Finite Group Databasebeta

← C2×C3×#5834 #11902696

Subgroups — C2×C3×#5834 #11902696

Order 1152.

3917 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×#41C211yesmaximal, characteristic, complemented
2C22×C3×#239C211yesmaximal, characteristic, complemented
3C3×#5834C2116yesmaximal, direct product factor
4C2×C3×Q8×Dic3C211yesmaximal, characteristic, complemented
5C2×C3×#940C213yesmaximal, complemented
6C2×C3×#938C213yesmaximal, complemented
7C2×C3×#930C213yesmaximal, complemented
8C2×C3×C4×#67C213yesmaximal, complemented
9C2×C3×#550—31nomaximal
10C2×#5834C311yesmaximal, characteristic, direct product factor
11C2×C32×#41—28no—
12C2×C32×#41C2218yescomplemented
13C22×C32×Q8C2211yescharacteristic
14C22×C32×D4C2213yes—
15C23×C32×C4C2213yes—
16C22×C3×#67C2213yes—
17C22×C3×#67C2216yes—
18C2×C3×#239—28no—
19C2×C3×#239C2218yescomplemented
20C22×C3×D12C2213yes—
21C22×C3×C4×D3C2213yes—
22C2×C3×#217C2213yes—
23C3×Q8×Dic3C2218yescomplemented
24C3×#940C22124yescomplemented
25C3×#938C22124yescomplemented
26C3×#930C22124yescomplemented
27C3×C4×#67C22124yescomplemented
28C2×C3×(C2×D3)⋊C4C2213yes—
29C2×C3×C4⋊Dic3C2213yes—
30C2×C3×Dic3⋊C4C2211yescharacteristic
31C2×C3×C4×Dic3C2213yes—
32C22×C3×#41—21no—
33C22×C3×#41—31no—
34C22×C3×#41D311yescharacteristic, complemented
35C22×C3×#41C2×C311yescharacteristic, complemented
36C22×#239C2×C311yescharacteristic, complemented
37C3×#550—316no—
38C2×C3×#127—33no—
39C2×C3×#121—33no—
40C2×C3×#121—33no—
41C2×C3×C4×Q8—31no—
42C2×C3×C4×D4—33no—
43#5834C2×C3116yesdirect product factor
44C2×Q8×Dic3C2×C311yescharacteristic, complemented
45C2×#940C2×C313yescomplemented
46C2×#938C2×C313yescomplemented
47C2×#930C2×C313yescomplemented
48C2×C4×#67C2×C313yescomplemented
49C24×C32C2313yes—
50C23×C3×D3C2313yes—
51C32×#41—232no—
52C2×C32×Q8D418yescomplemented
53C2×C32×Q8C2314yes—
54C2×C32×D4—212no—
55C2×C32×D4C23112yes—
56C22×C32×C4—212no—
57C22×C32×C4C2311yescharacteristic
58C22×C32×C4C23112yes—
59C22×C32×C4C2313yes—
60C2×C3×#67—212no—
61C2×C3×#67—412no—
62C2×C3×#67C23112yes—
63C2×C3×#67C23124yes—
64C22×C3×Dic3C2313yes—
65C22×C3×Dic3C2311yescharacteristic
66C22×C3×Dic3C2311yescharacteristic
67C3×#239—232no—
68C2×C3×D12—212no—
69C2×C3×D12C23112yes—
70C2×C3×C4×D3—212no—
71C2×C3×C4×D3C23112yes—
72C3×#217C23112yes—
73C3×(C2×D3)⋊C4C23112yes—
74C3×C4⋊Dic3C23112yes—
75C3×Dic3⋊C4C2314yes—
76C3×C4×Dic3C23112yes—
77C2×#550—31noSylow
78C2×C3×#41—216no—
79C2×C3×#41—28no—
80C2×C3×#41—28no—
81C2×C3×#41—28no—
82C2×C3×#41—38no—
83C2×C3×#41—68no—
84C2×C3×#41C2×D318yescomplemented
