C2×D4—31no—
381C2×D4—61no—
382C22×C4—21no—
383C22×C4—61no—
384C22×C4—61no—
385C22×C4C22×C3×D311yescharacteristic
386Q16—64no—
387SD16—64no—
388M16—21no—
389M16—21no—
390M16—21no—
391M16—32no—
392M16—64no—
393M16—61no—
394M16C3×#6712yes—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—31no—
398C2×C8—31no—
399C2×C8—64no—
400C2×C8—61no—
401C2×C8—61no—
402C2×C8—61no—
403C4⋊C4—61no—
404C4⋊C4—61no—
405C22×C3—21no—
406C22×C3—21no—
407C22×C3—21no—
408C22×C3—21no—
409C22×C3—21no—
410C22×C3—21no—
411C22×C3—21no—
412C22×C3—41no—
413C22×C3—41no—
414C22×C3—62no—
415C22×C3—121no—
416C22×C3—121no—
417C22×C3#88311yescharacteristic
418C22×C3#88411yescharacteristic
419C22×C3C4×#6711yescharacteristic
420C22×C3C3×C4×D411yescharacteristic
421C22×C3C3×#12111yescharacteristic
422C22×C3C3×#12311yescharacteristic
423C2×D3—22no—
424C2×D3—41no—
425C2×D3—41no—
426C3×C4—21no—
427C3×C4—21no—
428C3×C4—21no—
429C3×C4—21no—
430C3×C4—44no—
431C3×C4—42no—
432C3×C4—42no—
433C3×C4—62no—
434C3×C4—241no—
435C3×C4D3×C22⋊C411yescharacteristic
436C3×C4#67⋊C411yescharacteristic
437C3×C4#93711yescharacteristic
438C3×C4#93811yescharacteristic
439C3×C4C2×C3×C22⋊C411yescharacteristic
440C3×C4C3×C4×D411yescharacteristic
441C3×C4C3×#12011yescharacteristic
442C3×C4C3×#12111yescharacteristic
443Dic3—22no—
444Dic3—81no—
445C32#190811yescharacteristic, complemented, Sylow
446C23—61no—
447C23C3×#23511yescharacteristic
448Q8—22no—
449Q8—22no—
450Q8—32no—
451Q8—41no—
452Q8—41no—
453Q8—61no—
454D4—32no—
455D4—62no—
456D4—62no—
457D4—61no—
458C2×C4—22no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—61no—
462C2×C4—62no—
463C2×C4—62no—
464C2×C4—121no—
465C2×C4—241no—
466C2×C4C2×C3×C4×D311yescharacteristic
467C2×C4C3×D3×D411yescharacteristic
468C2×C4C3×D3×D411yescharacteristic
469C2×C4C3×D3×D411yescharacteristic
470C2×C4C3×#23711yescharacteristic
471C2×C4C2×C3×#6711yescharacteristic
472C8—22no—
473C8—41no—
474C8—62no—
475C8—61no—
476C8—61no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—21no—
481C2×C3—21no—
482C2×C3—21no—
483C2×C3—21no—
484C2×C3—41no—
485C2×C3—41no—
486C2×C3—122no—
487C2×C3#475411yescharacteristic
488C2×C3#475511yescharacteristic
489C2×C3#495611yescharacteristic
490C2×C3C3×#38611yescharacteristic
491C2×C3C3×#47111yescharacteristic
492C2×C3C3×#47311yescharacteristic
493D3—42no—
494C22—21no—
495C22—21no—
496C22—62no—
497C22—121no—
498C22—121no—
499C22C3×#88311yescharacteristic
500C22C3×#88411yescharacteristic
501C22C3×C4×#6711yescharacteristic
502C4—42no—
503C4—62no—
504C4—241no—
505C4C3×D3×C22⋊C411yescharacteristic
506C4C3×#67⋊C411yescharacteristic
507C4C3×#93711yescharacteristic
508C4C3×#93811yescharacteristic
509C3—21no—
510C3#7012311yescharacteristic, direct product factor
511C3C3×#190811yescharacteristic, complemented
512C2—21no—
513C2—21no—
514C2—122no—
515C2C3×#475411yescharacteristic
516C2C3×#475511yescharacteristic
517C2C3×#495611yescharacteristic
subgroupG/H
centerC22×C3#884
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC2×C32×#101C2
socleC22×C32#121
prime radicalC2×C3×#235C22
cosocleC22×C3×C4C23×C3
hypercenterC2×C3×#101D3
hypocenterC3C3×#1908
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×#101

Complements

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

NH ≅ G/Ntype
3. C2×C32×#101C2semidirect
4. C2×C3×#835C2semidirect
6. C3×#4558C2semidirect
7. C3×#4554C2semidirect
9. #70123C3direct (N × H)
16. C3×#942C4semidirect
37. C2×C3×#101D3semidirect
38. C2×C3×#101C2×C3semidirect
39. C2×#835C2×C3semidirect
43. #4558C2×C3semidirect
44. #4554C2×C3semidirect
106. #942C3×C4semidirect
161. C2×#101C3×D3semidirect
445. C32#1908semidirect
510. C3#70123direct (N × H)
511. C3C3×#1908semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9422
C2×C3×#9072
C2×C32×#1012
C2×C3×#8352
C3×#51332
C3×#45582
C3×#45542
C3×#19083
#701233