Finite Group Databasebeta

← C3×#85380 #11922666

Subgroups — C3×#85380 #11922666

Order 1152.

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

normal subgroups only

This group has 635 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×#985C211yesmaximal, characteristic, complemented
2C22×C3×GL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C3×#548⋊C3C211yesmaximal, characteristic, complemented
4C3×#5427C211yesmaximal, characteristic, complemented
5C3×#5419C211yesmaximal, characteristic, complemented
6C3×#5395C211yesmaximal, characteristic, complemented
7C3×C4×GL(2,𝔽3)C211yesmaximal, characteristic, complemented
8#85380C311yesmaximal, characteristic, direct product factor
9C3×#3282—31nomaximal
10C2×C3×#41⋊C3C2211yescharacteristic, complemented
11C22×C3×SL(2,𝔽3)C2211yescharacteristic
12C3×#985—22no—
13C2×C3×GL(2,𝔽3)—22no—
14C2×C3×GL(2,𝔽3)C2211yescharacteristic
15C2×C3×GL(2,𝔽3)C2211yescharacteristic
16C2×C3×#226C2211yescharacteristic, complemented
17C3×D3×C22⋊C4—41nomaximal
18C3×C4×SL(2,𝔽3)C2211yescharacteristic, complemented
19C3×#859C2211yescharacteristic, complemented
20C2×#985C2×C311yescharacteristic, complemented
21C22×GL(2,𝔽3)C2×C311yescharacteristic, complemented
22C2×C3×#135—31no—
23C22×C3×SD16—31no—
24C3×#548D311yescharacteristic, complemented
25C3×#546—31no—
26#548⋊C3—21no—
27#548⋊C3C2×C311yescharacteristic, complemented
28#5427C2×C311yescharacteristic, complemented
29#5419C2×C311yescharacteristic, complemented
30#5395C2×C311yescharacteristic, complemented
31C4×GL(2,𝔽3)C2×C311yescharacteristic, complemented
32C3×#495—31no—
33C3×#465—31no—
34C3×#465—31no—
35C3×#464—31no—
36C3×#463—31no—
37C3×#452—31no—
38C3×#451—31no—
39C3×#449—31no—
40C3×#447—31no—
41C3×C4×SD16—31no—
42C3×C8×D4—31no—
43C23×C3×D3—41no—
44C2×C3×C4×D3—41no—
45C2×C3×C4×D3—41no—
46C3×#41⋊C3—21no—
47C2×C3×SL(2,𝔽3)—21no—
48C2×C3×SL(2,𝔽3)C2311yescharacteristic
49C3×GL(2,𝔽3)—21no—
50C3×GL(2,𝔽3)D412yescomplemented
51C3×#226—21no—
52C32×C22⋊C4—41no—
53C3×#217—41no—
54C3×(C2×D3)⋊C4—41no—
55C3×(C2×D3)⋊C4—41no—
56#3282—31noSylow
57C2×C3×#41—31no—
58C2×C3×#41C2×D311yescharacteristic, complemented
59C22×C3×Q8C2×D311yescharacteristic
60C22×C3×D4—31no—
61C2×#41⋊C3—21no—
62C2×#41⋊C3C22×C311yescharacteristic, complemented
63C22×SL(2,𝔽3)—21no—
64C22×SL(2,𝔽3)C22×C311yescharacteristic
65#985—22no—
66C2×GL(2,𝔽3)—22no—
67C2×GL(2,𝔽3)C22×C311yescharacteristic
68C2×GL(2,𝔽3)C22×C311yescharacteristic
69C2×#226C22×C311yescharacteristic, complemented
70C3×#135—62no—
71C3×#135—62no—
72C2×C3×Q16—31no—
73C2×C3×Q16—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—
83C22×C3×C8—31no—
