Finite Group Databasebeta

← C3×#10591509 #13158532

Subgroups — C3×#10591509 #13158532

Order 1728.

1046 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 1046 conjugacy classes of subgroups — too many to draw. Only its 128 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
1C32×#1004C211yesmaximal, characteristic, complemented
2C32×#940C211yesmaximal, characteristic, complemented
3C3×#63823C212yesmaximal, complemented
4C3×#63660C211yesmaximal, characteristic, complemented
5C3×#63569C212yesmaximal, complemented
6C3×#5241—31nomaximal
7C3×#5077—31nomaximal
8#10591509C311yesmaximal, characteristic, direct product factor
9C2×C33×D4C2211yescharacteristic
10C2×C32×#67—22no—
11C2×C32×#67—22no—
12C32×#237—22no—
13C32×D3×D4—22no—
14C32×#235C2212yes—
15C2×C32×D12C2211yescharacteristic, complemented
16C3×#3748C2212yescomplemented
17C32×C4×Dic3C2211yescharacteristic, complemented
18C3×#3715—22no—
19C3×#3713—22no—
20C32×H(5,𝔽2)—31no—
21C3×#1004—21no—
22C3×#1004D311yescharacteristic, complemented
23C3×#1004C2×C311yescharacteristic, complemented
24C32×#127—31no—
25C3×#949—32no—
26C3×#940—21no—
27C3×#940D311yescharacteristic, complemented
28C3×#940C2×C311yescharacteristic, complemented
29C3×#932—32no—
30#63823C2×C312yescomplemented
31#63660C2×C311yescharacteristic, complemented
32C3×#837—32no—
33C3×#833—31no—
34C3×#833—31no—
35C3×#833—61no—
36C3×#805—32no—
37#63569C2×C312yescomplemented
38C23×C33D412yes—
39C22×C32×D3—22no—
40C22×C32×D3—21no—
41C33×D4—22no—
42C2×C33×C4C2311yescharacteristic
43C32×#67—24no—
44C32×#67—22no—
45C32×#67—44no—
46C2×C32×Dic3—22no—
47C2×C32×Dic3—21no—
48C32×D12—22no—
49C32×D12D412yescomplemented
50C32×C4×D3—22no—
51C32×Dic6D412yescomplemented
52C3×C3⋊(C3⋊C8)—22no—
53C3×#453—91no—
54#5241—31no—
55#5077—31no—
56C32×#41—32no—
57C32×#41—62no—
58C2×C32×D4—21no—
59C2×C32×D4—21no—
60C2×C32×D4—21no—
61C2×C32×D4—31no—
62C2×C32×D4—41no—
63C2×C32×D4—62no—
64C2×C32×D4—62no—
65C2×C32×D4—62no—
66C2×C32×D4C2×D311yescharacteristic
67C2×C32×D4C2×D311yescharacteristic
68C2×C32×D4C22×C311yescharacteristic
69C2×C3×#67—24no—
70C2×C3×#67—22no—
71C2×C3×#67—22no—
72C2×C3×#67—24no—
73C2×C3×#67—22no—
74C2×C3×#67—22no—
75C3×#237—24no—
76C3×#237—22no—
77C3×#237—22no—
78C3×D3×D4—24no—
79C3×D3×D4—22no—
80C3×D3×D4—22no—
81C3×#235—22no—
82C3×#235C2×D312yes—
83C3×#235C22×C312yes—
84C2×C3×D12—21no—
85C2×C3×D12C2×D311yescharacteristic, complemented
86C2×C3×D12C22×C311yescharacteristic, complemented
87C32×C42—31no—
88#3748C22×C312yescomplemented
