Finite Group Databasebeta

← C2×C3×#5923 #11923699

Subgroups — C2×C3×#5923 #11923699

Order 1152.

1557 conjugacy classes of subgroups; only the 204 normal subgroups are shown.

normal subgroups only

This group has 1557 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
1C22×C3×#41⋊C3C211yesmaximal, characteristic, complemented
2C3×#5923C218yesmaximal, direct product factor
3C2×C3×#986C211yesmaximal, characteristic, complemented
4C2×C3×#985C212yesmaximal, complemented
5C2×C3×#984C211yesmaximal, characteristic, complemented
6C2×C3×#983C212yesmaximal, complemented
7C2×#5923C311yesmaximal, characteristic, direct product factor
8C2×C3×#575—31nomaximal
9C2×C3×#235—41nomaximal
10C2×C3×#41⋊C3C2211yescharacteristic
11C2×C3×#41⋊C3C2211yescharacteristic, complemented
12C2×C3×#41⋊C3C2214yescomplemented
13C22×C3×SL(2,𝔽3)C2211yescharacteristic, complemented
14C3×#986C2214yescomplemented
15C3×#985C2218yescomplemented
16C3×#984C2214yescomplemented
17C3×#983C2218yescomplemented
18C2×C3×GL(2,𝔽3)C2212yescomplemented
19C2×C3×#226C2212yescomplemented
20C22×C3×#41—31no—
21C22×C3×#41D311yescharacteristic, complemented
22C22×#41⋊C3—21no—
23C22×#41⋊C3C2×C311yescharacteristic, complemented
24#5923C2×C318yesdirect product factor
25C2×#986C2×C311yescharacteristic, complemented
26C2×#985C2×C312yescomplemented
27C2×#984C2×C311yescharacteristic, complemented
28C2×#983C2×C312yescomplemented
29C3×#575—38no—
30C3×#575—38no—
31C2×C3×#137—31no—
32C2×C3×#137—31no—
33C2×C3×#137—32no—
34C2×C3×AGL(1,ℤ/8)—32no—
35C2×C3×AGL(1,ℤ/8)—31no—
36C2×C3×AGL(1,ℤ/8)—31no—
37C2×C3×#135—32no—
38C2×C3×#135—32no—
39C22×C3×M16—31no—
40C22×C32×C4—41no—
41C2×C3×#67—42no—
42C3×#235—48no—
43C2×C3×D12—41no—
44C2×C3×C4×D3—42no—
45C2×C3×Dic6—41no—
46C3×#41⋊C3C2312yes—
47C3×#41⋊C3C2312yescomplemented
48C2×C3×SL(2,𝔽3)C2311yescharacteristic
49C2×C3×SL(2,𝔽3)C2312yescomplemented
50C3×GL(2,𝔽3)C2314yescomplemented
51C3×#226C2314yescomplemented
52C2×#575—31noSylow
53C2×C3×#41—31no—
54C2×C3×#41—31no—
55C2×C3×#41—34no—
56C2×C3×#41—32no—
57C2×C3×#41—32no—
58C2×C3×#41—38no—
59C2×C3×#41—64no—
60C2×C3×#41—61no—
61C2×C3×#41—61no—
62C2×C3×#41C2×D311yescharacteristic
63C2×C3×#41C2×D311yescharacteristic
64C2×C3×#41C2×D314yescomplemented
65C22×C3×Q8—31no—
66C22×C3×Q8C2×D311yescharacteristic
67C22×C3×D4—31no—
68C22×C3×D4—31no—
69C22×C3×D4—61no—
70C23×C3×C4—31no—
71C23×C3×C4—61no—
72C2×#235—41no—
73C2×#41⋊C3—21no—
74C2×#41⋊C3—21no—
