Finite Group Databasebeta

← C2×A4×#217 #11923409

Subgroups — C2×A4×#217 #11923409

Order 1152.

2118 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 2118 conjugacy classes of subgroups — too many to draw. Only its 195 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
1C24×C3×A4C211yesmaximal, characteristic
2C22×Dic3×A4C212yesmaximal, complemented
3A4×#217C214yesmaximal, direct product factor
4C23×#217C311yesmaximal, characteristic, complemented
5C2×A4×C22⋊C4—31nomaximal
6C23×C3×A4—24no—
7C23×C3×A4C414yescomplemented
8C23×C3×A4C2212yes—
9C23×C3×A4C2211yescharacteristic
10C2×Dic3×A4—24no—
11C2×Dic3×A4C2214yescomplemented
12C2×C3×#217—41nomaximal
13C26×C3C2×C311yescharacteristic
14C24×A4—21no—
15C24×A4D311yescharacteristic
16C24×Dic3C2×C312yescomplemented
17C22×C4×A4—32no—
18C22×#217—31no—
19C22×#217—32no—
20C22×#217—34no—
21C22×#217—31no—
22C22×#217C2×C314yescomplemented
23A4×C22⋊C4—34no—
24C24×C32—41no—
25C22×C3×A4—28no—
26C22×C3×A4—24no—
27C22×C3×A4C2×C412yes—
28C22×C3×A4C2×C414yescomplemented
29C22×C3×A4D414yes—
30C22×C3×A4C2311yescharacteristic
31C22×C3×Dic3—42no—
32Dic3×A4—24no—
33C3×#217—44no—
34C23×C22⋊C4—31noSylow
35C25×C3—24no—
36C25×C3—34no—
37C25×C3—32no—
38C25×C3—31no—
39C25×C3—31no—
40C25×C3—64no—
41C25×C3C3×C414yescomplemented
42C25×C3C22×C312yes—
43C25×C3C22×C311yescharacteristic
44C23×A4—28no—
45C23×A4—24no—
46C23×A4—24no—
47C23×A4—22no—
48C23×A4—21no—
49C23×A4Dic314yescomplemented
50C23×A4C2×D312yes—
51C23×A4C2×D311yescharacteristic
52C23×Dic3—24no—
53C23×Dic3—32no—
54C23×Dic3—32no—
55C23×Dic3—34no—
56C23×Dic3—64no—
57C23×Dic3C22×C314yescomplemented
58C2×C4×A4—34no—
59C2×C4×A4—64no—
60C2×C3×C22⋊C4—121no—
61C2×#217—32no—
62C2×#217—34no—
63C2×#217—38no—
64C2×#217—31no—
65C2×#217—32no—
66C2×#217—34no—
67C2×#217—34no—
68C2×#217—34no—
69C2×#217—38no—
70C2×#217A411yescharacteristic, direct product factor
71C23×C32—44no—
72C23×C32—41no—
73C23×C32—42no—
74C23×C32—84no—
75C2×C3×A4—24no—
76C2×C3×A4C22⋊C414yescomplemented
77C2×C3×A4C22×C411yescharacteristic
78C2×C3×A4C2×D412yes—
79C2×C3×Dic3—44no—
80C2×C3×Dic3—84no—
81C26C3×D311yescharacteristic
82C24×C4—32no—
83C22×C22⋊C4—34no—
84C22×C22⋊C4—91no—
85C22×C22⋊C4—94no—
86C22×C22⋊C4—92no—
87C22×C22⋊C4—91no—
88C24×C3—28no—
89C24×C3—24no—
90C24×C3—32no—
91C24×C3—31no—
92C24×C3—34no—
93C24×C3—31no—
94C24×C3—38no—
95C24×C3—34no—
96C24×C3—32no—
97C24×C3—38no—
98C24×C3—34no—
99C24×C3—34no—
100C24×C3—32no—
101C24×C3—34no—
102C24×C3—34no—
103C24×C3—32no—
104C24×C3—32no—
105C24×C3—31no—
106C24×C3—41no—
107C24×C3—64no—
108C24×C3—64no—
109C24×C3—68no—
110C24×C3—64no—
111C24×C3—68no—
112C24×C3—616no—
113C24×C3—64no—
114C24×C3—64no—
115C24×C3—68no—
116C24×C3—616no—
117C24×C3—81no—
