Finite Group Databasebeta

← C3×#10596512 #13181031

Subgroups — C3×#10596512 #13181031

Order 1728.

1162 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 1162 conjugacy classes of subgroups — too many to draw. Only its 174 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
1C3×#64306C211yesmaximal, characteristic, complemented
2C2×C3×#3802C211yesmaximal, characteristic, complemented
3C3×#64286C211yesmaximal, characteristic, complemented
4C3×#64241C214yesmaximal, complemented
5C3×#64231C212yesmaximal, complemented
6C3×#64230C214yesmaximal, complemented
7C3×#64221C212yesmaximal, complemented
8#10596512C311yesmaximal, characteristic, direct product factor
9C22×C3×D32C2212yes—
10C2×C3×GO+(4,𝔽2)C2218yescomplemented
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C32×#235—21no—
13C2×C32×D12—21no—
14C3×#3811C2212yes—
15C2×C3×#625—21no—
16C3×#3805—21no—
17C3×#3802—21no—
18C3×#3802—21no—
19C3×#3802C2212yes—
20C3×#3802C2212yes—
21C3×#3802C2211yescharacteristic, complemented
22C3×C4×D32C2212yescomplemented
23C3×#3798—21no—
24C3×#3797C2211yescharacteristic, complemented
25C3×#3796—21no—
26C3×C22⋊#157C2212yes—
27C3×C4⋊#157C2212yescomplemented
28C3×C4×#157C2212yescomplemented
29C3×D32⋊C4C2218yescomplemented
30C3×D3×#41—61no—
31C2×C3×D3×D4—61no—
32#64306C2×C311yescharacteristic, complemented
33C2×#3802C2×C311yescharacteristic, complemented
34#64286C2×C311yescharacteristic, complemented
35#64241C2×C314yescomplemented
36#64231C2×C312yescomplemented
37#64230C2×C314yescomplemented
38#64221C2×C312yescomplemented
39C22×C3×C3⋊D3C2311yescharacteristic
40C22×C32×D3—22no—
41C2×C3×D32—22no—
42C2×C3×D32—44no—
43C2×C3×D32C2312yes—
44C2×C3×D32C2314yes—
45C2×C3×#157C2314yes—
46C3×GO+(4,𝔽2)—48no—
47C2×C33×C4D411yescharacteristic, complemented
48C2×C3×C3⋊Dic3D411yescharacteristic, complemented
49C3×C4×C3⋊D3D411yescharacteristic, complemented
50C3×C4×C3⋊D3D411yescharacteristic, complemented
51C3×C4×C3⋊D3C2311yescharacteristic
52C3×C4×C3⋊D3C2311yescharacteristic
53C32×#67—22no—
54C32×D12—21no—
55C32×D12—22no—
56C32×D12—22no—
57C32×C4×D3—22no—
58C32×Dic6—21no—
59C3×#627—21no—
60C3×#626—22no—
61C3×#625—21no—
62C3×#625—22no—
63C3×#625—22no—
64C3×#624C2312yes—
65C3×D3×Dic3—22no—
66C3×#535—91nomaximal
67C23×C3×D3—62no—
68C22×D32C22×C312yes—
69C2×GO+(4,𝔽2)C22×C318yescomplemented
70C32×#41—61no—
71C2×C32×D4—61no—
72C2×C4×C3⋊D3C22×C311yescharacteristic
73C2×C3×#67—62no—
74C3×#239—61no—
75C3×D3×Q8—61no—
76C3×#237—61no—
77C3×#237—62no—
78C3×D3×D4—62no—
79C3×D3×D4—62no—
80C3×D3×D4—61no—
81C3×D3×D4—62no—
82C3×D3×D4—62no—
83C3×D3×D4—62no—
84C3×#235—21no—
85C3×#235—21no—
86C3×#235—41no—
87C3×#235—62no—
88C2×C3×D12—21no—