85C2×C3×#41C22×C318yescomplemented
86C22×C3×Q8—21no—
87C22×C3×Q8C2×D311yescharacteristic
88C22×C3×Q8C22×C311yescharacteristic
89C22×C3×D4—23no—
90C22×C3×D4—33no—
91C22×C3×D4—36no—
92C22×C3×D4—33no—
93C22×C3×D4C2×D313yes—
94C22×C3×D4C22×C313yes—
95C23×C3×C4—23no—
96C23×C3×C4—33no—
97C23×C3×C4C2×D313yes—
98C23×C3×C4C22×C313yes—
99C22×#67C22×C313yes—
100C22×#67C22×C316yes—
101C2×#239—28no—
102C2×#239C22×C318yescomplemented
103C22×D12C22×C313yes—
104C22×C4×D3C22×C313yes—
105C3×#127—324no—
106C3×#121—324no—
107C3×#121—324no—
108C3×C4×Q8—38no—
109C3×C4×D4—324no—
110C2×C3×C4⋊C4—31no—
111C2×C3×C4⋊C4—33no—
112C2×C3×C22⋊C4—33no—
113C2×C3×C22⋊C4—33no—
114C2×C3×C42—33no—
115C2×#217C22×C313yes—
116Q8×Dic3C22×C318yescomplemented
117#940C22×C3124yescomplemented
118#938C22×C3124yescomplemented
119#930C22×C3124yescomplemented
120C4×#67C22×C3124yescomplemented
121C2×(C2×D3)⋊C4C22×C313yes—
122C2×C4⋊Dic3C22×C313yes—
123C2×Dic3⋊C4C22×C311yescharacteristic
124C2×C4×Dic3C22×C313yes—
125C23×C32—26no—
126C23×C32—26no—
127C23×C32—26no—
128C23×C32C2416yes—
129C23×C32C2411yescharacteristic
130C22×C3×D3—26no—
131C22×C3×D3—26no—
132C22×C3×D3—26no—
133C22×C3×D3C2416yes—
134C32×Q8C2×D4116yescomplemented
135C32×D4—224no—
136C2×C32×C4—22no—
137C2×C32×C4—224no—
138C2×C32×C4C2×D4112yes—
139C2×C32×C4C2412yes—
140C2×C32×C4C2416yes—
141C3×#67—224no—
142C3×#67—424no—
143C2×C3×Dic3—22no—
144C2×C3×Dic3—26no—
145C2×C3×Dic3#4114yescomplemented
146C2×C3×Dic3C2416yes—
147C2×C3×Dic3C2412yes—
148C2×C3×Dic3C2412yes—
149C3×D12—224no—
150C3×C4×D3—224no—
151C22×#41—31no—
152C22×#41C3×D311yescharacteristic, complemented
153#550—316no—
154C2×#127—33no—
155C2×#121—33no—
156C2×#121—33no—
157C2×C4×Q8—31no—
158C2×C4×D4—33no—
159C24×C3—23no—
160C24×C3—33no—
161C24×C3C22×D313yes—
162C24×C3C23×C313yes—
163C23×D3C23×C313yes—
164C3×#41—264no—
165C3×#41—232no—
166C3×#41—232no—
167C3×#41—632no—
168C2×C3×Q8—28no—
169C2×C3×Q8—24no—
170C2×C3×Q8#6718yescomplemented
171C2×C3×Q8C3×D418yescomplemented
172C2×C3×Q8C22×D314yes—
173C2×C3×Q8C23×C314yes—
174C2×C3×D4—224no—
175C2×C3×D4—212no—
176C2×C3×D4—212no—
177C2×C3×D4—212no—
178C2×C3×D4—312no—
179C2×C3×D4—324no—
180C2×C3×D4—312no—
181C2×C3×D4—612no—
182C2×C3×D4—612no—
183C2×C3×D4—1212no—
184C2×C3×D4C22×D3112yes—
185C2×C3×D4C23×C3112yes—
186C22×C3×C4—224no—
187C22×C3×C4—212no—
188C22×C3×C4—212no—
189C22×C3×C4—21no—
190C22×C3×C4—23no—
191C22×C3×C4—212no—
192C22×C3×C4—31no—
193C22×C3×C4—33no—
194C22×C3×C4—31no—
195C22×C3×C4—312no—
196C22×C3×C4—612no—
197C22×C3×C4C22×D3112yes—
198C22×C3×C4C22×D311yescharacteristic
