Finite Group Databasebeta

← C2×A4×#625 #13180912

Subgroups — C2×A4×#625 #13180912

Order 1728.

1482 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 1482 conjugacy classes of subgroups — too many to draw. Only its 120 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
1C22×C3×D3×A4C212yesmaximal, complemented
2C2×A4×C3⋊Dic3C211yesmaximal, characteristic, complemented
3A4×#625C214yesmaximal, direct product factor
4C23×#625C311yesmaximal, characteristic, complemented
5C2×A4×#67—32nomaximal
6C22×C32×A4C2211yescharacteristic
7C2×C3×D3×A4—24no—
8C2×C3×D3×A4C2214yescomplemented
9C2×C3×#625—41nomaximal
10A4×C3⋊Dic3C2212yescomplemented
11C24×C3×D3C2×C312yescomplemented
12C23×C3×A4—32no—
13C22×D3×A4—22no—
14C22×D3×A4D312yescomplemented
15C23×C3⋊Dic3C2×C311yescharacteristic, complemented
16C22×#625—31no—
17C22×#625—32no—
18C22×#625—31no—
19C22×#625—34no—
20C22×#625C2×C314yescomplemented
21A4×#67—38no—
22C2×Dic3×A4—32no—
23C2×Dic3×A4—61no—
24C22×C32×D3—42no—
25C2×C32×A4D412yescomplemented
26C2×C32×A4C2311yescharacteristic
27C3×D3×A4—24no—
28C2×C3×C3⋊Dic3—41no—
29C3×#625—44no—
30C23×#67—32no—
31C2×D4×A4—91no—
32C24×C32C22×C311yescharacteristic
33C23×C3×D3—24no—
34C23×C3×D3—32no—
35C23×C3×D3—32no—
36C23×C3×D3—34no—
37C23×C3×D3—64no—
38C23×C3×D3C22×C314yescomplemented
39C22×C3×A4—22no—
40C22×C3×A4—21no—
41C22×C3×A4—34no—
42C22×C3×A4—41no—
43C22×C3×A4—64no—
44C22×C3×A4C2×D312yes—
45C2×D3×A4—28no—
46C2×D3×A4—24no—
47C2×D3×A4—24no—
48C2×D3×A4C2×D314yescomplemented
49C22×C3⋊Dic3—31no—
50C22×C3⋊Dic3—31no—
51C22×C3⋊Dic3—32no—
52C22×C3⋊Dic3C22×C312yescomplemented
53C2×C3×#67—122no—
54C2×#625—32no—
55C2×#625—38no—
56C2×#625—34no—
57C2×#625—32no—
58C2×#625—31no—
59C2×#625—34no—
60C2×#625—34no—
61C2×#625—38no—
62C2×#625—34no—
63C2×#625A411yescharacteristic, direct product factor
64Dic3×A4—34no—
65Dic3×A4—62no—
66C22×C33—41no—
67C2×C32×D3—44no—
68C2×C32×D3—84no—
69C32×A4C2×D411yescharacteristic, complemented
70C3×C3⋊Dic3—42no—
71C25×C3—32no—
72C24×D3C3×D312yescomplemented
73C23×A4—32no—
74C23×A4—62no—
75C22×#67—38no—
76C22×#67—92no—
77C22×#67—92no—
78C22×#67—98no—
79C22×#67—92no—
80C22×#67—92no—
81C23×Dic3—32no—
82C23×Dic3—61no—
83D4×A4—94no—
84C2×C4×A4—91no—
85C23×C32—31no—
86C23×C32—32no—
87C23×C32—31no—
88C23×C32—122no—
89C23×C32C3×D412yescomplemented
90C23×C32C23×C311yescharacteristic
91C22×C3×D3—24no—
92C22×C3×D3—34no—
93C22×C3×D3—32no—
94C22×C3×D3—34no—
95C22×C3×D3—34no—
96C22×C3×D3—34no—
97C22×C3×D3—42no—
98C22×C3×D3—64no—
99C22×C3×D3—64no—
100C22×C3×D3—68no—
101C22×C3×D3—68no—
102C22×C3×D3—64no—
103C22×C3×D3—64no—
104C22×C3×D3—64no—
105C22×C3×D3—82no—
106C22×C3×D3C2×A412yescomplemented
