Finite Group Databasebeta

← C3×#87282 #11905939

Subgroups — C3×#87282 #11905939

Order 1152.

3945 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×H(5,𝔽2)C211yesmaximal, characteristic, complemented
2C2×C3×#1009C211yesmaximal, characteristic, complemented
3C3×#5834C214yesmaximal, complemented
4C3×#5832C213yesmaximal, complemented
5C3×#5830C213yesmaximal, complemented
6C3×Dic3×#41C211yesmaximal, characteristic, complemented
7C3×#5805C213yesmaximal, complemented
8C2×C3×#940C213yesmaximal, complemented
9C3×D4×#67C2112yesmaximal, complemented
10C3×#3498—31nomaximal
11#87282C311yesmaximal, characteristic, direct product factor
12C32×H(5,𝔽2)—28no—
13C2×C32×#41C2211yescharacteristic
14C2×C32×#41C2213yescomplemented
15C2×C32×#41C2212yescomplemented
16C22×C32×D4C2216yescomplemented
17C22×C32×D4C2213yescomplemented
18C22×C3×#67C22112yescomplemented
19C3×#1009—28no—
20C2×C3×#239C2212yescomplemented
21C2×C3×D3×D4C2216yescomplemented
22C2×C3×#235C2213yescomplemented
23C22×C3×D12C2213yescomplemented
24C3×#953C22112yescomplemented
25C3×#947C2213yescomplemented
26C3×Q8×Dic3C2212yescomplemented
27C3×#944C2211yescharacteristic, complemented
28C3×#940C22112yescomplemented
29C3×#940C2213yescomplemented
30C3×#940C2216yescomplemented
31C3×#939C22112yescomplemented
32C3×#938C22112yescomplemented
33C3×#937C22112yescomplemented
34C3×#936C2213yescomplemented
35C3×D4×Dic3C2216yescomplemented
36C3×#930C22112yescomplemented
37C3×C4×#67C22112yescomplemented
38C3×#926C2213yescomplemented
39C2×C3×C4×Dic3C2213yescomplemented
40C2×C3×H(5,𝔽2)—21no—
41C2×C3×H(5,𝔽2)—31no—
42C2×C3×H(5,𝔽2)D311yescharacteristic, complemented
43C2×C3×H(5,𝔽2)C2×C311yescharacteristic, complemented
44C2×#1009C2×C311yescharacteristic, complemented
45C3×#550—34no—
46C3×D42—312no—
47C3×#535—33no—
48C3×#535—33no—
49C3×#532—33no—
50C2×C3×#127—33no—
51C3×C4×#41—31no—
52#5834C2×C314yescomplemented
53#5832C2×C313yescomplemented
54#5830C2×C313yescomplemented
55Dic3×#41C2×C311yescharacteristic, complemented
56#5805C2×C313yescomplemented
57C2×#940C2×C313yescomplemented
58D4×#67C2×C3112yescomplemented
59C24×C32C2316yes—
60C23×C3×D3C2316yes—
61C32×#41—28no—
62C32×#41—212no—
63C32×#41D418yescomplemented
64C2×C32×Q8C2311yescharacteristic
65C2×C32×Q8C2311yescharacteristic
66C2×C32×D4—224no—
67C2×C32×D4—212no—
68C2×C32×D4C2313yes—
69C2×C32×D4C2316yes—
70C2×C32×D4C2316yes—
71C2×C32×D4C2313yes—
72C22×C32×C4C2316yes—
73C22×C32×C4C2313yes—
74C2×C3×#67—424no—
75C2×C3×#67C23112yes—
76C2×C3×#67C2316yes—
77C2×C3×#67C23124yes—
78C22×C3×Dic3C2316yes—
79C3×#239—28no—
80C3×D3×D4—224no—
81C3×#235—212no—