89C2×C3×D12—21no—
90C2×C3×D12—41no—
91C2×C3×C4×D3—62no—
92C2×C3×C4×D3—61no—
93C2×C3×C4×D3—61no—
94#3811C22×C312yes—
95C2×#625—21no—
96#3805—21no—
97#3802—21no—
98#3802—21no—
99#3802C22×C312yes—
100#3802C22×C312yes—
101#3802C22×C311yescharacteristic, complemented
102C4×D32C22×C312yescomplemented
103#3798—21no—
104#3797C22×C311yescharacteristic, complemented
105#3796—21no—
106C22⋊#157C22×C312yes—
107C4⋊#157C22×C312yescomplemented
108C4×#157C22×C312yescomplemented
109D32⋊C4C22×C318yescomplemented
110C22×C33C2×D411yescharacteristic
111C2×C3×C3⋊D3—22no—
112C2×C3×C3⋊D3C2×D411yescharacteristic
113C2×C3×C3⋊D3C2411yescharacteristic
114C2×C32×D3—22no—
115C2×C32×D3—24no—
116C2×C32×D3—44no—
117C3×D32—24no—
118C3×D32—28no—
119C3×D32—44no—
120C3×#157—24no—
121C33×C4C2×D411yescharacteristic
122C33×C4C2×D411yescharacteristic
123C3×C3⋊Dic3C2×D411yescharacteristic
124C3×C3⋊Dic3C2×D411yescharacteristic
125C32×Dic3—22no—
126C2×C3×#41—91no—
127C22×C3×D4—91no—
128D3×#41—61no—
129C2×D3×D4—61no—
130C3×#127—92no—
131C3×#124—92no—
132C3×#121—94no—
133C3×#120—94no—
134C3×#117—91no—
135C23×C32—62no—
136C22×C3⋊D3C23×C311yescharacteristic
137C22×C3×D3—22no—
138C22×C3×D3—22no—
139C22×C3×D3—42no—
140C22×C3×D3—64no—
141C22×C3×D3—62no—
142C22×C3×D3—61no—
143C22×C3×D3—61no—
144C22×C3×D3—64no—
145C22×C3×D3—128no—
146C2×D32—22no—
147C2×D32—44no—
148C2×D32C23×C312yes—
149C2×D32C23×C314yes—
150C2×#157C23×C314yes—
151GO+(4,𝔽2)—48no—
152C32×Q8—61no—
153C32×D4—62no—
154C32×D4—62no—
155C32×D4—61no—
156C32×D4—62no—
157C2×C32×C4—21no—
158C2×C32×C4—21no—
159C2×C32×C4—41no—
160C2×C32×C4—41no—
161C2×C32×C4—62no—
162C2×C32×C4C3×D411yescharacteristic, complemented
163C2×C3⋊Dic3C3×D411yescharacteristic, complemented
164C4×C3⋊D3C3×D411yescharacteristic, complemented
165C4×C3⋊D3C3×D411yescharacteristic, complemented
166C4×C3⋊D3C23×C311yescharacteristic
167C4×C3⋊D3C23×C311yescharacteristic
168C3×#67—22no—
169C3×#67—22no—
170C3×#67—42no—
171C3×#67—62no—
172C3×#67—62no—
173C3×#67—64no—
174C3×#67—64no—
175C2×C3×Dic3—61no—
176C2×C3×Dic3—61no—
177C2×C3×Dic3—62no—
178C3×D12—22no—
179C3×D12—22no—
180C3×D12—21no—
181C3×D12—21no—
182C3×D12—22no—
183C3×D12—22no—
184C3×D12—42no—
185C3×D12—42no—
186C3×D12—41no—
187C3×D12—62no—
188C3×C4×D3—22no—
189C3×C4×D3—22no—
190C3×C4×D3—42no—
191C3×C4×D3—62no—
192C3×C4×D3—61no—
193C3×C4×D3—61no—
194C3×C4×D3—61no—
195C3×C4×D3—61no—
196C3×C4×D3—61no—
197C3×C4×D3—61no—
198C3×C4×D3—61no—
199C3×C4×D3—61no—
200C3×C4×D3—62no—
201C3×Dic6—21no—
202C3×Dic6—21no—
203C3×Dic6—41no—
204C3×Dic6—62no—
205#627—21no—
206#626—22no—
207#625—21no—
208#625—22no—
209#625—22no—
210#624C23×C312yes—
211D3×Dic3—22no—
212#535—91noSylow
