Finite Group Databasebeta

← C3×#10593490 #13166857

Subgroups — C3×#10593490 #13166857

Order 1728.

876 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 876 conjugacy classes of subgroups — too many to draw. Only its 94 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
2C2×C3×C22⋊#157C211yesmaximal, characteristic, complemented
3C3×#63974C211yesmaximal, characteristic, complemented
4C3×#63738C214yesmaximal, complemented
5#10593490C311yesmaximal, characteristic, direct product factor
6C23×C3×C3⋊D3C2211yescharacteristic
7C22×C3×D32C411yescharacteristic, complemented
8C22×C3×#157—21no—
9C2×C32×#67—21no—
10C3×#3811—21no—
11C3×#3811—21no—
12C3×#3811C412yescomplemented
13C3×#3811C2212yes—
14C2×C3×#626—21no—
15C2×C3×#624—21no—
16C2×C3×#624C411yescharacteristic, complemented
17C3×C22⋊#157—21no—
18C3×C22⋊#157C2212yescomplemented
19C32×#217—21no—
20C3×#3722C2212yescomplemented
21C2×C3×D3×D4—61no—
22C2×#3811C2×C311yescharacteristic
23C2×C22⋊#157C2×C311yescharacteristic, complemented
24C3×D3×C22⋊C4—61no—
25#63974C2×C311yescharacteristic, complemented
26#63738C2×C314yescomplemented
27C23×C33D411yescharacteristic
28C22×C3×C3⋊D3—42no—
29C22×C3×C3⋊D3C2×C411yescharacteristic
30C22×C3×C3⋊D3C2×C411yescharacteristic
31C22×C3×C3⋊D3D411yescharacteristic
32C22×C3×C3⋊D3D411yescharacteristic
33C22×C3×C3⋊D3D411yescharacteristic
34C22×C3×C3⋊D3C2311yescharacteristic
35C22×C32×D3—21no—
36C2×C3×D32—21no—
37C2×C3×D32—42no—
38C2×C3×D32C2×C412yescomplemented
39C2×C3×#157—22no—
40C2×C3×#157—42no—
41C32×#67—22no—
42C32×#67—22no—
43C2×C32×Dic3—21no—
44C2×C32×Dic3—22no—
45C3×#626—22no—
46C3×#626—22no—
47C3×#624—21no—
48C3×#624—42no—
49C3×#624C2×C412yescomplemented
50C2×C3×#99—91nomaximal
51C23×C3⋊D3C22×C311yescharacteristic
52C23×C3×D3—61no—
53C23×C3×D3—61no—
54C23×C3×D3—61no—
55C22×D32C3×C411yescharacteristic, complemented
56C22×#157—21no—
57C2×C32×D4—61no—
58C2×C3×#67—21no—
59C2×C3×#67—21no—
60C2×C3×#67—41no—
61C2×C3×#67—61no—
62C3×D3×D4—62no—
63C3×D3×D4—62no—
64C3×D3×D4—62no—
65C3×D3×D4—62no—
66C2×C3×D12—61no—
67C2×C3×C4×D3—61no—
68C2×C3×C4×D3—62no—
69#3811—21no—
70#3811—21no—
71#3811C3×C412yescomplemented
72#3811C22×C312yes—
73C2×#626—21no—
74C2×#624—21no—
75C2×#624C3×C411yescharacteristic, complemented
76C22⋊#157—21no—
77C22⋊#157C22×C312yescomplemented
78C32×C22⋊C4—61no—
79C3×#217—21no—
80C3×#217—21no—
81C3×#217—41no—
82C3×(C2×D3)⋊C4—62no—
83#3722C22×C312yescomplemented
84C22×C33—41no—
85C22×C33C22⋊C411yescharacteristic
86C22×C33C22⋊C411yescharacteristic
87C22×C33C2×D411yescharacteristic