82C2×C3×D12—212no—
83C2×C3×D12C2316yes—
84C2×C3×D12C2313yes—
85C2×C3×C4×D3C2316yes—
86C2×C3×Dic6C2311yescharacteristic
87C3×#217C2316yes—
88C3×#217C23112yes—
89C3×(C2×D3)⋊C4C23112yes—
90C3×C4⋊Dic3C2316yes—
91C3×Dic3⋊C4C2314yes—
92C3×C4×Dic3C2311yescharacteristic
93C3×C4×Dic3C2313yes—
94C3×C4×Dic3C2316yes—
95#3498—31noSylow
96C3×H(5,𝔽2)—216no—
97C3×H(5,𝔽2)—28no—
98C3×H(5,𝔽2)—28no—
99C3×H(5,𝔽2)—68no—
100C2×C3×#41—21no—
101C2×C3×#41—23no—
102C2×C3×#41—22no—
103C2×C3×#41—32no—
104C2×C3×#41—33no—
105C2×C3×#41C2×D311yescharacteristic
106C2×C3×#41C2×D313yescomplemented
107C2×C3×#41C2×D312yescomplemented
108C2×C3×#41C22×C311yescharacteristic
109C2×C3×#41C22×C313yescomplemented
110C2×C3×#41C22×C312yescomplemented
111C22×C3×D4—23no—
112C22×C3×D4—26no—
113C22×C3×D4—312no—
114C22×C3×D4—36no—
115C22×C3×D4—33no—
116C22×C3×D4C2×D313yescomplemented
117C22×C3×D4C2×D316yescomplemented
118C22×C3×D4C22×C316yescomplemented
119C22×C3×D4C22×C313yescomplemented
120C22×#67C22×C3112yescomplemented
121#1009—28no—
122C2×#239C22×C312yescomplemented
123C2×D3×D4C22×C316yescomplemented
124C2×#235C22×C313yescomplemented
125C22×D12C22×C313yescomplemented
126C3×C4⋊Q8—31no—
127C3×#127—33no—
128C3×#127—36no—
129C3×#127—312no—
130C3×#124—33no—
131C3×#124—33no—
132C3×#121—312no—
133C3×#121—312no—
134C3×#121—312no—
135C3×#120—312no—
136C3×#120—312no—
137C3×C4×Q8—32no—
138C3×C4×D4—36no—
139C3×C4×D4—312no—
140C3×#117—33no—
141C2×C3×C42—33no—
142#953C22×C3112yescomplemented
143#947C22×C313yescomplemented
144Q8×Dic3C22×C312yescomplemented
145#944C22×C311yescharacteristic, complemented
146#940C22×C3112yescomplemented
147#940C22×C313yescomplemented
148#940C22×C316yescomplemented
149#939C22×C3112yescomplemented
150#938C22×C3112yescomplemented
151#937C22×C3112yescomplemented
152#936C22×C313yescomplemented
153D4×Dic3C22×C316yescomplemented
154#930C22×C3112yescomplemented
155C4×#67C22×C3112yescomplemented
156#926C22×C313yescomplemented
157C2×C4×Dic3C22×C313yescomplemented
158C23×C32—212no—
159C23×C32—412no—
160C23×C32C2416yes—
161C23×C32C2413yes—
162C22×C3×D3—212no—
163C22×C3×D3—412no—
164C22×C3×D3C2416yes—
165C32×Q8—22no—
166C32×Q8C2×D414yescomplemented
167C32×D4—212no—
168C32×D4—26no—
169C32×D4—212no—
170C32×D4C2×D4112yescomplemented
171C2×C32×C4—212no—
172C2×C32×C4C2×D4112yescomplemented
173C2×C32×C4C2412yes—
174C2×C32×C4C2411yescharacteristic
175C2×C32×C4C2413yes—
176C3×#67—212no—
177C3×#67—224no—
178C3×#67—424no—
179C2×C3×Dic3—212no—
180C2×C3×Dic3C2416yes—
181C2×C3×Dic3C2412yes—
182C2×C3×Dic3C2412yes—
183C3×D12—26no—
184C3×D12—212no—
185C3×C4×D3—212no—
186C3×Dic6—22no—