213C2×C33—21no—
214C2×C33C22×D411yescharacteristic
215C3×C3⋊D3—21no—
216C3×C3⋊D3H(5,𝔽2)12yes—
217C32×D3—44no—
218C32×D3—42no—
219C24×C3—92no—
220C23×D3—62no—
221C3×#41—61no—
222C3×#41—61no—
223C3×#41—121no—
224C3×#41—181no—
225C3×#41—181no—
226C3×#41—181no—
227C2×C3×Q8—91no—
228C2×C3×D4—61no—
229C2×C3×D4—61no—
230C2×C3×D4—92no—
231C2×C3×D4—92no—
232C2×C3×D4—92no—
233C2×C3×D4—91no—
234C2×C3×D4—98no—
235C2×C3×D4—121no—
236C2×C3×D4—181no—
237C2×C3×D4—181no—
238C2×C3×D4—181no—
239C22×C3×C4—92no—
240C22×C3×C4—91no—
241C2×#67—62no—
242#239—61no—
243D3×Q8—61no—
244#237—61no—
245#237—62no—
246D3×D4—62no—
247D3×D4—62no—
248D3×D4—61no—
249D3×D4—62no—
250D3×D4—62no—
251D3×D4—62no—
252#235—21no—
253#235—62no—
254C2×D12—21no—
255C2×C4×D3—62no—
256C2×C4×D3—61no—
257C2×C4×D3—61no—
258C3×C4⋊C4—92no—
259C3×C22⋊C4—98no—
260C3×C22⋊C4—92no—
261C3×C42—92no—
262C22×C32—21no—
263C22×C32—21no—
264C22×C32—41no—
265C22×C32—41no—
266C22×C32—62no—
267C22×C32—64no—
268C22×C32—124no—
269C22×C32C2×C3×D411yescharacteristic
270C2×C3⋊D3—22no—
271C2×C3⋊D3C2×C3×D411yescharacteristic
272C2×C3⋊D3C24×C311yescharacteristic
273C2×C3×D3—22no—
274C2×C3×D3—24no—
275C2×C3×D3—22no—
276C2×C3×D3—24no—
277C2×C3×D3—42no—
278C2×C3×D3—44no—
279C2×C3×D3—44no—
280C2×C3×D3—44no—
281C2×C3×D3—61no—
282C2×C3×D3—61no—
283C2×C3×D3—61no—
284C2×C3×D3—61no—
285C2×C3×D3—64no—
286C2×C3×D3—84no—
287C2×C3×D3—124no—
288C2×C3×D3—128no—
289C2×C3×D3—122no—
290C2×C3×D3—128no—
291D32—24no—
292D32—28no—
293D32—44no—
294#157—24no—
295C32×C4—21no—
296C32×C4—21no—
297C32×C4—21no—
298C32×C4—21no—
299C32×C4—41no—
300C32×C4—41no—
301C32×C4—41no—
302C32×C4—41no—
303C32×C4—62no—
304C32×C4C2×C3×D411yescharacteristic
305C32×C4C2×C3×D411yescharacteristic
306C3⋊Dic3C2×C3×D411yescharacteristic
307C3⋊Dic3C2×C3×D411yescharacteristic
308C3×Dic3—22no—
309C3×Dic3—22no—
310C3×Dic3—42no—
311C3×Dic3—61no—
312C3×Dic3—61no—
313C3×Dic3—61no—
314C3×Dic3—61no—
315C2×#41—91no—
316C22×D4—91no—
317#127—92no—
318#124—92no—
319#121—94no—
320#120—94no—
321#117—91no—
322C33#53511yescharacteristic, complemented, Sylow
323C23×C3—62no—
324C23×C3—62no—
325C23×C3—94no—
326C23×C3—92no—
327C23×C3—91no—
328C23×C3—122no—
329C23×C3—182no—
330C23×C3—364no—
331C22×D3—22no—
332C22×D3—62no—
333C22×D3—64no—
334C22×D3—61no—
335C22×D3—61no—
336C22×D3—64no—
337C22×D3—128no—
338C3×Q8—61no—
339C3×Q8—61no—
340C3×Q8—121no—
341C3×Q8—181no—
342C3×D4—62no—
343C3×D4—62no—
344C3×D4—61no—
345C3×D4—62no—
346C3×D4—62no—
347C3×D4—62no—
348C3×D4—62no—
349C3×D4—61no—
350C3×D4—122no—
351C3×D4—122no—
352C3×D4—121no—
353C3×D4—122no—
354C3×D4—182no—
355C3×D4—182no—
356C3×D4—182no—
357C3×D4—181no—
