Finite Group Databasebeta

← C3×#3486⋊C3 #11919736

Subgroups — C3×#3486⋊C3 #11919736

Order 1152.

988 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 988 conjugacy classes of subgroups — too many to draw. Only its 166 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
2C23×C3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C3×D4×SL(2,𝔽3)C212yesmaximal, complemented
4C3×#548⋊C3C212yesmaximal, complemented
5C3×#519⋊C3C211yesmaximal, characteristic, complemented
6C3×#3486C311yesmaximal, characteristic, complemented
7#3486⋊C3C313yesmaximal, direct product factor
8C3×2−1+4⋊C3—22no—
9C2×C3×#41⋊C3C2211yescharacteristic, complemented
10C22×C3×SL(2,𝔽3)—22no—
11C22×C3×SL(2,𝔽3)C2212yes—
12C22×C3×SL(2,𝔽3)C2211yescharacteristic
13C22×C3×SL(2,𝔽3)C2211yescharacteristic, complemented
14C32×#120—41nomaximal
15C3×C4×SL(2,𝔽3)C2212yescomplemented
16C2×C3×2−1+4C2×C311yescharacteristic, complemented
17C23×C3×Q8C2×C311yescharacteristic, complemented
18C2×2−1+4⋊C3C2×C313yescomplemented
19C23×SL(2,𝔽3)C2×C313yescomplemented
20C3×D4×Q8—32no—
21C3×D4×Q8C2×C312yescomplemented
22C3×#548—32no—
23C3×#548C2×C312yescomplemented
24C3×#536—31no—
25C3×#536—31no—
26C3×#525—31no—
27C3×#525—31no—
28C3×#519C2×C311yescharacteristic, complemented
29D4×SL(2,𝔽3)C2×C316yescomplemented
30#548⋊C3C2×C316yescomplemented
31#519⋊C3C2×C313yescomplemented
32C24×C32—41no—
33C2×C32×D4—41no—
34C2×C32×D4—42no—
35C3×#41⋊C3—21no—
36C2×C3×SL(2,𝔽3)—22no—
37C2×C3×SL(2,𝔽3)—21no—
38C2×C3×SL(2,𝔽3)D412yescomplemented
39C2×C3×SL(2,𝔽3)C2311yescharacteristic
40C32×C22⋊C4—41no—
41C32×C22⋊C4—42no—
42#3486C3211yescharacteristic, complemented, Sylow
43C3×2−1+4—22no—
44C3×2−1+4—62no—
45C2×C3×#41—31no—
46C2×C3×#41—31no—
47C2×C3×#41—31no—
48C2×C3×#41C22×C311yescharacteristic, complemented
49C22×C3×Q8—22no—
50C22×C3×Q8—31no—
51C22×C3×Q8—31no—
52C22×C3×Q8—32no—
53C22×C3×Q8—62no—
54C22×C3×Q8C22×C312yes—
55C22×C3×Q8C22×C311yescharacteristic
56C22×C3×Q8C22×C311yescharacteristic, complemented
57C23×C3×C4—31no—
582−1+4⋊C3—26no—
59C2×#41⋊C3C22×C313yescomplemented
60C22×SL(2,𝔽3)—26no—
61C22×SL(2,𝔽3)C22×C316yes—
62C22×SL(2,𝔽3)C22×C313yes—
63C22×SL(2,𝔽3)C22×C313yescomplemented
64C3×C4⋊Q8—32no—
65C3×C4⋊Q8—32no—
66C3×#124—32no—
67C3×#124—32no—
68C3×#123—32no—
69C3×#123—31no—
70C3×#123—31no—
71C3×C22⋊Q8—34no—
72C3×C22⋊Q8—32no—
73C3×C22⋊Q8—32no—
74C3×C22⋊Q8—32no—
75C3×C22⋊Q8—32no—
76C3×#121—32no—
77C3×#121—32no—
78C3×#120—31no—
79C3×#120—43no—
80C3×#120A411yescharacteristic
81C3×C4×Q8—32no—
82C3×C4×Q8C22×C312yescomplemented
83C3×C4×D4—34no—
84C3×C4×D4—32no—
85C3×C4×D4—32no—
86C3×#117—31no—
87C3×#117—31no—
88C4×SL(2,𝔽3)C22×C316yescomplemented
89C23×C32—41no—
90C23×C32—42no—
91C23×C32—41no—
92C23×C32—84no—
93C23×C32—82no—
94C32×D4—82no—
95C32×D4—84no—
96C2×C32×C4—41no—
97C2×C32×C4—42no—
98C3×SL(2,𝔽3)C2×D411yescharacteristic
99C2×2−1+4C2×C3211yescharacteristic, complemented
100C23×Q8C2×C3211yescharacteristic, complemented
101D4×Q8—32no—
102D4×Q8C2×C3212yescomplemented
103#548—32no—
104#548C2×C3212yescomplemented
105#536—31no—
106#536—31no—
107#525—31no—
108#525—31no—
109#519C2×C3211yescharacteristic, complemented
110C24×C3—43no—
