Finite Group Databasebeta

← C3×#85539 #11922819

Subgroups — C3×#85539 #11922819

Order 1152.

667 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 667 conjugacy classes of subgroups — too many to draw. Only its 76 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×C3×2−1+4⋊C3C211yesmaximal, characteristic, complemented
2C2×C3×#983C211yesmaximal, characteristic, complemented
3C3×#5427C211yesmaximal, characteristic, complemented
4C3×#5425C211yesmaximal, characteristic, complemented
5C3×#5421C211yesmaximal, characteristic, complemented
6C3×#5419C211yesmaximal, characteristic, complemented
7C2×C3×#859C211yesmaximal, characteristic, complemented
8#85539C311yesmaximal, characteristic, direct product factor
9C3×#3013—31nomaximal
10C3×2−1+4⋊C3—22no—
11C2×C3×#41⋊C3C2211yescharacteristic
12C22×C3×SL(2,𝔽3)C2211yescharacteristic
13C22×C3×SL(2,𝔽3)C2211yescharacteristic
14C3×#983—22no—
15C2×C3×GL(2,𝔽3)C2211yescharacteristic
16C2×C3×#226C2211yescharacteristic
17C3×#939—41nomaximal
18C3×#859C2211yescharacteristic
19C3×#859C2211yescharacteristic
20C2×C3×2−1+4D311yescharacteristic, complemented
21C2×2−1+4⋊C3—21no—
22C2×2−1+4⋊C3C2×C311yescharacteristic, complemented
23C2×#983C2×C311yescharacteristic, complemented
24C2×C3×#137—31no—
25C2×C3×#135—31no—
26C3×#537—31no—
27#5427C2×C311yescharacteristic, complemented
28#5425C2×C311yescharacteristic, complemented
29#5421C2×C311yescharacteristic, complemented
30#5419C2×C311yescharacteristic, complemented
31C2×#859C2×C311yescharacteristic, complemented
32C3×#452—31no—
33C3×#452—31no—
34C3×#451—31no—
35C3×#451—31no—
36C3×#449—31no—
37C3×#449—31no—
38C3×#448—31no—
39C3×#448—31no—
40C3×#417—31no—
41C2×C3×#103—31no—
42C3×#408—31no—
43C2×C32×D4—41no—
44C2×C3×#67—41no—
45C2×C3×#67—41no—
46C22×C3×Dic3—41no—
47C3×#41⋊C3—21no—
48C2×C3×SL(2,𝔽3)—21no—
49C2×C3×SL(2,𝔽3)D412yescomplemented
50C2×C3×SL(2,𝔽3)C2311yescharacteristic
51C3×GL(2,𝔽3)—21no—
52C3×#226—21no—
53C3×#217—41no—
54C3×(C2×D3)⋊C4—41no—
55C3×Dic3⋊C4—41no—
56#3013—31noSylow
57C3×2−1+4—22no—
58C3×2−1+4—62no—
59C2×C3×#41—31no—
60C2×C3×#41—31no—
61C2×C3×#41—31no—
62C2×C3×#41—31no—
63C2×C3×#41C2×D311yescharacteristic
64C22×C3×Q8—31no—
65C22×C3×Q8C2×D311yescharacteristic
66C22×C3×Q8C2×D311yescharacteristic
672−1+4⋊C3—24no—
682−1+4⋊C3—22no—
69C2×#41⋊C3—21no—
70C2×#41⋊C3C22×C311yescharacteristic
71C22×SL(2,𝔽3)—21no—
72C22×SL(2,𝔽3)—21no—
73C22×SL(2,𝔽3)C22×C311yescharacteristic
74C22×SL(2,𝔽3)C22×C311yescharacteristic
75#983—22no—
76C2×GL(2,𝔽3)C22×C311yescharacteristic
77C2×#226C22×C311yescharacteristic
78C3×#137—62no—
79C3×#137—62no—
80C3×#135—64no—
81C2×C3×Q16—31no—
82C2×C3×Q16—31no—
83C2×C3×Q16—31no—
84C2×C3×SD16—31no—
85C2×C3×SD16—31no—
86C2×C3×SD16—31no—
87C2×C3×SD16—31no—
88C2×C3×D8—31no—
89C2×C3×M16—31no—
90C22×C3×C8—31no—
91C3×C22⋊Q8—31no—
92C3×C22⋊Q8—31no—
93C3×C22⋊Q8—61no—
94C3×#121—31no—
95C3×#121—31no—
96C3×#121—61no—
