Finite Group Databasebeta

← C2×A4×#624 #13180911

Subgroups — C2×A4×#624 #13180911

Order 1728.

1530 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 1530 conjugacy classes of subgroups — too many to draw. Only its 144 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×A4×C3⋊D3C211yesmaximal, characteristic
2C2×C3×Dic3×A4C212yesmaximal, complemented
3A4×#624C214yesmaximal, direct product factor
4C23×#624C311yesmaximal, characteristic, complemented
5C2×C4×D3×A4—32nomaximal
6C22×C32×A4C2211yescharacteristic
7C2×A4×C3⋊D3C414yescomplemented
8C2×A4×C3⋊D3C2212yes—
9C2×C3×#624—41nomaximal
10C3×Dic3×A4C2214yescomplemented
11C24×C3⋊D3C2×C311yescharacteristic
12C22×D3×A4—32no—
13C22×D3×A4—61no—
14C23×C3×Dic3C2×C312yescomplemented
15C2×C3×C4×A4—32no—
16C22×#624—31no—
17C22×#624—31no—
18C22×#624—32no—
19C22×#624—34no—
20C22×#624C2×C314yescomplemented
21C2×Dic3×A4—22no—
22C2×Dic3×A4D312yescomplemented
23C4×D3×A4—38no—
24C22×C3×C3⋊D3—41no—
25C2×C32×A4C2×C412yescomplemented
26C2×C32×A4C2311yescharacteristic
27A4×C3⋊D3C2×C414yescomplemented
28C2×C32×Dic3—42no—
29C3×#624—44no—
30C23×C4×D3—32no—
31C22×C4×A4—91no—
32C24×C32C22×C311yescharacteristic
33C23×C3⋊D3—31no—
34C23×C3⋊D3—31no—
35C23×C3⋊D3—32no—
36C23×C3⋊D3—34no—
37C23×C3⋊D3C3×C414yescomplemented
38C23×C3⋊D3C22×C312yes—
39C22×C3×A4—22no—
40C22×C3×A4—21no—
41C22×C3×A4—41no—
42C22×C3×A4C2×D312yes—
43C2×D3×A4—38no—
44C2×D3×A4—34no—
45C2×D3×A4—64no—
46C2×D3×A4—62no—
47C22×C3×Dic3—32no—
48C22×C3×Dic3—32no—
49C22×C3×Dic3—34no—
50C22×C3×Dic3C22×C314yescomplemented
51C2×C3×C4×D3—122no—
52C3×C4×A4—34no—
53C2×#624—32no—
54C2×#624—34no—
55C2×#624—38no—
56C2×#624—31no—
57C2×#624—32no—
58C2×#624—34no—
59C2×#624—34no—
60C2×#624—34no—
61C2×#624—38no—
62C2×#624A411yescharacteristic, direct product factor
63Dic3×A4—24no—
64Dic3×A4C2×D314yescomplemented
65C22×C33—41no—
66C2×C3×C3⋊D3—44no—
67C2×C3×C3⋊D3—42no—
68C32×A4C22×C411yescharacteristic, complemented
69C32×Dic3—44no—
70C24×D3—32no—
71C24×D3—61no—
72C23×A4—91no—
73C23×C3×C4—32no—
74C23×Dic3C3×D312yescomplemented
75C22×C4×D3—38no—
76C22×C4×D3—92no—
77C22×C4×D3—98no—
78C22×C4×D3—92no—
79C22×C4×D3—92no—
80C22×C4×D3—92no—
81C2×C4×A4—32no—
82C2×C4×A4—62no—
83C2×C4×A4—94no—
84C23×C32—32no—
85C23×C32—31no—
86C23×C32—31no—
87C23×C32C2×C3×C412yescomplemented
88C23×C32C23×C311yescharacteristic
89C22×C3⋊D3—34no—
90C22×C3⋊D3—38no—
91C22×C3⋊D3—32no—
92C22×C3⋊D3—31no—
93C22×C3⋊D3—34no—
94C22×C3⋊D3—34no—
95C22×C3⋊D3—32no—
96C22×C3⋊D3—32no—
97C22×C3⋊D3—34no—
98C22×C3⋊D3—38no—
99C22×C3⋊D3—34no—
100C22×C3⋊D3—32no—
101C22×C3⋊D3C2×C3×C414yescomplemented
102C22×C3⋊D3C2×A411yescharacteristic
103C22×C3×D3—122no—
104C22×C3×D3—241no—
105C2×C3×A4—24no—
