Finite Group Databasebeta

← C2×C3×#5414 #11922771

Subgroups — C2×C3×#5414 #11922771

Order 1152.

768 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 768 conjugacy classes of subgroups — too many to draw. Only its 88 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×D4×A4C211yesmaximal, characteristic
2C2×C3×A4⋊Q8C211yesmaximal, characteristic, complemented
3C3×#5414C214yesmaximal, direct product factor
4C2×C3×A4⋊C8C211yesmaximal, characteristic, complemented
5C2×#5414C311yesmaximal, characteristic, direct product factor
6C2×C3×#450—31nomaximal
7C23×C3×A4—21no—
8C3×D4×A4—21no—
9C3×D4×A4C2212yes—
10C2×C3×C4×A4C2211yescharacteristic
11C2×C3×SL(2,ℤ/4)—21no—
12C3×A4⋊Q8—21no—
13C3×A4⋊Q8C2212yescomplemented
14C2×C3×#214—41nomaximal
15C3×A4⋊C8C2212yescomplemented
16C23×C3×D4D311yescharacteristic
17C2×D4×A4—21no—
18C2×D4×A4C2×C311yescharacteristic
19C2×A4⋊Q8C2×C311yescharacteristic, complemented
20C22×C3×SD16—31no—
21C22×C3×SD16—31no—
22C2×C3×C22⋊Q8—31no—
23#5414C2×C314yesdirect product factor
24C2×A4⋊C8C2×C311yescharacteristic, complemented
25C3×#450—34no—
26C3×#450—34no—
27C2×C3×#102—31no—
28C2×C3×#102—31no—
29C2×C3×C22⋊C8—31no—
30C22×C3×A4—22no—
31C22×C3×A4—42no—
32C22×C3×A4D411yescharacteristic
33C2×C32×D4—41no—
34C2×C3×Dic6—41no—
35C3×C4×A4D411yescharacteristic
36C3×C4×A4C2311yescharacteristic
37C3×SL(2,ℤ/4)—22no—
38C3×#214—44no—
39C2×C3×C3⋊C8—41no—
40C2×#450—31noSylow
41C25×C3—21no—
42C23×A4—22no—
43C23×A4—21no—
44C22×C3×Q8—31no—
45C22×C3×D4—21no—
46C22×C3×D4—31no—
47C22×C3×D4—31no—
48C22×C3×D4—32no—
49C22×C3×D4—61no—
50C22×C3×D4—61no—
51C22×C3×D4C2×D312yes—
52C23×C3×C4C2×D311yescharacteristic
53D4×A4—22no—
54D4×A4—21no—
55D4×A4—22no—
56D4×A4C22×C312yes—
57C2×C4×A4—21no—
58C2×C4×A4C22×C311yescharacteristic
59C2×SL(2,ℤ/4)—21no—
60A4⋊Q8—21no—
61A4⋊Q8C22×C312yescomplemented
62C2×C3×SD16—34no—
63C2×C3×SD16—34no—
64C2×C3×SD16—62no—
65C2×C3×SD16—62no—
66C2×C3×SD16—62no—
67C2×C3×SD16—61no—
68C2×C3×SD16—61no—
69C22×C3×C8—31no—
70C22×C3×C8—31no—
71C3×C22⋊Q8—32no—
72C3×C22⋊Q8—32no—
73C3×C22⋊Q8—61no—
74C3×C22⋊Q8—61no—
75C2×C3×C4⋊C4—31no—
76C2×C3×C4⋊C4—61no—
77C2×C3×C22⋊C4—61no—
78C2×#214—41no—
79A4⋊C8C22×C312yescomplemented
80C3×#102—34no—
81C3×#102—34no—
82C3×C22⋊C8—32no—
83C3×C22⋊C8—32no—
84C23×C32—81no—
85C2×C3×A4—42no—
86C2×C3×A4SD1612yescomplemented
87C2×C3×A4C2×D411yescharacteristic
88C32×D4—42no—
89C32×D4—81no—
90C2×C32×C4—41no—
91C2×C3×Dic3—81no—
92C3×Dic6—42no—
93C3×Dic6—81no—
94C3×C3⋊C8—42no—
95C23×D4C3×D311yescharacteristic
96C22×SD16—31no—
