Finite Group Databasebeta

← C3×#10593421 #13166783

Subgroups — C3×#10593421 #13166783

Order 1728.

670 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 670 conjugacy classes of subgroups — too many to draw. Only its 70 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×#3798C211yesmaximal, characteristic, complemented
2C2×C3×#3775C211yesmaximal, characteristic, complemented
3C2×C3×#3774C211yesmaximal, characteristic, complemented
4C3×#63912C211yesmaximal, characteristic, complemented
5C3×#63778C211yesmaximal, characteristic, complemented
6C3×#63740C211yesmaximal, characteristic, complemented
7C3×#63739C211yesmaximal, characteristic, complemented
8#10593421C311yesmaximal, characteristic, direct product factor
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×C32×C4×D3—21no—
11C2×C3×#627C2211yescharacteristic
12C2×C3×#626—21no—
13C2×C3×#626—21no—
14C2×C3×#625C2211yescharacteristic
15C2×C3×#625C2211yescharacteristic
16C3×#3798—22no—
17C3×#3798—22no—
18C2×C3×#622C2211yescharacteristic
19C2×C3×#622C2211yescharacteristic
20C3×#3775—22no—
21C3×#3774—22no—
22C32×(C2×D3)⋊C4—21no—
23C3×#3724C2211yescharacteristic
24C2×C3×#235—61no—
25C2×#3798C2×C311yescharacteristic, complemented
26C2×#3775C2×C311yescharacteristic, complemented
27C2×#3774C2×C311yescharacteristic, complemented
28C3×#884—61no—
29#63912C2×C311yescharacteristic, complemented
30#63778C2×C311yescharacteristic, complemented
31#63740C2×C311yescharacteristic, complemented
32#63739C2×C311yescharacteristic, complemented
33C22×C3×C3⋊D3D411yescharacteristic
34C22×C32×D3—21no—
35C22×C32×D3—21no—
36C2×C33×C4D411yescharacteristic
37C2×C3×C3⋊Dic3C2311yescharacteristic
38C3×C4×C3⋊D3—21no—
39C3×C4×C3⋊D3—21no—
40C2×C32×Dic3—21no—
41C2×C32×Dic3—21no—
42C32×C4×D3—24no—
43C3×#627—21no—
44C3×#627D412yescomplemented
45C3×#626—24no—
46C3×#626—42no—
47C3×#625—21no—
48C3×#625—21no—
49C3×#625—21no—
50C3×#625D412yescomplemented
51C3×#622—21no—
52C3×#622—21no—
53C3×#449—91nomaximal
54C22×C32×C4—61no—
55C2×C4×C3⋊D3C22×C311yescharacteristic
56C2×C3×#67—61no—
57C2×C3×#67—61no—
58C2×C3×#67—61no—
59C2×C3×#67—61no—
60C3×#235—64no—
61C3×#235—64no—
62C2×C3×D12—61no—
63C2×C3×D12—61no—
64C2×C3×C4×D3—21no—
65C2×C3×C4×D3—21no—
66C2×C3×C4×D3—41no—
67C2×C3×C4×D3—61no—
68C2×C3×C4×D3—61no—
69C2×C3×Dic6—61no—
70C2×#627C22×C311yescharacteristic
71C2×#626—21no—
72C2×#626—21no—
73C2×#625C22×C311yescharacteristic
74C2×#625C22×C311yescharacteristic
75#3798—22no—
76#3798—22no—
77C2×#622C22×C311yescharacteristic
78C2×#622C22×C311yescharacteristic
79#3775—22no—
80#3774—22no—
81C32×C22⋊C4—61no—
82C3×(C2×D3)⋊C4—21no—
83C3×(C2×D3)⋊C4—21no—
84C3×(C2×D3)⋊C4—41no—
85C3×Dic3⋊C4—61no—
86#3724C22×C311yescharacteristic
87C22×C33C2×D411yescharacteristic
88C2×C3×C3⋊D3—21no—
89C2×C3×C3⋊D3—21no—
90C2×C3×C3⋊D3—21no—
