Finite Group Databasebeta

← C3×#33016 #10651623

Subgroups — C3×#33016 #10651623

Order 768.

1292 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#520C211yesmaximal, characteristic
2C2×C3×#483C211yesmaximal, characteristic, complemented
3C2×C3×#452C211yesmaximal, characteristic
4C3×#2625C211yesmaximal, characteristic, complemented
5C3×#1878C212yesmaximal, complemented
6C3×#1877C212yesmaximal, complemented
7C3×#1865C212yesmaximal, complemented
8C3×#1864C212yesmaximal, complemented
9C3×#1824C211yesmaximal, characteristic, complemented
10C3×#1818C211yesmaximal, characteristic, complemented
11C3×#1788C211yesmaximal, characteristic
12#33016C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2211yescharacteristic
14C22×C3×Q16C2211yescharacteristic
15C22×C3×SD16C2211yescharacteristic
16C2×C3×C22⋊Q8C2211yescharacteristic
17C3×#520—24no—
18C3×#520—24no—
19C2×C3×C4×Q8C2212yes—
20C2×C3×C4×D4—22no—
21C2×C3×C4×D4C2212yes—
22C2×C3×#117—21no—
23C2×C3×#117C2211yescharacteristic
24C22×C3×C4⋊C4—21no—
25C22×C3×C4⋊C4C2211yescharacteristic
26C3×#483—24no—
27C3×#452—24no—
28C2×C3×#106C2212yes—
29C2×C3×#103C2212yes—
30C2×C3×#103C2211yescharacteristic
31C2×C3×#103C2212yes—
32C2×C3×#102C2212yes—
33C2×C3×#102C2211yescharacteristic
34C2×C3×C22⋊C8C2211yescharacteristic
35C2×C3×C22⋊C8C2211yescharacteristic
36C3×#400—21no—
37C3×#398—21no—
38C3×#397—22no—
39C3×#391C2212yes—
40C3×#389C2212yes—
41C3×#380C2211yescharacteristic
42C3×#340C2212yes—
43C3×#340C2212yes—
44C3×#336C2212yes—
45C3×#336C2212yes—
46C2×#520C2×C311yescharacteristic
47C2×#483C2×C311yescharacteristic, complemented
48C2×#452C2×C311yescharacteristic
49#2625C2×C311yescharacteristic, complemented
50#1878C2×C312yescomplemented
51#1877C2×C312yescomplemented
52#1865C2×C312yescomplemented
53#1864C2×C312yescomplemented
54#1824C2×C311yescharacteristic, complemented
55#1818C2×C311yescharacteristic, complemented
56#1788C2×C311yescharacteristic
57C2×C3×#41—24no—
58C2×C3×#41—24no—
59C2×C3×#41—22no—
60C2×C3×#41D414yes—
61C22×C3×Q8C2311yescharacteristic
62C22×C3×Q8C2311yescharacteristic
63C22×C3×D4—21no—
64C22×C3×D4C2311yescharacteristic
65C23×C3×C4—21no—
66C23×C3×C4—21no—
67C23×C3×C4—22no—
68C23×C3×C4C2311yescharacteristic
69C2×C3×Q16—22no—
70C2×C3×Q16—24no—
71C2×C3×SD16—22no—
72C2×C3×SD16—24no—
73C22×C3×C8C2311yescharacteristic
74C22×C3×C8C2311yescharacteristic
75C22×C3×C8C2312yes—
76C3×C22⋊Q8—22no—
77C3×C22⋊Q8—22no—
78C3×C4×Q8—24no—
79C3×C4×Q8—24no—
80C3×C4×D4—216no—
81C3×C4×D4—24no—
82C3×C4×D4—24no—
83C3×#117—28no—
84C3×#117—22no—
85C3×#117—22no—
86C2×C3×C4⋊C4—28no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—22no—
89C2×C3×C4⋊C4—22no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—22no—
92C2×C3×C4⋊C4D411yescharacteristic
93C2×C3×C4⋊C4D411yescharacteristic
