Finite Group Databasebeta

← C3×#10593457 #13166826

Subgroups — C3×#10593457 #13166826

Order 1728.

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

normal subgroups only

This group has 784 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
1C3×D32×D4C211yesmaximal, characteristic, complemented
2C3×#64230C211yesmaximal, characteristic, complemented
3C3×#64225C211yesmaximal, characteristic, complemented
4C3×#63926C211yesmaximal, characteristic, complemented
5C3×#63782C211yesmaximal, characteristic, complemented
6C3×#63729C211yesmaximal, characteristic, complemented
7C3×D32⋊C8C211yesmaximal, characteristic, complemented
8#10593457C311yesmaximal, characteristic, direct product factor
9C22×C3×D32—21no—
10C2×C3×GO+(4,𝔽2)—21no—
11C3×D4×C3⋊D3C2211yescharacteristic
12C32×D3×D4—21no—
13C3×#3811—21no—
14C3×D3×#67—22no—
15C3×#3802C2211yescharacteristic
16C3×#3802C2211yescharacteristic, complemented
17C3×D3×D12—21no—
18C3×C4×D32C2211yescharacteristic
19C3×C4⋊#157C2211yescharacteristic, complemented
20C3×#3787C2211yescharacteristic, complemented
21C3×#3774—22no—
22C3×D32⋊C4—21no—
23C32×#213—21no—
24C3×#3714—21no—
25C3×#3710C2211yescharacteristic, complemented
26C2×C3×D3×D4—61no—
27D32×D4C2×C311yescharacteristic, complemented
28#64230C2×C311yescharacteristic, complemented
29#64225C2×C311yescharacteristic, complemented
30C3×D3×D8—61no—
31#63926C2×C311yescharacteristic, complemented
32#63782C2×C311yescharacteristic, complemented
33#63729C2×C311yescharacteristic, complemented
34D32⋊C8C2×C311yescharacteristic, complemented
35C22×C3×C3⋊D3—21no—
36C22×C32×D3—22no—
37C2×C3×D32—22no—
38C2×C3×D32—21no—
39C2×C3×D32—21no—
40C2×C3×D32—42no—
41C2×C3×D32—42no—
42C2×C3×D32D411yescharacteristic
43C2×C3×#157—21no—
44C3×GO+(4,𝔽2)—42no—
45C33×D4D411yescharacteristic, complemented
46C3×#638—21no—
47C3×C4⋊(C3⋊D3)D411yescharacteristic, complemented
48C3×C4×C3⋊D3C2311yescharacteristic
49C32×#67—22no—
50C32×D12—21no—
51C32×D12—21no—
52C32×C4×D3—21no—
53C3×#626—22no—
54C3×#625—21no—
55C3×#625D412yescomplemented
56C3×#624D411yescharacteristic
57C3×D3×Dic3—21no—
58C3×#622—21no—
59C32×C3⋊C8—21no—
60C3×#447—91nomaximal
61C23×C3×D3—62no—
62C22×D32—21no—
63C2×GO+(4,𝔽2)—21no—
64C2×C32×D4—61no—
65D4×C3⋊D3C22×C311yescharacteristic
66C2×C3×#67—62no—
67C3×D3×D4—21no—
68C3×D3×D4—21no—
69C3×D3×D4—41no—
70C3×D3×D4—61no—
71C3×D3×D4—61no—
72C3×D3×D4—61no—
73C3×D3×D4—62no—
74C3×D3×D4—61no—
75C3×D3×D4—62no—
76C3×D3×D4—61no—
77C2×C3×D12—61no—
78C2×C3×C4×D3—61no—
79#3811—21no—
80D3×#67—22no—
81#3802C22×C311yescharacteristic
82#3802C22×C311yescharacteristic, complemented
83D3×D12—21no—
84C4×D32C22×C311yescharacteristic
85C4⋊#157C22×C311yescharacteristic, complemented
86#3787C22×C311yescharacteristic, complemented
87#3774—22no—
88D32⋊C4—21no—
89C32×D8—61no—
90C3×#213—21no—
91C3×#213—21no—
92C3×#213—41no—
93C3×#213—61no—
94C3×D24—61no—
95C3×D3×C8—61no—
96#3714—21no—
