Finite Group Databasebeta

← C2×C3×#64230 #13181027

Subgroups — C2×C3×#64230 #13181027

Order 1728.

1432 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1432 conjugacy classes of subgroups — too many to draw. Only its 210 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
1C22×C3×GO+(4,𝔽2)C212yesmaximal, complemented
2C2×C3×#3802C211yesmaximal, characteristic, complemented
3C2×C3×C4×D32C211yesmaximal, characteristic, complemented
4C2×C3×C4⋊#157C211yesmaximal, characteristic, complemented
5C2×C3×D32⋊C4C212yesmaximal, complemented
6C3×#64230C218yesmaximal, direct product factor
7C2×#64230C311yesmaximal, characteristic, direct product factor
8C22×C3×D32C2212yes—
9C22×C3×D32C2211yescharacteristic
10C22×C3×#157C2212yes—
11C2×C3×GO+(4,𝔽2)—28no—
12C2×C3×GO+(4,𝔽2)C2218yescomplemented
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14C2×C32×D12—21no—
15C2×C32×C4×D3—21no—
16C2×C3×#625—21no—
17C2×C3×#624C2211yescharacteristic
18C2×C3×D3×Dic3—21no—
19C3×#3802—22no—
20C3×#3802C2214yescomplemented
21C3×C4×D32—22no—
22C3×C4×D32C2214yescomplemented
23C3×C4⋊#157C2214yescomplemented
24C3×D32⋊C4C2218yescomplemented
25C22×GO+(4,𝔽2)C2×C312yescomplemented
26C2×C3×D3×D4—61no—
27C22×C3×C4×D3—61no—
28C2×#3802C2×C311yescharacteristic, complemented
29C2×C4×D32C2×C311yescharacteristic, complemented
30C2×C4⋊#157C2×C311yescharacteristic, complemented
31C2×D32⋊C4C2×C312yescomplemented
32#64230C2×C318yesdirect product factor
33C22×C3×C3⋊D3C2311yescharacteristic
34C22×C32×D3—21no—
35C22×C32×D3—22no—
36C2×C3×D32—21no—
37C2×C3×D32—22no—
38C2×C3×D32—24no—
39C2×C3×D32—24no—
40C2×C3×D32—22no—
41C2×C3×D32D414yescomplemented
42C2×C3×D32C2314yes—
43C2×C3×D32C2312yes—
44C2×C3×#157—24no—
45C2×C3×#157C2314yes—
46C3×GO+(4,𝔽2)—216no—
47C2×C33×C4D411yescharacteristic, complemented
48C2×C3×C3⋊Dic3D411yescharacteristic, complemented
49C3×C4×C3⋊D3D412yescomplemented
50C3×C4×C3⋊D3C2312yes—
51C2×C32×Dic3—21no—
52C32×D12—24no—
53C32×C4×D3—24no—
54C3×#625—24no—
55C3×#624—21no—
56C3×#624C2312yes—
57C3×D3×Dic3—24no—
58C2×C3×#121—91nomaximal
59C23×C3×D3—61no—
60C23×C3×D3—62no—
61C22×D32C22×C312yes—
62C22×D32C22×C311yescharacteristic
63C22×#157C22×C312yes—
64C2×GO+(4,𝔽2)—28no—
65C2×GO+(4,𝔽2)C22×C318yescomplemented
66C2×C32×D4—61no—
67C22×C32×C4—61no—
68C2×C4×C3⋊D3C22×C311yescharacteristic
69C2×C3×#67—62no—
70C22×C3×Dic3—61no—
71C3×D3×D4—64no—
72C3×D3×D4—64no—
73C2×C3×D12—21no—
74C2×C3×D12—21no—
75C2×C3×D12—41no—
76C2×C3×C4×D3—21no—
77C2×C3×C4×D3—21no—
78C2×C3×C4×D3—41no—
79C2×C3×C4×D3—64no—
80C2×C3×C4×D3—64no—
81C2×C3×C4×D3—61no—
82C2×C3×C4×D3—61no—
83C2×C3×C4×D3—61no—
84C2×C3×C4×D3—61no—
85C2×#625—21no—