75C2×#41⋊C3—24no—
76C2×#41⋊C3C22×C311yescharacteristic
77C2×#41⋊C3C22×C311yescharacteristic, complemented
78C2×#41⋊C3C22×C314yescomplemented
79C22×SL(2,𝔽3)—21no—
80C22×SL(2,𝔽3)C22×C311yescharacteristic, complemented
81#986C22×C314yescomplemented
82#985C22×C318yescomplemented
83#984C22×C314yescomplemented
84#983C22×C318yescomplemented
85C2×GL(2,𝔽3)C22×C312yescomplemented
86C2×#226C22×C312yescomplemented
87C3×#137—38no—
88C3×#137—34no—
89C3×#137—38no—
90C3×#137—34no—
91C3×#137—38no—
92C3×AGL(1,ℤ/8)—316no—
93C3×AGL(1,ℤ/8)—34no—
94C3×AGL(1,ℤ/8)—38no—
95C3×AGL(1,ℤ/8)—34no—
96C3×#135—38no—
97C3×#135—316no—
98C3×#135—38no—
99C2×C3×Q16—32no—
100C2×C3×Q16—32no—
101C2×C3×SD16—32no—
102C2×C3×SD16—32no—
103C2×C3×SD16—32no—
104C2×C3×SD16—32no—
105C2×C3×D8—32no—
106C2×C3×D8—32no—
107C2×C3×M16—32no—
108C2×C3×M16—31no—
109C2×C3×M16—31no—
110C2×C3×M16—38no—
111C22×C3×C8—32no—
112C23×C32—41no—
113C22×C3×D3—42no—
114C2×C32×C4—41no—
115C2×C32×C4—44no—
116C2×C32×C4—41no—
117C3×#67—48no—
118C2×C3×Dic3—42no—
119C3×D12—44no—
120C3×C4×D3—48no—
121C3×Dic6—44no—
122C3×SL(2,𝔽3)C2411yescharacteristic
123C22×#41—31no—
124C22×#41C3×D311yescharacteristic, complemented
125#575—38no—
126#575—38no—
127C2×#137—31no—
128C2×#137—31no—
129C2×#137—32no—
130C2×AGL(1,ℤ/8)—32no—
131C2×AGL(1,ℤ/8)—31no—
132C2×AGL(1,ℤ/8)—31no—
133C2×#135—32no—
134C2×#135—32no—
135C22×M16—31no—
136C24×C3—31no—
137C24×C3—61no—
138C3×#41—32no—
139C3×#41—34no—
140C3×#41—32no—
141C3×#41—34no—
142C3×#41—38no—
143C3×#41—38no—
144C3×#41—62no—
145C3×#41—62no—
146C3×#41—64no—
147C3×#41—68no—
148C3×#41—64no—
149C3×#41—68no—
150C3×#41C22×D312yes—
151C3×#41C22×D312yes—
152C2×C3×Q8—31no—
153C2×C3×Q8—32no—
154C2×C3×Q8—32no—
155C2×C3×Q8—34no—
156C2×C3×Q8—62no—
157C2×C3×Q8—61no—
158C2×C3×Q8C22×D311yescharacteristic
159C2×C3×Q8C22×D312yes—
160C2×C3×D4—31no—
161C2×C3×D4—31no—
162C2×C3×D4—34no—
163C2×C3×D4—32no—
164C2×C3×D4—34no—
165C2×C3×D4—61no—
166C2×C3×D4—62no—
167C2×C3×D4—62no—
168C2×C3×D4—68no—
169C2×C3×D4—62no—
170C2×C3×D4—62no—
171C2×C3×D4—61no—
172C22×C3×C4—31no—
173C22×C3×C4—31no—
174C22×C3×C4—34no—
175C22×C3×C4—31no—
176C22×C3×C4—34no—
177C22×C3×C4—31no—
178C22×C3×C4—31no—
179C22×C3×C4—41no—
180C22×C3×C4—62no—
181C22×C3×C4—68no—
182C22×C3×C4—61no—
183C22×C3×C4—62no—
184C22×C3×C4—81no—
185C22×C3×C4S411yescharacteristic
186C2×#67—42no—
187#235—48no—
188C2×D12—41no—
189C2×C4×D3—42no—
190C2×Dic6—41no—
191#41⋊C3—22no—
192#41⋊C3—22no—