111C24×C3C2×A411yescharacteristic
112C3×#41—21no—
113C3×#41—62no—
114C3×#41—62no—
115C3×#41—62no—
116C3×#41—62no—
117C3×#41—62no—
118C3×#41—62no—
119C3×#41—61no—
120C2×C3×Q8—22no—
121C2×C3×Q8—21no—
122C2×C3×Q8—32no—
123C2×C3×Q8—31no—
124C2×C3×Q8—31no—
125C2×C3×Q8—32no—
126C2×C3×Q8—32no—
127C2×C3×Q8—31no—
128C2×C3×Q8—62no—
129C2×C3×Q8—62no—
130C2×C3×Q8—62no—
131C2×C3×Q8—64no—
132C2×C3×Q8—62no—
133C2×C3×Q8—62no—
134C2×C3×Q8—61no—
135C2×C3×Q8—64no—
136C2×C3×Q8C3×D412yescomplemented
137C2×C3×Q8C23×C311yescharacteristic
138C2×C3×D4—31no—
139C2×C3×D4—32no—
140C2×C3×D4—31no—
141C2×C3×D4—31no—
142C2×C3×D4—43no—
143C2×C3×D4—46no—
144C2×C3×D4C2×A412yes—
145C2×C3×D4C2×A411yescharacteristic
146C22×C3×C4—31no—
147C22×C3×C4—32no—
148C22×C3×C4—32no—
149C22×C3×C4—32no—
150C22×C3×C4—31no—
151C22×C3×C4—31no—
152C22×C3×C4—31no—
153C22×C3×C4—62no—
154C22×C3×C4—62no—
155#41⋊C3—23no—
156C2×SL(2,𝔽3)—26no—
157C2×SL(2,𝔽3)—23no—
158C2×SL(2,𝔽3)C3×D416yescomplemented
159C2×SL(2,𝔽3)C23×C313yes—
160C3×C4⋊C4—32no—
161C3×C4⋊C4—34no—
162C3×C4⋊C4—32no—
163C3×C4⋊C4—32no—
164C3×C4⋊C4—32no—
165C3×C22⋊C4—34no—
166C3×C22⋊C4—31no—
167C3×C22⋊C4—32no—
168C3×C22⋊C4—32no—
169C3×C22⋊C4—32no—
170C3×C22⋊C4—43no—
171C3×C22⋊C4—46no—
172C3×C22⋊C4C2×A411yescharacteristic
173C3×C22⋊C4C2×A412yes—
174C3×C42—32no—
175C3×C42—32no—
176C22×C32—42no—
177C22×C32—44no—
178C22×C32—41no—
179C22×C32—82no—
180C22×C32—82no—
181C22×C32—82no—
182C22×C32—82no—
183C22×C32—81no—
184C22×C32—88no—
185C32×C4—82no—
186C32×C4—81no—
1872−1+4—22no—
1882−1+4—62no—
189C2×#41—31no—
190C2×#41—31no—
191C2×#41—31no—
192C2×#41C22×C3211yescharacteristic, complemented
193C22×Q8—22no—
194C22×Q8—31no—
195C22×Q8—31no—
196C22×Q8—32no—
197C22×Q8—62no—
198C22×Q8C22×C3212yes—
199C22×Q8C22×C3211yescharacteristic
200C22×Q8C22×C3211yescharacteristic, complemented
201C23×C4—31no—
202C4⋊Q8—32no—
203C4⋊Q8—32no—
204#124—32no—
205#124—32no—
206#123—32no—
207#123—31no—
208#123—31no—
209C22⋊Q8—34no—
210C22⋊Q8—32no—
211C22⋊Q8—32no—
212C22⋊Q8—32no—
213C22⋊Q8—32no—
214#121—32no—
215#121—32no—
216#120—31no—
217#120C3×A411yescharacteristic
218C4×Q8—32no—
219C4×Q8C22×C3212yescomplemented
220C4×D4—34no—
221C4×D4—32no—
222C4×D4—32no—
223#117—31no—
224#117—31no—
225C23×C3—22no—
226C23×C3—24no—
227C23×C3—31no—
228C23×C3—43no—
229C23×C3—46no—
230C23×C3—43no—
231C23×C3—86no—
232C23×C3—812no—
233C23×C3C22×A411yescharacteristic
234C23×C3C22×A411yescharacteristic
235C23×C3C22×A412yes—
236C3×Q8—32no—
237C3×Q8—31no—
238C3×Q8—32no—
239C3×Q8—62no—
240C3×Q8—62no—
241C3×Q8—61no—
242C3×Q8—62no—
243C3×Q8—61no—
244C3×Q8—64no—
245C3×Q8C2×C3×D411yescharacteristic
246C3×D4—22no—
247C3×D4—24no—
248C3×D4—62no—
249C3×D4—61no—
250C3×D4—61no—
251C3×D4—62no—
252C3×D4—64no—
253C3×D4—86no—
254C3×D4—812no—
255C2×C3×C4—32no—
256C2×C3×C4—32no—
257C2×C3×C4—31no—
258C2×C3×C4—31no—
259C2×C3×C4—32no—
260C2×C3×C4—32no—
261C2×C3×C4—31no—
262C2×C3×C4—46no—
263C2×C3×C4—43no—
264C2×C3×C4—61no—
265C2×C3×C4—61no—
266C2×C3×C4—62no—
267C2×C3×C4—64no—
268C2×C3×C4—62no—
269C2×C3×C4—61no—
270C2×C3×C4—61no—
271C2×C3×C4—64no—
272C2×C3×C4—62no—
273C2×C3×C4C22×A412yes—
274C2×C3×C4C22×A411yescharacteristic
275SL(2,𝔽3)C2×C3×D413yes—