106C2×C3×A4—22no—
107C2×C3×A4—22no—
108C2×C3×A4—21no—
109C2×C3×A4—42no—
110C2×C3×A4—41no—
111C2×C3×A4C4×D314yescomplemented
112C2×C3×A4C22×D312yes—
113D3×A4—38no—
114D3×A4—64no—
115C2×C32×C4—122no—
116C2×C3×Dic3—34no—
117C2×C3×Dic3—32no—
118C2×C3×Dic3—34no—
119C2×C3×Dic3—34no—
120C2×C3×Dic3—34no—
121C2×C3×Dic3—42no—
122C2×C3×Dic3—82no—
123C2×C3×Dic3C2×A412yescomplemented
124C3×C4×D3—128no—
125#624—38no—
126#624—34no—
127#624—38no—
128#624C2×A414yesdirect product factor
129C24×C4—91noSylow
130C2×C33—42no—
131C2×C33—41no—
132C3×C3⋊D3—44no—
133C24×C3—21no—
134C24×C3C2×C3×D312yes—
135C23×D3—38no—
136C23×D3—34no—
137C23×D3—64no—
138C23×D3—62no—
139C23×D3—92no—
140C23×D3—94no—
141C23×D3—92no—
142C23×D3—98no—
143C23×D3—181no—
144C23×D3—182no—
145C23×D3—181no—
146C23×D3—184no—
147C22×A4—22no—
148C22×A4—41no—
149C22×A4—94no—
150C22×A4—92no—
151C22×A4D3211yescharacteristic
152C22×C3×C4—34no—
153C22×C3×C4—92no—
154C22×C3×C4—94no—
155C22×C3×C4—92no—
156C22×C3×C4—361no—
157C22×Dic3—32no—
158C22×Dic3—32no—
159C22×Dic3—34no—
160C22×Dic3C2×C3×D314yescomplemented
161C2×C4×D3—32no—
162C2×C4×D3—98no—
163C2×C4×D3—98no—
164C2×C4×D3—92no—
165C2×C4×D3—92no—
166C2×C4×D3—98no—
167C2×C4×D3—98no—
168C2×C4×D3—94no—
169C2×C4×D3—98no—
170C2×C4×D3—92no—
171C2×C4×D3—98no—
172C2×C4×D3—98no—
173C2×C4×D3—98no—
174C4×A4—34no—
175C4×A4—64no—
176C22×C32—32no—
177C22×C32—31no—
178C22×C32—32no—
179C22×C32—32no—
180C22×C32—32no—
181C22×C32—31no—
182C22×C32—31no—
183C22×C32—42no—
184C22×C32—81no—
185C22×C32—82no—
186C22×C32—161no—
187C22×C32C22×C3×C411yescharacteristic, complemented
188C22×C32C22×A411yescharacteristic
189C2×C3⋊D3—34no—
190C2×C3⋊D3—38no—
191C2×C3⋊D3—34no—
192C2×C3⋊D3—32no—
193C2×C3⋊D3—34no—
194C2×C3⋊D3—38no—
195C2×C3⋊D3—34no—
196C2×C3⋊D3—34no—
197C2×C3⋊D3C4×A414yescomplemented
198C2×C3⋊D3C22×A412yes—
199C2×C3×D3—124no—
200C2×C3×D3—128no—
201C2×C3×D3—244no—
202C2×C3×D3—242no—
203C3×A4—22no—
204C3×A4—21no—
205C3×A4—41no—
206C3×A4C2×C4×D312yescomplemented
207C32×C4—124no—
208C3×Dic3—34no—
209C3×Dic3—44no—
210C3×Dic3—84no—
211C3×Dic3C22×A414yescomplemented
212C25—91no—
213C23×C4—32no—
214C23×C4—94no—
215C23×C4—271no—
216C23×C4—272no—
217C23×C4—274no—
218C23×C4—271no—
219C33—41noSylow
220C23×C3—22no—
221C23×C3—21no—
222C23×C3—32no—
223C23×C3—34no—
224C23×C3—32no—
225C23×C3—61no—
226C23×C3—61no—
227C23×C3—62no—
228C23×C3—361no—
229C23×C3C3×C4×D314yescomplemented
230C23×C3C22×C3×D312yes—
231C22×D3—32no—
232C22×D3—38no—
233C22×D3—61no—
234C22×D3—64no—
235C22×D3—94no—
236C22×D3—94no—
237C22×D3—98no—
238C22×D3—98no—
239C22×D3—94no—
240C22×D3—916no—
241C22×D3—98no—
242C22×D3—92no—
243C22×D3—94no—
244C22×D3—916no—
245C22×D3—98no—
