Finite Group Databasebeta

← C2×D3×A42 #13181818

Subgroups — C2×D3×A42 #13181818

Order 1728.

1234 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 1234 conjugacy classes of subgroups — too many to draw. Only its 77 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
1C2×C3×A42C211yesmaximal, characteristic, complemented
2D3×A42C212yesmaximal, direct product factor
3C2×D3×C22⋊A4C312yesmaximal, complemented
4C23×D3×A4C312yesmaximal, complemented
5C22×A42—31nomaximal
6C2×C3×D3×A4—42nomaximal
7C3×A42C2211yescharacteristic, complemented
8C2×C3×C22⋊A4C2×C312yescomplemented
9C23×C3×A4C2×C312yescomplemented
10D3×C22⋊A4C2×C314yescomplemented
11C22×D3×A4—32no—
12C22×D3×A4—32no—
13C22×D3×A4—34no—
14C22×D3×A4C2×C314yescomplemented
15C2×A42—22no—
16C2×A42—22no—
17C2×A42—32no—
18C2×A42D311yescharacteristic, direct product factor
19C2×C32×A4—42no—
20C3×D3×A4—44no—
21C25×D3C3211yescharacteristic, complemented
22C22×C22⋊A4—32no—
23C24×A4—32no—
24C23×C3×D3—42no—
25C3×C22⋊A4C22×C312yescomplemented
26C22×C3×A4—32no—
27C22×C3×A4—32no—
28C22×C3×A4—122no—
29C22×C3×A4C22×C312yescomplemented
30C2×D3×A4—34no—
31C2×D3×A4—32no—
32C2×D3×A4—34no—
33C2×D3×A4—34no—
34C2×D3×A4—34no—
35C2×D3×A4—44no—
36C2×D3×A4—122no—
37C2×D3×A4A412yesdirect product factor
38A42—22no—
39A42—22no—
40A42C2×D311yescharacteristic, direct product factor
41C2×C32×D3—161no—
42C32×A4—42no—
43C25×C3C2×C3211yescharacteristic, complemented
44C24×D3—32no—
45C24×D3—32no—
46C24×D3—34no—
47C24×D3—91no—
48C24×D3—92no—
49C24×D3—91no—
50C24×D3C2×C3212yescomplemented
51C2×C22⋊A4—22no—
52C2×C22⋊A4—34no—
53C2×C22⋊A4C3×D312yescomplemented
54C23×A4—22no—
55C23×A4—34no—
56C23×A4—92no—
57C23×A4—94no—
58C23×A4—92no—
59C23×A4C3×D312yescomplemented
60C23×C32—42no—
61C22×C3×D3—44no—
62C22×C3×D3—122no—
63C22×C3×D3—124no—
64C22×C3×D3—122no—
65C2×C3×A4—32no—
66C2×C3×A4—32no—
67C2×C3×A4—42no—
68C2×C3×A4—44no—
69C2×C3×A4—84no—
70C2×C3×A4—82no—
71C2×C3×A4—82no—
72C2×C3×A4—122no—
73C2×C3×A4—124no—
74C2×C3×A4C2×A412yescomplemented
75D3×A4—34no—
76D3×A4—48no—
77D3×A4—124no—
78D3×A4C2×A414yesdirect product factor
79C26—31noSylow
80C2×C33—161no—
81C32×D3—162no—
82C24×C3—32no—
83C24×C3—32no—
84C24×C3—91no—
85C24×C3—91no—
86C24×C3—122no—
87C24×C3C22×C3211yescharacteristic, complemented
88C23×D3—34no—
89C23×D3—32no—
90C23×D3—32no—
91C23×D3—34no—
92C23×D3—34no—
93C23×D3—34no—
94C23×D3—91no—
95C23×D3—92no—
96C23×D3—94no—
97C23×D3—94no—
98C23×D3—92no—
99C23×D3—92no—
100C23×D3—94no—
101C23×D3—92no—
102C23×D3—92no—
103C23×D3—94no—
104C23×D3—98no—
105C23×D3—94no—
106C23×D3—98no—
107C23×D3—94no—
108C23×D3—92no—
109C23×D3—92no—
110C23×D3—94no—
111C23×D3—91no—
112C23×D3—92no—
113C23×D3C3×A412yescomplemented
114C22⋊A4—22no—
115C22⋊A4C2×C3×D312yescomplemented
116C22×A4—22no—
117C22×A4—32no—
118C22×A4—32no—
119C22×A4—32no—
120C22×A4—62no—
121C22×A4—62no—
122C22×A4—94no—
123C22×A4—94no—
124C22×A4—94no—
125C22×A4—94no—
126C22×A4—92no—
127C22×A4—124no—
128C22×A4—362no—
129C22×A4C2×C3×D312yescomplemented
130C22×C32—42no—
131C22×C32—122no—
132C22×C32—122no—
133C22×C32—481no—
134C2×C3×D3—42no—
135C2×C3×D3—124no—
136C2×C3×D3—124no—