193#41⋊C3C23×C312yes—
194#41⋊C3C23×C312yescomplemented
195C2×SL(2,𝔽3)—21no—
196C2×SL(2,𝔽3)—22no—
197C2×SL(2,𝔽3)C23×C311yescharacteristic
198C2×SL(2,𝔽3)C23×C312yescomplemented
199GL(2,𝔽3)C23×C314yescomplemented
200#226C23×C314yescomplemented
201C3×Q16—38no—
202C3×Q16—34no—
203C3×Q16—34no—
204C3×SD16—38no—
205C3×SD16—38no—
206C3×SD16—38no—
207C3×SD16—34no—
208C3×SD16—34no—
209C3×D8—38no—
210C3×D8—38no—
211C3×M16—38no—
212C3×M16—34no—
213C3×M16—34no—
214C2×C3×C8—32no—
215C2×C3×C8—32no—
216C2×C3×C8—38no—
217C22×C32—41no—
218C22×C32—42no—
219C22×C32—82no—
220C2×C3×D3—44no—
221C2×C3×D3—84no—
222C32×C4—42no—
223C32×C4—42no—
224C3×Dic3—44no—
225C2×#41—31no—
226C2×#41—31no—
227C2×#41—34no—
228C2×#41—32no—
229C2×#41—32no—
230C2×#41—38no—
231C2×#41—64no—
232C2×#41—61no—
233C2×#41—61no—
234C2×#41C2×C3×D311yescharacteristic
235C2×#41C2×C3×D311yescharacteristic
236C2×#41C2×C3×D314yescomplemented
237C22×Q8—31no—
238C22×Q8C2×C3×D311yescharacteristic
239C22×D4—31no—
240C22×D4—31no—
241C22×D4—61no—
242C23×C4—31no—
243C23×C4—61no—
244#137—38no—
245#137—34no—
246#137—38no—
247#137—34no—
248#137—38no—
249AGL(1,ℤ/8)—316no—
250AGL(1,ℤ/8)—34no—
251AGL(1,ℤ/8)—38no—
252AGL(1,ℤ/8)—34no—
253#135—38no—
254#135—316no—
255#135—38no—
256C2×Q16—32no—
257C2×Q16—32no—
258C2×SD16—32no—
259C2×SD16—32no—
260C2×SD16—32no—
261C2×SD16—32no—
262C2×D8—32no—
263C2×D8—32no—
264C2×M16—32no—
265C2×M16—31no—
266C2×M16—31no—
267C2×M16—38no—
268C22×C8—32no—
269C23×C3—31no—
270C23×C3—31no—
271C23×C3—34no—
272C23×C3—41no—
273C23×C3—64no—
274C23×C3—62no—
275C23×C3—81no—
276C23×C3—122no—
277C23×C3—124no—
278C23×C3C2×S411yescharacteristic
279C22×D3—42no—
280C3×Q8—32no—
281C3×Q8—31no—
282C3×Q8—34no—
283C3×Q8—61no—
284C3×Q8—62no—
285C3×Q8—64no—
286C3×Q8C23×D311yescharacteristic
287C3×D4—32no—
288C3×D4—32no—
289C3×D4—34no—
290C3×D4—64no—
291C3×D4—61no—
292C3×D4—61no—
293C3×D4—62no—
294C3×D4—68no—
295C3×D4—68no—
296C3×D4—64no—
297C2×C3×C4—31no—
298C2×C3×C4—32no—
299C2×C3×C4—31no—
300C2×C3×C4—32no—
301C2×C3×C4—32no—
302C2×C3×C4—32no—
303C2×C3×C4—34no—
304C2×C3×C4—32no—
305C2×C3×C4—34no—
306C2×C3×C4—32no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4—44no—
310C2×C3×C4—68no—
311C2×C3×C4—64no—
312C2×C3×C4—62no—
313C2×C3×C4—68no—
314C2×C3×C4—81no—
315C2×C3×C4—81no—
316C2×C3×C4—84no—
317C2×C3×C4C2×S411yescharacteristic
318C2×C3×C4C2×S414yes—
319C2×C3×C4C2×S411yescharacteristic
320#67—48no—
321C2×Dic3—42no—
