Finite Group Databasebeta

← C3×#3489⋊C3 #11919742

Subgroups — C3×#3489⋊C3 #11919742

Order 1152.

902 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 902 conjugacy classes of subgroups — too many to draw. Only its 162 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×C3×2−1+4⋊C3C211yesmaximal, characteristic, complemented
2C22×C3×#41⋊C3C211yesmaximal, characteristic, complemented
3C3×D4×SL(2,𝔽3)C212yesmaximal, complemented
4C3×#548⋊C3C212yesmaximal, complemented
5C3×#520⋊C3C211yesmaximal, characteristic, complemented
6C3×#3489C311yesmaximal, characteristic, complemented
7#3489⋊C3C313yesmaximal, direct product factor
8C3×2−1+4⋊C3—22no—
9C2×C3×#41⋊C3—22no—
10C2×C3×#41⋊C3C2211yescharacteristic
11C2×C3×#41⋊C3C2211yescharacteristic
12C22×C3×SL(2,𝔽3)C2211yescharacteristic
13C22×C3×SL(2,𝔽3)C2212yescomplemented
14C32×#121—41nomaximal
15C3×C4×SL(2,𝔽3)C2212yescomplemented
16C2×C3×2−1+4C2×C311yescharacteristic, complemented
17C22×C3×#41C2×C311yescharacteristic, complemented
18C2×2−1+4⋊C3C2×C313yescomplemented
19C22×#41⋊C3C2×C313yescomplemented
20C3×D4×Q8C2×C312yescomplemented
21C3×#548C2×C312yescomplemented
22C3×#547—32no—
23C3×#546—32no—
24C3×#537—31no—
25C3×#536—31no—
26C3×#525—31no—
27C2×C3×C22⋊Q8—31no—
28C3×#520C2×C311yescharacteristic, complemented
29D4×SL(2,𝔽3)C2×C316yescomplemented
30#548⋊C3C2×C316yescomplemented
31#520⋊C3C2×C313yescomplemented
32C2×C32×D4—41no—
33C2×C32×D4—42no—
34C22×C32×C4—41no—
35C3×#41⋊C3—21no—
36C3×#41⋊C3D412yescomplemented
37C2×C3×SL(2,𝔽3)—21no—
38C2×C3×SL(2,𝔽3)—22no—
39C2×C3×SL(2,𝔽3)C2311yescharacteristic
40C32×C4⋊C4—41no—
41C32×C22⋊C4—42no—
42#3489C3211yescharacteristic, complemented, Sylow
43C3×2−1+4—22no—
44C3×2−1+4—62no—
45C2×C3×#41—22no—
46C2×C3×#41—31no—
47C2×C3×#41—32no—
48C2×C3×#41—31no—
49C2×C3×#41—31no—
50C2×C3×#41—62no—
51C2×C3×#41C22×C311yescharacteristic
52C2×C3×#41C22×C311yescharacteristic
53C22×C3×Q8—31no—
54C22×C3×Q8C22×C311yescharacteristic
55C22×C3×Q8C22×C312yes—
56C22×C3×D4—31no—
57C23×C3×C4—31no—
582−1+4⋊C3—26no—
59C2×#41⋊C3—26no—
60C2×#41⋊C3C22×C313yes—
61C2×#41⋊C3C22×C313yes—
62C22×SL(2,𝔽3)C22×C313yes—
63C22×SL(2,𝔽3)C22×C316yescomplemented
64C3×C4⋊Q8—32no—
65C3×#124—32no—
66C3×#123—34no—
67C3×#123—32no—
68C3×C22⋊Q8—32no—
69C3×C22⋊Q8—31no—
70C3×C22⋊Q8—31no—
71C3×C22⋊Q8—32no—
72C3×C22⋊Q8—32no—
73C3×C22⋊Q8—32no—
74C3×#121—32no—
75C3×#121—34no—
76C3×#121—31no—
77C3×#121—43no—
78C3×#121A411yescharacteristic
79C3×#120—32no—
80C3×C4×Q8C22×C312yes—
81C3×C4×D4—32no—
82C3×C4×D4—32no—
83C3×C4×D4—32no—
84C3×#117—31no—
85C2×C3×C4⋊C4—32no—
86C2×C3×C4⋊C4—31no—
87C2×C3×C22⋊C4—32no—
88C4×SL(2,𝔽3)C22×C316yescomplemented
89C23×C32—41no—
90C23×C32—42no—
91C32×D4—82no—
92C32×D4—84no—
93C2×C32×C4—41no—
94C2×C32×C4—41no—
95C2×C32×C4—42no—
96C2×C32×C4—82no—
97C3×SL(2,𝔽3)C2×D411yescharacteristic
98C2×2−1+4C2×C3211yescharacteristic, complemented
99C22×#41C2×C3211yescharacteristic, complemented
100D4×Q8C2×C3212yescomplemented
101#548C2×C3212yescomplemented
102#547—32no—
103#546—32no—
104#537—31no—
105#536—31no—
106#525—31no—
107C2×C22⋊Q8—31no—
108#520C2×C3211yescharacteristic, complemented
109C24×C3—31no—
110C3×#41—21no—
111C3×#41—32no—
112C3×#41—62no—
113C3×#41—62no—
114C3×#41—62no—