86C2×#624C22×C311yescharacteristic
87C2×D3×Dic3—21no—
88#3802—22no—
89#3802C22×C314yescomplemented
90C4×D32—22no—
91C4×D32C22×C314yescomplemented
92C4⋊#157C22×C314yescomplemented
93D32⋊C4C22×C318yescomplemented
94C22×C33C2×D411yescharacteristic
95C2×C3×C3⋊D3C2×D411yescharacteristic
96C2×C3×C3⋊D3C2×D412yes—
97C2×C3×C3⋊D3#4112yes—
98C2×C3×C3⋊D3C2411yescharacteristic
99C2×C32×D3—24no—
100C2×C32×D3—22no—
101C2×C32×D3—24no—
102C2×C32×D3—44no—
103C3×D32—24no—
104C3×D32—22no—
105C3×D32—24no—
106C3×D32—24no—
107C3×D32—44no—
108C3×D32C2×D414yescomplemented
109C3×#157—24no—
110C33×C4C2×D412yescomplemented
111C3×C3⋊Dic3C2×D412yescomplemented
112C32×Dic3—22no—
113C22×C3×D4—91no—
114C22×C3×D4—92no—
115C23×C3×C4—91no—
116C2×D3×D4—61no—
117C22×C4×D3—61no—
118C3×#121—98no—
119C2×C3×C4⋊C4—91no—
120C2×C3×C22⋊C4—92no—
121C23×C32—61no—
122C23×C32—62no—
123C22×C3⋊D3C23×C311yescharacteristic
124C22×C3×D3—21no—
125C22×C3×D3—21no—
126C22×C3×D3—22no—
127C22×C3×D3—22no—
128C22×C3×D3—41no—
129C22×C3×D3—42no—
130C22×C3×D3—64no—
131C22×C3×D3—62no—
132C22×C3×D3—64no—
133C22×C3×D3—61no—
134C22×C3×D3—61no—
135C22×C3×D3—64no—
136C22×C3×D3—62no—
137C22×C3×D3—64no—
138C22×C3×D3—124no—
139C22×C3×D3—124no—
140C2×D32—21no—
141C2×D32—22no—
142C2×D32—22no—
143C2×D32—24no—
144C2×D32—24no—
145C2×D32C3×D414yescomplemented
146C2×D32C23×C314yes—
147C2×D32C23×C312yes—
148C2×#157—24no—
149C2×#157C23×C314yes—
150GO+(4,𝔽2)—216no—
151C32×D4—64no—
152C2×C32×C4—21no—
153C2×C32×C4—21no—
154C2×C32×C4—41no—
155C2×C32×C4—41no—
156C2×C32×C4—64no—
157C2×C32×C4—61no—
158C2×C32×C4C3×D411yescharacteristic, complemented
159C2×C3⋊Dic3C3×D411yescharacteristic, complemented
160C4×C3⋊D3C3×D412yescomplemented
161C4×C3⋊D3C23×C312yes—
162C3×#67—68no—
163C2×C3×Dic3—21no—
164C2×C3×Dic3—21no—
165C2×C3×Dic3—41no—
166C2×C3×Dic3—61no—
167C2×C3×Dic3—61no—
168C2×C3×Dic3—64no—
169C3×D12—24no—
170C3×D12—24no—
171C3×D12—44no—
172C3×C4×D3—24no—
173C3×C4×D3—24no—
174C3×C4×D3—44no—
175C3×C4×D3—62no—
176C3×C4×D3—62no—
177C3×C4×D3—62no—
178C3×C4×D3—62no—
179C3×C4×D3—62no—
180C3×C4×D3—62no—
181C3×C4×D3—64no—
182#625—24no—
183#624—21no—
184#624C23×C312yes—
185D3×Dic3—24no—
186C2×#121—91noSylow
187C2×C33#12112yescomplemented
188C2×C33C22×D411yescharacteristic
189C3×C3⋊D3#12112yescomplemented
190C3×C3⋊D3C22×D411yescharacteristic
191C3×C3⋊D3C2×#4111yescharacteristic
192C32×D3—24no—
193C32×D3—44no—
194C24×C3—91no—
195C24×C3—92no—
196C23×D3—61no—
197C23×D3—62no—
198C2×C3×D4—61no—
199C2×C3×D4—61no—
200C2×C3×D4—94no—
201C2×C3×D4—98no—
202C2×C3×D4—121no—
203C2×C3×D4—182no—
204C2×C3×D4—181no—
205C2×C3×D4—188no—
206C22×C3×C4—61no—
207C22×C3×C4—61no—