107C2×C3×A4—24no—
108C2×C3×A4—22no—
109C2×C3×A4—22no—
110C2×C3×A4—21no—
111C2×C3×A4—42no—
112C2×C3×A4—41no—
113C2×C3×A4—64no—
114C2×C3×A4#6714yescomplemented
115C2×C3×A4C22×D312yes—
116D3×A4—28no—
117D3×A4—24no—
118C2×C3⋊Dic3—32no—
119C2×C3⋊Dic3—31no—
120C2×C3⋊Dic3—32no—
121C2×C3⋊Dic3—32no—
122C2×C3⋊Dic3—32no—
123C2×C3⋊Dic3C2×A411yescharacteristic, complemented
124C3×#67—128no—
125C2×C3×Dic3—122no—
126C2×C3×Dic3—241no—
127#625—38no—
128#625—38no—
129#625—34no—
130#625C2×A414yesdirect product factor
131C23×D4—91noSylow
132C2×C33—42no—
133C2×C33—41no—
134C32×D3—84no—
135C24×C3—21no—
136C24×C3—34no—
137C24×C3—64no—
138C24×C3—92no—
139C24×C3—94no—
140C24×C3—92no—
141C24×C3—184no—
142C24×C3C2×C3×D312yes—
143C23×D3—24no—
144C23×D3—32no—
145C23×D3—32no—
146C23×D3—34no—
147C23×D3—64no—
148C23×D3C2×C3×D314yescomplemented
149C22×A4—22no—
150C22×A4—34no—
151C22×A4—41no—
152C22×A4—64no—
153C22×A4—68no—
154C22×A4—64no—
155C22×A4D3211yescharacteristic
156C2×C3×D4—361no—
157C2×#67—32no—
158C2×#67—92no—
159C2×#67—98no—
160C2×#67—98no—
161C2×#67—92no—
162C2×#67—98no—
163C2×#67—98no—
164C2×#67—98no—
165C2×#67—94no—
166C2×#67—92no—
167C2×#67—98no—
168C2×#67—98no—
169C2×#67—98no—
170C22×Dic3—34no—
171C22×Dic3—62no—
172C22×Dic3—92no—
173C22×Dic3—94no—
174C22×Dic3—92no—
175C22×Dic3—181no—
176C22×Dic3—182no—
177C22×Dic3—181no—
178C4×A4—92no—
179C22×C32—32no—
180C22×C32—31no—
181C22×C32—32no—
182C22×C32—32no—
183C22×C32—32no—
184C22×C32—31no—
185C22×C32—31no—
186C22×C32—42no—
187C22×C32—81no—
188C22×C32—82no—
189C22×C32—124no—
190C22×C32—161no—
191C22×C32—244no—
192C22×C32C2×C3×D411yescharacteristic, complemented
193C22×C32C22×A411yescharacteristic
194C2×C3×D3—24no—
195C2×C3×D3—34no—
196C2×C3×D3—44no—
197C2×C3×D3—64no—
198C2×C3×D3—64no—
199C2×C3×D3—68no—
200C2×C3×D3—68no—
201C2×C3×D3—64no—
202C2×C3×D3—64no—
203C2×C3×D3—64no—
204C2×C3×D3—84no—
205C2×C3×D3—88no—
206C2×C3×D3—84no—
207C2×C3×D3C22×A414yescomplemented
208C3×A4—22no—
209C3×A4—21no—
210C3×A4—41no—
211C3×A4C2×#6712yescomplemented
212C3⋊Dic3—32no—
213C3⋊Dic3C22×A412yescomplemented
214C3×Dic3—124no—
215C3×Dic3—242no—
216C25—32no—
217C22×D4—94no—
218C22×D4—271no—
219C22×D4—271no—
220C22×D4—274no—
221C22×D4—272no—
222C23×C4—91no—
223C33—41noSylow
224C23×C3—22no—
225C23×C3—21no—
226C23×C3—32no—
227C23×C3—34no—
228C23×C3—32no—
229C23×C3—32no—
230C23×C3—61no—
231C23×C3—61no—
232C23×C3—62no—
233C23×C3—64no—
234C23×C3—94no—
235C23×C3—94no—
236C23×C3—94no—
237C23×C3—94no—
238C23×C3—92no—
239C23×C3—122no—
240C23×C3—184no—
241C23×C3—184no—
242C23×C3—184no—
243C23×C3—188no—
244C23×C3—184no—
245C23×C3—184no—
246C23×C3—188no—
247C23×C3—242no—
