Finite Group Databasebeta

← C3×#85384 #11922671

Subgroups — C3×#85384 #11922671

Order 1152.

591 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 591 conjugacy classes of subgroups — too many to draw. Only its 68 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
2C2×C3×#983C211yesmaximal, characteristic, complemented
3C3×#548⋊C3C211yesmaximal, characteristic, complemented
4C3×#5425C211yesmaximal, characteristic, complemented
5C3×#5421C211yesmaximal, characteristic, complemented
6C3×#5395C211yesmaximal, characteristic, complemented
7C3×C4×GL(2,𝔽3)C211yesmaximal, characteristic, complemented
8#85384C311yesmaximal, characteristic, direct product factor
9C3×#3275—31nomaximal
10C2×C3×#41⋊C3C2211yescharacteristic
11C22×C3×SL(2,𝔽3)C2211yescharacteristic, complemented
12C3×#986—22no—
13C3×#983—22no—
14C2×C3×GL(2,𝔽3)C2211yescharacteristic
15C2×C3×GL(2,𝔽3)C2211yescharacteristic
16C2×C3×#226C2211yescharacteristic, complemented
17C3×#884—41nomaximal
18C3×C4×SL(2,𝔽3)C2211yescharacteristic, complemented
19C3×#859C2211yescharacteristic, complemented
20C2×#986C2×C311yescharacteristic, complemented
21C2×#983C2×C311yescharacteristic, complemented
22C2×C3×#137—31no—
23C2×C3×AGL(1,ℤ/8)—31no—
24C3×#548D311yescharacteristic, complemented
25C3×#546—31no—
26#548⋊C3—21no—
27#548⋊C3C2×C311yescharacteristic, complemented
28#5425C2×C311yescharacteristic, complemented
29#5421C2×C311yescharacteristic, complemented
30#5395C2×C311yescharacteristic, complemented
31C4×GL(2,𝔽3)C2×C311yescharacteristic, complemented
32C3×#495—31no—
33C3×#470—31no—
34C3×#469—31no—
35C3×#464—31no—
36C3×#463—31no—
37C3×#452—31no—
38C3×#450—31no—
39C3×#449—31no—
40C3×#448—31no—
41C3×C4×SD16—31no—
42C3×#436—31no—
43C2×C3×#67—41no—
44C2×C3×#67—41no—
45C2×C3×D12—41no—
46C2×C3×C4×D3—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)D412yescomplemented
52C3×#226—21no—
53C32×C22⋊C4—41no—
54C3×(C2×D3)⋊C4—41no—
55C3×Dic3⋊C4—41no—
56#3275—31noSylow
57C2×C3×#41—31no—
58C2×C3×#41C2×D311yescharacteristic
59C22×C3×Q8C2×D311yescharacteristic
60C22×C3×D4—31no—
61C2×#41⋊C3—21no—
62C2×#41⋊C3C22×C311yescharacteristic
63C22×SL(2,𝔽3)—21no—
64C22×SL(2,𝔽3)C22×C311yescharacteristic, complemented
65#986—22no—
66#983—22no—
67C2×GL(2,𝔽3)C22×C311yescharacteristic
68C2×GL(2,𝔽3)C22×C311yescharacteristic
69C2×#226C22×C311yescharacteristic, complemented
70C3×#137—62no—
71C3×#137—62no—
72C3×AGL(1,ℤ/8)—62no—
73C3×AGL(1,ℤ/8)—62no—
74C2×C3×Q16—31no—
75C2×C3×Q16—31no—
76C2×C3×SD16—31no—
77C2×C3×SD16—31no—
78C2×C3×SD16—31no—
79C2×C3×SD16—31no—
80C2×C3×SD16—31no—
81C2×C3×D8—31no—
82C2×C3×D8—31no—
83C2×C3×M16—31no—
84C2×C3×M16—31no—
85C3×#124—31no—
86C3×#124—31no—
87C3×#123—61no—
88C3×C22⋊Q8—31no—
89C3×C22⋊Q8—31no—
90C3×#121—31no—
91C3×#121—31no—
92C3×#121—61no—
93C3×#120—61no—
