Finite Group Databasebeta

← C3×#10595223 #13174767

Subgroups — C3×#10595223 #13174767

Order 1728.

1830 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×C3⋊D3C211yesmaximal, characteristic
2C32×D4×Dic3C212yesmaximal, complemented
3C3×#63974C212yesmaximal, complemented
4C2×C3×#3722C212yesmaximal, complemented
5C3×#63952C214yesmaximal, complemented
6C3×#63894C212yesmaximal, complemented
7C3×#63874C211yesmaximal, characteristic, complemented
8C3×#63849C211yesmaximal, characteristic, complemented
9C3×#5545—32nomaximal
10#10595223C311yesmaximal, characteristic, direct product factor
11C23×C3×C3⋊D3C2212yes—
12C2×C33×D4C2211yescharacteristic
13C2×C3×#638C2212yes—
14C3×D4×C3⋊D3C418yescomplemented
15C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
16C2×C3×C4×C3⋊D3C2211yescharacteristic
17C22×C32×Dic3C2214yescomplemented
18C2×C3×#624C2214yescomplemented
19C32×#217C2214yescomplemented
20C32×C4⋊Dic3C2212yescomplemented
21C32×C4×Dic3C2212yescomplemented
22C3×#3724C2212yescomplemented
23C3×#3722C2211yescharacteristic
24C3×#3722C2212yes—
25C3×#3722C2211yescharacteristic
26C3×#3722C2214yescomplemented
27C3×Dic32C2212yescomplemented
28C2×D4×C3⋊D3C2×C311yescharacteristic
29C2×C3×D3×D4—32no—
30C2×C3×D3×D4—61no—
31C32×C4×D4—32no—
32C3×D4×Dic3—22no—
33C3×D4×Dic3D312yescomplemented
34C3×D4×Dic3C2×C312yescomplemented
35C3×C4×#67—34no—
36C2×C3×(C2×D3)⋊C4—34no—
37C3×#893—32no—
38C3×#892—32no—
39C3×#67⋊C4—34no—
40C3×D3×C22⋊C4—34no—
41C3×C4×D12—32no—
42C3×#872—32no—
43#63974C2×C312yescomplemented
44C2×#3722C2×C312yescomplemented
45#63952C2×C314yescomplemented
46#63894C2×C312yescomplemented
47#63874C2×C311yescharacteristic, complemented
48#63849C2×C311yescharacteristic, complemented
49C23×C33C2312yes—
50C22×C3×C3⋊D3—44no—
51C22×C3×C3⋊D3C2×C418yescomplemented
52C22×C3×C3⋊D3C2311yescharacteristic
53C22×C3×C3⋊D3C2312yes—
54C33×D4C2×C414yescomplemented
55C2×C33×C4C2311yescharacteristic
56C3×#638C2×C418yescomplemented
57C2×C3×C3⋊Dic3C2311yescharacteristic
58C3×C4⋊(C3⋊D3)C2×C414yescomplemented
59C3×C4×C3⋊D3C2×C414yescomplemented
60C2×C32×Dic3—44no—
61C2×C32×Dic3C2312yes—
62C2×C32×Dic3C2314yes—
63C2×C32×Dic3C2312yes—
64C3×#624—44no—
65C3×#518—91no—
66#5545—32no—
67C23×C3⋊D3C22×C312yes—
68C23×C3×D3—34no—
69C23×C3×D3—62no—
70C2×C32×D4—21no—
71C2×C32×D4—22no—
72C2×C32×D4—41no—
73C2×C32×D4C2×D312yes—
