Finite Group Databasebeta

← C3×#10593406 #13166767

Subgroups — C3×#10593406 #13166767

Order 1728.

656 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 90 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×C4⋊#157C211yesmaximal, characteristic, complemented
2C2×C3×#3773C211yesmaximal, characteristic, complemented
3C2×C3×D32⋊C4C211yesmaximal, characteristic
4C2×C3×D32⋊C4C211yesmaximal, characteristic
5C3×#63921C211yesmaximal, characteristic
6C3×#63883C211yesmaximal, characteristic, complemented
7C3×#63737C211yesmaximal, characteristic, complemented
8#10593406C311yesmaximal, 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×C4⋊#157—22no—
17C3×#3773—22no—
18C3×D32⋊C4—22no—
19C3×D32⋊C4—22no—
20C32×(C2×D3)⋊C4—21no—
21C32×Dic3⋊C4—21no—
22C3×Dic3⋊Dic3—21no—
23C3×(C2×D3)⋊Dic3—21no—
24C2×C4⋊#157C2×C311yescharacteristic, complemented
25C2×#3773C2×C311yescharacteristic, complemented
26C2×D32⋊C4C2×C311yescharacteristic
27C2×D32⋊C4C2×C311yescharacteristic
28C3×D3×C4⋊C4—61no—
29C3×D3×C22⋊C4—61no—
30#63921C2×C311yescharacteristic
31#63883C2×C311yescharacteristic, complemented
32#63737C2×C311yescharacteristic, complemented
33C22×C3×C3⋊D3C2311yescharacteristic
34C22×C32×D3—21no—
35C2×C3×D32—21no—
36C2×C3×D32—21no—
37C2×C3×D32—21no—
38C2×C3×D32—21no—
39C2×C3×D32—21no—
40C2×C3×D32Q812yes—
41C2×C3×#157—21no—
42C2×C3×#157—21no—
43C2×C3×#157—21no—
44C2×C3×#157—21no—
45C2×C3×#157—21no—
46C2×C3×#157D412yes—
47C2×C33×C4D411yescharacteristic
48C2×C3×C3⋊Dic3D411yescharacteristic
49C3×C4×C3⋊D3—21no—
50C3×C4×C3⋊D3—21no—
51C2×C32×Dic3—21no—
52C2×C32×Dic3—21no—
53C2×C32×Dic3—21no—
54C3×#624—21no—
55C3×#624—21no—
56C3×#624—21no—
57C3×#624—21no—
58C3×#624—21no—
59C3×#624D412yes—
60C3×#396—91nomaximal
61C23×C3×D3—61no—
62C22×D32C22×C311yescharacteristic
63C22×#157C22×C311yescharacteristic
64C22×#157C22×C311yescharacteristic
65C2×C4×C3⋊D3C22×C311yescharacteristic
66C2×C3×C4×D3—61no—
67C2×C3×C4×D3—61no—
68C2×C3×C4×D3—61no—
69C2×C3×C4×D3—61no—
70C2×C3×C4×D3—61no—
71C2×#624C22×C311yescharacteristic
72C2×#624C22×C311yescharacteristic
73C2×#624C22×C311yescharacteristic
74C4⋊#157—22no—
75#3773—22no—
76D32⋊C4—22no—
77D32⋊C4—22no—
78C32×C4⋊C4—61no—
79C32×C22⋊C4—61no—
80C3×#217—61no—
81C3×(C2×D3)⋊C4—21no—
82C3×(C2×D3)⋊C4—21no—
83C3×(C2×D3)⋊C4—41no—
84C3×(C2×D3)⋊C4—61no—
85C3×C4⋊Dic3—61no—
86C3×Dic3⋊C4—21no—
87C3×Dic3⋊C4—21no—
88C3×Dic3⋊C4—41no—
89C3×Dic3⋊C4—61no—
90Dic3⋊Dic3—21no—
91(C2×D3)⋊Dic3—21no—
92C22×C33C2×D411yescharacteristic
93C2×C3×C3⋊D3C2×D411yescharacteristic
94C2×C3×C3⋊D3C2×D411yescharacteristic
95C2×C3×C3⋊D3C2×Q811yescharacteristic
96C2×C3×C3⋊D3#4111yescharacteristic
97C2×C3×C3⋊D3#4111yescharacteristic
98C2×C3×C3⋊D3#4111yescharacteristic
99C2×C32×D3—22no—
100C2×C32×D3—41no—
101C2×C32×D3—41no—
102C3×D32—22no—
103C3×D32—22no—
104C3×D32—41no—
105C3×D32—41no—
106C3×#157—22no—
107C3×#157—41no—
108C33×C4—21no—