94C2×C3×C4⋊C4D414yes—
95C2×C3×C4⋊C4C2312yes—
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C4⋊C4C2312yes—
98C2×C3×C22⋊C4—22no—
99C2×C3×C22⋊C4—21no—
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4D412yes—
102C2×C3×C42—22no—
103C2×C3×C42C2312yes—
104C3×#106—24no—
105C3×#103—22no—
106C3×#103—24no—
107C3×#103—24no—
108C3×#102—22no—
109C3×#102—24no—
110C3×C22⋊C8—22no—
111C3×C22⋊C8—22no—
112C3×#95—21no—
113C3×#95—22no—
114C3×#95—22no—
115C3×#95—22no—
116C22×#41C22×C311yescharacteristic
117C22×Q16C22×C311yescharacteristic
118C22×SD16C22×C311yescharacteristic
119C2×C22⋊Q8C22×C311yescharacteristic
120#520—24no—
121#520—24no—
122C2×C4×Q8C22×C312yes—
123C2×C4×D4—22no—
124C2×C4×D4C22×C312yes—
125C2×#117—21no—
126C2×#117C22×C311yescharacteristic
127C22×C4⋊C4—21no—
128C22×C4⋊C4C22×C311yescharacteristic
129#483—24no—
130#452—24no—
131C2×#106C22×C312yes—
132C2×#103C22×C312yes—
133C2×#103C22×C311yescharacteristic
134C2×#103C22×C312yes—
135C2×#102C22×C311yescharacteristic
136C2×#102C22×C312yes—
137C2×C22⋊C8C22×C311yescharacteristic
138C2×C22⋊C8C22×C311yescharacteristic
139#400—21no—
140#398—21no—
141#397—22no—
142#391C22×C312yes—
143#389C22×C312yes—
144#380C22×C311yescharacteristic
145#340C22×C312yes—
146#340C22×C312yes—
147#336C22×C312yes—
148#336C22×C312yes—
149C24×C3—21no—
150C24×C3C2×D411yescharacteristic
151C3×#41—216no—
152C3×#41—28no—
153C3×#41—28no—
154C2×C3×Q8—21no—
155C2×C3×Q8—22no—
156C2×C3×Q8—21no—
157C2×C3×Q8—21no—
158C2×C3×Q8—22no—
159C2×C3×Q8—22no—
160C2×C3×Q8C2×D412yes—
161C2×C3×Q8#4114yes—
162C2×C3×D4—28no—
163C2×C3×D4—24no—
164C2×C3×D4—21no—
165C2×C3×D4—22no—
166C2×C3×D4C2×D412yes—
167C2×C3×D4#4114yes—
168C22×C3×C4—28no—
169C22×C3×C4—21no—
170C22×C3×C4—24no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—28no—
175C22×C3×C4—24no—
176C22×C3×C4—22no—
177C22×C3×C4—21no—
178C22×C3×C4—21no—
179C22×C3×C4—84no—
180C22×C3×C4—82no—
181C22×C3×C4C2×D412yes—
182C22×C3×C4C2×D412yes—
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D412yes—
185C22×C3×C4C2×D412yes—
186C22×C3×C4C2411yescharacteristic
187C3×Q16—44no—
188C3×SD16—44no—
189C2×C3×C8—21no—
190C2×C3×C8—22no—
191C2×C3×C8—21no—
192C2×C3×C8—22no—
193C2×C3×C8—22no—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C3×C4⋊C4—28no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—24no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—42no—
206C3×C4⋊C4—82no—
207C3×C4⋊C4C2×D414yes—
208C3×C22⋊C4—28no—
209C3×C22⋊C4—24no—
210C3×C22⋊C4—42no—
211C3×C22⋊C4—81no—
212C3×C42—28no—
213C3×C42—22no—
214C3×C42—22no—
215C2×#41—24no—
216C2×#41—24no—
217C2×#41—22no—
218C2×#41C3×D414yes—
219C22×Q8C23×C311yescharacteristic
220C22×Q8C23×C311yescharacteristic
221C22×D4—21no—
222C22×D4C23×C311yescharacteristic
