Finite Group Databasebeta

← C2×A4×#638 #13181476

Subgroups — C2×A4×#638 #13181476

Order 1728.

2158 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 2158 conjugacy classes of subgroups — too many to draw. Only its 177 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
1C23×C32×A4C211yesmaximal, characteristic, complemented
2C22×A4×C3⋊D3C211yesmaximal, characteristic, complemented
3A4×#638C214yesmaximal, direct product factor
4C2×A4×C3⋊Dic3C211yesmaximal, characteristic, complemented
5C23×#638C311yesmaximal, characteristic, complemented
6C2×A4×#67—34nomaximal
7C22×C32×A4—22no—
8C22×C32×A4C2211yescharacteristic
9C22×C32×A4C2212yescomplemented
10C2×A4×C3⋊D3—22no—
11C2×A4×C3⋊D3C2212yescomplemented
12C2×C3×#638—41nomaximal
13A4×C3⋊Dic3C2212yescomplemented
14C25×C32C2×C311yescharacteristic, complemented
15C24×C3⋊D3C2×C311yescharacteristic, complemented
16C23×C3×A4—24no—
17C23×C3×A4D314yescomplemented
18C22×D3×A4—34no—
19C22×#638—31no—
20C22×#638—31no—
21C22×#638—31no—
22C22×#638—34no—
23C22×#638—31no—
24C22×#638C2×C314yescomplemented
25C23×C3⋊Dic3C2×C311yescharacteristic, complemented
26A4×#67—316no—
27C2×Dic3×A4—34no—
28C23×C33—41no—
29C22×C3×C3⋊D3—41no—
30C2×C32×A4—22no—
31C2×C32×A4D412yescomplemented
32C2×C32×A4C2311yescharacteristic
33A4×C3⋊D3—22no—
34C3×#638—44no—
35C2×C3×C3⋊Dic3—41no—
36C23×#67—34no—
37C2×D4×A4—91no—
38C24×C32—22no—
39C24×C32—31no—
40C24×C32—32no—
41C24×C32—31no—
42C24×C32—62no—
43C24×C32C22×C311yescharacteristic
44C24×C32C22×C312yescomplemented
45C23×C3⋊D3—22no—
46C23×C3⋊D3—31no—
47C23×C3⋊D3—31no—
48C23×C3⋊D3—32no—
49C23×C3⋊D3—62no—
50C23×C3⋊D3C22×C312yescomplemented
51C22×C3×A4—216no—
52C22×C3×A4—24no—
53C22×C3×A4—28no—
54C22×C3×A4—28no—
55C22×C3×A4C2×D314yes—
56C22×C3×A4C2×D318yescomplemented
57C2×D3×A4—38no—
58C2×D3×A4—68no—
59C2×#638—34no—
60C2×#638—34no—
61C2×#638—32no—
62C2×#638—34no—
63C2×#638—31no—
64C2×#638—31no—
65C2×#638—34no—
66C2×#638—34no—
67C2×#638—34no—
68C2×#638—34no—
69C2×#638—34no—
70C2×#638—31no—
71C2×#638A411yescharacteristic, direct product factor
72C22×C3⋊Dic3—31no—
73C22×C3⋊Dic3—31no—
74C22×C3⋊Dic3—32no—
75C22×C3⋊Dic3C22×C312yescomplemented
76C2×C3×#67—124no—
77Dic3×A4—38no—
78C22×C33—41no—
79C22×C33—42no—
80C22×C33—82no—
81C2×C3×C3⋊D3—42no—
82C2×C3×C3⋊D3—82no—
83C32×A4C2×D411yescharacteristic, complemented
84C3×C3⋊Dic3—42no—
85C25×C3C3×D314yescomplemented
86C24×D3—34no—
87C23×A4—24no—
88C23×A4—91no—
89C23×A4C3⋊D311yescharacteristic, complemented
90C22×#67—316no—
91C22×#67—94no—
92C22×#67—94no—
93C22×#67—916no—
94C22×#67—94no—
95C22×#67—94no—
96C23×Dic3—34no—
97D4×A4—94no—
98C2×C4×A4—91no—
99C23×C32—22no—
100C23×C32—31no—
101C23×C32—32no—
102C23×C32—31no—
103C23×C32—32no—
104C23×C32—32no—
105C23×C32—31no—
106C23×C32—32no—
107C23×C32—32no—
108C23×C32—44no—
109C23×C32—62no—
110C23×C32—62no—
111C23×C32—62no—
112C23×C32—64no—
113C23×C32—62no—
114C23×C32—62no—
115C23×C32—64no—
116C23×C32—84no—
117C23×C32C3×D412yescomplemented