88C2×C3×C3⋊D3—22no—
89C2×C3×C3⋊D3—22no—
90C2×C3×C3⋊D3—22no—
91C2×C3×C3⋊D3—41no—
92C2×C3×C3⋊D3—41no—
93C2×C3×C3⋊D3—41no—
94C2×C3×C3⋊D3C22⋊C411yescharacteristic
95C2×C3×C3⋊D3C22⋊C411yescharacteristic
96C2×C3×C3⋊D3C22×C411yescharacteristic
97C2×C3×C3⋊D3C2×D411yescharacteristic
98C2×C32×D3—22no—
99C2×C32×D3—42no—
100C3×D32—42no—
101C3×D32—42no—
102C3×#157—42no—
103C32×Dic3—22no—
104C32×Dic3—42no—
105C22×C3×D4—91no—
106C2×D3×D4—61no—
107C2×C3×C22⋊C4—91no—
108C2×C3×C22⋊C4—91no—
109D3×C22⋊C4—61no—
110C3×#99—94no—
111C23×C32—21no—
112C23×C32—21no—
113C23×C32—41no—
114C23×C32—41no—
115C23×C32—61no—
116C23×C32C3×D411yescharacteristic
117C22×C3⋊D3—42no—
118C22×C3⋊D3C2×C3×C411yescharacteristic
119C22×C3⋊D3C2×C3×C411yescharacteristic
120C22×C3⋊D3C3×D411yescharacteristic
121C22×C3⋊D3C3×D411yescharacteristic
122C22×C3⋊D3C3×D411yescharacteristic
123C22×C3⋊D3C23×C311yescharacteristic
124C22×C3×D3—21no—
125C22×C3×D3—21no—
126C22×C3×D3—41no—
127C22×C3×D3—62no—
128C22×C3×D3—61no—
129C22×C3×D3—61no—
130C22×C3×D3—61no—
131C22×C3×D3—61no—
132C22×C3×D3—61no—
133C22×C3×D3—61no—
134C22×C3×D3—61no—
135C22×C3×D3—61no—
136C22×C3×D3—61no—
137C22×C3×D3—61no—
138C22×C3×D3—61no—
139C22×C3×D3—61no—
140C22×C3×D3—62no—
141C22×C3×D3—124no—
142C22×C3×D3—124no—
143C22×C3×D3—124no—
144C2×D32—21no—
145C2×D32—42no—
146C2×D32C2×C3×C412yescomplemented
147C2×#157—22no—
148C2×#157—42no—
149C32×D4—62no—
150C32×D4—62no—
151C2×C32×C4—61no—
152C2×C32×C4—62no—
153C3×#67—22no—
154C3×#67—22no—
155C3×#67—22no—
156C3×#67—22no—
157C3×#67—42no—
158C3×#67—42no—
159C3×#67—62no—
160C3×#67—62no—
161C2×C3×Dic3—21no—
162C2×C3×Dic3—21no—
163C2×C3×Dic3—22no—
164C2×C3×Dic3—22no—
165C2×C3×Dic3—41no—
166C2×C3×Dic3—42no—
167C3×D12—62no—
168C3×D12—62no—
169C3×C4×D3—62no—
170C3×C4×D3—62no—
171C3×C4×D3—124no—
172#626—22no—
173#626—22no—
174#624—21no—
175#624—42no—
176#624C2×C3×C412yescomplemented
177C2×#99—91noSylow
178C2×C33—21no—
179C2×C33—21no—
180C2×C33—21no—
181C2×C33C2×C22⋊C411yescharacteristic
182C3×C3⋊D3—21no—
183C3×C3⋊D3—21no—
184C3×C3⋊D3—21no—
185C3×C3⋊D3#9912yescomplemented
186C32×D3—42no—
187C24×C3—91no—
188C24×C3—91no—
189C23×D3—61no—
190C23×D3—61no—
191C23×D3—61no—
192C2×C3×D4—61no—
193C2×C3×D4—61no—
194C2×C3×D4—92no—
195C2×C3×D4—92no—
196C2×C3×D4—121no—
197C2×C3×D4—181no—
198C2×C3×D4—181no—
199C2×C3×D4—181no—
200C22×C3×C4—91no—
201C22×C3×C4—181no—
202C22×C3×C4—181no—
203C2×#67—21no—
204C2×#67—61no—
205D3×D4—62no—
206D3×D4—62no—
207D3×D4—62no—
208D3×D4—62no—
209C2×D12—61no—
210C2×C4×D3—61no—
211C2×C4×D3—62no—
