Finite Group Databasebeta

← C2×C8×A42 #72247644

Subgroups — C2×C8×A42 #72247644

Order 2304.

1481 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 1481 conjugacy classes of subgroups — too many to draw. Only its 121 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×C4×A42C211yesmaximal, characteristic
2C8×A42C212yesmaximal, direct product factor
3C2×C8×C22⋊A4C312yesmaximal, complemented
4C23×C8×A4C312yesmaximal, complemented
5C22×A42C411yescharacteristic
6C4×A42C411yescharacteristic
7C4×A42C2211yescharacteristic
8C2×C3×C8×A4—42nomaximal
9C2×C4×C22⋊A4C2×C312yes—
10C23×C4×A4C2×C312yes—
11C8×C22⋊A4C2×C314yescomplemented
12C22×C8×A4—32no—
13C22×C8×A4—32no—
14C22×C8×A4—34no—
15C22×C8×A4C2×C314yescomplemented
16C2×A42C812yesdirect product factor
17C2×A42C2×C411yescharacteristic
18C2×C3×C4×A4—42no—
19C3×C8×A4—44no—
20C25×C8C3211yescharacteristic, complemented, Sylow
21C22×C22⋊A4C3×C412yes—
22C24×A4C3×C412yes—
23C4×C22⋊A4C3×C412yes—
24C4×C22⋊A4C22×C312yes—
25C22×C4×A4—32no—
26C22×C4×A4—32no—
27C22×C4×A4—32no—
28C22×C4×A4—32no—
29C22×C4×A4C3×C412yes—
30C22×C4×A4C22×C312yes—
31C23×C3×C8—42no—
32C2×C8×A4—34no—
33C2×C8×A4—32no—
34C2×C8×A4—34no—
35C2×C8×A4—34no—
36C2×C8×A4—34no—
37C2×C8×A4—44no—
38C2×C8×A4—122no—
39C2×C8×A4A412yesdirect product factor
40C22×C3×A4—42no—
41A42C2×C811yescharacteristic, direct product factor
42C3×C4×A4—42no—
43C3×C4×A4—42no—
44C2×C32×C8—161no—
45C25×C4C2×C3211yescharacteristic
46C24×C8—32no—
47C24×C8—32no—
48C24×C8—34no—
49C24×C8—91no—
50C24×C8—92no—
51C24×C8—91no—
52C24×C8C2×C3212yescomplemented
53C2×C22⋊A4C3×C814yescomplemented
54C2×C22⋊A4C2×C3×C412yes—
55C23×A4—32no—
56C23×A4—34no—
57C23×A4—32no—
58C23×A4C3×C814yescomplemented
59C23×A4C2×C3×C412yes—
60C23×C3×C4—42no—
61C2×C4×A4—34no—
62C2×C4×A4—32no—
63C2×C4×A4—32no—
64C2×C4×A4—32no—
65C2×C4×A4—32no—
66C2×C4×A4—32no—
67C2×C4×A4—32no—
68C2×C4×A4—32no—
69C2×C4×A4—44no—
70C2×C4×A4—122no—
71C2×C4×A4C2×A412yes—
72C22×C3×C8—44no—
73C22×C3×C8—122no—
74C22×C3×C8—124no—
75C22×C3×C8—122no—
76C8×A4—34no—
77C8×A4—48no—
78C8×A4—124no—
79C8×A4C2×A414yesdirect product factor
80C2×C3×A4—44no—
81C2×C3×A4—42no—
82C2×C32×C4—161no—
83C32×C8—162no—
84C26C32×C411yescharacteristic
85C24×C4—32no—
86C24×C4—32no—
87C24×C4—32no—
88C24×C4—32no—
89C24×C4—91no—
90C24×C4—91no—
91C24×C4—91no—
92C24×C4—91no—
93C24×C4C32×C411yescharacteristic
94C24×C4C22×C3211yescharacteristic
95C23×C8—34no—
96C23×C8—32no—
97C23×C8—32no—
98C23×C8—34no—
99C23×C8—34no—
100C23×C8—34no—
101C23×C8—91no—
102C23×C8—92no—
103C23×C8—94no—
104C23×C8—94no—
105C23×C8—92no—
106C23×C8—92no—
107C23×C8—94no—
108C23×C8—92no—
109C23×C8—92no—
110C23×C8—94no—
111C23×C8—98no—
