Finite Group Databasebeta

← C3×#3516⋊C3 #11919758

Subgroups — C3×#3516⋊C3 #11919758

Order 1152.

652 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#557⋊C3C211yesmaximal, characteristic, complemented
2C3×D4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C3×#548⋊C3C211yesmaximal, characteristic, complemented
4C3×#548⋊C3C211yesmaximal, characteristic, complemented
5C3×#520⋊C3C211yesmaximal, characteristic, complemented
6C3×#519⋊C3C211yesmaximal, characteristic, complemented
7C2×C3×C4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
8C3×#3516C311yesmaximal, characteristic, complemented
9#3516⋊C3C313yesmaximal, direct product factor
10C2×C3×#41⋊C3C2211yescharacteristic
11C22×C3×SL(2,𝔽3)C2211yescharacteristic
12C22×C3×SL(2,𝔽3)C2211yescharacteristic
13C32×#123—41nomaximal
14C3×C4×SL(2,𝔽3)C2211yescharacteristic
15C3×C4×SL(2,𝔽3)C2211yescharacteristic
16C3×C4×SL(2,𝔽3)C2211yescharacteristic
17C3×C4×SL(2,𝔽3)C2211yescharacteristic
18C3×#557C2×C311yescharacteristic, complemented
19C3×#556—31no—
20C3×#554—31no—
21C3×#553—31no—
22C3×#552—31no—
23C3×D4×Q8C2×C311yescharacteristic
24C3×#548C2×C311yescharacteristic, complemented
25C3×#548C2×C311yescharacteristic, complemented
26C3×#542—31no—
27C3×#541—31no—
28C3×#529—31no—
29C3×#529—31no—
30C3×#520C2×C311yescharacteristic, complemented
31C3×#519C2×C311yescharacteristic, complemented
32C2×C3×C4×Q8C2×C311yescharacteristic, complemented
33#557⋊C3C2×C313yescomplemented
34D4×SL(2,𝔽3)C2×C313yescomplemented
35#548⋊C3C2×C313yescomplemented
36#548⋊C3C2×C313yescomplemented
37#520⋊C3C2×C313yescomplemented
38#519⋊C3C2×C313yescomplemented
39C2×C4×SL(2,𝔽3)C2×C313yescomplemented
40C2×C32×D4—41no—
41C22×C32×C4—41no—
42C3×#41⋊C3—21no—
43C2×C3×SL(2,𝔽3)—21no—
44C2×C3×SL(2,𝔽3)—21no—
45C2×C3×SL(2,𝔽3)C2311yescharacteristic
46C32×C4⋊C4—41no—
47C32×C4⋊C4—41no—
48C32×C22⋊C4—41no—
49C32×C22⋊C4—41no—
50C32×C22⋊C4—41no—
51#3516C3211yescharacteristic, complemented, Sylow
52C2×C3×#41C22×C311yescharacteristic
53C22×C3×Q8C22×C311yescharacteristic
54C22×C3×Q8C22×C311yescharacteristic
55C2×#41⋊C3C22×C313yes—
56C22×SL(2,𝔽3)C22×C313yes—
57C22×SL(2,𝔽3)C22×C313yes—
58C3×C4⋊Q8—31no—
59C3×C4⋊Q8—31no—
60C3×#126—32no—
61C3×#126—32no—
62C3×#126—31no—
63C3×#126—31no—
64C3×#125—31no—
65C3×#125—31no—
66C3×#125—32no—
67C3×#125—31no—
68C3×#124—31no—
69C3×#124—31no—
