Finite Group Databasebeta

← C3×#10593399 #13166760

Subgroups — C3×#10593399 #13166760

Order 1728.

684 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 102 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×#3773C211yesmaximal, characteristic, complemented
2C2×C3×D32⋊C4C211yesmaximal, characteristic
3C2×C3×D32⋊C4C211yesmaximal, characteristic
4C3×#63921C211yesmaximal, characteristic
5C3×C4×#624C211yesmaximal, characteristic, complemented
6C3×#63781C211yesmaximal, characteristic, complemented
7C3×#63737C211yesmaximal, characteristic, complemented
8#10593399C311yesmaximal, characteristic, direct product factor
9C22×C3×D32C2211yescharacteristic
10C22×C3×#157C2211yescharacteristic
11C22×C3×#157C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C3×#624C2211yescharacteristic
14C2×C3×#624C2211yescharacteristic
15C2×C3×#624C2211yescharacteristic
16C3×#3773—22no—
17C3×D32⋊C4—22no—
18C3×D32⋊C4C414yescomplemented
19C32×(C2×D3)⋊C4—21no—
20C32×C4×Dic3—21no—
21C3×(C2×D3)⋊Dic3—21no—
22C3×Dic32—21no—
23C2×#3773C2×C311yescharacteristic, complemented
24C2×D32⋊C4C2×C311yescharacteristic
25C2×D32⋊C4C2×C311yescharacteristic
26C3×D3×C22⋊C4—61no—
27C3×C42×D3—61no—
28#63921C2×C311yescharacteristic
29C4×#624C2×C311yescharacteristic, complemented
30#63781C2×C311yescharacteristic, complemented
31#63737C2×C311yescharacteristic, complemented
32C22×C3×C3⋊D3C2311yescharacteristic
33C22×C32×D3—21no—
34C2×C3×D32—21no—
35C2×C3×D32—21no—
36C2×C3×D32—21no—
37C2×C3×D32—21no—
38C2×C3×D32—21no—
39C2×C3×D32C2×C412yes—
40C2×C3×#157—21no—
41C2×C3×#157—21no—
42C2×C3×#157—21no—
43C2×C3×#157—21no—
44C2×C3×#157—21no—
45C2×C3×#157C2×C412yes—
46C2×C33×C4D411yescharacteristic
47C2×C3×C3⋊Dic3D411yescharacteristic
48C3×C4×C3⋊D3—21no—
49C3×C4×C3⋊D3—21no—
50C2×C32×Dic3—21no—
51C2×C32×Dic3—21no—
52C2×C32×Dic3—21no—
53C3×#624—21no—
54C3×#624—21no—
55C3×#624—21no—
56C3×#624—21no—
57C3×#624C2×C412yes—
58C3×#624D412yes—
59C3×#388—91nomaximal
60C23×C3×D3—61no—
61C22×D32C22×C311yescharacteristic
62C22×#157C22×C311yescharacteristic
63C22×#157C22×C311yescharacteristic
64C2×C4×C3⋊D3C22×C311yescharacteristic
65C2×C3×C4×D3—61no—
66C2×C3×C4×D3—61no—
67C2×C3×C4×D3—61no—
68C2×C3×C4×D3—61no—
69C2×C3×C4×D3—61no—
70C2×#624C22×C311yescharacteristic
71C2×#624C22×C311yescharacteristic
72C2×#624C22×C311yescharacteristic
73#3773—22no—
74D32⋊C4—22no—
75D32⋊C4C3×C414yescomplemented
76C32×C22⋊C4—61no—
77C32×C42—61no—
78C3×#217—61no—
79C3×(C2×D3)⋊C4—21no—