74C2×C32×D4C22×C311yescharacteristic
75C22×C32×C4—34no—
76C2×#638C22×C312yes—
77D4×C3⋊D3C3×C418yescomplemented
78C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
79C2×C4×C3⋊D3C22×C311yescharacteristic
80C2×C3×#67—34no—
81C2×C3×#67—62no—
82C22×C3×Dic3—24no—
83C22×C3×Dic3C2×D314yescomplemented
84C22×C3×Dic3C22×C314yescomplemented
85C3×D3×D4—316no—
86C3×D3×D4—68no—
87C2×C3×D12—32no—
88C2×C3×D12—61no—
89C2×C3×C4×D3—32no—
90C2×C3×C4×D3—34no—
91C2×C3×C4×D3—34no—
92C2×C3×C4×D3—61no—
93C2×#624C22×C314yescomplemented
94C32×C4⋊C4—32no—
95C32×C22⋊C4—34no—
96C32×C42—32no—
97C3×#217—24no—
98C3×#217C2×D314yescomplemented
99C3×#217C22×C314yescomplemented
100C3×(C2×D3)⋊C4—32no—
101C3×(C2×D3)⋊C4—34no—
102C3×(C2×D3)⋊C4—34no—
103C3×(C2×D3)⋊C4—32no—
104C3×(C2×D3)⋊C4—34no—
105C3×C4⋊Dic3—22no—
106C3×C4⋊Dic3C2×D312yescomplemented
107C3×C4⋊Dic3C22×C312yescomplemented
108C3×Dic3⋊C4—32no—
109C3×Dic3⋊C4—32no—
110C3×C4×Dic3—22no—
111C3×C4×Dic3—34no—
112C3×C4×Dic3C2×D312yescomplemented
113C3×C4×Dic3C22×C312yescomplemented
114#3724C22×C312yescomplemented
115#3722C22×C312yes—
116#3722C22×C311yescharacteristic
117#3722C22×C311yescharacteristic
118#3722C22×C314yescomplemented
119Dic32C22×C312yescomplemented
120C22×C33—24no—
121C22×C33C22×C414yes—
122C22×C33C2411yescharacteristic
123C2×C3×C3⋊D3—22no—
124C2×C3×C3⋊D3—24no—
125C2×C3×C3⋊D3—48no—
126C2×C3×C3⋊D3C22×C412yes—
127C2×C3×C3⋊D3C22×C414yes—
128C33×C4C22×C412yes—
129C3×C3⋊Dic3C22×C412yes—
130C32×Dic3—22no—
131C32×Dic3—22no—
132C32×Dic3—24no—
133C22×C3×D4—91no—
134C2×D3×D4—32no—
135C2×D3×D4—61no—
136C3×C4×D4—32no—
137C3×C4×D4—32no—
138C3×C4×D4—62no—
139C3×C4×D4—92no—
140C3×C4×D4—94no—
141C3×#117—91no—
142C3×#117—91no—
143C2×C3×C22⋊C4—92no—
144C2×C3×C22⋊C4—92no—
145D4×Dic3C3×D312yescomplemented
146C4×#67—34no—
147C2×(C2×D3)⋊C4—34no—
148#893—32no—
149#892—32no—
150#67⋊C4—34no—
151D3×C22⋊C4—34no—
152C4×D12—32no—
153#872—32no—
154C23×C32—22no—
155C23×C32—24no—
156C23×C32—42no—
157C23×C32C22×D314yes—
158C23×C32C23×C312yes—
159C22×C3⋊D3—44no—
160C22×C3⋊D3C2×C3×C418yescomplemented
161C22×C3⋊D3C23×C311yescharacteristic
162C22×C3⋊D3C23×C312yes—
163C22×C3×D3—32no—
164C22×C3×D3—34no—
165C22×C3×D3—316no—
166C22×C3×D3—61no—
167C22×C3×D3—68no—
168C22×C3×D3—62no—
169C22×C3×D3—124no—
170C22×C3×D3—124no—
171C22×C3×D3—128no—
172C32×D4—24no—
173C32×D4—28no—
174C32×D4—44no—