208C22×C3×C4—91no—
209C22×C3×C4—94no—
210C22×C3×C4—91no—
211C22×C3×C4—92no—
212C22×C3×C4—121no—
213C22×C3×C4—182no—
214C22×C3×C4—181no—
215C2×#67—62no—
216C22×Dic3—61no—
217D3×D4—64no—
218D3×D4—64no—
219C2×D12—21no—
220C2×C4×D3—21no—
221C2×C4×D3—64no—
222C2×C4×D3—61no—
223C2×C4×D3—61no—
224C2×C4×D3—61no—
225C2×C4×D3—64no—
226C2×C4×D3—61no—
227C3×C4⋊C4—94no—
228C3×C22⋊C4—98no—
229C22×C32—21no—
230C22×C32—21no—
231C22×C32—41no—
232C22×C32—41no—
233C22×C32—62no—
234C22×C32—64no—
235C22×C32—64no—
236C22×C32—124no—
237C22×C32C2×C3×D411yescharacteristic
238C2×C3⋊D3C2×C3×D411yescharacteristic
239C2×C3⋊D3C2×C3×D412yes—
240C2×C3⋊D3C3×#4112yes—
241C2×C3⋊D3C24×C311yescharacteristic
242C2×C3×D3—22no—
243C2×C3×D3—24no—
244C2×C3×D3—24no—
245C2×C3×D3—22no—
246C2×C3×D3—24no—
247C2×C3×D3—24no—
248C2×C3×D3—42no—
249C2×C3×D3—44no—
250C2×C3×D3—44no—
251C2×C3×D3—44no—
252C2×C3×D3—44no—
253C2×C3×D3—64no—
254C2×C3×D3—61no—
255C2×C3×D3—61no—
256C2×C3×D3—62no—
257C2×C3×D3—62no—
258C2×C3×D3—62no—
259C2×C3×D3—62no—
260C2×C3×D3—61no—
261C2×C3×D3—61no—
262C2×C3×D3—68no—
263C2×C3×D3—64no—
264C2×C3×D3—84no—
265C2×C3×D3—124no—
266C2×C3×D3—124no—
267C2×C3×D3—128no—
268D32—24no—
269D32—22no—
270D32—24no—
271D32—24no—
272D32—44no—
273D32C2×C3×D414yescomplemented
274#157—24no—
275C32×C4—22no—
276C32×C4—22no—
277C32×C4—42no—
278C32×C4—42no—
279C32×C4—62no—
280C32×C4C2×C3×D412yescomplemented
281C3⋊Dic3C2×C3×D412yescomplemented
282C3×Dic3—22no—
283C3×Dic3—22no—
284C3×Dic3—42no—
285C3×Dic3—62no—
286C3×Dic3—62no—
287C22×D4—91no—
288C22×D4—92no—
289C23×C4—91no—
290#121—98no—
291C2×C4⋊C4—91no—
292C2×C22⋊C4—92no—
293C33C2×#12111yescharacteristic, complemented, Sylow
294C23×C3—61no—
295C23×C3—62no—
296C23×C3—61no—
297C23×C3—62no—
298C23×C3—92no—
299C23×C3—94no—
300C23×C3—94no—
301C23×C3—91no—
302C23×C3—121no—
303C23×C3—122no—
304C23×C3—181no—
305C23×C3—182no—
306C23×C3—182no—
307C23×C3—184no—
308C23×C3—184no—
309C22×D3—21no—
310C22×D3—22no—
311C22×D3—64no—
312C22×D3—62no—
313C22×D3—64no—
314C22×D3—61no—
315C22×D3—61no—
316C22×D3—62no—
317C22×D3—64no—
318C22×D3—64no—
319C22×D3—124no—
320C22×D3—124no—
321C3×D4—64no—
322C3×D4—64no—
323C3×D4—124no—
324C3×D4—184no—
325C3×D4—1816no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—41no—
329C2×C3×C4—41no—
330C2×C3×C4—64no—
331C2×C3×C4—61no—
332C2×C3×C4—61no—
333C2×C3×C4—64no—
334C2×C3×C4—92no—
335C2×C3×C4—91no—
336C2×C3×C4—92no—
337C2×C3×C4—92no—
338C2×C3×C4—94no—
339C2×C3×C4—124no—
340C2×C3×C4—121no—
341C2×C3×C4—181no—
342C2×C3×C4—184no—
343C2×C3×C4—184no—
344C2×C3×C4GO+(4,𝔽2)11yescharacteristic, complemented
345#67—68no—
346C2×Dic3—21no—
347C2×Dic3—61no—
