Finite Group Databasebeta

← C3×#3488⋊C3 #11919740

Subgroups — C3×#3488⋊C3 #11919740

Order 1152.

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

normal subgroups only

This group has 894 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×#994C211yesmaximal, characteristic, complemented
2C22×C3×#41⋊C3C211yesmaximal, characteristic, complemented
3C3×C4⋊(#41⋊C3)C212yesmaximal, complemented
4C3×#548⋊C3C212yesmaximal, complemented
5C3×#520⋊C3C211yesmaximal, characteristic, complemented
6C3×#3488C311yesmaximal, characteristic, complemented
7#3488⋊C3C313yesmaximal, direct product factor
8C3×#994—22no—
9C2×C3×#41⋊C3—22no—
10C2×C3×#41⋊C3C2211yescharacteristic
11C2×C3×#41⋊C3C2211yescharacteristic, complemented
12C2×C3×#41⋊C3C2212yescomplemented
13C22×C3×SL(2,𝔽3)C2211yescharacteristic, complemented
14C32×C22⋊Q8—41nomaximal
15C3×C4×SL(2,𝔽3)C2212yescomplemented
16C2×C3×H(5,𝔽2)C2×C311yescharacteristic, complemented
17C22×C3×#41C2×C311yescharacteristic
18C2×#994C2×C313yescomplemented
19C22×#41⋊C3C2×C313yescomplemented
20C3×#550C2×C312yescomplemented
21C3×#548C2×C312yes—
22C3×#546—32no—
23C3×D42—32no—
24C3×#537—31no—
25C3×#535—31no—
26C3×#525—31no—
27C2×C3×#121—31no—
28C3×#520C2×C311yescharacteristic, complemented
29C4⋊(#41⋊C3)C2×C316yescomplemented
30#548⋊C3C2×C316yescomplemented
31#520⋊C3C2×C313yescomplemented
32C2×C32×Q8—41no—
33C22×C32×C4—41no—
34C3×#41⋊C3—21no—
35C3×#41⋊C3—22no—
36C3×#41⋊C3D412yescomplemented
37C2×C3×SL(2,𝔽3)—21no—
38C2×C3×SL(2,𝔽3)C2311yescharacteristic
39C32×C4⋊C4—41no—
40C32×C4⋊C4—42no—
41C32×C22⋊C4—42no—
42#3488C3211yescharacteristic, complemented, Sylow
43C3×H(5,𝔽2)—22no—
44C3×H(5,𝔽2)—62no—
45C2×C3×#41—22no—
46C2×C3×#41—31no—
47C2×C3×#41—31no—
48C2×C3×#41—31no—
49C2×C3×#41—62no—
50C2×C3×#41C22×C311yescharacteristic
51C2×C3×#41C22×C312yes—
52C2×C3×#41C22×C311yescharacteristic
53C22×C3×Q8C22×C311yescharacteristic
54C22×C3×D4—31no—
55C22×C3×D4—32no—
56C22×C3×D4—31no—
57C22×C3×D4—32no—
58C23×C3×C4—31no—
59#994—26no—
60C2×#41⋊C3—26no—
61C2×#41⋊C3C22×C313yes—
62C2×#41⋊C3C22×C313yescomplemented
63C2×#41⋊C3C22×C316yescomplemented
64C22×SL(2,𝔽3)C22×C313yescomplemented
65C3×#127—32no—
66C3×#124—32no—
67C3×#123—32no—
68C3×C22⋊Q8—31no—
69C3×C22⋊Q8—32no—
70C3×C22⋊Q8—43no—
71C3×C22⋊Q8A411yescharacteristic
72C3×#121—32no—
73C3×#121—32no—
74C3×#121—34no—
75C3×#121—31no—
76C3×#121—31no—
77C3×#121—32no—
78C3×#121—32no—
79C3×#120—34no—
80C3×#120—32no—
81C3×C4×Q8C22×C312yes—
82C3×C4×D4—32no—
83C3×C4×D4—32no—
84C3×C4×D4—32no—
85C3×#117—31no—
86C2×C3×C4⋊C4—31no—
87C2×C3×C22⋊C4—32no—
88C4×SL(2,𝔽3)C22×C316yescomplemented