84C22×C3×C8—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, complemented
95C3×C4×D4—31no—
96C3×C4×D4—31no—
97C2×C3×C22⋊C4—61no—
98D3×C22⋊C4—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—
116C22×C3×D3—44no—
117C22×C3×D3—82no—
118C22×C3×D3—82no—
119C2×C32×C4—41no—
120C2×C32×C4—41no—
121C2×C3×Dic3—41no—
122C2×C3×Dic3—41no—
123C3×C4×D3—82no—
124C3×C4×D3—82no—
125C3×SL(2,𝔽3)C2×D411yescharacteristic
126C2×#135—31no—
127C22×SD16—31no—
128#548C3×D311yescharacteristic, complemented
129#546—31no—
130#495—31no—
131#465—31no—
132#465—31no—
133#464—31no—
134#463—31no—
135#452—31no—
136#451—31no—
137#449—31no—
138#447—31no—
139C4×SD16—31no—
140C8×D4—31no—
141C24×C3—61no—
142C23×D3—41no—
143C3×#41—21no—
144C3×#41—61no—
145C3×#41—62no—
146C3×#41—121no—
147C2×C3×Q8—21no—
148C2×C3×Q8—31no—
149C2×C3×Q8—31no—
150C2×C3×Q8—61no—
151C2×C3×Q8C22×D311yescharacteristic
152C2×C3×D4—31no—
153C2×C3×D4—31no—
154C2×C3×D4—31no—
155C2×C3×D4—31no—
156C2×C3×D4—61no—
157C2×C3×D4—62no—
158C2×C3×D4—61no—
159C2×C3×D4—61no—
160C2×C3×D4—121no—
161C22×C3×C4—31no—
162C22×C3×C4—31no—
163C22×C3×C4—61no—
164C22×C3×C4—61no—
165C2×C4×D3—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—61no—
177C3×Q16—61no—
178C3×Q16—121no—
179C3×SD16—32no—
180C3×SD16—32no—
181C3×SD16—62no—
182C3×SD16—62no—
183C3×SD16—62no—
184C3×SD16—61no—
185C3×SD16—61no—
186C3×D8—62no—
187C3×D8—121no—
188C2×C3×C8—31no—
189C2×C3×C8—31no—
190C2×C3×C8—31no—
191C2×C3×C8—31no—
192C2×C3×C8—62no—
193C2×C3×C8—62no—
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#217—41no—
210(C2×D3)⋊C4—41no—
211(C2×D3)⋊C4—41no—
212C22×C32—42no—
213C22×C32—41no—
214C22×C32—81no—
215C22×C32—81no—
216C2×C3×D3—42no—
217C2×C3×D3—42no—
218C2×C3×D3—42no—
219C2×C3×D3—42no—
220C2×C3×D3—81no—
221C2×C3×D3—81no—
222C2×C3×D3—88no—
223C32×C4—81no—
224C32×C4—81no—
225C3×Dic3—81no—
226C3×Dic3—81no—
227C2×#41—31no—
228C2×#41C2×C3×D311yescharacteristic, complemented
229C22×Q8C2×C3×D311yescharacteristic
230C22×D4—31no—
231#135—62no—
232#135—62no—
233C2×Q16—31no—
234C2×Q16—31no—
235C2×SD16—31no—
236C2×SD16—31no—
237C2×SD16—31no—
238C2×SD16—31no—
239C2×SD16—31no—
240C2×SD16—62no—
241C2×SD16—62no—
242C2×D8—31no—
243C2×D8—31no—
244C22×C8—31no—
245C22×C8—31no—
246#124—31no—
247#124—31no—
248#123—61no—
249C22⋊Q8—31no—
250C22⋊Q8—31no—
251#121—31no—
252#121—31no—
253#121—61no—
254#120—61no—
255C4×Q8C2×C3×D311yescharacteristic, complemented
256C4×D4—31no—
257C4×D4—31no—
258C2×C22⋊C4—61no—
259#106—31no—
260C4⋊C8—31no—
261#103—31no—
262#103—31no—
263#103—31no—
264#102—31no—
265#102—31no—
266#102—31no—
267C22⋊C8—31no—
268C22⋊C8—31no—
269C4×C8—31no—
270C23×C3—31no—
271C23×C3—41no—
272C23×C3—61no—
