Finite Group Databasebeta

← C3×#87489 #11923700

Subgroups — C3×#87489 #11923700

Order 1152.

1760 conjugacy classes of subgroups; only the 204 normal subgroups are shown.

normal subgroups only

This group has 1760 conjugacy classes of subgroups — too many to draw. Only its 204 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
1C3×A4×#41C211yesmaximal, characteristic, complemented
2C3×#5919C212yesmaximal, complemented
3C3×D4×S4C216yesmaximal, complemented
4C3×#5912C213yesmaximal, complemented
5C2×C3×C4⋊S4C213yesmaximal, complemented
6#87489C311yesmaximal, characteristic, direct product factor
7C3×#3488—31nomaximal
8C22×C3×S4C2216yescomplemented
9C3×#1009—41nomaximal
10C3×Q8×A4C2211yescharacteristic, complemented
11C3×D4×A4C2213yescomplemented
12C2×C3×C4×A4C2213yescomplemented
13C3×GL(2,ℤ/4)C2216yescomplemented
14C3×C4⋊S4C2213yescomplemented
15C3×C4⋊S4C2216yescomplemented
16C3×C4×S4C2216yescomplemented
17C3×A4⋊Q8C2211yescharacteristic, complemented
18C2×C3×H(5,𝔽2)—31no—
19C22×C3×#41D311yescharacteristic, complemented
20A4×#41—21no—
21A4×#41C2×C311yescharacteristic, complemented
22#5919C2×C312yescomplemented
23D4×S4C2×C316yescomplemented
24#5912C2×C313yescomplemented
25C2×C4⋊S4C2×C313yescomplemented
26C3×#550—32no—
27C3×#548—32no—
28C3×#546—36no—
29C3×D42—36no—
30C3×#537—33no—
31C3×#535—33no—
32C3×#525—33no—
33C2×C3×#121—33no—
34C3×#520—31no—
35C22×C3×A4C2313yes—
36C2×C3×S4—46no—
37C2×C3×S4C2316yes—
38C32×#41—41no—
39C3×#239—42no—
40C3×D3×D4—46no—
41C3×#235—43no—
42C2×C3×D12—43no—
43C3×C4×A4C2311yescharacteristic
44C3×C4×A4C2313yes—
45C3×SL(2,ℤ/4)C2312yes—
46#3488—31noSylow
47C22×S4C22×C316yescomplemented
48C3×H(5,𝔽2)—61no—
49C3×H(5,𝔽2)—63no—
50C3×H(5,𝔽2)—63no—
51C3×H(5,𝔽2)—61no—
52C2×C3×#41—31no—
53C2×C3×#41—33no—
54C2×C3×#41—33no—
55C2×C3×#41—31no—
56C2×C3×#41—33no—
57C2×C3×#41—32no—
58C22×C3×Q8C2×D311yescharacteristic, complemented
59C22×C3×D4—33no—
60C22×C3×D4—36no—
61C22×C3×D4—36no—
62C22×C3×D4C2×D313yescomplemented
63C23×C3×C4C2×D313yescomplemented
64#1009—41no—
65Q8×A4—21no—
66Q8×A4C22×C311yescharacteristic, complemented
67D4×A4—23no—
68D4×A4C22×C313yescomplemented
69C2×C4×A4—23no—
70C2×C4×A4C22×C313yescomplemented
71GL(2,ℤ/4)C22×C316yescomplemented
72C4⋊S4C22×C313yescomplemented
73C4⋊S4C22×C316yescomplemented
74C4×S4C22×C316yescomplemented
75A4⋊Q8C22×C311yescharacteristic, complemented
76C3×#127—36no—
77C3×#124—36no—
78C3×#123—36no—
79C3×C22⋊Q8—33no—
80C3×C22⋊Q8—36no—
81C3×C22⋊Q8—31no—
82C3×#121—36no—
83C3×#121—36no—
84C3×#121—33no—
85C3×#121—312no—
86C3×#121—36no—
87C3×#121—33no—
88C3×#121—36no—
89C3×#120—312no—
90C3×#120—36no—
91C3×C4×Q8—32no—
92C3×C4×D4—36no—
93C3×C4×D4—36no—
94C3×C4×D4—36no—
95C3×#117—33no—
96C2×C3×C4⋊C4—33no—
97C2×C3×C22⋊C4—36no—
98C22×C3×D3—46no—
99C2×C3×A4—23no—
100C2×C3×A4C2411yescharacteristic
101C3×S4—26no—
102C32×Q8—41no—
103C32×D4—43no—
104C2×C32×C4—43no—
105C3×#67—46no—
106C3×D12—43no—
107C3×D12—46no—
108C3×C4×D3—46no—
109C3×Dic6—41no—
110C2×H(5,𝔽2)—31no—
111C22×#41C3×D311yescharacteristic, complemented
112#550—32no—
113#548—32no—
114#546—36no—
115D42—36no—
116#537—33no—
117#535—33no—
118#525—33no—
119C2×#121—33no—
120#520—31no—
121C24×C3—36no—
122C24×C3C22×D313yes—