97C22×SD16—31no—
98C2×C22⋊Q8—31no—
99#450—34no—
100#450—34no—
101C2×#102—31no—
102C2×#102—31no—
103C2×C22⋊C8—31no—
104C24×C3—22no—
105C24×C3—42no—
106C24×C3—61no—
107C24×C3—61no—
108C24×C3—62no—
109C24×C3—122no—
110C24×C3#6711yescharacteristic
111C22×A4—24no—
112C22×A4—22no—
113C22×A4—21no—
114C22×A4—44no—
115C22×A4—42no—
116C22×A4C3×D411yescharacteristic
117C2×C3×Q8—32no—
118C2×C3×Q8—61no—
119C2×C3×Q8—61no—
120C2×C3×Q8—61no—
121C2×C3×Q8—61no—
122C2×C3×Q8—61no—
123C2×C3×D4—32no—
124C2×C3×D4—32no—
125C2×C3×D4—32no—
126C2×C3×D4—32no—
127C2×C3×D4—32no—
128C2×C3×D4—31no—
129C2×C3×D4—41no—
130C2×C3×D4—61no—
131C2×C3×D4—61no—
132C2×C3×D4—61no—
133C2×C3×D4—62no—
134C2×C3×D4—61no—
135C2×C3×D4—61no—
136C2×C3×D4—62no—
137C2×C3×D4—64no—
138C2×C3×D4—81no—
139C2×C3×D4S411yescharacteristic
140C22×C3×C4—31no—
141C22×C3×C4—31no—
142C22×C3×C4—31no—
143C22×C3×C4—31no—
144C22×C3×C4—61no—
145C22×C3×C4—61no—
146C22×C3×C4#6711yescharacteristic
147C22×C3×C4C22×D311yescharacteristic
148C2×Dic6—41no—
149C4×A4—21no—
150C4×A4—21no—
151C4×A4C3×D411yescharacteristic
152C4×A4C23×C311yescharacteristic
153SL(2,ℤ/4)—22no—
154C3×SD16—68no—
155C3×SD16—64no—
156C3×SD16—64no—
157C2×C3×C8—32no—
158C2×C3×C8—32no—
159C2×C3×C8—61no—
160C2×C3×C8—61no—
161C2×C3×C8—61no—
162C2×C3×C8—61no—
163C3×C4⋊C4—32no—
164C3×C4⋊C4—61no—
165C3×C4⋊C4—62no—
166C3×C4⋊C4—62no—
167C3×C22⋊C4—62no—
168C3×C22⋊C4—62no—
169#214—44no—
170C2×C3⋊C8—41no—
171C22×C32—41no—
172C22×C32—82no—
173C22×C32—162no—
174C3×A4C2×SD1611yescharacteristic, complemented
175C32×C4—41no—
176C32×C4—41no—
177C3×Dic3—82no—
178C25—21no—
179C22×Q8—31no—
180C22×D4—21no—
181C22×D4—31no—
182C22×D4—31no—
183C22×D4—32no—
184C22×D4—61no—
185C22×D4—61no—
186C22×D4C2×C3×D312yes—
187C23×C4C2×C3×D311yescharacteristic
188C2×SD16—34no—
189C2×SD16—34no—
190C2×SD16—62no—
191C2×SD16—62no—
192C2×SD16—62no—
193C2×SD16—61no—
194C2×SD16—61no—
195C22×C8—31no—
196C22×C8—31no—
197C22⋊Q8—32no—
198C22⋊Q8—32no—
199C22⋊Q8—61no—
200C22⋊Q8—61no—
201C2×C4⋊C4—31no—
202C2×C4⋊C4—61no—
203C2×C22⋊C4—61no—
204#102—34no—
205#102—34no—
206C22⋊C8—32no—
207C22⋊C8—32no—
208C23×C3—21no—
209C23×C3—31no—
210C23×C3—31no—
211C23×C3—32no—
212C23×C3—42no—
213C23×C3—62no—
214C23×C3—62no—
215C23×C3—62no—
216C23×C3—62no—
217C23×C3—61no—
218C23×C3—81no—
219C23×C3—82no—
220C23×C3—122no—
221C23×C3—122no—
222C23×C3—122no—
223C23×C3—124no—
224C23×C3—122no—
225C23×C3—122no—
226C23×C3—124no—
227C23×C3#21412yescomplemented
228C23×C3C2×#6711yescharacteristic
229C2×A4—22no—
230C2×A4—21no—