115C3×#41—62no—
116C3×#41—62no—
117C3×#41—64no—
118C3×#41—64no—
119C3×#41—62no—
120C3×#41—61no—
121C3×#41C3×D412yescomplemented
122C2×C3×Q8—21no—
123C2×C3×Q8—22no—
124C2×C3×Q8—31no—
125C2×C3×Q8—32no—
126C2×C3×Q8—31no—
127C2×C3×Q8—62no—
128C2×C3×Q8—62no—
129C2×C3×Q8—61no—
130C2×C3×Q8—62no—
131C2×C3×Q8C23×C311yescharacteristic
132C2×C3×D4—31no—
133C2×C3×D4—31no—
134C2×C3×D4—32no—
135C2×C3×D4—32no—
136C2×C3×D4—32no—
137C2×C3×D4—43no—
138C2×C3×D4—46no—
139C2×C3×D4—62no—
140C2×C3×D4—62no—
141C2×C3×D4C2×A412yes—
142C2×C3×D4C2×A411yescharacteristic
143C22×C3×C4—31no—
144C22×C3×C4—32no—
145C22×C3×C4—32no—
146C22×C3×C4—31no—
147C22×C3×C4—32no—
148C22×C3×C4—31no—
149C22×C3×C4—31no—
150C22×C3×C4—31no—
151C22×C3×C4—43no—
152C22×C3×C4—62no—
153C22×C3×C4—62no—
154C22×C3×C4C2×A411yescharacteristic
155#41⋊C3—23no—
156#41⋊C3C3×D416yescomplemented
157C2×SL(2,𝔽3)—23no—
158C2×SL(2,𝔽3)—26no—
159C2×SL(2,𝔽3)C23×C313yes—
160C3×C4⋊C4—32no—
161C3×C4⋊C4—34no—
162C3×C4⋊C4—32no—
163C3×C4⋊C4—31no—
164C3×C4⋊C4—32no—
165C3×C4⋊C4—43no—
166C3×C4⋊C4C2×A411yescharacteristic
167C3×C22⋊C4—34no—
168C3×C22⋊C4—32no—
169C3×C22⋊C4—32no—
170C3×C22⋊C4—32no—
171C3×C22⋊C4—32no—
172C3×C22⋊C4—46no—
173C3×C22⋊C4C2×A412yes—
174C3×C42—32no—
175C22×C32—42no—
176C22×C32—41no—
177C22×C32—81no—
178C22×C32—81no—
179C22×C32—82no—
180C22×C32—82no—
181C22×C32—82no—
182C32×C4—42no—
183C32×C4—81no—
184C32×C4—82no—
1852−1+4—22no—
1862−1+4—62no—
187C2×#41—22no—
188C2×#41—31no—
189C2×#41—32no—
190C2×#41—31no—
191C2×#41—31no—
192C2×#41—62no—
193C2×#41C22×C3211yescharacteristic
194C2×#41C22×C3211yescharacteristic
195C22×Q8—31no—
196C22×Q8C22×C3212yes—
197C22×Q8C22×C3211yescharacteristic
198C22×D4—31no—
199C23×C4—31no—
200C4⋊Q8—32no—
201#124—32no—
202#123—34no—
203#123—32no—
204C22⋊Q8—32no—
205C22⋊Q8—31no—
206C22⋊Q8—31no—
207C22⋊Q8—32no—
208C22⋊Q8—32no—
209C22⋊Q8—32no—
210#121—32no—
211#121—34no—
212#121—31no—
213#121C3×A411yescharacteristic
214#120—32no—
215C4×Q8C22×C3212yes—
216C4×D4—32no—
217C4×D4—32no—
218C4×D4—32no—
219#117—31no—
220C2×C4⋊C4—32no—
221C2×C4⋊C4—31no—
222C2×C22⋊C4—32no—
223C23×C3—31no—
224C23×C3—31no—
225C23×C3—43no—
226C23×C3—46no—
227C23×C3—62no—
228C23×C3—62no—
229C23×C3—121no—
230C23×C3C22×A412yes—
231C23×C3C22×A411yescharacteristic
232C3×Q8—31no—
233C3×Q8—62no—
234C3×Q8—61no—
235C3×Q8—62no—
236C3×Q8—61no—
237C3×Q8—62no—
238C3×Q8C2×C3×D411yescharacteristic
239C3×D4—24no—
240C3×D4—22no—
241C3×D4—32no—
242C3×D4—32no—
243C3×D4—61no—
244C3×D4—61no—
245C3×D4—64no—
246C3×D4—64no—
247C3×D4—86no—
248C3×D4—812no—
249C3×D4—122no—
250C2×C3×C4—22no—
251C2×C3×C4—31no—
252C2×C3×C4—32no—
253C2×C3×C4—32no—
254C2×C3×C4—32no—
255C2×C3×C4—32no—
256C2×C3×C4—31no—
257C2×C3×C4—43no—
258C2×C3×C4—46no—
259C2×C3×C4—43no—
260C2×C3×C4—61no—
261C2×C3×C4—61no—
262C2×C3×C4—62no—
263C2×C3×C4—61no—
264C2×C3×C4—61no—
265C2×C3×C4—62no—
266C2×C3×C4—62no—
267C2×C3×C4—64no—
268C2×C3×C4—62no—
269C2×C3×C4—86no—
270C2×C3×C4—122no—
271C2×C3×C4C22×A412yes—
272C2×C3×C4C22×A411yescharacteristic
273C2×C3×C4C22×A411yescharacteristic
274SL(2,𝔽3)C2×C3×D413yes—
275C2×C32—41no—
276C2×C32—41no—
277C2×C32—41no—
278C2×C32—82no—
279C2×C32—162no—
280C24—31no—
281#41—21no—
282#41—32no—
283#41—62no—
284#41—62no—