123C22×A4—23no—
124C22×A4C23×C313yes—
125C2×S4—46no—
126C2×S4C23×C316yes—
127C3×#41—31no—
128C3×#41—33no—
129C3×#41—33no—
130C3×#41—41no—
131C3×#41—61no—
132C3×#41—63no—
133C3×#41—63no—
134C3×#41—63no—
135C3×#41—63no—
136C3×#41—63no—
137C3×#41—66no—
138C3×#41—62no—
139C3×#41—63no—
140C3×#41—81no—
141C3×#41S411yescharacteristic, complemented
142C2×C3×Q8—31no—
143C2×C3×Q8—33no—
144C2×C3×Q8—31no—
145C2×C3×D4—33no—
146C2×C3×D4—36no—
147C2×C3×D4—33no—
148C2×C3×D4—36no—
149C2×C3×D4—33no—
150C2×C3×D4—36no—
151C2×C3×D4—36no—
152C2×C3×D4—36no—
153C2×C3×D4—312no—
154C2×C3×D4—63no—
155C2×C3×D4—63no—
156C2×C3×D4—63no—
157C2×C3×D4—612no—
158C2×C3×D4—66no—
159C2×C3×D4—66no—
160C2×C3×D4—63no—
161C2×C3×D4—126no—
162C2×C3×D4—126no—
163C22×C3×C4—33no—
164C22×C3×C4—33no—
165C22×C3×C4—33no—
166C22×C3×C4—33no—
167C22×C3×C4—36no—
168C22×C3×C4—36no—
169C22×C3×C4C22×D313yes—
170C22×C3×C4C22×D311yescharacteristic
171#239—42no—
172D3×D4—46no—
173#235—43no—
174C2×D12—43no—
175C4×A4—21no—
176C4×A4—23no—
177C4×A4C23×C311yescharacteristic
178C4×A4C23×C313yes—
179SL(2,ℤ/4)C23×C312yes—
180C3×C4⋊C4—33no—
181C3×C4⋊C4—36no—
182C3×C4⋊C4—31no—
183C3×C4⋊C4—32no—
184C3×C4⋊C4—36no—
185C3×C22⋊C4—36no—
186C3×C22⋊C4—312no—
187C3×C22⋊C4—36no—
188C3×C22⋊C4—36no—
189C3×C22⋊C4—36no—
190C3×C22⋊C4—32no—
191C3×C42—36no—
192C22×C32—43no—
193C2×C3×D3—46no—
194C2×C3×D3—166no—
195C3×A4H(5,𝔽2)11yescharacteristic, complemented
196C32×C4—41no—
197C32×C4—43no—
198C3×Dic3—42no—
199H(5,𝔽2)—61no—
200H(5,𝔽2)—63no—
201H(5,𝔽2)—63no—
202H(5,𝔽2)—61no—
203C2×#41—31no—
204C2×#41—33no—
205C2×#41—33no—
206C2×#41—31no—
207C2×#41—33no—
208C2×#41—32no—
209C22×Q8C2×C3×D311yescharacteristic, complemented
210C22×D4—33no—
211C22×D4—36no—
212C22×D4—36no—
213C22×D4C2×C3×D313yescomplemented
214C23×C4C2×C3×D313yescomplemented
215#127—36no—
216#124—36no—
217#123—36no—
218C22⋊Q8—33no—
219C22⋊Q8—36no—
220C22⋊Q8—31no—
221#121—36no—
222#121—36no—
223#121—33no—
224#121—312no—
225#121—36no—
226#121—33no—
227#121—36no—
228#120—312no—
229#120—36no—
230C4×Q8—32no—
231C4×D4—36no—
232C4×D4—36no—
233C4×D4—36no—
234#117—33no—
235C2×C4⋊C4—33no—
236C2×C22⋊C4—36no—
237C23×C3—23no—
238C23×C3—33no—
239C23×C3—33no—
240C23×C3—36no—
241C23×C3—63no—
242C23×C3—66no—
243C23×C3—66no—
244C23×C3—126no—
245C23×C3—126no—
246C23×C3C23×D311yescharacteristic
247C22×D3—46no—
248C2×A4—23no—
249C2×A4—21no—
250C2×A4—43no—
251C2×A4C24×C311yescharacteristic
252S4—26no—
253C3×Q8—33no—
254C3×Q8—41no—
255C3×Q8—61no—
256C3×Q8—61no—
257C3×Q8—61no—
258C3×Q8—81no—
259C3×Q8C2×S411yescharacteristic, complemented
260C3×D4—36no—
261C3×D4—33no—
262C3×D4—43no—
263C3×D4—63no—
264C3×D4—63no—
265C3×D4—66no—
266C3×D4—66no—
267C3×D4—66no—
268C3×D4—66no—
269C3×D4—66no—
270C3×D4—63no—
271C3×D4—66no—
272C3×D4—83no—
273C3×D4—126no—
274C3×D4—1212no—
275C3×D4C2×S413yescomplemented
276C2×C3×C4—33no—
277C2×C3×C4—33no—
278C2×C3×C4—31no—
279C2×C3×C4—33no—
280C2×C3×C4—33no—
281C2×C3×C4—31no—
282C2×C3×C4—33no—
283C2×C3×C4—32no—
284C2×C3×C4—36no—
285C2×C3×C4—32no—
286C2×C3×C4—43no—
287C2×C3×C4—63no—
288C2×C3×C4—66no—
289C2×C3×C4—66no—
290C2×C3×C4—83no—
291C2×C3×C4—126no—
292C2×C3×C4C2×S413yescomplemented
293#67—46no—
294D12—43no—
295D12—46no—
296C4×D3—46no—