94C3×C4×Q8C2×D311yescharacteristic
95C3×C4×D4—31no—
96C3×C4×D4—31no—
97C2×C3×C22⋊C4—61no—
98#884—41no—
99C4×SL(2,𝔽3)—21no—
100C4×SL(2,𝔽3)C22×C311yescharacteristic, complemented
101#859C22×C311yescharacteristic, complemented
102C3×#106—31no—
103C3×C4⋊C8—31no—
104C3×#103—31no—
105C3×#103—31no—
106C3×#103—31no—
107C3×#102—31no—
108C3×#102—31no—
109C3×#102—31no—
110C3×C22⋊C8—31no—
111C3×C22⋊C8—31no—
112C3×C4×C8—31no—
113C23×C32—41no—
114C22×C3×D3—41no—
115C22×C3×D3—41no—
116C2×C32×C4—41no—
117C2×C32×C4—41no—
118C3×#67—82no—
119C3×#67—82no—
120C2×C3×Dic3—41no—
121C2×C3×Dic3—41no—
122C3×D12—82no—
123C3×C4×D3—82no—
124C3×SL(2,𝔽3)C2×D411yescharacteristic
125C2×#137—31no—
126C2×AGL(1,ℤ/8)—31no—
127#548C3×D311yescharacteristic, complemented
128#546—31no—
129#495—31no—
130#470—31no—
131#469—31no—
132#464—31no—
133#463—31no—
134#452—31no—
135#450—31no—
136#449—31no—
137#448—31no—
138C4×SD16—31no—
139#436—31no—
140C24×C3—61no—
141C3×#41—21no—
142C3×#41—61no—
143C3×#41—62no—
144C3×#41—121no—
145C2×C3×Q8—21no—
146C2×C3×Q8—31no—
147C2×C3×Q8—31no—
148C2×C3×Q8—61no—
149C2×C3×Q8C22×D311yescharacteristic
150C2×C3×D4—31no—
151C2×C3×D4—31no—
152C2×C3×D4—31no—
153C2×C3×D4—31no—
154C2×C3×D4—62no—
155C2×C3×D4—61no—
156C2×C3×D4—61no—
157C2×C3×D4—61no—
158C2×C3×D4—121no—
159C22×C3×C4—31no—
160C22×C3×C4—31no—
161C22×C3×C4—61no—
162C22×C3×C4—61no—
163C2×#67—41no—
164C2×#67—41no—
165C2×D12—41no—
166C2×C4×D3—41no—
167#41⋊C3—21no—
168#41⋊C3—41no—
169C2×SL(2,𝔽3)—21no—
170C2×SL(2,𝔽3)—21no—
171C2×SL(2,𝔽3)—41no—
172C2×SL(2,𝔽3)C23×C311yescharacteristic
173GL(2,𝔽3)—21no—
174GL(2,𝔽3)C3×D412yescomplemented
175#226—21no—
176C3×Q16—62no—
177C3×Q16—61no—
178C3×Q16—61no—
179C3×SD16—32no—
180C3×SD16—32no—
181C3×SD16—62no—
182C3×SD16—61no—
183C3×SD16—61no—
184C3×SD16—121no—
185C3×SD16—121no—
186C3×D8—62no—
187C3×D8—62no—
188C3×M16—62no—
189C3×M16—62no—
190C2×C3×C8—31no—
191C2×C3×C8—31no—
192C2×C3×C8—31no—
193C2×C3×C8—31no—
194C3×C4⋊C4—31no—
195C3×C4⋊C4—31no—
196C3×C4⋊C4—31no—
197C3×C4⋊C4—61no—
198C3×C22⋊C4—31no—
199C3×C22⋊C4—41no—
200C3×C22⋊C4—61no—
201C3×C22⋊C4—61no—
202C3×C22⋊C4—61no—
203C3×C22⋊C4—61no—
204C3×C22⋊C4—61no—
205C3×C22⋊C4—61no—
206C3×C22⋊C4—81no—
207C3×C22⋊C4S411yescharacteristic
208C3×C42—31no—
209(C2×D3)⋊C4—41no—
210Dic3⋊C4—41no—
211C22×C32—41no—
212C22×C32—81no—
213C22×C32—81no—
214C22×C32—81no—
215C2×C3×D3—42no—
216C2×C3×D3—81no—
217C2×C3×D3—81no—
218C2×C3×D3—81no—
219C2×C3×D3—81no—
220C2×C3×D3—81no—
221C32×C4—81no—
222C32×C4—81no—
223C3×Dic3—42no—
224C3×Dic3—81no—
225C2×#41—31no—
226C2×#41C2×C3×D311yescharacteristic
227C22×Q8C2×C3×D311yescharacteristic
228C22×D4—31no—
229#137—62no—
230#137—62no—
231AGL(1,ℤ/8)—62no—
232AGL(1,ℤ/8)—62no—
233C2×Q16—31no—
234C2×Q16—31no—
235C2×SD16—31no—
236C2×SD16—31no—
237C2×SD16—31no—