97#3710C22×C311yescharacteristic, complemented
98C22×C33—21no—
99C2×C3×C3⋊D3—21no—
100C2×C3×C3⋊D3—41no—
101C2×C3×C3⋊D3—41no—
102C2×C3×C3⋊D3C2×D411yescharacteristic
103C2×C32×D3—21no—
104C2×C32×D3—22no—
105C2×C32×D3—44no—
106C2×C32×D3—41no—
107C3×D32—21no—
108C3×D32—44no—
109C3×D32—41no—
110C3×D32—41no—
111C3×D32—41no—
112C3×D32—41no—
113C3×D32D812yescomplemented
114C3×#157—41no—
115C33×C4C2×D411yescharacteristic
116C3×C3⋊Dic3C2×D411yescharacteristic
117C32×Dic3—21no—
118C22×C3×D4—91no—
119C2×D3×D4—61no—
120C2×C3×D8—91no—
121C2×C3×D8—91no—
122C3×#121—91no—
123D3×D8—61no—
124C3×#102—91no—
125C3×#102—91no—
126C3×C22⋊C8—91no—
127C23×C32—62no—
128C22×C3⋊D3—21no—
129C22×C3×D3—22no—
130C22×C3×D3—22no—
131C22×C3×D3—42no—
132C22×C3×D3—61no—
133C22×C3×D3—62no—
134C22×C3×D3—62no—
135C22×C3×D3—62no—
136C22×C3×D3—62no—
137C22×C3×D3—121no—
138C22×C3×D3—121no—
139C22×C3×D3—124no—
140C22×C3×D3—124no—
141C2×D32—22no—
142C2×D32—21no—
143C2×D32—21no—
144C2×D32—42no—
145C2×D32—42no—
146C2×D32C3×D411yescharacteristic
147C2×#157—21no—
148GO+(4,𝔽2)—42no—
149C32×D4—21no—
150C32×D4—21no—
151C32×D4—41no—
152C32×D4—41no—
153C32×D4—61no—
154C32×D4—61no—
155C32×D4—62no—
156C32×D4C3×D411yescharacteristic, complemented
157C2×C32×C4—61no—
158#638—21no—
159C4⋊(C3⋊D3)C3×D411yescharacteristic, complemented
160C4×C3⋊D3C23×C311yescharacteristic
161C3×#67—22no—
162C3×#67—22no—
163C3×#67—42no—
164C3×#67—62no—
165C3×#67—62no—
166C3×#67—62no—
167C3×#67—121no—
168C3×#67—121no—
169C2×C3×Dic3—61no—
170C3×D12—21no—
171C3×D12—21no—
172C3×D12—21no—
173C3×D12—21no—
174C3×D12—41no—
175C3×D12—41no—
176C3×D12—61no—
177C3×D12—61no—
178C3×D12—62no—
179C3×C4×D3—21no—
180C3×C4×D3—21no—
181C3×C4×D3—41no—
182C3×C4×D3—61no—
183C3×C4×D3—61no—
184C3×C4×D3—61no—
185C3×C4×D3—61no—
186#626—22no—
187#625—21no—
188#625C3×D412yescomplemented
189#624C3×D411yescharacteristic
190D3×Dic3—21no—
191#622—21no—
192C32×C8—61no—
193C3×C3⋊C8—21no—
194C3×C3⋊C8—21no—
195C3×C3⋊C8—41no—
196#447—91noSylow
197C2×C33—41no—
198C2×C33#12011yescharacteristic
199C3×C3⋊D3—41no—
200C3×C3⋊D3C2×D811yescharacteristic
201C3×C3⋊D3AGL(1,ℤ/8)11yescharacteristic
202C32×D3—22no—
203C32×D3—42no—
204C32×D3—81no—
205C24×C3—181no—
206C23×D3—62no—
207C2×C3×D4—61no—
208C2×C3×D4—61no—
209C2×C3×D4—91no—
210C2×C3×D4—91no—
211C2×C3×D4—91no—
212C2×C3×D4—121no—
213C2×C3×D4—181no—
214C2×C3×D4—181no—
215C2×C3×D4—182no—
216C2×C3×D4—181no—
217C22×C3×C4—91no—
218C2×#67—62no—
219D3×D4—21no—
220D3×D4—61no—
221D3×D4—61no—
222D3×D4—61no—
223D3×D4—61no—
224D3×D4—61no—
225D3×D4—62no—
226D3×D4—62no—
227C2×D12—61no—
228C2×C4×D3—61no—
229C3×D8—61no—
230C3×D8—61no—
231C3×D8—121no—
232C3×D8—181no—
233C3×D8—182no—
234C2×C3×C8—91no—
235C2×C3×C8—91no—
236C3×C4⋊C4—91no—
237C3×C22⋊C4—181no—
238#213—21no—
239#213—61no—
240D24—61no—
241D3×C8—61no—
242C22×C32—22no—
243C22×C32—21no—
244C22×C32—41no—
245C22×C32—42no—
246C22×C32—61no—
247C22×C32—62no—
248C22×C32—81no—
249C22×C32—121no—