297Dic6—41no—
298C2×C32—41no—
299C2×C32—83no—
300C3×D3—86no—
301C24—36no—
302C24C22×C3×D313yes—
303#41—31no—
304#41—33no—
305#41—33no—
306#41—61no—
307#41—63no—
308#41—63no—
309#41—63no—
310#41—63no—
311#41—63no—
312#41—66no—
313#41—62no—
314#41—63no—
315#41C3×S411yescharacteristic, complemented
316C2×Q8—31no—
317C2×Q8—33no—
318C2×Q8—31no—
319C2×D4—33no—
320C2×D4—36no—
321C2×D4—33no—
322C2×D4—36no—
323C2×D4—33no—
324C2×D4—36no—
325C2×D4—36no—
326C2×D4—36no—
327C2×D4—312no—
328C2×D4—63no—
329C2×D4—63no—
330C2×D4—63no—
331C2×D4—612no—
332C2×D4—66no—
333C2×D4—66no—
334C2×D4—63no—
335C2×D4—126no—
336C2×D4—126no—
337C22×C4—33no—
338C22×C4—33no—
339C22×C4—33no—
340C22×C4—33no—
341C22×C4—36no—
342C22×C4—36no—
343C22×C4C22×C3×D313yes—
344C22×C4C22×C3×D311yescharacteristic
345C4⋊C4—33no—
346C4⋊C4—36no—
347C4⋊C4—31no—
348C4⋊C4—32no—
349C4⋊C4—36no—
350C22⋊C4—36no—
351C22⋊C4—312no—
352C22⋊C4—36no—
353C22⋊C4—36no—
354C22⋊C4—36no—
355C22⋊C4—32no—
356C42—36no—
357C22×C3—31no—
358C22×C3—31no—
359C22×C3—33no—
360C22×C3—43no—
361C22×C3—63no—
362C22×C3—63no—
363C22×C3—63no—
364C22×C3—63no—
365C22×C3—66no—
366C22×C3—66no—
367C22×C3—66no—
368C22×C3—83no—
369C22×C3—246no—
370C22×C3—246no—
371C22×C3#100911yescharacteristic, complemented
372C22×C3C22×S413yes—
373C2×D3—46no—
374C2×D3—166no—
375A4—21no—
376A4C3×H(5,𝔽2)11yescharacteristic, complemented
377C3×C4—31no—
378C3×C4—33no—
379C3×C4—41no—
380C3×C4—43no—
381C3×C4—66no—
382C3×C4—62no—
383C3×C4—62no—
384C3×C4—81no—
385C3×C4—83no—
386C3×C4C22×S411yescharacteristic
387C3×C4C22×S413yes—
388Dic3—42no—
389C32—41noSylow
390C23—23no—
391C23—33no—
392C23—33no—
393C23—36no—
394C23—63no—
395C23—66no—
396C23—66no—
397C23—126no—
398C23—126no—
399C23C23×C3×D311yescharacteristic
400Q8—33no—
401Q8—61no—
402Q8—61no—
403Q8—61no—
404Q8C2×C3×S411yescharacteristic, complemented
405D4—36no—
406D4—33no—
407D4—63no—
408D4—63no—
409D4—66no—
410D4—66no—
411D4—66no—
412D4—66no—
413D4—66no—
414D4—63no—
415D4—66no—
416D4—1212no—
417D4—126no—
418D4C2×C3×S413yescomplemented
419C2×C4—33no—
420C2×C4—33no—
421C2×C4—31no—
422C2×C4—33no—
423C2×C4—33no—
424C2×C4—31no—
425C2×C4—33no—
426C2×C4—32no—
427C2×C4—36no—
428C2×C4—32no—
429C2×C4—63no—
430C2×C4—66no—
431C2×C4—66no—
432C2×C4—126no—
433C2×C4C2×C3×S413yescomplemented
434C2×C3—23no—
435C2×C3—31no—
436C2×C3—31no—
437C2×C3—41no—
438C2×C3—63no—
439C2×C3—83no—
440C2×C3—81no—
441C2×C3—126no—
442C2×C3—163no—
443C2×C3C23×S411yescharacteristic
444D3—86no—
445C22—31no—
446C22—31no—
447C22—33no—
448C22—63no—
449C22—63no—
450C22—63no—
451C22—63no—
452C22—66no—
453C22—66no—
454C22—66no—
455C22—246no—
456C22—246no—
457C22C3×#100911yescharacteristic, complemented
458C22C22×C3×S413yes—
459C4—31no—
460C4—33no—
461C4—66no—
462C4—62no—
463C4—62no—
464C4C22×C3×S411yescharacteristic
465C4C22×C3×S413yes—
466C3—41no—
467C3—81no—
468C3#8748911yescharacteristic, direct product factor
469C2—23no—
470C2—31no—
471C2—31no—
472C2—63no—
473C2—126no—
474C2C23×C3×S411yescharacteristic
subgroupG/H
centerC2×C3C23×S4
commutator, cosocleC2×A4C24×C3
FrattiniC2C23×C3×S4
FittingC22×C3×#41D3
socleC23×C3C23×D3
hypercenterC3×#41S4
hypocenterA4C3×H(5,𝔽2)
LCSG ▷ C2×A4 ▷ A4
UCSC1 ◁ C2×C3 ◁ C3×#41

