Finite Group Databasebeta

← C3×#29633 #10648240

Subgroups — C3×#29633 #10648240

Order 768.

1260 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#536C211yesmaximal, characteristic
2C2×C3×#452C211yesmaximal, characteristic
3C2×C3×#451C211yesmaximal, characteristic, complemented
4C3×#2625C211yesmaximal, characteristic, complemented
5C3×#2043C212yesmaximal, complemented
6C3×#2042C212yesmaximal, complemented
7C3×#2016C212yesmaximal, complemented
8C3×#2015C212yesmaximal, complemented
9C3×#1818C211yesmaximal, characteristic, complemented
10C3×#1788C211yesmaximal, characteristic
11C3×#1787C211yesmaximal, characteristic, complemented
12#29633C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2211yescharacteristic
14C23×C3×Q8C2211yescharacteristic
15C22×C3×Q16C2212yes—
16C22×C3×Q16C2211yescharacteristic
17C22×C3×SD16C2211yescharacteristic
18C3×#536—24no—
19C3×#536—24no—
20C2×C3×C4⋊Q8C2212yes—
21C2×C3×#124C2212yes—
22C2×C3×#123—22no—
23C2×C3×C22⋊Q8—21no—
24C2×C3×C22⋊Q8—21no—
25C2×C3×C22⋊Q8C2211yescharacteristic
26C2×C3×#117C2211yescharacteristic
27C3×#452—24no—
28C3×#451—24no—
29C2×C3×#103C2212yes—
30C2×C3×#103C2211yescharacteristic
31C2×C3×#103C2212yes—
32C2×C3×#103C2212yes—
33C2×C3×#103C2212yes—
34C2×C3×#102C2211yescharacteristic
35C2×C3×#102C2212yes—
36C2×C3×C22⋊C8C2211yescharacteristic
37C2×C3×C22⋊C8C2211yescharacteristic
38C3×#400—21no—
39C3×#398—21no—
40C3×#397—22no—
41C3×#391C2212yes—
42C3×#389C2212yes—
43C3×#380C2211yescharacteristic
44C3×#336C2212yes—
45C3×#336C2212yes—
46C2×#536C2×C311yescharacteristic
47C2×#452C2×C311yescharacteristic
48C2×#451C2×C311yescharacteristic, complemented
49#2625C2×C311yescharacteristic, complemented
50#2043C2×C312yescomplemented
51#2042C2×C312yescomplemented
52#2016C2×C312yescomplemented
53#2015C2×C312yescomplemented
54#1818C2×C311yescharacteristic, complemented
55#1788C2×C311yescharacteristic
56#1787C2×C311yescharacteristic, complemented
57C2×C3×#41—22no—
58C2×C3×#41—42no—
59C2×C3×#41—42no—
60C2×C3×#41D414yes—
61C22×C3×Q8—21no—
62C22×C3×Q8—22no—
63C22×C3×Q8—42no—
64C22×C3×Q8—42no—
65C22×C3×Q8D414yes—
66C22×C3×Q8C2311yescharacteristic
67C22×C3×Q8C2312yes—
68C22×C3×Q8C2311yescharacteristic
69C22×C3×D4—21no—
70C22×C3×D4C2311yescharacteristic
71C23×C3×C4—21no—
72C23×C3×C4—21no—
73C23×C3×C4C2311yescharacteristic
74C2×C3×Q16—24no—
75C2×C3×Q16—22no—
76C2×C3×Q16—42no—
77C2×C3×Q16—42no—
78C2×C3×Q16—42no—
79C2×C3×SD16—22no—
80C2×C3×SD16—42no—
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2312yes—
84C3×C4⋊Q8—24no—
85C3×C4⋊Q8—24no—
86C3×#124—24no—
87C3×#124—24no—
88C3×#123—216no—
89C3×C22⋊Q8—28no—
90C3×C22⋊Q8—28no—
91C3×C22⋊Q8—22no—
92C3×C22⋊Q8—22no—
93C3×#117—22no—
94C3×#117—22no—
95C2×C3×C4⋊C4—22no—
96C2×C3×C4⋊C4—22no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4—22no—
99C2×C3×C4⋊C4D411yescharacteristic