250C22×C32—124no—
251C2×C3⋊D3—21no—
252C2×C3⋊D3—41no—
253C2×C3⋊D3—41no—
254C2×C3⋊D3C2×C3×D411yescharacteristic
255C2×C3×D3—21no—
256C2×C3×D3—21no—
257C2×C3×D3—22no—
258C2×C3×D3—22no—
259C2×C3×D3—41no—
260C2×C3×D3—42no—
261C2×C3×D3—44no—
262C2×C3×D3—44no—
263C2×C3×D3—41no—
264C2×C3×D3—41no—
265C2×C3×D3—62no—
266C2×C3×D3—61no—
267C2×C3×D3—61no—
268C2×C3×D3—62no—
269C2×C3×D3—62no—
270C2×C3×D3—84no—
271C2×C3×D3—81no—
272C2×C3×D3—121no—
273C2×C3×D3—124no—
274C2×C3×D3—122no—
275C2×C3×D3—122no—
276C2×C3×D3—124no—
277C2×C3×D3—122no—
278C2×C3×D3—122no—
279C2×C3×D3—241no—
280C2×C3×D3—241no—
281C2×C3×D3—241no—
282C2×C3×D3—241no—
283D32—21no—
284D32—44no—
285D32—41no—
286D32—41no—
287D32—41no—
288D32—41no—
289D32C3×D812yescomplemented
290#157—41no—
291C32×C4—21no—
292C32×C4—21no—
293C32×C4—41no—
294C32×C4—41no—
295C32×C4—61no—
296C32×C4C2×C3×D411yescharacteristic
297C3⋊Dic3C2×C3×D411yescharacteristic
298C3×Dic3—21no—
299C3×Dic3—21no—
300C3×Dic3—41no—
301C3×Dic3—61no—
302C3×Dic3—61no—
303C22×D4—91no—
304C2×D8—91no—
305C2×D8—91no—
306#121—91no—
307#102—91no—
308#102—91no—
309C22⋊C8—91no—
310C33#44711yescharacteristic, complemented, Sylow
311C23×C3—62no—
312C23×C3—62no—
313C23×C3—91no—
314C23×C3—122no—
315C23×C3—181no—
316C23×C3—181no—
317C23×C3—182no—
318C23×C3—181no—
319C23×C3—362no—
320C23×C3—362no—
321C22×D3—22no—
322C22×D3—61no—
323C22×D3—62no—
324C22×D3—62no—
325C22×D3—62no—
326C22×D3—62no—
327C22×D3—121no—
328C22×D3—121no—
329C22×D3—124no—
330C22×D3—124no—
331C3×D4—21no—
332C3×D4—21no—
333C3×D4—41no—
334C3×D4—41no—
335C3×D4—61no—
336C3×D4—61no—
337C3×D4—62no—
338C3×D4—62no—
339C3×D4—61no—
340C3×D4—61no—
341C3×D4—92no—
342C3×D4—91no—
343C3×D4—121no—
344C3×D4—121no—
345C3×D4—122no—
346C3×D4—182no—
347C3×D4—181no—
348C3×D4—181no—
349C3×D4—362no—
350C3×D4GO+(4,𝔽2)11yescharacteristic, complemented
351C2×C3×C4—61no—
352C2×C3×C4—61no—
353C2×C3×C4—91no—
354C2×C3×C4—91no—
355C2×C3×C4—121no—
356C2×C3×C4—181no—
357C2×C3×C4—181no—
358#67—22no—
359#67—62no—
360#67—62no—
361#67—62no—
362#67—121no—
363#67—121no—
364C2×Dic3—61no—
365D12—21no—
366D12—21no—
367D12—61no—
368D12—61no—
369D12—62no—
370C4×D3—21no—
371C4×D3—61no—
372C4×D3—61no—
373C4×D3—61no—
374C4×D3—61no—
375C3×C8—61no—
376C3×C8—61no—
377C3×C8—121no—
378C3×C8—181no—
379C3⋊C8—21no—
380C2×C32—21no—
381C2×C32—21no—
382C2×C32—41no—
383C2×C32—41no—
384C2×C32—42no—
385C2×C32—41no—
386C2×C32—62no—
387C2×C32—81no—
388C2×C32—82no—
389C2×C32—122no—
390C2×C32—161no—
391C2×C32—241no—
392C2×C32C3×#12011yescharacteristic
393C3⋊D3—41no—
394C3⋊D3C2×C3×D811yescharacteristic
395C3⋊D3C3×AGL(1,ℤ/8)11yescharacteristic
396C3×D3—22no—
397C3×D3—22no—
398C3×D3—42no—
399C3×D3—42no—
400C3×D3—42no—
401C3×D3—61no—
402C3×D3—61no—
403C3×D3—61no—
404C3×D3—61no—
405C3×D3—82no—
406C3×D3—81no—
407C3×D3—81no—
408C3×D3—122no—
409C3×D3—161no—
410C3×D3—241no—
411C24—181no—
412C2×D4—61no—
413C2×D4—91no—
414C2×D4—91no—
415C2×D4—91no—