348C2×Dic3—61no—
349C2×Dic3—64no—
350D12—24no—
351C4×D3—24no—
352C4×D3—62no—
353C4×D3—62no—
354C4×D3—62no—
355C4×D3—62no—
356C4×D3—62no—
357C4×D3—62no—
358C4×D3—64no—
359C2×C32—21no—
360C2×C32—21no—
361C2×C32—22no—
362C2×C32—22no—
363C2×C32—42no—
364C2×C32—42no—
365C2×C32—41no—
366C2×C32—41no—
367C2×C32—64no—
368C2×C32—124no—
369C2×C32C3×#12112yescomplemented
370C2×C32C22×C3×D411yescharacteristic
371C3⋊D3C3×#12112yescomplemented
372C3⋊D3C22×C3×D411yescharacteristic
373C3⋊D3C2×C3×#4111yescharacteristic
374C3×D3—24no—
375C3×D3—24no—
376C3×D3—44no—
377C3×D3—44no—
378C3×D3—44no—
379C3×D3—62no—
380C3×D3—62no—
381C3×D3—61no—
382C3×D3—61no—
383C3×D3—61no—
384C3×D3—61no—
385C3×D3—84no—
386C24—91no—
387C24—92no—
388C2×D4—61no—
389C2×D4—94no—
390C2×D4—98no—
391C2×D4—182no—
392C2×D4—181no—
393C2×D4—188no—
394C22×C4—61no—
395C22×C4—91no—
396C22×C4—94no—
397C22×C4—91no—
398C22×C4—92no—
399C22×C4—182no—
400C22×C4—181no—
401C4⋊C4—94no—
402C22⋊C4—98no—
403C22×C3—21no—
404C22×C3—21no—
405C22×C3—41no—
406C22×C3—41no—
407C22×C3—62no—
408C22×C3—64no—
409C22×C3—64no—
410C22×C3—62no—
411C22×C3—64no—
412C22×C3—64no—
413C22×C3—94no—
414C22×C3—91no—
415C22×C3—91no—
416C22×C3—92no—
417C22×C3—92no—
418C22×C3—122no—
419C22×C3—124no—
420C22×C3—124no—
421C22×C3—124no—
422C22×C3—124no—
423C22×C3—182no—
424C22×C3—184no—
425C22×C3—184no—
426C22×C3—184no—
427C22×C3—244no—
428C22×C3—364no—
429C22×C3C2×GO+(4,𝔽2)11yescharacteristic
430C2×D3—22no—
431C2×D3—24no—
432C2×D3—24no—
433C2×D3—44no—
434C2×D3—64no—
435C2×D3—61no—
436C2×D3—61no—
437C2×D3—61no—
438C2×D3—61no—
439C2×D3—62no—
440C2×D3—62no—
441C2×D3—62no—
442C2×D3—62no—
443C2×D3—68no—
444C2×D3—64no—
445C2×D3—124no—
446C2×D3—124no—
447C2×D3—128no—
448C3×C4—22no—
449C3×C4—22no—
450C3×C4—42no—
451C3×C4—42no—
452C3×C4—62no—
453C3×C4—62no—
454C3×C4—92no—
455C3×C4—122no—
456C3×C4—184no—
457C3×C4C2×GO+(4,𝔽2)12yescomplemented
458Dic3—22no—
459Dic3—62no—
460Dic3—62no—
461C32—21no—
462C32—21no—
463C32—41no—
464C32—41no—
465C32C2×C3×#12111yescharacteristic, complemented
466C23—61no—
467C23—62no—
468C23—92no—
469C23—94no—
470C23—94no—
471C23—91no—
472C23—181no—
473C23—182no—
474C23—182no—
475C23—184no—
476C23—184no—
477D4—64no—
478D4—184no—
479D4—1816no—
480C2×C4—61no—
481C2×C4—64no—
482C2×C4—92no—
483C2×C4—91no—
484C2×C4—92no—
485C2×C4—92no—
486C2×C4—94no—
487C2×C4—181no—
488C2×C4—184no—
489C2×C4—184no—
490C2×C4C3×GO+(4,𝔽2)11yescharacteristic, complemented
491C2×C3—21no—
492C2×C3—21no—
493C2×C3—22no—
494C2×C3—22no—
495C2×C3—41no—
496C2×C3—41no—
497C2×C3—42no—
498C2×C3—42no—
499C2×C3—64no—
500C2×C3—64no—
501C2×C3—91no—
502C2×C3—91no—
503C2×C3—92no—
504C2×C3—124no—