285#41—62no—
286#41—62no—
287#41—62no—
288#41—64no—
289#41—64no—
290#41—62no—
291#41—61no—
292#41C32×D412yescomplemented
293C2×Q8—21no—
294C2×Q8—22no—
295C2×Q8—31no—
296C2×Q8—32no—
297C2×Q8—31no—
298C2×Q8—62no—
299C2×Q8—62no—
300C2×Q8—61no—
301C2×Q8—62no—
302C2×Q8C23×C3211yescharacteristic
303C2×D4—31no—
304C2×D4—31no—
305C2×D4—32no—
306C2×D4—32no—
307C2×D4—32no—
308C2×D4—62no—
309C2×D4—62no—
310C2×D4C2×C3×A412yes—
311C2×D4C2×C3×A411yescharacteristic
312C22×C4—31no—
313C22×C4—32no—
314C22×C4—32no—
315C22×C4—31no—
316C22×C4—32no—
317C22×C4—31no—
318C22×C4—31no—
319C22×C4—31no—
320C22×C4—62no—
321C22×C4—62no—
322C22×C4C2×C3×A411yescharacteristic
323C4⋊C4—32no—
324C4⋊C4—34no—
325C4⋊C4—32no—
326C4⋊C4—31no—
327C4⋊C4—32no—
328C4⋊C4C2×C3×A411yescharacteristic
329C22⋊C4—34no—
330C22⋊C4—32no—
331C22⋊C4—32no—
332C22⋊C4—32no—
333C22⋊C4—32no—
334C22⋊C4C2×C3×A412yes—
335C42—32no—
336C22×C3—21no—
337C22×C3—21no—
338C22×C3—22no—
339C22×C3—22no—
340C22×C3—22no—
341C22×C3—32no—
342C22×C3—46no—
343C22×C3—43no—
344C22×C3—61no—
345C22×C3—61no—
346C22×C3—61no—
347C22×C3—61no—
348C22×C3—61no—
349C22×C3—83no—
350C22×C3—83no—
351C22×C3—86no—
352C22×C3—86no—
353C22×C3—86no—
354C22×C3—124no—
355C22×C32−1+4⋊C312yescomplemented
356C22×C3C23×A411yescharacteristic
357C3×C4—21no—
358C3×C4—22no—
359C3×C4—31no—
360C3×C4—31no—
361C3×C4—46no—
362C3×C4—61no—
363C3×C4—62no—
364C3×C4—62no—
365C3×C4—83no—
366C3×C4—86no—
367C3×C4D4×A412yes—
368C32—41noSylow
369C23—31no—
370C23—31no—
371C23—62no—
372C23—62no—
373C23—121no—
374C23C22×C3×A412yes—
375C23C22×C3×A411yescharacteristic
376Q8—31no—
377Q8—62no—
378Q8—61no—
379Q8—62no—
380Q8—61no—
381Q8—62no—
382Q8C2×C32×D411yescharacteristic
383D4—24no—
384D4—22no—
385D4—32no—
386D4—32no—
387D4—61no—
388D4—61no—
389D4—64no—
390D4—64no—
391D4—122no—
392C2×C4—22no—
393C2×C4—31no—
394C2×C4—32no—
395C2×C4—32no—
396C2×C4—32no—
397C2×C4—32no—
398C2×C4—31no—
399C2×C4—61no—
400C2×C4—61no—
401C2×C4—62no—
402C2×C4—61no—
403C2×C4—61no—
404C2×C4—62no—
405C2×C4—62no—
406C2×C4—64no—
407C2×C4—62no—
408C2×C4—122no—
409C2×C4C22×C3×A412yes—
410C2×C4C22×C3×A411yescharacteristic
411C2×C4C22×C3×A411yescharacteristic
412C2×C3—22no—
413C2×C3—42no—
414C2×C3—43no—
415C2×C3—43no—
416C2×C3—43no—
417C2×C3—62no—
418C2×C3—86no—
419C2×C3—121no—
420C2×C3—166no—
421C2×C3C2×D4×A411yescharacteristic
422C2×C3C2×2−1+4⋊C311yescharacteristic
423C2×C3#594511yescharacteristic
424C22—21no—
425C22—21no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—32no—
430C22—61no—
431C22—61no—
432C22—61no—
433C22—61no—
434C22—61no—
435C22—124no—
436C22C3×2−1+4⋊C312yescomplemented
437C22C23×C3×A411yescharacteristic
438C4—21no—
439C4—22no—
440C4—31no—
441C4—31no—
442C4—61no—
443C4—62no—
444C4—62no—
445C4C3×D4×A412yes—
446C3—43no—
447C3#3489⋊C311yescharacteristic, direct product factor
448C2—22no—
449C2—42no—
450C2—62no—
451C2—121no—
452C2C2×C3×D4×A411yescharacteristic
453C2C2×C3×2−1+4⋊C311yescharacteristic
454C2C3×#594511yescharacteristic
subgroupG/H
center, socleC22×C3C23×A4
commutator, cosocleC2×Q8C23×C32
FrattiniC22C23×C3×A4
FittingC3×#3489C3
hypercenterC3×#121A4
hypocenterQ8C2×C32×D4
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C22×C3 ◁ C3×#121

