Finite Group Databasebeta

← C3×#10590581 #13157575

Subgroups — C3×#10590581 #13157575

Order 1728.

662 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×D3×C4⋊C4C211yesmaximal, characteristic
2C3×#63841C211yesmaximal, characteristic
3C2×C3×#3711C211yesmaximal, characteristic
4C3×#63634C211yesmaximal, characteristic, complemented
5C32×#808C211yesmaximal, characteristic, complemented
6C3×#63575C211yesmaximal, characteristic, complemented
7C3×#63574C211yesmaximal, characteristic, complemented
8C3×#4964—31nomaximal
9C3×#4861—31nomaximal
10#10590581C311yesmaximal, characteristic, direct product factor
11C2×C32×C4×D3—21no—
12C2×C32×C4×D3C2211yescharacteristic
13C33×C4⋊C4C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C32×Dic3⋊C4—21no—
19C2×C32×C3⋊C8C2211yescharacteristic
20C3×(C2×D3)⋊Dic3—21no—
21C3×Dic32—21no—
22C3×#3711C414yescomplemented
23C2×C32×C4⋊C4—31no—
24C3×#926—31no—
25C2×C3×#208—31no—
26C2×C3×ΓL(1,𝔽25)—31no—
27C3×#892—31no—
28C3×D3×C4⋊C4—21no—
29C3×D3×C4⋊C4D311yescharacteristic
30C3×D3×C4⋊C4C2×C311yescharacteristic
31#63841C2×C311yescharacteristic
32C2×#3711C2×C311yescharacteristic
33C32×#106—31no—
34#63634C2×C311yescharacteristic, complemented
35C3×#818—31no—
36C3×#808—21no—
37C3×#808—31no—
38C3×#808—31no—
39C3×#808D311yescharacteristic, complemented
40C3×#808C2×C311yescharacteristic, complemented
41C3×#807—31no—
42C3×#807—31no—
43C3×#807—31no—
44C3×#807—61no—
45#63575C2×C311yescharacteristic, complemented
46#63574C2×C311yescharacteristic, complemented
47C22×C32×D3D411yescharacteristic
48C2×C33×C4—21no—
49C2×C33×C4C2311yescharacteristic
50C2×C3×C3⋊Dic3—21no—
51C2×C32×Dic3—21no—
52C2×C32×Dic3—21no—
53C2×C32×Dic3D411yescharacteristic
54C32×C4×D3—42no—
55C32×C4×D3C2×C412yes—
56C32×C4×D3Q812yes—
57C3×C3⋊(C3⋊C8)C2×C412yescomplemented
58C32×C3⋊C8C2×C412yescomplemented
59C3×#428—91no—
60#4964—31no—
61#4861—31no—
62C22×C32×C4—31no—
63C22×C32×C4—61no—
64C2×C3×C4×D3—21no—
65C2×C3×C4×D3—22no—
66C2×C3×C4×D3—21no—
67C2×C3×C4×D3—21no—
68C2×C3×C4×D3C2×D311yescharacteristic
69C2×C3×C4×D3C22×C311yescharacteristic
70C2×C32×C8—31no—
71C32×C4⋊C4—21no—
72C32×C4⋊C4—21no—
73C32×C4⋊C4—21no—
74C32×C4⋊C4—31no—
75C32×C4⋊C4—31no—
76C32×C4⋊C4—41no—
77C32×C4⋊C4—61no—
78C32×C4⋊C4C2×D311yescharacteristic
79C32×C4⋊C4C2×D311yescharacteristic
80C32×C4⋊C4C22×C311yescharacteristic
81C4⋊(C3⋊Dic3)C22×C311yescharacteristic
82C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
83C3×#217—61no—
84C3×(C2×D3)⋊C4—61no—
85C3×C4⋊Dic3—21no—
86C3×C4⋊Dic3—21no—
87C3×C4⋊Dic3—31no—
88C3×C4⋊Dic3—31no—
89C3×C4⋊Dic3—61no—
90C3×C4⋊Dic3C2×D311yescharacteristic
