Finite Group Databasebeta

← C2×C4×A4×C3⋊D3 #13181457

Subgroups — C2×C4×A4×C3⋊D3 #13181457

Order 1728.

2030 conjugacy classes of subgroups; only the 201 normal subgroups are shown.

normal subgroups only

This group has 2030 conjugacy classes of subgroups — too many to draw. Only its 201 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×C32×C4×A4C211yesmaximal, characteristic, complemented
3C2×A4×C3⋊Dic3C211yesmaximal, characteristic, complemented
4C4×A4×C3⋊D3C214yesmaximal, direct product factor
5C23×C4×C3⋊D3C311yesmaximal, characteristic, complemented
6C2×C4×D3×A4—34nomaximal
7C22×C32×A4C2211yescharacteristic
8C2×A4×C3⋊D3C414yesdirect product factor
9C2×A4×C3⋊D3C2212yes—
10C2×C3×C4×C3⋊D3—41nomaximal
11C32×C4×A4C2212yescomplemented
12A4×C3⋊Dic3C2212yescomplemented
13C24×C3⋊D3C2×C311yescharacteristic
14C22×D3×A4—34no—
15C23×C32×C4C2×C311yescharacteristic, complemented
16C23×C3⋊Dic3C2×C311yescharacteristic, complemented
17C22×C4×C3⋊D3—31no—
18C22×C4×C3⋊D3—31no—
19C22×C4×C3⋊D3—31no—
20C22×C4×C3⋊D3—31no—
21C22×C4×C3⋊D3—34no—
22C22×C4×C3⋊D3C2×C314yescomplemented
23C2×C3×C4×A4—24no—
24C2×C3×C4×A4D314yescomplemented
25C2×Dic3×A4—34no—
26C4×D3×A4—316no—
27C22×C3×C3⋊D3—41no—
28C2×C32×A4C2×C412yescomplemented
29C2×C32×A4C2311yescharacteristic
30A4×C3⋊D3C2×C414yesdirect product factor
31C2×C33×C4—41no—
32C2×C3×C3⋊Dic3—41no—
33C3×C4×C3⋊D3—44no—
34C23×C4×D3—34no—
35C22×C4×A4—91no—
36C24×C32C22×C311yescharacteristic
37C23×C3⋊D3—31no—
38C23×C3⋊D3—31no—
39C23×C3⋊D3—32no—
40C23×C3⋊D3—34no—
41C23×C3⋊D3C3×C414yescomplemented
42C23×C3⋊D3C22×C312yes—
43C22×C3×A4—24no—
44C22×C3×A4C2×D314yes—
45C2×D3×A4—316no—
46C2×D3×A4—38no—
47C22×C32×C4—31no—
48C22×C32×C4—31no—
49C22×C32×C4—32no—
50C22×C32×C4C22×C312yescomplemented
51C22×C3⋊Dic3—31no—
52C22×C3⋊Dic3—31no—
53C22×C3⋊Dic3—32no—
54C22×C3⋊Dic3C22×C312yescomplemented
55C2×C4×C3⋊D3—32no—
56C2×C4×C3⋊D3—34no—
57C2×C4×C3⋊D3—34no—
58C2×C4×C3⋊D3—34no—
59C2×C4×C3⋊D3—31no—
60C2×C4×C3⋊D3—31no—
61C2×C4×C3⋊D3—31no—
62C2×C4×C3⋊D3—34no—
63C2×C4×C3⋊D3—34no—
64C2×C4×C3⋊D3—34no—
65C2×C4×C3⋊D3—34no—
66C2×C4×C3⋊D3—34no—
67C2×C4×C3⋊D3A411yescharacteristic, direct product factor
68C2×C3×C4×D3—124no—
69C3×C4×A4—28no—
70C3×C4×A4C2×D318yescomplemented
71Dic3×A4—38no—
72C22×C33—41no—
73C2×C3×C3⋊D3—44no—
74C2×C3×C3⋊D3—42no—
75C32×A4C22×C411yescharacteristic, complemented
76C33×C4—42no—
77C3×C3⋊Dic3—42no—
78C24×D3—34no—
79C23×A4—91no—
80C23×C3×C4C3×D314yescomplemented
81C23×Dic3—34no—
82C22×C4×D3—316no—
83C22×C4×D3—94no—
84C22×C4×D3—916no—
85C22×C4×D3—94no—
86C22×C4×D3—94no—
87C22×C4×D3—94no—
88C2×C4×A4—24no—
89C2×C4×A4—91no—
90C2×C4×A4—94no—
91C2×C4×A4C3⋊D311yescharacteristic, direct product factor
92C23×C32—32no—
93C23×C32—31no—
94C23×C32—31no—