70C3×#124—31no—
71C3×#123—32no—
72C3×#123—31no—
73C3×#123—43no—
74C3×#123A411yescharacteristic
75C3×C22⋊Q8—32no—
76C3×C22⋊Q8—31no—
77C3×C22⋊Q8—31no—
78C3×C22⋊Q8—31no—
79C3×C22⋊Q8—31no—
80C3×C22⋊Q8—31no—
81C3×#121—31no—
82C3×#121—31no—
83C3×#121—31no—
84C3×C4×Q8—31no—
85C3×C4×Q8—31no—
86C3×C4×Q8—31no—
87C3×C4×Q8—31no—
88C3×C4×Q8—31no—
89C3×C4×Q8C22×C311yescharacteristic
90C3×C4×Q8C22×C311yescharacteristic
91C3×C4×Q8C22×C311yescharacteristic
92C3×C4×Q8C22×C311yescharacteristic
93C3×C4×D4—32no—
94C3×C4×D4—31no—
95C3×C4×D4—31no—
96C3×C4×D4—31no—
97C3×C4×D4—31no—
98C3×C4×D4—31no—
99C3×#117—32no—
100C3×#117—31no—
101C3×#117—31no—
102C3×#117—31no—
103C2×C3×C4⋊C4—31no—
104C2×C3×C4⋊C4—31no—
105C2×C3×C42—31no—
106C4×SL(2,𝔽3)C22×C313yes—
107C4×SL(2,𝔽3)C22×C313yes—
108C4×SL(2,𝔽3)C22×C313yes—
109C4×SL(2,𝔽3)C22×C313yes—
110C23×C32—41no—
111C23×C32—41no—
112C32×D4—82no—
113C2×C32×C4—41no—
114C2×C32×C4—41no—
115C2×C32×C4—41no—
116C2×C32×C4—41no—
117C2×C32×C4—41no—
118C2×C32×C4—82no—
119C3×SL(2,𝔽3)#4111yescharacteristic
120#557C2×C3211yescharacteristic, complemented
121#556—31no—
122#554—31no—
123#553—31no—
124#552—31no—
125D4×Q8C2×C3211yescharacteristic
126#548C2×C3211yescharacteristic, complemented
127#548C2×C3211yescharacteristic, complemented
128#542—31no—
129#541—31no—
130#529—31no—
131#529—31no—
132#520C2×C3211yescharacteristic, complemented
133#519C2×C3211yescharacteristic, complemented
134C2×C4×Q8C2×C3211yescharacteristic, complemented
135C3×#41—21no—
136C3×#41—61no—
137C2×C3×Q8—21no—
138C2×C3×Q8—21no—
139C2×C3×Q8—31no—
140C2×C3×Q8—31no—
141C2×C3×Q8—31no—
142C2×C3×Q8—61no—
143C2×C3×Q8—61no—
144C2×C3×Q8C23×C311yescharacteristic
145C2×C3×D4—31no—
146C2×C3×D4—31no—
147C2×C3×D4—31no—
148C2×C3×D4—43no—
149C2×C3×D4C2×A411yescharacteristic
150C22×C3×C4—31no—
151C22×C3×C4—31no—
152C22×C3×C4—32no—
153C22×C3×C4—31no—
154C22×C3×C4—31no—
155C22×C3×C4—43no—
156C22×C3×C4C2×A411yescharacteristic
157#41⋊C3—23no—
158C2×SL(2,𝔽3)—23no—
159C2×SL(2,𝔽3)—23no—
160C2×SL(2,𝔽3)C23×C313yes—
161C3×C4⋊C4—31no—
162C3×C4⋊C4—31no—
163C3×C4⋊C4—31no—
164C3×C4⋊C4—31no—
165C3×C4⋊C4—32no—
166C3×C4⋊C4—32no—
167C3×C4⋊C4—32no—
168C3×C4⋊C4—31no—
169C3×C4⋊C4—31no—
170C3×C4⋊C4—31no—
171C3×C4⋊C4—31no—
172C3×C4⋊C4—31no—
173C3×C4⋊C4—31no—
174C3×C4⋊C4—31no—
175C3×C4⋊C4—31no—
176C3×C4⋊C4—43no—
177C3×C4⋊C4—43no—