91C3×C4⋊Dic3C2×D311yescharacteristic
92C3×C4⋊Dic3C22×C311yescharacteristic
93C3×C4⋊Dic3C22×C311yescharacteristic
94C3×Dic3⋊C4—22no—
95C3×Dic3⋊C4—21no—
96C3×Dic3⋊C4—21no—
97C3×C4×Dic3—61no—
98C3×C4×Dic3—61no—
99C3×#208—34no—
100C2×C3×C3⋊C8—21no—
101C2×C3×C3⋊C8—31no—
102C2×C3×C3⋊C8—31no—
103C2×C3×C3⋊C8—61no—
104C2×C3×C3⋊C8C2×D311yescharacteristic
105C2×C3×C3⋊C8C22×C311yescharacteristic
106C3×ΓL(1,𝔽25)—34no—
107(C2×D3)⋊Dic3—21no—
108Dic32—21no—
109#3711C3×C414yescomplemented
110C22×C33C2×D411yescharacteristic
111C2×C32×D3—21no—
112C2×C32×D3—21no—
113C2×C32×D3C4⋊C412yes—
114C33×C4—41no—
115C33×C4C22×C411yescharacteristic
116C33×C4C2×Q811yescharacteristic
117C3×C3⋊Dic3—41no—
118C32×Dic3—41no—
119C32×Dic3—41no—
120C32×Dic3C4⋊C412yes—
121C2×C3×M16—91no—
122C3×#117—91no—
123C2×C3×C4⋊C4—31no—
124C2×C3×C4⋊C4—31no—
125C2×C3×C4⋊C4—61no—
126#926—31no—
127C2×#208—31no—
128C2×ΓL(1,𝔽25)—31no—
129#892—31no—
130D3×C4⋊C4C3×D311yescharacteristic
131C3×#107—91no—
132C3×#107—91no—
133C3×#106—31no—
134C3×#106—31no—
135C3×#106—61no—
136C3×#106—91no—
137#818—31no—
138#808—31no—
139#808—31no—
140#808C3×D311yescharacteristic, complemented
141#807—31no—
142#807—31no—
143#807—31no—
144#807—61no—
145C23×C32—31no—
146C22×C3×D3—21no—
147C22×C3×D3#6711yescharacteristic
148C22×C3×D3C3×D411yescharacteristic
149C2×C32×C4—21no—
150C2×C32×C4—22no—
151C2×C32×C4—21no—
152C2×C32×C4—21no—
153C2×C32×C4—21no—
154C2×C32×C4—21no—
155C2×C32×C4—21no—
156C2×C32×C4—31no—
157C2×C32×C4—32no—
158C2×C32×C4—32no—
159C2×C32×C4—41no—
160C2×C32×C4—41no—
161C2×C32×C4—41no—
162C2×C32×C4—42no—
163C2×C32×C4—61no—
164C2×C32×C4—61no—
165C2×C32×C4—122no—
166C2×C32×C4C22×D311yescharacteristic
167C2×C32×C4C22×D311yescharacteristic
168C2×C32×C4C23×C311yescharacteristic
169C2×C3⋊Dic3—21no—
170C2×C3×Dic3—22no—
171C2×C3×Dic3—21no—
172C2×C3×Dic3—21no—
173C2×C3×Dic3—21no—
174C2×C3×Dic3—21no—
175C2×C3×Dic3—21no—
176C2×C3×Dic3—41no—
177C2×C3×Dic3—61no—
178C2×C3×Dic3—61no—
179C2×C3×Dic3—121no—
180C2×C3×Dic3#6711yescharacteristic
181C2×C3×Dic3C3×D411yescharacteristic
182C3×C4×D3—22no—
183C3×C4×D3—22no—
184C3×C4×D3—44no—
185C3×C4×D3—42no—
186C3×C4×D3—42no—
187C3×C4×D3Dic612yes—
188C3×C4×D3C4×D312yes—
189C3×C4×D3C2×C3×C412yes—
190C3×C4×D3C3×Q812yes—
191C32×C8—32no—
192C3⋊(C3⋊C8)C2×C3×C412yescomplemented
193C3×C3⋊C8—22no—
194C3×C3⋊C8—32no—
195C3×C3⋊C8—32no—
196C3×C3⋊C8—62no—
197C3×C3⋊C8C4×D312yescomplemented
198C3×C3⋊C8C2×C3×C412yescomplemented
199#428—91noSylow
200C2×C33C2×C4⋊C411yescharacteristic
201C2×C33AGL(1,ℤ/8)11yescharacteristic
202C2×C33#13711yescharacteristic