238C2×SD16—31no—
239C2×SD16—31no—
240C2×D8—31no—
241C2×D8—31no—
242C2×M16—31no—
243C2×M16—31no—
244#124—31no—
245#124—31no—
246#123—61no—
247C22⋊Q8—31no—
248C22⋊Q8—31no—
249#121—31no—
250#121—31no—
251#121—61no—
252#120—61no—
253C4×Q8C2×C3×D311yescharacteristic
254C4×D4—31no—
255C4×D4—31no—
256C2×C22⋊C4—61no—
257#106—31no—
258C4⋊C8—31no—
259#103—31no—
260#103—31no—
261#103—31no—
262#102—31no—
263#102—31no—
264#102—31no—
265C22⋊C8—31no—
266C22⋊C8—31no—
267C4×C8—31no—
268C23×C3—31no—
269C23×C3—41no—
270C23×C3—61no—
271C23×C3—61no—
272C23×C3—81no—
273C23×C3—122no—
274C23×C3—122no—
275C23×C3—241no—
276C23×C3C2×S411yescharacteristic
277C22×D3—41no—
278C22×D3—41no—
279C3×Q8—31no—
280C3×Q8—61no—
281C3×Q8—61no—
282C3×Q8—61no—
283C3×Q8—61no—
284C3×Q8D3×D411yescharacteristic
285C3×D4—32no—
286C3×D4—61no—
287C3×D4—61no—
288C3×D4—61no—
289C3×D4—61no—
290C3×D4—61no—
291C3×D4—122no—
292C3×D4—121no—
293C3×D4—122no—
294C3×D4—241no—
295C2×C3×C4—31no—
296C2×C3×C4—31no—
297C2×C3×C4—31no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—61no—
301C2×C3×C4—61no—
302C2×C3×C4—61no—
303C2×C3×C4—61no—
304C2×C3×C4—61no—
305C2×C3×C4—61no—
306C2×C3×C4—81no—
307C2×C3×C4—81no—
308C2×C3×C4—122no—
309C2×C3×C4—241no—
310C2×C3×C4C2×S411yescharacteristic
311C2×C3×C4C2×S411yescharacteristic
312#67—82no—
313#67—82no—
314C2×Dic3—41no—
315C2×Dic3—41no—
316D12—82no—
317C4×D3—82no—
318SL(2,𝔽3)—21no—
319SL(2,𝔽3)C2×C3×D411yescharacteristic
320C3×C8—32no—
321C3×C8—61no—
322C3×C8—61no—
323C3×C8—61no—
324C2×C32—41no—
325C2×C32—41no—
326C2×C32—41no—
327C2×C32—161no—
328C3×D3—82no—
329C3×D3—161no—
330C24—61no—
331#41—21no—
332#41—61no—
333#41—62no—
334#41—121no—
335C2×Q8—21no—
336C2×Q8—31no—
337C2×Q8—31no—
338C2×Q8—61no—
339C2×Q8C22×C3×D311yescharacteristic
340C2×D4—31no—
341C2×D4—31no—
342C2×D4—31no—
343C2×D4—31no—
344C2×D4—62no—
345C2×D4—61no—
346C2×D4—61no—
347C2×D4—61no—
348C2×D4—121no—
349C22×C4—31no—
350C22×C4—31no—
351C22×C4—61no—
352C22×C4—61no—
353Q16—62no—
354Q16—61no—
355Q16—61no—
356SD16—32no—
357SD16—32no—
358SD16—62no—
359SD16—61no—
360SD16—61no—
361SD16—121no—
362SD16—121no—
363D8—62no—
364D8—62no—
365M16—62no—
366M16—62no—
367C2×C8—31no—
368C2×C8—31no—
369C2×C8—31no—
370C2×C8—31no—
371C4⋊C4—31no—
372C4⋊C4—31no—
373C4⋊C4—31no—
374C4⋊C4—61no—
375C22⋊C4—31no—
376C22⋊C4—61no—
377C22⋊C4—61no—
378C22⋊C4—61no—
379C22⋊C4—61no—
380C22⋊C4—61no—
381C22⋊C4—61no—
382C22⋊C4C3×S411yescharacteristic
383C42—31no—
384C22×C3—21no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—41no—
388C22×C3—61no—
389C22×C3—62no—
390C22×C3—61no—
391C22×C3—61no—
392C22×C3—81no—
393C22×C3—81no—
394C22×C3—81no—
395C22×C3—82no—
396C22×C3—81no—
397C22×C3—121no—
398C22×C3—121no—
399C22×C3—121no—
400C22×C3—121no—
401C22×C3—121no—
402C22×C3—161no—
403C22×C3—161no—