187C2×H(5,𝔽2)—31no—
188C2×H(5,𝔽2)C3×D311yescharacteristic, complemented
189#550—34no—
190D42—312no—
191#535—33no—
192#535—33no—
193#532—33no—
194C2×#127—33no—
195C4×#41—31no—
196C24×C3—26no—
197C24×C3—36no—
198C24×C3C22×D316yes—
199C24×C3C23×C316yes—
200C23×D3C23×C316yes—
201C3×#41—216no—
202C3×#41—224no—
203C3×#41—28no—
204C3×#41—28no—
205C3×#41—212no—
206C3×#41—212no—
207C3×#41—28no—
208C3×#41—68no—
209C3×#41—612no—
210C3×#41#6718yescomplemented
211C3×#41C3×D418yescomplemented
212C2×C3×Q8—21no—
213C2×C3×Q8—21no—
214C2×C3×Q8—31no—
215C2×C3×Q8C22×D311yescharacteristic
216C2×C3×Q8C22×D311yescharacteristic
217C2×C3×Q8C23×C311yescharacteristic
218C2×C3×Q8C23×C311yescharacteristic
219C2×C3×D4—248no—
220C2×C3×D4—224no—
221C2×C3×D4—224no—
222C2×C3×D4—224no—
223C2×C3×D4—212no—
224C2×C3×D4—212no—
225C2×C3×D4—26no—
226C2×C3×D4—26no—
227C2×C3×D4—23no—
228C2×C3×D4—23no—
229C2×C3×D4—312no—
230C2×C3×D4—36no—
231C2×C3×D4—324no—
232C2×C3×D4—36no—
233C2×C3×D4—33no—
234C2×C3×D4—624no—
235C2×C3×D4—612no—
236C2×C3×D4—1224no—
237C2×C3×D4C22×D316yes—
238C2×C3×D4C22×D316yes—
239C2×C3×D4C22×D313yes—
240C2×C3×D4C22×D313yes—
241C2×C3×D4C23×C313yes—
242C2×C3×D4C23×C316yes—
243C2×C3×D4C23×C316yes—
244C2×C3×D4C23×C313yes—
245C22×C3×C4—26no—
246C22×C3×C4—23no—
247C22×C3×C4—36no—
248C22×C3×C4—36no—
249C22×C3×C4C22×D316yes—
250C22×C3×C4C22×D313yes—
251C22×C3×C4C23×C316yes—
252C22×C3×C4C23×C313yes—
253C2×#67—424no—
254C2×#67C23×C3112yes—
255C2×#67C23×C316yes—
256C2×#67C23×C3124yes—
257C22×Dic3C23×C316yes—
258#239—28no—
259D3×D4—224no—
260#235—212no—
261C2×D12—212no—
262C2×D12C23×C316yes—
263C2×D12C23×C313yes—
264C2×C4×D3C23×C316yes—
265C2×Dic6C23×C311yescharacteristic
266C3×C4⋊C4—34no—
267C3×C4⋊C4—36no—
268C3×C22⋊C4—36no—
269C3×C22⋊C4—312no—
270C3×C22⋊C4—312no—
271C3×C42—31no—
272C3×C42—33no—
273C3×C42—36no—
274#217C23×C316yes—
275#217C23×C3112yes—
276(C2×D3)⋊C4C23×C3112yes—
277C4⋊Dic3C23×C316yes—
278Dic3⋊C4C23×C314yes—
279C4×Dic3C23×C311yescharacteristic
280C4×Dic3C23×C313yes—
281C4×Dic3C23×C316yes—
282C22×C32—26no—
283C22×C32—26no—
284C22×C32—212no—
285C22×C32—812no—
286C22×C32C22×D416yes—
287C22×C32C2511yescharacteristic
288C2×C3×D3—26no—
289C2×C3×D3—212no—
290C2×C3×D3—812no—
291C32×C4—22no—
292C32×C4C22×D412yes—
293C32×C4C22×D416yes—
294C3×Dic3—22no—
295C3×Dic3—26no—
296C3×Dic3H(5,𝔽2)14yescomplemented
297H(5,𝔽2)—28no—
298H(5,𝔽2)—68no—
299C2×#41—32no—
300C2×#41—33no—
301C2×#41C2×C3×D311yescharacteristic
302C2×#41C2×C3×D313yescomplemented
303C2×#41C2×C3×D312yescomplemented
304C22×D4—312no—
305C22×D4—36no—
306C22×D4—33no—
307C22×D4C2×C3×D313yescomplemented