199C22×C3×C4C22×D313yes—
200C22×C3×C4C23×C311yescharacteristic
201C22×C3×C4C23×C3112yes—
202C22×C3×C4C23×C313yes—
203C2×#67—212no—
204C2×#67—412no—
205C2×#67C23×C3112yes—
206C2×#67C23×C3124yes—
207C22×Dic3C23×C313yes—
208C22×Dic3C23×C311yescharacteristic
209C22×Dic3C23×C311yescharacteristic
210#239—232no—
211C2×D12—212no—
212C2×D12C23×C3112yes—
213C2×C4×D3—212no—
214C2×C4×D3C23×C3112yes—
215C3×C4⋊C4—34no—
216C3×C4⋊C4—312no—
217C3×C22⋊C4—312no—
218C3×C22⋊C4—312no—
219C3×C42—312no—
220#217C23×C3112yes—
221(C2×D3)⋊C4C23×C3112yes—
222C4⋊Dic3C23×C3112yes—
223Dic3⋊C4C23×C314yes—
224C4×Dic3C23×C3112yes—
225C22×C32—26no—
226C22×C32—26no—
227C22×C32—26no—
228C22×C32—412no—
229C22×C32C22×D412yes—
230C22×C32C2×#4112yes—
231C22×C32H(5,𝔽2)12yes—
232C22×C32C2511yescharacteristic
233C2×C3×D3—26no—
234C2×C3×D3—26no—
235C2×C3×D3—26no—
236C2×C3×D3—412no—
237C32×C4—22no—
238C32×C4C22×D4112yes—
239C3×Dic3—22no—
240C3×Dic3—26no—
241C3×Dic3C2×#4114yescomplemented
242C2×#41—28no—
243C2×#41—38no—
244C2×#41—68no—
245C2×#41C2×C3×D318yescomplemented
246C22×Q8C2×C3×D311yescharacteristic
247C22×D4—33no—
248C22×D4—36no—
249C22×D4—33no—
250C22×D4C2×C3×D313yes—
251C23×C4—33no—
252C23×C4C2×C3×D313yes—
253#127—324no—
254#121—324no—
255#121—324no—
256C4×Q8—38no—
257C4×D4—324no—
258C2×C4⋊C4—31no—
259C2×C4⋊C4—33no—
260C2×C22⋊C4—33no—
261C2×C22⋊C4—33no—
262C2×C42—33no—
263C23×C3—212no—
264C23×C3—26no—
265C23×C3—26no—
266C23×C3—26no—
267C23×C3—26no—
268C23×C3—26no—
269C23×C3—26no—
270C23×C3—21no—
271C23×C3—26no—
272C23×C3—36no—
273C23×C3—46no—
274C23×C3—46no—
275C23×C3—66no—
276C23×C3—66no—
277C23×C3—66no—
278C23×C3C23×D316yes—
279C23×C3C23×D311yescharacteristic
280C23×C3C24×C316yes—
281C23×C3C24×C311yescharacteristic
282C22×D3—26no—
283C22×D3—26no—
284C22×D3—26no—
285C22×D3C24×C316yes—
286C3×Q8—216no—
287C3×Q8C2×#67116yescomplemented
288C3×Q8C2×C3×D4116yescomplemented
289C3×D4—248no—
290C3×D4—224no—
291C3×D4—224no—
292C3×D4—624no—
293C3×D4—624no—
294C3×D4—1224no—
295C2×C3×C4—24no—
296C2×C3×C4—248no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—224no—
300C2×C3×C4—224no—
301C2×C3×C4—22no—
302C2×C3×C4—26no—
303C2×C3×C4—212no—
304C2×C3×C4—34no—
305C2×C3×C4—32no—
306C2×C3×C4—36no—
307C2×C3×C4—32no—
308C2×C3×C4—66no—
309C2×C3×C4—62no—
310C2×C3×C4—624no—
311C2×C3×C4C2×#67112yes—
312C2×C3×C4C2×C3×D4112yes—
313C2×C3×C4C23×D312yes—
314C2×C3×C4C23×D316yes—
315C2×C3×C4C24×C312yes—
316C2×C3×C4C24×C316yes—
317#67—224no—
318#67—424no—
319C2×Dic3—22no—
320C2×Dic3—26no—
321C2×Dic3C3×#4114yescomplemented
322C2×Dic3C24×C316yes—
