Finite Group Databasebeta

← C3×#29631 #10648238

Subgroups — C3×#29631 #10648238

Order 768.

1302 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#536C211yesmaximal, characteristic, complemented
2C2×C3×#449C211yesmaximal, characteristic
3C2×C3×#448C211yesmaximal, characteristic, complemented
4C3×#2624C211yesmaximal, characteristic, complemented
5C3×#2043C212yesmaximal, complemented
6C3×#2042C212yesmaximal, complemented
7C3×#2013C212yesmaximal, complemented
8C3×#2012C212yesmaximal, complemented
9C3×#1818C211yesmaximal, characteristic, complemented
10C3×#1788C211yesmaximal, characteristic, complemented
11C3×#1787C211yesmaximal, characteristic, complemented
12#29631C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2211yescharacteristic
14C23×C3×Q8C2211yescharacteristic
15C22×C3×SD16C2212yes—
16C22×C3×SD16C2211yescharacteristic
17C22×C3×D8C2211yescharacteristic
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×#121C2211yescharacteristic
26C2×C3×#117C2211yescharacteristic
27C3×#449—24no—
28C3×#448—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×#399—21no—
39C3×#397—21no—
40C3×#394—22no—
41C3×#390C2212yes—
42C3×#389C2212yes—
43C3×#380C2211yescharacteristic
44C3×#336C2212yes—
45C3×#336C2212yes—
46C2×#536C2×C311yescharacteristic, complemented
47C2×#449C2×C311yescharacteristic
48C2×#448C2×C311yescharacteristic, complemented
49#2624C2×C311yescharacteristic, complemented
50#2043C2×C312yescomplemented
51#2042C2×C312yescomplemented
52#2013C2×C312yescomplemented
53#2012C2×C312yescomplemented
54#1818C2×C311yescharacteristic, complemented
55#1788C2×C311yescharacteristic, complemented
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×Q8C2312yes—
67C22×C3×Q8C2311yescharacteristic
68C22×C3×D4—21no—
69C22×C3×D4—21no—
70C22×C3×D4C2311yescharacteristic
71C22×C3×D4C2311yescharacteristic
72C23×C3×C4—21no—
73C23×C3×C4—21no—
74C23×C3×C4C2311yescharacteristic
75C2×C3×SD16—24no—
76C2×C3×SD16—22no—
77C2×C3×SD16—42no—
78C2×C3×SD16—42no—
79C2×C3×SD16—42no—
80C2×C3×D8—22no—
81C2×C3×D8—42no—
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2312yes—
85C3×C4⋊Q8—24no—
86C3×C4⋊Q8—24no—
87C3×#124—24no—
88C3×#124—24no—
89C3×#123—216no—
90C3×C22⋊Q8—28no—
91C3×C22⋊Q8—28no—
92C3×#121—22no—
93C3×#121—22no—
94C3×#117—22no—
95C3×#117—22no—
96C2×C3×C4⋊C4—22no—
97C2×C3×C4⋊C4—22no—
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⋊C4—22no—
108C2×C3×C22⋊C4—22no—
109C2×C3×C22⋊C4D412yes—
110C2×C3×C42C2312yes—
111C3×#103—24no—
112C3×#103—22no—
113C3×#103—42no—
114C3×#103—42no—
115C3×#103—42no—
116C3×#102—22no—
117C3×#102—42no—
118C3×C22⋊C8—22no—
119C3×C22⋊C8—22no—
120C3×#95—21no—
121C3×#95—22no—
122C22×#41C22×C311yescharacteristic
123C23×Q8C22×C311yescharacteristic
124C22×SD16C22×C312yes—
125C22×SD16C22×C311yescharacteristic
126C22×D8C22×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×#121C22×C311yescharacteristic
135C2×#117C22×C311yescharacteristic
136#449—24no—
137#448—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#399—21no—
148#397—21no—
149#394—22no—
150#390C22×C312yes—
151#389C22×C312yes—
152#380C22×C311yescharacteristic
153#336C22×C312yes—
154#336C22×C312yes—
155C24×C3—21no—
156C24×C3—21no—
157C24×C3C2×D411yescharacteristic
158C3×#41—44no—
159C3×#41—44no—
160C3×#41—48no—
161C2×C3×Q8—24no—
162C2×C3×Q8—22no—
163C2×C3×Q8—21no—
164C2×C3×Q8—22no—
165C2×C3×Q8—22no—
166C2×C3×Q8—22no—
167C2×C3×Q8—22no—
168C2×C3×Q8—22no—
169C2×C3×Q8—22no—