308C22×D4C2×C3×D316yescomplemented
309C4⋊Q8—31no—
310#127—33no—
311#127—36no—
312#127—312no—
313#124—33no—
314#124—33no—
315#121—312no—
316#121—312no—
317#121—312no—
318#120—312no—
319#120—312no—
320C4×Q8—32no—
321C4×D4—36no—
322C4×D4—312no—
323#117—33no—
324C2×C42—33no—
325C23×C3—224no—
326C23×C3—212no—
327C23×C3—212no—
328C23×C3—26no—
329C23×C3—23no—
330C23×C3—36no—
331C23×C3—412no—
332C23×C3—412no—
333C23×C3—612no—
334C23×C3—812no—
335C23×C3—1212no—
336C23×C3C23×D316yes—
337C23×C3C23×D313yes—
338C23×C3C24×C316yes—
339C23×C3C24×C313yes—
340C22×D3—212no—
341C22×D3—412no—
342C22×D3C24×C316yes—
343C3×Q8—24no—
344C3×Q8—22no—
345C3×Q8—22no—
346C3×Q8—24no—
347C3×Q8—62no—
348C3×Q8C2×#6714yescomplemented
349C3×Q8C2×C3×D414yescomplemented
350C3×D4—224no—
351C3×D4—224no—
352C3×D4—212no—
353C3×D4—212no—
354C3×D4—212no—
355C3×D4—212no—
356C3×D4—26no—
357C3×D4—26no—
358C3×D4—212no—
359C3×D4—212no—
360C3×D4—624no—
361C3×D4—612no—
362C3×D4—612no—
363C3×D4—66no—
364C3×D4—1224no—
365C3×D4C2×#67112yescomplemented
366C3×D4C2×C3×D4112yescomplemented
367C2×C3×C4—224no—
368C2×C3×C4—212no—
369C2×C3×C4—212no—
370C2×C3×C4—22no—
371C2×C3×C4—21no—
372C2×C3×C4—23no—
373C2×C3×C4—212no—
374C2×C3×C4—32no—
375C2×C3×C4—36no—
376C2×C3×C4—32no—
377C2×C3×C4—612no—
378C2×C3×C4—612no—
379C2×C3×C4C2×#67112yescomplemented
380C2×C3×C4C2×C3×D4112yescomplemented
381C2×C3×C4C23×D312yes—
382C2×C3×C4C23×D311yescharacteristic
383C2×C3×C4C23×D313yes—
384C2×C3×C4C24×C312yes—
385C2×C3×C4C24×C311yescharacteristic
386C2×C3×C4C24×C313yes—
387#67—212no—
388#67—224no—
389#67—424no—
390C2×Dic3—212no—
391C2×Dic3C24×C316yes—
392C2×Dic3C24×C312yes—
393C2×Dic3C24×C312yes—
394D12—26no—
395D12—212no—
396C4×D3—212no—
397Dic6—22no—
398C2×C32—26no—
399C2×C32—46no—
400C2×C32C23×D411yescharacteristic
401C2×C32C2×H(5,𝔽2)12yes—
402C3×D3—46no—
403C24—36no—
404C24C22×C3×D316yes—
405#41—28no—
406#41—212no—
407#41—68no—
408#41—612no—
409#41C3×#6718yescomplemented
410C2×Q8—31no—
411C2×Q8C22×C3×D311yescharacteristic
412C2×Q8C22×C3×D311yescharacteristic
413C2×D4—224no—
414C2×D4—212no—
415C2×D4—312no—
416C2×D4—36no—
417C2×D4—324no—
418C2×D4—36no—
419C2×D4—33no—
420C2×D4—624no—
421C2×D4—612no—
422C2×D4—1224no—
423C2×D4C22×C3×D316yes—
424C2×D4C22×C3×D316yes—
425C2×D4C22×C3×D313yes—
426C2×D4C22×C3×D313yes—
427C22×C4—36no—
428C22×C4—36no—
429C22×C4C22×C3×D316yes—
430C22×C4C22×C3×D313yes—
431C4⋊C4—36no—
432C4⋊C4—34no—
433C22⋊C4—36no—
434C22⋊C4—312no—
435C22⋊C4—312no—
436C42—31no—
437C42—33no—
438C42—36no—
439C22×C3—26no—
440C22×C3—26no—
441C22×C3—212no—
442C22×C3—212no—
443C22×C3—26no—
444C22×C3—26no—
445C22×C3—212no—