109C3×C3⋊Dic3—21no—
110C32×Dic3—22no—
111C32×Dic3—41no—
112C32×Dic3—41no—
113C2×C3×C4⋊C4—91no—
114C2×C3×C4⋊C4—91no—
115C2×C3×C4⋊C4—91no—
116C2×C3×C22⋊C4—91no—
117C2×C3×C22⋊C4—91no—
118C2×C3×C22⋊C4—91no—
119D3×C4⋊C4—61no—
120D3×C22⋊C4—61no—
121C3×#95—91no—
122C23×C32—61no—
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—62no—
131C22×C3×D3—121no—
132C22×C3×D3—121no—
133C22×C3×D3—121no—
134C22×C3×D3—121no—
135C2×D32—21no—
136C2×D32—21no—
137C2×D32—21no—
138C2×D32—21no—
139C2×D32—21no—
140C2×D32C3×Q812yes—
141C2×#157—21no—
142C2×#157—21no—
143C2×#157—21no—
144C2×#157—21no—
145C2×#157—21no—
146C2×#157C3×D412yes—
147C2×C32×C4—21no—
148C2×C32×C4—21no—
149C2×C32×C4—41no—
150C2×C32×C4—41no—
151C2×C32×C4—61no—
152C2×C32×C4—61no—
153C2×C32×C4—61no—
154C2×C32×C4C3×D411yescharacteristic
155C2×C3⋊Dic3C3×D411yescharacteristic
156C4×C3⋊D3—21no—
157C4×C3⋊D3—21no—
158C2×C3×Dic3—21no—
159C2×C3×Dic3—21no—
160C2×C3×Dic3—21no—
161C2×C3×Dic3—21no—
162C2×C3×Dic3—21no—
163C2×C3×Dic3—21no—
164C2×C3×Dic3—41no—
165C2×C3×Dic3—41no—
166C2×C3×Dic3—41no—
167C2×C3×Dic3—61no—
168C2×C3×Dic3—61no—
169C3×C4×D3—62no—
170C3×C4×D3—62no—
171C3×C4×D3—121no—
172C3×C4×D3—121no—
173C3×C4×D3—121no—
174C3×C4×D3—121no—
175C3×C4×D3—121no—
176C3×C4×D3—121no—
177C3×C4×D3—121no—
178C3×C4×D3—121no—
179#624—21no—
180#624—21no—
181#624—21no—
182#624—21no—
183#624—21no—
184#624C3×D412yes—
185#396—91noSylow
186C2×C33#12111yescharacteristic
187C2×C33#12111yescharacteristic
188C2×C33C22⋊Q811yescharacteristic
189C3×C3⋊D3#12111yescharacteristic
190C3×C3⋊D3C22⋊Q811yescharacteristic
191C3×C3⋊D3C22⋊Q811yescharacteristic
192C3×C3⋊D3#12611yescharacteristic
193C32×D3—42no—
194C24×C3—91no—
195C23×D3—61no—
196C22×C3×C4—91no—
197C22×C3×C4—91no—
198C22×C3×C4—91no—
199C22×C3×C4—91no—
200C22×C3×C4—91no—
201C22×C3×C4—91no—
202C2×C4×D3—61no—
203C2×C4×D3—61no—
204C2×C4×D3—61no—
205C2×C4×D3—61no—
206C2×C4×D3—61no—
207C3×C4⋊C4—61no—
208C3×C4⋊C4—61no—
209C3×C4⋊C4—121no—
210C3×C4⋊C4—181no—
211C3×C4⋊C4—182no—
212C3×C4⋊C4—182no—
213C3×C22⋊C4—61no—
214C3×C22⋊C4—61no—
215C3×C22⋊C4—121no—
216C3×C22⋊C4—181no—
217C3×C22⋊C4—182no—
218C3×C22⋊C4—182no—
219#217—61no—
220(C2×D3)⋊C4—21no—
221(C2×D3)⋊C4—61no—
222C4⋊Dic3—61no—
223Dic3⋊C4—21no—
224Dic3⋊C4—61no—
225C22×C32—21no—
226C22×C32—21no—
227C22×C32—41no—
228C22×C32—41no—
229C22×C32—62no—
230C22×C32—121no—
231C22×C32—121no—
232C22×C32C2×C3×D411yescharacteristic
233C2×C3⋊D3C2×C3×D411yescharacteristic
234C2×C3⋊D3C2×C3×D411yescharacteristic
235C2×C3⋊D3C2×C3×Q811yescharacteristic
236C2×C3⋊D3C3×#4111yescharacteristic
237C2×C3⋊D3C3×#4111yescharacteristic
238C2×C3⋊D3C3×#4111yescharacteristic
239C2×C3×D3—22no—
240C2×C3×D3—22no—
241C2×C3×D3—42no—
242C2×C3×D3—41no—
243C2×C3×D3—41no—
244C2×C3×D3—41no—
245C2×C3×D3—41no—
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—61no—
256C2×C3×D3—61no—
257C2×C3×D3—61no—
258C2×C3×D3—81no—
259C2×C3×D3—81no—