112C23×C8—94no—
113C23×C8—98no—
114C23×C8—94no—
115C23×C8—92no—
116C23×C8—92no—
117C23×C8—94no—
118C23×C8—91no—
119C23×C8—92no—
120C23×C8C3×A412yescomplemented
121C24×C3—42no—
122C22⋊A4C2×C3×C812yescomplemented
123C22×A4—34no—
124C22×A4—32no—
125C22×A4—34no—
126C22×A4—34no—
127C22×A4—34no—
128C22×A4—32no—
129C22×A4—32no—
130C22×A4—44no—
131C22×A4—122no—
132C22×A4C2×C3×C812yescomplemented
133C22×A4C4×A412yes—
134C22×C3×C4—42no—
135C22×C3×C4—42no—
136C22×C3×C4—122no—
137C22×C3×C4—122no—
138C22×C3×C4—122no—
139C22×C3×C4—122no—
140C4×A4—32no—
141C4×A4—32no—
142C4×A4—44no—
143C4×A4—44no—
144C4×A4—122no—
145C4×A4—122no—
146C4×A4C4×A412yes—
147C4×A4C22×A412yes—
148C2×C3×C8—42no—
149C2×C3×C8—124no—
150C2×C3×C8—124no—
151C2×C3×C8—122no—
152C2×C3×C8—124no—
153C2×C3×C8—124no—
154C2×C3×C8—162no—
155C22×C32—161no—
156C3×A4—42no—
157C32×C4—161no—
158C32×C4—161no—
159C25—32no—
160C25—34no—
161C25—32no—
162C25—91no—
163C25—91no—
164C25—92no—
165C25C32×C812yescomplemented
166C25C2×C32×C411yescharacteristic
167C23×C4—32no—
168C23×C4—34no—
169C23×C4—32no—
170C23×C4—32no—
171C23×C4—32no—
172C23×C4—32no—
173C23×C4—32no—
174C23×C4—32no—
175C23×C4—32no—
176C23×C4—91no—
177C23×C4—91no—
178C23×C4—91no—
179C23×C4—92no—
180C23×C4—92no—
181C23×C4—92no—
182C23×C4—92no—
183C23×C4—92no—
184C23×C4—92no—
185C23×C4—92no—
186C23×C4—92no—
187C23×C4—92no—
188C23×C4—92no—
189C23×C4—94no—
190C23×C4—94no—
191C23×C4—94no—
192C23×C4—94no—
193C23×C4—94no—
194C23×C4—94no—
195C23×C4—94no—
196C23×C4—91no—
197C23×C4—91no—
198C23×C4—92no—
199C23×C4—92no—
200C23×C4—92no—
201C23×C4—91no—
202C23×C4—91no—
203C23×C4—91no—
204C23×C4C2×C3×A412yes—
205C22×C8—32no—
206C22×C8—32no—
207C22×C8—34no—
208C22×C8—34no—
209C22×C8—34no—
210C22×C8—92no—
211C22×C8—94no—
212C22×C8—91no—
213C22×C8—92no—
214C22×C8—92no—
215C22×C8—94no—
216C22×C8—98no—
217C22×C8—94no—
218C22×C8—94no—
219C22×C8—98no—
220C22×C8—94no—
221C22×C8—94no—
222C22×C8—94no—
223C22×C8—94no—
224C22×C8—94no—
225C22×C8—94no—
226C22×C8—94no—
227C22×C8—98no—
228C22×C8—94no—
229C22×C8—916no—
230C22×C8—92no—
231C22×C8—91no—
232C22×C8—94no—
233C22×C8—92no—
234C22×C8—98no—
235C22×C8—98no—
236C22×C8—94no—
237C22×C8—92no—
238C22×C8—94no—
239C22×C8—92no—
240C22×C8C2×C3×A414yescomplemented
241C23×C3—42no—
242C23×C3—44no—
243C23×C3—122no—
244C23×C3—122no—
245C23×C3—124no—
246C2×A4—34no—
247C2×A4—32no—
248C2×A4—32no—
249C2×A4—44no—
250C2×A4—48no—
251C2×A4—122no—
252C2×A4—124no—
253C2×A4C8×A414yesdirect product factor
254C2×A4C2×C4×A412yes—
255C2×C3×C4—42no—
256C2×C3×C4—122no—
257C2×C3×C4—122no—
258C2×C3×C4—122no—
259C2×C3×C4—122no—
260C2×C3×C4—122no—
261C2×C3×C4—124no—
262C2×C3×C4—122no—
263C2×C3×C4—122no—
