Finite Group Databasebeta

← C2×C3×#5415 #11922772

Subgroups — C2×C3×#5415 #11922772

Order 1152.

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

normal subgroups only

This group has 836 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, complemented
2C2×C3×C4⋊S4C211yesmaximal, characteristic, complemented
3C3×#5415C214yesmaximal, direct product factor
4C2×C3×A4⋊C8C211yesmaximal, characteristic, complemented
5C2×#5415C311yesmaximal, characteristic, direct product factor
6C2×C3×#447—31nomaximal
7C23×C3×A4—21no—
8C22×C3×S4—21no—
9C3×D4×A4—21no—
10C3×D4×A4C2212yescomplemented
11C2×C3×C4×A4C2211yescharacteristic
12C3×C4⋊S4—21no—
13C3×C4⋊S4C2212yescomplemented
14C2×C3×#213—41nomaximal
15C3×A4⋊C8C2212yescomplemented
16C23×C3×D4D311yescharacteristic, complemented
17C2×D4×A4—21no—
18C2×D4×A4C2×C311yescharacteristic, complemented
19C2×C4⋊S4C2×C311yescharacteristic, complemented
20C22×C3×D8—31no—
21C22×C3×D8—31no—
22C2×C3×#121—31no—
23#5415C2×C314yesdirect product factor
24C2×A4⋊C8C2×C311yescharacteristic, complemented
25C3×#447—34no—
26C3×#447—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×C3×S4—22no—
34C2×C3×S4—42no—
35C2×C32×D4—41no—
36C2×C3×D12—41no—
37C3×C4×A4D411yescharacteristic
38C3×C4×A4C2311yescharacteristic
39C3×#213—44no—
40C2×C3×C3⋊C8—41no—
41C2×#447—31noSylow
42C25×C3—21no—
43C23×A4—22no—
44C23×A4—21no—
45C22×S4—21no—
46C22×C3×D4—21no—
47C22×C3×D4—31no—
48C22×C3×D4—31no—
49C22×C3×D4—32no—
50C22×C3×D4—31no—
51C22×C3×D4—61no—
52C22×C3×D4—61no—
53C22×C3×D4—61no—
54C22×C3×D4C2×D312yescomplemented
55C23×C3×C4C2×D311yescharacteristic
56D4×A4—22no—
57D4×A4—21no—
58D4×A4—22no—
59D4×A4C22×C312yescomplemented
60C2×C4×A4—21no—
61C2×C4×A4C22×C311yescharacteristic
62C4⋊S4—21no—
63C4⋊S4C22×C312yescomplemented
64C2×C3×D8—34no—
65C2×C3×D8—34no—
66C2×C3×D8—62no—
67C2×C3×D8—62no—
68C2×C3×D8—62no—
69C2×C3×D8—61no—
70C2×C3×D8—61no—
71C22×C3×C8—31no—
72C22×C3×C8—31no—
73C3×#121—32no—
74C3×#121—32no—
75C3×#121—61no—
76C3×#121—61no—
77C2×C3×C4⋊C4—31no—
78C2×C3×C22⋊C4—61no—
79C2×#213—41no—
80A4⋊C8C22×C312yescomplemented
81C3×#102—34no—
82C3×#102—34no—
83C3×C22⋊C8—32no—
84C3×C22⋊C8—32no—
85C23×C32—81no—
86C22×C3×D3—81no—
87C2×C3×A4—42no—
88C2×C3×A4D812yescomplemented
89C2×C3×A4C2×D411yescharacteristic
90C3×S4—42no—
91C32×D4—42no—
92C32×D4—81no—
93C2×C32×C4—41no—
94C3×D12—42no—
95C3×D12—81no—
96C3×C3⋊C8—42no—
97C23×D4C3×D311yescharacteristic, complemented
98C22×D8—31no—
99C22×D8—31no—
100C2×#121—31no—
101#447—34no—
102#447—34no—
103C2×#102—31no—
104C2×#102—31no—
105C2×C22⋊C8—31no—
106C24×C3—22no—
107C24×C3—42no—
108C24×C3—61no—