322D12—44no—
323C4×D3—48no—
324Dic6—44no—
325SL(2,𝔽3)—21no—
326SL(2,𝔽3)C24×C311yescharacteristic
327C3×C8—34no—
328C3×C8—34no—
329C2×C32—42no—
330C2×C32—41no—
331C2×C32—82no—
332C3×D3—84no—
333C24—31no—
334C24—61no—
335#41—32no—
336#41—34no—
337#41—32no—
338#41—34no—
339#41—38no—
340#41—38no—
341#41—62no—
342#41—62no—
343#41—64no—
344#41—68no—
345#41—64no—
346#41—68no—
347#41C22×C3×D312yes—
348#41C22×C3×D312yes—
349C2×Q8—31no—
350C2×Q8—32no—
351C2×Q8—32no—
352C2×Q8—34no—
353C2×Q8—62no—
354C2×Q8—61no—
355C2×Q8C22×C3×D311yescharacteristic
356C2×Q8C22×C3×D312yes—
357C2×D4—31no—
358C2×D4—31no—
359C2×D4—34no—
360C2×D4—32no—
361C2×D4—34no—
362C2×D4—61no—
363C2×D4—62no—
364C2×D4—62no—
365C2×D4—68no—
366C2×D4—62no—
367C2×D4—62no—
368C2×D4—61no—
369C22×C4—31no—
370C22×C4—31no—
371C22×C4—31no—
372C22×C4—34no—
373C22×C4—31no—
374C22×C4—31no—
375C22×C4—34no—
376C22×C4—62no—
377C22×C4—68no—
378C22×C4—61no—
379C22×C4—62no—
380C22×C4C3×S411yescharacteristic
381Q16—38no—
382Q16—34no—
383Q16—34no—
384SD16—38no—
385SD16—38no—
386SD16—38no—
387SD16—34no—
388SD16—34no—
389D8—38no—
390D8—38no—
391M16—38no—
392M16—34no—
393M16—34no—
394C2×C8—32no—
395C2×C8—32no—
396C2×C8—38no—
397C22×C3—22no—
398C22×C3—32no—
399C22×C3—32no—
400C22×C3—41no—
401C22×C3—42no—
402C22×C3—62no—
403C22×C3—62no—
404C22×C3—64no—
405C22×C3—82no—
406C22×C3—84no—
407C22×C3—82no—
408C22×C3—81no—
409C22×C3—124no—
410C22×C3—128no—
411C22×C3—124no—
412C22×C3C22×S412yes—
413C22×C3C22×S411yescharacteristic
414C2×D3—44no—
415C2×D3—84no—
416C3×C4—31no—
417C3×C4—32no—
418C3×C4—31no—
419C3×C4—42no—
420C3×C4—42no—
421C3×C4—64no—
422C3×C4—82no—
423C3×C4—82no—
424C3×C4C22×S412yes—
425C3×C4C22×S412yes—
426Dic3—44no—
427C32—41noSylow
428C23—31no—
429C23—31no—
430C23—34no—
431C23—64no—
432C23—62no—
433C23—122no—
434C23—124no—
435C23C2×C3×S411yescharacteristic
436Q8—32no—
437Q8—31no—
438Q8—34no—
439Q8—61no—
440Q8—62no—
441Q8—64no—
442Q8C23×C3×D311yescharacteristic
443D4—32no—
444D4—32no—
445D4—34no—
446D4—64no—
447D4—61no—
448D4—61no—
449D4—62no—
450D4—68no—
451D4—68no—
452D4—64no—
453C2×C4—31no—
454C2×C4—32no—
455C2×C4—31no—
456C2×C4—32no—
457C2×C4—32no—
458C2×C4—32no—
459C2×C4—34no—
460C2×C4—32no—
461C2×C4—34no—
462C2×C4—32no—
463C2×C4—68no—
464C2×C4—64no—
465C2×C4—62no—
466C2×C4—68no—
467C2×C4C2×C3×S411yescharacteristic
468C2×C4C2×C3×S411yescharacteristic
469C2×C4C2×C3×S414yes—
470C8—34no—
471C8—34no—
472C2×C3—22no—