264C2×C3×C4—162no—
265C3×C8—44no—
266C3×C8—124no—
267C3×C8—164no—
268C2×C32—161no—
269C2×C32—162no—
270C24—32no—
271C24—34no—
272C24—34no—
273C24—32no—
274C24—32no—
275C24—32no—
276C24—34no—
277C24—34no—
278C24—91no—
279C24—92no—
280C24—91no—
281C24—92no—
282C24—92no—
283C24—92no—
284C24—94no—
285C24—94no—
286C24—92no—
287C24—94no—
288C24—92no—
289C24—91no—
290C24—91no—
291C24—92no—
292C24—94no—
293C24—92no—
294C24—94no—
295C24—98no—
296C24—94no—
297C24—98no—
298C24—94no—
299C24C2×C32×C811yescharacteristic, complemented
300C24C3×C4×A412yes—
301C22×C4—32no—
302C22×C4—32no—
303C22×C4—32no—
304C22×C4—32no—
305C22×C4—32no—
306C22×C4—32no—
307C22×C4—32no—
308C22×C4—32no—
309C22×C4—91no—
310C22×C4—92no—
311C22×C4—91no—
312C22×C4—91no—
313C22×C4—92no—
314C22×C4—91no—
315C22×C4—91no—
316C22×C4—92no—
317C22×C4—94no—
318C22×C4—94no—
319C22×C4—94no—
320C22×C4—94no—
321C22×C4—94no—
322C22×C4—94no—
323C22×C4—94no—
324C22×C4—92no—
325C22×C4—92no—
326C22×C4—92no—
327C22×C4—92no—
328C22×C4—92no—
329C22×C4—92no—
330C22×C4—92no—
331C22×C4—92no—
332C22×C4—92no—
333C22×C4—92no—
334C22×C4—92no—
335C22×C4—92no—
336C22×C4—92no—
337C22×C4—92no—
338C22×C4—98no—
339C22×C4—92no—
340C22×C4—92no—
341C22×C4—98no—
342C22×C4—98no—
343C22×C4—92no—
344C22×C4—91no—
345C22×C4—92no—
346C22×C4—92no—
347C22×C4—92no—
348C22×C4—92no—
349C22×C4—94no—
350C22×C4—94no—
351C22×C4—94no—
352C22×C4—94no—
353C22×C4—94no—
354C22×C4—92no—
355C22×C4—92no—
356C22×C4—91no—
357C22×C4—91no—
358C22×C4C3×C4×A412yes—
359C22×C4C22×C3×A412yes—
360C2×C8—34no—
361C2×C8—32no—
362C2×C8—34no—
363C2×C8—34no—
364C2×C8—34no—
365C2×C8—92no—
366C2×C8—92no—
367C2×C8—94no—
368C2×C8—94no—
369C2×C8—98no—
370C2×C8—98no—
371C2×C8—94no—
372C2×C8—94no—
373C2×C8—91no—
374C2×C8—92no—
375C2×C8—92no—
376C2×C8—94no—
377C2×C8—94no—
378C2×C8A4211yescharacteristic, direct product factor
379C22×C3—42no—
380C22×C3—42no—
381C22×C3—122no—
382C22×C3—122no—
383C22×C3—124no—
384C22×C3—124no—
385C22×C3—122no—
386C22×C3—124no—
387C22×C3—124no—
388C22×C3—162no—
389A4—44no—
390A4—122no—
391A4C2×C8×A412yesdirect product factor
392C3×C4—42no—
393C3×C4—42no—
394C3×C4—122no—
395C3×C4—122no—
396C3×C4—162no—
397C3×C4—162no—
398C32—161noSylow
399C23—32no—
400C23—34no—
401C23—32no—
402C23—32no—
403C23—34no—
404C23—32no—
405C23—34no—
406C23—32no—
407C23—91no—
408C23—91no—
409C23—92no—
410C23—91no—
411C23—92no—
412C23—94no—
413C23—92no—
414C23—92no—
415C23—92no—
416C23—92no—
417C23—94no—
418C23—92no—
419C23—94no—
420C23—94no—
421C23—94no—
422C23—94no—
423C23—94no—
424C23—94no—
425C23—94no—
426C23—94no—
427C23—94no—
428C23—98no—
429C23—98no—
430C23—92no—
431C23—94no—
432C23—92no—
433C23—91no—