109C24×C3—61no—
110C24×C3—62no—
111C24×C3—61no—
112C24×C3—122no—
113C24×C3#6711yescharacteristic
114C22×A4—24no—
115C22×A4—22no—
116C22×A4—21no—
117C22×A4—44no—
118C22×A4—42no—
119C22×A4C3×D411yescharacteristic
120C2×S4—22no—
121C2×S4—42no—
122C2×C3×D4—32no—
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—61no—
138C2×C3×D4—61no—
139C2×C3×D4—61no—
140C2×C3×D4—61no—
141C2×C3×D4—64no—
142C2×C3×D4—62no—
143C2×C3×D4—62no—
144C2×C3×D4—61no—
145C2×C3×D4—81no—
146C2×C3×D4—122no—
147C2×C3×D4—122no—
148C2×C3×D4S411yescharacteristic, complemented
149C22×C3×C4—31no—
150C22×C3×C4—31no—
151C22×C3×C4—31no—
152C22×C3×C4—31no—
153C22×C3×C4—61no—
154C22×C3×C4#6711yescharacteristic
155C22×C3×C4C22×D311yescharacteristic
156C2×D12—41no—
157C4×A4—21no—
158C4×A4—21no—
159C4×A4C3×D411yescharacteristic
160C4×A4C23×C311yescharacteristic
161C3×D8—68no—
162C3×D8—64no—
163C3×D8—64no—
164C2×C3×C8—32no—
165C2×C3×C8—32no—
166C2×C3×C8—61no—
167C2×C3×C8—61no—
168C2×C3×C8—61no—
169C2×C3×C8—61no—
170C3×C4⋊C4—32no—
171C3×C4⋊C4—61no—
172C3×C22⋊C4—62no—
173C3×C22⋊C4—62no—
174#213—44no—
175C2×C3⋊C8—41no—
176C22×C32—41no—
177C22×C32—82no—
178C22×C32—162no—
179C2×C3×D3—82no—
180C2×C3×D3—162no—
181C3×A4C2×D811yescharacteristic, complemented
182C32×C4—41no—
183C32×C4—41no—
184C25—21no—
185C22×D4—21no—
186C22×D4—31no—
187C22×D4—31no—
188C22×D4—32no—
189C22×D4—31no—
190C22×D4—61no—
191C22×D4—61no—
192C22×D4—61no—
193C22×D4C2×C3×D312yescomplemented
194C23×C4C2×C3×D311yescharacteristic
195C2×D8—34no—
196C2×D8—34no—
197C2×D8—62no—
198C2×D8—62no—
199C2×D8—62no—
200C2×D8—61no—
201C2×D8—61no—
202C22×C8—31no—
203C22×C8—31no—
204#121—32no—
205#121—32no—
206#121—61no—
207#121—61no—
208C2×C4⋊C4—31no—
209C2×C22⋊C4—61no—
210#102—34no—
211#102—34no—
212C22⋊C8—32no—
213C22⋊C8—32no—
214C23×C3—21no—
215C23×C3—31no—
216C23×C3—31no—
217C23×C3—32no—
218C23×C3—42no—
219C23×C3—62no—
220C23×C3—62no—
221C23×C3—62no—
222C23×C3—62no—
223C23×C3—62no—
224C23×C3—61no—
225C23×C3—81no—
226C23×C3—82no—
227C23×C3—122no—
228C23×C3—122no—
229C23×C3—122no—
230C23×C3—124no—
231C23×C3—122no—
232C23×C3—122no—
233C23×C3—121no—
234C23×C3—122no—
235C23×C3—122no—
236C23×C3—124no—
237C23×C3—121no—
238C23×C3#21312yescomplemented
239C23×C3C2×#6711yescharacteristic
240C22×D3—81no—
241C2×A4—22no—
242C2×A4—21no—
243C2×A4—44no—
244C2×A4—42no—
245C2×A4C3×D812yescomplemented
246C2×A4C2×C3×D411yescharacteristic
247S4—42no—
248C3×D4—21no—
249C3×D4—34no—
250C3×D4—32no—
251C3×D4—42no—
252C3×D4—62no—
253C3×D4—62no—
254C3×D4—62no—
255C3×D4—62no—
256C3×D4—61no—
257C3×D4—62no—
258C3×D4—81no—
259C3×D4—82no—
260C3×D4—82no—