248C23×C3C3×#6714yescomplemented
249C23×C3C22×C3×D312yes—
250C22×D3—24no—
251C22×D3—34no—
252C22×D3—32no—
253C22×D3—34no—
254C22×D3—34no—
255C22×D3—34no—
256C22×D3—64no—
257C22×D3—64no—
258C22×D3—68no—
259C22×D3—68no—
260C22×D3—64no—
261C22×D3—64no—
262C22×D3—64no—
263C22×D3D3×A412yescomplemented
264C2×A4—24no—
265C2×A4—22no—
266C2×A4—42no—
267C2×A4—41no—
268C2×A4—68no—
269C2×A4—64no—
270C2×A4#62512yesdirect product factor
271C2×A4C2×D3211yescharacteristic
272C3×D4—364no—
273C2×C3×C4—361no—
274#67—38no—
275#67—98no—
276#67—98no—
277#67—916no—
278#67—98no—
279C2×Dic3—32no—
280C2×Dic3—61no—
281C2×Dic3—94no—
282C2×Dic3—94no—
283C2×Dic3—94no—
284C2×Dic3—92no—
285C2×Dic3—94no—
286C2×Dic3—182no—
287C2×Dic3—182no—
288C2×Dic3—181no—
289C2×Dic3—182no—
290C2×Dic3—182no—
291C2×C32—32no—
292C2×C32—31no—
293C2×C32—31no—
294C2×C32—44no—
295C2×C32—42no—
296C2×C32—81no—
297C2×C32—82no—
298C2×C32—84no—
299C2×C32—82no—
300C2×C32—161no—
301C2×C32—162no—
302C2×C32—244no—
303C2×C32D4×A412yescomplemented
304C2×C32C23×A411yescharacteristic
305C3×D3—24no—
306C3×D3—64no—
307C3×D3—84no—
308C3×D3—88no—
309C24—34no—
310C24—64no—
311C24—92no—
312C24—94no—
313C24—92no—
314C24—184no—
315C24C3×D3211yescharacteristic
316C2×D4—91no—
317C2×D4—271no—
318C2×D4—274no—
319C2×D4—274no—
320C2×D4—272no—
321C2×D4—274no—
322C2×D4—278no—
323C2×D4—272no—
324C2×D4—274no—
325C2×D4—278no—
326C22×C4—92no—
327C22×C4—271no—
328C22×C4—272no—
329C22×C4—271no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3—34no—
333C22×C3—32no—
334C22×C3—34no—
335C22×C3—34no—
336C22×C3—34no—
337C22×C3—32no—
338C22×C3—32no—
339C22×C3—34no—
340C22×C3—41no—
341C22×C3—61no—
342C22×C3—61no—
343C22×C3—62no—
344C22×C3—62no—
345C22×C3—62no—
346C22×C3—61no—
347C22×C3—62no—
348C22×C3—64no—
349C22×C3—82no—
350C22×C3—94no—
351C22×C3—124no—
352C22×C3—161no—
353C22×C3—184no—
354C22×C3—184no—
355C22×C3—188no—
356C22×C3—184no—
357C22×C3—184no—
358C22×C3—188no—
359C22×C3—184no—
360C22×C3—244no—
361C22×C3—244no—
362C22×C3—248no—
363C22×C3C2×C3×#6712yescomplemented
364C22×C3C2×D3×A412yes—
365C2×D3—24no—
366C2×D3—34no—
367C2×D3—64no—
368C2×D3—68no—
369C2×D3—64no—
370C2×D3—68no—
371C2×D3—64no—
372C2×D3—64no—
373C2×D3—64no—
374C2×D3C2×D3×A414yescomplemented
375A4—22no—
376A4—41no—
377A4C2×#62511yescharacteristic, direct product factor
378C3×C4—362no—
379Dic3—34no—
380Dic3—62no—
381Dic3—94no—
382Dic3—182no—
383C32—42no—
384C32—81no—
385C32—82no—
386C32—161no—
387C32C2×D4×A411yescharacteristic, complemented
388C23—31no—
389C23—32no—
390C23—31no—
391C23—32no—
392C23—64no—
393C23—94no—
394C23—94no—
395C23—94no—
396C23—94no—
397C23—92no—
398C23—184no—
399C23—184no—
400C23—184no—