212C3×C22⋊C4—61no—
213C3×C22⋊C4—61no—
214C3×C22⋊C4—92no—
215C3×C22⋊C4—92no—
216C3×C22⋊C4—121no—
217C3×C22⋊C4—181no—
218#217—21no—
219(C2×D3)⋊C4—62no—
220C22×C32—21no—
221C22×C32—21no—
222C22×C32—21no—
223C22×C32—21no—
224C22×C32—21no—
225C22×C32—21no—
226C22×C32—41no—
227C22×C32—41no—
228C22×C32—41no—
229C22×C32—41no—
230C22×C32—41no—
231C22×C32—41no—
232C22×C32—41no—
233C22×C32—42no—
234C22×C32—42no—
235C22×C32—62no—
236C22×C32—82no—
237C22×C32—82no—
238C22×C32—122no—
239C22×C32C3×C22⋊C411yescharacteristic
240C22×C32C3×C22⋊C411yescharacteristic
241C22×C32C2×C3×D411yescharacteristic
242C2×C3⋊D3—22no—
243C2×C3⋊D3—22no—
244C2×C3⋊D3—22no—
245C2×C3⋊D3—41no—
246C2×C3⋊D3—41no—
247C2×C3⋊D3—41no—
248C2×C3⋊D3C3×C22⋊C411yescharacteristic
249C2×C3⋊D3C3×C22⋊C411yescharacteristic
250C2×C3⋊D3C22×C3×C411yescharacteristic
251C2×C3⋊D3C2×C3×D411yescharacteristic
252C2×C3×D3—22no—
253C2×C3×D3—22no—
254C2×C3×D3—42no—
255C2×C3×D3—42no—
256C2×C3×D3—42no—
257C2×C3×D3—61no—
258C2×C3×D3—61no—
259C2×C3×D3—61no—
260C2×C3×D3—61no—
261C2×C3×D3—61no—
262C2×C3×D3—61no—
263C2×C3×D3—61no—
264C2×C3×D3—61no—
265C2×C3×D3—64no—
266C2×C3×D3—64no—
267C2×C3×D3—82no—
268C2×C3×D3—122no—
269C2×C3×D3—122no—
270C2×C3×D3—122no—
271C2×C3×D3—122no—
272C2×C3×D3—124no—
273C2×C3×D3—124no—
274C2×C3×D3—122no—
275C2×C3×D3—122no—
276C2×C3×D3—122no—
277C2×C3×D3—122no—
278C2×C3×D3—122no—
279C2×C3×D3—122no—
280D32—42no—
281D32—42no—
282#157—42no—
283C32×C4—62no—
284C32×C4—122no—
285C3×Dic3—22no—
286C3×Dic3—22no—
287C3×Dic3—42no—
288C3×Dic3—42no—
289C3×Dic3—42no—
290C3×Dic3—82no—
291C22×D4—91no—
292C2×C22⋊C4—91no—
293C2×C22⋊C4—91no—
294#99—94no—
295C33C2×#9911yescharacteristic, complemented, Sylow
296C23×C3—21no—
297C23×C3—21no—
298C23×C3—41no—
299C23×C3—41no—
300C23×C3—61no—
301C23×C3—61no—
302C23×C3—92no—
303C23×C3—91no—
304C23×C3—91no—
305C23×C3—91no—
306C23×C3—91no—
307C23×C3—91no—
308C23×C3—91no—
309C23×C3—121no—
310C23×C3—181no—
311C23×C3—362no—
312C23×C3—362no—
313C23×C3GO+(4,𝔽2)11yescharacteristic
314C22×D3—21no—
315C22×D3—62no—
316C22×D3—61no—
317C22×D3—61no—
318C22×D3—61no—
319C22×D3—61no—
320C22×D3—61no—
321C22×D3—61no—
322C22×D3—61no—
323C22×D3—61no—
324C22×D3—61no—
325C22×D3—61no—
326C22×D3—61no—
327C22×D3—61no—
328C22×D3—62no—
329C22×D3—124no—
330C22×D3—124no—
331C22×D3—124no—
332C3×D4—62no—
333C3×D4—62no—
334C3×D4—62no—
335C3×D4—62no—
336C3×D4—122no—
337C3×D4—122no—
338C3×D4—182no—
339C3×D4—182no—
340C2×C3×C4—61no—
341C2×C3×C4—61no—
342C2×C3×C4—62no—
343C2×C3×C4—62no—
344C2×C3×C4—92no—
345C2×C3×C4—121no—