91C2×C32×D3—22no—
92C2×C32×D3—41no—
93C2×C32×D3—41no—
94C2×C32×D3—41no—
95C2×C32×D3—41no—
96C2×C32×D3—41no—
97C33×C4—21no—
98C3×C3⋊Dic3C2×D411yescharacteristic
99C3×C3⋊Dic3C2×D411yescharacteristic
100C32×Dic3—22no—
101C32×Dic3—41no—
102C2×C3×#41—91no—
103C2×#235—61no—
104C2×C3×SD16—91no—
105C2×C3×D8—91no—
106C3×#121—91no—
107#884—61no—
108C3×#103—91no—
109C3×#102—91no—
110C3×C22⋊C8—91no—
111C23×C32—61no—
112C23×C32—61no—
113C22×C3⋊D3C3×D411yescharacteristic
114C22×C3×D3—21no—
115C22×C3×D3—21no—
116C22×C3×D3—21no—
117C22×C3×D3—21no—
118C22×C3×D3—41no—
119C22×C3×D3—41no—
120C22×C3×D3—61no—
121C22×C3×D3—61no—
122C2×C32×C4—21no—
123C2×C32×C4—21no—
124C2×C32×C4—41no—
125C2×C32×C4—41no—
126C2×C32×C4—64no—
127C2×C32×C4—61no—
128C2×C32×C4—61no—
129C2×C32×C4C3×D411yescharacteristic
130C2×C3⋊Dic3C23×C311yescharacteristic
131C4×C3⋊D3—21no—
132C4×C3⋊D3—21no—
133C3×#67—62no—
134C3×#67—64no—
135C3×#67—62no—
136C3×#67—121no—
137C3×#67—121no—
138C3×#67—122no—
139C2×C3×Dic3—21no—
140C2×C3×Dic3—21no—
141C2×C3×Dic3—21no—
142C2×C3×Dic3—21no—
143C2×C3×Dic3—41no—
144C2×C3×Dic3—41no—
145C2×C3×Dic3—61no—
146C2×C3×Dic3—61no—
147C3×D12—64no—
148C3×D12—122no—
149C3×C4×D3—24no—
150C3×C4×D3—24no—
151C3×C4×D3—44no—
152C3×C4×D3—62no—
153C3×C4×D3—62no—
154C3×C4×D3—121no—
155C3×C4×D3—121no—
156C3×Dic6—62no—
157C3×Dic6—62no—
158#627—21no—
159#627C3×D412yescomplemented
160#626—24no—
161#626—42no—
162#625—21no—
163#625—21no—
164#625—21no—
165#625C3×D412yescomplemented
166#622—21no—
167#622—21no—
168#449—91noSylow
169C2×C33#12011yescharacteristic
170C2×C33#13511yescharacteristic
171C2×C33AGL(1,ℤ/8)11yescharacteristic
172C3×C3⋊D3—41no—
173C32×D3—42no—
174C32×D3—81no—
175C3×#41—182no—
176C3×#41—182no—
177C2×C3×Q8—91no—
178C2×C3×D4—91no—
179C2×C3×D4—91no—
180C2×C3×D4—181no—
181C2×C3×D4—181no—
182C22×C3×C4—61no—
183C22×C3×C4—61no—
184C22×C3×C4—91no—
185C22×C3×C4—121no—
186C2×#67—61no—
187C2×#67—61no—
188C2×#67—61no—
189C2×#67—61no—
190#235—64no—
191#235—64no—
192C2×D12—61no—
193C2×D12—61no—
194C2×C4×D3—21no—
195C2×C4×D3—61no—
196C2×C4×D3—61no—
197C2×Dic6—61no—
198C3×SD16—182no—
199C3×D8—182no—
200C2×C3×C8—91no—
201C2×C3×C8—91no—
202C3×C4⋊C4—91no—
203C3×C22⋊C4—61no—
204C3×C22⋊C4—61no—
205C3×C22⋊C4—121no—
206(C2×D3)⋊C4—21no—
207Dic3⋊C4—61no—
208C22×C32—21no—
209C22×C32—21no—
210C22×C32—41no—
211C22×C32—41no—
212C22×C32—62no—
213C22×C32—121no—
214C22×C32—121no—
215C22×C32—121no—
216C22×C32—121no—
217C22×C32—121no—
218C22×C32C2×C3×D411yescharacteristic
219C2×C3⋊D3—21no—
220C2×C3⋊D3—21no—
221C2×C3⋊D3—21no—
222C2×C3×D3—22no—
223C2×C3×D3—22no—
224C2×C3×D3—42no—
225C2×C3×D3—41no—
226C2×C3×D3—41no—
227C2×C3×D3—42no—
228C2×C3×D3—42no—
229C2×C3×D3—41no—
230C2×C3×D3—41no—
231C2×C3×D3—41no—
232C2×C3×D3—41no—
233C2×C3×D3—41no—
234C2×C3×D3—42no—
235C2×C3×D3—41no—
236C2×C3×D3—41no—