Complements

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

NH ≅ G/Ntype
1. C3×A4×#41C2semidirect
2. C3×#5919C2semidirect
3. C3×D4×S4C2semidirect
4. C3×#5912C2semidirect
5. C2×C3×C4⋊S4C2semidirect
6. #87489C3direct (N × H)
8. C22×C3×S4C22semidirect
10. C3×Q8×A4C22semidirect
11. C3×D4×A4C22semidirect
12. C2×C3×C4×A4C22semidirect
13. C3×GL(2,ℤ/4)C22semidirect
14. C3×C4⋊S4C22semidirect
15. C3×C4⋊S4C22semidirect
16. C3×C4×S4C22semidirect
17. C3×A4⋊Q8C22semidirect
19. C22×C3×#41D3semidirect
21. A4×#41C2×C3semidirect
22. #5919C2×C3semidirect
23. D4×S4C2×C3semidirect
24. #5912C2×C3semidirect
25. C2×C4⋊S4C2×C3semidirect
47. C22×S4C22×C3semidirect
58. C22×C3×Q8C2×D3semidirect
62. C22×C3×D4C2×D3semidirect
63. C23×C3×C4C2×D3semidirect
66. Q8×A4C22×C3semidirect
68. D4×A4C22×C3semidirect
70. C2×C4×A4C22×C3semidirect
71. GL(2,ℤ/4)C22×C3semidirect
72. C4⋊S4C22×C3semidirect
73. C4⋊S4C22×C3semidirect
74. C4×S4C22×C3semidirect
75. A4⋊Q8C22×C3semidirect
111. C22×#41C3×D3semidirect
141. C3×#41S4semidirect
195. C3×A4H(5,𝔽2)semidirect
209. C22×Q8C2×C3×D3semidirect
213. C22×D4C2×C3×D3semidirect
214. C23×C4C2×C3×D3semidirect
259. C3×Q8C2×S4semidirect
275. C3×D4C2×S4semidirect
292. C2×C3×C4C2×S4semidirect
315. #41C3×S4semidirect
371. C22×C3#1009semidirect
376. A4C3×H(5,𝔽2)semidirect
404. Q8C2×C3×S4semidirect
418. D4C2×C3×S4semidirect
433. C2×C4C2×C3×S4semidirect
457. C22C3×#1009semidirect
468. C3#87489direct (N × H)

Maximal subgroups

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

subgroupindex
C3×A4×#412
C3×#59192
C3×D4×S42
C3×#59122
C2×C3×C4⋊S42
#874893
C3×#34883
C3×#10094