Finite Group Databasebeta

← C3×#10593514 #13166878

Subgroups — C3×#10593514 #13166878

Order 1728.

1152 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 1152 conjugacy classes of subgroups — too many to draw. Only its 122 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3811C211yesmaximal, characteristic
2C22×C3×#624C211yesmaximal, characteristic, complemented
3C2×C3×C22⋊#157C211yesmaximal, characteristic, complemented
4C2×C3×D32⋊C4C212yesmaximal, complemented
5C3×#63737C212yesmaximal, complemented
6#10593514C311yesmaximal, characteristic, direct product factor
7C23×C3×C3⋊D3C2211yescharacteristic
8C22×C3×D32C2211yescharacteristic
9C22×C3×#157C2212yes—
10C2×C32×#67—21no—
11C22×C32×Dic3—21no—
12C3×#3811—22no—
13C3×#3811C414yescomplemented
14C2×C3×#626—21no—
15C2×C3×#624—21no—
16C2×C3×#624—24no—
17C2×C3×#624C2211yescharacteristic
18C2×C3×#624C2212yes—
19C3×C22⋊#157—22no—
20C3×D32⋊C4—24no—
21C2×C3×D3×D4—61no—
22C22×C3×C4×D3—61no—
23C2×#3811C2×C311yescharacteristic
24C22×#624C2×C311yescharacteristic, complemented
25C2×C22⋊#157C2×C311yescharacteristic, complemented
26C2×D32⋊C4C2×C312yescomplemented
27#63737C2×C312yescomplemented
28C23×C33D411yescharacteristic
29C22×C3×C3⋊D3—22no—
30C22×C3×C3⋊D3—22no—
31C22×C3×C3⋊D3C2×C412yes—
32C22×C3×C3⋊D3D411yescharacteristic
33C22×C3×C3⋊D3D412yes—
34C22×C3×C3⋊D3C2311yescharacteristic
35C22×C32×D3—21no—
36C2×C3×D32—22no—
37C2×C3×D32—22no—
38C2×C3×D32—21no—
39C2×C3×D32C2×C412yescomplemented
40C2×C3×#157—22no—
41C2×C3×#157—22no—
42C2×C3×#157—22no—
43C32×#67—24no—
44C2×C32×Dic3—24no—
45C2×C32×Dic3—22no—
46C2×C32×Dic3—21no—
47C3×#626—24no—
48C3×#624—24no—
49C3×#624—22no—
50C3×#624—21no—
51C3×#624C2×C412yescomplemented
52C3×#624D414yescomplemented
53C3×#386—91nomaximal
54C23×C3⋊D3C22×C311yescharacteristic
55C23×C3×D3—61no—
56C23×C3×D3—61no—
57C23×C3×D3—61no—
58C22×D32C22×C311yescharacteristic
59C22×#157C22×C312yes—
60C2×C32×D4—61no—
61C22×C32×C4—61no—
62C2×C3×#67—21no—
63C2×C3×#67—21no—
64C2×C3×#67—41no—
65C2×C3×#67—61no—
66C22×C3×Dic3—21no—
67C22×C3×Dic3—21no—
68C22×C3×Dic3—41no—
69C3×D3×D4—64no—
70C3×D3×D4—64no—
71C2×C3×D12—61no—
72C2×C3×C4×D3—62no—
73C2×C3×C4×D3—61no—
74C2×C3×C4×D3—68no—
75C2×C3×C4×D3—62no—
76#3811—22no—
77#3811C3×C414yescomplemented
78C2×#626—21no—
79C2×#624—21no—
80C2×#624—24no—
81C2×#624C22×C311yescharacteristic
82C2×#624C22×C312yes—
83C22⋊#157—22no—
84D32⋊C4—24no—
85C22×C33—21no—
86C22×C33—21no—
87C22×C33C22⋊C412yescomplemented
88C22×C33C2×D411yescharacteristic
89C2×C3×C3⋊D3—21no—
90C2×C3×C3⋊D3—21no—