178C3×C4⋊C4C2×A411yescharacteristic
179C3×C4⋊C4C2×A411yescharacteristic
180C3×C22⋊C4—32no—
181C3×C22⋊C4—32no—
182C3×C22⋊C4—31no—
183C3×C22⋊C4—31no—
184C3×C22⋊C4—31no—
185C3×C22⋊C4—31no—
186C3×C22⋊C4—31no—
187C3×C22⋊C4—31no—
188C3×C22⋊C4—31no—
189C3×C22⋊C4—43no—
190C3×C22⋊C4—43no—
191C3×C22⋊C4—43no—
192C3×C22⋊C4C2×A411yescharacteristic
193C3×C22⋊C4C2×A411yescharacteristic
194C3×C22⋊C4C2×A411yescharacteristic
195C3×C42—32no—
196C3×C42—31no—
197C3×C42—31no—
198C3×C42—31no—
199C3×C42—31no—
200C3×C42—31no—
201C3×C42—31no—
202C22×C32—42no—
203C22×C32—41no—
204C22×C32—81no—
205C22×C32—81no—
206C22×C32—81no—
207C22×C32—81no—
208C22×C32—81no—
209C32×C4—81no—
210C32×C4—81no—
211C32×C4—81no—
212C32×C4—81no—
213C32×C4—81no—
214C2×#41C22×C3211yescharacteristic
215C22×Q8C22×C3211yescharacteristic
216C22×Q8C22×C3211yescharacteristic
217C4⋊Q8—31no—
218C4⋊Q8—31no—
219#126—32no—
220#126—32no—
221#126—31no—
222#126—31no—
223#125—31no—
224#125—31no—
225#125—32no—
226#125—31no—
227#124—31no—
228#124—31no—
229#124—31no—
230#123—32no—
231#123—31no—
232#123C3×A411yescharacteristic
233C22⋊Q8—32no—
234C22⋊Q8—31no—
235C22⋊Q8—31no—
236C22⋊Q8—31no—
237C22⋊Q8—31no—
238C22⋊Q8—31no—
239#121—31no—
240#121—31no—
241#121—31no—
242C4×Q8—31no—
243C4×Q8—31no—
244C4×Q8—31no—
245C4×Q8—31no—
246C4×Q8—31no—
247C4×Q8C22×C3211yescharacteristic
248C4×Q8C22×C3211yescharacteristic
249C4×Q8C22×C3211yescharacteristic
250C4×Q8C22×C3211yescharacteristic
251C4×D4—32no—
252C4×D4—31no—
253C4×D4—31no—
254C4×D4—31no—
255C4×D4—31no—
256C4×D4—31no—
257#117—32no—
258#117—31no—
259#117—31no—
260#117—31no—
261C2×C4⋊C4—31no—
262C2×C4⋊C4—31no—
263C2×C42—31no—
264C23×C3—31no—
265C23×C3—43no—
266C23×C3—43no—
267C23×C3C22×A411yescharacteristic
268C23×C3C22×A411yescharacteristic
269C3×Q8—31no—
270C3×Q8—61no—
271C3×Q8—61no—
272C3×Q8—61no—
273C3×Q8—61no—
274C3×Q8—121no—
275C3×Q8C3×#4111yescharacteristic
276C3×D4—22no—
277C3×D4—61no—
278C3×D4—61no—
279C3×D4—62no—
280C3×D4—86no—
281C3×D4—121no—
282C2×C3×C4—22no—
283C2×C3×C4—31no—
284C2×C3×C4—31no—
285C2×C3×C4—31no—
286C2×C3×C4—31no—
287C2×C3×C4—31no—
288C2×C3×C4—31no—
289C2×C3×C4—31no—
290C2×C3×C4—43no—
291C2×C3×C4—43no—
292C2×C3×C4—43no—
293C2×C3×C4—43no—
294C2×C3×C4—43no—
295C2×C3×C4—61no—
296C2×C3×C4—61no—
297C2×C3×C4—61no—
298C2×C3×C4—61no—
299C2×C3×C4—61no—
300C2×C3×C4—61no—
301C2×C3×C4—86no—