175C32×D4C4×D318yescomplemented
176C32×D4C2×C3×C414yescomplemented
177C2×C32×C4—21no—
178C2×C32×C4—22no—
179C2×C32×C4—34no—
180C2×C32×C4—32no—
181C2×C32×C4—32no—
182C2×C32×C4—41no—
183C2×C32×C4—124no—
184C2×C32×C4C22×D312yes—
185C2×C32×C4C23×C311yescharacteristic
186#638C2×C3×C418yescomplemented
187C2×C3⋊Dic3C23×C311yescharacteristic
188C4⋊(C3⋊D3)C2×C3×C414yescomplemented
189C4×C3⋊D3C2×C3×C414yescomplemented
190C3×#67—316no—
191C3×#67—68no—
192C2×C3×Dic3—22no—
193C2×C3×Dic3—22no—
194C2×C3×Dic3—24no—
195C2×C3×Dic3—32no—
196C2×C3×Dic3—44no—
197C2×C3×Dic3—44no—
198C2×C3×Dic3—48no—
199C2×C3×Dic3—61no—
200C2×C3×Dic3C22×D314yes—
201C2×C3×Dic3C22×D312yes—
202C2×C3×Dic3C22×D312yes—
203C2×C3×Dic3C23×C314yes—
204C2×C3×Dic3C23×C312yes—
205C2×C3×Dic3C23×C312yes—
206C3×D12—38no—
207C3×D12—64no—
208C3×C4×D3—38no—
209C3×C4×D3—64no—
210C3×C4×D3—124no—
211C3×C4×D3—124no—
212#624—44no—
213#518—91noSylow
214C2×C33—24no—
215C2×C33C23×C411yescharacteristic
216C2×C33H(5,𝔽2)12yes—
217C3×C3⋊D3—22no—
218C3×C3⋊D3—24no—
219C24×C3—92no—
220C23×D3—34no—
221C23×D3—62no—
222C2×C3×D4—22no—
223C2×C3×D4—21no—
224C2×C3×D4—41no—
225C2×C3×D4—92no—
226C2×C3×D4—98no—
227C2×C3×D4—91no—
228C2×C3×D4D3211yescharacteristic
229C2×C3×D4C2×C3×D312yes—
230C22×C3×C4—34no—
231C22×C3×C4—34no—
232C22×C3×C4—64no—
233C22×C3×C4—94no—
234C22×C3×C4—91no—
235C2×#67—34no—
236C2×#67—62no—
237C22×Dic3C2×C3×D314yescomplemented
238D3×D4—316no—
239D3×D4—68no—
240C2×D12—32no—
241C2×D12—61no—
242C2×C4×D3—32no—
243C2×C4×D3—34no—
244C2×C4×D3—34no—
245C2×C4×D3—61no—
246C3×C4⋊C4—32no—
247C3×C4⋊C4—32no—
248C3×C4⋊C4—62no—
249C3×C4⋊C4—92no—
250C3×C22⋊C4—34no—
251C3×C22⋊C4—34no—
252C3×C22⋊C4—64no—
253C3×C22⋊C4—91no—
254C3×C22⋊C4—91no—
255C3×C22⋊C4—92no—
256C3×C22⋊C4—94no—
257C3×C42—32no—
258C3×C42—32no—
259C3×C42—62no—
260C3×C42—92no—
261#217C2×C3×D314yescomplemented
262(C2×D3)⋊C4—32no—
263(C2×D3)⋊C4—32no—
264(C2×D3)⋊C4—34no—
265(C2×D3)⋊C4—34no—
266(C2×D3)⋊C4—34no—
267C4⋊Dic3C2×C3×D312yescomplemented
268Dic3⋊C4—32no—
269Dic3⋊C4—32no—
270C4×Dic3—34no—
271C4×Dic3C2×C3×D312yescomplemented
272C22×C32—24no—
273C22×C32—24no—
274C22×C32—24no—
275C22×C32—21no—
276C22×C32—22no—
277C22×C32—24no—
278C22×C32—28no—
279C22×C32—41no—
280C22×C32—44no—
281C22×C32—44no—
282C22×C32—44no—
283C22×C32—44no—
284C22×C32—84no—
285C22×C32C2×C4×D318yes—
286C22×C32C22×C3×C414yes—