91C2×C3×C3⋊D3—24no—
92C2×C3×C3⋊D3—24no—
93C2×C3×C3⋊D3C22⋊C412yescomplemented
94C2×C3×C3⋊D3C22×C411yescharacteristic
95C2×C3×C3⋊D3C2×D412yes—
96C2×C3×C3⋊D3C2×D411yescharacteristic
97C2×C3×C3⋊D3#4112yes—
98C2×C32×D3—22no—
99C2×C32×D3—41no—
100C2×C32×D3—41no—
101C3×D32—24no—
102C3×D32—42no—
103C3×#157—42no—
104C32×Dic3—22no—
105C32×Dic3—24no—
106C22×C3×D4—91no—
107C23×C3×C4—91no—
108C2×D3×D4—61no—
109C22×C4×D3—61no—
110C2×C3×C22⋊C4—91no—
111C2×C3×C22⋊C4—92no—
112C3×#95—92no—
113C23×C32—21no—
114C23×C32—21no—
115C23×C32—41no—
116C23×C32—41no—
117C23×C32—61no—
118C23×C32C3×D411yescharacteristic
119C22×C3⋊D3—22no—
120C22×C3⋊D3—22no—
121C22×C3⋊D3C2×C3×C412yes—
122C22×C3⋊D3C3×D411yescharacteristic
123C22×C3⋊D3C3×D412yes—
124C22×C3⋊D3C23×C311yescharacteristic
125C22×C3×D3—21no—
126C22×C3×D3—21no—
127C22×C3×D3—41no—
128C22×C3×D3—62no—
129C22×C3×D3—61no—
130C22×C3×D3—62no—
131C22×C3×D3—62no—
132C22×C3×D3—61no—
133C22×C3×D3—61no—
134C22×C3×D3—62no—
135C22×C3×D3—61no—
136C22×C3×D3—62no—
137C22×C3×D3—62no—
138C22×C3×D3—64no—
139C22×C3×D3—64no—
140C22×C3×D3—122no—
141C22×C3×D3—122no—
142C22×C3×D3—122no—
143C22×C3×D3—122no—
144C2×D32—21no—
145C2×D32—22no—
146C2×D32—22no—
147C2×D32C2×C3×C412yescomplemented
148C2×#157—22no—
149C2×#157—22no—
150C2×#157—22no—
151C32×D4—64no—
152C2×C32×C4—64no—
153C2×C32×C4—62no—
154C2×C32×C4—61no—
155C3×#67—24no—
156C3×#67—24no—
157C3×#67—44no—
158C3×#67—64no—
159C2×C3×Dic3—24no—
160C2×C3×Dic3—24no—
161C2×C3×Dic3—22no—
162C2×C3×Dic3—21no—
163C2×C3×Dic3—22no—
164C2×C3×Dic3—21no—
165C2×C3×Dic3—42no—
166C2×C3×Dic3—44no—
167C2×C3×Dic3—41no—
168C3×D12—64no—
169C3×C4×D3—62no—
170C3×C4×D3—64no—
171C3×C4×D3—64no—
172C3×C4×D3—68no—
173C3×C4×D3—62no—
174#626—24no—
175#624—24no—
176#624—22no—
177#624—21no—
178#624C2×C3×C412yescomplemented
179#624C3×D414yescomplemented
180#386—91noSylow
181C2×C33—22no—
182C2×C33C2×C22⋊C411yescharacteristic
183C2×C33#12011yescharacteristic
184C2×C33#12311yescharacteristic
185C3×C3⋊D3—22no—
186C3×C3⋊D3C4×D412yes—
187C3×C3⋊D3#12112yes—
188C32×D3—42no—
189C24×C3—91no—
190C24×C3—91no—
191C23×D3—61no—
192C23×D3—61no—
193C23×D3—61no—
194C2×C3×D4—61no—
195C2×C3×D4—61no—
196C2×C3×D4—94no—
197C2×C3×D4—121no—
198C2×C3×D4—181no—
199C2×C3×D4—182no—
200C22×C3×C4—61no—
201C22×C3×C4—61no—
202C22×C3×C4—91no—
203C22×C3×C4—92no—
204C22×C3×C4—92no—
205C22×C3×C4—121no—
206C22×C3×C4—184no—
207C22×C3×C4—181no—
208C2×#67—21no—
209C2×#67—61no—
210C22×Dic3—21no—
211D3×D4—64no—
212D3×D4—64no—
213C2×D12—61no—
214C2×C4×D3—62no—
215C2×C4×D3—61no—