170C2×C3×Q8—48no—
171C2×C3×Q8—44no—
172C2×C3×Q8—44no—
173C2×C3×Q8—44no—
174C2×C3×Q8C2×D414yes—
175C2×C3×Q8C2×D412yes—
176C2×C3×D4—24no—
177C2×C3×D4—21no—
178C2×C3×D4—21no—
179C2×C3×D4—21no—
180C2×C3×D4—22no—
181C2×C3×D4—22no—
182C2×C3×D4—22no—
183C2×C3×D4—22no—
184C2×C3×D4—44no—
185C2×C3×D4—42no—
186C2×C3×D4—82no—
187C2×C3×D4C2×D412yes—
188C22×C3×C4—22no—
189C22×C3×C4—22no—
190C22×C3×C4—24no—
191C22×C3×C4—22no—
192C22×C3×C4—21no—
193C22×C3×C4—24no—
194C22×C3×C4—22no—
195C22×C3×C4—21no—
196C22×C3×C4—44no—
197C22×C3×C4—44no—
198C22×C3×C4C2×D412yes—
199C22×C3×C4C2×D412yes—
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D412yes—
202C22×C3×C4C2×D412yes—
203C22×C3×C4C2411yescharacteristic
204C3×SD16—48no—
205C3×SD16—44no—
206C3×D8—44no—
207C2×C3×C8—21no—
208C2×C3×C8—21no—
209C2×C3×C8—22no—
210C2×C3×C8—22no—
211C2×C3×C8—22no—
212C2×C3×C8—22no—
213C2×C3×C8—22no—
214C3×C4⋊C4—28no—
215C3×C4⋊C4—28no—
216C3×C4⋊C4—22no—
217C3×C4⋊C4—22no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—22no—
221C3×C4⋊C4—22no—
222C3×C4⋊C4—82no—
223C3×C22⋊C4—28no—
224C3×C22⋊C4—28no—
225C3×C22⋊C4—24no—
226C3×C22⋊C4—42no—
227C3×C22⋊C4—82no—
228C3×C22⋊C4—82no—
229C3×C22⋊C4—81no—
230C3×C42—22no—
231C3×C42—22no—
232C2×#41—22no—
233C2×#41—42no—
234C2×#41—42no—
235C2×#41C3×D414yes—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8—42no—
239C22×Q8—42no—
240C22×Q8C3×D414yes—
241C22×Q8C23×C312yes—
242C22×Q8C23×C311yescharacteristic
243C22×D4—21no—
244C22×D4—21no—
245C22×D4C23×C311yescharacteristic
246C22×D4C23×C311yescharacteristic
247C23×C4—21no—
248C23×C4—21no—
249C23×C4C23×C311yescharacteristic
250C2×SD16—24no—
251C2×SD16—22no—
252C2×SD16—42no—
253C2×SD16—42no—
254C2×SD16—42no—
255C2×D8—22no—
256C2×D8—42no—
257C22×C8C23×C311yescharacteristic
258C22×C8C23×C311yescharacteristic
259C22×C8C23×C312yes—
260C4⋊Q8—24no—
261C4⋊Q8—24no—
262#124—24no—
263#124—24no—
264#123—216no—
265C22⋊Q8—28no—
266C22⋊Q8—28no—
267#121—22no—
268#121—22no—
269#117—22no—
270#117—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4C3×D411yescharacteristic
275C2×C4⋊C4C3×D411yescharacteristic
276C2×C4⋊C4C23×C311yescharacteristic
277C2×C4⋊C4C23×C312yes—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4C3×D412yes—
285C2×C42C23×C312yes—
286#103—24no—
287#103—22no—
288#103—42no—
289#103—42no—
290#103—42no—
291#102—22no—
292#102—42no—
293C22⋊C8—22no—
294C22⋊C8—22no—
295#95—21no—
296#95—22no—
297C23×C3—22no—
298C23×C3—21no—
299C23×C3—22no—
300C23×C3—21no—
301C23×C3—22no—
302C23×C3—42no—
303C23×C3—42no—
304C23×C3—41no—
305C23×C3—41no—
306C23×C3—41no—
307C23×C3—42no—
308C23×C3—41no—
309C23×C3—42no—
310C23×C3—41no—
311C23×C3#12012yes—
312C23×C3C22×D411yescharacteristic
313C3×Q8—24no—
314C3×Q8—22no—
315C3×Q8—24no—
316C3×Q8—22no—
317C3×Q8—48no—
318C3×Q8—42no—
319C3×Q8—42no—
320C3×Q8—42no—
321C3×D4—22no—
322C3×D4—22no—
323C3×D4—48no—
324C3×D4—42no—
325C3×D4—41no—
326C3×D4—41no—
327C3×D4—42no—
328C3×D4—84no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—24no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—24no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—48no—
342C2×C3×C4—48no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4—42no—
346C2×C3×C4—41no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4—42no—
351C2×C3×C4#12012yes—
352C2×C3×C4C22×D411yescharacteristic
353C2×C3×C4C22×D411yescharacteristic
354C2×C3×C4H(5,𝔽2)12yes—
355C2×C3×C4H(5,𝔽2)12yes—
356C3×C8—41no—
357C3×C8—41no—