97C3×#121—61no—
98C3×#121—61no—
99C2×C3×C4⋊C4—31no—
100#939—41no—
101#859C22×C311yescharacteristic
102#859C22×C311yescharacteristic
103C3×#103—31no—
104C3×#103—31no—
105C3×#103—31no—
106C3×#103—31no—
107C3×#103—31no—
108C3×#103—31no—
109C3×#102—31no—
110C3×#102—31no—
111C3×C22⋊C8—31no—
112C3×C22⋊C8—31no—
113C3×C22⋊C8—31no—
114C3×C22⋊C8—31no—
115C23×C32—41no—
116C23×C32—41no—
117C22×C3×D3—41no—
118C32×D4—82no—
119C2×C32×C4—41no—
120C3×#67—82no—
121C3×#67—82no—
122C2×C3×Dic3—41no—
123C2×C3×Dic3—41no—
124C2×C3×Dic3—41no—
125C2×C3×Dic3—82no—
126C3×SL(2,𝔽3)C2×D411yescharacteristic
127C2×2−1+4C3×D311yescharacteristic, complemented
128C2×#137—31no—
129C2×#135—31no—
130#537—31no—
131#452—31no—
132#452—31no—
133#451—31no—
134#451—31no—
135#449—31no—
136#449—31no—
137#448—31no—
138#448—31no—
139#417—31no—
140C2×#103—31no—
141#408—31no—
142C3×#41—21no—
143C3×#41—34no—
144C3×#41—62no—
145C3×#41—62no—
146C3×#41—62no—
147C3×#41—62no—
148C3×#41—62no—
149C3×#41—62no—
150C3×#41—62no—
151C3×#41—61no—
152C2×C3×Q8—21no—
153C2×C3×Q8—31no—
154C2×C3×Q8—31no—
155C2×C3×Q8—31no—
156C2×C3×Q8—32no—
157C2×C3×Q8—31no—
158C2×C3×Q8—62no—
159C2×C3×Q8—62no—
160C2×C3×Q8—61no—
161C2×C3×Q8#6712yescomplemented
162C2×C3×Q8C22×D311yescharacteristic
163C2×C3×D4—31no—
164C2×C3×D4—31no—
165C2×C3×D4—31no—
166C2×C3×D4—31no—
167C2×C3×D4—41no—
168C2×C3×D4—61no—
169C2×C3×D4—61no—
170C2×C3×D4—61no—
171C2×C3×D4—81no—
172C2×C3×D4S411yescharacteristic
173C22×C3×C4—31no—
174C22×C3×C4—31no—
175C22×C3×C4—31no—
176C22×C3×C4—61no—
177C22×C3×C4—61no—
178C22×C3×C4—61no—
179C2×#67—41no—
180C2×#67—41no—
181C22×Dic3—41no—
182#41⋊C3—22no—
183#41⋊C3—21no—
184C2×SL(2,𝔽3)—22no—
185C2×SL(2,𝔽3)—21no—
186C2×SL(2,𝔽3)—22no—
187C2×SL(2,𝔽3)—21no—
188C2×SL(2,𝔽3)C3×D412yescomplemented
189C2×SL(2,𝔽3)C23×C311yescharacteristic
190GL(2,𝔽3)—21no—
191#226—21no—
192C3×Q16—62no—
193C3×Q16—62no—
194C3×Q16—61no—
195C3×Q16—61no—
196C3×SD16—62no—
197C3×SD16—62no—
198C3×SD16—62no—
199C3×SD16—61no—
200C3×SD16—61no—
201C3×D8—62no—
202C3×M16—62no—
203C2×C3×C8—31no—
204C2×C3×C8—31no—
205C2×C3×C8—31no—
206C2×C3×C8—31no—
207C2×C3×C8—62no—
208C3×C4⋊C4—31no—
209C3×C4⋊C4—31no—
210C3×C4⋊C4—61no—
211C3×C4⋊C4—61no—
212C3×C4⋊C4—61no—
213C3×C22⋊C4—61no—
214C3×C22⋊C4—61no—
215C3×C22⋊C4—61no—
216C3×C22⋊C4—61no—
217#217—41no—
218(C2×D3)⋊C4—41no—
219Dic3⋊C4—41no—
220C22×C32—42no—
221C22×C32—41no—
222C22×C32—81no—
223C22×C32—81no—
224C22×C32—81no—
225C22×C32—81no—
226C22×C32—81no—
227C2×C3×D3—81no—
228C2×C3×D3—81no—
229C2×C3×D3—81no—
230C32×C4—81no—
231C3×Dic3—42no—
232C3×Dic3—81no—
233C3×Dic3—81no—
2342−1+4—22no—
2352−1+4—62no—
236C2×#41—31no—
237C2×#41—31no—
238C2×#41—31no—
239C2×#41—31no—
240C2×#41C2×C3×D311yescharacteristic
241C22×Q8—31no—
242C22×Q8C2×C3×D311yescharacteristic
243C22×Q8C2×C3×D311yescharacteristic
244#137—62no—
245#137—62no—
246#135—64no—
247C2×Q16—31no—
248C2×Q16—31no—
249C2×Q16—31no—
250C2×SD16—31no—