118C23×C32C2×A411yescharacteristic, complemented
119C23×C32C23×C311yescharacteristic
120C22×C3⋊D3—22no—
121C22×C3⋊D3—32no—
122C22×C3⋊D3—31no—
123C22×C3⋊D3—32no—
124C22×C3⋊D3—32no—
125C22×C3⋊D3—32no—
126C22×C3⋊D3—62no—
127C22×C3⋊D3—62no—
128C22×C3⋊D3—64no—
129C22×C3⋊D3—64no—
130C22×C3⋊D3—62no—
131C22×C3⋊D3—62no—
132C22×C3⋊D3—62no—
133C22×C3⋊D3C2×A411yescharacteristic, complemented
134C22×C3×D3—124no—
135C2×C3×A4—216no—
136C2×C3×A4—28no—
137C2×C3×A4—28no—
138C2×C3×A4—24no—
139C2×C3×A4#6718yescomplemented
140C2×C3×A4C22×D314yes—
141D3×A4—68no—
142#638—38no—
143#638—34no—
144#638—34no—
145#638—34no—
146#638C2×A414yesdirect product factor
147C2×C3⋊Dic3—32no—
148C2×C3⋊Dic3—31no—
149C2×C3⋊Dic3—32no—
150C2×C3⋊Dic3—32no—
151C2×C3⋊Dic3—32no—
152C2×C3⋊Dic3C2×A411yescharacteristic, complemented
153C3×#67—1216no—
154C2×C3×Dic3—124no—
155C23×D4—91noSylow
156C2×C33—42no—
157C2×C33—41no—
158C2×C33—82no—
159C3×C3⋊D3—82no—
160C24×C3—28no—
161C24×C3—38no—
162C24×C3—34no—
163C24×C3—34no—
164C24×C3—68no—
165C24×C3C2×C3×D314yes—
166C24×C3C2×C3×D318yescomplemented
167C23×D3—38no—
168C23×D3—68no—
169C23×D3—94no—
170C23×D3—98no—
171C23×D3—94no—
172C23×D3—188no—
173C22×A4—216no—
174C22×A4—22no—
175C22×A4—24no—
176C22×A4—28no—
177C22×A4—92no—
178C22×A4—182no—
179C22×A4C2×C3⋊D311yescharacteristic
180C22×A4C2×C3⋊D312yescomplemented
181C2×C3×D4—361no—
182C2×#67—34no—
183C2×#67—94no—
184C2×#67—916no—
185C2×#67—916no—
186C2×#67—94no—
187C2×#67—916no—
188C2×#67—916no—
189C2×#67—916no—
190C2×#67—98no—
191C2×#67—94no—
192C2×#67—916no—
193C2×#67—916no—
194C2×#67—916no—
195C22×Dic3—38no—
196C22×Dic3—94no—
197C22×Dic3—98no—
198C22×Dic3—94no—
199C4×A4—92no—
200C22×C32—22no—
201C22×C32—32no—
202C22×C32—31no—
203C22×C32—32no—
204C22×C32—32no—
205C22×C32—32no—
206C22×C32—31no—
207C22×C32—31no—
208C22×C32—32no—
209C22×C32—44no—
210C22×C32—48no—
211C22×C32—62no—
212C22×C32—62no—
213C22×C32—64no—
214C22×C32—62no—
215C22×C32—62no—
216C22×C32—64no—
217C22×C32—62no—
218C22×C32—88no—
219C22×C32—816no—
220C22×C32—84no—
221C22×C32—88no—
222C22×C32C2×C3×D411yescharacteristic, complemented
223C22×C32C22×A411yescharacteristic
224C22×C32C22×A412yescomplemented
225C2×C3⋊D3—22no—
226C2×C3⋊D3—32no—
227C2×C3⋊D3—62no—
228C2×C3⋊D3—62no—
229C2×C3⋊D3—64no—
230C2×C3⋊D3—64no—
231C2×C3⋊D3—62no—
232C2×C3⋊D3—62no—
233C2×C3⋊D3—62no—
234C2×C3⋊D3C22×A412yescomplemented
235C2×C3×D3—128no—
236C2×C3×D3—248no—
237C3×A4—24no—
238C3×A4C2×#6714yescomplemented
239C3⋊Dic3—32no—
240C3⋊Dic3C22×A412yescomplemented
241C3×Dic3—128no—
242C25—91no—
243C25C3×C3⋊D311yescharacteristic, complemented
244C22×D4—94no—
245C22×D4—271no—
246C22×D4—271no—
247C22×D4—274no—
248C22×D4—271no—
249C22×D4—271no—
250C23×C4—91no—
251C33—41noSylow
252C23×C3—28no—
253C23×C3—34no—
254C23×C3—38no—
255C23×C3—34no—
256C23×C3—38no—
257C23×C3—38no—
258C23×C3—34no—
259C23×C3—38no—
260C23×C3—38no—
261C23×C3—41no—
262C23×C3—68no—
263C23×C3—68no—
264C23×C3—68no—
265C23×C3—616no—
266C23×C3—68no—