100C2×C3×C4⋊C4D411yescharacteristic
101C2×C3×C4⋊C4C2312yes—
102C2×C3×C4⋊C4C2311yescharacteristic
103C2×C3×C22⋊C4—22no—
104C2×C3×C22⋊C4—22no—
105C2×C3×C22⋊C4—21no—
106C2×C3×C22⋊C4—21no—
107C2×C3×C22⋊C4D412yes—
108C2×C3×C42C2312yes—
109C3×#103—24no—
110C3×#103—22no—
111C3×#103—42no—
112C3×#103—42no—
113C3×#103—42no—
114C3×#102—22no—
115C3×#102—42no—
116C3×C22⋊C8—22no—
117C3×C22⋊C8—22no—
118C3×#95—21no—
119C3×#95—22no—
120C3×#95—22no—
121C3×#95—22no—
122C22×#41C22×C311yescharacteristic
123C23×Q8C22×C311yescharacteristic
124C22×Q16C22×C312yes—
125C22×Q16C22×C311yescharacteristic
126C22×SD16C22×C311yescharacteristic
127#536—24no—
128#536—24no—
129C2×C4⋊Q8C22×C312yes—
130C2×#124C22×C312yes—
131C2×#123—22no—
132C2×C22⋊Q8—21no—
133C2×C22⋊Q8—21no—
134C2×C22⋊Q8C22×C311yescharacteristic
135C2×#117C22×C311yescharacteristic
136#452—24no—
137#451—24no—
138C2×#103C22×C312yes—
139C2×#103C22×C311yescharacteristic
140C2×#103C22×C312yes—
141C2×#103C22×C312yes—
142C2×#103C22×C312yes—
143C2×#102C22×C311yescharacteristic
144C2×#102C22×C312yes—
145C2×C22⋊C8C22×C311yescharacteristic
146C2×C22⋊C8C22×C311yescharacteristic
147#400—21no—
148#398—21no—
149#397—22no—
150#391C22×C312yes—
151#389C22×C312yes—
152#380C22×C311yescharacteristic
153#336C22×C312yes—
154#336C22×C312yes—
155C24×C3—21no—
156C24×C3C2×D411yescharacteristic
157C3×#41—44no—
158C3×#41—44no—
159C3×#41—48no—
160C2×C3×Q8—24no—
161C2×C3×Q8—22no—
162C2×C3×Q8—21no—
163C2×C3×Q8—21no—
164C2×C3×Q8—21no—
165C2×C3×Q8—22no—
166C2×C3×Q8—22no—
167C2×C3×Q8—22no—
168C2×C3×Q8—22no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C2×C3×Q8—22no—
173C2×C3×Q8—48no—
174C2×C3×Q8—44no—
175C2×C3×Q8—44no—
176C2×C3×Q8—44no—
177C2×C3×Q8C2×D414yes—
178C2×C3×Q8C2×D412yes—
179C2×C3×D4—24no—
180C2×C3×D4—21no—
181C2×C3×D4—22no—
182C2×C3×D4—22no—
183C2×C3×D4—44no—
184C2×C3×D4C2×D412yes—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—24no—
188C22×C3×C4—22no—
189C22×C3×C4—21no—
190C22×C3×C4—24no—
191C22×C3×C4—22no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—44no—
195C22×C3×C4—44no—
196C22×C3×C4C2×D412yes—
197C22×C3×C4C2×D412yes—
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D412yes—
200C22×C3×C4C2×D412yes—
201C22×C3×C4C2411yescharacteristic
202C3×Q16—48no—
203C3×Q16—44no—
204C3×SD16—44no—
205C2×C3×C8—21no—
206C2×C3×C8—21no—
207C2×C3×C8—22no—
208C2×C3×C8—22no—
209C2×C3×C8—22no—
210C2×C3×C8—22no—
211C2×C3×C8—22no—
212C3×C4⋊C4—28no—
213C3×C4⋊C4—28no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4—22no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—22no—
219C3×C4⋊C4—22no—
220C3×C4⋊C4—42no—
221C3×C4⋊C4—82no—
222C3×C22⋊C4—28no—
223C3×C22⋊C4—28no—
224C3×C22⋊C4—24no—
225C3×C22⋊C4—42no—
226C3×C22⋊C4—81no—
227C3×C42—22no—
228C3×C42—22no—
229C2×#41—22no—
230C2×#41—42no—
231C2×#41—42no—
232C2×#41C3×D414yes—
233C22×Q8—21no—