416C2×D4—181no—
417C2×D4—181no—
418C2×D4—182no—
419C2×D4—181no—
420C22×C4—91no—
421D8—61no—
422D8—181no—
423D8—182no—
424C2×C8—91no—
425C2×C8—91no—
426C4⋊C4—91no—
427C22⋊C4—181no—
428C22×C3—22no—
429C22×C3—21no—
430C22×C3—41no—
431C22×C3—42no—
432C22×C3—61no—
433C22×C3—62no—
434C22×C3—61no—
435C22×C3—62no—
436C22×C3—81no—
437C22×C3—92no—
438C22×C3—91no—
439C22×C3—121no—
440C22×C3—122no—
441C22×C3—121no—
442C22×C3—124no—
443C22×C3—124no—
444C22×C3—121no—
445C22×C3—181no—
446C22×C3—181no—
447C22×C3—241no—
448C22×C3—244no—
449C22×C3—361no—
450C22×C3—361no—
451C22×C3—364no—
452C22×C3—361no—
453C22×C3—361no—
454C22×C3—361no—
455C22×C3—361no—
456C2×D3—21no—
457C2×D3—22no—
458C2×D3—44no—
459C2×D3—41no—
460C2×D3—62no—
461C2×D3—61no—
462C2×D3—61no—
463C2×D3—62no—
464C2×D3—62no—
465C2×D3—121no—
466C2×D3—124no—
467C2×D3—122no—
468C2×D3—122no—
469C2×D3—124no—
470C2×D3—122no—
471C2×D3—122no—
472C2×D3—241no—
473C2×D3—241no—
474C2×D3—241no—
475C2×D3—241no—
476C3×C4—21no—
477C3×C4—21no—
478C3×C4—41no—
479C3×C4—41no—
480C3×C4—61no—
481C3×C4—61no—
482C3×C4—91no—
483C3×C4—121no—
484C3×C4—361no—
485C3×C4C2×GO+(4,𝔽2)11yescharacteristic
486Dic3—21no—
487Dic3—61no—
488Dic3—61no—
489C32—21no—
490C32—21no—
491C32—41no—
492C32—41no—
493C32C3×#44711yescharacteristic, complemented
494C23—62no—
495C23—91no—
496C23—181no—
497C23—181no—
498C23—182no—
499C23—181no—
500C23—362no—
501C23—362no—
502D4—62no—
503D4—61no—
504D4—61no—
505D4—92no—
506D4—91no—
507D4—182no—
508D4—181no—
509D4—181no—
510D4—362no—
511D4C3×GO+(4,𝔽2)11yescharacteristic, complemented
512C2×C4—61no—
513C2×C4—91no—
514C2×C4—91no—
515C2×C4—181no—
516C2×C4—181no—
517C8—61no—
518C8—181no—
519C2×C3—21no—
520C2×C3—21no—
521C2×C3—42no—
522C2×C3—41no—
523C2×C3—41no—
524C2×C3—41no—
525C2×C3—62no—
526C2×C3—62no—
527C2×C3—81no—
528C2×C3—82no—
529C2×C3—91no—
530C2×C3—91no—
531C2×C3—122no—
532C2×C3—122no—
533C2×C3—122no—
534C2×C3—161no—
535C2×C3—242no—
536C2×C3—241no—
537C2×C3—241no—
538C2×C3—361no—
539C2×C3—481no—
540C2×C3#6424111yescharacteristic
541D3—22no—
542D3—42no—
543D3—61no—
544D3—61no—
545D3—61no—
546D3—61no—
547D3—81no—
548D3—122no—
549D3—241no—
550C22—21no—
551C22—61no—
552C22—62no—
553C22—92no—
554C22—91no—
555C22—124no—
556C22—121no—
557C22—181no—
558C22—181no—
559C22—361no—
560C22—361no—
561C22—364no—
562C22—361no—
563C22—361no—
564C22—361no—
565C22—361no—
566C4—61no—
567C4—91no—
568C4—361no—
569C4C2×C3×GO+(4,𝔽2)11yescharacteristic
570C3—21no—
571C3—21no—
572C3—41no—
573C3—41no—
574C3#1059345711yescharacteristic, direct product factor
575C2—41no—
576C2—62no—
577C2—91no—
578C2—91no—
579C2—122no—
580C2—241no—
581C2—361no—
582C2C3×#6424111yescharacteristic
subgroupG/H
centerC2×C3#64241
commutator, cosocleC4×C3⋊D3C23×C3
FrattiniC4C2×C3×GO+(4,𝔽2)
FittingC33×D4D4
socleC2×C33#120
hypercenterC3×D4GO+(4,𝔽2)
hypocenterC32C3×#447
LCSG ▷ C4×C3⋊D3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×D4