251C2×SD16—31no—
252C2×SD16—31no—
253C2×SD16—31no—
254C2×D8—31no—
255C2×M16—31no—
256C22×C8—31no—
257C22⋊Q8—31no—
258C22⋊Q8—31no—
259C22⋊Q8—61no—
260#121—31no—
261#121—31no—
262#121—61no—
263#121—61no—
264#121—61no—
265C2×C4⋊C4—31no—
266#103—31no—
267#103—31no—
268#103—31no—
269#103—31no—
270#103—31no—
271#103—31no—
272#102—31no—
273#102—31no—
274C22⋊C8—31no—
275C22⋊C8—31no—
276C22⋊C8—31no—
277C22⋊C8—31no—
278C23×C3—31no—
279C23×C3—41no—
280C23×C3—41no—
281C23×C3—61no—
282C23×C3—81no—
283C23×C3—81no—
284C23×C3C2×S411yescharacteristic
285C23×C3C2×S411yescharacteristic
286C22×D3—41no—
287C3×Q8—32no—
288C3×Q8—31no—
289C3×Q8—32no—
290C3×Q8—62no—
291C3×Q8—61no—
292C3×Q8—61no—
293C3×Q8—61no—
294C3×Q8—61no—
295C3×Q8C2×#6711yescharacteristic
296C3×D4—22no—
297C3×D4—32no—
298C3×D4—62no—
299C3×D4—61no—
300C3×D4—61no—
301C3×D4—61no—
302C3×D4—62no—
303C3×D4—82no—
304C3×D4—84no—
305C3×D4—122no—
306C3×D4—241no—
307C3×D4—241no—
308C2×C3×C4—32no—
309C2×C3×C4—32no—
310C2×C3×C4—31no—
311C2×C3×C4—31no—
312C2×C3×C4—31no—
313C2×C3×C4—41no—
314C2×C3×C4—61no—
315C2×C3×C4—61no—
316C2×C3×C4—61no—
317C2×C3×C4—61no—
318C2×C3×C4—64no—
319C2×C3×C4—61no—
320C2×C3×C4—61no—
321C2×C3×C4—61no—
322C2×C3×C4—81no—
323C2×C3×C4—122no—
324C2×C3×C4—241no—
325C2×C3×C4C2×S411yescharacteristic
326#67—82no—
327#67—82no—
328C2×Dic3—41no—
329C2×Dic3—41no—
330C2×Dic3—41no—
331C2×Dic3—82no—
332SL(2,𝔽3)—21no—
333SL(2,𝔽3)C2×C3×D411yescharacteristic
334C3×C8—61no—
335C3×C8—61no—
336C3×C8—61no—
337C3×C8—61no—
338C2×C32—41no—
339C2×C32—41no—
340C2×C32—41no—
341C2×C32—82no—
342C2×C32—161no—
343C3×D3—161no—
344#41—21no—
345#41—34no—
346#41—62no—
347#41—62no—
348#41—62no—
349#41—62no—
350#41—62no—
351#41—62no—
352#41—62no—
353#41—61no—
354C2×Q8—21no—
355C2×Q8—31no—
356C2×Q8—31no—
357C2×Q8—31no—
358C2×Q8—32no—
359C2×Q8—31no—
360C2×Q8—62no—
361C2×Q8—62no—
362C2×Q8—61no—
363C2×Q8C3×#6712yescomplemented
364C2×Q8C22×C3×D311yescharacteristic
365C2×D4—31no—
366C2×D4—31no—
367C2×D4—31no—
368C2×D4—31no—
369C2×D4—61no—
370C2×D4—61no—
371C2×D4—61no—
372C2×D4C3×S411yescharacteristic
373C22×C4—31no—
374C22×C4—31no—
375C22×C4—31no—
376C22×C4—61no—
377C22×C4—61no—
378C22×C4—61no—
379Q16—62no—
380Q16—62no—
381Q16—61no—
382Q16—61no—
383SD16—62no—
384SD16—62no—
385SD16—62no—
386SD16—61no—
387SD16—61no—
388D8—62no—
389M16—62no—
390C2×C8—31no—
391C2×C8—31no—
392C2×C8—31no—
393C2×C8—31no—
394C2×C8—62no—
395C4⋊C4—31no—
396C4⋊C4—31no—
397C4⋊C4—61no—
398C4⋊C4—61no—
399C4⋊C4—61no—
400C22⋊C4—61no—
401C22⋊C4—61no—
402C22⋊C4—61no—
403C22⋊C4—61no—
404C22×C3—21no—
405C22×C3—21no—
406C22×C3—21no—
407C22×C3—21no—
408C22×C3—21no—
409C22×C3—42no—
410C22×C3—41no—
411C22×C3—61no—
412C22×C3—61no—
413C22×C3—61no—
414C22×C3—81no—
415C22×C3—81no—
416C22×C3—81no—
417C22×C3—81no—
418C22×C3—81no—
419C22×C3—82no—
420C22×C3—82no—
421C22×C3—81no—
422C22×C3—121no—
423C22×C3—121no—
424C22×C3—121no—