95C23×C32C2×C3×C412yescomplemented
96C23×C32C23×C311yescharacteristic
97C22×C3⋊D3—34no—
98C22×C3⋊D3—38no—
99C22×C3⋊D3—32no—
100C22×C3⋊D3—31no—
101C22×C3⋊D3—34no—
102C22×C3⋊D3—34no—
103C22×C3⋊D3—32no—
104C22×C3⋊D3—32no—
105C22×C3⋊D3—34no—
106C22×C3⋊D3—38no—
107C22×C3⋊D3—34no—
108C22×C3⋊D3—32no—
109C22×C3⋊D3C2×C3×C414yescomplemented
110C22×C3⋊D3C2×A411yescharacteristic
111C22×C3×D3—124no—
112C2×C3×A4—28no—
113C2×C3×A4—24no—
114C2×C3×A4C4×D318yescomplemented
115C2×C3×A4C22×D314yes—
116D3×A4—316no—
117C2×C32×C4—32no—
118C2×C32×C4—31no—
119C2×C32×C4—32no—
120C2×C32×C4—32no—
121C2×C32×C4—32no—
122C2×C32×C4—44no—
123C2×C32×C4—84no—
124C2×C32×C4C2×A411yescharacteristic, complemented
125C2×C3⋊Dic3—32no—
126C2×C3⋊Dic3—31no—
127C2×C3⋊Dic3—32no—
128C2×C3⋊Dic3—32no—
129C2×C3⋊Dic3—32no—
130C2×C3⋊Dic3C2×A411yescharacteristic, complemented
131C4×C3⋊D3—38no—
132C4×C3⋊D3—34no—
133C4×C3⋊D3—34no—
134C4×C3⋊D3—34no—
135C4×C3⋊D3C2×A414yesdirect product factor
136C2×C3×Dic3—124no—
137C3×C4×D3—1216no—
138C24×C4—91noSylow
139C2×C33—42no—
140C2×C33—41no—
141C3×C3⋊D3—44no—
142C24×C3C2×C3×D314yes—
143C23×D3—316no—
144C23×D3—38no—
145C23×D3—94no—
146C23×D3—98no—
147C23×D3—94no—
148C23×D3—916no—
149C22×A4—24no—
150C22×A4—94no—
151C22×A4—92no—
152C22×A4C2×C3⋊D311yescharacteristic
153C22×C3×C4—34no—
154C22×C3×C4—34no—
155C22×C3×C4—38no—
156C22×C3×C4—361no—
157C22×C3×C4C2×C3×D318yescomplemented
158C22×Dic3—38no—
159C22×Dic3—94no—
160C22×Dic3—98no—
161C22×Dic3—94no—
162C2×C4×D3—34no—
163C2×C4×D3—916no—
164C2×C4×D3—916no—
165C2×C4×D3—94no—
166C2×C4×D3—94no—
167C2×C4×D3—916no—
168C2×C4×D3—916no—
169C2×C4×D3—98no—
170C2×C4×D3—916no—
171C2×C4×D3—94no—
172C2×C4×D3—916no—
173C2×C4×D3—916no—
174C2×C4×D3—916no—
175C4×A4—28no—
176C4×A4—92no—
177C4×A4C2×C3⋊D312yesdirect product factor
178C22×C32—32no—
179C22×C32—31no—
180C22×C32—32no—
181C22×C32—32no—
182C22×C32—32no—
183C22×C32—31no—
184C22×C32—31no—
185C22×C32—44no—
186C22×C32—84no—
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×A414yesdirect product factor
198C2×C3⋊D3C22×A412yes—
199C2×C3×D3—128no—
200C2×C3×D3—1216no—
201C3×A4—24no—
202C3×A4C2×C4×D314yescomplemented
203C32×C4—32no—
204C32×C4—48no—
205C32×C4—88no—
206C32×C4C22×A412yescomplemented
207C3⋊Dic3—32no—
208C3⋊Dic3C22×A412yescomplemented
209C3×Dic3—128no—
210C25—91no—
211C23×C4—91no—
212C23×C4—94no—
213C23×C4—271no—
214C23×C4—271no—
215C23×C4—274no—
216C23×C4—271no—
217C23×C4—271no—
218C23×C4C3×C3⋊D311yescharacteristic, complemented
219C33—41noSylow
220C23×C3—34no—
221C23×C3—38no—
222C23×C3—34no—
223C23×C3—361no—
224C23×C3C3×C4×D318yescomplemented
225C23×C3C22×C3×D314yes—
226C22×D3—34no—
227C22×D3—316no—
228C22×D3—98no—
229C22×D3—98no—
230C22×D3—916no—
231C22×D3—916no—
232C22×D3—98no—
233C22×D3—932no—