118C24×C3C2×C3×C412yes—
119C24×C3C2×C3×C414yescomplemented
120C24×C3C3×D414yes—
121C24×C3C2×A411yescharacteristic
122C24×C3C23×C311yescharacteristic
123C22×A4—216no—
124C22×A4—28no—
125C22×A4—28no—
126C22×A4—24no—
127C22×A4—22no—
128C22×A4—24no—
129C22×A4—24no—
130C22×A4—21no—
131C22×A4C2×Dic312yes—
132C22×A4C2×Dic314yescomplemented
133C22×A4#6714yes—
134C22×A4C22×D311yescharacteristic
135C22×C3×C4—122no—
136C22×Dic3—24no—
137C22×Dic3—34no—
138C22×Dic3—32no—
139C22×Dic3—34no—
140C22×Dic3—34no—
141C22×Dic3—34no—
142C22×Dic3—64no—
143C22×Dic3—64no—
144C22×Dic3—68no—
145C22×Dic3—68no—
146C22×Dic3—64no—
147C22×Dic3—64no—
148C22×Dic3—64no—
149C22×Dic3C2×A412yescomplemented
150C4×A4—64no—
151C3×C22⋊C4—124no—
152#217—38no—
153#217—34no—
154#217—38no—
155#217C2×A414yesdirect product factor
156C22×C32—42no—
157C22×C32—44no—
158C22×C32—44no—
159C22×C32—41no—
160C22×C32—84no—
161C22×C32—88no—
162C3×A4C2×C22⋊C411yescharacteristic, complemented
163C3×Dic3—84no—
164C25—24no—
165C25—32no—
166C25—31no—
167C25—34no—
168C25—31no—
169C25—64no—
170C25C3×Dic314yescomplemented
171C25C2×C3×D312yes—
172C25C2×C3×D311yescharacteristic
173C23×C4—34no—
174C23×C4—64no—
175C23×C4—92no—
176C23×C4—94no—
177C23×C4—92no—
178C23×C4—184no—
179C2×C22⋊C4—31no—
180C2×C22⋊C4—94no—
181C2×C22⋊C4—94no—
182C2×C22⋊C4—92no—
183C2×C22⋊C4—98no—
184C2×C22⋊C4—92no—
185C2×C22⋊C4—94no—
186C2×C22⋊C4—91no—
187C2×C22⋊C4—98no—
188C2×C22⋊C4—94no—
189C23×C3—24no—
190C23×C3—24no—
191C23×C3—34no—
192C23×C3—32no—
193C23×C3—32no—
194C23×C3—31no—
195C23×C3—34no—
196C23×C3—38no—
197C23×C3—34no—
198C23×C3—32no—
199C23×C3—38no—
200C23×C3—32no—
201C23×C3—32no—
202C23×C3—34no—
203C23×C3—34no—
204C23×C3—31no—
205C23×C3—31no—
206C23×C3—34no—
207C23×C3—38no—
208C23×C3—34no—
209C23×C3—34no—
210C23×C3—38no—
211C23×C3—34no—
212C23×C3—34no—
213C23×C3—32no—
214C23×C3—42no—
215C23×C3—41no—
216C23×C3—44no—
217C23×C3—64no—
218C23×C3—68no—
219C23×C3—64no—
220C23×C3—68no—
221C23×C3—616no—
222C23×C3—68no—
223C23×C3—64no—
224C23×C3—616no—
225C23×C3—68no—
226C23×C3—64no—
227C23×C3—68no—
228C23×C3—64no—
229C23×C3—68no—
230C23×C3—616no—
231C23×C3—68no—
232C23×C3—632no—
233C23×C3—68no—
234C23×C3—616no—
235C23×C3—64no—
236C23×C3—84no—
237C23×C3—88no—
238C23×C3—82no—
239C23×C3—81no—
240C23×C3—84no—
241C23×C3C3×C22⋊C414yescomplemented
242C23×C3C4×A414yescomplemented
243C23×C3C22×C3×C411yescharacteristic
244C23×C3C2×C3×D412yes—
245C23×C3C22×A412yes—
246C23×C3C22×A411yescharacteristic
247C2×A4—28no—
248C2×A4—24no—
249C2×A4—24no—
250C2×A4—21no—
251C2×A4—22no—
252C2×A4#21714yesdirect product factor
253C2×A4C22×Dic311yescharacteristic
254C2×A4C2×#6712yes—
255C2×C3×C4—124no—
256C2×C3×C4—244no—
257C2×Dic3—24no—
258C2×Dic3—34no—