261C3×D4—121no—
262C3×D4—124no—
263C3×D4—122no—
264C3×D4—122no—
265C3×D4—121no—
266C3×D4C2×S412yescomplemented
267C2×C3×C4—31no—
268C2×C3×C4—31no—
269C2×C3×C4—31no—
270C2×C3×C4—31no—
271C2×C3×C4—31no—
272C2×C3×C4—31no—
273C2×C3×C4—31no—
274C2×C3×C4—41no—
275C2×C3×C4—61no—
276C2×C3×C4—62no—
277C2×C3×C4—81no—
278C2×C3×C4—122no—
279C2×C3×C4C2×S411yescharacteristic
280D12—42no—
281D12—81no—
282C3×C8—62no—
283C3×C8—62no—
284C3⋊C8—42no—
285C2×C32—42no—
286C2×C32—41no—
287C2×C32—162no—
288C3×D3—162no—
289C24—22no—
290C24—42no—
291C24—61no—
292C24—61no—
293C24—62no—
294C24—61no—
295C24—122no—
296C24C3×#6711yescharacteristic
297C2×D4—32no—
298C2×D4—32no—
299C2×D4—32no—
300C2×D4—32no—
301C2×D4—32no—
302C2×D4—32no—
303C2×D4—31no—
304C2×D4—61no—
305C2×D4—61no—
306C2×D4—61no—
307C2×D4—62no—
308C2×D4—61no—
309C2×D4—61no—
310C2×D4—62no—
311C2×D4—61no—
312C2×D4—61no—
313C2×D4—61no—
314C2×D4—61no—
315C2×D4—64no—
316C2×D4—62no—
317C2×D4—62no—
318C2×D4—61no—
319C2×D4—122no—
320C2×D4—122no—
321C2×D4C3×S411yescharacteristic, complemented
322C22×C4—31no—
323C22×C4—31no—
324C22×C4—31no—
325C22×C4—31no—
326C22×C4—61no—
327C22×C4C3×#6711yescharacteristic
328C22×C4C22×C3×D311yescharacteristic
329D8—68no—
330D8—64no—
331D8—64no—
332C2×C8—32no—
333C2×C8—32no—
334C2×C8—61no—
335C2×C8—61no—
336C2×C8—61no—
337C2×C8—61no—
338C4⋊C4—32no—
339C4⋊C4—61no—
340C22⋊C4—62no—
341C22⋊C4—62no—
342C22×C3—22no—
343C22×C3—32no—
344C22×C3—31no—
345C22×C3—32no—
346C22×C3—32no—
347C22×C3—31no—
348C22×C3—31no—
349C22×C3—42no—
350C22×C3—41no—
351C22×C3—61no—
352C22×C3—62no—
353C22×C3—82no—
354C22×C3—84no—
355C22×C3—81no—
356C22×C3—122no—
357C22×C3—122no—
358C22×C3—124no—
359C22×C3—122no—
360C22×C3—122no—
361C22×C3—124no—
362C22×C3—122no—
363C22×C3—122no—
364C22×C3—122no—
365C22×C3—122no—
366C22×C3—162no—
367C22×C3—164no—
368C22×C3—242no—
369C22×C3—242no—
370C22×C3C2×#21311yescharacteristic, complemented
371C22×C3GL(2,ℤ/4)11yescharacteristic
372C2×D3—82no—
373C2×D3—162no—
374A4—21no—
375A4C2×C3×D811yescharacteristic, complemented
376C3×C4—31no—
377C3×C4—31no—
378C3×C4—41no—
379C3×C4—41no—
380C3×C4—81no—
381C3×C4—81no—
382C3×C4—122no—
383C3×C4GL(2,ℤ/4)11yescharacteristic
384C3×C4C22×S411yescharacteristic
385C32—41noSylow
386C23—21no—
387C23—31no—
388C23—31no—
389C23—32no—
390C23—42no—
391C23—62no—
392C23—62no—
393C23—62no—
394C23—62no—
395C23—62no—
396C23—61no—
397C23—122no—
398C23—122no—
399C23—122no—
400C23—124no—
401C23—122no—
402C23—122no—
403C23—121no—
404C23—122no—
405C23—122no—
406C23—124no—
407C23—121no—
408C23C3×#21312yescomplemented
409C23C2×C3×#6711yescharacteristic