89C3×#215—62no—
90C3×#214—62no—
91C3×#213—62no—
92C3×#213—62no—
93C3×C4×Dic3—21no—
94C3×C4×Dic3C2×D311yescharacteristic, complemented
95C3×C4×Dic3C22×C311yescharacteristic, complemented
96C3×#208—32no—
97C3×#208—32no—
98C3×#208—62no—
99#3715—22no—
100#3713—22no—
101C22×C33—22no—
102C22×C33—42no—
103C22×C33C2×D411yescharacteristic
104C2×C32×D3—22no—
105C2×C32×D3—42no—
106C2×C32×D3—41no—
107C2×C32×D3—82no—
108C33×C4C2×D412yes—
109C32×Dic3—22no—
110C32×Dic3—22no—
111C3×H(5,𝔽2)—31no—
112C3×H(5,𝔽2)—31no—
113C3×H(5,𝔽2)—61no—
114#1004C3×D311yescharacteristic, complemented
115C3×AGL(1,ℤ/8)—92no—
116C3×#127—31no—
117C3×#127—31no—
118C3×#127—61no—
119#949—32no—
120#940C3×D311yescharacteristic, complemented
121#932—32no—
122C3×#104—92no—
123C3×#100—91no—
124#837—32no—
125#833—31no—
126#833—31no—
127#833—61no—
128#805—32no—
129C23×C32—22no—
130C23×C32—22no—
131C23×C32—22no—
132C23×C32—42no—
133C23×C32—62no—
134C23×C32—61no—
135C23×C32#6712yes—
136C23×C32#6712yes—
137C23×C32C3×D412yes—
138C22×C3×D3—24no—
139C22×C3×D3—22no—
140C22×C3×D3—22no—
141C22×C3×D3—22no—
142C22×C3×D3—21no—
143C22×C3×D3—21no—
144C32×Q8—32no—
145C32×D4—22no—
146C32×D4—24no—
147C32×D4—22no—
148C32×D4—22no—
149C32×D4—32no—
150C32×D4—42no—
151C32×D4—42no—
152C32×D4—44no—
153C32×D4—64no—
154C32×D4—62no—
155C32×D4—62no—
156C32×D4—124no—
157C2×C32×C4—21no—
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4—41no—
161C2×C32×C4—62no—
162C2×C32×C4—61no—
163C2×C32×C4—62no—
164C2×C32×C4C22×D311yescharacteristic
165C2×C32×C4C22×D311yescharacteristic
166C2×C32×C4C23×C311yescharacteristic
167C3×#67—24no—
168C3×#67—28no—
169C3×#67—24no—
170C3×#67—24no—
171C3×#67—22no—
172C3×#67—22no—
173C3×#67—44no—
174C3×#67—44no—
175C3×#67—48no—
176C2×C3×Dic3—22no—
177C2×C3×Dic3—24no—
178C2×C3×Dic3—22no—
179C2×C3×Dic3—22no—
180C2×C3×Dic3—21no—
181C2×C3×Dic3—21no—
182C3×D12—22no—
183C3×D12—22no—
184C3×D12—22no—
185C3×D12—42no—
186C3×D12#6712yescomplemented
187C3×D12C3×D412yescomplemented
188C3×C4×D3—24no—
189C3×C4×D3—22no—
190C3×C4×D3—22no—
191C3×Dic6—22no—
192C3×Dic6#6712yescomplemented
193C3×Dic6C3×D412yescomplemented
194C3⋊(C3⋊C8)—22no—
195C3×C3⋊C8—62no—
196C3×C3⋊C8—62no—
197C3×C3⋊C8—122no—
198#453—91noSylow
199C2×C33—21no—
200C2×C33—42no—
201C2×C33#12011yescharacteristic
202C32×D3—42no—
203C32×D3—81no—
204C3×#41—32no—
205C3×#41—32no—
206C3×#41—62no—
207C3×#41—62no—
208C3×#41—62no—
209C3×#41—64no—
210C2×C3×D4—21no—
211C2×C3×D4—21no—
212C2×C3×D4—21no—
213C2×C3×D4—31no—
214C2×C3×D4—31no—