89C23×C32—41no—
90C32×Q8—82no—
91C2×C32×C4—41no—
92C2×C32×C4—42no—
93C2×C32×C4—41no—
94C2×C32×C4—42no—
95C2×C32×C4—82no—
96C3×SL(2,𝔽3)C2×D411yescharacteristic, complemented
97C2×H(5,𝔽2)C2×C3211yescharacteristic, complemented
98C22×#41C2×C3211yescharacteristic
99#550C2×C3212yescomplemented
100#548C2×C3212yes—
101#546—32no—
102D42—32no—
103#537—31no—
104#535—31no—
105#525—31no—
106C2×#121—31no—
107#520C2×C3211yescharacteristic, complemented
108C24×C3—32no—
109C24×C3—31no—
110C3×#41—21no—
111C3×#41—22no—
112C3×#41—32no—
113C3×#41—62no—
114C3×#41—62no—
115C3×#41—62no—
116C3×#41—62no—
117C3×#41—62no—
118C3×#41—62no—
119C3×#41—61no—
120C3×#41—62no—
121C3×#41C3×D412yes—
122C2×C3×Q8—21no—
123C2×C3×Q8—31no—
124C2×C3×Q8—43no—
125C2×C3×Q8—61no—
126C2×C3×Q8C2×A411yescharacteristic
127C2×C3×Q8C23×C311yescharacteristic
128C2×C3×D4—31no—
129C2×C3×D4—32no—
130C2×C3×D4—31no—
131C2×C3×D4—32no—
132C2×C3×D4—32no—
133C2×C3×D4—34no—
134C2×C3×D4—32no—
135C2×C3×D4—31no—
136C2×C3×D4—32no—
137C2×C3×D4—62no—
138C2×C3×D4—62no—
139C2×C3×D4—62no—
140C2×C3×D4—62no—
141C2×C3×D4—64no—
142C2×C3×D4—64no—
143C2×C3×D4—124no—
144C22×C3×C4—31no—
145C22×C3×C4—31no—
146C22×C3×C4—32no—
147C22×C3×C4—31no—
148C22×C3×C4—31no—
149C22×C3×C4—32no—
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⋊C3—26no—
157#41⋊C3C3×D416yescomplemented
158C2×SL(2,𝔽3)—23no—
159C2×SL(2,𝔽3)C23×C313yes—
160C3×C4⋊C4—32no—
161C3×C4⋊C4—31no—
162C3×C4⋊C4—32no—
163C3×C4⋊C4—43no—
164C3×C4⋊C4—46no—
165C3×C4⋊C4C2×A411yescharacteristic
166C3×C4⋊C4C2×A412yes—
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—
179C32×C4—42no—
180C32×C4—82no—
181C32×C4—81no—
182C32×C4—82no—
183H(5,𝔽2)—22no—
184H(5,𝔽2)—62no—
185C2×#41—22no—
186C2×#41—31no—
187C2×#41—31no—
188C2×#41—31no—
189C2×#41—62no—
190C2×#41C22×C3212yes—
191C2×#41C22×C3211yescharacteristic
192C2×#41C22×C3211yescharacteristic
193C22×Q8C22×C3211yescharacteristic
194C22×D4—31no—
195C22×D4—32no—
196C22×D4—31no—
197C22×D4—32no—
198C23×C4—31no—
199#127—32no—
200#124—32no—
201#123—32no—
202C22⋊Q8—31no—
203C22⋊Q8—32no—
204C22⋊Q8C3×A411yescharacteristic
205#121—32no—
206#121—32no—
207#121—34no—
208#121—31no—
209#121—31no—
210#121—32no—
211#121—32no—
212#120—34no—
213#120—32no—
214C4×Q8C22×C3212yes—
215C4×D4—32no—
216C4×D4—32no—
217C4×D4—32no—
218#117—31no—
219C2×C4⋊C4—31no—
220C2×C22⋊C4—32no—
221C23×C3—31no—
222C23×C3—32no—
223C23×C3—31no—
224C23×C3—43no—
225C23×C3—62no—
226C23×C3—64no—
227C23×C3—62no—
228C23×C3—121no—
229C23×C3—122no—
230C23×C3—122no—
231C23×C3C22×A411yescharacteristic
232C3×Q8—22no—
233C3×Q8—31no—