234C22×D3—916no—
235C22×D3—94no—
236C22×D3—98no—
237C22×D3—932no—
238C22×D3—916no—
239C22×D3—916no—
240C2×A4—28no—
241C2×A4—24no—
242C2×A4—94no—
243C2×A4C4×C3⋊D312yesdirect product factor
244C2×A4C22×C3⋊D311yescharacteristic
245C2×C3×C4—38no—
246C2×C3×C4—34no—
247C2×C3×C4—38no—
248C2×C3×C4—38no—
249C2×C3×C4—38no—
250C2×C3×C4—41no—
251C2×C3×C4—84no—
252C2×C3×C4—364no—
253C2×C3×C4—361no—
254C2×C3×C4D3×A414yescomplemented
255C2×Dic3—34no—
256C2×Dic3—98no—
257C2×Dic3—98no—
258C2×Dic3—98no—
259C2×Dic3—94no—
260C2×Dic3—98no—
261C4×D3—316no—
262C4×D3—916no—
263C4×D3—916no—
264C4×D3—932no—
265C4×D3—916no—
266C2×C32—32no—
267C2×C32—31no—
268C2×C32—31no—
269C2×C32—48no—
270C2×C32—44no—
271C2×C32—88no—
272C2×C32—84no—
273C2×C32C2×C4×A412yescomplemented
274C2×C32C23×A411yescharacteristic
275C3⋊D3—34no—
276C3⋊D3C2×C4×A414yesdirect product factor
277C3×D3—1216no—
278C24—94no—
279C24—92no—
280C24—271no—
281C24—272no—
282C24—271no—
283C24—274no—
284C24C2×C3×C3⋊D311yescharacteristic
285C22×C4—31no—
286C22×C4—31no—
287C22×C4—32no—
288C22×C4—91no—
289C22×C4—92no—
290C22×C4—274no—
291C22×C4—271no—
292C22×C4—274no—
293C22×C4—271no—
294C22×C4—274no—
295C22×C4—274no—
296C22×C4—272no—
297C22×C4—274no—
298C22×C4—274no—
299C22×C4—274no—
300C22×C4—274no—
301C22×C4—271no—
302C22×C4—271no—
303C22×C4—272no—
304C22×C4—271no—
305C22×C4C2×C3×C3⋊D312yescomplemented
306C22×C3—38no—
307C22×C3—34no—
308C22×C3—38no—
309C22×C3—38no—
310C22×C3—38no—
311C22×C3—34no—
312C22×C3—34no—
313C22×C3—41no—
314C22×C3—84no—
315C22×C3—362no—
316C22×C3—364no—
317C22×C3C2×C3×C4×D314yescomplemented
318C22×C3C2×D3×A414yes—
319C2×D3—316no—
320C2×D3—38no—
321C2×D3—916no—
322C2×D3—916no—
323C2×D3—932no—
324C2×D3—932no—
325C2×D3—916no—
326C2×D3—916no—
327C2×D3—98no—
328C2×D3—916no—
329A4—24no—
330A4C2×C4×C3⋊D311yescharacteristic, direct product factor
331C3×C4—38no—
332C3×C4—42no—
333C3×C4—88no—
334C3×C4—362no—
335C3×C4C2×D3×A418yescomplemented
336Dic3—38no—
337Dic3—98no—
338C32—44no—
339C32—84no—
340C32C22×C4×A411yescharacteristic, complemented
341C23—31no—
342C23—32no—
343C23—31no—
344C23—91no—
345C23—94no—
346C23—272no—
347C23—272no—
348C23—274no—
349C23—274no—
350C23—272no—
351C23—278no—
352C23—274no—
353C23—274no—
354C23—272no—
355C23—278no—
356C23—274no—
357C23—271no—
358C23C3×C4×C3⋊D312yescomplemented
359C23C22×C3×C3⋊D311yescharacteristic
360C2×C4—32no—
361C2×C4—31no—
362C2×C4—32no—
363C2×C4—32no—
364C2×C4—32no—
365C2×C4—91no—
366C2×C4—94no—
367C2×C4—274no—
368C2×C4—274no—
369C2×C4—278no—
370C2×C4—272no—
371C2×C4—272no—
372C2×C4—272no—
373C2×C4—271no—
374C2×C4—274no—
375C2×C4—272no—
376C2×C4A4×C3⋊D311yescharacteristic, direct product factor
377C2×C3—34no—
378C2×C3—38no—
379C2×C3—34no—
380C2×C3—42no—
381C2×C3—41no—