260C2×C3×D3—122no—
261C2×C3×D3—122no—
262C2×C3×D3—122no—
263C2×C3×D3—122no—
264D32—22no—
265D32—22no—
266D32—41no—
267D32—41no—
268#157—22no—
269#157—41no—
270C32×C4—21no—
271C32×C4—41no—
272C32×C4—41no—
273C32×C4—62no—
274C32×C4—81no—
275C32×C4—81no—
276C32×C4—121no—
277C32×C4—121no—
278C3⋊Dic3—21no—
279C3×Dic3—22no—
280C3×Dic3—22no—
281C3×Dic3—42no—
282C3×Dic3—41no—
283C3×Dic3—41no—
284C3×Dic3—41no—
285C3×Dic3—41no—
286C3×Dic3—81no—
287C3×Dic3—81no—
288C3×Dic3—121no—
289C3×Dic3—121no—
290C2×C4⋊C4—91no—
291C2×C4⋊C4—91no—
292C2×C4⋊C4—91no—
293C2×C22⋊C4—91no—
294C2×C22⋊C4—91no—
295C2×C22⋊C4—91no—
296#95—91no—
297C33#39611yescharacteristic, complemented, Sylow
298C23×C3—61no—
299C23×C3—61no—
300C23×C3—92no—
301C23×C3—91no—
302C23×C3—121no—
303C23×C3—181no—
304C23×C3—181no—
305C23×C3—181no—
306C23×C3—181no—
307C23×C3—181no—
308C22×D3—21no—
309C22×D3—62no—
310C22×D3—61no—
311C22×D3—61no—
312C22×D3—62no—
313C22×D3—121no—
314C22×D3—121no—
315C22×D3—121no—
316C22×D3—121no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—41no—
320C2×C3×C4—41no—
321C2×C3×C4—61no—
322C2×C3×C4—61no—
323C2×C3×C4—61no—
324C2×C3×C4—61no—
325C2×C3×C4—61no—
326C2×C3×C4—61no—
327C2×C3×C4—92no—
328C2×C3×C4—92no—
329C2×C3×C4—91no—
330C2×C3×C4—121no—
331C2×C3×C4—121no—
332C2×C3×C4—121no—
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×C4—181no—
343C2×C3×C4—181no—
344C2×C3×C4—181no—
345C2×C3×C4GO+(4,𝔽2)11yescharacteristic
346C2×Dic3—21no—
347C2×Dic3—21no—
348C2×Dic3—21no—
349C2×Dic3—61no—
350C2×Dic3—61no—
351C4×D3—62no—
352C4×D3—62no—
353C4×D3—121no—
354C4×D3—121no—
355C4×D3—121no—
356C4×D3—121no—
357C4×D3—121no—
358C4×D3—121no—
359C4×D3—121no—
360C4×D3—121no—
361C2×C32—21no—
362C2×C32—21no—
363C2×C32—21no—
364C2×C32—21no—
365C2×C32—21no—
366C2×C32—21no—
367C2×C32—41no—
368C2×C32—41no—
369C2×C32—41no—
370C2×C32—41no—
371C2×C32—41no—
372C2×C32—41no—
373C2×C32—122no—
374C2×C32C3×#12111yescharacteristic
375C2×C32C3×#12111yescharacteristic
376C2×C32C3×C22⋊Q811yescharacteristic
377C3⋊D3C3×#12111yescharacteristic
378C3⋊D3C3×C22⋊Q811yescharacteristic
379C3⋊D3C3×C22⋊Q811yescharacteristic
380C3⋊D3C3×#12611yescharacteristic
381C3×D3—42no—
382C3×D3—42no—
383C3×D3—61no—
384C3×D3—61no—
385C3×D3—61no—
386C3×D3—61no—
387C3×D3—61no—
388C3×D3—61no—
389C3×D3—61no—
390C3×D3—61no—
391C3×D3—82no—
392C24—91no—
393C22×C4—91no—
394C22×C4—91no—
395C22×C4—91no—
396C22×C4—91no—
397C22×C4—91no—
398C22×C4—91no—
399C4⋊C4—61no—
400C4⋊C4—181no—
401C4⋊C4—182no—
402C4⋊C4—182no—
403C22⋊C4—61no—
404C22⋊C4—181no—
405C22⋊C4—182no—
406C22⋊C4—182no—
407C22×C3—21no—
408C22×C3—21no—
409C22×C3—41no—
410C22×C3—41no—
411C22×C3—62no—
412C22×C3—62no—
413C22×C3—91no—
414C22×C3—91no—
415C22×C3—91no—
416C22×C3—91no—
417C22×C3—91no—
418C22×C3—91no—
419C22×C3—122no—
420C22×C3—121no—
421C22×C3—121no—
422C22×C3—121no—
423C22×C3—121no—
424C22×C3—182no—
425C22×C3—182no—
426C22×C3—241no—