446C22×C3—26no—
447C22×C3—21no—
448C22×C3—46no—
449C22×C3—412no—
450C22×C3—612no—
451C22×C3—66no—
452C22×C3—812no—
453C22×C3—812no—
454C22×C3—824no—
455C22×C3—2412no—
456C22×C3C22×#6716yes—
457C22×C3C22×C3×D416yes—
458C22×C3C24×D311yescharacteristic
459C22×C3C25×C311yescharacteristic
460C2×D3—26no—
461C2×D3—212no—
462C2×D3—812no—
463C3×C4—24no—
464C3×C4—22no—
465C3×C4—22no—
466C3×C4—26no—
467C3×C4—22no—
468C3×C4—34no—
469C3×C4—66no—
470C3×C4—62no—
471C3×C4C22×#6716yes—
472C3×C4C22×#6712yes—
473C3×C4C22×C3×D412yes—
474C3×C4C22×C3×D416yes—
475Dic3—22no—
476Dic3—26no—
477Dic3C3×H(5,𝔽2)14yescomplemented
478C32#349811yescharacteristic, complemented, Sylow
479C23—212no—
480C23—36no—
481C23—412no—
482C23—612no—
483C23—1212no—
484C23C23×C3×D316yes—
485C23C23×C3×D313yes—
486Q8—22no—
487Q8—62no—
488Q8C2×C3×#6714yescomplemented
489D4—212no—
490D4—212no—
491D4—26no—
492D4—624no—
493D4—612no—
494D4—612no—
495D4—66no—
496D4—1224no—
497D4C2×C3×#67112yescomplemented
498C2×C4—212no—
499C2×C4—32no—
500C2×C4—36no—
501C2×C4—32no—
502C2×C4—612no—
503C2×C4—612no—
504C2×C4C2×C3×#67112yescomplemented
505C2×C4C23×C3×D312yes—
506C2×C4C23×C3×D311yescharacteristic
507C2×C4C23×C3×D313yes—
508C2×C3—26no—
509C2×C3—26no—
510C2×C3—22no—
511C2×C3—21no—
512C2×C3—46no—
513C2×C3—412no—
514C2×C3—46no—
515C2×C3—46no—
516C2×C3—126no—
517C2×C3D3×H(5,𝔽2)12yes—
518C2×C3C23×#6711yescharacteristic
519C2×C3C23×C3×D411yescharacteristic
520C2×C3C2×C3×H(5,𝔽2)12yes—
521D3—46no—
522C22—26no—
523C22—212no—
524C22—26no—
525C22—612no—
526C22—66no—
527C22—812no—
528C22—2412no—
529C22C22×C3×#6716yes—
530C22C24×C3×D311yescharacteristic
531C4—22no—
532C4—34no—
533C4—66no—
534C4—62no—
535C4C22×C3×#6716yes—
536C4C22×C3×#6712yes—
537C3—21no—
538C3#8728211yescharacteristic, direct product factor
539C3C3×#349811yescharacteristic, complemented
540C2—26no—
541C2—46no—
542C2—126no—
543C2C3×D3×H(5,𝔽2)12yes—
544C2C23×C3×#6711yescharacteristic
subgroupG/H
centerC22×C3C24×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC2×C32×H(5,𝔽2)C2
socleC22×C32C25
hypercenterC2×C3×H(5,𝔽2)D3
hypocenterC3C3×#3498
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C2×C3×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×C32×H(5,𝔽2)C2semidirect
2. C2×C3×#1009C2semidirect
3. C3×#5834C2semidirect
4. C3×#5832C2semidirect
5. C3×#5830C2semidirect
6. C3×Dic3×#41C2semidirect
7. C3×#5805C2semidirect
8. C2×C3×#940C2semidirect
9. C3×D4×#67C2semidirect
11. #87282C3direct (N × H)
14. C2×C32×#41C22semidirect
15. C2×C32×#41C22semidirect
16. C22×C32×D4C22semidirect
17. C22×C32×D4C22semidirect
18. C22×C3×#67C22semidirect