259C2×Dic3—64no—
260C2×Dic3—68no—
261C2×Dic3—64no—
262C2×Dic3—68no—
263C2×Dic3—64no—
264C2×Dic3—64no—
265C2×Dic3—64no—
266C2×Dic3C22×A414yescomplemented
267C2×C32—41no—
268C2×C32—42no—
269C2×C32—44no—
270C2×C32—84no—
271C24—28no—
272C24—24no—
273C24—31no—
274C24—34no—
275C24—32no—
276C24—31no—
277C24—32no—
278C24—38no—
279C24—34no—
280C24—32no—
281C24—38no—
282C24—34no—
283C24—34no—
284C24—32no—
285C24—32no—
286C24—31no—
287C24—34no—
288C24—34no—
289C24—64no—
290C24—64no—
291C24—68no—
292C24—64no—
293C24—68no—
294C24—616no—
295C24—64no—
296C24—64no—
297C24—68no—
298C24—616no—
299C24C2×C3×Dic312yes—
300C24C2×C3×Dic314yescomplemented
301C24C3×#6714yes—
302C24D3×A411yescharacteristic
303C24C22×C3×D311yescharacteristic
304C22×C4—32no—
305C22×C4—64no—
306C22×C4—94no—
307C22×C4—94no—
308C22×C4—94no—
309C22×C4—92no—
310C22×C4—94no—
311C22×C4—184no—
312C22×C4—184no—
313C22×C4—188no—
314C22×C4—188no—
315C22×C4—184no—
316C22×C4—184no—
317C22×C4—184no—
318C22⋊C4—34no—
319C22⋊C4—98no—
320C22⋊C4—98no—
321C22⋊C4—94no—
322C22×C3—24no—
323C22×C3—28no—
324C22×C3—34no—
325C22×C3—31no—
326C22×C3—32no—
327C22×C3—38no—
328C22×C3—32no—
329C22×C3—34no—
330C22×C3—32no—
331C22×C3—31no—
332C22×C3—31no—
333C22×C3—34no—
334C22×C3—32no—
335C22×C3—32no—
336C22×C3—34no—
337C22×C3—38no—
338C22×C3—34no—
339C22×C3—34no—
340C22×C3—42no—
341C22×C3—44no—
342C22×C3—44no—
343C22×C3—41no—
344C22×C3—64no—
345C22×C3—616no—
346C22×C3—68no—
347C22×C3—64no—
348C22×C3—64no—
349C22×C3—64no—
350C22×C3—616no—
351C22×C3—68no—
352C22×C3—64no—
353C22×C3—68no—
354C22×C3—84no—
355C22×C3—88no—
356C22×C3—88no—
357C22×C3—816no—
358C22×C3—82no—
359C22×C3—84no—
360C22×C3—84no—
361C22×C3—81no—
362C22×C3C2×C3×C22⋊C411yescharacteristic, complemented
363C22×C3C2×C4×A412yes—
364C22×C3C2×C4×A414yescomplemented
365C22×C3D4×A414yes—
366C22×C3C23×A411yescharacteristic
367A4—21no—
368A4C2×#21711yescharacteristic, direct product factor
369C3×C4—244no—
370Dic3—24no—
371Dic3—64no—
372C32—41noSylow
373C23—24no—
374C23—24no—
375C23—34no—
376C23—32no—
377C23—32no—
378C23—31no—
379C23—34no—
380C23—38no—
381C23—34no—
382C23—32no—
383C23—38no—
384C23—32no—
385C23—32no—
386C23—34no—
387C23—34no—
388C23—31no—
389C23—31no—
390C23—38no—
391C23—34no—
392C23—34no—
393C23—34no—
394C23—34no—
395C23—38no—
396C23—34no—
397C23—32no—
398C23—64no—
399C23—68no—
400C23—64no—
401C23—68no—
402C23—68no—
403C23—616no—
404C23—64no—
405C23—616no—
406C23—68no—
407C23—64no—
408C23—68no—
409C23—64no—
410C23—68no—
411C23—616no—
412C23—68no—
413C23—632no—
414C23—68no—
415C23—616no—
416C23—64no—
417C23C3×#21714yescomplemented
418C23Dic3×A414yescomplemented
419C23C22×C3×Dic311yescharacteristic
420C23C2×C3×#6712yes—
421C23C2×D3×A412yes—