203C32×D3—22no—
204C22×C3×C4—31no—
205C22×C3×C4—31no—
206C22×C3×C4—61no—
207C22×C3×C4—61no—
208C22×C3×C4—61no—
209C22×C3×C4—62no—
210C2×C4×D3—21no—
211C2×C4×D3C2×C3×D311yescharacteristic
212C3×M16—94no—
213C2×C3×C8—31no—
214C2×C3×C8—31no—
215C2×C3×C8—61no—
216C2×C3×C8—91no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—31no—
221C3×C4⋊C4—31no—
222C3×C4⋊C4—31no—
223C3×C4⋊C4—31no—
224C3×C4⋊C4—41no—
225C3×C4⋊C4—61no—
226C3×C4⋊C4—61no—
227C3×C4⋊C4—61no—
228C3×C4⋊C4—61no—
229C3×C4⋊C4—62no—
230C3×C4⋊C4—91no—
231C3×C4⋊C4D3211yescharacteristic
232C3×C4⋊C4C2×C3×D311yescharacteristic
233C3×C4⋊C4C2×C3×D311yescharacteristic
234C3×C22⋊C4—181no—
235C3×C42—181no—
236#217—61no—
237(C2×D3)⋊C4—61no—
238C4⋊Dic3—31no—
239C4⋊Dic3—31no—
240C4⋊Dic3—61no—
241C4⋊Dic3C2×C3×D311yescharacteristic
242C4⋊Dic3C2×C3×D311yescharacteristic
243Dic3⋊C4—21no—
244C4×Dic3—61no—
245C4×Dic3—61no—
246#208—34no—
247C2×C3⋊C8—31no—
248C2×C3⋊C8—31no—
249C2×C3⋊C8—61no—
250C2×C3⋊C8C2×C3×D311yescharacteristic
251ΓL(1,𝔽25)—34no—
252C22×C32—21no—
253C22×C32—21no—
254C22×C32—21no—
255C22×C32—32no—
256C22×C32—41no—
257C22×C32—61no—
258C22×C32—61no—
259C22×C32D3×D411yescharacteristic
260C22×C32C2×#6711yescharacteristic
261C22×C32C2×C3×D411yescharacteristic
262C2×C3×D3—22no—
263C2×C3×D3—22no—
264C2×C3×D3—21no—
265C2×C3×D3—21no—
266C2×C3×D3—21no—
267C2×C3×D3—21no—
268C2×C3×D3—22no—
269C2×C3×D3Dic3⋊C412yes—
270C2×C3×D3C3×C4⋊C412yes—
271C32×C4—21no—
272C32×C4—21no—
273C32×C4—21no—
274C32×C4—21no—
275C32×C4—21no—
276C32×C4—21no—
277C32×C4—32no—
278C32×C4—41no—
279C32×C4—41no—
280C32×C4—41no—
281C32×C4—42no—
282C32×C4—41no—
283C32×C4—41no—
284C32×C4—81no—
285C32×C4—81no—
286C32×C4—82no—
287C32×C4—121no—
288C32×C4—121no—
289C32×C4C2×Dic611yescharacteristic
290C32×C4C2×C4×D311yescharacteristic
291C32×C4C2×C4×D311yescharacteristic
292C32×C4D3×Q811yescharacteristic
293C32×C4C22×C3×C411yescharacteristic
294C32×C4C2×C3×Q811yescharacteristic
295C3⋊Dic3—41no—
296C3×Dic3—22no—
297C3×Dic3—42no—
298C3×Dic3—41no—
299C3×Dic3—41no—
300C3×Dic3—41no—
301C3×Dic3—41no—
302C3×Dic3—81no—
303C3×Dic3—121no—
304C3×Dic3—121no—
305C3×Dic3—241no—
306C3×Dic3Dic3⋊C412yes—
307C3×Dic3C3×C4⋊C412yes—
308C2×M16—91no—
309#117—91no—
310C2×C4⋊C4—31no—
311#107—91no—
312#107—91no—
313#106—31no—
314#106—91no—
315C33#42811yescharacteristic, complemented, Sylow
316C23×C3—31no—
317C23×C3—31no—
318C23×C3—61no—
319C22×D3C3×#6711yescharacteristic
320C2×C3×C4—22no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—31no—
328C2×C3×C4—31no—
329C2×C3×C4—32no—
330C2×C3×C4—32no—
331C2×C3×C4—32no—