404C22×C3—244no—
405C22×C3C22×S411yescharacteristic
406C2×D3—42no—
407C2×D3—81no—
408C2×D3—81no—
409C2×D3—81no—
410C2×D3—81no—
411C2×D3—81no—
412C3×C4—21no—
413C3×C4—21no—
414C3×C4—31no—
415C3×C4—31no—
416C3×C4—61no—
417C3×C4—61no—
418C3×C4—81no—
419C3×C4—81no—
420C3×C4—121no—
421C3×C4—121no—
422C3×C4—161no—
423C3×C4—161no—
424Dic3—42no—
425Dic3—81no—
426C32—41noSylow
427C23—31no—
428C23—61no—
429C23—61no—
430C23—122no—
431C23—122no—
432C23—241no—
433C23C2×C3×S411yescharacteristic
434Q8—31no—
435Q8—61no—
436Q8—61no—
437Q8—61no—
438Q8—61no—
439Q8C3×D3×D411yescharacteristic
440D4—32no—
441D4—61no—
442D4—61no—
443D4—61no—
444D4—61no—
445D4—61no—
446D4—122no—
447D4—121no—
448D4—122no—
449D4—241no—
450C2×C4—31no—
451C2×C4—31no—
452C2×C4—31no—
453C2×C4—61no—
454C2×C4—61no—
455C2×C4—61no—
456C2×C4—61no—
457C2×C4—61no—
458C2×C4—61no—
459C2×C4—122no—
460C2×C4—241no—
461C2×C4C2×C3×S411yescharacteristic
462C2×C4C2×C3×S411yescharacteristic
463C8—32no—
464C8—61no—
465C8—61no—
466C8—61no—
467C2×C3—41no—
468C2×C3—41no—
469C2×C3—41no—
470C2×C3—41no—
471C2×C3—81no—
472C2×C3—81no—
473C2×C3—81no—
474C2×C3—121no—
475C2×C3—122no—
476C2×C3—161no—
477C2×C3—162no—
478C2×C3—241no—
479C2×C3D4×S411yescharacteristic
480C2×C3#592311yescharacteristic
481C2×C3#592611yescharacteristic
482D3—82no—
483D3—161no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—61no—
488C22—62no—
489C22—61no—
490C22—61no—
491C22—121no—
492C22—121no—
493C22—121no—
494C22—121no—
495C22—121no—
496C22—244no—
497C22C22×C3×S411yescharacteristic
498C4—21no—
499C4—21no—
500C4—31no—
501C4—31no—
502C4—61no—
503C4—61no—
504C4—121no—
505C4—121no—
506C3—41no—
507C3—81no—
508C3#8538411yescharacteristic, direct product factor
509C2—41no—
510C2—121no—
511C2—122no—
512C2—241no—
513C2C3×D4×S411yescharacteristic
514C2C3×#592311yescharacteristic
515C2C3×#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. C2×C3×#983C2semidirect
3. C3×#548⋊C3C2semidirect
4. C3×#5425C2semidirect
5. C3×#5421C2semidirect
6. C3×#5395C2semidirect
7. C3×C4×GL(2,𝔽3)C2semidirect
8. #85384C3direct (N × H)
11. C22×C3×SL(2,𝔽3)C22semidirect
16. C2×C3×#226C22semidirect
18. C3×C4×SL(2,𝔽3)C22semidirect
19. C3×#859C22semidirect
20. C2×#986C2×C3semidirect
21. C2×#983C2×C3semidirect
24. C3×#548D3semidirect
27. #548⋊C3C2×C3semidirect
28. #5425C2×C3semidirect
29. #5421C2×C3semidirect
30. #5395C2×C3semidirect
31. C4×GL(2,𝔽3)C2×C3semidirect
51. C3×GL(2,𝔽3)D4semidirect
64. C22×SL(2,𝔽3)C22×C3semidirect
69. C2×#226C22×C3semidirect
100. C4×SL(2,𝔽3)C22×C3semidirect
101. #859C22×C3semidirect
127. #548C3×D3semidirect
174. GL(2,𝔽3)C3×D4semidirect
508. C3#85384direct (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
C2×C3×#9832
C3×#548⋊C32
C3×#54252
C3×#54212
C3×#53952
C3×C4×GL(2,𝔽3)2
#853843
C3×#32753
C3×#8844