Finite Group Databasebeta

← C2×C3×#63738 #13166856

Subgroups — C2×C3×#63738 #13166856

Order 1728.

896 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 896 conjugacy classes of subgroups — too many to draw. Only its 98 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
3C2×C3×#3722C211yesmaximal, characteristic, complemented
4C3×#63738C214yesmaximal, direct product factor
5C2×#63738C311yesmaximal, characteristic, direct product factor
6C23×C3×C3⋊D3C2211yescharacteristic
7C22×C3×D32C411yescharacteristic, complemented
8C22×C3×#157—21no—
9C2×C32×#67—21no—
10C22×C32×Dic3—21no—
11C3×#3811—21no—
12C3×#3811—21no—
13C3×#3811C412yescomplemented
14C3×#3811C2212yes—
15C2×C3×#626—21no—
16C2×C3×#624C411yescharacteristic, complemented
17C3×C22⋊#157—21no—
18C3×C22⋊#157C2212yescomplemented
19C3×#3722—21no—
20C3×#3722C2212yescomplemented
21C2×C3×D3×D4—61no—
22C2×#3811C2×C311yescharacteristic
23C2×C22⋊#157C2×C311yescharacteristic, complemented
24C2×C3×(C2×D3)⋊C4—61no—
25C2×#3722C2×C311yescharacteristic, complemented
26#63738C2×C314yesdirect product factor
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—
45C2×C32×Dic3—42no—
46C3×#626—22no—
47C3×#626—22no—
48C3×#624—21no—
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—
58C22×C32×C4—61no—
59C2×C3×#67—21no—
60C2×C3×#67—21no—
61C2×C3×#67—41no—
62C2×C3×#67—61no—
63C22×C3×Dic3—21no—
64C22×C3×Dic3—21no—
65C22×C3×Dic3—41no—
66C3×D3×D4—62no—
67C3×D3×D4—62no—
68C3×D3×D4—62no—
69C3×D3×D4—62no—
70C2×C3×D12—61no—
71C2×C3×C4×D3—61no—
72#3811—21no—
73#3811—21no—
74#3811C3×C412yescomplemented
75#3811C22×C312yes—
76C2×#626—21no—
77C2×#624C3×C411yescharacteristic, complemented
78C22⋊#157—21no—
79C22⋊#157C22×C312yescomplemented
80C3×(C2×D3)⋊C4—62no—
81C3×(C2×D3)⋊C4—62no—
82#3722—21no—
83#3722C22×C312yescomplemented
84C22×C33—22no—
85C22×C33C22⋊C411yescharacteristic
86C22×C33C22⋊C411yescharacteristic
87C22×C33C2×D411yescharacteristic
88C2×C3×C3⋊D3—22no—
89C2×C3×C3⋊D3—22no—
90C2×C3×C3⋊D3—41no—
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—
109C2×(C2×D3)⋊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—122no—
144C22×C3×D3—122no—
145C2×D32—21no—
146C2×D32—42no—
147C2×D32C2×C3×C412yescomplemented
148C2×#157—22no—
149C2×#157—42no—
150C32×D4—62no—
151C32×D4—62no—
152C2×C32×C4—61no—
153C2×C32×C4—62no—
154C2×C32×C4—122no—
155C3×#67—22no—
156C3×#67—22no—
157C3×#67—22no—
158C3×#67—22no—
159C3×#67—42no—
160C3×#67—42no—
161C3×#67—62no—
162C3×#67—62no—
163C2×C3×Dic3—21no—
164C2×C3×Dic3—21no—
165C2×C3×Dic3—22no—
166C2×C3×Dic3—22no—
167C2×C3×Dic3—42no—
168C2×C3×Dic3—41no—
169C2×C3×Dic3—42no—
170C2×C3×Dic3—42no—
171C2×C3×Dic3—44no—
172C3×D12—62no—
173C3×D12—62no—
174C3×C4×D3—62no—
175C3×C4×D3—62no—
176#626—22no—
177#626—22no—
178#624—21no—
179#624C2×C3×C412yescomplemented
180C2×#99—91noSylow
181C2×C33—21no—
182C2×C33—21no—
