Finite Group Databasebeta

← C3×#85382 #11922669

Subgroups — C3×#85382 #11922669

Order 1152.

675 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 675 conjugacy classes of subgroups — too many to draw. Only its 72 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×#986C211yesmaximal, characteristic, complemented
2C22×C3×GL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C3×#548⋊C3C211yesmaximal, characteristic, complemented
4C3×#5427C211yesmaximal, characteristic, complemented
5C3×#5421C211yesmaximal, characteristic, complemented
6C3×#5394C211yesmaximal, characteristic, complemented
7C3×C4⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
8#85382C311yesmaximal, characteristic, direct product factor
9C3×#3265—31nomaximal
10C2×C3×#41⋊C3C2211yescharacteristic, complemented
11C22×C3×SL(2,𝔽3)C2211yescharacteristic, complemented
12C3×#986—22no—
13C2×C3×GL(2,𝔽3)—22no—
14C2×C3×GL(2,𝔽3)C2211yescharacteristic
15C2×C3×GL(2,𝔽3)C2211yescharacteristic, complemented
16C2×C3×GL(2,𝔽3)C2211yescharacteristic, complemented
17C3×#882—41nomaximal
18C3×C4×SL(2,𝔽3)C2211yescharacteristic, complemented
19C3×#859C2211yescharacteristic, complemented
20C2×#986C2×C311yescharacteristic, complemented
21C22×GL(2,𝔽3)C2×C311yescharacteristic, complemented
22C2×C3×AGL(1,ℤ/8)—31no—
23C22×C3×SD16—31no—
24C3×#548D311yescharacteristic, complemented
25C3×D42—31no—
26#548⋊C3—21no—
27#548⋊C3C2×C311yescharacteristic, complemented
28#5427C2×C311yescharacteristic, complemented
29#5421C2×C311yescharacteristic, complemented
30#5394C2×C311yescharacteristic, complemented
31C4⋊GL(2,𝔽3)C2×C311yescharacteristic, complemented
32C3×#496—31no—
33C3×#468—31no—
34C3×#466—31no—
35C3×#463—31no—
36C3×#460—31no—
37C3×#450—31no—
38C3×#449—31no—
39C3×#448—31no—
40C3×#447—31no—
41C3×#440—31no—
42C3×#435—31no—
43C23×C3×D3—41no—
44C2×C3×#67—41no—
45C2×C3×D12—41no—
46C2×C3×D12—41no—
47C3×#41⋊C3—21no—
48C2×C3×SL(2,𝔽3)—21no—
49C2×C3×SL(2,𝔽3)C2311yescharacteristic
50C3×GL(2,𝔽3)—21no—
51C3×GL(2,𝔽3)—21no—
52C3×GL(2,𝔽3)D412yescomplemented
53C32×C22⋊C4—41no—
54C3×(C2×D3)⋊C4—41no—
55C3×(C2×D3)⋊C4—41no—
56#3265—31noSylow
57C2×C3×#41C2×D311yescharacteristic, complemented
58C22×C3×Q8C2×D311yescharacteristic
59C22×C3×D4—31no—
60C22×C3×D4—31no—
61C22×C3×D4—61no—
62C2×#41⋊C3—21no—
63C2×#41⋊C3C22×C311yescharacteristic, complemented
64C22×SL(2,𝔽3)—21no—
65C22×SL(2,𝔽3)C22×C311yescharacteristic, complemented
66#986—22no—
67C2×GL(2,𝔽3)—22no—
68C2×GL(2,𝔽3)C22×C311yescharacteristic
