Finite Group Databasebeta

← C3×#72103 #11788668

Subgroups — C3×#72103 #11788668

Order 1152.

753 conjugacy classes of subgroups.

The subgroup lattice has 543 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#137C211yesmaximal, characteristic
2C2×C3×#948C211yesmaximal, characteristic
3C3×#5232C211yesmaximal, characteristic
4C2×C3×#836C211yesmaximal, characteristic, complemented
5C2×C3×#835C211yesmaximal, characteristic, complemented
6C3×#4558C211yesmaximal, characteristic, complemented
7C3×#4554C211yesmaximal, characteristic, complemented
8C3×#1908—31nomaximal
9#72103C311yesmaximal, characteristic, direct product factor
10C2×C32×#41C2211yescharacteristic
11C22×C32×Q8C2211yescharacteristic
12C32×#137—22no—
13C32×#137C414yescomplemented
14C2×C32×Q16—21no—
15C2×C32×SD16—21no—
16C2×C32×M16C2211yescharacteristic
17C3×#948—22no—
18C3×#948—22no—
19C2×C3×C4⋊Dic3C2211yescharacteristic
20C2×C3×#208—21no—
21C2×C3×#208C2211yescharacteristic
22C2×C3×#208C2211yescharacteristic
23C22×C3×C3⋊C8—21no—
24C22×C3×C3⋊C8C2211yescharacteristic
25C3×#836—22no—
26C3×#835—21no—
27C3×#835—21no—
28C3×#835—21no—
29C3×#832—21no—
30C2×C3×#137—21no—
31C2×C3×#137D311yescharacteristic
32C2×C3×#137C2×C311yescharacteristic
33C2×C3×#131—31no—
34C2×#948C2×C311yescharacteristic
35C3×#417—31no—
36C2×C3×#103—31no—
37C2×C3×#101—31no—
38#5232C2×C311yescharacteristic
39C2×#836C2×C311yescharacteristic, complemented
40C2×#835C2×C311yescharacteristic, complemented
41C3×#343—31no—
42C3×#341—31no—
43#4558C2×C311yescharacteristic, complemented
44#4554C2×C311yescharacteristic, complemented
45C32×#41—21no—
46C32×#41—21no—
47C32×#41C2×C412yes—
48C32×#41D412yes—
49C2×C32×Q8—21no—
50C2×C32×Q8—21no—
51C2×C32×Q8—21no—
52C2×C32×Q8—21no—
53C2×C32×Q8—21no—
54C2×C32×Q8C2×C412yes—
55C2×C32×Q8D411yescharacteristic
56C2×C32×D4—21no—
57C2×C32×D4D411yescharacteristic
58C22×C32×C4—21no—
59C22×C32×C4—21no—
60C22×C32×C4C2311yescharacteristic
61C22×C3×Dic3—21no—
62C32×Q16—24no—
63C32×SD16—24no—
64C32×M16—21no—
65C32×M16C2×C412yes—
66C2×C32×C8—21no—
67C3×C4⋊Dic3—21no—
68C3×C4⋊Dic3—21no—
69C3×#208—24no—
70C3×#208—21no—
71C3×#208—21no—
72C3×#208—21no—
73C3×#208D412yes—
74C2×C3×C3⋊C8—24no—
75C2×C3×C3⋊C8—21no—
76C2×C3×C3⋊C8—21no—
77C2×C3×C3⋊C8—21no—
78C2×C3×C3⋊C8—21no—
79C2×C3×C3⋊C8D411yescharacteristic
80C2×C3×C3⋊C8D411yescharacteristic
81#1908—31noSylow
82C2×C3×#41—21no—
83C2×C3×#41C2×D311yescharacteristic
84C2×C3×#41C22×C311yescharacteristic
85C22×C3×Q8—21no—
86C22×C3×Q8C2×D311yescharacteristic
87C22×C3×Q8C22×C311yescharacteristic
88C3×#137—24no—
89C3×#137—22no—
90C3×#137—22no—
91C3×#137—24no—
92C3×#137Dic314yescomplemented
93C3×#137C3×C414yescomplemented
94C2×C3×Q16—22no—
95C2×C3×Q16—21no—
96C2×C3×Q16—21no—
97C2×C3×SD16—22no—
98C2×C3×SD16—21no—
99C2×C3×SD16—21no—
100C3×#131—62no—
101C3×#131—62no—
102C2×C3×M16—21no—
103C2×C3×M16—31no—
104C2×C3×M16—31no—
105C2×C3×M16—61no—
106C2×C3×M16C2×D311yescharacteristic
107C2×C3×M16C22×C311yescharacteristic
108C22×C3×C8—31no—
109C22×C3×C8—61no—
110C2×C3×C4⋊C4—31no—
111#948—22no—
112#948—22no—
113C2×C4⋊Dic3C22×C311yescharacteristic
114C2×#208—21no—