183C2×C33#9912yescomplemented
184C2×C33C2×C22⋊C411yescharacteristic
185C3×C3⋊D3—21no—
186C3×C3⋊D3—21no—
187C3×C3⋊D3—21no—
188C3×C3⋊D3—21no—
189C32×D3—42no—
190C24×C3—91no—
191C24×C3—91no—
192C23×D3—61no—
193C23×D3—61no—
194C23×D3—61no—
195C2×C3×D4—61no—
196C2×C3×D4—61no—
197C2×C3×D4—92no—
198C2×C3×D4—92no—
199C2×C3×D4—121no—
200C2×C3×D4—181no—
201C2×C3×D4—181no—
202C2×C3×D4—181no—
203C22×C3×C4—61no—
204C22×C3×C4—61no—
205C22×C3×C4—91no—
206C22×C3×C4—121no—
207C22×C3×C4—181no—
208C2×#67—21no—
209C2×#67—61no—
210C22×Dic3—21no—
211D3×D4—62no—
212D3×D4—62no—
213D3×D4—62no—
214D3×D4—62no—
215C2×D12—61no—
216C2×C4×D3—61no—
217C3×C22⋊C4—92no—
218C3×C22⋊C4—92no—
219C3×C22⋊C4—181no—
220C3×C22⋊C4—181no—
221(C2×D3)⋊C4—62no—
222(C2×D3)⋊C4—62no—
223C22×C32—22no—
224C22×C32—22no—
225C22×C32—22no—
226C22×C32—21no—
227C22×C32—21no—
228C22×C32—21no—
229C22×C32—21no—
230C22×C32—21no—
231C22×C32—21no—
232C22×C32—41no—
233C22×C32—41no—
234C22×C32—41no—
235C22×C32—41no—
236C22×C32—41no—
237C22×C32—41no—
238C22×C32—42no—
239C22×C32—42no—
240C22×C32—42no—
241C22×C32—62no—
242C22×C32—82no—
243C22×C32—122no—
244C22×C32C3×C22⋊C411yescharacteristic
245C22×C32C3×C22⋊C411yescharacteristic
246C22×C32C2×C3×D411yescharacteristic
247C2×C3⋊D3—22no—
248C2×C3⋊D3—22no—
249C2×C3⋊D3—41no—
250C2×C3⋊D3—41no—
251C2×C3⋊D3—41no—
252C2×C3⋊D3—41no—
253C2×C3⋊D3C3×C22⋊C411yescharacteristic
254C2×C3⋊D3C3×C22⋊C411yescharacteristic
255C2×C3⋊D3C22×C3×C411yescharacteristic
256C2×C3⋊D3C2×C3×D411yescharacteristic
257C2×C3×D3—22no—
258C2×C3×D3—22no—
259C2×C3×D3—42no—
260C2×C3×D3—42no—
261C2×C3×D3—42no—
262C2×C3×D3—61no—
263C2×C3×D3—61no—
264C2×C3×D3—61no—
265C2×C3×D3—61no—
266C2×C3×D3—61no—
267C2×C3×D3—61no—
268C2×C3×D3—61no—
269C2×C3×D3—61no—
270C2×C3×D3—64no—
271C2×C3×D3—82no—
272C2×C3×D3—122no—
273C2×C3×D3—122no—
274C2×C3×D3—122no—
275C2×C3×D3—124no—
276C2×C3×D3—124no—
277C2×C3×D3—122no—
278C2×C3×D3—122no—
279C2×C3×D3—122no—
280C2×C3×D3—122no—
281C2×C3×D3—121no—
282C2×C3×D3—121no—
283C2×C3×D3—121no—
284C2×C3×D3—121no—
285C2×C3×D3—122no—
286C2×C3×D3—122no—
287D32—42no—
288D32—42no—
289#157—42no—
290C32×C4—62no—
291C32×C4—122no—
292C3×Dic3—22no—
293C3×Dic3—22no—
294C3×Dic3—42no—
295C3×Dic3—42no—
296C3×Dic3—42no—
297C3×Dic3—44no—
298C22×D4—91no—
299C2×C22⋊C4—91no—
300C2×C22⋊C4—91no—
301#99—94no—
302C33C2×#9911yescharacteristic, complemented, Sylow
303C23×C3—21no—
304C23×C3—21no—
305C23×C3—41no—
306C23×C3—41no—
307C23×C3—61no—
308C23×C3—61no—
309C23×C3—92no—
310C23×C3—91no—
311C23×C3—91no—
312C23×C3—91no—
313C23×C3—91no—
314C23×C3—91no—
315C23×C3—91no—
316C23×C3—121no—
317C23×C3—181no—
318C23×C3—362no—
319C23×C3—362no—
320C23×C3GO+(4,𝔽2)11yescharacteristic
321C22×D3—21no—
322C22×D3—62no—
323C22×D3—61no—
324C22×D3—61no—
325C22×D3—61no—
326C22×D3—61no—
327C22×D3—61no—
328C22×D3—61no—
329C22×D3—61no—
330C22×D3—61no—
331C22×D3—61no—
332C22×D3—61no—