237C2×C3×D3—41no—
238C2×C3×D3—62no—
239C2×C3×D3—62no—
240C2×C3×D3—81no—
241C2×C3×D3—81no—
242C2×C3×D3—121no—
243C2×C3×D3—121no—
244C2×C3×D3—121no—
245C2×C3×D3—121no—
246C32×C4—22no—
247C32×C4—21no—
248C32×C4—41no—
249C32×C4—42no—
250C32×C4—62no—
251C32×C4—81no—
252C32×C4—121no—
253C3⋊Dic3C2×C3×D411yescharacteristic
254C3⋊Dic3C2×C3×D411yescharacteristic
255C3×Dic3—22no—
256C3×Dic3—22no—
257C3×Dic3—42no—
258C3×Dic3—41no—
259C3×Dic3—41no—
260C3×Dic3—61no—
261C3×Dic3—61no—
262C3×Dic3—61no—
263C3×Dic3—61no—
264C3×Dic3—81no—
265C2×#41—91no—
266C2×SD16—91no—
267C2×D8—91no—
268#121—91no—
269#103—91no—
270#102—91no—
271C22⋊C8—91no—
272C33#44911yescharacteristic, complemented, Sylow
273C23×C3—61no—
274C23×C3—61no—
275C23×C3—61no—
276C23×C3—61no—
277C23×C3—91no—
278C23×C3—121no—
279C23×C3—121no—
280C22×D3—21no—
281C22×D3—21no—
282C22×D3—61no—
283C22×D3—61no—
284C3×Q8—92no—
285C3×Q8—181no—
286C3×D4—92no—
287C3×D4—184no—
288C3×D4—181no—
289C3×D4—181no—
290C3×D4—181no—
291C3×D4—362no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—41no—
295C2×C3×C4—41no—
296C2×C3×C4—64no—
297C2×C3×C4—64no—
298C2×C3×C4—61no—
299C2×C3×C4—61no—
300C2×C3×C4—61no—
301C2×C3×C4—61no—
302C2×C3×C4—91no—
303C2×C3×C4—124no—
304C2×C3×C4—121no—
305C2×C3×C4—121no—
306C2×C3×C4—181no—
307C2×C3×C4—181no—
308C2×C3×C4GO+(4,𝔽2)11yescharacteristic
309#67—62no—
310#67—64no—
311#67—62no—
312#67—121no—
313#67—121no—
314#67—122no—
315C2×Dic3—21no—
316C2×Dic3—21no—
317C2×Dic3—61no—
318C2×Dic3—61no—
319D12—64no—
320D12—122no—
321C4×D3—24no—
322C4×D3—62no—
323C4×D3—62no—
324C4×D3—121no—
325C4×D3—121no—
326Dic6—62no—
327Dic6—62no—
328C3×C8—181no—
329C3×C8—181no—
330C2×C32—21no—
331C2×C32—21no—
332C2×C32—21no—
333C2×C32—21no—
334C2×C32—21no—
335C2×C32—21no—
336C2×C32—41no—
337C2×C32—41no—
338C2×C32—41no—
339C2×C32—41no—
340C2×C32—41no—
341C2×C32—41no—
342C2×C32—122no—
343C2×C32—241no—
344C2×C32C3×#12011yescharacteristic
345C2×C32C3×#13511yescharacteristic
346C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
347C3⋊D3—41no—
348C3×D3—42no—
349C3×D3—44no—
350C3×D3—42no—
351C3×D3—81no—
352C3×D3—82no—
353C3×D3—81no—
354C3×D3—122no—
355C3×D3—122no—
356#41—182no—
357#41—182no—
358C2×Q8—91no—
359C2×D4—91no—
360C2×D4—91no—
361C2×D4—181no—
362C2×D4—181no—
363C22×C4—61no—
364C22×C4—91no—
365SD16—182no—
366D8—182no—
367C2×C8—91no—
368C2×C8—91no—
369C4⋊C4—91no—
370C22⋊C4—61no—
371C22×C3—21no—
372C22×C3—21no—
373C22×C3—41no—
374C22×C3—41no—
375C22×C3—62no—
376C22×C3—62no—
377C22×C3—122no—
378C22×C3—121no—
379C22×C3—121no—
380C22×C3—121no—
381C22×C3—121no—
382C22×C3—121no—
383C22×C3—122no—
384C22×C3—122no—
385C22×C3—122no—
386C22×C3—121no—
387C22×C3—121no—
388C22×C3—121no—
389C22×C3—121no—
390C22×C3—121no—
391C22×C3—181no—
392C22×C3—181no—
393C22×C3—181no—
394C22×C3—241no—