473C2×C3—41no—
474C2×C3—42no—
475C2×C3—62no—
476C2×C3—62no—
477C2×C3—82no—
478C2×C3—84no—
479C2×C3—81no—
480C2×C3—82no—
481C2×C3—124no—
482C2×C3#592312yesdirect product factor
483C2×C3C23×S411yescharacteristic
484D3—84no—
485C22—22no—
486C22—32no—
487C22—32no—
488C22—62no—
489C22—62no—
490C22—64no—
491C22—124no—
492C22—128no—
493C22—124no—
494C22C22×C3×S412yes—
495C22C22×C3×S411yescharacteristic
496C4—31no—
497C4—32no—
498C4—31no—
499C4—64no—
500C4C22×C3×S412yes—
501C4C22×C3×S412yes—
502C3—41no—
503C3—81no—
504C3C2×#592311yescharacteristic, direct product factor
505C2—22no—
506C2—62no—
507C2—62no—
508C2—124no—
509C2C3×#592312yesdirect product factor
510C2C23×C3×S411yescharacteristic
subgroupG/H
centerC2×C3×C4C2×S4
commutator, cosocle, hypocenterSL(2,𝔽3)C24×C3
FrattiniC2C23×C3×S4
FittingC22×C3×#41D3
socleC22×C3C22×S4
hypercenterC22×C3×C4S4
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C2×C3×C4 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×#41⋊C3C2semidirect
2. C3×#5923C2direct (N × H)
3. C2×C3×#986C2semidirect
4. C2×C3×#985C2semidirect
5. C2×C3×#984C2semidirect
6. C2×C3×#983C2semidirect
7. C2×#5923C3direct (N × H)
11. C2×C3×#41⋊C3C22semidirect
12. C2×C3×#41⋊C3C22semidirect
13. C22×C3×SL(2,𝔽3)C22semidirect
14. C3×#986C22semidirect
15. C3×#985C22semidirect
16. C3×#984C22semidirect
17. C3×#983C22semidirect
18. C2×C3×GL(2,𝔽3)C22semidirect
19. C2×C3×#226C22semidirect
21. C22×C3×#41D3semidirect
23. C22×#41⋊C3C2×C3semidirect
24. #5923C2×C3direct (N × H)
25. C2×#986C2×C3semidirect
26. C2×#985C2×C3semidirect
27. C2×#984C2×C3semidirect
28. C2×#983C2×C3semidirect
47. C3×#41⋊C3C23semidirect
49. C2×C3×SL(2,𝔽3)C23semidirect
50. C3×GL(2,𝔽3)C23semidirect
51. C3×#226C23semidirect
64. C2×C3×#41C2×D3semidirect
77. C2×#41⋊C3C22×C3semidirect
78. C2×#41⋊C3C22×C3semidirect
80. C22×SL(2,𝔽3)C22×C3semidirect
81. #986C22×C3semidirect
82. #985C22×C3semidirect
83. #984C22×C3semidirect
84. #983C22×C3semidirect
85. C2×GL(2,𝔽3)C22×C3semidirect
86. C2×#226C22×C3semidirect
124. C22×#41C3×D3semidirect
194. #41⋊C3C23×C3semidirect
198. C2×SL(2,𝔽3)C23×C3semidirect
199. GL(2,𝔽3)C23×C3semidirect
200. #226C23×C3semidirect
236. C2×#41C2×C3×D3semidirect
482. C2×C3#5923direct (N × H)
504. C3C2×#5923direct (N × H)
509. C2C3×#5923direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#41⋊C32
C3×#59232
C2×C3×#9862
C2×C3×#9852
C2×C3×#9842
C2×C3×#9832
C2×#59233
C2×C3×#5753
C2×C3×#2354