382C2×C3—88no—
383C2×C3—84no—
384C2×C3—364no—
385C2×C3C4×D3×A418yescomplemented
386C2×C3C22×D3×A414yes—
387D3—316no—
388D3—916no—
389C22—32no—
390C22—31no—
391C22—32no—
392C22—32no—
393C22—32no—
394C22—31no—
395C22—31no—
396C22—94no—
397C22—92no—
398C22—274no—
399C22—274no—
400C22—278no—
401C22—274no—
402C22—274no—
403C22—278no—
404C22—274no—
405C22—272no—
406C22C2×C3×C4×C3⋊D311yescharacteristic, complemented
407C22C2×A4×C3⋊D311yescharacteristic
408C4—32no—
409C4—92no—
410C4—272no—
411C4C2×A4×C3⋊D312yesdirect product factor
412C3—41no—
413C3—84no—
414C3C2×C4×D3×A414yescomplemented
415C2—31no—
416C2—32no—
417C2—31no—
418C2—94no—
419C2—274no—
420C2C4×A4×C3⋊D312yesdirect product factor
421C2C22×A4×C3⋊D311yescharacteristic
subgroupG/H
center, hypercenterC2×C4A4×C3⋊D3
commutator, hypocenterC22×C32C22×C3×C4
FrattiniC2C22×A4×C3⋊D3
FittingC23×C32×C4C2×C3
socleC24×C32C22×C3
prime radicalC22×A4×C3⋊D3C2
cosocleC23×C32C23×C3
LCSG ▷ C22×C32
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×C32×C4×A4C2semidirect
3. C2×A4×C3⋊Dic3C2semidirect
4. C4×A4×C3⋊D3C2direct (N × H)
5. C23×C4×C3⋊D3C3semidirect
8. C2×A4×C3⋊D3C4direct (N × H)
11. C32×C4×A4C22semidirect
12. A4×C3⋊Dic3C22semidirect
15. C23×C32×C4C2×C3semidirect
16. C23×C3⋊Dic3C2×C3semidirect
22. C22×C4×C3⋊D3C2×C3semidirect
24. C2×C3×C4×A4D3semidirect
28. C2×C32×A4C2×C4semidirect
30. A4×C3⋊D3C2×C4direct (N × H)
41. C23×C3⋊D3C3×C4semidirect
50. C22×C32×C4C22×C3semidirect
54. C22×C3⋊Dic3C22×C3semidirect
67. C2×C4×C3⋊D3A4direct (N × H)
70. C3×C4×A4C2×D3semidirect
75. C32×A4C22×C4semidirect
80. C23×C3×C4C3×D3semidirect
91. C2×C4×A4C3⋊D3direct (N × H)
95. C23×C32C2×C3×C4semidirect
109. C22×C3⋊D3C2×C3×C4semidirect
114. C2×C3×A4C4×D3semidirect
124. C2×C32×C4C2×A4semidirect
130. C2×C3⋊Dic3C2×A4semidirect
135. C4×C3⋊D3C2×A4direct (N × H)
157. C22×C3×C4C2×C3×D3semidirect
177. C4×A4C2×C3⋊D3direct (N × H)
187. C22×C32C22×C3×C4semidirect
197. C2×C3⋊D3C4×A4direct (N × H)
202. C3×A4C2×C4×D3semidirect
206. C32×C4C22×A4semidirect
208. C3⋊Dic3C22×A4semidirect
218. C23×C4C3×C3⋊D3semidirect
224. C23×C3C3×C4×D3semidirect
243. C2×A4C4×C3⋊D3direct (N × H)
254. C2×C3×C4D3×A4semidirect
273. C2×C32C2×C4×A4semidirect
276. C3⋊D3C2×C4×A4direct (N × H)
305. C22×C4C2×C3×C3⋊D3semidirect
317. C22×C3C2×C3×C4×D3semidirect
330. A4C2×C4×C3⋊D3direct (N × H)
335. C3×C4C2×D3×A4semidirect
340. C32C22×C4×A4semidirect
358. C23C3×C4×C3⋊D3semidirect
376. C2×C4A4×C3⋊D3direct (N × H)
385. C2×C3C4×D3×A4semidirect
406. C22C2×C3×C4×C3⋊D3semidirect
411. C4C2×A4×C3⋊D3direct (N × H)
414. C3C2×C4×D3×A4semidirect
420. C2C4×A4×C3⋊D3direct (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×C32×C4×A42
C2×A4×C3⋊Dic32
C4×A4×C3⋊D32
C23×C4×C3⋊D33
C2×C4×D3×A43
C2×C3×C4×C3⋊D34