395C22×C3—241no—
396C22×C3C2×GO+(4,𝔽2)11yescharacteristic
397C2×D3—22no—
398C2×D3—41no—
399C2×D3—41no—
400C2×D3—41no—
401C2×D3—41no—
402C2×D3—41no—
403C2×D3—62no—
404C2×D3—62no—
405C2×D3—121no—
406C2×D3—121no—
407C2×D3—121no—
408C2×D3—121no—
409C3×C4—22no—
410C3×C4—21no—
411C3×C4—41no—
412C3×C4—42no—
413C3×C4—62no—
414C3×C4—62no—
415C3×C4—81no—
416C3×C4—91no—
417C3×C4—91no—
418C3×C4—122no—
419C3×C4—121no—
420C3×C4—121no—
421C3×C4—241no—
422Dic3—22no—
423Dic3—41no—
424Dic3—61no—
425Dic3—61no—
426Dic3—61no—
427Dic3—61no—
428C32—21no—
429C32—21no—
430C32—41no—
431C32—41no—
432C32C3×#44911yescharacteristic, complemented
433C23—61no—
434C23—61no—
435C23—91no—
436Q8—92no—
437Q8—181no—
438D4—92no—
439D4—184no—
440D4—181no—
441D4—181no—
442D4—181no—
443D4—362no—
444C2×C4—64no—
445C2×C4—61no—
446C2×C4—61no—
447C2×C4—91no—
448C2×C4—181no—
449C2×C4—181no—
450C2×C4C3×GO+(4,𝔽2)11yescharacteristic
451C8—181no—
452C8—181no—
453C2×C3—21no—
454C2×C3—21no—
455C2×C3—21no—
456C2×C3—21no—
457C2×C3—21no—
458C2×C3—21no—
459C2×C3—41no—
460C2×C3—41no—
461C2×C3—41no—
462C2×C3—41no—
463C2×C3—41no—
464C2×C3—41no—
465C2×C3—122no—
466C2×C3—124no—
467C2×C3—122no—
468C2×C3—241no—
469C2×C3—242no—
470C2×C3—241no—
471C2×C3—361no—
472C2×C3#6422311yescharacteristic
473C2×C3#6422411yescharacteristic
474C2×C3#6424111yescharacteristic
475D3—42no—
476D3—81no—
477D3—122no—
478D3—122no—
479C22—62no—
480C22—121no—
481C22—121no—
482C22—121no—
483C22—121no—
484C22—121no—
485C22—181no—
486C22—181no—
487C22—181no—
488C22C2×C3×GO+(4,𝔽2)11yescharacteristic
489C4—21no—
490C4—62no—
491C4—91no—
492C4—91no—
493C4—121no—
494C3—21no—
495C3—21no—
496C3—41no—
497C3—41no—
498C3#1059342111yescharacteristic, direct product factor
499C2—122no—
500C2—241no—
501C2—361no—
502C2C3×#6422311yescharacteristic
503C2C3×#6422411yescharacteristic
504C2C3×#6424111yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊Dic3C23×C3
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C2×D4
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#449
LCSG ▷ C2×C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3798C2semidirect
2. C2×C3×#3775C2semidirect
3. C2×C3×#3774C2semidirect
4. C3×#63912C2semidirect
5. C3×#63778C2semidirect
6. C3×#63740C2semidirect
7. C3×#63739C2semidirect
8. #10593421C3direct (N × H)
25. C2×#3798C2×C3semidirect
26. C2×#3775C2×C3semidirect
27. C2×#3774C2×C3semidirect
29. #63912C2×C3semidirect
30. #63778C2×C3semidirect
31. #63740C2×C3semidirect
32. #63739C2×C3semidirect
44. C3×#627D4semidirect
50. C3×#625D4semidirect
159. #627C3×D4semidirect
165. #625C3×D4semidirect
272. C33#449semidirect
432. C32C3×#449semidirect
498. C3#10593421direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37982
C2×C3×#37752
C2×C3×#37742
C3×#639122
C3×#637782
C3×#637402
C3×#637392
#105934213
C3×#4499