80C3×(C2×D3)⋊C4—21no—
81C3×(C2×D3)⋊C4—41no—
82C3×(C2×D3)⋊C4—61no—
83C3×C4×Dic3—21no—
84C3×C4×Dic3—21no—
85C3×C4×Dic3—41no—
86C3×C4×Dic3—61no—
87C3×C4×Dic3—61no—
88(C2×D3)⋊Dic3—21no—
89Dic32—21no—
90C22×C33C2×D411yescharacteristic
91C2×C3×C3⋊D3C22×C411yescharacteristic
92C2×C3×C3⋊D3C2×D411yescharacteristic
93C2×C3×C3⋊D3#4111yescharacteristic
94C2×C3×C3⋊D3#4111yescharacteristic
95C2×C3×C3⋊D3#4111yescharacteristic
96C2×C3×C3⋊D3#4111yescharacteristic
97C2×C32×D3—22no—
98C2×C32×D3—41no—
99C2×C32×D3—41no—
100C3×D32—22no—
101C3×D32—22no—
102C3×D32—41no—
103C3×D32—41no—
104C3×#157—22no—
105C3×#157—41no—
106C33×C4—21no—
107C3×C3⋊Dic3—21no—
108C32×Dic3—22no—
109C32×Dic3—22no—
110C32×Dic3—41no—
111C2×C3×C4⋊C4—91no—
112C2×C3×C22⋊C4—91no—
113C2×C3×C22⋊C4—91no—
114C2×C3×C22⋊C4—91no—
115C2×C3×C42—91no—
116D3×C22⋊C4—61no—
117C42×D3—61no—
118C3×#95—91no—
119C3×#95—91no—
120C23×C32—61no—
121C22×C3⋊D3C23×C311yescharacteristic
122C22×C3×D3—21no—
123C22×C3×D3—21no—
124C22×C3×D3—41no—
125C22×C3×D3—62no—
126C22×C3×D3—61no—
127C22×C3×D3—61no—
128C22×C3×D3—62no—
129C22×C3×D3—121no—
130C22×C3×D3—121no—
131C22×C3×D3—121no—
132C22×C3×D3—121no—
133C2×D32—21no—
134C2×D32—21no—
135C2×D32—21no—
136C2×D32—21no—
137C2×D32—21no—
138C2×D32C2×C3×C412yes—
139C2×#157—21no—
140C2×#157—21no—
141C2×#157—21no—
142C2×#157—21no—
143C2×#157—21no—
144C2×#157C2×C3×C412yes—
145C2×C32×C4—21no—
146C2×C32×C4—21no—
147C2×C32×C4—41no—
148C2×C32×C4—41no—
149C2×C32×C4—61no—
150C2×C32×C4—61no—
151C2×C32×C4—61no—
152C2×C32×C4C3×D411yescharacteristic
153C2×C3⋊Dic3C3×D411yescharacteristic
154C4×C3⋊D3—21no—
155C4×C3⋊D3—21no—
156C2×C3×Dic3—21no—
157C2×C3×Dic3—21no—
158C2×C3×Dic3—21no—
159C2×C3×Dic3—21no—
160C2×C3×Dic3—21no—
161C2×C3×Dic3—21no—
162C2×C3×Dic3—41no—
163C2×C3×Dic3—41no—
164C2×C3×Dic3—41no—
165C2×C3×Dic3—61no—
166C2×C3×Dic3—61no—
167C3×C4×D3—62no—
168C3×C4×D3—62no—
169C3×C4×D3—62no—
170C3×C4×D3—62no—
171C3×C4×D3—62no—
172C3×C4×D3—62no—
173C3×C4×D3—121no—
174C3×C4×D3—121no—
175C3×C4×D3—121no—
176C3×C4×D3—121no—
177#624—21no—
178#624—21no—
179#624—21no—
180#624—21no—
181#624C2×C3×C412yes—
182#624C3×D412yes—
183#388—91noSylow
184C2×C33C4×D411yescharacteristic
185C2×C33#12111yescharacteristic
186C2×C33#12311yescharacteristic
187C3×C3⋊D3#11711yescharacteristic
188C3×C3⋊D3C4×D411yescharacteristic
189C3×C3⋊D3#12411yescharacteristic
190C3×C3⋊D3#12611yescharacteristic
191C32×D3—42no—
192C24×C3—91no—
193C23×D3—61no—
194C22×C3×C4—91no—