216C2×C4×D3—68no—
217C2×C4×D3—62no—
218C3×C22⋊C4—184no—
219C3×C22⋊C4—182no—
220C22×C32—22no—
221C22×C32—22no—
222C22×C32—21no—
223C22×C32—21no—
224C22×C32—21no—
225C22×C32—21no—
226C22×C32—22no—
227C22×C32—22no—
228C22×C32—42no—
229C22×C32—41no—
230C22×C32—41no—
231C22×C32—42no—
232C22×C32—41no—
233C22×C32—41no—
234C22×C32—42no—
235C22×C32—42no—
236C22×C32—62no—
237C22×C32—81no—
238C22×C32—81no—
239C22×C32—121no—
240C22×C32—121no—
241C22×C32C3×C22⋊C412yescomplemented
242C22×C32C2×C3×D411yescharacteristic
243C2×C3⋊D3—21no—
244C2×C3⋊D3—21no—
245C2×C3⋊D3—24no—
246C2×C3⋊D3—24no—
247C2×C3⋊D3C3×C22⋊C412yescomplemented
248C2×C3⋊D3C22×C3×C411yescharacteristic
249C2×C3⋊D3C2×C3×D412yes—
250C2×C3⋊D3C2×C3×D411yescharacteristic
251C2×C3⋊D3C3×#4112yes—
252C2×C3×D3—22no—
253C2×C3×D3—22no—
254C2×C3×D3—42no—
255C2×C3×D3—41no—
256C2×C3×D3—41no—
257C2×C3×D3—41no—
258C2×C3×D3—41no—
259C2×C3×D3—62no—
260C2×C3×D3—61no—
261C2×C3×D3—61no—
262C2×C3×D3—62no—
263C2×C3×D3—61no—
264C2×C3×D3—61no—
265C2×C3×D3—62no—
266C2×C3×D3—62no—
267C2×C3×D3—62no—
268C2×C3×D3—62no—
269C2×C3×D3—62no—
270C2×C3×D3—68no—
271C2×C3×D3—68no—
272C2×C3×D3—62no—
273C2×C3×D3—81no—
274C2×C3×D3—81no—
275C2×C3×D3—124no—
276C2×C3×D3—124no—
277C2×C3×D3—124no—
278C2×C3×D3—121no—
279C2×C3×D3—121no—
280C2×C3×D3—124no—
281D32—24no—
282D32—42no—
283#157—42no—
284C32×C4—64no—
285C32×C4—62no—
286C3×Dic3—24no—
287C3×Dic3—22no—
288C3×Dic3—22no—
289C3×Dic3—24no—
290C3×Dic3—44no—
291C3×Dic3—42no—
292C22×D4—91no—
293C23×C4—91no—
294C2×C22⋊C4—91no—
295C2×C22⋊C4—92no—
296#95—92no—
297C33#38611yescharacteristic, complemented, Sylow
298C23×C3—21no—
299C23×C3—21no—
300C23×C3—41no—
301C23×C3—41no—
302C23×C3—61no—
303C23×C3—61no—
304C23×C3—92no—
305C23×C3—91no—
306C23×C3—91no—
307C23×C3—92no—
308C23×C3—92no—
309C23×C3—121no—
310C23×C3—182no—
311C23×C3—182no—
312C23×C3—181no—
313C23×C3—182no—
314C23×C3—182no—
315C23×C3GO+(4,𝔽2)11yescharacteristic
316C22×D3—21no—
317C22×D3—62no—
318C22×D3—62no—
319C22×D3—61no—
320C22×D3—62no—
321C22×D3—61no—
322C22×D3—61no—
323C22×D3—62no—
324C22×D3—61no—
325C22×D3—62no—
326C22×D3—62no—
327C22×D3—64no—
328C22×D3—64no—
329C22×D3—122no—
330C22×D3—122no—
331C22×D3—122no—
332C22×D3—122no—
333C3×D4—64no—
334C3×D4—64no—
335C3×D4—124no—
336C3×D4—184no—
337C2×C3×C4—64no—
338C2×C3×C4—62no—
339C2×C3×C4—61no—
340C2×C3×C4—64no—
341C2×C3×C4—62no—
342C2×C3×C4—61no—
343C2×C3×C4—94no—
344C2×C3×C4—92no—
345C2×C3×C4—124no—
346C2×C3×C4—122no—
347C2×C3×C4—121no—
348C2×C3×C4—182no—
349C2×C3×C4—181no—
350C2×C3×C4—184no—