346C2×C3×C4—122no—
347C2×C3×C4—181no—
348C2×C3×C4—182no—
349C2×C3×C4—362no—
350C2×C3×C4—362no—
351#67—22no—
352#67—22no—
353#67—62no—
354#67—62no—
355C2×Dic3—21no—
356C2×Dic3—22no—
357D12—62no—
358D12—62no—
359C4×D3—62no—
360C4×D3—62no—
361C4×D3—124no—
362C2×C32—22no—
363C2×C32—22no—
364C2×C32—21no—
365C2×C32—21no—
366C2×C32—21no—
367C2×C32—21no—
368C2×C32—21no—
369C2×C32—41no—
370C2×C32—41no—
371C2×C32—41no—
372C2×C32—41no—
373C2×C32—41no—
374C2×C32—41no—
375C2×C32—42no—
376C2×C32—42no—
377C2×C32—81no—
378C2×C32—81no—
379C2×C32—81no—
380C2×C32—81no—
381C2×C32—122no—
382C2×C32C2×C3×C22⋊C411yescharacteristic
383C3⋊D3—21no—
384C3⋊D3—21no—
385C3⋊D3—21no—
386C3⋊D3C3×#9912yescomplemented
387C3×D3—42no—
388C3×D3—42no—
389C3×D3—62no—
390C3×D3—62no—
391C3×D3—62no—
392C3×D3—62no—
393C3×D3—82no—
394C3×D3—121no—
395C3×D3—121no—
396C3×D3—121no—
397C3×D3—121no—
398C24—91no—
399C24—91no—
400C2×D4—61no—
401C2×D4—92no—
402C2×D4—92no—
403C2×D4—181no—
404C2×D4—181no—
405C2×D4—181no—
406C22×C4—91no—
407C22×C4—181no—
408C22×C4—181no—
409C22⋊C4—61no—
410C22⋊C4—92no—
411C22⋊C4—92no—
412C22⋊C4—181no—
413C22×C3—21no—
414C22×C3—21no—
415C22×C3—21no—
416C22×C3—21no—
417C22×C3—21no—
418C22×C3—21no—
419C22×C3—41no—
420C22×C3—42no—
421C22×C3—42no—
422C22×C3—41no—
423C22×C3—41no—
424C22×C3—41no—
425C22×C3—41no—
426C22×C3—41no—
427C22×C3—41no—
428C22×C3—62no—
429C22×C3—62no—
430C22×C3—82no—
431C22×C3—82no—
432C22×C3—91no—
433C22×C3—91no—
434C22×C3—91no—
435C22×C3—91no—
436C22×C3—122no—
437C22×C3—122no—
438C22×C3—122no—
439C22×C3—182no—
440C22×C3—182no—
441C22×C3—182no—
442C22×C3—242no—
443C22×C3—361no—
444C22×C3—361no—
445C22×C3—361no—
446C22×C3—362no—
447C22×C3—362no—
448C22×C3D32⋊C411yescharacteristic
449C22×C3D32⋊C411yescharacteristic
450C22×C3C2×GO+(4,𝔽2)11yescharacteristic
451C2×D3—22no—
452C2×D3—42no—
453C2×D3—61no—
454C2×D3—61no—
455C2×D3—61no—
456C2×D3—61no—
457C2×D3—61no—
458C2×D3—61no—
459C2×D3—61no—
460C2×D3—61no—
461C2×D3—64no—
462C2×D3—64no—
463C2×D3—122no—
464C2×D3—122no—
465C2×D3—122no—
466C2×D3—122no—
467C2×D3—124no—
468C2×D3—124no—
469C2×D3—122no—
470C2×D3—122no—
471C2×D3—122no—
472C2×D3—122no—
473C2×D3—122no—
474C2×D3—122no—
475C3×C4—62no—
476C3×C4—62no—
477C3×C4—122no—
478C3×C4—122no—
479C3×C4—122no—
480C3×C4—242no—
481C3×C4—362no—
482Dic3—22no—
483Dic3—42no—
484C32—21no—
485C32—21no—
486C32—41no—
487C32—41no—
488C32C2×C3×#9911yescharacteristic, complemented
489C23—61no—
490C23—92no—
491C23—91no—
492C23—91no—
493C23—91no—
494C23—91no—
495C23—91no—
496C23—91no—