195C22×C3×C4—91no—
196C22×C3×C4—91no—
197C22×C3×C4—91no—
198C22×C3×C4—91no—
199C22×C3×C4—91no—
200C2×C4×D3—61no—
201C2×C4×D3—61no—
202C2×C4×D3—61no—
203C2×C4×D3—61no—
204C2×C4×D3—61no—
205C3×C4⋊C4—182no—
206C3×C22⋊C4—61no—
207C3×C22⋊C4—61no—
208C3×C22⋊C4—94no—
209C3×C22⋊C4—121no—
210C3×C22⋊C4—181no—
211C3×C22⋊C4—182no—
212C3×C42—61no—
213C3×C42—61no—
214C3×C42—121no—
215C3×C42—181no—
216#217—61no—
217(C2×D3)⋊C4—21no—
218(C2×D3)⋊C4—61no—
219C4×Dic3—21no—
220C4×Dic3—61no—
221C4×Dic3—61no—
222C22×C32—21no—
223C22×C32—21no—
224C22×C32—41no—
225C22×C32—41no—
226C22×C32—62no—
227C22×C32—121no—
228C22×C32—121no—
229C22×C32C2×C3×D411yescharacteristic
230C2×C3⋊D3C22×C3×C411yescharacteristic
231C2×C3⋊D3C2×C3×D411yescharacteristic
232C2×C3⋊D3C3×#4111yescharacteristic
233C2×C3⋊D3C3×#4111yescharacteristic
234C2×C3⋊D3C3×#4111yescharacteristic
235C2×C3⋊D3C3×#4111yescharacteristic
236C2×C3×D3—22no—
237C2×C3×D3—22no—
238C2×C3×D3—42no—
239C2×C3×D3—41no—
240C2×C3×D3—41no—
241C2×C3×D3—41no—
242C2×C3×D3—41no—
243C2×C3×D3—61no—
244C2×C3×D3—61no—
245C2×C3×D3—61no—
246C2×C3×D3—61no—
247C2×C3×D3—61no—
248C2×C3×D3—61no—
249C2×C3×D3—61no—
250C2×C3×D3—61no—
251C2×C3×D3—61no—
252C2×C3×D3—61no—
253C2×C3×D3—61no—
254C2×C3×D3—61no—
255C2×C3×D3—81no—
256C2×C3×D3—81no—
257C2×C3×D3—122no—
258C2×C3×D3—122no—
259C2×C3×D3—122no—
260C2×C3×D3—122no—
261D32—22no—
262D32—22no—
263D32—41no—
264D32—41no—
265#157—22no—
266#157—41no—
267C32×C4—22no—
268C32×C4—21no—
269C32×C4—41no—
270C32×C4—42no—
271C32×C4—62no—
272C32×C4—62no—
273C32×C4—81no—
274C32×C4—121no—
275C3⋊Dic3—21no—
276C3×Dic3—22no—
277C3×Dic3—22no—
278C3×Dic3—22no—
279C3×Dic3—22no—
280C3×Dic3—42no—
281C3×Dic3—42no—
282C3×Dic3—41no—
283C3×Dic3—41no—
284C3×Dic3—62no—
285C3×Dic3—81no—
286C3×Dic3—121no—
287C2×C4⋊C4—91no—
288C2×C22⋊C4—91no—
289C2×C22⋊C4—91no—
290C2×C22⋊C4—91no—
291C2×C42—91no—
292#95—91no—
293#95—91no—
294C33#38811yescharacteristic, complemented, Sylow
295C23×C3—61no—
296C23×C3—61no—
297C23×C3—92no—
298C23×C3—91no—
299C23×C3—121no—
300C23×C3—181no—
301C23×C3—181no—
302C23×C3—181no—
303C23×C3—181no—
304C23×C3—181no—
305C22×D3—21no—
306C22×D3—62no—
307C22×D3—61no—
308C22×D3—61no—
309C22×D3—62no—
310C22×D3—121no—
311C22×D3—121no—
312C22×D3—121no—
313C22×D3—121no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—41no—
317C2×C3×C4—41no—
318C2×C3×C4—61no—
319C2×C3×C4—61no—
320C2×C3×C4—61no—
321C2×C3×C4—61no—
322C2×C3×C4—61no—
323C2×C3×C4—61no—
324C2×C3×C4—92no—