234C22×Q8—22no—
235C22×Q8—42no—
236C22×Q8—42no—
237C22×Q8C3×D414yes—
238C22×Q8C23×C311yescharacteristic
239C22×Q8C23×C312yes—
240C22×Q8C23×C311yescharacteristic
241C22×D4—21no—
242C22×D4C23×C311yescharacteristic
243C23×C4—21no—
244C23×C4—21no—
245C23×C4C23×C311yescharacteristic
246C2×Q16—24no—
247C2×Q16—22no—
248C2×Q16—42no—
249C2×Q16—42no—
250C2×Q16—42no—
251C2×SD16—22no—
252C2×SD16—42no—
253C22×C8C23×C311yescharacteristic
254C22×C8C23×C311yescharacteristic
255C22×C8C23×C312yes—
256C4⋊Q8—24no—
257C4⋊Q8—24no—
258#124—24no—
259#124—24no—
260#123—216no—
261C22⋊Q8—28no—
262C22⋊Q8—28no—
263C22⋊Q8—22no—
264C22⋊Q8—22no—
265#117—22no—
266#117—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4C3×D411yescharacteristic
272C2×C4⋊C4C3×D411yescharacteristic
273C2×C4⋊C4C23×C311yescharacteristic
274C2×C4⋊C4C23×C312yes—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4C3×D412yes—
280C2×C42C23×C312yes—
281#103—24no—
282#103—22no—
283#103—42no—
284#103—42no—
285#103—42no—
286#102—22no—
287#102—42no—
288C22⋊C8—22no—
289C22⋊C8—22no—
290#95—21no—
291#95—22no—
292#95—22no—
293#95—22no—
294C23×C3—22no—
295C23×C3—21no—
296C23×C3—22no—
297C23×C3—21no—
298C23×C3—22no—
299C23×C3—42no—
300C23×C3—41no—
301C23×C3—42no—
302C23×C3—41no—
303C23×C3#12012yes—
304C23×C3C22×D411yescharacteristic
305C3×Q8—24no—
306C3×Q8—22no—
307C3×Q8—24no—
308C3×Q8—22no—
309C3×Q8—48no—
310C3×Q8—42no—
311C3×Q8—42no—
312C3×Q8—42no—
313C3×Q8—41no—
314C3×Q8—41no—
315C3×Q8—42no—
316C3×D4—22no—
317C3×D4—22no—
318C3×D4—48no—
319C3×D4—42no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—24no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—24no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—24no—
330C2×C3×C4—22no—
331C2×C3×C4—48no—
332C2×C3×C4—48no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—41no—
342C2×C3×C4—42no—
343C2×C3×C4—41no—
344C2×C3×C4#12012yes—
345C2×C3×C4C22×D411yescharacteristic
346C2×C3×C4C22×D411yescharacteristic
347C2×C3×C4H(5,𝔽2)12yes—
348C2×C3×C4H(5,𝔽2)12yes—
349C3×C8—41no—
350C3×C8—41no—
351C3×C8—42no—
352C24—21no—
353C24C2×C3×D411yescharacteristic
354#41—44no—
355#41—44no—
356#41—48no—
357C2×Q8—24no—
358C2×Q8—22no—
359C2×Q8—21no—
360C2×Q8—21no—
361C2×Q8—21no—
362C2×Q8—22no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—48no—
371C2×Q8—44no—
372C2×Q8—44no—
373C2×Q8—44no—
374C2×Q8C2×C3×D414yes—
375C2×Q8C2×C3×D412yes—
376C2×D4—24no—
377C2×D4—21no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—44no—
381C2×D4C2×C3×D412yes—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4C2×C3×D412yes—
394C22×C4C2×C3×D412yes—
395C22×C4C2×C3×D411yescharacteristic
396C22×C4C2×C3×D412yes—
397C22×C4C2×C3×D412yes—
398C22×C4C24×C311yescharacteristic
399Q16—48no—
400Q16—44no—
401SD16—44no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C4⋊C4—28no—