358C3×D4—368no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—62no—
364C2×C3×C4—62no—
365C2×C3×C4—92no—
366C2×C3×C4—91no—
367C2×C3×C4—91no—
368C2×C3×C4—91no—
369C2×C3×C4—91no—
370C2×C3×C4—91no—
371C2×C3×C4—94no—
372C2×C3×C4—122no—
373C2×C3×C4—182no—
374C2×C3×C4GO+(4,𝔽2)11yescharacteristic, complemented
375#67—22no—
376#67—62no—
377#67—62no—
378#67—64no—
379#67—64no—
380C2×Dic3—61no—
381C2×Dic3—61no—
382C2×Dic3—62no—
383D12—22no—
384D12—22no—
385D12—21no—
386D12—62no—
387C4×D3—22no—
388C4×D3—62no—
389C4×D3—61no—
390C4×D3—61no—
391C4×D3—61no—
392C4×D3—61no—
393C4×D3—61no—
394C4×D3—61no—
395C4×D3—61no—
396C4×D3—61no—
397C4×D3—62no—
398Dic6—21no—
399Dic6—62no—
400C2×C32—22no—
401C2×C32—21no—
402C2×C32—21no—
403C2×C32—21no—
404C2×C32—41no—
405C2×C32—41no—
406C2×C32—41no—
407C2×C32—42no—
408C2×C32—81no—
409C2×C32—124no—
410C2×C32—122no—
411C2×C32C22×C3×D411yescharacteristic
412C3⋊D3—21no—
413C3⋊D3C3×H(5,𝔽2)12yes—
414C3×D3—44no—
415C3×D3—42no—
416C3×D3—44no—
417C3×D3—42no—
418C3×D3—62no—
419C3×D3—62no—
420C3×D3—62no—
421C3×D3—82no—
422C3×D3—84no—
423C3×D3—121no—
424C24—92no—
425#41—61no—
426#41—181no—
427#41—181no—
428#41—181no—
429C2×Q8—91no—
430C2×D4—61no—
431C2×D4—92no—
432C2×D4—92no—
433C2×D4—92no—
434C2×D4—91no—
435C2×D4—98no—
436C2×D4—181no—
437C2×D4—181no—
438C2×D4—181no—
439C22×C4—92no—
440C22×C4—91no—
441C4⋊C4—92no—
442C22⋊C4—92no—
443C22⋊C4—98no—
444C42—92no—
445C22×C3—21no—
446C22×C3—21no—
447C22×C3—41no—
448C22×C3—41no—
449C22×C3—62no—
450C22×C3—64no—
451C22×C3—62no—
452C22×C3—64no—
453C22×C3—91no—
454C22×C3—91no—
455C22×C3—122no—
456C22×C3—124no—
457C22×C3—124no—
458C22×C3—124no—
459C22×C3—182no—
460C22×C3—184no—
461C22×C3—188no—
462C22×C3—244no—
463C22×C3—364no—
464C22×C3C2×GO+(4,𝔽2)11yescharacteristic
465C2×D3—22no—
466C2×D3—24no—
467C2×D3—44no—
468C2×D3—61no—
469C2×D3—61no—
470C2×D3—61no—
471C2×D3—61no—
472C2×D3—64no—
473C2×D3—128no—
474C2×D3—124no—
475C2×D3—122no—
476C2×D3—128no—
477C3×C4—21no—
478C3×C4—21no—
479C3×C4—21no—
480C3×C4—21no—
481C3×C4—41no—
482C3×C4—41no—
483C3×C4—41no—
484C3×C4—41no—
485C3×C4—62no—
486C3×C4—62no—
487C3×C4—91no—
488C3×C4—91no—
489C3×C4—122no—
490C3×C4—184no—
491C3×C4C2×GO+(4,𝔽2)11yescharacteristic
492C3×C4C2×GO+(4,𝔽2)11yescharacteristic
493Dic3—22no—
494Dic3—61no—
495Dic3—61no—
496Dic3—61no—
497Dic3—61no—
498C32—21no—
499C32—21no—
500C32—41no—
501C32—41no—
502C32C3×#53511yescharacteristic, complemented
503C23—62no—
504C23—94no—
505C23—92no—
506C23—91no—
507C23—182no—
508C23—364no—
509Q8—61no—
510Q8—181no—
511D4—62no—
512D4—62no—
513D4—62no—
514D4—61no—
515D4—182no—
516D4—182no—
517D4—182no—
518D4—181no—
519D4—368no—
520C2×C4—62no—
521C2×C4—92no—
522C2×C4—91no—