287C22×C32C23×D312yes—
288C22×C32C24×C311yescharacteristic
289C2×C3⋊D3—22no—
290C2×C3⋊D3—24no—
291C2×C3⋊D3—48no—
292C2×C3⋊D3C22×C3×C412yes—
293C2×C3⋊D3C22×C3×C414yes—
294C2×C3×D3—34no—
295C2×C3×D3—38no—
296C2×C3×D3—62no—
297C2×C3×D3—64no—
298C2×C3×D3—64no—
299C2×C3×D3—64no—
300C2×C3×D3—62no—
301C2×C3×D3—64no—
302C2×C3×D3—62no—
303C2×C3×D3—124no—
304C2×C3×D3—1216no—
305C2×C3×D3—1216no—
306C32×C4—22no—
307C32×C4—24no—
308C32×C4—42no—
309C32×C4—62no—
310C32×C4—62no—
311C32×C4—64no—
312C32×C4C2×C4×D314yes—
313C32×C4C22×C3×C412yes—
314C3⋊Dic3C22×C3×C412yes—
315C3×Dic3—24no—
316C3×Dic3—24no—
317C3×Dic3—22no—
318C3×Dic3—22no—
319C3×Dic3—22no—
320C3×Dic3—24no—
321C3×Dic3—22no—
322C3×Dic3—34no—
323C3×Dic3—42no—
324C3×Dic3—44no—
325C3×Dic3—62no—
326C22×D4—91no—
327C4×D4—32no—
328C4×D4—92no—
329C4×D4—94no—
330#117—91no—
331#117—91no—
332C2×C22⋊C4—92no—
333C2×C22⋊C4—92no—
334C33#51811yescharacteristic, complemented, Sylow
335C23×C3—24no—
336C23×C3—22no—
337C23×C3—42no—
338C23×C3—91no—
339C23×C3—92no—
340C23×C3—98no—
341C23×C3—364no—
342C23×C3C2×D3212yes—
343C23×C3C22×C3×D314yes—
344C22×D3—32no—
345C22×D3—34no—
346C22×D3—316no—
347C22×D3—61no—
348C22×D3—68no—
349C22×D3—62no—
350C22×D3—124no—
351C22×D3—124no—
352C22×D3—128no—
353C3×D4—28no—
354C3×D4—24no—
355C3×D4—44no—
356C3×D4—98no—
357C3×D4—94no—
358C3×D4#62414yescomplemented
359C3×D4C3×C4×D318yescomplemented
360C2×C3×C4—22no—
361C2×C3×C4—21no—
362C2×C3×C4—32no—
363C2×C3×C4—32no—
364C2×C3×C4—34no—
365C2×C3×C4—34no—
366C2×C3×C4—32no—
367C2×C3×C4—32no—
368C2×C3×C4—41no—
369C2×C3×C4—62no—
370C2×C3×C4—62no—
371C2×C3×C4—64no—
372C2×C3×C4—91no—
373C2×C3×C4—94no—
374C2×C3×C4—124no—
375C2×C3×C4—124no—
376C2×C3×C4—128no—
377C2×C3×C4—364no—
378C2×C3×C4C2×D3211yescharacteristic
379C2×C3×C4C22×C3×D312yes—
380#67—316no—
381#67—68no—
382C2×Dic3—32no—
383C2×Dic3—44no—
384C2×Dic3—61no—
385C2×Dic3C22×C3×D314yes—
386C2×Dic3C22×C3×D312yes—
387C2×Dic3C22×C3×D312yes—
388D12—38no—
389D12—64no—
390C4×D3—38no—
391C4×D3—64no—
392C4×D3—124no—
393C4×D3—124no—
394C2×C32—28no—
395C2×C32—24no—
396C2×C32—22no—
397C2×C32—21no—
398C2×C32—22no—
399C2×C32—22no—
400C2×C32—22no—
401C2×C32—41no—
402C2×C32—42no—
403C2×C32—44no—
404C2×C32—48no—
405C2×C32—84no—
406C2×C32C22×C4×D312yes—
407C2×C32#100412yes—
408C2×C32#100912yes—
409C2×C32C23×C3×C411yescharacteristic
410C2×C32C3×H(5,𝔽2)12yes—