20. C2×C3×#239C22semidirect
21. C2×C3×D3×D4C22semidirect
22. C2×C3×#235C22semidirect
23. C22×C3×D12C22semidirect
24. C3×#953C22semidirect
25. C3×#947C22semidirect
26. C3×Q8×Dic3C22semidirect
27. C3×#944C22semidirect
28. C3×#940C22semidirect
29. C3×#940C22semidirect
30. C3×#940C22semidirect
31. C3×#939C22semidirect
32. C3×#938C22semidirect
33. C3×#937C22semidirect
34. C3×#936C22semidirect
35. C3×D4×Dic3C22semidirect
36. C3×#930C22semidirect
37. C3×C4×#67C22semidirect
38. C3×#926C22semidirect
39. C2×C3×C4×Dic3C22semidirect
42. C2×C3×H(5,𝔽2)D3semidirect
43. C2×C3×H(5,𝔽2)C2×C3semidirect
44. C2×#1009C2×C3semidirect
52. #5834C2×C3semidirect
53. #5832C2×C3semidirect
54. #5830C2×C3semidirect
55. Dic3×#41C2×C3semidirect
56. #5805C2×C3semidirect
57. C2×#940C2×C3semidirect
58. D4×#67C2×C3semidirect
63. C32×#41D4semidirect
106. C2×C3×#41C2×D3semidirect
107. C2×C3×#41C2×D3semidirect
109. C2×C3×#41C22×C3semidirect
110. C2×C3×#41C22×C3semidirect
116. C22×C3×D4C2×D3semidirect
117. C22×C3×D4C2×D3semidirect
118. C22×C3×D4C22×C3semidirect
119. C22×C3×D4C22×C3semidirect
120. C22×#67C22×C3semidirect
122. C2×#239C22×C3semidirect
123. C2×D3×D4C22×C3semidirect
124. C2×#235C22×C3semidirect
125. C22×D12C22×C3semidirect
142. #953C22×C3semidirect
143. #947C22×C3semidirect
144. Q8×Dic3C22×C3semidirect
145. #944C22×C3semidirect
146. #940C22×C3semidirect
147. #940C22×C3semidirect
148. #940C22×C3semidirect
149. #939C22×C3semidirect
150. #938C22×C3semidirect
151. #937C22×C3semidirect
152. #936C22×C3semidirect
153. D4×Dic3C22×C3semidirect
154. #930C22×C3semidirect
155. C4×#67C22×C3semidirect
156. #926C22×C3semidirect
157. C2×C4×Dic3C22×C3semidirect
166. C32×Q8C2×D4semidirect
170. C32×D4C2×D4semidirect
172. C2×C32×C4C2×D4semidirect
188. C2×H(5,𝔽2)C3×D3semidirect
210. C3×#41#67semidirect
211. C3×#41C3×D4semidirect
296. C3×Dic3H(5,𝔽2)semidirect
302. C2×#41C2×C3×D3semidirect
303. C2×#41C2×C3×D3semidirect
307. C22×D4C2×C3×D3semidirect
308. C22×D4C2×C3×D3semidirect
348. C3×Q8C2×#67semidirect
349. C3×Q8C2×C3×D4semidirect
365. C3×D4C2×#67semidirect
366. C3×D4C2×C3×D4semidirect
379. C2×C3×C4C2×#67semidirect
380. C2×C3×C4C2×C3×D4semidirect
409. #41C3×#67semidirect
477. Dic3C3×H(5,𝔽2)semidirect
478. C32#3498semidirect
488. Q8C2×C3×#67semidirect
497. D4C2×C3×#67semidirect
504. C2×C4C2×C3×#67semidirect
538. C3#87282direct (N × H)
539. C3C3×#3498semidirect

Maximal subgroups

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

subgroupindex
C2×C32×H(5,𝔽2)2
C2×C3×#10092
C3×#58342
C3×#58322
C3×#58302
C3×Dic3×#412
C3×#58052
C2×C3×#9402
C3×D4×#672
C3×#34983
#872823