323C2×Dic3C24×C312yes—
324C2×Dic3C24×C312yes—
325D12—224no—
326C4×D3—224no—
327C2×C32—46no—
328C2×C32#55014yescomplemented
329C2×C32C23×D411yescharacteristic
330C2×C32C22×#4111yescharacteristic
331C2×C32C2×H(5,𝔽2)11yescharacteristic
332C3×D3—46no—
333C24—33no—
334C24C22×C3×D313yes—
335#41—232no—
336#41—632no—
337C2×Q8C3×#6718yescomplemented
338C2×Q8C22×C3×D314yes—
339C2×D4—212no—
340C2×D4—312no—
341C2×D4—324no—
342C2×D4—312no—
343C2×D4—612no—
344C2×D4—612no—
345C2×D4—1212no—
346C2×D4C22×C3×D3112yes—
347C22×C4—212no—
348C22×C4—31no—
349C22×C4—33no—
350C22×C4—31no—
351C22×C4—312no—
352C22×C4—612no—
353C22×C4C22×C3×D311yescharacteristic
354C22×C4C22×C3×D3112yes—
355C22×C4C22×C3×D313yes—
356C4⋊C4—34no—
357C4⋊C4—312no—
358C22⋊C4—312no—
359C22⋊C4—312no—
360C42—312no—
361C22×C3—212no—
362C22×C3—26no—
363C22×C3—26no—
364C22×C3—26no—
365C22×C3—26no—
366C22×C3—26no—
367C22×C3—26no—
368C22×C3—22no—
369C22×C3—22no—
370C22×C3—22no—
371C22×C3—21no—
372C22×C3—46no—
373C22×C3—46no—
374C22×C3—424no—
375C22×C3—412no—
376C22×C3—412no—
377C22×C3—66no—
378C22×C3—66no—
379C22×C3—66no—
380C22×C3—1212no—
381C22×C3D3×#4112yes—
382C22×C3#100912yes—
383C22×C3C22×#6712yes—
384C22×C3C22×C3×D412yes—
385C22×C3C2×C3×#4112yes—
386C22×C3C3×H(5,𝔽2)12yes—
387C22×C3C24×D311yescharacteristic
388C22×C3C25×C311yescharacteristic
389C2×D3—26no—
390C2×D3—26no—
391C2×D3—26no—
392C2×D3—412no—
393C3×C4—24no—
394C3×C4—22no—
395C3×C4—22no—
396C3×C4—212no—
397C3×C4—34no—
398C3×C4—66no—
399C3×C4—62no—
400C3×C4C22×#67112yes—
401C3×C4C22×C3×D4112yes—
402Dic3—22no—
403Dic3—26no—
404Dic3C2×C3×#4114yescomplemented
405C32C2×#55011yescharacteristic, complemented, Sylow
406C23—26no—
407C23—26no—
408C23—26no—
409C23—36no—
410C23—66no—
411C23—66no—
412C23—66no—
413C23C23×C3×D316yes—
414C23C23×C3×D311yescharacteristic
415Q8C2×C3×#67116yescomplemented
416D4—224no—
417D4—624no—
418D4—624no—
419D4—1224no—
420C2×C4—22no—
421C2×C4—224no—
422C2×C4—34no—
423C2×C4—32no—
424C2×C4—36no—
425C2×C4—32no—
426C2×C4—66no—
427C2×C4—62no—
428C2×C4—624no—
429C2×C4C2×C3×#67112yes—
430C2×C4C23×C3×D312yes—
431C2×C4C23×C3×D316yes—
432C2×C3—24no—
433C2×C3—21no—
434C2×C3—21no—
435C2×C3—21no—
436C2×C3—412no—
437C2×C3—46no—
438C2×C3—46no—
439C2×C3—126no—
440C2×C3#583414yesdirect product factor
441C2×C3C3×#55014yescomplemented
442C2×C3C2×D3×#4111yescharacteristic
443C2×C3C2×#100911yescharacteristic
444C2×C3C23×#6711yescharacteristic
445C2×C3C23×C3×D411yescharacteristic
446C2×C3C22×C3×#4111yescharacteristic
447C2×C3C2×C3×H(5,𝔽2)11yescharacteristic
448D3—46no—
449C22—26no—