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—22no—
226C23×C4C23×C311yescharacteristic
227C2×Q16—22no—
228C2×Q16—24no—
229C2×SD16—22no—
230C2×SD16—24no—
231C22×C8C23×C311yescharacteristic
232C22×C8C23×C311yescharacteristic
233C22×C8C23×C312yes—
234C22⋊Q8—22no—
235C22⋊Q8—22no—
236C4×Q8—24no—
237C4×Q8—24no—
238C4×D4—216no—
239C4×D4—24no—
240C4×D4—24no—
241#117—28no—
242#117—22no—
243#117—22no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4C3×D411yescharacteristic
251C2×C4⋊C4C3×D411yescharacteristic
252C2×C4⋊C4C3×D414yes—
253C2×C4⋊C4C23×C311yescharacteristic
254C2×C4⋊C4C23×C312yes—
255C2×C4⋊C4C23×C312yes—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4C3×D412yes—
260C2×C42—22no—
261C2×C42C23×C312yes—
262#106—24no—
263#103—22no—
264#103—24no—
265#103—24no—
266#102—22no—
267#102—24no—
268C22⋊C8—22no—
269C22⋊C8—22no—
270#95—21no—
271#95—22no—
272#95—22no—
273#95—22no—
274C23×C3—22no—
275C23×C3—22no—
276C23×C3—24no—
277C23×C3—21no—
278C23×C3—22no—
279C23×C3—21no—
280C23×C3—42no—
281C23×C3—41no—
282C23×C3—41no—
283C23×C3#12012yes—
284C23×C3C22×D411yescharacteristic
285C3×Q8—22no—
286C3×Q8—24no—
287C3×Q8—41no—
288C3×Q8—41no—
289C3×Q8—42no—
290C3×Q8#12314yes—
291C3×D4—216no—
292C3×D4—22no—
293C3×D4—24no—
294C3×D4#12314yes—
295C2×C3×C4—216no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—24no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—24no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—24no—
305C2×C3×C4—22no—
306C2×C3×C4—24no—
307C2×C3×C4—22no—
308C2×C3×C4—44no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—41no—
315C2×C3×C4—42no—
316C2×C3×C4—41no—
317C2×C3×C4—84no—
318C2×C3×C4—88no—
319C2×C3×C4#12012yes—
320C2×C3×C4C22×D411yescharacteristic
321C2×C3×C4C22×D411yescharacteristic
322C2×C3×C4C2×#4112yes—
323C2×C3×C4C2×#4112yes—
324C3×C8—41no—
325C3×C8—41no—
326C3×C8—42no—
327C24—21no—
328C24C2×C3×D411yescharacteristic
329#41—216no—
330#41—28no—
331#41—28no—
332C2×Q8—21no—
333C2×Q8—22no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8C2×C3×D412yes—
339C2×Q8C3×#4114yes—
340C2×D4—28no—
341C2×D4—24no—
342C2×D4—21no—
343C2×D4—22no—
344C2×D4C2×C3×D412yes—
345C2×D4C3×#4114yes—
346C22×C4—28no—
347C22×C4—21no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—28no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—84no—
358C22×C4—82no—
359C22×C4C2×C3×D412yes—
360C22×C4C2×C3×D412yes—
361C22×C4C2×C3×D411yescharacteristic
362C22×C4C2×C3×D412yes—
363C22×C4C2×C3×D412yes—
364C22×C4C24×C311yescharacteristic
365Q16—44no—
366SD16—44no—
367C2×C8—21no—
368C2×C8—22no—
369C2×C8—21no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—22no—
374C4⋊C4—28no—
375C4⋊C4—22no—
376C4⋊C4—24no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—42no—
384C4⋊C4—82no—
385C4⋊C4C2×C3×D414yes—
386C22⋊C4—28no—