434C23—92no—
435C23—98no—
436C23—94no—
437C23—94no—
438C23—98no—
439C23—94no—
440C23—94no—
441C23—92no—
442C23—98no—
443C23—916no—
444C23—91no—
445C23C3×C8×A414yescomplemented
446C23C2×C3×C4×A412yes—
447C2×C4—34no—
448C2×C4—32no—
449C2×C4—32no—
450C2×C4—32no—
451C2×C4—32no—
452C2×C4—32no—
453C2×C4—32no—
454C2×C4—32no—
455C2×C4—91no—
456C2×C4—91no—
457C2×C4—92no—
458C2×C4—92no—
459C2×C4—92no—
460C2×C4—92no—
461C2×C4—92no—
462C2×C4—94no—
463C2×C4—94no—
464C2×C4—94no—
465C2×C4—94no—
466C2×C4—94no—
467C2×C4—94no—
468C2×C4—91no—
469C2×C4—91no—
470C2×C4—91no—
471C2×C4—91no—
472C2×C4—91no—
473C2×C4—92no—
474C2×C4—92no—
475C2×C4—92no—
476C2×C4—92no—
477C2×C4C2×A4211yescharacteristic
478C8—34no—
479C8—92no—
480C8C2×A4212yesdirect product factor
481C2×C3—44no—
482C2×C3—42no—
483C2×C3—122no—
484C2×C3—122no—
485C2×C3—124no—
486C2×C3—164no—
487C2×C3—162no—
488C22—32no—
489C22—34no—
490C22—34no—
491C22—34no—
492C22—34no—
493C22—32no—
494C22—32no—
495C22—32no—
496C22—91no—
497C22—92no—
498C22—94no—
499C22—91no—
500C22—92no—
501C22—94no—
502C22—92no—
503C22—92no—
504C22—94no—
505C22—92no—
506C22—94no—
507C22—98no—
508C22—94no—
509C22—98no—
510C22—92no—
511C22—94no—
512C22—94no—
513C22—92no—
514C22—92no—
515C22—91no—
516C22—91no—
517C22C2×C3×C8×A412yescomplemented
518C22C4×A4211yescharacteristic
519C4—32no—
520C4—32no—
521C4—91no—
522C4—91no—
523C4C4×A4211yescharacteristic
524C4C22×A4211yescharacteristic
525C3—42no—
526C3—162no—
527C2—32no—
528C2—32no—
529C2—34no—
530C2—91no—
531C2—92no—
532C2—91no—
533C2C8×A4212yesdirect product factor
534C2C2×C4×A4211yescharacteristic
subgroupG/H
center, hypercenterC2×C8A42
commutator, hypocenterC24C2×C32×C8
FrattiniC4C22×A42
FittingC25×C8C32
socleC26C32×C4
prime radicalC22×A42C4
cosocleC24×C4C22×C32
LCSG ▷ C24
UCSC1 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C8×A42C2direct (N × H)
3. C2×C8×C22⋊A4C3semidirect
4. C23×C8×A4C3semidirect
11. C8×C22⋊A4C2×C3semidirect
15. C22×C8×A4C2×C3semidirect
16. C2×A42C8direct (N × H)
20. C25×C8C32semidirect
39. C2×C8×A4A4direct (N × H)
41. A42C2×C8direct (N × H)
52. C24×C8C2×C32semidirect
53. C2×C22⋊A4C3×C8semidirect
58. C23×A4C3×C8semidirect
79. C8×A4C2×A4direct (N × H)
120. C23×C8C3×A4semidirect
122. C22⋊A4C2×C3×C8semidirect
132. C22×A4C2×C3×C8semidirect
165. C25C32×C8semidirect
240. C22×C8C2×C3×A4semidirect
253. C2×A4C8×A4direct (N × H)
299. C24C2×C32×C8semidirect
378. C2×C8A42direct (N × H)
391. A4C2×C8×A4direct (N × H)
445. C23C3×C8×A4semidirect
480. C8C2×A42direct (N × H)
517. C22C2×C3×C8×A4semidirect
533. C2C8×A42direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×A422
C8×A422
C2×C8×C22⋊A43
C23×C8×A43
C2×C3×C8×A44