302C2×C3×C4—121no—
303C2×C3×C4—122no—
304C2×C3×C4C22×A411yescharacteristic
305C2×C3×C4C22×A411yescharacteristic
306C2×C3×C4C22×A411yescharacteristic
307C2×C3×C4C22×A411yescharacteristic
308C2×C3×C4C22×A411yescharacteristic
309SL(2,𝔽3)C3×#4113yes—
310C2×C32—41no—
311C2×C32—41no—
312C2×C32—41no—
313C2×C32—82no—
314C2×C32—161no—
315#41—21no—
316#41—61no—
317C2×Q8—21no—
318C2×Q8—21no—
319C2×Q8—31no—
320C2×Q8—31no—
321C2×Q8—31no—
322C2×Q8—61no—
323C2×Q8—61no—
324C2×Q8C23×C3211yescharacteristic
325C2×D4—31no—
326C2×D4—31no—
327C2×D4—31no—
328C2×D4C2×C3×A411yescharacteristic
329C22×C4—31no—
330C22×C4—31no—
331C22×C4—32no—
332C22×C4—31no—
333C22×C4—31no—
334C22×C4C2×C3×A411yescharacteristic
335C4⋊C4—31no—
336C4⋊C4—31no—
337C4⋊C4—31no—
338C4⋊C4—31no—
339C4⋊C4—32no—
340C4⋊C4—32no—
341C4⋊C4—32no—
342C4⋊C4—31no—
343C4⋊C4—31no—
344C4⋊C4—31no—
345C4⋊C4—31no—
346C4⋊C4—31no—
347C4⋊C4—31no—
348C4⋊C4—31no—
349C4⋊C4—31no—
350C4⋊C4C2×C3×A411yescharacteristic
351C4⋊C4C2×C3×A411yescharacteristic
352C22⋊C4—32no—
353C22⋊C4—32no—
354C22⋊C4—31no—
355C22⋊C4—31no—
356C22⋊C4—31no—
357C22⋊C4—31no—
358C22⋊C4—31no—
359C22⋊C4—31no—
360C22⋊C4—31no—
361C22⋊C4C2×C3×A411yescharacteristic
362C22⋊C4C2×C3×A411yescharacteristic
363C22⋊C4C2×C3×A411yescharacteristic
364C42—32no—
365C42—31no—
366C42—31no—
367C42—31no—
368C42—31no—
369C42—31no—
370C42—31no—
371C22×C3—21no—
372C22×C3—21no—
373C22×C3—21no—
374C22×C3—21no—
375C22×C3—21no—
376C22×C3—46no—
377C22×C3—43no—
378C22×C3—61no—
379C22×C3—61no—
380C22×C3—61no—
381C22×C3—83no—
382C22×C3—83no—
383C22×C3—83no—
384C22×C3—83no—
385C22×C3—83no—
386C22×C32−1+4⋊C312yes—
387C22×C3C23×A411yescharacteristic
388C3×C4—21no—
389C3×C4—21no—
390C3×C4—21no—
391C3×C4—21no—
392C3×C4—21no—
393C3×C4—31no—
394C3×C4—31no—
395C3×C4—61no—
396C3×C4—61no—
397C3×C4—61no—
398C3×C4—61no—
399C3×C4—61no—
400C3×C4—61no—
401C3×C4—83no—
402C3×C4—83no—
403C3×C4—83no—
404C3×C4—83no—
405C3×C4—83no—
406C32—41noSylow
407C23—31no—
408C23C22×C3×A411yescharacteristic
409C23C22×C3×A411yescharacteristic
410Q8—31no—
411Q8—61no—
412Q8—61no—
413Q8—61no—
414Q8—61no—
415Q8—121no—
416Q8C32×#4111yescharacteristic
417D4—22no—
418D4—61no—
419D4—61no—
420D4—62no—
421D4—121no—
422C2×C4—22no—
423C2×C4—31no—
424C2×C4—31no—
425C2×C4—31no—
426C2×C4—31no—
427C2×C4—31no—
428C2×C4—31no—
429C2×C4—31no—
430C2×C4—61no—