333C22×D3—61no—
334C22×D3—61no—
335C22×D3—62no—
336C22×D3—124no—
337C22×D3—124no—
338C22×D3—122no—
339C22×D3—122no—
340C3×D4—62no—
341C3×D4—62no—
342C3×D4—62no—
343C3×D4—62no—
344C3×D4—122no—
345C3×D4—122no—
346C3×D4—182no—
347C3×D4—182no—
348C2×C3×C4—61no—
349C2×C3×C4—61no—
350C2×C3×C4—62no—
351C2×C3×C4—62no—
352C2×C3×C4—92no—
353C2×C3×C4—122no—
354C2×C3×C4—121no—
355C2×C3×C4—122no—
356C2×C3×C4—122no—
357C2×C3×C4—124no—
358C2×C3×C4—181no—
359C2×C3×C4—182no—
360C2×C3×C4—362no—
361#67—22no—
362#67—22no—
363#67—62no—
364#67—62no—
365C2×Dic3—21no—
366C2×Dic3—22no—
367C2×Dic3—42no—
368D12—62no—
369D12—62no—
370C4×D3—62no—
371C4×D3—62no—
372C2×C32—21no—
373C2×C32—21no—
374C2×C32—22no—
375C2×C32—21no—
376C2×C32—21no—
377C2×C32—21no—
378C2×C32—21no—
379C2×C32—22no—
380C2×C32—22no—
381C2×C32—42no—
382C2×C32—42no—
383C2×C32—41no—
384C2×C32—41no—
385C2×C32—41no—
386C2×C32—41no—
387C2×C32—41no—
388C2×C32—41no—
389C2×C32—42no—
390C2×C32—81no—
391C2×C32—81no—
392C2×C32—122no—
393C2×C32C3×#9912yescomplemented
394C2×C32C2×C3×C22⋊C411yescharacteristic
395C3⋊D3—21no—
396C3⋊D3—21no—
397C3⋊D3—21no—
398C3⋊D3—21no—
399C3×D3—42no—
400C3×D3—42no—
401C3×D3—62no—
402C3×D3—62no—
403C3×D3—82no—
404C3×D3—121no—
405C3×D3—121no—
406C3×D3—121no—
407C3×D3—121no—
408C3×D3—121no—
409C3×D3—121no—
410C24—91no—
411C24—91no—
412C2×D4—61no—
413C2×D4—92no—
414C2×D4—92no—
415C2×D4—181no—
416C2×D4—181no—
417C2×D4—181no—
418C22×C4—61no—
419C22×C4—91no—
420C22×C4—181no—
421C22⋊C4—92no—
422C22⋊C4—92no—
423C22⋊C4—181no—
424C22⋊C4—181no—
425C22×C3—22no—
426C22×C3—22no—
427C22×C3—22no—
428C22×C3—21no—
429C22×C3—21no—
430C22×C3—21no—
431C22×C3—21no—
432C22×C3—21no—
433C22×C3—21no—
434C22×C3—42no—
435C22×C3—42no—
436C22×C3—42no—
437C22×C3—41no—
438C22×C3—41no—
439C22×C3—41no—
440C22×C3—41no—
441C22×C3—41no—
442C22×C3—41no—
443C22×C3—62no—
444C22×C3—62no—
445C22×C3—82no—
446C22×C3—91no—
447C22×C3—91no—
448C22×C3—91no—
449C22×C3—91no—
450C22×C3—122no—
451C22×C3—122no—
452C22×C3—122no—
453C22×C3—182no—
454C22×C3—182no—
455C22×C3—242no—
456C22×C3—361no—
457C22×C3—361no—
458C22×C3—361no—
459C22×C3—361no—
460C22×C3—362no—
461C22×C3—362no—
462C22×C3D32⋊C411yescharacteristic
463C22×C3D32⋊C411yescharacteristic
464C22×C3C2×GO+(4,𝔽2)11yescharacteristic
465C2×D3—22no—
466C2×D3—42no—
467C2×D3—61no—
468C2×D3—61no—
469C2×D3—61no—
470C2×D3—61no—
471C2×D3—61no—
472C2×D3—61no—
473C2×D3—61no—
474C2×D3—61no—
475C2×D3—64no—
476C2×D3—122no—
477C2×D3—122no—
478C2×D3—122no—
479C2×D3—124no—
480C2×D3—124no—
481C2×D3—122no—
482C2×D3—122no—
483C2×D3—122no—
484C2×D3—122no—
485C2×D3—121no—
486C2×D3—121no—
487C2×D3—121no—
488C2×D3—121no—
489C2×D3—122no—
490C2×D3—122no—
491C3×C4—62no—
492C3×C4—62no—
493C3×C4—122no—
494C3×C4—122no—
495C3×C4—122no—
496C3×C4—124no—
497C3×C4—362no—
498Dic3—22no—
499Dic3—42no—
500C32—21no—
501C32—21no—
502C32—41no—