115C2×#208C22×C311yescharacteristic
116C2×#208C22×C311yescharacteristic
117C22×C3⋊C8—21no—
118C22×C3⋊C8C22×C311yescharacteristic
119C3×#103—61no—
120C3×#103—61no—
121C3×#103—61no—
122C3×#102—61no—
123C3×#101—62no—
124#836—22no—
125#835—21no—
126#835—21no—
127#835—21no—
128#832—21no—
129C23×C32—21no—
130C23×C32#4111yescharacteristic
131C32×Q8—22no—
132C32×Q8—21no—
133C32×Q8—22no—
134C32×Q8—41no—
135C32×Q8—41no—
136C32×Q8C22⋊C412yes—
137C32×D4—22no—
138C32×D4—22no—
139C32×D4—21no—
140C32×D4C22⋊C412yes—
141C2×C32×C4—21no—
142C2×C32×C4—22no—
143C2×C32×C4—22no—
144C2×C32×C4—22no—
145C2×C32×C4—41no—
146C2×C32×C4—41no—
147C2×C32×C4C22×C411yescharacteristic
148C2×C32×C4C2×D411yescharacteristic
149C2×C32×C4C2×D411yescharacteristic
150C2×C32×C4C2×D411yescharacteristic
151C2×C32×C4C2×D411yescharacteristic
152C2×C32×C4#4111yescharacteristic
153C2×C3×Dic3—41no—
154C2×C3×Dic3—81no—
155C32×C8—22no—
156C3×C3⋊C8—21no—
157C3×C3⋊C8—22no—
158C3×C3⋊C8—21no—
159C3×C3⋊C8—41no—
160C2×#137C3×D311yescharacteristic
161C2×#131—31no—
162#417—31no—
163C2×#103—31no—
164C2×#101—31no—
165#343—31no—
166#341—31no—
167C3×#41—21no—
168C3×#41—21no—
169C3×#41—21no—
170C3×#41—21no—
171C3×#41—22no—
172C3×#41—22no—
173C3×#41—41no—
174C3×#41—41no—
175C3×#41C2×Dic312yes—
176C3×#41#6712yes—
177C3×#41C2×C3×C412yes—
178C3×#41C3×D412yes—
179C2×C3×Q8—21no—
180C2×C3×Q8—21no—
181C2×C3×Q8—22no—
182C2×C3×Q8—22no—
183C2×C3×Q8—21no—
184C2×C3×Q8—21no—
185C2×C3×Q8—21no—
186C2×C3×Q8—21no—
187C2×C3×Q8—21no—
188C2×C3×Q8—21no—
189C2×C3×Q8—21no—
190C2×C3×Q8—21no—
191C2×C3×Q8—22no—
192C2×C3×Q8—21no—
193C2×C3×Q8—41no—
194C2×C3×Q8—41no—
195C2×C3×Q8—41no—
196C2×C3×Q8C2×Dic312yes—
197C2×C3×Q8#6711yescharacteristic
198C2×C3×Q8C2×C3×C412yes—
199C2×C3×Q8C3×D411yescharacteristic
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4—41no—
204C2×C3×D4#6711yescharacteristic
205C2×C3×D4C3×D411yescharacteristic
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—41no—
212C22×C3×C4—41no—
213C22×C3×C4—61no—
214C22×C3×C4C22×D311yescharacteristic
215C22×C3×C4C23×C311yescharacteristic
216C22×Dic3—21no—
217C3×Q16—28no—
218C3×Q16—24no—
219C3×Q16—24no—
220C3×SD16—28no—
221C3×SD16—24no—
222C3×SD16—24no—
223C3×M16—22no—
224C3×M16—21no—
225C3×M16—21no—
226C3×M16—22no—
227C3×M16—32no—
228C3×M16—64no—
229C3×M16—61no—
230C3×M16—61no—
231C3×M16—61no—
232C3×M16C2×Dic312yes—
233C3×M16C2×C3×C412yes—
234C2×C3×C8—22no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—31no—
238C2×C3×C8—31no—
239C2×C3×C8—64no—
240C2×C3×C8—61no—
241C2×C3×C8—61no—
242C2×C3×C8—61no—
243C2×C3×C8—61no—
244C3×C4⋊C4—61no—
245C3×C4⋊C4—61no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248#208—24no—
249#208—21no—
250#208—21no—
251#208—21no—
252#208C3×D412yes—
253C2×C3⋊C8—24no—
254C2×C3⋊C8—21no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8C3×D411yescharacteristic
259C2×C3⋊C8C3×D411yescharacteristic
260C22×C32—21no—
261C22×C32—21no—
262C22×C32—22no—
263C22×C32—41no—
264C22×C32—41no—
265C22×C32C4×D411yescharacteristic
266C22×C32#12111yescharacteristic
267C22×C32#12311yescharacteristic
268C32×C4—22no—
269C32×C4—42no—
270C32×C4C2×C22⋊C411yescharacteristic