431C2×C4—61no—
432C2×C4—61no—
433C2×C4—61no—
434C2×C4—61no—
435C2×C4—61no—
436C2×C4—121no—
437C2×C4—122no—
438C2×C4C22×C3×A411yescharacteristic
439C2×C4C22×C3×A411yescharacteristic
440C2×C4C22×C3×A411yescharacteristic
441C2×C4C22×C3×A411yescharacteristic
442C2×C4C22×C3×A411yescharacteristic
443C2×C3—22no—
444C2×C3—41no—
445C2×C3—43no—
446C2×C3—43no—
447C2×C3—43no—
448C2×C3—86no—
449C2×C3—121no—
450C2×C3—163no—
451C2×C3A4×#4111yescharacteristic
452C2×C3C2×2−1+4⋊C311yescharacteristic
453C2×C3#594511yescharacteristic
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—61no—
460C22—61no—
461C22—61no—
462C22C3×2−1+4⋊C312yes—
463C22C23×C3×A411yescharacteristic
464C4—21no—
465C4—21no—
466C4—21no—
467C4—21no—
468C4—21no—
469C4—31no—
470C4—31no—
471C4—61no—
472C4—61no—
473C4—61no—
474C4—61no—
475C4—61no—
476C4—61no—
477C3—43no—
478C3#3516⋊C311yescharacteristic, direct product factor
479C2—22no—
480C2—41no—
481C2—121no—
482C2C3×A4×#4111yescharacteristic
483C2C2×C3×2−1+4⋊C311yescharacteristic
484C2C3×#594511yescharacteristic
subgroupG/H
center, socleC22×C3C23×A4
commutator, cosocleC2×Q8C23×C32
FrattiniC22C23×C3×A4
FittingC3×#3516C3
hypercenterC3×#123A4
hypocenterQ8C32×#41
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C22×C3 ◁ C3×#123

Complements

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

NH ≅ G/Ntype
1. C3×#557⋊C3C2semidirect
2. C3×D4×SL(2,𝔽3)C2semidirect
3. C3×#548⋊C3C2semidirect
4. C3×#548⋊C3C2semidirect
5. C3×#520⋊C3C2semidirect
6. C3×#519⋊C3C2semidirect
7. C2×C3×C4×SL(2,𝔽3)C2semidirect
8. C3×#3516C3semidirect
9. #3516⋊C3C3direct (N × H)
18. C3×#557C2×C3semidirect
24. C3×#548C2×C3semidirect
25. C3×#548C2×C3semidirect
30. C3×#520C2×C3semidirect
31. C3×#519C2×C3semidirect
32. C2×C3×C4×Q8C2×C3semidirect
33. #557⋊C3C2×C3semidirect
34. D4×SL(2,𝔽3)C2×C3semidirect
35. #548⋊C3C2×C3semidirect
36. #548⋊C3C2×C3semidirect
37. #520⋊C3C2×C3semidirect
38. #519⋊C3C2×C3semidirect
39. C2×C4×SL(2,𝔽3)C2×C3semidirect
51. #3516C32semidirect
120. #557C2×C32semidirect
126. #548C2×C32semidirect
127. #548C2×C32semidirect
132. #520C2×C32semidirect
133. #519C2×C32semidirect
134. C2×C4×Q8C2×C32semidirect
478. C3#3516⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#557⋊C32
C3×D4×SL(2,𝔽3)2
C3×#548⋊C32
C3×#520⋊C32
C3×#519⋊C32
C2×C3×C4×SL(2,𝔽3)2
C3×#35163
#3516⋊C33
C32×#1234