267C23×C3—68no—
268C23×C3—616no—
269C23×C3—84no—
270C23×C3—361no—
271C23×C3C3×#6718yescomplemented
272C23×C3D3×A414yescomplemented
273C23×C3C22×C3×D314yes—
274C22×D3—34no—
275C22×D3—68no—
276C22×D3—98no—
277C22×D3—98no—
278C22×D3—98no—
279C22×D3—94no—
280C22×D3—98no—
281C22×D3—188no—
282C22×D3—188no—
283C22×D3—1816no—
284C22×D3—1816no—
285C22×D3—188no—
286C22×D3—188no—
287C22×D3—188no—
288C2×A4—216no—
289C2×A4—22no—
290C2×A4—28no—
291C2×A4—24no—
292C2×A4—182no—
293C2×A4#63812yesdirect product factor
294C2×A4C22×C3⋊D311yescharacteristic
295C3×D4—364no—
296C2×C3×C4—361no—
297#67—316no—
298#67—916no—
299#67—916no—
300#67—932no—
301#67—916no—
302C2×Dic3—34no—
303C2×Dic3—98no—
304C2×Dic3—98no—
305C2×Dic3—98no—
306C2×Dic3—94no—
307C2×Dic3—98no—
308C2×C32—22no—
309C2×C32—32no—
310C2×C32—31no—
311C2×C32—31no—
312C2×C32—48no—
313C2×C32—44no—
314C2×C32—62no—
315C2×C32—88no—
316C2×C32—816no—
317C2×C32—88no—
318C2×C32—84no—
319C2×C32D4×A412yescomplemented
320C2×C32C23×A411yescharacteristic
321C3⋊D3—22no—
322C3⋊D3—62no—
323C3×D3—248no—
324C24—22no—
325C24—32no—
326C24—31no—
327C24—31no—
328C24—62no—
329C24—92no—
330C24—182no—
331C24—271no—
332C24—272no—
333C24—271no—
334C24—542no—
335C24C2×C3×C3⋊D311yescharacteristic
336C24C2×C3×C3⋊D312yescomplemented
337C2×D4—91no—
338C2×D4—271no—
339C2×D4—274no—
340C2×D4—274no—
341C2×D4—271no—
342C2×D4—274no—
343C2×D4—274no—
344C2×D4—274no—
345C2×D4—272no—
346C2×D4—271no—
347C2×D4—274no—
348C2×D4—274no—
349C2×D4—274no—
350C22×C4—92no—
351C22×C4—271no—
352C22×C4—272no—
353C22×C4—271no—
354C22×C3—28no—
355C22×C3—38no—
356C22×C3—34no—
357C22×C3—38no—
358C22×C3—38no—
359C22×C3—38no—
360C22×C3—34no—
361C22×C3—34no—
362C22×C3—38no—
363C22×C3—41no—
364C22×C3—42no—
365C22×C3—68no—
366C22×C3—616no—
367C22×C3—68no—
368C22×C3—68no—
369C22×C3—68no—
370C22×C3—616no—
371C22×C3—68no—
372C22×C3—82no—
373C22×C3—816no—
374C22×C3—84no—
375C22×C3—88no—
376C22×C3—362no—
377C22×C3—722no—
378C22×C3C2×C3×#6714yescomplemented
379C22×C3C2×D3×A414yes—
380C22×C3C2×D3×A418yescomplemented
381C2×D3—38no—
382C2×D3—68no—
383C2×D3—98no—
384C2×D3—188no—
385C2×D3—188no—
386C2×D3—1816no—
387C2×D3—1816no—
388C2×D3—188no—
389C2×D3—188no—
390C2×D3—188no—
391A4—24no—
392A4C2×#63811yescharacteristic, direct product factor
393C3×C4—362no—
394Dic3—38no—
395Dic3—98no—
396C32—44no—
397C32—84no—
398C32C2×D4×A411yescharacteristic, complemented
399C23—22no—
400C23—31no—
401C23—32no—
402C23—31no—
403C23—32no—
404C23—32no—
405C23—31no—
406C23—32no—
407C23—32no—
408C23—62no—
409C23—62no—
410C23—62no—
411C23—64no—
412C23—62no—
413C23—62no—
414C23—64no—
415C23—91no—
416C23—182no—
417C23—272no—
418C23—272no—
419C23—272no—
420C23—272no—
421C23—271no—
422C23—542no—
423C23—542no—
424C23—542no—
425C23—544no—
426C23—542no—
427C23—542no—
428C23—544no—
429C23C3×#63812yescomplemented
430C23A4×C3⋊D311yescharacteristic, complemented
431C23C22×C3×C3⋊D311yescharacteristic
432D4—94no—
433D4—274no—
434D4—274no—
435D4—278no—