422C23C2×D3×A411yescharacteristic
423C2×C4—34no—
424C2×C4—64no—
425C2×C4—94no—
426C2×C4—184no—
427C2×C4—184no—
428C2×C4—188no—
429C2×C4—188no—
430C2×C4—184no—
431C2×C4—184no—
432C2×C4—184no—
433C2×C3—24no—
434C2×C3—34no—
435C2×C3—31no—
436C2×C3—32no—
437C2×C3—31no—
438C2×C3—41no—
439C2×C3—42no—
440C2×C3—44no—
441C2×C3—64no—
442C2×C3—84no—
443C2×C3—88no—
444C2×C3—81no—
445C2×C3—82no—
446C2×C3—84no—
447C2×C3A4×C22⋊C414yescomplemented
448C2×C3C22×C4×A411yescharacteristic
449C2×C3C2×D4×A412yes—
450C22—24no—
451C22—28no—
452C22—34no—
453C22—31no—
454C22—32no—
455C22—38no—
456C22—32no—
457C22—34no—
458C22—31no—
459C22—32no—
460C22—31no—
461C22—32no—
462C22—34no—
463C22—32no—
464C22—34no—
465C22—34no—
466C22—38no—
467C22—34no—
468C22—64no—
469C22—64no—
470C22—616no—
471C22—68no—
472C22—64no—
473C22—64no—
474C22—616no—
475C22—68no—
476C22—64no—
477C22—68no—
478C22C2×C3×#21711yescharacteristic, complemented
479C22C2×Dic3×A412yes—
480C22C2×Dic3×A414yescomplemented
481C22A4×#6714yes—
482C22C22×D3×A411yescharacteristic
483C4—64no—
484C4—184no—
485C3—41no—
486C3—81no—
487C3C2×A4×C22⋊C411yescharacteristic, complemented
488C2—24no—
489C2—31no—
490C2—34no—
491C2—31no—
492C2—32no—
493C2—64no—
494C2A4×#21714yesdirect product factor
495C2C22×Dic3×A411yescharacteristic
496C2C2×A4×#6712yes—
subgroupG/H
centerC23C2×D3×A4
commutatorC23×C3C22×C3×C4
FrattiniC22C22×D3×A4
FittingC26×C3C2×C3
socleC25×C3C22×C3
prime radicalC24×C3×A4C2
cosocleC24×C3C23×C3
hypercenterC24D3×A4
hypocenterC22×C3C2×C3×C22⋊C4
LCSG ▷ C23×C3 ▷ C22×C3
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C22×Dic3×A4C2semidirect
3. A4×#217C2direct (N × H)
4. C23×#217C3semidirect
7. C23×C3×A4C4semidirect
11. C2×Dic3×A4C22semidirect
16. C24×Dic3C2×C3semidirect
22. C22×#217C2×C3semidirect
28. C22×C3×A4C2×C4semidirect
41. C25×C3C3×C4semidirect
49. C23×A4Dic3semidirect
57. C23×Dic3C22×C3semidirect
70. C2×#217A4direct (N × H)
76. C2×C3×A4C22⋊C4semidirect
119. C24×C3C2×C3×C4semidirect
132. C22×A4C2×Dic3semidirect
149. C22×Dic3C2×A4semidirect
155. #217C2×A4direct (N × H)
162. C3×A4C2×C22⋊C4semidirect
170. C25C3×Dic3semidirect
241. C23×C3C3×C22⋊C4semidirect
242. C23×C3C4×A4semidirect
252. C2×A4#217direct (N × H)
266. C2×Dic3C22×A4semidirect
300. C24C2×C3×Dic3semidirect
362. C22×C3C2×C3×C22⋊C4semidirect
364. C22×C3C2×C4×A4semidirect
368. A4C2×#217direct (N × H)
417. C23C3×#217semidirect
418. C23Dic3×A4semidirect
447. C2×C3A4×C22⋊C4semidirect
478. C22C2×C3×#217semidirect
480. C22C2×Dic3×A4semidirect
487. C3C2×A4×C22⋊C4semidirect
494. C2A4×#217direct (N × H)

Maximal subgroups

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

subgroupindex
C24×C3×A42
C22×Dic3×A42
A4×#2172
C23×#2173
C2×A4×C22⋊C43
C2×C3×#2174