Complements

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

NH ≅ G/Ntype
1. C3×D32×D4C2semidirect
2. C3×#64230C2semidirect
3. C3×#64225C2semidirect
4. C3×#63926C2semidirect
5. C3×#63782C2semidirect
6. C3×#63729C2semidirect
7. C3×D32⋊C8C2semidirect
8. #10593457C3direct (N × H)
16. C3×#3802C22semidirect
19. C3×C4⋊#157C22semidirect
20. C3×#3787C22semidirect
25. C3×#3710C22semidirect
27. D32×D4C2×C3semidirect
28. #64230C2×C3semidirect
29. #64225C2×C3semidirect
31. #63926C2×C3semidirect
32. #63782C2×C3semidirect
33. #63729C2×C3semidirect
34. D32⋊C8C2×C3semidirect
45. C33×D4D4semidirect
47. C3×C4⋊(C3⋊D3)D4semidirect
55. C3×#625D4semidirect
82. #3802C22×C3semidirect
85. C4⋊#157C22×C3semidirect
86. #3787C22×C3semidirect
97. #3710C22×C3semidirect
113. C3×D32D8semidirect
156. C32×D4C3×D4semidirect
159. C4⋊(C3⋊D3)C3×D4semidirect
188. #625C3×D4semidirect
289. D32C3×D8semidirect
310. C33#447semidirect
350. C3×D4GO+(4,𝔽2)semidirect
493. C32C3×#447semidirect
511. D4C3×GO+(4,𝔽2)semidirect
574. C3#10593457direct (N × H)

Maximal subgroups

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

subgroupindex
C3×D32×D42
C3×#642302
C3×#642252
C3×#639262
C3×#637822
C3×#637292
C3×D32⋊C82
#105934573
C3×#4479