69C2×GL(2,𝔽3)C22×C311yescharacteristic, complemented
70C2×GL(2,𝔽3)C22×C311yescharacteristic, complemented
71C3×AGL(1,ℤ/8)—62no—
72C3×AGL(1,ℤ/8)—62no—
73C2×C3×SD16—31no—
74C2×C3×SD16—31no—
75C2×C3×SD16—31no—
76C2×C3×SD16—31no—
77C2×C3×SD16—31no—
78C2×C3×SD16—31no—
79C2×C3×SD16—62no—
80C2×C3×SD16—62no—
81C2×C3×D8—31no—
82C2×C3×D8—31no—
83C2×C3×D8—31no—
84C2×C3×M16—31no—
85C22×C3×C8—31no—
86C3×#127—31no—
87C3×#124—31no—
88C3×C22⋊Q8—31no—
89C3×#121—31no—
90C3×#121—31no—
91C3×#121—31no—
92C3×#121—61no—
93C3×#120—61no—
94C3×#120—61no—
95C3×C4×Q8C2×D311yescharacteristic, complemented
96C3×C4×D4—31no—
97C3×C4×D4—31no—
98#882—41no—
99C4×SL(2,𝔽3)—21no—
100C4×SL(2,𝔽3)C22×C311yescharacteristic, complemented
101#859C22×C311yescharacteristic, complemented
102C3×#107—31no—
103C3×C4⋊C8—31no—
104C3×#103—31no—
105C3×#103—31no—
106C3×#102—31no—
107C3×#102—31no—
108C3×#102—31no—
109C3×#102—31no—
110C3×C22⋊C8—31no—
111C3×C22⋊C8—31no—
112C3×#97—31no—
113C23×C32—41no—
114C22×C3×D3—41no—
115C22×C3×D3—41no—
116C22×C3×D3—41no—
117C22×C3×D3—82no—
118C22×C3×D3—82no—
119C22×C3×D3—82no—
120C2×C32×C4—41no—
121C2×C32×C4—41no—
122C3×#67—82no—
123C2×C3×Dic3—41no—
124C3×D12—82no—
125C3×D12—82no—
126C3×SL(2,𝔽3)C2×D411yescharacteristic, complemented
127C2×AGL(1,ℤ/8)—31no—
128C22×SD16—31no—
129#548C3×D311yescharacteristic, complemented
130D42—31no—
131#496—31no—
132#468—31no—
133#466—31no—
134#463—31no—
135#460—31no—
136#450—31no—
137#449—31no—
138#448—31no—
139#447—31no—
140#440—31no—
141#435—31no—
142C24×C3—61no—
143C24×C3—61no—
144C23×D3—41no—
145C3×#41—21no—
146C3×#41—61no—
147C2×C3×Q8—21no—
148C2×C3×Q8—31no—
149C2×C3×Q8—61no—
150C2×C3×Q8C22×D311yescharacteristic
151C2×C3×D4—31no—
152C2×C3×D4—31no—
153C2×C3×D4—31no—
154C2×C3×D4—31no—
155C2×C3×D4—31no—
156C2×C3×D4—61no—
157C2×C3×D4—61no—
158C2×C3×D4—62no—
159C2×C3×D4—62no—
160C2×C3×D4—61no—
161C2×C3×D4—61no—
162C2×C3×D4—61no—
163C2×C3×D4—61no—
164C2×C3×D4—121no—
165C2×C3×D4—121no—
166C2×C3×D4—124no—
167C22×C3×C4—31no—
168C22×C3×C4—31no—
169C22×C3×C4—61no—
170C2×#67—41no—
171C2×D12—41no—
172C2×D12—41no—
173#41⋊C3—21no—
174#41⋊C3—41no—
175C2×SL(2,𝔽3)—21no—
176C2×SL(2,𝔽3)—21no—
177C2×SL(2,𝔽3)—41no—
178C2×SL(2,𝔽3)C23×C311yescharacteristic
179GL(2,𝔽3)—21no—
180GL(2,𝔽3)—21no—
181GL(2,𝔽3)C3×D412yescomplemented
182C3×SD16—32no—
183C3×SD16—32no—
184C3×SD16—62no—
185C3×SD16—62no—
186C3×SD16—61no—
187C3×SD16—61no—
188C3×SD16—61no—
189C3×SD16—61no—
190C3×SD16—121no—
191C3×D8—62no—
192C3×D8—62no—
193C3×D8—121no—
194C3×M16—62no—
195C2×C3×C8—31no—