410C4⋊C4—28no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—42no—
418C4⋊C4—82no—
419C22⋊C4—28no—
420C22⋊C4—28no—
421C22⋊C4—24no—
422C22⋊C4—42no—
423C22⋊C4—81no—
424C42—22no—
425C42—22no—
426C22×C3—22no—
427C22×C3—22no—
428C22×C3—22no—
429C22×C3—42no—
430C22×C3—42no—
431C22×C3—42no—
432C22×C3—42no—
433C22×C3—42no—
434C22×C3—82no—
435C22×C3—82no—
436C22×C3C2×#12011yescharacteristic
437C22×C3#53512yes—
438C22×C3#57711yescharacteristic
439C22×C3#57811yescharacteristic
440C22×C3#57812yes—
441C3×C4—22no—
442C3×C4—42no—
443C3×C4—44no—
444C3×C4—42no—
445C3×C4—42no—
446C3×C4—81no—
447C3×C4—82no—
448C3×C4—81no—
449C3×C4#53411yescharacteristic
450C3×C4#53411yescharacteristic
451C3×C4#53412yes—
452C23—21no—
453C23—22no—
454C23—21no—
455C23—22no—
456C23—22no—
457C23—41no—
458C23—42no—
459C23—41no—
460C23—42no—
461C23C3×#12012yes—
462C23C22×C3×D411yescharacteristic
463Q8—24no—
464Q8—22no—
465Q8—24no—
466Q8—22no—
467Q8—48no—
468Q8—42no—
469Q8—42no—
470Q8—42no—
471Q8—41no—
472Q8—41no—
473Q8—42no—
474D4—22no—
475D4—22no—
476D4—48no—
477D4—42no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—48no—
490C2×C4—48no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4C3×#12012yes—
503C2×C4C22×C3×D411yescharacteristic
504C2×C4C22×C3×D411yescharacteristic
505C2×C4C3×H(5,𝔽2)12yes—
506C2×C4C3×H(5,𝔽2)12yes—
507C8—41no—
508C8—41no—
509C8—42no—
510C2×C3—42no—
511C2×C3—82no—
512C2×C3#261311yescharacteristic
513C2×C3#301111yescharacteristic
514C2×C3#301311yescharacteristic
515C2×C3#322312yes—
516C2×C3#322512yes—
517C22—22no—
518C22—22no—
519C22—22no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—82no—
526C22—82no—
527C22C2×C3×#12011yescharacteristic
528C22C3×#53512yes—
529C22C3×#57711yescharacteristic
530C22C3×#57811yescharacteristic
531C22C3×#57812yes—
532C4—22no—
533C4—42no—
534C4—42no—
535C4—42no—
536C4—44no—
537C4—82no—
538C4—81no—
539C4—81no—
540C4C3×#53411yescharacteristic
541C4C3×#53411yescharacteristic
542C4C3×#53412yes—
543C3#2963311yescharacteristic, direct product factor, Sylow
544C2—42no—
545C2—82no—
546C2C3×#261311yescharacteristic
547C2C3×#301111yescharacteristic
548C2C3×#301311yescharacteristic
549C2C3×#322312yes—
550C2C3×#322512yes—
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
3. C2×C3×#451C2semidirect
4. C3×#2625C2semidirect
5. C3×#2043C2semidirect
6. C3×#2042C2semidirect
7. C3×#2016C2semidirect
8. C3×#2015C2semidirect
9. C3×#1818C2semidirect
11. C3×#1787C2semidirect
12. #29633C3direct (N × H)
48. C2×#451C2×C3semidirect
49. #2625C2×C3semidirect
50. #2043C2×C3semidirect
51. #2042C2×C3semidirect
52. #2016C2×C3semidirect
53. #2015C2×C3semidirect
54. #1818C2×C3semidirect
56. #1787C2×C3semidirect
543. C3#29633direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5362
C2×C3×#4522
C2×C3×#4512
C3×#26252
C3×#20432
C3×#20422
C3×#20162
C3×#20152
C3×#18182
C3×#17882
C3×#17872
#296333