325C2×C3×C4—92no—
326C2×C3×C4—92no—
327C2×C3×C4—91no—
328C2×C3×C4—121no—
329C2×C3×C4—121no—
330C2×C3×C4—121no—
331C2×C3×C4—181no—
332C2×C3×C4—181no—
333C2×C3×C4—181no—
334C2×C3×C4—181no—
335C2×C3×C4—181no—
336C2×C3×C4—181no—
337C2×C3×C4—181no—
338C2×C3×C4—181no—
339C2×C3×C4—181no—
340C2×C3×C4—181no—
341C2×C3×C4—181no—
342C2×C3×C4GO+(4,𝔽2)11yescharacteristic
343C2×Dic3—21no—
344C2×Dic3—21no—
345C2×Dic3—21no—
346C2×Dic3—61no—
347C2×Dic3—61no—
348C4×D3—62no—
349C4×D3—62no—
350C4×D3—62no—
351C4×D3—62no—
352C4×D3—62no—
353C4×D3—62no—
354C4×D3—121no—
355C4×D3—121no—
356C4×D3—121no—
357C4×D3—121no—
358C2×C32—21no—
359C2×C32—21no—
360C2×C32—21no—
361C2×C32—21no—
362C2×C32—21no—
363C2×C32—21no—
364C2×C32—41no—
365C2×C32—41no—
366C2×C32—41no—
367C2×C32—41no—
368C2×C32—41no—
369C2×C32—41no—
370C2×C32—122no—
371C2×C32C3×C4×D411yescharacteristic
372C2×C32C3×#12111yescharacteristic
373C2×C32C3×#12311yescharacteristic
374C3⋊D3C3×#11711yescharacteristic
375C3⋊D3C3×C4×D411yescharacteristic
376C3⋊D3C3×#12411yescharacteristic
377C3⋊D3C3×#12611yescharacteristic
378C3×D3—42no—
379C3×D3—42no—
380C3×D3—61no—
381C3×D3—61no—
382C3×D3—61no—
383C3×D3—61no—
384C3×D3—61no—
385C3×D3—61no—
386C3×D3—61no—
387C3×D3—61no—
388C3×D3—82no—
389C24—91no—
390C22×C4—91no—
391C22×C4—91no—
392C22×C4—91no—
393C22×C4—91no—
394C22×C4—91no—
395C22×C4—91no—
396C4⋊C4—182no—
397C22⋊C4—61no—
398C22⋊C4—94no—
399C22⋊C4—181no—
400C22⋊C4—182no—
401C42—61no—
402C42—181no—
403C22×C3—21no—
404C22×C3—21no—
405C22×C3—41no—
406C22×C3—41no—
407C22×C3—62no—
408C22×C3—62no—
409C22×C3—91no—
410C22×C3—91no—
411C22×C3—91no—
412C22×C3—91no—
413C22×C3—91no—
414C22×C3—91no—
415C22×C3—122no—
416C22×C3—121no—
417C22×C3—121no—
418C22×C3—121no—
419C22×C3—121no—
420C22×C3—182no—
421C22×C3—182no—
422C22×C3—241no—
423C22×C3—241no—
424C22×C3—361no—
425C22×C3—361no—
426C22×C3C2×GO+(4,𝔽2)11yescharacteristic
427C2×D3—22no—
428C2×D3—41no—
429C2×D3—41no—
430C2×D3—61no—
431C2×D3—61no—
432C2×D3—61no—
433C2×D3—61no—
434C2×D3—61no—
435C2×D3—61no—
436C2×D3—61no—
437C2×D3—61no—
438C2×D3—61no—
439C2×D3—61no—
440C2×D3—61no—
441C2×D3—61no—
442C2×D3—122no—
443C2×D3—122no—
444C2×D3—122no—
445C2×D3—122no—
446C3×C4—22no—
447C3×C4—21no—
448C3×C4—41no—
449C3×C4—42no—
450C3×C4—62no—
451C3×C4—62no—
452C3×C4—62no—
453C3×C4—62no—
454C3×C4—81no—
455C3×C4—122no—
456C3×C4—122no—
457C3×C4—121no—
458C3×C4—121no—
459C3×C4—182no—
460C3×C4—181no—
461C3×C4—241no—
462C3×C4—361no—
463Dic3—22no—