246C22×D3—98no—
247C22×D3—182no—
248C22×D3—184no—
249C22×D3—182no—
250C22×D3—184no—
251C22×D3—181no—
252C22×D3—182no—
253C22×D3—188no—
254C22×D3—184no—
255C22×D3—182no—
256C22×D3—188no—
257C22×D3—184no—
258C22×D3—184no—
259C2×A4—24no—
260C2×A4—22no—
261C2×A4—42no—
262C2×A4—41no—
263C2×A4—94no—
264C2×A4#62412yesdirect product factor
265C2×A4C2×D3211yescharacteristic
266C2×C3×C4—32no—
267C2×C3×C4—94no—
268C2×C3×C4—94no—
269C2×C3×C4—94no—
270C2×C3×C4—92no—
271C2×C3×C4—94no—
272C2×C3×C4—122no—
273C2×C3×C4—242no—
274C2×C3×C4—364no—
275C2×Dic3—34no—
276C2×Dic3—32no—
277C2×Dic3—34no—
278C2×Dic3—34no—
279C2×Dic3—34no—
280C2×Dic3D3×A412yescomplemented
281C4×D3—38no—
282C4×D3—98no—
283C4×D3—98no—
284C4×D3—916no—
285C4×D3—98no—
286C2×C32—32no—
287C2×C32—31no—
288C2×C32—31no—
289C2×C32—44no—
290C2×C32—42no—
291C2×C32—81no—
292C2×C32—82no—
293C2×C32—84no—
294C2×C32—82no—
295C2×C32—161no—
296C2×C32—162no—
297C2×C32C2×C4×A412yescomplemented
298C2×C32C23×A411yescharacteristic
299C3⋊D3—34no—
300C3⋊D3C2×C4×A414yescomplemented
301C3×D3—128no—
302C3×D3—244no—
303C24—94no—
304C24—92no—
305C24—271no—
306C24—272no—
307C24—271no—
308C24—274no—
309C24C3×D3211yescharacteristic
310C22×C4—34no—
311C22×C4—91no—
312C22×C4—92no—
313C22×C4—94no—
314C22×C4—92no—
315C22×C4—272no—
316C22×C4—274no—
317C22×C4—274no—
318C22×C4—271no—
319C22×C4—278no—
320C22×C4—274no—
321C22×C4—272no—
322C22×C4—278no—
323C22×C4—274no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—34no—
327C22×C3—32no—
328C22×C3—34no—
329C22×C3—34no—
330C22×C3—34no—
331C22×C3—32no—
332C22×C3—32no—
333C22×C3—41no—
334C22×C3—61no—
335C22×C3—62no—
336C22×C3—61no—
337C22×C3—62no—
338C22×C3—62no—
339C22×C3—61no—
340C22×C3—62no—
341C22×C3—82no—
342C22×C3—161no—
343C22×C3—362no—
344C22×C3—364no—
345C22×C3C2×C3×C4×D312yescomplemented
346C22×C3C2×D3×A412yes—
347C2×D3—38no—
348C2×D3—34no—
349C2×D3—64no—
350C2×D3—62no—
351C2×D3—98no—
352C2×D3—98no—
353C2×D3—916no—
354C2×D3—916no—
355C2×D3—98no—
356C2×D3—98no—
357C2×D3—94no—
358C2×D3—98no—
359C2×D3—184no—
360C2×D3—184no—
361C2×D3—188no—
362C2×D3—182no—
363C2×D3—184no—
364C2×D3—184no—
365C2×D3—188no—
366C2×D3—184no—
367A4—22no—
368A4—41no—
369A4C2×#62411yescharacteristic, direct product factor
370C3×C4—34no—
371C3×C4—94no—
372C3×C4—124no—
373C3×C4—244no—
374Dic3—34no—
375Dic3C2×D3×A414yescomplemented
376C32—42no—
377C32—81no—
378C32—82no—
379C32—161no—
380C32C22×C4×A411yescharacteristic, complemented
381C23—31no—
382C23—32no—
383C23—31no—
384C23—91no—
385C23—94no—
386C23—272no—
387C23—272no—
388C23—274no—
389C23—274no—
390C23—272no—
391C23—278no—
392C23—274no—
393C23—274no—
394C23—272no—
395C23—278no—
396C23—274no—
397C23—271no—