196C2×C3×C8—31no—
197C2×C3×C8—31no—
198C2×C3×C8—31no—
199C2×C3×C8—62no—
200C3×C4⋊C4—31no—
201C3×C4⋊C4—31no—
202C3×C4⋊C4—31no—
203C3×C22⋊C4—31no—
204C3×C22⋊C4—41no—
205C3×C22⋊C4—61no—
206C3×C22⋊C4—61no—
207C3×C22⋊C4—61no—
208C3×C22⋊C4—61no—
209C3×C22⋊C4—81no—
210C3×C22⋊C4S411yescharacteristic
211C3×C42—31no—
212(C2×D3)⋊C4—41no—
213(C2×D3)⋊C4—41no—
214C22×C32—42no—
215C22×C32—41no—
216C22×C32—81no—
217C22×C32—81no—
218C2×C3×D3—42no—
219C2×C3×D3—42no—
220C2×C3×D3—81no—
221C2×C3×D3—81no—
222C2×C3×D3—81no—
223C2×C3×D3—81no—
224C2×C3×D3—81no—
225C2×C3×D3—81no—
226C2×C3×D3—81no—
227C2×C3×D3—88no—
228C32×C4—81no—
229C32×C4—81no—
230C3×Dic3—81no—
231C2×#41C2×C3×D311yescharacteristic, complemented
232C22×Q8C2×C3×D311yescharacteristic
233C22×D4—31no—
234C22×D4—31no—
235C22×D4—61no—
236AGL(1,ℤ/8)—62no—
237AGL(1,ℤ/8)—62no—
238C2×SD16—31no—
239C2×SD16—31no—
240C2×SD16—31no—
241C2×SD16—31no—
242C2×SD16—31no—
243C2×SD16—31no—
244C2×SD16—62no—
245C2×SD16—62no—
246C2×D8—31no—
247C2×D8—31no—
248C2×D8—31no—
249C2×M16—31no—
250C22×C8—31no—
251#127—31no—
252#124—31no—
253C22⋊Q8—31no—
254#121—31no—
255#121—31no—
256#121—31no—
257#121—61no—
258#120—61no—
259#120—61no—
260C4×Q8C2×C3×D311yescharacteristic, complemented
261C4×D4—31no—
262C4×D4—31no—
263#107—31no—
264C4⋊C8—31no—
265#103—31no—
266#103—31no—
267#102—31no—
268#102—31no—
269#102—31no—
270#102—31no—
271C22⋊C8—31no—
272C22⋊C8—31no—
273#97—31no—
274C23×C3—31no—
275C23×C3—41no—
276C23×C3—61no—
277C23×C3—61no—
278C23×C3—61no—
279C23×C3—81no—
280C23×C3—122no—
281C23×C3—122no—
282C23×C3—122no—
283C23×C3—122no—
284C23×C3—241no—
285C23×C3—241no—
286C23×C3C2×S411yescharacteristic
287C22×D3—41no—
288C22×D3—41no—
289C22×D3—41no—
290C22×D3—82no—
291C22×D3—82no—
292C22×D3—82no—
293C3×Q8—31no—
294C3×Q8—61no—
295C3×Q8—61no—
296C3×Q8D3×D411yescharacteristic
297C3×D4—32no—
298C3×D4—61no—
299C3×D4—61no—
300C3×D4—61no—
301C3×D4—61no—
302C3×D4—61no—
303C3×D4—61no—
304C3×D4—61no—
305C3×D4—62no—
306C3×D4—122no—
307C3×D4—122no—
308C3×D4—122no—
309C3×D4—122no—
310C3×D4—122no—
311C3×D4—241no—
312C2×C3×C4—31no—
313C2×C3×C4—31no—
314C2×C3×C4—31no—
315C2×C3×C4—41no—
316C2×C3×C4—41no—
317C2×C3×C4—61no—
318C2×C3×C4—61no—
319C2×C3×C4—61no—
320C2×C3×C4—61no—
321C2×C3×C4—61no—
322C2×C3×C4—81no—
323C2×C3×C4—81no—
324C2×C3×C4—122no—
325C2×C3×C4C2×S411yescharacteristic
326C2×C3×C4C2×S411yescharacteristic
327#67—82no—
328C2×Dic3—41no—
329D12—82no—
330D12—82no—
331SL(2,𝔽3)—21no—
332SL(2,𝔽3)C2×C3×D411yescharacteristic, complemented
333C3×C8—32no—