271C32×C4C4×D411yescharacteristic
272C32×C4#12011yescharacteristic
273C32×C4#12111yescharacteristic
274C3×Dic3—81no—
275C2×#41C2×C3×D311yescharacteristic
276C22×Q8C2×C3×D311yescharacteristic
277#137—22no—
278#137C3×Dic314yescomplemented
279C2×Q16—21no—
280C2×SD16—21no—
281#131—62no—
282#131—62no—
283C2×M16—31no—
284C2×M16—31no—
285C2×M16—61no—
286C2×M16C2×C3×D311yescharacteristic
287C22×C8—31no—
288C22×C8—61no—
289C2×C4⋊C4—31no—
290#103—61no—
291#103—61no—
292#103—61no—
293#102—61no—
294#101—62no—
295C23×C3—21no—
296C23×C3—21no—
297C23×C3—21no—
298C23×C3—41no—
299C23×C3#23711yescharacteristic
300C23×C3C3×#4111yescharacteristic
301C3×Q8—21no—
302C3×Q8—21no—
303C3×Q8—24no—
304C3×Q8—22no—
305C3×Q8—22no—
306C3×Q8—22no—
307C3×Q8—22no—
308C3×Q8—22no—
309C3×Q8—42no—
310C3×Q8—41no—
311C3×Q8—42no—
312C3×Q8—42no—
313C3×Q8—41no—
314C3×Q8—41no—
315C3×Q8—41no—
316C3×Q8—41no—
317C3×Q8#21712yes—
318C3×Q8C3×C22⋊C412yes—
319C3×D4—22no—
320C3×D4—22no—
321C3×D4—22no—
322C3×D4—22no—
323C3×D4—21no—
324C3×D4—21no—
325C3×D4—22no—
326C3×D4—41no—
327C3×D4—42no—
328C3×D4—42no—
329C3×D4#21712yes—
330C3×D4C3×C22⋊C412yes—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—42no—
346C2×C3×C4—42no—
347C2×C3×C4—41no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4—42no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—121no—
356C2×C3×C4—241no—
357C2×C3×C4D3×D411yescharacteristic
358C2×C3×C4D3×D411yescharacteristic
359C2×C3×C4#23711yescharacteristic
360C2×C3×C4C22×Dic311yescharacteristic
361C2×C3×C4C2×#6711yescharacteristic
362C2×C3×C4C2×#6711yescharacteristic
363C2×C3×C4C22×C3×C411yescharacteristic
364C2×C3×C4C2×C3×D411yescharacteristic
365C2×C3×C4C2×C3×D411yescharacteristic
366C2×C3×C4C2×C3×D411yescharacteristic
367C2×C3×C4C2×C3×D411yescharacteristic
368C2×C3×C4C3×#4111yescharacteristic
369C2×Dic3—41no—
370C2×Dic3—81no—
371C3×C8—24no—
372C3×C8—22no—
373C3×C8—22no—
374C3×C8—61no—
375C3×C8—61no—
376C3×C8—62no—
377C3×C8—121no—
378C3⋊C8—21no—
379C3⋊C8—22no—
380C3⋊C8—21no—
381C3⋊C8—41no—
382C2×C32—21no—
383C2×C32—21no—
384C2×C32—42no—
385C2×C32#38611yescharacteristic
386C2×C32#47111yescharacteristic
387C2×C32#47311yescharacteristic
388#41—21no—
389#41—21no—
390#41C2×C3×Dic312yes—
391#41C3×#6712yes—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8C2×C3×Dic312yes—
398C2×Q8C3×#6711yescharacteristic
399C2×D4—21no—
400C2×D4C3×#6711yescharacteristic
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—61no—
404C22×C4C22×C3×D311yescharacteristic
405Q16—24no—
406SD16—24no—
407M16—21no—
408M16—32no—
409M16—64no—
410M16—61no—
411M16—61no—
412M16—61no—
413M16C2×C3×Dic312yes—
414C2×C8—21no—
415C2×C8—31no—
416C2×C8—31no—
417C2×C8—64no—
418C2×C8—61no—
419C2×C8—61no—
420C2×C8—61no—
421C2×C8—61no—
422C4⋊C4—61no—
423C4⋊C4—61no—
424C22×C3—21no—
425C22×C3—21no—
426C22×C3—21no—
427C22×C3—21no—
428C22×C3—22no—
429C22×C3—22no—
430C22×C3—21no—
431C22×C3—21no—
432C22×C3—21no—
433C22×C3—41no—
434C22×C3—41no—
435C22×C3—42no—
436C22×C3—41no—
437C22×C3—41no—
438C22×C3—41no—
439C22×C3—41no—
440C22×C3—81no—
441C22×C3—81no—
442C22×C3D4×Dic311yescharacteristic