505C2×C3—124no—
506C2×C3—124no—
507C2×C3—244no—
508C2×C3#6423012yesdirect product factor
509C2×C3C22×GO+(4,𝔽2)11yescharacteristic
510D3—24no—
511D3—44no—
512D3—62no—
513D3—61no—
514D3—61no—
515D3—62no—
516D3—61no—
517D3—61no—
518C22—62no—
519C22—64no—
520C22—64no—
521C22—94no—
522C22—91no—
523C22—91no—
524C22—92no—
525C22—92no—
526C22—124no—
527C22—182no—
528C22—184no—
529C22—184no—
530C22—184no—
531C22—364no—
532C22C2×C3×GO+(4,𝔽2)11yescharacteristic
533C4—62no—
534C4—92no—
535C4—184no—
536C4C2×C3×GO+(4,𝔽2)12yescomplemented
537C3—21no—
538C3—21no—
539C3—41no—
540C3—41no—
541C3C2×#6423011yescharacteristic, direct product factor
542C2—64no—
543C2—91no—
544C2—91no—
545C2—92no—
546C2—124no—
547C2C3×#6423012yesdirect product factor
548C2C22×C3×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊D3C24×C3
FrattiniC2C22×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C2×D4
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C2×C3×#121
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. C22×C3×GO+(4,𝔽2)C2semidirect
2. C2×C3×#3802C2semidirect
3. C2×C3×C4×D32C2semidirect
4. C2×C3×C4⋊#157C2semidirect
5. C2×C3×D32⋊C4C2semidirect
6. C3×#64230C2direct (N × H)
7. C2×#64230C3direct (N × H)
12. C2×C3×GO+(4,𝔽2)C22semidirect
20. C3×#3802C22semidirect
22. C3×C4×D32C22semidirect
23. C3×C4⋊#157C22semidirect
24. C3×D32⋊C4C22semidirect
25. C22×GO+(4,𝔽2)C2×C3semidirect
28. C2×#3802C2×C3semidirect
29. C2×C4×D32C2×C3semidirect
30. C2×C4⋊#157C2×C3semidirect
31. C2×D32⋊C4C2×C3semidirect
32. #64230C2×C3direct (N × H)
41. C2×C3×D32D4semidirect
47. C2×C33×C4D4semidirect
48. C2×C3×C3⋊Dic3D4semidirect
49. C3×C4×C3⋊D3D4semidirect
65. C2×GO+(4,𝔽2)C22×C3semidirect
89. #3802C22×C3semidirect
91. C4×D32C22×C3semidirect
92. C4⋊#157C22×C3semidirect
93. D32⋊C4C22×C3semidirect
108. C3×D32C2×D4semidirect
110. C33×C4C2×D4semidirect
111. C3×C3⋊Dic3C2×D4semidirect
145. C2×D32C3×D4semidirect
158. C2×C32×C4C3×D4semidirect
159. C2×C3⋊Dic3C3×D4semidirect
160. C4×C3⋊D3C3×D4semidirect
187. C2×C33#121semidirect
189. C3×C3⋊D3#121semidirect
273. D32C2×C3×D4semidirect
280. C32×C4C2×C3×D4semidirect
281. C3⋊Dic3C2×C3×D4semidirect
293. C33C2×#121semidirect
344. C2×C3×C4GO+(4,𝔽2)semidirect
369. C2×C32C3×#121semidirect
371. C3⋊D3C3×#121semidirect
457. C3×C4C2×GO+(4,𝔽2)semidirect
465. C32C2×C3×#121semidirect
490. C2×C4C3×GO+(4,𝔽2)semidirect
508. C2×C3#64230direct (N × H)
536. C4C2×C3×GO+(4,𝔽2)semidirect
541. C3C2×#64230direct (N × H)
547. C2C3×#64230direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×GO+(4,𝔽2)2
C2×C3×#38022
C2×C3×C4×D322
C2×C3×C4⋊#1572
C2×C3×D32⋊C42
C3×#642302
C2×#642303
C2×C3×#1219