276C2×C32—41no—
277C2×C32—42no—
278C2×C32—82no—
279C2×C32—84no—
280C2×C32—161no—
281C24C2×C3×A411yescharacteristic
282#41—21no—
283#41—62no—
284#41—62no—
285#41—62no—
286#41—62no—
287#41—62no—
288#41—62no—
289#41—61no—
290C2×Q8—22no—
291C2×Q8—21no—
292C2×Q8—32no—
293C2×Q8—31no—
294C2×Q8—31no—
295C2×Q8—32no—
296C2×Q8—32no—
297C2×Q8—31no—
298C2×Q8—62no—
299C2×Q8—62no—
300C2×Q8—62no—
301C2×Q8—64no—
302C2×Q8—62no—
303C2×Q8—62no—
304C2×Q8—61no—
305C2×Q8—64no—
306C2×Q8C32×D412yescomplemented
307C2×Q8C23×C3211yescharacteristic
308C2×D4—31no—
309C2×D4—32no—
310C2×D4—31no—
311C2×D4—31no—
312C2×D4C2×C3×A412yes—
313C2×D4C2×C3×A411yescharacteristic
314C22×C4—31no—
315C22×C4—32no—
316C22×C4—32no—
317C22×C4—32no—
318C22×C4—31no—
319C22×C4—31no—
320C22×C4—31no—
321C22×C4—62no—
322C22×C4—62no—
323C4⋊C4—32no—
324C4⋊C4—34no—
325C4⋊C4—32no—
326C4⋊C4—32no—
327C4⋊C4—32no—
328C22⋊C4—34no—
329C22⋊C4—31no—
330C22⋊C4—32no—
331C22⋊C4—32no—
332C22⋊C4—32no—
333C22⋊C4C2×C3×A411yescharacteristic
334C22⋊C4C2×C3×A412yes—
335C42—32no—
336C42—32no—
337C22×C3—28no—
338C22×C3—22no—
339C22×C3—22no—
340C22×C3—22no—
341C22×C3—21no—
342C22×C3—22no—
343C22×C3—46no—
344C22×C3—412no—
345C22×C3—43no—
346C22×C3—61no—
347C22×C3—62no—
348C22×C3—86no—
349C22×C3—86no—
350C22×C3—86no—
351C22×C3—86no—
352C22×C3—83no—
353C22×C3—824no—
354C22×C3D4×A412yes—
355C22×C32−1+4⋊C314yescomplemented
356C22×C3C23×A411yescharacteristic
357C3×C4—21no—
358C3×C4—22no—
359C3×C4—32no—
360C3×C4—31no—
361C3×C4—31no—
362C3×C4—61no—
363C3×C4—62no—
364C3×C4—62no—
365C3×C4—86no—
366C3×C4—83no—
367C32—41noSylow
368C23—22no—
369C23—24no—
370C23—31no—
371C23C22×C3×A411yescharacteristic
372C23C22×C3×A411yescharacteristic
373C23C22×C3×A412yes—
374Q8—32no—
375Q8—31no—
376Q8—32no—
377Q8—62no—
378Q8—62no—
379Q8—61no—
380Q8—62no—
381Q8—61no—
382Q8—64no—
383Q8C2×C32×D411yescharacteristic
384D4—22no—
385D4—24no—
386D4—62no—
387D4—61no—
388D4—61no—
389D4—62no—
390D4—64no—
391C2×C4—32no—
392C2×C4—32no—
393C2×C4—31no—
394C2×C4—31no—
395C2×C4—32no—
396C2×C4—32no—
397C2×C4—31no—
398C2×C4—61no—
399C2×C4—61no—
400C2×C4—62no—
401C2×C4—62no—
402C2×C4—64no—
403C2×C4—61no—
404C2×C4—61no—
405C2×C4—64no—
406C2×C4—62no—
407C2×C4C22×C3×A412yes—
408C2×C4C22×C3×A411yescharacteristic
409C2×C3—22no—
410C2×C3—24no—
411C2×C3—41no—
412C2×C3—43no—
413C2×C3—46no—
414C2×C3—86no—
415C2×C3—812no—
416C2×C3—121no—
417C2×C3—163no—
418C2×C3C2×D4×A411yescharacteristic
419C2×C3C2×2−1+4⋊C312yes—
420C22—28no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—21no—
425C22—22no—
426C22—61no—
427C22—62no—
428C22C3×D4×A412yes—
429C22C3×2−1+4⋊C314yescomplemented
430C22C23×C3×A411yescharacteristic
431C4—21no—
432C4—22no—
433C4—31no—
434C4—32no—
435C4—31no—
436C4—61no—
437C4—62no—
438C4—62no—
439C3—43no—
440C3#3486⋊C311yescharacteristic, direct product factor
441C2—22no—
442C2—24no—
443C2—41no—
444C2—121no—
445C2C2×C3×D4×A411yescharacteristic
446C2C2×C3×2−1+4⋊C312yes—
subgroupG/H
center, socleC22×C3C23×A4
commutator, cosocleC2×Q8C23×C32
FrattiniC22C23×C3×A4
FittingC3×#3486C3
hypercenterC3×#120A4
hypocenterQ8C2×C32×D4
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C22×C3 ◁ C3×#120