215C2×C3×D4—41no—
216C2×C3×D4—61no—
217C2×C3×D4—62no—
218C2×C3×D4—62no—
219C2×C3×D4—64no—
220C2×C3×D4—64no—
221C2×C3×D4—62no—
222C2×C3×D4—62no—
223C2×C3×D4—64no—
224C2×C3×D4—62no—
225C2×C3×D4—62no—
226C2×C3×D4D3211yescharacteristic
227C2×C3×D4C2×C3×D311yescharacteristic
228C2×C3×D4C2×C3×D311yescharacteristic
229C2×#67—22no—
230C2×#67—22no—
231#237—22no—
232D3×D4—22no—
233#235C2×C3×D312yes—
234C2×D12C2×C3×D311yescharacteristic, complemented
235C3×SD16—182no—
236C3×D8—182no—
237C3×M16—92no—
238C3×C42—31no—
239C3×C42—31no—
240C3×C42—61no—
241#215—62no—
242#214—62no—
243#213—62no—
244#213—62no—
245C4×Dic3C2×C3×D311yescharacteristic, complemented
246#208—32no—
247#208—32no—
248#208—62no—
249C22×C32—22no—
250C22×C32—24no—
251C22×C32—22no—
252C22×C32—22no—
253C22×C32—21no—
254C22×C32—21no—
255C22×C32—21no—
256C22×C32—41no—
257C22×C32—42no—
258C22×C32—42no—
259C22×C32—44no—
260C22×C32—42no—
261C22×C32—42no—
262C22×C32—42no—
263C22×C32—44no—
264C22×C32—62no—
265C22×C32—82no—
266C22×C32—82no—
267C22×C32—84no—
268C22×C32—122no—
269C22×C32—121no—
270C22×C32—242no—
271C22×C32C2×#6711yescharacteristic
272C22×C32C2×#6711yescharacteristic
273C22×C32C2×C3×D411yescharacteristic
274C2×C3×D3—24no—
275C2×C3×D3—22no—
276C2×C3×D3—22no—
277C2×C3×D3—42no—
278C2×C3×D3—44no—
279C2×C3×D3—42no—
280C2×C3×D3—42no—
281C2×C3×D3—41no—
282C2×C3×D3—41no—
283C2×C3×D3—84no—
284C2×C3×D3—82no—
285C2×C3×D3—82no—
286C32×C4—22no—
287C32×C4—22no—
288C32×C4—22no—
289C32×C4—42no—
290C32×C4—62no—
291C32×C4—62no—
292C32×C4D3×D412yes—
293C32×C4C2×#6712yes—
294C32×C4C2×C3×D412yes—
295C3×Dic3—24no—
296C3×Dic3—24no—
297C3×Dic3—22no—
298C3×Dic3—22no—
299C3×Dic3—22no—
300C3×Dic3—22no—
301H(5,𝔽2)—31no—
302AGL(1,ℤ/8)—92no—
303#127—31no—
304#104—92no—
305#100—91no—
306C33#45311yescharacteristic, complemented, Sylow
307C23×C3—22no—
308C23×C3—22no—
309C23×C3—22no—
310C23×C3—42no—
311C23×C3—64no—
312C23×C3—62no—
313C23×C3—62no—
314C23×C3—62no—
315C23×C3—61no—
316C23×C3—61no—
317C23×C3#62512yes—
318C23×C3C3×#6712yes—
319C23×C3C3×#6712yes—
320C22×D3—22no—
321C22×D3—21no—
322C3×Q8—32no—
323C3×Q8—32no—
324C3×Q8—62no—
325C3×D4—24no—
326C3×D4—22no—
327C3×D4—22no—
328C3×D4—22no—
329C3×D4—32no—
330C3×D4—32no—
331C3×D4—42no—
332C3×D4—42no—
333C3×D4—44no—
334C3×D4—62no—
335C3×D4—68no—
336C3×D4—64no—
337C3×D4—64no—
338C3×D4—64no—
339C3×D4—62no—
340C3×D4—62no—
341C3×D4—62no—
342C3×D4—62no—
343C3×D4—122no—
344C3×D4—124no—
345C3×D4—124no—
346C3×D4—128no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—41no—