137C2×C3×D3—122no—
138C2×C3×D3—124no—
139C2×C3×D3—124no—
140C2×C3×D3—162no—
141C3×A4—42no—
142C3×A4—44no—
143C3×A4—84no—
144C3×A4—82no—
145C3×A4—82no—
146C3×A4—122no—
147C3×A4C22×A412yescomplemented
148C25—32no—
149C25—92no—
150C25—94no—
151C25—92no—
152C25—271no—
153C25—271no—
154C25—272no—
155C25C32×D311yescharacteristic, complemented
156C33—161noSylow
157C23×C3—32no—
158C23×C3—32no—
159C23×C3—32no—
160C23×C3—42no—
161C23×C3—82no—
162C23×C3—91no—
163C23×C3—91no—
164C23×C3—92no—
165C23×C3—92no—
166C23×C3—92no—
167C23×C3—92no—
168C23×C3—91no—
169C23×C3—94no—
170C23×C3—124no—
171C23×C3—362no—
172C23×C3—362no—
173C23×C3—364no—
174C23×C3C2×C3×A412yescomplemented
175C22×D3—32no—
176C22×D3—32no—
177C22×D3—34no—
178C22×D3—34no—
179C22×D3—34no—
180C22×D3—92no—
181C22×D3—94no—
182C22×D3—91no—
183C22×D3—92no—
184C22×D3—92no—
185C22×D3—94no—
186C22×D3—98no—
187C22×D3—94no—
188C22×D3—94no—
189C22×D3—98no—
190C22×D3—94no—
191C22×D3—94no—
192C22×D3—94no—
193C22×D3—94no—
194C22×D3—94no—
195C22×D3—94no—
196C22×D3—94no—
197C22×D3—98no—
198C22×D3—94no—
199C22×D3—916no—
200C22×D3—92no—
201C22×D3—91no—
202C22×D3—94no—
203C22×D3—92no—
204C22×D3—98no—
205C22×D3—98no—
206C22×D3—94no—
207C22×D3—92no—
208C22×D3—94no—
209C22×D3—92no—
210C22×D3C2×C3×A414yescomplemented
211C2×A4—22no—
212C2×A4—34no—
213C2×A4—32no—
214C2×A4—32no—
215C2×A4—44no—
216C2×A4—62no—
217C2×A4—62no—
218C2×A4—84no—
219C2×A4—94no—
220C2×A4—122no—
221C2×A4—128no—
222C2×A4—242no—
223C2×A4—364no—
224C2×A4D3×A412yesdirect product factor
225C2×C32—42no—
226C2×C32—122no—
227C2×C32—122no—
228C2×C32—161no—
229C2×C32—162no—
230C2×C32—322no—
231C2×C32—322no—
232C2×C32—482no—
233C3×D3—44no—
234C3×D3—124no—
235C3×D3—164no—
236C24—32no—
237C24—32no—
238C24—32no—
239C24—91no—
240C24—94no—
241C24—94no—
242C24—94no—
243C24—92no—
244C24—94no—
245C24—91no—
246C24—92no—
247C24—272no—
248C24—272no—
249C24—271no—
250C24—272no—
251C24—274no—
252C24—272no—
253C24—272no—
254C24—274no—
255C24—274no—
256C24—272no—
257C24—271no—
258C24—274no—
259C24—272no—
260C24—272no—
261C24—274no—
262C24—274no—
263C24—278no—
264C24—278no—
265C24—274no—
266C24C2×C32×D311yescharacteristic, complemented
267C22×C3—32no—
268C22×C3—32no—
269C22×C3—32no—
270C22×C3—42no—
271C22×C3—82no—
272C22×C3—91no—
273C22×C3—92no—
274C22×C3—91no—
275C22×C3—92no—
276C22×C3—92no—
277C22×C3—94no—
278C22×C3—92no—
279C22×C3—91no—
280C22×C3—122no—
281C22×C3—122no—
282C22×C3—122no—
283C22×C3—242no—
284C22×C3—242no—
285C22×C3—362no—
286C22×C3—364no—
287C22×C3—364no—
288C22×C3—364no—
289C22×C3—364no—
290C22×C3—482no—
291C22×C3C22×C3×A412yescomplemented
292C2×D3—34no—
293C2×D3—32no—
294C2×D3—34no—
295C2×D3—34no—
296C2×D3—34no—
297C2×D3—92no—
298C2×D3—92no—
299C2×D3—94no—
300C2×D3—94no—
301C2×D3—98no—
302C2×D3—98no—
303C2×D3—94no—
304C2×D3—94no—
305C2×D3—91no—
306C2×D3—92no—
307C2×D3—92no—
308C2×D3—94no—
309C2×D3—94no—
310C2×D3A4211yescharacteristic, direct product factor
311A4—22no—
312A4—44no—
313A4—84no—
314A4—122no—
315A4—242no—
316A4C2×D3×A412yesdirect product factor
317C32—42no—
318C32—161no—
319C32—162no—
320C32—322no—
321C32—322no—