Complements

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

NH ≅ G/Ntype
1. C2×C3×2−1+4⋊C3C2semidirect
2. C23×C3×SL(2,𝔽3)C2semidirect
3. C3×D4×SL(2,𝔽3)C2semidirect
4. C3×#548⋊C3C2semidirect
5. C3×#519⋊C3C2semidirect
6. C3×#3486C3semidirect
7. #3486⋊C3C3direct (N × H)
9. C2×C3×#41⋊C3C22semidirect
13. C22×C3×SL(2,𝔽3)C22semidirect
15. C3×C4×SL(2,𝔽3)C22semidirect
16. C2×C3×2−1+4C2×C3semidirect
17. C23×C3×Q8C2×C3semidirect
18. C2×2−1+4⋊C3C2×C3semidirect
19. C23×SL(2,𝔽3)C2×C3semidirect
21. C3×D4×Q8C2×C3semidirect
23. C3×#548C2×C3semidirect
28. C3×#519C2×C3semidirect
29. D4×SL(2,𝔽3)C2×C3semidirect
30. #548⋊C3C2×C3semidirect
31. #519⋊C3C2×C3semidirect
38. C2×C3×SL(2,𝔽3)D4semidirect
42. #3486C32semidirect
48. C2×C3×#41C22×C3semidirect
56. C22×C3×Q8C22×C3semidirect
59. C2×#41⋊C3C22×C3semidirect
63. C22×SL(2,𝔽3)C22×C3semidirect
82. C3×C4×Q8C22×C3semidirect
88. C4×SL(2,𝔽3)C22×C3semidirect
99. C2×2−1+4C2×C32semidirect
100. C23×Q8C2×C32semidirect
102. D4×Q8C2×C32semidirect
104. #548C2×C32semidirect
109. #519C2×C32semidirect
136. C2×C3×Q8C3×D4semidirect
158. C2×SL(2,𝔽3)C3×D4semidirect
192. C2×#41C22×C32semidirect
200. C22×Q8C22×C32semidirect
219. C4×Q8C22×C32semidirect
306. C2×Q8C32×D4semidirect
355. C22×C32−1+4⋊C3semidirect
429. C22C3×2−1+4⋊C3semidirect
440. C3#3486⋊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
C23×C3×SL(2,𝔽3)2
C3×D4×SL(2,𝔽3)2
C3×#548⋊C32
C3×#519⋊C32
C3×#34863
#3486⋊C33
C32×#1204