436D4—274no—
437C2×C4—91no—
438C2×C4—272no—
439C2×C4—272no—
440C2×C4—272no—
441C2×C4—271no—
442C2×C4—272no—
443C2×C3—28no—
444C2×C3—34no—
445C2×C3—38no—
446C2×C3—34no—
447C2×C3—41no—
448C2×C3—42no—
449C2×C3—68no—
450C2×C3—82no—
451C2×C3—816no—
452C2×C3—84no—
453C2×C3—88no—
454C2×C3—722no—
455C2×C3A4×#6718yescomplemented
456C2×C3C22×D3×A414yes—
457D3—68no—
458D3—188no—
459C22—22no—
460C22—32no—
461C22—31no—
462C22—32no—
463C22—31no—
464C22—31no—
465C22—32no—
466C22—32no—
467C22—32no—
468C22—62no—
469C22—62no—
470C22—64no—
471C22—62no—
472C22—62no—
473C22—64no—
474C22—62no—
475C22—92no—
476C22—182no—
477C22—272no—
478C22—542no—
479C22—542no—
480C22—544no—
481C22—542no—
482C22—542no—
483C22—544no—
484C22—542no—
485C22C2×C3×#63811yescharacteristic, complemented
486C22C2×A4×C3⋊D311yescharacteristic
487C22C2×A4×C3⋊D312yescomplemented
488C4—92no—
489C4—272no—
490C3—41no—
491C3—84no—
492C3C2×A4×#6714yescomplemented
493C2—22no—
494C2—31no—
495C2—32no—
496C2—31no—
497C2—62no—
498C2—182no—
499C2—542no—
500C2A4×#63812yesdirect product factor
501C2C22×A4×C3⋊D311yescharacteristic
subgroupG/H
centerC22C2×A4×C3⋊D3
commutator, cosocleC23×C32C23×C3
FrattiniC2C22×A4×C3⋊D3
FittingC25×C32C2×C3
socleC24×C32C22×C3
hypercenterC23A4×C3⋊D3
hypocenterC22×C32C2×C3×D4
LCSG ▷ C23×C32 ▷ C22×C32
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C23×C32×A4C2semidirect
2. C22×A4×C3⋊D3C2semidirect
3. A4×#638C2direct (N × H)
4. C2×A4×C3⋊Dic3C2semidirect
5. C23×#638C3semidirect
9. C22×C32×A4C22semidirect
11. C2×A4×C3⋊D3C22semidirect
13. A4×C3⋊Dic3C22semidirect
14. C25×C32C2×C3semidirect
15. C24×C3⋊D3C2×C3semidirect
17. C23×C3×A4D3semidirect
24. C22×#638C2×C3semidirect
25. C23×C3⋊Dic3C2×C3semidirect
31. C2×C32×A4D4semidirect
44. C24×C32C22×C3semidirect
50. C23×C3⋊D3C22×C3semidirect
56. C22×C3×A4C2×D3semidirect
71. C2×#638A4direct (N × H)
75. C22×C3⋊Dic3C22×C3semidirect
83. C32×A4C2×D4semidirect
85. C25×C3C3×D3semidirect
89. C23×A4C3⋊D3semidirect
117. C23×C32C3×D4semidirect
118. C23×C32C2×A4semidirect
133. C22×C3⋊D3C2×A4semidirect
139. C2×C3×A4#67semidirect
146. #638C2×A4direct (N × H)
152. C2×C3⋊Dic3C2×A4semidirect
166. C24×C3C2×C3×D3semidirect
180. C22×A4C2×C3⋊D3semidirect
222. C22×C32C2×C3×D4semidirect
224. C22×C32C22×A4semidirect
234. C2×C3⋊D3C22×A4semidirect
238. C3×A4C2×#67semidirect
240. C3⋊Dic3C22×A4semidirect
243. C25C3×C3⋊D3semidirect
271. C23×C3C3×#67semidirect
272. C23×C3D3×A4semidirect
293. C2×A4#638direct (N × H)
319. C2×C32D4×A4semidirect
336. C24C2×C3×C3⋊D3semidirect
378. C22×C3C2×C3×#67semidirect
380. C22×C3C2×D3×A4semidirect
392. A4C2×#638direct (N × H)
398. C32C2×D4×A4semidirect
429. C23C3×#638semidirect
430. C23A4×C3⋊D3semidirect
455. C2×C3A4×#67semidirect
485. C22C2×C3×#638semidirect
487. C22C2×A4×C3⋊D3semidirect
492. C3C2×A4×#67semidirect
500. C2A4×#638direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C32×A42
C22×A4×C3⋊D32
A4×#6382
C2×A4×C3⋊Dic32
C23×#6383
C2×A4×#673
C2×C3×#6384