322C23—32no—
323C23—32no—
324C23—32no—
325C23—34no—
326C23—91no—
327C23—91no—
328C23—92no—
329C23—92no—
330C23—92no—
331C23—92no—
332C23—94no—
333C23—92no—
334C23—92no—
335C23—94no—
336C23—91no—
337C23—94no—
338C23—94no—
339C23—271no—
340C23—272no—
341C23—274no—
342C23—272no—
343C23—274no—
344C23—272no—
345C23—274no—
346C23—272no—
347C23—271no—
348C23—274no—
349C23—274no—
350C23—274no—
351C23—274no—
352C23—274no—
353C23—274no—
354C23—274no—
355C23—274no—
356C23—278no—
357C23—278no—
358C23—274no—
359C23—272no—
360C23—274no—
361C23—278no—
362C23—278no—
363C23—274no—
364C23—272no—
365C23—272no—
366C23—2716no—
367C23—274no—
368C23—278no—
369C23C3×D3×A412yescomplemented
370C2×C3—32no—
371C2×C3—32no—
372C2×C3—42no—
373C2×C3—82no—
374C2×C3—91no—
375C2×C3—91no—
376C2×C3—122no—
377C2×C3—122no—
378C2×C3—124no—
379C2×C3—162no—
380C2×C3—242no—
381C2×C3—242no—
382C2×C3—322no—
383C2×C3—364no—
384C2×C3—484no—
385C2×C3C2×A4211yescharacteristic, complemented
386D3—34no—
387D3—92no—
388D3C2×A4212yesdirect product factor
389C22—32no—
390C22—32no—
391C22—31no—
392C22—32no—
393C22—91no—
394C22—92no—
395C22—94no—
396C22—91no—
397C22—94no—
398C22—92no—
399C22—92no—
400C22—94no—
401C22—94no—
402C22—94no—
403C22—92no—
404C22—91no—
405C22—92no—
406C22—272no—
407C22—274no—
408C22—274no—
409C22—272no—
410C22—274no—
411C22—274no—
412C22—272no—
413C22—272no—
414C22—278no—
415C22—278no—
416C22—274no—
417C22—274no—
418C22—271no—
419C22C2×C3×D3×A412yescomplemented
420C3—42no—
421C3—82no—
422C3—162no—
423C3—322no—
424C3C22×A4211yescharacteristic, complemented
425C2—32no—
426C2—32no—
427C2—32no—
428C2—91no—
429C2—91no—
430C2—94no—
431C2—272no—
432C2D3×A4211yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2D3×A42
commutator, cosocle, hypocenterC24×C3C22×C32
FrattiniC1G
Fitting, socleC25×C3C2×C32
LCSG ▷ C24×C3
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C2×C3×A42C2semidirect
2. D3×A42C2direct (N × H)
3. C2×D3×C22⋊A4C3semidirect
4. C23×D3×A4C3semidirect
7. C3×A42C22semidirect
8. C2×C3×C22⋊A4C2×C3semidirect
9. C23×C3×A4C2×C3semidirect
10. D3×C22⋊A4C2×C3semidirect
14. C22×D3×A4C2×C3semidirect
18. C2×A42D3direct (N × H)
21. C25×D3C32semidirect
25. C3×C22⋊A4C22×C3semidirect
29. C22×C3×A4C22×C3semidirect
37. C2×D3×A4A4direct (N × H)
40. A42C2×D3direct (N × H)
43. C25×C3C2×C32semidirect
50. C24×D3C2×C32semidirect
53. C2×C22⋊A4C3×D3semidirect
59. C23×A4C3×D3semidirect
74. C2×C3×A4C2×A4semidirect
78. D3×A4C2×A4direct (N × H)
87. C24×C3C22×C32semidirect
113. C23×D3C3×A4semidirect
115. C22⋊A4C2×C3×D3semidirect
129. C22×A4C2×C3×D3semidirect
147. C3×A4C22×A4semidirect
155. C25C32×D3semidirect
174. C23×C3C2×C3×A4semidirect
210. C22×D3C2×C3×A4semidirect
224. C2×A4D3×A4direct (N × H)
266. C24C2×C32×D3semidirect
291. C22×C3C22×C3×A4semidirect
310. C2×D3A42direct (N × H)
316. A4C2×D3×A4direct (N × H)
369. C23C3×D3×A4semidirect
385. C2×C3C2×A42semidirect
388. D3C2×A42direct (N × H)
419. C22C2×C3×D3×A4semidirect
424. C3C22×A42semidirect
432. C2D3×A42direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×A422
D3×A422
C2×D3×C22⋊A43
C23×D3×A43
C22×A423
C2×C3×D3×A44