411C3⋊D3—22no—
412C3⋊D3—24no—
413C3×D3—64no—
414C3×D3—68no—
415C3×D3—64no—
416C3×D3—124no—
417C24—92no—
418C2×D4—92no—
419C2×D4—98no—
420C2×D4—91no—
421C2×D4C3×D3211yescharacteristic
422C22×C4—34no—
423C22×C4—94no—
424C22×C4—91no—
425C4⋊C4—32no—
426C4⋊C4—92no—
427C22⋊C4—34no—
428C22⋊C4—91no—
429C22⋊C4—91no—
430C22⋊C4—92no—
431C22⋊C4—94no—
432C42—32no—
433C42—92no—
434C22×C3—24no—
435C22×C3—24no—
436C22×C3—24no—
437C22×C3—21no—
438C22×C3—22no—
439C22×C3—24no—
440C22×C3—28no—
441C22×C3—44no—
442C22×C3—44no—
443C22×C3—44no—
444C22×C3—41no—
445C22×C3—44no—
446C22×C3—84no—
447C22×C3—92no—
448C22×C3—94no—
449C22×C3—184no—
450C22×C3—182no—
451C22×C3—368no—
452C22×C3C2×#62414yes—
453C22×C3C2×C3×C4×D318yes—
454C22×C3C22×D3211yescharacteristic
455C22×C3C23×C3×D312yes—
456C2×D3—34no—
457C2×D3—38no—
458C2×D3—62no—
459C2×D3—64no—
460C2×D3—64no—
461C2×D3—64no—
462C2×D3—62no—
463C2×D3—64no—
464C2×D3—62no—
465C2×D3—124no—
466C2×D3—1216no—
467C2×D3—1216no—
468C3×C4—22no—
469C3×C4—24no—
470C3×C4—42no—
471C3×C4—64no—
472C3×C4—62no—
473C3×C4—64no—
474C3×C4—62no—
475C3×C4—62no—
476C3×C4—64no—
477C3×C4—62no—
478C3×C4—92no—
479C3×C4—124no—
480C3×C4—122no—
481C3×C4C2×#62412yes—
482C3×C4C2×C3×C4×D314yes—
483Dic3—24no—
484Dic3—22no—
485Dic3—22no—
486Dic3—34no—
487Dic3—62no—
488C32—22no—
489C32—21no—
490C32—41no—
491C32#554512yescomplemented
492C32C3×#51811yescharacteristic, complemented
493C23—91no—
494C23—92no—
495C23—98no—
496C23—364no—
497C23C2×C3×D3212yes—
498D4—98no—
499D4—94no—
500D4C3×#62414yescomplemented
501C2×C4—32no—
502C2×C4—32no—
503C2×C4—34no—
504C2×C4—91no—
505C2×C4—94no—
506C2×C4—124no—
507C2×C4—364no—
508C2×C4C2×C3×D3211yescharacteristic
509C2×C3—28no—
510C2×C3—24no—
511C2×C3—21no—
512C2×C3—22no—
513C2×C3—22no—
514C2×C3—22no—
515C2×C3—22no—
516C2×C3—44no—
517C2×C3—48no—
518C2×C3—41no—
519C2×C3—42no—
520C2×C3—84no—
521C2×C3—182no—
522C2×C3—184no—
523C2×C3#6431412yes—
524C2×C3C22×#62411yescharacteristic
525C2×C3C22×C3×C4×D312yes—
526C2×C3C3×#100412yes—
527C2×C3C3×#100912yes—
528D3—64no—
529D3—64no—
530D3—68no—
531D3—124no—
532C22—24no—
533C22—92no—
534C22—94no—
535C22—184no—
536C22—182no—
537C22—368no—
538C22C2×C3×#62414yes—
539C22C22×C3×D3211yescharacteristic
540C4—62no—
541C4—64no—
542C4—62no—
543C4—92no—
544C4C2×C3×#62412yes—
545C3—21no—
546C3—22no—
547C3—41no—