425C22×C3—161no—
426C22×C3—161no—
427C22×C3—161no—
428C22×C3—161no—
429C22×C3GL(2,ℤ/4)12yes—
430C22×C3C22×S411yescharacteristic
431C2×D3—81no—
432C2×D3—81no—
433C2×D3—81no—
434C3×C4—21no—
435C3×C4—31no—
436C3×C4—31no—
437C3×C4—32no—
438C3×C4—61no—
439C3×C4—81no—
440C3×C4—82no—
441C3×C4—121no—
442C3×C4—121no—
443C3×C4—121no—
444Dic3—42no—
445Dic3—81no—
446Dic3—81no—
447C32—41noSylow
448C23—31no—
449C23—61no—
450C23C2×C3×S411yescharacteristic
451C23C2×C3×S411yescharacteristic
452Q8—32no—
453Q8—31no—
454Q8—32no—
455Q8—62no—
456Q8—61no—
457Q8—61no—
458Q8—61no—
459Q8—61no—
460Q8C2×C3×#6711yescharacteristic
461D4—22no—
462D4—32no—
463D4—62no—
464D4—61no—
465D4—61no—
466D4—61no—
467D4—62no—
468D4—122no—
469D4—241no—
470D4—241no—
471C2×C4—32no—
472C2×C4—32no—
473C2×C4—31no—
474C2×C4—31no—
475C2×C4—31no—
476C2×C4—61no—
477C2×C4—61no—
478C2×C4—61no—
479C2×C4—61no—
480C2×C4—64no—
481C2×C4—61no—
482C2×C4—61no—
483C2×C4—61no—
484C2×C4—122no—
485C2×C4—241no—
486C2×C4C2×C3×S411yescharacteristic
487C8—61no—
488C8—61no—
489C8—61no—
490C8—61no—
491C2×C3—22no—
492C2×C3—41no—
493C2×C3—41no—
494C2×C3—41no—
495C2×C3—41no—
496C2×C3—82no—
497C2×C3—81no—
498C2×C3—81no—
499C2×C3—81no—
500C2×C3—121no—
501C2×C3—162no—
502C2×C3—161no—
503C2×C3—162no—
504C2×C3—241no—
505C2×C3#592611yescharacteristic
506C2×C3#592711yescharacteristic
507C2×C3C2×GL(2,ℤ/4)11yescharacteristic
508D3—161no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—61no—
515C22—61no—
516C22—61no—
517C22—121no—
518C22—121no—
519C22—121no—
520C22C3×GL(2,ℤ/4)12yes—
521C22C22×C3×S411yescharacteristic
522C4—21no—
523C4—31no—
524C4—31no—
525C4—32no—
526C4—61no—
527C4—121no—
528C4—121no—
529C4—121no—
530C3—41no—
531C3—81no—
532C3#8553911yescharacteristic, direct product factor
533C2—22no—
534C2—41no—
535C2—121no—
536C2—241no—
537C2C3×#592611yescharacteristic
538C2C3×#592711yescharacteristic
539C2C2×C3×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, socleC22×C3C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23×C3
FrattiniC22C22×C3×S4
FittingC2×C3×2−1+4D3
hypercenterC2×C3×D4S4
hypocenterSL(2,𝔽3)C2×C3×D4
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×2−1+4⋊C3C2semidirect
2. C2×C3×#983C2semidirect
3. C3×#5427C2semidirect
4. C3×#5425C2semidirect
5. C3×#5421C2semidirect
6. C3×#5419C2semidirect
7. C2×C3×#859C2semidirect
8. #85539C3direct (N × H)
20. C2×C3×2−1+4D3semidirect
22. C2×2−1+4⋊C3C2×C3semidirect
23. C2×#983C2×C3semidirect
27. #5427C2×C3semidirect
28. #5425C2×C3semidirect
29. #5421C2×C3semidirect
30. #5419C2×C3semidirect
31. C2×#859C2×C3semidirect
49. C2×C3×SL(2,𝔽3)D4semidirect
127. C2×2−1+4C3×D3semidirect
161. C2×C3×Q8#67semidirect
188. C2×SL(2,𝔽3)C3×D4semidirect
363. C2×Q8C3×#67semidirect
532. C3#85539direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×2−1+4⋊C32
C2×C3×#9832
C3×#54272
C3×#54252
C3×#54212
C3×#54192
C2×C3×#8592
#855393
C3×#30133
C3×#9394