443C22×C3#93511yescharacteristic
444C22×C3#93911yescharacteristic
445C22×C3C3×C4×D411yescharacteristic
446C22×C3C3×#12111yescharacteristic
447C22×C3C3×#12311yescharacteristic
448C3×C4—22no—
449C3×C4—22no—
450C3×C4—21no—
451C3×C4—21no—
452C3×C4—21no—
453C3×C4—21no—
454C3×C4—42no—
455C3×C4—44no—
456C3×C4—42no—
457C3×C4—42no—
458C3×C4—241no—
459C3×C4D4×Dic311yescharacteristic
460C3×C4#93711yescharacteristic
461C3×C4#93911yescharacteristic
462C3×C4C2×#21711yescharacteristic
463C3×C4C2×C3×C22⋊C411yescharacteristic
464C3×C4C3×C4×D411yescharacteristic
465C3×C4C3×#12011yescharacteristic
466C3×C4C3×#12111yescharacteristic
467Dic3—81no—
468C32#190811yescharacteristic, complemented, Sylow
469C23—21no—
470C23C3×#23711yescharacteristic
471Q8—21no—
472Q8—22no—
473Q8—22no—
474Q8—41no—
475Q8—41no—
476Q8C3×#21712yes—
477D4—22no—
478D4—22no—
479D4—21no—
480D4C3×#21712yes—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—121no—
488C2×C4—241no—
489C2×C4C3×D3×D411yescharacteristic
490C2×C4C3×D3×D411yescharacteristic
491C2×C4C3×#23711yescharacteristic
492C2×C4C22×C3×Dic311yescharacteristic
493C2×C4C2×C3×#6711yescharacteristic
494C2×C4C2×C3×#6711yescharacteristic
495C8—22no—
496C8—61no—
497C8—61no—
498C8—62no—
499C8—121no—
500C2×C3—21no—
501C2×C3—21no—
502C2×C3—21no—
503C2×C3—21no—
504C2×C3—21no—
505C2×C3—21no—
506C2×C3—21no—
507C2×C3—41no—
508C2×C3—41no—
509C2×C3—42no—
510C2×C3—42no—
511C2×C3—82no—
512C2×C3#520311yescharacteristic
513C2×C3#520411yescharacteristic
514C2×C3#522011yescharacteristic
515C2×C3C3×#38611yescharacteristic
516C2×C3C3×#47111yescharacteristic
517C2×C3C3×#47311yescharacteristic
518C22—21no—
519C22—21no—
520C22—22no—
521C22—41no—
522C22—41no—
523C22C3×D4×Dic311yescharacteristic
524C22C3×#93511yescharacteristic
525C22C3×#93911yescharacteristic
526C4—22no—
527C4—42no—
528C4—241no—
529C4C3×D4×Dic311yescharacteristic
530C4C3×#93711yescharacteristic
531C4C3×#93911yescharacteristic
532C4C2×C3×#21711yescharacteristic
533C3—21no—
534C3#7210311yescharacteristic, direct product factor
535C3C3×#190811yescharacteristic, complemented
536C2—21no—
537C2—21no—
538C2—42no—
539C2C3×#520311yescharacteristic
540C2C3×#520411yescharacteristic
541C2C3×#522011yescharacteristic
subgroupG/H
centerC22×C3#939
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC2×C32×#137C2
socleC22×C32#121
prime radicalC2×C32×#41C22
cosocleC22×C3×C4C23×C3
hypercenterC2×C3×#137D3
hypocenterC3C3×#1908
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×#137

Complements

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

NH ≅ G/Ntype
4. C2×C3×#836C2semidirect
5. C2×C3×#835C2semidirect
6. C3×#4558C2semidirect
7. C3×#4554C2semidirect
9. #72103C3direct (N × H)
13. C32×#137C4semidirect
39. C2×#836C2×C3semidirect
40. C2×#835C2×C3semidirect
43. #4558C2×C3semidirect
44. #4554C2×C3semidirect
92. C3×#137Dic3semidirect
93. C3×#137C3×C4semidirect
278. #137C3×Dic3semidirect
468. C32#1908semidirect
534. C3#72103direct (N × H)
535. C3C3×#1908semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#1372
C2×C3×#9482
C3×#52322
C2×C3×#8362
C2×C3×#8352
C3×#45582
C3×#45542
C3×#19083
#721033