334C3×C8—61no—
335C3×C8—61no—
336C3×C8—61no—
337C2×C32—41no—
338C2×C32—41no—
339C2×C32—41no—
340C2×C32—82no—
341C3×D3—82no—
342C3×D3—82no—
343C3×D3—161no—
344C24—61no—
345C24—61no—
346#41—21no—
347#41—61no—
348C2×Q8—21no—
349C2×Q8—31no—
350C2×Q8—61no—
351C2×Q8C22×C3×D311yescharacteristic
352C2×D4—31no—
353C2×D4—31no—
354C2×D4—31no—
355C2×D4—31no—
356C2×D4—31no—
357C2×D4—61no—
358C2×D4—61no—
359C2×D4—62no—
360C2×D4—62no—
361C2×D4—61no—
362C2×D4—61no—
363C2×D4—61no—
364C2×D4—61no—
365C2×D4—121no—
366C2×D4—121no—
367C2×D4—124no—
368C22×C4—31no—
369C22×C4—31no—
370C22×C4—61no—
371SD16—32no—
372SD16—32no—
373SD16—62no—
374SD16—62no—
375SD16—61no—
376SD16—61no—
377SD16—61no—
378SD16—61no—
379SD16—121no—
380D8—62no—
381D8—62no—
382D8—121no—
383M16—62no—
384C2×C8—31no—
385C2×C8—31no—
386C2×C8—31no—
387C2×C8—31no—
388C2×C8—62no—
389C4⋊C4—31no—
390C4⋊C4—31no—
391C4⋊C4—31no—
392C22⋊C4—31no—
393C22⋊C4—61no—
394C22⋊C4—61no—
395C22⋊C4—61no—
396C22⋊C4—61no—
397C22⋊C4C3×S411yescharacteristic
398C42—31no—
399C22×C3—21no—
400C22×C3—21no—
401C22×C3—42no—
402C22×C3—41no—
403C22×C3—61no—
404C22×C3—62no—
405C22×C3—61no—
406C22×C3—61no—
407C22×C3—81no—
408C22×C3—81no—
409C22×C3—82no—
410C22×C3—81no—
411C22×C3—121no—
412C22×C3—121no—
413C22×C3—121no—
414C22×C3—121no—
415C22×C3—121no—
416C22×C3—121no—
417C22×C3—121no—
418C22×C3—121no—
419C22×C3—161no—
420C22×C3—161no—
421C22×C3—244no—
422C22×C3—244no—
423C22×C3#98612yescomplemented
424C22×C3C22×S411yescharacteristic
425C2×D3—42no—
426C2×D3—42no—
427C2×D3—81no—
428C2×D3—81no—
429C2×D3—81no—
430C2×D3—81no—
431C2×D3—81no—
432C2×D3—81no—
433C2×D3—81no—
434C2×D3—88no—
435C3×C4—21no—
436C3×C4—21no—
437C3×C4—31no—
438C3×C4—31no—
439C3×C4—61no—
440C3×C4—61no—
441C3×C4—81no—
442C3×C4—81no—
443C3×C4—121no—
444C3×C4—161no—
445C3×C4—161no—
446Dic3—81no—
447C32—41noSylow
448C23—31no—
449C23—61no—
450C23—61no—
451C23—61no—
452C23—122no—
453C23—122no—
454C23—122no—
455C23—122no—
456C23—241no—
457C23—241no—
458C23C2×C3×S411yescharacteristic
459Q8—31no—
460Q8—61no—
461Q8—61no—
462Q8C3×D3×D411yescharacteristic
463D4—32no—
464D4—61no—
465D4—61no—
466D4—61no—
467D4—61no—
468D4—61no—
469D4—61no—
470D4—61no—
471D4—62no—
472D4—122no—
473D4—122no—
474D4—122no—
475D4—122no—
476D4—122no—
477D4—241no—
478C2×C4—31no—
479C2×C4—31no—
480C2×C4—31no—
481C2×C4—61no—
482C2×C4—61no—
483C2×C4—61no—
484C2×C4—61no—
485C2×C4—61no—
486C2×C4—122no—
487C2×C4C2×C3×S411yescharacteristic
488C2×C4C2×C3×S411yescharacteristic
489C8—32no—
490C8—61no—
491C8—61no—