398C23C3×#62412yescomplemented
399C23C2×C3×D3211yescharacteristic
400C2×C4—32no—
401C2×C4—94no—
402C2×C4—94no—
403C2×C4—94no—
404C2×C4—94no—
405C2×C4—92no—
406C2×C4—94no—
407C2×C4—278no—
408C2×C4—274no—
409C2×C4—278no—
410C2×C3—21no—
411C2×C3—22no—
412C2×C3—32no—
413C2×C3—34no—
414C2×C3—32no—
415C2×C3—41no—
416C2×C3—42no—
417C2×C3—61no—
418C2×C3—62no—
419C2×C3—61no—
420C2×C3—82no—
421C2×C3—84no—
422C2×C3—161no—
423C2×C3—162no—
424C2×C3—364no—
425C2×C3C4×D3×A414yescomplemented
426C2×C3C22×D3×A412yes—
427D3—38no—
428D3—64no—
429D3—98no—
430D3—184no—
431C22—32no—
432C22—31no—
433C22—32no—
434C22—31no—
435C22—31no—
436C22—32no—
437C22—32no—
438C22—92no—
439C22—94no—
440C22—274no—
441C22—274no—
442C22—278no—
443C22—274no—
444C22—274no—
445C22—278no—
446C22—272no—
447C22—274no—
448C22C2×C3×#62411yescharacteristic, complemented
449C22D32×A411yescharacteristic
450C4—34no—
451C4—94no—
452C3—21no—
453C3—41no—
454C3—82no—
455C3—161no—
456C3C2×C4×D3×A412yescomplemented
457C2—31no—
458C2—32no—
459C2—31no—
460C2—94no—
461C2—274no—
462C2A4×#62412yesdirect product factor
463C2C2×D32×A411yescharacteristic
subgroupG/H
center, hypercenterC22D32×A4
commutator, hypocenterC22×C32C22×C3×C4
FrattiniC2C2×D32×A4
Fitting, socleC24×C32C22×C3
prime radicalC22×A4×C3⋊D3C2
cosocleC23×C32C23×C3
LCSG ▷ C22×C32
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C2×C3×Dic3×A4C2semidirect
3. A4×#624C2direct (N × H)
4. C23×#624C3semidirect
7. C2×A4×C3⋊D3C4semidirect
10. C3×Dic3×A4C22semidirect
14. C23×C3×Dic3C2×C3semidirect
20. C22×#624C2×C3semidirect
22. C2×Dic3×A4D3semidirect
25. C2×C32×A4C2×C4semidirect
27. A4×C3⋊D3C2×C4semidirect
37. C23×C3⋊D3C3×C4semidirect
50. C22×C3×Dic3C22×C3semidirect
62. C2×#624A4direct (N × H)
64. Dic3×A4C2×D3semidirect
68. C32×A4C22×C4semidirect
74. C23×Dic3C3×D3semidirect
87. C23×C32C2×C3×C4semidirect
101. C22×C3⋊D3C2×C3×C4semidirect
111. C2×C3×A4C4×D3semidirect
123. C2×C3×Dic3C2×A4semidirect
128. #624C2×A4direct (N × H)
160. C22×Dic3C2×C3×D3semidirect
187. C22×C32C22×C3×C4semidirect
197. C2×C3⋊D3C4×A4semidirect
206. C3×A4C2×C4×D3semidirect
211. C3×Dic3C22×A4semidirect
229. C23×C3C3×C4×D3semidirect
264. C2×A4#624direct (N × H)
280. C2×Dic3D3×A4semidirect
297. C2×C32C2×C4×A4semidirect
300. C3⋊D3C2×C4×A4semidirect
345. C22×C3C2×C3×C4×D3semidirect
369. A4C2×#624direct (N × H)
375. Dic3C2×D3×A4semidirect
380. C32C22×C4×A4semidirect
398. C23C3×#624semidirect
425. C2×C3C4×D3×A4semidirect
448. C22C2×C3×#624semidirect
456. C3C2×C4×D3×A4semidirect
462. C2A4×#624direct (N × H)

Maximal subgroups

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

subgroupindex
C22×A4×C3⋊D32
C2×C3×Dic3×A42
A4×#6242
C23×#6243
C2×C4×D3×A43
C2×C3×#6244