231C2×A4—44no—
232C2×A4—42no—
233C2×A4C3×SD1612yescomplemented
234C2×A4C2×C3×D411yescharacteristic
235C3×Q8—62no—
236C3×Q8—62no—
237C3×Q8—121no—
238C3×Q8—121no—
239C3×D4—21no—
240C3×D4—34no—
241C3×D4—32no—
242C3×D4—42no—
243C3×D4—62no—
244C3×D4—62no—
245C3×D4—62no—
246C3×D4—61no—
247C3×D4—81no—
248C3×D4—82no—
249C3×D4—82no—
250C3×D4C2×S412yes—
251C2×C3×C4—31no—
252C2×C3×C4—31no—
253C2×C3×C4—31no—
254C2×C3×C4—31no—
255C2×C3×C4—31no—
256C2×C3×C4—31no—
257C2×C3×C4—31no—
258C2×C3×C4—41no—
259C2×C3×C4—61no—
260C2×C3×C4—62no—
261C2×C3×C4—62no—
262C2×C3×C4—81no—
263C2×C3×C4—122no—
264C2×C3×C4—121no—
265C2×C3×C4—121no—
266C2×C3×C4C2×S411yescharacteristic
267C2×Dic3—81no—
268Dic6—42no—
269Dic6—81no—
270C3×C8—62no—
271C3×C8—62no—
272C3⋊C8—42no—
273C2×C32—42no—
274C2×C32—41no—
275C2×C32—162no—
276C24—22no—
277C24—42no—
278C24—61no—
279C24—61no—
280C24—62no—
281C24—122no—
282C24C3×#6711yescharacteristic
283C2×Q8—32no—
284C2×Q8—61no—
285C2×Q8—61no—
286C2×Q8—61no—
287C2×Q8—61no—
288C2×Q8—61no—
289C2×D4—32no—
290C2×D4—32no—
291C2×D4—32no—
292C2×D4—32no—
293C2×D4—32no—
294C2×D4—31no—
295C2×D4—61no—
296C2×D4—61no—
297C2×D4—61no—
298C2×D4—62no—
299C2×D4—61no—
300C2×D4—61no—
301C2×D4—62no—
302C2×D4—64no—
303C2×D4C3×S411yescharacteristic
304C22×C4—31no—
305C22×C4—31no—
306C22×C4—31no—
307C22×C4—31no—
308C22×C4—61no—
309C22×C4—61no—
310C22×C4C3×#6711yescharacteristic
311C22×C4C22×C3×D311yescharacteristic
312SD16—68no—
313SD16—64no—
314SD16—64no—
315C2×C8—32no—
316C2×C8—32no—
317C2×C8—61no—
318C2×C8—61no—
319C2×C8—61no—
320C2×C8—61no—
321C4⋊C4—32no—
322C4⋊C4—61no—
323C4⋊C4—62no—
324C4⋊C4—62no—
325C22⋊C4—62no—
326C22⋊C4—62no—
327C22×C3—22no—
328C22×C3—32no—
329C22×C3—31no—
330C22×C3—32no—
331C22×C3—32no—
332C22×C3—31no—
333C22×C3—31no—
334C22×C3—42no—
335C22×C3—41no—
336C22×C3—61no—
337C22×C3—62no—
338C22×C3—82no—
339C22×C3—84no—
340C22×C3—81no—
341C22×C3—122no—
342C22×C3—122no—
343C22×C3—124no—
344C22×C3—122no—
345C22×C3—122no—
346C22×C3—124no—
347C22×C3—122no—
348C22×C3—162no—
349C22×C3—164no—
350C22×C3C2×#21411yescharacteristic, complemented
351C22×C3GL(2,ℤ/4)11yescharacteristic
352A4—21no—
353A4C2×C3×SD1611yescharacteristic, complemented
354C3×C4—31no—
355C3×C4—31no—
356C3×C4—41no—
357C3×C4—41no—
358C3×C4—81no—
359C3×C4—81no—
360C3×C4—122no—
361C3×C4—122no—
362C3×C4GL(2,ℤ/4)11yescharacteristic
363C3×C4C22×S411yescharacteristic
364Dic3—82no—
365C32—41noSylow
366C23—21no—
367C23—31no—
368C23—31no—
369C23—32no—
370C23—42no—
371C23—62no—
372C23—62no—
373C23—62no—
374C23—62no—