492C8—61no—
493C2×C3—22no—
494C2×C3—41no—
495C2×C3—41no—
496C2×C3—41no—
497C2×C3—82no—
498C2×C3—81no—
499C2×C3—81no—
500C2×C3—81no—
501C2×C3—121no—
502C2×C3—122no—
503C2×C3—162no—
504C2×C3—241no—
505C2×C3—241no—
506C2×C3D4×S411yescharacteristic
507C2×C3C2×#98611yescharacteristic
508C2×C3#592611yescharacteristic
509D3—82no—
510D3—82no—
511D3—161no—
512C22—21no—
513C22—21no—
514C22—61no—
515C22—62no—
516C22—61no—
517C22—61no—
518C22—121no—
519C22—121no—
520C22—121no—
521C22—121no—
522C22—121no—
523C22—121no—
524C22—121no—
525C22—121no—
526C22—244no—
527C22—244no—
528C22C3×#98612yescomplemented
529C22C22×C3×S411yescharacteristic
530C4—21no—
531C4—21no—
532C4—31no—
533C4—31no—
534C4—61no—
535C4—61no—
536C4—121no—
537C3—41no—
538C3—81no—
539C3#8538211yescharacteristic, direct product factor
540C2—22no—
541C2—121no—
542C2—122no—
543C2—241no—
544C2—241no—
545C2C3×D4×S411yescharacteristic
546C2C2×C3×#98611yescharacteristic
547C2C3×#592611yescharacteristic
subgroupG/H
center, socleC22×C3C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23×C3
FrattiniC22C22×C3×S4
FittingC3×#548D3
hypercenterC3×C22⋊C4S4
hypocenterSL(2,𝔽3)C2×C3×D4
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#986C2semidirect
2. C22×C3×GL(2,𝔽3)C2semidirect
3. C3×#548⋊C3C2semidirect
4. C3×#5427C2semidirect
5. C3×#5421C2semidirect
6. C3×#5394C2semidirect
7. C3×C4⋊GL(2,𝔽3)C2semidirect
8. #85382C3direct (N × H)
10. C2×C3×#41⋊C3C22semidirect
11. C22×C3×SL(2,𝔽3)C22semidirect
15. C2×C3×GL(2,𝔽3)C22semidirect
16. C2×C3×GL(2,𝔽3)C22semidirect
18. C3×C4×SL(2,𝔽3)C22semidirect
19. C3×#859C22semidirect
20. C2×#986C2×C3semidirect
21. C22×GL(2,𝔽3)C2×C3semidirect
24. C3×#548D3semidirect
27. #548⋊C3C2×C3semidirect
28. #5427C2×C3semidirect
29. #5421C2×C3semidirect
30. #5394C2×C3semidirect
31. C4⋊GL(2,𝔽3)C2×C3semidirect
52. C3×GL(2,𝔽3)D4semidirect
57. C2×C3×#41C2×D3semidirect
63. C2×#41⋊C3C22×C3semidirect
65. C22×SL(2,𝔽3)C22×C3semidirect
69. C2×GL(2,𝔽3)C22×C3semidirect
70. C2×GL(2,𝔽3)C22×C3semidirect
95. C3×C4×Q8C2×D3semidirect
100. C4×SL(2,𝔽3)C22×C3semidirect
101. #859C22×C3semidirect
126. C3×SL(2,𝔽3)C2×D4semidirect
129. #548C3×D3semidirect
181. GL(2,𝔽3)C3×D4semidirect
231. C2×#41C2×C3×D3semidirect
260. C4×Q8C2×C3×D3semidirect
332. SL(2,𝔽3)C2×C3×D4semidirect
423. C22×C3#986semidirect
528. C22C3×#986semidirect
539. C3#85382direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#9862
C22×C3×GL(2,𝔽3)2
C3×#548⋊C32
C3×#54272
C3×#54212
C3×#53942
C3×C4⋊GL(2,𝔽3)2
#853823
C3×#32653
C3×#8824