548C3#1059522311yescharacteristic, direct product factor
549C3C3×#554512yescomplemented
550C2—24no—
551C2—182no—
552C2—184no—
553C2C3×#6431412yes—
554C2C22×C3×#62411yescharacteristic
subgroupG/H
centerC22×C3C22×D32
commutatorC2×C32C23×C3×C4
FrattiniC22C22×C3×D32
FittingC2×C33×D4C22
socleC22×C33C24
prime radicalC2×C3×D4×C3⋊D3C2
cosocleC22×C32C24×C3
hypercenterC2×C3×D4D32
hypocenterC32C3×#518
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
2. C32×D4×Dic3C2semidirect
3. C3×#63974C2semidirect
4. C2×C3×#3722C2semidirect
5. C3×#63952C2semidirect
6. C3×#63894C2semidirect
7. C3×#63874C2semidirect
8. C3×#63849C2semidirect
10. #10595223C3direct (N × H)
14. C3×D4×C3⋊D3C4semidirect
17. C22×C32×Dic3C22semidirect
18. C2×C3×#624C22semidirect
19. C32×#217C22semidirect
20. C32×C4⋊Dic3C22semidirect
21. C32×C4×Dic3C22semidirect
22. C3×#3724C22semidirect
26. C3×#3722C22semidirect
27. C3×Dic32C22semidirect
33. C3×D4×Dic3D3semidirect
34. C3×D4×Dic3C2×C3semidirect
43. #63974C2×C3semidirect
44. C2×#3722C2×C3semidirect
45. #63952C2×C3semidirect
46. #63894C2×C3semidirect
47. #63874C2×C3semidirect
48. #63849C2×C3semidirect
51. C22×C3×C3⋊D3C2×C4semidirect
54. C33×D4C2×C4semidirect
56. C3×#638C2×C4semidirect
58. C3×C4⋊(C3⋊D3)C2×C4semidirect
59. C3×C4×C3⋊D3C2×C4semidirect
77. D4×C3⋊D3C3×C4semidirect
83. C22×C3×Dic3C2×D3semidirect
84. C22×C3×Dic3C22×C3semidirect
93. C2×#624C22×C3semidirect
98. C3×#217C2×D3semidirect
99. C3×#217C22×C3semidirect
106. C3×C4⋊Dic3C2×D3semidirect
107. C3×C4⋊Dic3C22×C3semidirect
112. C3×C4×Dic3C2×D3semidirect
113. C3×C4×Dic3C22×C3semidirect
114. #3724C22×C3semidirect
118. #3722C22×C3semidirect
119. Dic32C22×C3semidirect
145. D4×Dic3C3×D3semidirect
160. C22×C3⋊D3C2×C3×C4semidirect
175. C32×D4C4×D3semidirect
176. C32×D4C2×C3×C4semidirect
186. #638C2×C3×C4semidirect
188. C4⋊(C3⋊D3)C2×C3×C4semidirect
189. C4×C3⋊D3C2×C3×C4semidirect
237. C22×Dic3C2×C3×D3semidirect
261. #217C2×C3×D3semidirect
267. C4⋊Dic3C2×C3×D3semidirect
271. C4×Dic3C2×C3×D3semidirect
334. C33#518semidirect
358. C3×D4#624semidirect
359. C3×D4C3×C4×D3semidirect
491. C32#5545semidirect
492. C32C3×#518semidirect
500. D4C3×#624semidirect
548. C3#10595223direct (N × H)
549. C3C3×#5545semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D4×C3⋊D32
C32×D4×Dic32
C3×#639742
C2×C3×#37222
C3×#639522
C3×#638942
C3×#638742
C3×#638492
C3×#55453
#105952233