503C32—41no—
504C32C2×C3×#9911yescharacteristic, complemented
505C23—61no—
506C23—92no—
507C23—91no—
508C23—91no—
509C23—91no—
510C23—91no—
511C23—91no—
512C23—91no—
513C23—181no—
514C23—362no—
515C23—362no—
516C23C3×GO+(4,𝔽2)11yescharacteristic
517D4—62no—
518D4—62no—
519D4—182no—
520D4—182no—
521C2×C4—61no—
522C2×C4—62no—
523C2×C4—92no—
524C2×C4—122no—
525C2×C4—181no—
526C2×C4—182no—
527C2×C4—362no—
528C2×C3—21no—
529C2×C3—21no—
530C2×C3—22no—
531C2×C3—21no—
532C2×C3—21no—
533C2×C3—22no—
534C2×C3—22no—
535C2×C3—21no—
536C2×C3—21no—
537C2×C3—41no—
538C2×C3—41no—
539C2×C3—41no—
540C2×C3—41no—
541C2×C3—42no—
542C2×C3—42no—
543C2×C3—42no—
544C2×C3—41no—
545C2×C3—41no—
546C2×C3—81no—
547C2×C3—81no—
548C2×C3—122no—
549C2×C3—122no—
550C2×C3—181no—
551C2×C3—181no—
552C2×C3—181no—
553C2×C3—181no—
554C2×C3—242no—
555C2×C3#6373812yesdirect product factor
556C2×C3C2×D32⋊C411yescharacteristic
557D3—42no—
558D3—62no—
559D3—62no—
560D3—121no—
561D3—121no—
562D3—121no—
563D3—121no—
564D3—121no—
565D3—121no—
566C22—22no—
567C22—62no—
568C22—91no—
569C22—91no—
570C22—91no—
571C22—91no—
572C22—122no—
573C22—182no—
574C22—182no—
575C22—361no—
576C22—361no—
577C22—361no—
578C22—361no—
579C22—362no—
580C22—362no—
581C22C3×D32⋊C411yescharacteristic
582C22C3×D32⋊C411yescharacteristic
583C22C2×C3×GO+(4,𝔽2)11yescharacteristic
584C4—62no—
585C4—122no—
586C4—362no—
587C3—21no—
588C3—21no—
589C3—41no—
590C3—41no—
591C3C2×#6373811yescharacteristic, direct product factor
592C2—21no—
593C2—21no—
594C2—122no—
595C2—181no—
596C2—181no—
597C2—181no—
598C2—181no—
599C2C3×#6373812yesdirect product factor
600C2C2×C3×D32⋊C411yescharacteristic
subgroupG/H
centerC22×C3D32⋊C4
commutatorC2×C3⋊D3C22×C3×C4
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC23×C33D4
socleC22×C33C22⋊C4
prime radicalC2×C3×#3811C2
cosocleC22×C3⋊D3C23×C3
hypercenterC23×C3GO+(4,𝔽2)
hypocenterC32C2×C3×#99
LCSG ▷ C2×C3⋊D3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
2. C2×C3×C22⋊#157C2semidirect
3. C2×C3×#3722C2semidirect
4. C3×#63738C2direct (N × H)
5. C2×#63738C3direct (N × H)
7. C22×C3×D32C4semidirect
13. C3×#3811C4semidirect
16. C2×C3×#624C4semidirect
18. C3×C22⋊#157C22semidirect
20. C3×#3722C22semidirect
23. C2×C22⋊#157C2×C3semidirect
25. C2×#3722C2×C3semidirect
26. #63738C2×C3direct (N × H)
38. C2×C3×D32C2×C4semidirect
49. C3×#624C2×C4semidirect
55. C22×D32C3×C4semidirect
74. #3811C3×C4semidirect
77. C2×#624C3×C4semidirect
79. C22⋊#157C22×C3semidirect
83. #3722C22×C3semidirect
147. C2×D32C2×C3×C4semidirect
179. #624C2×C3×C4semidirect
183. C2×C33#99semidirect
302. C33C2×#99semidirect
393. C2×C32C3×#99semidirect
504. C32C2×C3×#99semidirect
555. C2×C3#63738direct (N × H)
591. C3C2×#63738direct (N × H)
599. C2C3×#63738direct (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
C2×C3×#37222
C3×#637382
C2×#637383
C2×C3×#999