273C23×C3—61no—
274C23×C3—81no—
275C23×C3—122no—
276C23×C3—122no—
277C23×C3—241no—
278C23×C3C2×S411yescharacteristic
279C22×D3—41no—
280C22×D3—41no—
281C22×D3—44no—
282C22×D3—82no—
283C22×D3—82no—
284C3×Q8—31no—
285C3×Q8—61no—
286C3×Q8—61no—
287C3×Q8—61no—
288C3×Q8—61no—
289C3×Q8D3×D411yescharacteristic
290C3×D4—32no—
291C3×D4—61no—
292C3×D4—61no—
293C3×D4—61no—
294C3×D4—61no—
295C3×D4—61no—
296C3×D4—62no—
297C3×D4—122no—
298C3×D4—122no—
299C3×D4—241no—
300C2×C3×C4—31no—
301C2×C3×C4—31no—
302C2×C3×C4—31no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—61no—
306C2×C3×C4—61no—
307C2×C3×C4—61no—
308C2×C3×C4—61no—
309C2×C3×C4—61no—
310C2×C3×C4—61no—
311C2×C3×C4—81no—
312C2×C3×C4—81no—
313C2×C3×C4—122no—
314C2×C3×C4—241no—
315C2×C3×C4C2×S411yescharacteristic
316C2×C3×C4C2×S411yescharacteristic
317C2×Dic3—41no—
318C2×Dic3—41no—
319C4×D3—82no—
320C4×D3—82no—
321SL(2,𝔽3)—21no—
322SL(2,𝔽3)C2×C3×D411yescharacteristic
323C3×C8—32no—
324C3×C8—61no—
325C3×C8—61no—
326C3×C8—61no—
327C2×C32—41no—
328C2×C32—41no—
329C2×C32—41no—
330C2×C32—82no—
331C3×D3—44no—
332C3×D3—82no—
333C24—61no—
334#41—21no—
335#41—61no—
336#41—62no—
337#41—121no—
338C2×Q8—21no—
339C2×Q8—31no—
340C2×Q8—31no—
341C2×Q8—61no—
342C2×Q8C22×C3×D311yescharacteristic
343C2×D4—31no—
344C2×D4—31no—
345C2×D4—31no—
346C2×D4—31no—
347C2×D4—61no—
348C2×D4—62no—
349C2×D4—61no—
350C2×D4—61no—
351C2×D4—121no—
352C22×C4—31no—
353C22×C4—31no—
354C22×C4—61no—
355C22×C4—61no—
356Q16—61no—
357Q16—61no—
358Q16—121no—
359SD16—32no—
360SD16—32no—
361SD16—62no—
362SD16—62no—
363SD16—62no—
364SD16—61no—
365SD16—61no—
366D8—62no—
367D8—121no—
368C2×C8—31no—
369C2×C8—31no—
370C2×C8—31no—
371C2×C8—31no—
372C2×C8—62no—
373C2×C8—62no—
374C4⋊C4—31no—
375C4⋊C4—31no—
376C4⋊C4—31no—
377C4⋊C4—61no—
378C22⋊C4—31no—
379C22⋊C4—61no—
380C22⋊C4—61no—
381C22⋊C4—61no—
382C22⋊C4—61no—
383C22⋊C4—61no—
384C22⋊C4—61no—
385C22⋊C4C3×S411yescharacteristic
386C42—31no—
387C22×C3—21no—
388C22×C3—21no—
389C22×C3—41no—
390C22×C3—42no—
391C22×C3—61no—
392C22×C3—62no—
393C22×C3—61no—
394C22×C3—61no—
395C22×C3—81no—
396C22×C3—81no—
397C22×C3—81no—
398C22×C3—82no—
399C22×C3—121no—
400C22×C3—121no—
401C22×C3—121no—
402C22×C3—121no—
403C22×C3—121no—
404C22×C3—161no—
405C22×C3—161no—
406C22×C3—244no—
407C22×C3#98512yescomplemented
408C22×C3C22×S411yescharacteristic
409C2×D3—42no—
410C2×D3—42no—
411C2×D3—42no—
412C2×D3—42no—
413C2×D3—81no—
414C2×D3—81no—
415C2×D3—88no—
416C3×C4—21no—
417C3×C4—21no—
418C3×C4—31no—
419C3×C4—31no—
420C3×C4—61no—
421C3×C4—61no—
422C3×C4—81no—
423C3×C4—81no—
424C3×C4—121no—
425C3×C4—121no—
426C3×C4—161no—
427C3×C4—161no—
428Dic3—81no—
429Dic3—81no—