401C23—188no—
402C23—184no—
403C23—184no—
404C23—188no—
405C23C3×#62512yescomplemented
406C23C2×C3×D3211yescharacteristic
407D4—94no—
408D4—274no—
409D4—278no—
410D4—278no—
411C2×C4—91no—
412C2×C4—272no—
413C2×C4—272no—
414C2×C4—272no—
415C2×C4—271no—
416C2×C4—272no—
417C2×C3—21no—
418C2×C3—22no—
419C2×C3—32no—
420C2×C3—34no—
421C2×C3—32no—
422C2×C3—41no—
423C2×C3—42no—
424C2×C3—61no—
425C2×C3—61no—
426C2×C3—62no—
427C2×C3—64no—
428C2×C3—82no—
429C2×C3—84no—
430C2×C3—161no—
431C2×C3—162no—
432C2×C3—184no—
433C2×C3—244no—
434C2×C3—248no—
435C2×C3A4×#6714yescomplemented
436C2×C3C22×D3×A412yes—
437D3—24no—
438D3—64no—
439C22—32no—
440C22—31no—
441C22—32no—
442C22—31no—
443C22—31no—
444C22—32no—
445C22—32no—
446C22—34no—
447C22—64no—
448C22—94no—
449C22—184no—
450C22—184no—
451C22—188no—
452C22—184no—
453C22—184no—
454C22—188no—
455C22—184no—
456C22C2×C3×#62511yescharacteristic, complemented
457C22D32×A411yescharacteristic
458C4—92no—
459C4—272no—
460C3—21no—
461C3—41no—
462C3—82no—
463C3—161no—
464C3C2×A4×#6712yescomplemented
465C2—31no—
466C2—32no—
467C2—31no—
468C2—64no—
469C2—184no—
470C2A4×#62512yesdirect product factor
471C2C2×D32×A411yescharacteristic
subgroupG/H
center, hypercenterC22D32×A4
commutator, cosocleC23×C32C23×C3
FrattiniC2C2×D32×A4
Fitting, socleC24×C32C22×C3
hypocenterC22×C32C2×C3×D4
LCSG ▷ C23×C32 ▷ C22×C32
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C22×C3×D3×A4C2semidirect
2. C2×A4×C3⋊Dic3C2semidirect
3. A4×#625C2direct (N × H)
4. C23×#625C3semidirect
8. C2×C3×D3×A4C22semidirect
10. A4×C3⋊Dic3C22semidirect
11. C24×C3×D3C2×C3semidirect
14. C22×D3×A4D3semidirect
15. C23×C3⋊Dic3C2×C3semidirect
20. C22×#625C2×C3semidirect
25. C2×C32×A4D4semidirect
38. C23×C3×D3C22×C3semidirect
48. C2×D3×A4C2×D3semidirect
52. C22×C3⋊Dic3C22×C3semidirect
63. C2×#625A4direct (N × H)
69. C32×A4C2×D4semidirect
72. C24×D3C3×D3semidirect
89. C23×C32C3×D4semidirect
106. C22×C3×D3C2×A4semidirect
114. C2×C3×A4#67semidirect
123. C2×C3⋊Dic3C2×A4semidirect
130. #625C2×A4direct (N × H)
148. C23×D3C2×C3×D3semidirect
192. C22×C32C2×C3×D4semidirect
207. C2×C3×D3C22×A4semidirect
211. C3×A4C2×#67semidirect
213. C3⋊Dic3C22×A4semidirect
248. C23×C3C3×#67semidirect
263. C22×D3D3×A4semidirect
270. C2×A4#625direct (N × H)
303. C2×C32D4×A4semidirect
363. C22×C3C2×C3×#67semidirect
374. C2×D3C2×D3×A4semidirect
377. A4C2×#625direct (N × H)
387. C32C2×D4×A4semidirect
405. C23C3×#625semidirect
435. C2×C3A4×#67semidirect
456. C22C2×C3×#625semidirect
464. C3C2×A4×#67semidirect
470. C2A4×#625direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×D3×A42
C2×A4×C3⋊Dic32
A4×#6252
C23×#6253
C2×A4×#673
C2×C3×#6254