450C22—26no—
451C22—26no—
452C22—412no—
453C22—66no—
454C22—66no—
455C22—66no—
456C22—1212no—
457C22C3×D3×#4112yes—
458C22C3×#100912yes—
459C22C22×C3×#6712yes—
460C22C24×C3×D311yescharacteristic
461C4—22no—
462C4—34no—
463C4—66no—
464C4—62no—
465C4C22×C3×#67112yes—
466C3—21no—
467C3C2×#583411yescharacteristic, direct product factor
468C3C2×C3×#55011yescharacteristic, complemented
469C2—46no—
470C2—126no—
471C2C3×#583414yesdirect product factor
472C2C2×C3×D3×#4111yescharacteristic
473C2C2×C3×#100911yescharacteristic
474C2C23×C3×#6711yescharacteristic
subgroupG/H
centerC23×C3C23×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC22×C32×#41C2
socleC23×C32C24
hypercenterC22×C3×#41D3
hypocenterC3C2×C3×#550
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C22×C3×#41

Complements

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

NH ≅ G/Ntype
1. C22×C32×#41C2semidirect
2. C22×C3×#239C2semidirect
3. C3×#5834C2direct (N × H)
4. C2×C3×Q8×Dic3C2semidirect
5. C2×C3×#940C2semidirect
6. C2×C3×#938C2semidirect
7. C2×C3×#930C2semidirect
8. C2×C3×C4×#67C2semidirect
10. C2×#5834C3direct (N × H)
12. C2×C32×#41C22semidirect
19. C2×C3×#239C22semidirect
23. C3×Q8×Dic3C22semidirect
24. C3×#940C22semidirect
25. C3×#938C22semidirect
26. C3×#930C22semidirect
27. C3×C4×#67C22semidirect
34. C22×C3×#41D3semidirect
35. C22×C3×#41C2×C3semidirect
36. C22×#239C2×C3semidirect
43. #5834C2×C3direct (N × H)
44. C2×Q8×Dic3C2×C3semidirect
45. C2×#940C2×C3semidirect
46. C2×#938C2×C3semidirect
47. C2×#930C2×C3semidirect
48. C2×C4×#67C2×C3semidirect
52. C2×C32×Q8D4semidirect
84. C2×C3×#41C2×D3semidirect
85. C2×C3×#41C22×C3semidirect
102. C2×#239C22×C3semidirect
116. Q8×Dic3C22×C3semidirect
117. #940C22×C3semidirect
118. #938C22×C3semidirect
119. #930C22×C3semidirect
120. C4×#67C22×C3semidirect
134. C32×Q8C2×D4semidirect
145. C2×C3×Dic3#41semidirect
152. C22×#41C3×D3semidirect
170. C2×C3×Q8#67semidirect
171. C2×C3×Q8C3×D4semidirect
241. C3×Dic3C2×#41semidirect
245. C2×#41C2×C3×D3semidirect
287. C3×Q8C2×#67semidirect
288. C3×Q8C2×C3×D4semidirect
321. C2×Dic3C3×#41semidirect
328. C2×C32#550semidirect
337. C2×Q8C3×#67semidirect
404. Dic3C2×C3×#41semidirect
405. C32C2×#550semidirect
415. Q8C2×C3×#67semidirect
440. C2×C3#5834direct (N × H)
441. C2×C3C3×#550semidirect
467. C3C2×#5834direct (N × H)
468. C3C2×C3×#550semidirect
471. C2C3×#5834direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C32×#412
C22×C3×#2392
C3×#58342
C2×C3×Q8×Dic32
C2×C3×#9402
C2×C3×#9382
C2×C3×#9302
C2×C3×C4×#672
C2×C3×#5503
C2×#58343