497C23—181no—
498C23—362no—
499C23—362no—
500C23C3×GO+(4,𝔽2)11yescharacteristic
501D4—62no—
502D4—62no—
503D4—182no—
504D4—182no—
505C2×C4—61no—
506C2×C4—62no—
507C2×C4—92no—
508C2×C4—181no—
509C2×C4—182no—
510C2×C4—362no—
511C2×C4—362no—
512C2×C3—22no—
513C2×C3—22no—
514C2×C3—21no—
515C2×C3—21no—
516C2×C3—21no—
517C2×C3—21no—
518C2×C3—21no—
519C2×C3—41no—
520C2×C3—41no—
521C2×C3—41no—
522C2×C3—41no—
523C2×C3—42no—
524C2×C3—42no—
525C2×C3—41no—
526C2×C3—41no—
527C2×C3—81no—
528C2×C3—81no—
529C2×C3—81no—
530C2×C3—81no—
531C2×C3—92no—
532C2×C3—122no—
533C2×C3—122no—
534C2×C3—181no—
535C2×C3—181no—
536C2×C3—181no—
537C2×C3—242no—
538C2×C3C2×D32⋊C411yescharacteristic
539D3—42no—
540D3—62no—
541D3—62no—
542D3—62no—
543D3—62no—
544D3—121no—
545D3—121no—
546D3—121no—
547D3—121no—
548C22—41no—
549C22—62no—
550C22—91no—
551C22—91no—
552C22—91no—
553C22—91no—
554C22—122no—
555C22—182no—
556C22—182no—
557C22—182no—
558C22—361no—
559C22—361no—
560C22—361no—
561C22—362no—
562C22—362no—
563C22C3×D32⋊C411yescharacteristic
564C22C3×D32⋊C411yescharacteristic
565C22C2×C3×GO+(4,𝔽2)11yescharacteristic
566C4—62no—
567C4—122no—
568C4—362no—
569C3—21no—
570C3—21no—
571C3—41no—
572C3—41no—
573C3#1059349011yescharacteristic, direct product factor
574C2—21no—
575C2—21no—
576C2—21no—
577C2—92no—
578C2—122no—
579C2—181no—
580C2—181no—
581C2—181no—
582C2C2×C3×D32⋊C411yescharacteristic
subgroupG/H
centerC2×C3C2×D32⋊C4
commutatorC2×C3⋊D3C22×C3×C4
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC23×C33D4
socleC2×C33C2×C22⋊C4
prime radicalC2×C3×#3811C2
cosocleC22×C3⋊D3C23×C3
hypercenterC23×C3GO+(4,𝔽2)
hypocenterC32C2×C3×#99
LCSG ▷ C2×C3⋊D3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
2. C2×C3×C22⋊#157C2semidirect
3. C3×#63974C2semidirect
4. C3×#63738C2semidirect
5. #10593490C3direct (N × H)
7. C22×C3×D32C4semidirect
12. C3×#3811C4semidirect
16. C2×C3×#624C4semidirect
18. C3×C22⋊#157C22semidirect
20. C3×#3722C22semidirect
23. C2×C22⋊#157C2×C3semidirect
25. #63974C2×C3semidirect
26. #63738C2×C3semidirect
38. C2×C3×D32C2×C4semidirect
49. C3×#624C2×C4semidirect
55. C22×D32C3×C4semidirect
71. #3811C3×C4semidirect
75. C2×#624C3×C4semidirect
77. C22⋊#157C22×C3semidirect
83. #3722C22×C3semidirect
146. C2×D32C2×C3×C4semidirect
176. #624C2×C3×C4semidirect
185. C3×C3⋊D3#99semidirect
295. C33C2×#99semidirect
386. C3⋊D3C3×#99semidirect
488. C32C2×C3×#99semidirect
573. C3#10593490direct (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
C2×C3×C22⋊#1572
C3×#639742
C3×#637382
#105934903
C2×C3×#999