387C22⋊C4—24no—
388C22⋊C4—42no—
389C22⋊C4—81no—
390C42—28no—
391C42—22no—
392C42—22no—
393C22×C3—22no—
394C22×C3—24no—
395C22×C3—22no—
396C22×C3—22no—
397C22×C3—42no—
398C22×C3—44no—
399C22×C3—42no—
400C22×C3—44no—
401C22×C3—42no—
402C22×C3—42no—
403C22×C3C2×#12011yescharacteristic
404C22×C3#52512yes—
405C22×C3#57712yes—
406C22×C3#57711yescharacteristic
407C22×C3#57811yescharacteristic
408C3×C4—22no—
409C3×C4—24no—
410C3×C4—42no—
411C3×C4—44no—
412C3×C4—42no—
413C3×C4—81no—
414C3×C4—82no—
415C3×C4—81no—
416C3×C4C2×#12311yescharacteristic
417C3×C4C2×#12311yescharacteristic
418C3×C4#52512yes—
419C23—21no—
420C23—22no—
421C23—21no—
422C23—22no—
423C23—22no—
424C23—24no—
425C23—42no—
426C23—41no—
427C23—41no—
428C23C3×#12012yes—
429C23C22×C3×D411yescharacteristic
430Q8—22no—
431Q8—24no—
432Q8—41no—
433Q8—41no—
434Q8—42no—
435Q8C3×#12314yes—
436D4—216no—
437D4—22no—
438D4—24no—
439D4C3×#12314yes—
440C2×C4—216no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—24no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—84no—
463C2×C4—88no—
464C2×C4C3×#12012yes—
465C2×C4C22×C3×D411yescharacteristic
466C2×C4C22×C3×D411yescharacteristic
467C2×C4C2×C3×#4112yes—
468C2×C4C2×C3×#4112yes—
469C8—41no—
470C8—41no—
471C8—42no—
472C2×C3—42no—
473C2×C3—44no—
474C2×C3#284911yescharacteristic
475C2×C3#300911yescharacteristic
476C2×C3#301311yescharacteristic
477C2×C3#312012yes—
478C2×C3#312512yes—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—24no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—44no—
487C22—42no—
488C22—44no—
489C22C2×C3×#12011yescharacteristic
490C22C3×#52512yes—
491C22C3×#57712yes—
492C22C3×#57711yescharacteristic
493C22C3×#57811yescharacteristic
494C4—22no—
495C4—24no—
496C4—42no—
497C4—44no—
498C4—42no—
499C4—81no—
500C4—81no—
501C4—82no—
502C4C2×C3×#12311yescharacteristic
503C4C2×C3×#12311yescharacteristic
504C4C3×#52512yes—
505C3#3301611yescharacteristic, direct product factor, Sylow
506C2—42no—
507C2—44no—
508C2C3×#284911yescharacteristic
509C2C3×#300911yescharacteristic
510C2C3×#301311yescharacteristic
511C2C3×#312012yes—
512C2C3×#312512yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#483C2semidirect
4. C3×#2625C2semidirect
5. C3×#1878C2semidirect
6. C3×#1877C2semidirect
7. C3×#1865C2semidirect
8. C3×#1864C2semidirect
9. C3×#1824C2semidirect
10. C3×#1818C2semidirect
12. #33016C3direct (N × H)
47. C2×#483C2×C3semidirect
49. #2625C2×C3semidirect
50. #1878C2×C3semidirect
51. #1877C2×C3semidirect
52. #1865C2×C3semidirect
53. #1864C2×C3semidirect
54. #1824C2×C3semidirect
55. #1818C2×C3semidirect
505. C3#33016direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5202
C2×C3×#4832
C2×C3×#4522
C3×#26252
C3×#18782
C3×#18772
C3×#18652
C3×#18642
C3×#18242
C3×#18182
C3×#17882
#330163