332C2×C3×C4—32no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—42no—
336C2×C3×C4—41no—
337C2×C3×C4—62no—
338C2×C3×C4—62no—
339C2×C3×C4—61no—
340C2×C3×C4—62no—
341C2×C3×C4—61no—
342C2×C3×C4—61no—
343C2×C3×C4—61no—
344C2×C3×C4—61no—
345C2×C3×C4—121no—
346C2×C3×C4—124no—
347C2×C3×C4—122no—
348C2×C3×C4—122no—
349C2×C3×C4—181no—
350C2×C3×C4C2×D3211yescharacteristic
351C2×C3×C4C22×C3×D311yescharacteristic
352C2×C3×C4C22×C3×D311yescharacteristic
353C2×Dic3—21no—
354C2×Dic3—21no—
355C2×Dic3—61no—
356C2×Dic3—61no—
357C2×Dic3—121no—
358C2×Dic3C3×#6711yescharacteristic
359C4×D3—42no—
360C4×D3C3×Dic612yes—
361C4×D3C3×C4×D312yes—
362C3×C8—32no—
363C3×C8—32no—
364C3×C8—62no—
365C3×C8—92no—
366C3⋊C8—32no—
367C3⋊C8—32no—
368C3⋊C8—62no—
369C3⋊C8C3×C4×D312yescomplemented
370C2×C32—21no—
371C2×C32—21no—
372C2×C32—21no—
373C2×C32—21no—
374C2×C32—21no—
375C2×C32—21no—
376C2×C32—21no—
377C2×C32—21no—
378C2×C32—21no—
379C2×C32—41no—
380C2×C32—41no—
381C2×C32—41no—
382C2×C32—62no—
383C2×C32D3×C4⋊C411yescharacteristic
384C2×C32#91411yescharacteristic
385C2×C32#91511yescharacteristic
386C2×C32C2×Dic3⋊C411yescharacteristic
387C2×C32#93211yescharacteristic
388C2×C32#94211yescharacteristic
389C2×C32C2×C3×C4⋊C411yescharacteristic
390C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
391C2×C32C3×#13711yescharacteristic
392C3×D3—24no—
393C3×D3—22no—
394C3×D3—22no—
395C22×C4—31no—
396C22×C4—61no—
397M16—94no—
398C2×C8—31no—
399C2×C8—91no—
400C4⋊C4—31no—
401C4⋊C4—31no—
402C4⋊C4—61no—
403C4⋊C4—91no—
404C4⋊C4C3×D3211yescharacteristic
405C22⋊C4—181no—
406C42—181no—
407C22×C3—21no—
408C22×C3—21no—
409C22×C3—21no—
410C22×C3—32no—
411C22×C3—32no—
412C22×C3—41no—
413C22×C3—62no—
414C22×C3—62no—
415C22×C3—62no—
416C22×C3—61no—
417C22×C3—61no—
418C22×C3—61no—
419C22×C3—61no—
420C22×C3D3×#6711yescharacteristic
421C22×C3C3×D3×D411yescharacteristic
422C22×C3C2×C3×#6711yescharacteristic
423C2×D3—21no—
424C2×D3—21no—
425C2×D3C3×Dic3⋊C412yes—
426C3×C4—21no—
427C3×C4—21no—
428C3×C4—21no—
429C3×C4—21no—
430C3×C4—21no—
431C3×C4—21no—
432C3×C4—32no—
433C3×C4—32no—
434C3×C4—42no—
435C3×C4—41no—
436C3×C4—41no—
437C3×C4—41no—
438C3×C4—41no—
439C3×C4—41no—
440C3×C4—62no—
441C3×C4—81no—
442C3×C4—81no—
443C3×C4—82no—
444C3×C4—122no—
445C3×C4—121no—
446C3×C4—121no—
447C3×C4—121no—
448C3×C4—121no—
449C3×C4—241no—
450C3×C4—361no—
451C3×C4D3×Dic611yescharacteristic
452C3×C4C4×D3211yescharacteristic
453C3×C4C2×C3×Dic611yescharacteristic
454C3×C4C2×C3×C4×D311yescharacteristic
455C3×C4C2×C3×C4×D311yescharacteristic
456C3×C4C3×D3×Q811yescharacteristic
457Dic3—41no—
458Dic3—41no—
459Dic3—121no—