464Dic3—22no—
465Dic3—41no—
466Dic3—62no—
467Dic3—121no—
468C32—21no—
469C32—21no—
470C32—41no—
471C32—41no—
472C32C3×#38811yescharacteristic, complemented
473C23—61no—
474C23—92no—
475C23—91no—
476C23—181no—
477C23—181no—
478C23—181no—
479C23—181no—
480C23—181no—
481C2×C4—61no—
482C2×C4—61no—
483C2×C4—61no—
484C2×C4—92no—
485C2×C4—92no—
486C2×C4—92no—
487C2×C4—91no—
488C2×C4—181no—
489C2×C4—181no—
490C2×C4—181no—
491C2×C4—181no—
492C2×C4—181no—
493C2×C4—181no—
494C2×C4—181no—
495C2×C4—181no—
496C2×C4—181no—
497C2×C4—181no—
498C2×C4—181no—
499C2×C4C3×GO+(4,𝔽2)11yescharacteristic
500C2×C3—21no—
501C2×C3—21no—
502C2×C3—21no—
503C2×C3—21no—
504C2×C3—21no—
505C2×C3—21no—
506C2×C3—41no—
507C2×C3—41no—
508C2×C3—41no—
509C2×C3—41no—
510C2×C3—41no—
511C2×C3—41no—
512C2×C3—91no—
513C2×C3—91no—
514C2×C3—91no—
515C2×C3—91no—
516C2×C3—122no—
517C2×C3—122no—
518C2×C3—242no—
519C2×C3C4×GO+(4,𝔽2)11yescharacteristic
520C2×C3#6423011yescharacteristic
521C2×C3#6423311yescharacteristic
522D3—42no—
523D3—61no—
524D3—61no—
525D3—61no—
526D3—61no—
527D3—61no—
528D3—61no—
529D3—61no—
530D3—61no—
531C22—62no—
532C22—91no—
533C22—91no—
534C22—91no—
535C22—91no—
536C22—91no—
537C22—91no—
538C22—121no—
539C22—121no—
540C22—182no—
541C22—182no—
542C22—361no—
543C22—361no—
544C22C2×C3×GO+(4,𝔽2)11yescharacteristic
545C4—21no—
546C4—62no—
547C4—62no—
548C4—121no—
549C4—182no—
550C4—181no—
551C4—361no—
552C3—21no—
553C3—21no—
554C3—41no—
555C3—41no—
556C3#1059339911yescharacteristic, direct product factor
557C2—91no—
558C2—91no—
559C2—91no—
560C2—91no—
561C2—122no—
562C2C3×C4×GO+(4,𝔽2)11yescharacteristic
563C2C3×#6423011yescharacteristic
564C2C3×#6423311yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutatorC2×C3⋊D3C22×C3×C4
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC22×C3×D32C22
cosocleC22×C3⋊D3C23×C3
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#388
LCSG ▷ C2×C3⋊D3 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3773C2semidirect
5. C3×C4×#624C2semidirect
6. C3×#63781C2semidirect
7. C3×#63737C2semidirect
8. #10593399C3direct (N × H)
18. C3×D32⋊C4C4semidirect
23. C2×#3773C2×C3semidirect
29. C4×#624C2×C3semidirect
30. #63781C2×C3semidirect
31. #63737C2×C3semidirect
75. D32⋊C4C3×C4semidirect
294. C33#388semidirect
472. C32C3×#388semidirect
556. C3#10593399direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37732
C2×C3×D32⋊C42
C3×#639212
C3×C4×#6242
C3×#637812
C3×#637372
#105933993
C3×#3889