375C23—61no—
376C23—122no—
377C23—122no—
378C23—122no—
379C23—124no—
380C23—122no—
381C23—122no—
382C23—124no—
383C23C3×#21412yescomplemented
384C23C2×C3×#6711yescharacteristic
385Q8—62no—
386Q8—62no—
387Q8—121no—
388Q8—121no—
389D4—21no—
390D4—34no—
391D4—32no—
392D4—62no—
393D4—62no—
394D4—62no—
395D4—61no—
396D4C2×C3×S412yes—
397C2×C4—31no—
398C2×C4—31no—
399C2×C4—31no—
400C2×C4—31no—
401C2×C4—31no—
402C2×C4—31no—
403C2×C4—31no—
404C2×C4—61no—
405C2×C4—62no—
406C2×C4—62no—
407C2×C4—121no—
408C2×C4—122no—
409C2×C4—121no—
410C2×C4C2×C3×S411yescharacteristic
411C8—62no—
412C8—62no—
413C2×C3—31no—
414C2×C3—31no—
415C2×C3—32no—
416C2×C3—42no—
417C2×C3—42no—
418C2×C3—41no—
419C2×C3—82no—
420C2×C3—81no—
421C2×C3—122no—
422C2×C3—162no—
423C2×C3—164no—
424C2×C3#541412yesdirect product factor
425C2×C3C2×GL(2,ℤ/4)11yescharacteristic
426C22—22no—
427C22—32no—
428C22—31no—
429C22—32no—
430C22—32no—
431C22—31no—
432C22—31no—
433C22—42no—
434C22—61no—
435C22—62no—
436C22—122no—
437C22—122no—
438C22—124no—
439C22—122no—
440C22—122no—
441C22—124no—
442C22—122no—
443C22C2×C3×#21411yescharacteristic, complemented
444C22C3×GL(2,ℤ/4)11yescharacteristic
445C4—31no—
446C4—31no—
447C4—122no—
448C4—122no—
449C4C3×GL(2,ℤ/4)11yescharacteristic
450C4C22×C3×S411yescharacteristic
451C3—41no—
452C3—81no—
453C3C2×#541411yescharacteristic, direct product factor
454C2—31no—
455C2—31no—
456C2—32no—
457C2—42no—
458C2—122no—
459C2C3×#541412yesdirect product factor
460C2C2×C3×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
centerC22×C3GL(2,ℤ/4)
commutator, cosocleC4×A4C23×C3
FrattiniC4C22×C3×S4
FittingC23×C3×D4D3
socleC24×C3#67
prime radicalC2×C3×D4×A4C2
hypercenterC2×C3×D4S4
hypocenterA4C2×C3×SD16
LCSG ▷ C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
2. C2×C3×A4⋊Q8C2semidirect
3. C3×#5414C2direct (N × H)
4. C2×C3×A4⋊C8C2semidirect
5. C2×#5414C3direct (N × H)
13. C3×A4⋊Q8C22semidirect
15. C3×A4⋊C8C22semidirect
19. C2×A4⋊Q8C2×C3semidirect
23. #5414C2×C3direct (N × H)
24. C2×A4⋊C8C2×C3semidirect
61. A4⋊Q8C22×C3semidirect
79. A4⋊C8C22×C3semidirect
86. C2×C3×A4SD16semidirect
174. C3×A4C2×SD16semidirect
227. C23×C3#214semidirect
233. C2×A4C3×SD16semidirect
350. C22×C3C2×#214semidirect
353. A4C2×C3×SD16semidirect
383. C23C3×#214semidirect
424. C2×C3#5414direct (N × H)
443. C22C2×C3×#214semidirect
453. C3C2×#5414direct (N × H)
459. C2C3×#5414direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D4×A42
C2×C3×A4⋊Q82
C3×#54142
C2×C3×A4⋊C82
C2×#54143
C2×C3×#4503
C2×C3×#2144