460Dic3—121no—
461Dic3—241no—
462Dic3C3×Dic3⋊C412yes—
463C32—21no—
464C32—21no—
465C32—21no—
466C32—41no—
467C32#486111yescharacteristic, complemented
468C32#496411yescharacteristic, complemented
469C32C3×#42811yescharacteristic, complemented
470C23—31no—
471C2×C4—21no—
472C2×C4—31no—
473C2×C4—32no—
474C2×C4—32no—
475C2×C4—61no—
476C2×C4—61no—
477C2×C4—122no—
478C2×C4—181no—
479C2×C4C2×C3×D3211yescharacteristic
480C8—32no—
481C8—92no—
482C2×C3—21no—
483C2×C3—21no—
484C2×C3—21no—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—41no—
492C2×C3—41no—
493C2×C3—41no—
494C2×C3—64no—
495C2×C3—62no—
496C2×C3—62no—
497C2×C3D3×Dic3⋊C411yescharacteristic
498C2×C3#6393411yescharacteristic
499C2×C3#6394611yescharacteristic
500C2×C3C3×D3×C4⋊C411yescharacteristic
501C2×C3C3×#91411yescharacteristic
502C2×C3C3×#91511yescharacteristic
503C2×C3C2×C3×Dic3⋊C411yescharacteristic
504C2×C3C3×#93211yescharacteristic
505C2×C3C3×#94211yescharacteristic
506D3—22no—
507C22—32no—
508C22—61no—
509C22—61no—
510C22C3×D3×#6711yescharacteristic
511C4—32no—
512C4—41no—
513C4—121no—
514C4—121no—
515C4—361no—
516C4C3×D3×Dic611yescharacteristic
517C4C3×C4×D3211yescharacteristic
518C3—21no—
519C3—21no—
520C3—21no—
521C3—41no—
522C3#1059058111yescharacteristic, direct product factor
523C3C3×#486111yescharacteristic, complemented
524C3C3×#496411yescharacteristic, complemented
525C2—62no—
526C2C3×D3×Dic3⋊C411yescharacteristic
527C2C3×#6393411yescharacteristic
528C2C3×#6394611yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC22×C32×D3D4
cosocleC2×C32×C4C23×C3
hypercenterC3×C4⋊C4D32
hypocenterC32C3×#428
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
4. C3×#63634C2semidirect
5. C32×#808C2semidirect
6. C3×#63575C2semidirect
7. C3×#63574C2semidirect
10. #10590581C3direct (N × H)
22. C3×#3711C4semidirect
34. #63634C2×C3semidirect
39. C3×#808D3semidirect
40. C3×#808C2×C3semidirect
45. #63575C2×C3semidirect
46. #63574C2×C3semidirect
57. C3×C3⋊(C3⋊C8)C2×C4semidirect
58. C32×C3⋊C8C2×C4semidirect
109. #3711C3×C4semidirect
140. #808C3×D3semidirect
192. C3⋊(C3⋊C8)C2×C3×C4semidirect
197. C3×C3⋊C8C4×D3semidirect
198. C3×C3⋊C8C2×C3×C4semidirect
315. C33#428semidirect
369. C3⋊C8C3×C4×D3semidirect
467. C32#4861semidirect
468. C32#4964semidirect
469. C32C3×#428semidirect
522. C3#10590581direct (N × H)
523. C3C3×#4861semidirect
524. C3C3×#4964semidirect

Maximal subgroups

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

subgroupindex
C32×D3×C4⋊C42
C3×#638412
C2×C3×#37112
C3×#636342
C32×#8082
C3×#635752
C3×#635742
C3×#49643
C3×#48613
#105905813