351C2×C3×C4—64no—
352C2×C3×C4—62no—
353C2×C3×C4—64no—
354C2×C3×C4—62no—
355C2×C3×C4—62no—
356C2×C3×C4—61no—
357C2×C3×C4—61no—
358C2×C3×C4—62no—
359C2×C3×C4—62no—
360C2×C3×C4C2×D3211yescharacteristic
361C2×C3×C4C22×C3×D311yescharacteristic
362C2×C3×C4C22×C3×D311yescharacteristic
363#67—24no—
364#67—22no—
365#67—44no—
366C2×Dic3—22no—
367C2×Dic3—21no—
368D12—22no—
369D12C3×#6712yescomplemented
370C4×D3—22no—
371Dic6C3×#6712yescomplemented
372C3×C8—182no—
373C3⋊C8—62no—
374C3⋊C8—62no—
375C3⋊C8—122no—
376C2×C32—21no—
377C2×C32—22no—
378C2×C32—21no—
379C2×C32—21no—
380C2×C32—21no—
381C2×C32—21no—
382C2×C32—21no—
383C2×C32—41no—
384C2×C32—41no—
385C2×C32—41no—
386C2×C32—42no—
387C2×C32—42no—
388C2×C32—44no—
389C2×C32—42no—
390C2×C32—42no—
391C2×C32—82no—
392C2×C32—82no—
393C2×C32—84no—
394C2×C32—122no—
395C2×C32—241no—
396C2×C32#93711yescharacteristic
397C2×C32#95311yescharacteristic
398C2×C32C3×#12011yescharacteristic
399C3×D3—44no—
400C3×D3—42no—
401C3×D3—42no—
402C3×D3—82no—
403C3×D3—81no—
404C3×D3—81no—
405#41—32no—
406#41—62no—
407C2×D4—31no—
408C2×D4—62no—
409C2×D4—62no—
410C2×D4—62no—
411C2×D4C3×D3211yescharacteristic
412SD16—182no—
413D8—182no—
414M16—92no—
415C42—31no—
416C22×C3—24no—
417C22×C3—22no—
418C22×C3—22no—
419C22×C3—22no—
420C22×C3—21no—
421C22×C3—21no—
422C22×C3—21no—
423C22×C3—42no—
424C22×C3—42no—
425C22×C3—44no—
426C22×C3—42no—
427C22×C3—42no—
428C22×C3—42no—
429C22×C3—44no—
430C22×C3—41no—
431C22×C3—64no—
432C22×C3—62no—
433C22×C3—62no—
434C22×C3—82no—
435C22×C3—82no—
436C22×C3—84no—
437C22×C3—124no—
438C22×C3—122no—
439C22×C3—122no—
440C22×C3—122no—
441C22×C3—121no—
442C22×C3—121no—
443C22×C3—244no—
444C22×C3—242no—
445C22×C3—242no—
446C22×C3C2×#62511yescharacteristic
447C22×C3C2×C3×#6711yescharacteristic
448C22×C3C2×C3×#6711yescharacteristic
449C2×D3—22no—
450C2×D3—42no—
451C2×D3—41no—
452C2×D3—82no—
453C3×C4—22no—
454C3×C4—22no—
455C3×C4—22no—
456C3×C4—42no—
457C3×C4—64no—
458C3×C4—64no—
459C3×C4—62no—
460C3×C4—62no—
461C3×C4—62no—
462C3×C4—62no—
463C3×C4D3×#6712yes—
464C3×C4C3×D3×D412yes—
465C3×C4C2×C3×#6712yes—
466Dic3—22no—
467Dic3—22no—
468C32—21no—
469C32—21no—
470C32—21no—
471C32—41no—
472C32#507711yescharacteristic, complemented
473C32#524111yescharacteristic, complemented
474C32C3×#45311yescharacteristic, complemented
475C23—62no—
476C23—61no—
477C23C3×#62512yes—
478Q8—32no—
479D4—22no—
480D4—32no—
481D4—64no—
482D4—62no—
483D4—62no—
484D4—124no—
485C2×C4—62no—