430C32—41noSylow
431C23—31no—
432C23—61no—
433C23—61no—
434C23—122no—
435C23—122no—
436C23—241no—
437C23C2×C3×S411yescharacteristic
438Q8—31no—
439Q8—61no—
440Q8—61no—
441Q8—61no—
442Q8—61no—
443Q8C3×D3×D411yescharacteristic
444D4—32no—
445D4—61no—
446D4—61no—
447D4—61no—
448D4—61no—
449D4—61no—
450D4—62no—
451D4—122no—
452D4—122no—
453D4—241no—
454C2×C4—31no—
455C2×C4—31no—
456C2×C4—31no—
457C2×C4—61no—
458C2×C4—61no—
459C2×C4—61no—
460C2×C4—61no—
461C2×C4—61no—
462C2×C4—61no—
463C2×C4—122no—
464C2×C4—241no—
465C2×C4C2×C3×S411yescharacteristic
466C2×C4C2×C3×S411yescharacteristic
467C8—32no—
468C8—61no—
469C8—61no—
470C8—61no—
471C2×C3—22no—
472C2×C3—41no—
473C2×C3—41no—
474C2×C3—41no—
475C2×C3—82no—
476C2×C3—81no—
477C2×C3—81no—
478C2×C3—81no—
479C2×C3—121no—
480C2×C3—122no—
481C2×C3—162no—
482C2×C3—241no—
483C2×C3D4×S411yescharacteristic
484C2×C3C2×#98511yescharacteristic
485C2×C3#592611yescharacteristic
486D3—44no—
487D3—82no—
488C22—21no—
489C22—21no—
490C22—61no—
491C22—62no—
492C22—61no—
493C22—61no—
494C22—121no—
495C22—121no—
496C22—121no—
497C22—121no—
498C22—121no—
499C22—244no—
500C22C3×#98512yescomplemented
501C22C22×C3×S411yescharacteristic
502C4—21no—
503C4—21no—
504C4—31no—
505C4—31no—
506C4—61no—
507C4—61no—
508C4—121no—
509C4—121no—
510C3—41no—
511C3—81no—
512C3#8538011yescharacteristic, direct product factor
513C2—22no—
514C2—121no—
515C2—122no—
516C2—241no—
517C2C3×D4×S411yescharacteristic
518C2C2×C3×#98511yescharacteristic
519C2C3×#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×#985C2semidirect
2. C22×C3×GL(2,𝔽3)C2semidirect
3. C3×#548⋊C3C2semidirect
4. C3×#5427C2semidirect
5. C3×#5419C2semidirect
6. C3×#5395C2semidirect
7. C3×C4×GL(2,𝔽3)C2semidirect
8. #85380C3direct (N × H)
10. C2×C3×#41⋊C3C22semidirect
16. C2×C3×#226C22semidirect
18. C3×C4×SL(2,𝔽3)C22semidirect
19. C3×#859C22semidirect
20. C2×#985C2×C3semidirect
21. C22×GL(2,𝔽3)C2×C3semidirect
24. C3×#548D3semidirect
27. #548⋊C3C2×C3semidirect
28. #5427C2×C3semidirect
29. #5419C2×C3semidirect
30. #5395C2×C3semidirect
31. C4×GL(2,𝔽3)C2×C3semidirect
50. C3×GL(2,𝔽3)D4semidirect
58. C2×C3×#41C2×D3semidirect
62. C2×#41⋊C3C22×C3semidirect
69. C2×#226C22×C3semidirect
94. C3×C4×Q8C2×D3semidirect
100. C4×SL(2,𝔽3)C22×C3semidirect
101. #859C22×C3semidirect
128. #548C3×D3semidirect
174. GL(2,𝔽3)C3×D4semidirect
228. C2×#41C2×C3×D3semidirect
255. C4×Q8C2×C3×D3semidirect
407. C22×C3#985semidirect
500. C22C3×#985semidirect
512. C3#85380direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#9852
C22×C3×GL(2,𝔽3)2
C3×#548⋊C32
C3×#54272
C3×#54192
C3×#53952
C3×C4×GL(2,𝔽3)2
#853803
C3×#32823
C3×D3×C22⋊C44