351C2×C3×C4—182no—
352C2×C3×C4—182no—
353C2×C3×C4—182no—
354#67—24no—
355#67—64no—
356C2×Dic3—24no—
357C2×Dic3—22no—
358C2×Dic3—21no—
359D12—64no—
360C4×D3—62no—
361C4×D3—64no—
362C4×D3—64no—
363C4×D3—68no—
364C4×D3—62no—
365C2×C32—24no—
366C2×C32—22no—
367C2×C32—21no—
368C2×C32—21no—
369C2×C32—21no—
370C2×C32—21no—
371C2×C32—21no—
372C2×C32—21no—
373C2×C32—41no—
374C2×C32—41no—
375C2×C32—41no—
376C2×C32—41no—
377C2×C32—41no—
378C2×C32—41no—
379C2×C32—42no—
380C2×C32—44no—
381C2×C32—82no—
382C2×C32—122no—
383C2×C32C2×C3×C22⋊C411yescharacteristic
384C2×C32C3×#12011yescharacteristic
385C2×C32C3×#12311yescharacteristic
386C3⋊D3—22no—
387C3⋊D3C3×C4×D412yes—
388C3⋊D3C3×#12112yes—
389C3×D3—42no—
390C3×D3—42no—
391C3×D3—62no—
392C3×D3—62no—
393C3×D3—62no—
394C3×D3—62no—
395C3×D3—64no—
396C3×D3—82no—
397C3×D3—122no—
398C24—91no—
399C24—91no—
400C2×D4—61no—
401C2×D4—94no—
402C2×D4—181no—
403C2×D4—182no—
404C22×C4—61no—
405C22×C4—91no—
406C22×C4—92no—
407C22×C4—92no—
408C22×C4—184no—
409C22×C4—181no—
410C22⋊C4—184no—
411C22⋊C4—182no—
412C22×C3—22no—
413C22×C3—22no—
414C22×C3—21no—
415C22×C3—21no—
416C22×C3—21no—
417C22×C3—21no—
418C22×C3—22no—
419C22×C3—22no—
420C22×C3—41no—
421C22×C3—41no—
422C22×C3—42no—
423C22×C3—42no—
424C22×C3—41no—
425C22×C3—41no—
426C22×C3—42no—
427C22×C3—42no—
428C22×C3—62no—
429C22×C3—62no—
430C22×C3—81no—
431C22×C3—81no—
432C22×C3—91no—
433C22×C3—91no—
434C22×C3—92no—
435C22×C3—92no—
436C22×C3—92no—
437C22×C3—122no—
438C22×C3—121no—
439C22×C3—121no—
440C22×C3—121no—
441C22×C3—121no—
442C22×C3—184no—
443C22×C3—184no—
444C22×C3—184no—
445C22×C3—181no—
446C22×C3—181no—
447C22×C3—241no—
448C22×C3—241no—
449C22×C3—362no—
450C22×C3D32⋊C412yescomplemented
451C22×C3C2×GO+(4,𝔽2)11yescharacteristic
452C2×D3—22no—
453C2×D3—41no—
454C2×D3—41no—
455C2×D3—62no—
456C2×D3—61no—
457C2×D3—61no—
458C2×D3—62no—
459C2×D3—61no—
460C2×D3—61no—
461C2×D3—62no—
462C2×D3—62no—
463C2×D3—62no—
464C2×D3—62no—
465C2×D3—62no—
466C2×D3—68no—
467C2×D3—68no—
468C2×D3—62no—
469C2×D3—124no—
470C2×D3—124no—
471C2×D3—124no—
472C2×D3—121no—
473C2×D3—121no—
474C2×D3—124no—
475C3×C4—64no—
476C3×C4—62no—
477C3×C4—62no—
478C3×C4—64no—
479C3×C4—124no—
480C3×C4—122no—
481C3×C4—362no—
482Dic3—24no—
483Dic3—22no—
484C32—21no—
485C32—21no—
486C32—41no—
487C32—41no—
488C32C3×#38611yescharacteristic, complemented
489C23—61no—
490C23—92no—
491C23—91no—
492C23—91no—
493C23—92no—
494C23—92no—
495C23—182no—
496C23—182no—
497C23—181no—
498C23—182no—
499C23—182no—
500C23C3×GO+(4,𝔽2)11yescharacteristic