486C2×C4—61no—
487C2×C4—62no—
488C2×C4C2×C3×D3211yescharacteristic
489C8—182no—
490C2×C3—22no—
491C2×C3—21no—
492C2×C3—21no—
493C2×C3—21no—
494C2×C3—21no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—41no—
498C2×C3—41no—
499C2×C3—42no—
500C2×C3—44no—
501C2×C3—42no—
502C2×C3—42no—
503C2×C3—42no—
504C2×C3—41no—
505C2×C3—82no—
506C2×C3—82no—
507C2×C3—84no—
508C2×C3—124no—
509C2×C3—122no—
510C2×C3—122no—
511C2×C3—242no—
512C2×C3—241no—
513C2×C3—241no—
514C2×C3#6397611yescharacteristic
515C2×C3C3×#93711yescharacteristic
516C2×C3C3×#95311yescharacteristic
517D3—42no—
518D3—81no—
519C22—22no—
520C22—42no—
521C22—62no—
522C22—122no—
523C22—121no—
524C22—242no—
525C22C2×C3×#62511yescharacteristic
526C4—62no—
527C4—62no—
528C4C3×D3×#6712yes—
529C3—21no—
530C3—21no—
531C3—21no—
532C3—41no—
533C3#1059150911yescharacteristic, direct product factor
534C3C3×#507711yescharacteristic, complemented
535C3C3×#524111yescharacteristic, complemented
536C2—21no—
537C2—42no—
538C2—122no—
539C2—241no—
540C2C3×#6397611yescharacteristic
subgroupG/H
centerC2×C3#63976
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×D4C22
socleC2×C33#120
hypercenterC2×C3×D4D32
hypocenterC32C3×#453
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
1. C32×#1004C2semidirect
2. C32×#940C2semidirect
3. C3×#63823C2semidirect
4. C3×#63660C2semidirect
5. C3×#63569C2semidirect
8. #10591509C3direct (N × H)
15. C2×C32×D12C22semidirect
16. C3×#3748C22semidirect
17. C32×C4×Dic3C22semidirect
22. C3×#1004D3semidirect
23. C3×#1004C2×C3semidirect
27. C3×#940D3semidirect
28. C3×#940C2×C3semidirect
30. #63823C2×C3semidirect
31. #63660C2×C3semidirect
37. #63569C2×C3semidirect
49. C32×D12D4semidirect
51. C32×Dic6D4semidirect
85. C2×C3×D12C2×D3semidirect
86. C2×C3×D12C22×C3semidirect
88. #3748C22×C3semidirect
94. C3×C4×Dic3C2×D3semidirect
95. C3×C4×Dic3C22×C3semidirect
114. #1004C3×D3semidirect
120. #940C3×D3semidirect
186. C3×D12#67semidirect
187. C3×D12C3×D4semidirect
192. C3×Dic6#67semidirect
193. C3×Dic6C3×D4semidirect
234. C2×D12C2×C3×D3semidirect
245. C4×Dic3C2×C3×D3semidirect
306. C33#453semidirect
369. D12C3×#67semidirect
371. Dic6C3×#67semidirect
472. C32#5077semidirect
473. C32#5241semidirect
474. C32C3×#453semidirect
533. C3#10591509direct (N × H)
534. C3C3×#5077semidirect
535. C3C3×#5241semidirect

Maximal subgroups

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

subgroupindex
C32×#10042
C32×#9402
C3×#638232
C3×#636602
C3×#635692
C3×#52413
C3×#50773
#105915093