427C22×C3—241no—
428C22×C3—361no—
429C22×C3—361no—
430C22×C3C2×GO+(4,𝔽2)11yescharacteristic
431C2×D3—22no—
432C2×D3—41no—
433C2×D3—41no—
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—61no—
443C2×D3—61no—
444C2×D3—61no—
445C2×D3—61no—
446C2×D3—122no—
447C2×D3—122no—
448C2×D3—122no—
449C2×D3—122no—
450C3×C4—21no—
451C3×C4—41no—
452C3×C4—41no—
453C3×C4—62no—
454C3×C4—62no—
455C3×C4—81no—
456C3×C4—81no—
457C3×C4—122no—
458C3×C4—121no—
459C3×C4—121no—
460C3×C4—121no—
461C3×C4—121no—
462C3×C4—182no—
463C3×C4—181no—
464C3×C4—241no—
465C3×C4—241no—
466C3×C4—361no—
467Dic3—22no—
468Dic3—41no—
469Dic3—41no—
470Dic3—121no—
471Dic3—121no—
472C32—21no—
473C32—21no—
474C32—41no—
475C32—41no—
476C32C3×#39611yescharacteristic, complemented
477C23—61no—
478C23—92no—
479C23—91no—
480C23—181no—
481C23—181no—
482C23—181no—
483C23—181no—
484C23—181no—
485C2×C4—61no—
486C2×C4—61no—
487C2×C4—61no—
488C2×C4—92no—
489C2×C4—92no—
490C2×C4—91no—
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×C4—181no—
500C2×C4—181no—
501C2×C4—181no—
502C2×C4—181no—
503C2×C4C3×GO+(4,𝔽2)11yescharacteristic
504C2×C3—21no—
505C2×C3—21no—
506C2×C3—21no—
507C2×C3—21no—
508C2×C3—21no—
509C2×C3—21no—
510C2×C3—41no—
511C2×C3—41no—
512C2×C3—41no—
513C2×C3—41no—
514C2×C3—41no—
515C2×C3—41no—
516C2×C3—91no—
517C2×C3—91no—
518C2×C3—91no—
519C2×C3—91no—
520C2×C3—122no—
521C2×C3—122no—
522C2×C3—242no—
523C2×C3D32⋊Q811yescharacteristic
524C2×C3#6423011yescharacteristic
525C2×C3#6424211yescharacteristic
526D3—42no—
527D3—61no—
528D3—61no—
529D3—61no—
530D3—61no—
531D3—61no—
532D3—61no—
533D3—61no—
534D3—61no—
535C22—62no—
536C22—91no—
537C22—91no—
538C22—91no—
539C22—91no—
540C22—91no—
541C22—91no—
542C22—121no—
543C22—121no—
544C22—182no—
545C22—182no—
546C22—361no—
547C22—361no—
548C22C2×C3×GO+(4,𝔽2)11yescharacteristic
549C4—21no—
550C4—62no—
551C4—121no—
552C4—121no—
553C4—182no—
554C4—181no—
555C4—361no—
556C3—21no—
557C3—21no—
558C3—41no—
559C3—41no—
560C3#1059340611yescharacteristic, direct product factor
561C2—91no—
562C2—91no—
563C2—91no—
564C2—91no—
565C2—122no—
566C2C3×D32⋊Q811yescharacteristic
567C2C3×#6423011yescharacteristic
568C2C3×#6424211yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC22×C3⋊D3C23×C3
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC22×C3×D32C22
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#396
LCSG ▷ C22×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×C4⋊#157C2semidirect
2. C2×C3×#3773C2semidirect
6. C3×#63883C2semidirect
7. C3×#63737C2semidirect
8. #10593406C3direct (N × H)
24. C2×C4⋊#157C2×C3semidirect
25. C2×#3773C2×C3semidirect
31. #63883C2×C3semidirect
32. #63737C2×C3semidirect
297. C33#396semidirect
476. C32C3×#396semidirect
560. C3#10593406direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4⋊#1572
C2×C3×#37732
C2×C3×D32⋊C42
C3×#639212
C3×#638832
C3×#637372
#105934063
C3×#3969