501D4—64no—
502D4—184no—
503C2×C4—64no—
504C2×C4—62no—
505C2×C4—61no—
506C2×C4—94no—
507C2×C4—92no—
508C2×C4—182no—
509C2×C4—181no—
510C2×C4—184no—
511C2×C4—182no—
512C2×C4—182no—
513C2×C4—182no—
514C2×C3—24no—
515C2×C3—22no—
516C2×C3—21no—
517C2×C3—21no—
518C2×C3—21no—
519C2×C3—21no—
520C2×C3—21no—
521C2×C3—21no—
522C2×C3—42no—
523C2×C3—44no—
524C2×C3—41no—
525C2×C3—41no—
526C2×C3—41no—
527C2×C3—41no—
528C2×C3—41no—
529C2×C3—41no—
530C2×C3—82no—
531C2×C3—92no—
532C2×C3—92no—
533C2×C3—122no—
534C2×C3—122no—
535C2×C3—182no—
536C2×C3—242no—
537C2×C3C2×D32⋊C411yescharacteristic
538C2×C3#6423311yescharacteristic
539C2×C3#6424111yescharacteristic
540D3—42no—
541D3—62no—
542D3—62no—
543D3—62no—
544D3—62no—
545D3—64no—
546D3—122no—
547C22—21no—
548C22—21no—
549C22—62no—
550C22—92no—
551C22—92no—
552C22—91no—
553C22—91no—
554C22—92no—
555C22—121no—
556C22—121no—
557C22—184no—
558C22—184no—
559C22—184no—
560C22—181no—
561C22—181no—
562C22—362no—
563C22C3×D32⋊C412yescomplemented
564C22C2×C3×GO+(4,𝔽2)11yescharacteristic
565C4—64no—
566C4—62no—
567C4—362no—
568C3—21no—
569C3—21no—
570C3—41no—
571C3—41no—
572C3#1059351411yescharacteristic, direct product factor
573C2—22no—
574C2—92no—
575C2—92no—
576C2—122no—
577C2—182no—
578C2C2×C3×D32⋊C411yescharacteristic
579C2C3×#6423311yescharacteristic
580C2C3×#6424111yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutatorC2×C3⋊D3C22×C3×C4
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC23×C33D4
socleC22×C33C2×D4
prime radicalC2×C3×#3811C2
cosocleC22×C3⋊D3C23×C3
hypercenterC23×C3GO+(4,𝔽2)
hypocenterC32C3×#386
LCSG ▷ C2×C3⋊D3 ▷ C32
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
2. C22×C3×#624C2semidirect
3. C2×C3×C22⋊#157C2semidirect
4. C2×C3×D32⋊C4C2semidirect
5. C3×#63737C2semidirect
6. #10593514C3direct (N × H)
13. C3×#3811C4semidirect
24. C22×#624C2×C3semidirect
25. C2×C22⋊#157C2×C3semidirect
26. C2×D32⋊C4C2×C3semidirect
27. #63737C2×C3semidirect
39. C2×C3×D32C2×C4semidirect
51. C3×#624C2×C4semidirect
52. C3×#624D4semidirect
77. #3811C3×C4semidirect
87. C22×C33C22⋊C4semidirect
93. C2×C3×C3⋊D3C22⋊C4semidirect
147. C2×D32C2×C3×C4semidirect
178. #624C2×C3×C4semidirect
179. #624C3×D4semidirect
241. C22×C32C3×C22⋊C4semidirect
247. C2×C3⋊D3C3×C22⋊C4semidirect
297. C33#386semidirect
450. C22×C3D32⋊C4semidirect
488. C32C3×#386semidirect
563. C22C3×D32⋊C4semidirect
572. C3#10593514direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38112
C22×C3×#6242
C2×C3×C22⋊#1572
C2×C3×D32⋊C42
C3×#637372
#105935143
C3×#3869