Complements

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

NH ≅ G/Ntype
1. C2×C3×2−1+4⋊C3C2semidirect
2. C22×C3×#41⋊C3C2semidirect
3. C3×D4×SL(2,𝔽3)C2semidirect
4. C3×#548⋊C3C2semidirect
5. C3×#520⋊C3C2semidirect
6. C3×#3489C3semidirect
7. #3489⋊C3C3direct (N × H)
13. C22×C3×SL(2,𝔽3)C22semidirect
15. C3×C4×SL(2,𝔽3)C22semidirect
16. C2×C3×2−1+4C2×C3semidirect
17. C22×C3×#41C2×C3semidirect
18. C2×2−1+4⋊C3C2×C3semidirect
19. C22×#41⋊C3C2×C3semidirect
20. C3×D4×Q8C2×C3semidirect
21. C3×#548C2×C3semidirect
28. C3×#520C2×C3semidirect
29. D4×SL(2,𝔽3)C2×C3semidirect
30. #548⋊C3C2×C3semidirect
31. #520⋊C3C2×C3semidirect
36. C3×#41⋊C3D4semidirect
42. #3489C32semidirect
63. C22×SL(2,𝔽3)C22×C3semidirect
88. C4×SL(2,𝔽3)C22×C3semidirect
98. C2×2−1+4C2×C32semidirect
99. C22×#41C2×C32semidirect
100. D4×Q8C2×C32semidirect
101. #548C2×C32semidirect
108. #520C2×C32semidirect
121. C3×#41C3×D4semidirect
156. #41⋊C3C3×D4semidirect
292. #41C32×D4semidirect
355. C22×C32−1+4⋊C3semidirect
436. C22C3×2−1+4⋊C3semidirect
447. C3#3489⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×2−1+4⋊C32
C22×C3×#41⋊C32
C3×D4×SL(2,𝔽3)2
C3×#548⋊C32
C3×#520⋊C32
C3×#34893
#3489⋊C33
C32×#1214