234C3×Q8—61no—
235C3×Q8—61no—
236C3×Q8—86no—
237C3×Q8C2×C3×D411yescharacteristic
238C3×D4—32no—
239C3×D4—32no—
240C3×D4—64no—
241C3×D4—62no—
242C3×D4—62no—
243C3×D4—61no—
244C3×D4—61no—
245C3×D4—64no—
246C3×D4—64no—
247C3×D4—62no—
248C3×D4—122no—
249C3×D4—124no—
250C2×C3×C4—22no—
251C2×C3×C4—31no—
252C2×C3×C4—32no—
253C2×C3×C4—32no—
254C2×C3×C4—32no—
255C2×C3×C4—31no—
256C2×C3×C4—43no—
257C2×C3×C4—46no—
258C2×C3×C4—46no—
259C2×C3×C4—43no—
260C2×C3×C4—61no—
261C2×C3×C4—61no—
262C2×C3×C4—62no—
263C2×C3×C4—62no—
264C2×C3×C4—62no—
265C2×C3×C4—61no—
266C2×C3×C4—61no—
267C2×C3×C4—62no—
268C2×C3×C4—86no—
269C2×C3×C4—122no—
270C2×C3×C4C22×A412yes—
271C2×C3×C4C22×A411yescharacteristic
272C2×C3×C4C22×A412yes—
273C2×C3×C4C22×A411yescharacteristic
274SL(2,𝔽3)C2×C3×D413yescomplemented
275C2×C32—41no—
276C2×C32—41no—
277C2×C32—41no—
278C2×C32—82no—
279C24—32no—
280C24—31no—
281#41—21no—
282#41—22no—
283#41—32no—
284#41—62no—
285#41—62no—
286#41—62no—
287#41—62no—
288#41—62no—
289#41—62no—
290#41—61no—
291#41—62no—
292#41C32×D412yes—
293C2×Q8—21no—
294C2×Q8—31no—
295C2×Q8—61no—
296C2×Q8C2×C3×A411yescharacteristic
297C2×Q8C23×C3211yescharacteristic
298C2×D4—31no—
299C2×D4—32no—
300C2×D4—31no—
301C2×D4—32no—
302C2×D4—32no—
303C2×D4—34no—
304C2×D4—32no—
305C2×D4—31no—
306C2×D4—32no—
307C2×D4—62no—
308C2×D4—62no—
309C2×D4—62no—
310C2×D4—62no—
311C2×D4—64no—
312C2×D4—64no—
313C2×D4—124no—
314C22×C4—31no—
315C22×C4—31no—
316C22×C4—32no—
317C22×C4—31no—
318C22×C4—31no—
319C22×C4—32no—
320C22×C4—31no—
321C22×C4—62no—
322C22×C4—62no—
323C22×C4C2×C3×A411yescharacteristic
324C4⋊C4—32no—
325C4⋊C4—31no—
326C4⋊C4—32no—
327C4⋊C4C2×C3×A411yescharacteristic
328C4⋊C4C2×C3×A412yes—
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—32no—
339C22×C3—46no—
340C22×C3—43no—
341C22×C3—61no—
342C22×C3—61no—
343C22×C3—61no—
344C22×C3—62no—
345C22×C3—62no—
346C22×C3—61no—
347C22×C3—61no—
348C22×C3—62no—
349C22×C3—83no—
350C22×C3—83no—
351C22×C3—124no—
352C22×C3—244no—
353C22×C3#99412yescomplemented
354C22×C3C23×A411yescharacteristic
355C3×C4—21no—
356C3×C4—22no—
357C3×C4—22no—
358C3×C4—31no—
359C3×C4—31no—
360C3×C4—46no—
361C3×C4—61no—
362C3×C4—62no—
363C3×C4—83no—
364C3×C4—86no—
365C3×C4—86no—
366C3×C4D4×A412yes—
367C32—41noSylow
368C23—31no—
369C23—32no—
370C23—31no—
371C23—62no—
372C23—64no—
373C23—62no—
374C23—121no—
375C23—122no—
376C23—122no—
377C23C22×C3×A411yescharacteristic
378Q8—22no—
379Q8—31no—
380Q8—61no—
381Q8—61no—
382Q8C2×C32×D411yescharacteristic
383D4—32no—
384D4—32no—
385D4—64no—
386D4—62no—
387D4—62no—
388D4—61no—
389D4—61no—
390D4—64no—