523C2×C4—91no—
524C2×C4—91no—
525C2×C4—91no—
526C2×C4—91no—
527C2×C4—94no—
528C2×C4—182no—
529C2×C4C3×GO+(4,𝔽2)11yescharacteristic, complemented
530C2×C3—22no—
531C2×C3—21no—
532C2×C3—21no—
533C2×C3—21no—
534C2×C3—41no—
535C2×C3—42no—
536C2×C3—41no—
537C2×C3—41no—
538C2×C3—81no—
539C2×C3—92no—
540C2×C3—124no—
541C2×C3—122no—
542C2×C3—124no—
543C2×C3—122no—
544C2×C3—181no—
545C2×C3—244no—
546C2×C3—242no—
547C2×C3C22×GO+(4,𝔽2)11yescharacteristic
548D3—44no—
549D3—42no—
550D3—62no—
551D3—62no—
552D3—62no—
553D3—121no—
554C22—62no—
555C22—64no—
556C22—91no—
557C22—91no—
558C22—124no—
559C22—182no—
560C22—184no—
561C22—188no—
562C22—364no—
563C22C2×C3×GO+(4,𝔽2)11yescharacteristic
564C4—62no—
565C4—91no—
566C4—91no—
567C4—184no—
568C4C2×C3×GO+(4,𝔽2)11yescharacteristic
569C4C2×C3×GO+(4,𝔽2)11yescharacteristic
570C3—21no—
571C3—21no—
572C3—41no—
573C3—41no—
574C3#1059651211yescharacteristic, direct product factor
575C2—21no—
576C2—92no—
577C2—124no—
578C2—122no—
579C2—181no—
580C2C22×C3×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
centerC2×C3C22×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊D3C24×C3
FrattiniC2C22×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC2×C33C22×D4
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#535
LCSG ▷ C2×C3⋊D3 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C3×#64306C2semidirect
2. C2×C3×#3802C2semidirect
3. C3×#64286C2semidirect
4. C3×#64241C2semidirect
5. C3×#64231C2semidirect
6. C3×#64230C2semidirect
7. C3×#64221C2semidirect
8. #10596512C3direct (N × H)
10. C2×C3×GO+(4,𝔽2)C22semidirect
21. C3×#3802C22semidirect
22. C3×C4×D32C22semidirect
24. C3×#3797C22semidirect
27. C3×C4⋊#157C22semidirect
28. C3×C4×#157C22semidirect
29. C3×D32⋊C4C22semidirect
32. #64306C2×C3semidirect
33. C2×#3802C2×C3semidirect
34. #64286C2×C3semidirect
35. #64241C2×C3semidirect
36. #64231C2×C3semidirect
37. #64230C2×C3semidirect
38. #64221C2×C3semidirect
47. C2×C33×C4D4semidirect
48. C2×C3×C3⋊Dic3D4semidirect
49. C3×C4×C3⋊D3D4semidirect
50. C3×C4×C3⋊D3D4semidirect
69. C2×GO+(4,𝔽2)C22×C3semidirect
101. #3802C22×C3semidirect
102. C4×D32C22×C3semidirect
104. #3797C22×C3semidirect
107. C4⋊#157C22×C3semidirect
108. C4×#157C22×C3semidirect
109. D32⋊C4C22×C3semidirect
162. C2×C32×C4C3×D4semidirect
163. C2×C3⋊Dic3C3×D4semidirect
164. C4×C3⋊D3C3×D4semidirect
165. C4×C3⋊D3C3×D4semidirect
322. C33#535semidirect
374. C2×C3×C4GO+(4,𝔽2)semidirect
502. C32C3×#535semidirect
529. C2×C4C3×GO+(4,𝔽2)semidirect
574. C3#10596512direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#643062
C2×C3×#38022
C3×#642862
C3×#642412
C3×#642312
C3×#642302
C3×#642212
#105965123
C3×#5359