410D4—21no—
411D4—34no—
412D4—32no—
413D4—62no—
414D4—62no—
415D4—62no—
416D4—62no—
417D4—61no—
418D4—62no—
419D4—121no—
420D4—124no—
421D4—122no—
422D4—122no—
423D4—121no—
424D4C2×C3×S412yescomplemented
425C2×C4—31no—
426C2×C4—31no—
427C2×C4—31no—
428C2×C4—31no—
429C2×C4—31no—
430C2×C4—31no—
431C2×C4—31no—
432C2×C4—61no—
433C2×C4—62no—
434C2×C4—122no—
435C2×C4C2×C3×S411yescharacteristic
436C8—62no—
437C8—62no—
438C2×C3—31no—
439C2×C3—31no—
440C2×C3—32no—
441C2×C3—42no—
442C2×C3—42no—
443C2×C3—41no—
444C2×C3—82no—
445C2×C3—81no—
446C2×C3—122no—
447C2×C3—162no—
448C2×C3—164no—
449C2×C3—242no—
450C2×C3#541512yesdirect product factor
451C2×C3C2×GL(2,ℤ/4)11yescharacteristic
452D3—162no—
453C22—22no—
454C22—32no—
455C22—31no—
456C22—32no—
457C22—32no—
458C22—31no—
459C22—31no—
460C22—42no—
461C22—61no—
462C22—62no—
463C22—122no—
464C22—122no—
465C22—124no—
466C22—122no—
467C22—122no—
468C22—124no—
469C22—122no—
470C22—122no—
471C22—122no—
472C22—122no—
473C22—242no—
474C22—242no—
475C22C2×C3×#21311yescharacteristic, complemented
476C22C3×GL(2,ℤ/4)11yescharacteristic
477C4—31no—
478C4—31no—
479C4—122no—
480C4C3×GL(2,ℤ/4)11yescharacteristic
481C4C22×C3×S411yescharacteristic
482C3—41no—
483C3—81no—
484C3C2×#541511yescharacteristic, direct product factor
485C2—31no—
486C2—31no—
487C2—32no—
488C2—42no—
489C2—122no—
490C2—242no—
491C2C3×#541512yesdirect product factor
492C2C2×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
hypercenterC2×C3×D4S4
hypocenterA4C2×C3×D8
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
1. C2×C3×D4×A4C2semidirect
2. C2×C3×C4⋊S4C2semidirect
3. C3×#5415C2direct (N × H)
4. C2×C3×A4⋊C8C2semidirect
5. C2×#5415C3direct (N × H)
10. C3×D4×A4C22semidirect
13. C3×C4⋊S4C22semidirect
15. C3×A4⋊C8C22semidirect
16. C23×C3×D4D3semidirect
18. C2×D4×A4C2×C3semidirect
19. C2×C4⋊S4C2×C3semidirect
23. #5415C2×C3direct (N × H)
24. C2×A4⋊C8C2×C3semidirect
54. C22×C3×D4C2×D3semidirect
59. D4×A4C22×C3semidirect
63. C4⋊S4C22×C3semidirect
80. A4⋊C8C22×C3semidirect
88. C2×C3×A4D8semidirect
97. C23×D4C3×D3semidirect
148. C2×C3×D4S4semidirect
181. C3×A4C2×D8semidirect
193. C22×D4C2×C3×D3semidirect
238. C23×C3#213semidirect
245. C2×A4C3×D8semidirect
266. C3×D4C2×S4semidirect
321. C2×D4C3×S4semidirect
370. C22×C3C2×#213semidirect
375. A4C2×C3×D8semidirect
408. C23C3×#213semidirect
424. D4C2×C3×S4semidirect
450. C2×C3#5415direct (N × H)
475. C22C2×C3×#213semidirect
484. C3C2×#5415direct (N × H)
491. C2C3×#5415direct (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×C4⋊S42
C3×#54152
C2×C3×A4⋊C82
C2×#54153
C2×C3×#4473
C2×C3×#2134