391D4—64no—
392D4—62no—
393D4—122no—
394D4—124no—
395C2×C4—22no—
396C2×C4—31no—
397C2×C4—32no—
398C2×C4—32no—
399C2×C4—32no—
400C2×C4—31no—
401C2×C4—61no—
402C2×C4—61no—
403C2×C4—62no—
404C2×C4—62no—
405C2×C4—62no—
406C2×C4—61no—
407C2×C4—61no—
408C2×C4—62no—
409C2×C4—122no—
410C2×C4C22×C3×A412yes—
411C2×C4C22×C3×A411yescharacteristic
412C2×C4C22×C3×A412yes—
413C2×C4C22×C3×A411yescharacteristic
414C2×C3—22no—
415C2×C3—43no—
416C2×C3—43no—
417C2×C3—43no—
418C2×C3—62no—
419C2×C3—86no—
420C2×C3—121no—
421C2×C3—122no—
422C2×C3C2×D4×A411yescharacteristic
423C2×C3C2×#99411yescharacteristic
424C2×C3#594511yescharacteristic
425C22—21no—
426C22—21no—
427C22—32no—
428C22—61no—
429C22—61no—
430C22—61no—
431C22—62no—
432C22—62no—
433C22—61no—
434C22—61no—
435C22—62no—
436C22—124no—
437C22—244no—
438C22C3×#99412yescomplemented
439C22C23×C3×A411yescharacteristic
440C4—21no—
441C4—22no—
442C4—22no—
443C4—31no—
444C4—31no—
445C4—61no—
446C4—62no—
447C4C3×D4×A412yes—
448C3—43no—
449C3#3488⋊C311yescharacteristic, direct product factor
450C2—22no—
451C2—62no—
452C2—121no—
453C2—122no—
454C2C2×C3×D4×A411yescharacteristic
455C2C2×C3×#99411yescharacteristic
456C2C3×#594511yescharacteristic
subgroupG/H
center, socleC22×C3C23×A4
commutator, cosocleC2×Q8C23×C32
FrattiniC22C23×C3×A4
FittingC3×#3488C3
hypercenterC3×C22⋊Q8A4
hypocenterQ8C2×C32×D4
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C22×C3 ◁ C3×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×#994C2semidirect
2. C22×C3×#41⋊C3C2semidirect
3. C3×C4⋊(#41⋊C3)C2semidirect
4. C3×#548⋊C3C2semidirect
5. C3×#520⋊C3C2semidirect
6. C3×#3488C3semidirect
7. #3488⋊C3C3direct (N × H)
11. C2×C3×#41⋊C3C22semidirect
12. C2×C3×#41⋊C3C22semidirect
13. C22×C3×SL(2,𝔽3)C22semidirect
15. C3×C4×SL(2,𝔽3)C22semidirect
16. C2×C3×H(5,𝔽2)C2×C3semidirect
18. C2×#994C2×C3semidirect
19. C22×#41⋊C3C2×C3semidirect
20. C3×#550C2×C3semidirect
28. C3×#520C2×C3semidirect
29. C4⋊(#41⋊C3)C2×C3semidirect
30. #548⋊C3C2×C3semidirect
31. #520⋊C3C2×C3semidirect
36. C3×#41⋊C3D4semidirect
42. #3488C32semidirect
62. C2×#41⋊C3C22×C3semidirect
63. C2×#41⋊C3C22×C3semidirect
64. C22×SL(2,𝔽3)C22×C3semidirect
88. C4×SL(2,𝔽3)C22×C3semidirect
96. C3×SL(2,𝔽3)C2×D4semidirect
97. C2×H(5,𝔽2)C2×C32semidirect
99. #550C2×C32semidirect
107. #520C2×C32semidirect
157. #41⋊C3C3×D4semidirect
274. SL(2,𝔽3)C2×C3×D4semidirect
353. C22×C3#994semidirect
438. C22C3×#994semidirect
449. C3#3488⋊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×#9942
C22×C3×#41⋊C32
C3×C4⋊(#41⋊C3)2
C3×#548⋊C32
C3×#520⋊C32
C3×#34883
#3488⋊C33
C32×C22⋊Q84