358C3×C8—42no—
359C24—21no—
360C24—21no—
361C24C2×C3×D411yescharacteristic
362#41—44no—
363#41—44no—
364#41—48no—
365C2×Q8—24no—
366C2×Q8—22no—
367C2×Q8—21no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—48no—
375C2×Q8—44no—
376C2×Q8—44no—
377C2×Q8—44no—
378C2×Q8C2×C3×D414yes—
379C2×Q8C2×C3×D412yes—
380C2×D4—24no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—44no—
389C2×D4—42no—
390C2×D4—82no—
391C2×D4C2×C3×D412yes—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4C2×C3×D412yes—
403C22×C4C2×C3×D412yes—
404C22×C4C2×C3×D411yescharacteristic
405C22×C4C2×C3×D412yes—
406C22×C4C2×C3×D412yes—
407C22×C4C24×C311yescharacteristic
408SD16—48no—
409SD16—44no—
410D8—44no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C4⋊C4—28no—
419C4⋊C4—28no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—82no—
427C22⋊C4—28no—
428C22⋊C4—28no—
429C22⋊C4—24no—
430C22⋊C4—42no—
431C22⋊C4—82no—
432C22⋊C4—82no—
433C22⋊C4—81no—
434C42—22no—
435C42—22no—
436C22×C3—22no—
437C22×C3—22no—
438C22×C3—22no—
439C22×C3—42no—
440C22×C3—42no—
441C22×C3—42no—
442C22×C3—42no—
443C22×C3—42no—
444C22×C3—82no—
445C22×C3—82no—
446C22×C3—81no—
447C22×C3—82no—
448C22×C3—82no—
449C22×C3—81no—
450C22×C3—81no—
451C22×C3C2×#12011yescharacteristic
452C22×C3#53512yes—
453C22×C3#57711yescharacteristic
454C22×C3#57712yes—
455C22×C3#57811yescharacteristic
456C3×C4—22no—
457C3×C4—42no—
458C3×C4—44no—
459C3×C4—42no—
460C3×C4—42no—
461C3×C4—82no—
462C3×C4—81no—
463C3×C4#53411yescharacteristic
464C3×C4#53411yescharacteristic
465C3×C4#53412yes—
466C23—21no—
467C23—22no—
468C23—21no—
469C23—22no—
470C23—22no—
471C23—42no—
472C23—41no—
473C23—41no—
474C23—42no—
475C23—41no—
476C23—42no—
477C23—41no—
478C23—41no—
479C23—42no—
480C23C3×#12012yes—
481C23C22×C3×D411yescharacteristic
482Q8—24no—
483Q8—22no—
484Q8—24no—
485Q8—22no—
486Q8—48no—
487Q8—42no—
488Q8—42no—
489Q8—42no—
490D4—22no—
491D4—22no—
492D4—48no—
493D4—42no—
494D4—41no—
495D4—41no—
496D4—42no—
497D4—84no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—48no—
511C2×C4—48no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4C3×#12012yes—
521C2×C4C22×C3×D411yescharacteristic
522C2×C4C22×C3×D411yescharacteristic
523C2×C4C3×H(5,𝔽2)12yes—
524C2×C4C3×H(5,𝔽2)12yes—
525C8—41no—
526C8—41no—
527C8—42no—
528C2×C3—42no—
529C2×C3—82no—
530C2×C3—161no—
531C2×C3#261311yescharacteristic
532C2×C3#301011yescharacteristic
533C2×C3#301311yescharacteristic
534C2×C3#322112yes—
535C2×C3#322512yes—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—42no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—82no—
545C22—82no—
546C22—82no—
547C22—82no—
548C22—81no—
549C22—81no—
550C22—81no—
551C22C2×C3×#12011yescharacteristic
552C22C3×#53512yes—
553C22C3×#57711yescharacteristic
554C22C3×#57712yes—
555C22C3×#57811yescharacteristic
556C4—22no—
557C4—42no—
558C4—42no—
559C4—42no—
560C4—44no—
561C4—82no—
562C4—81no—
563C4C3×#53411yescharacteristic
564C4C3×#53411yescharacteristic
565C4C3×#53412yes—
566C3#2963111yescharacteristic, direct product factor, Sylow
567C2—42no—
568C2—82no—
569C2—161no—
570C2C3×#261311yescharacteristic
571C2C3×#301011yescharacteristic
572C2C3×#301311yescharacteristic
573C2C3×#322112yes—
574C2C3×#322512yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#449C2
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
1. C2×C3×#536C2semidirect
3. C2×C3×#448C2semidirect
4. C3×#2624C2semidirect
5. C3×#2043C2semidirect
6. C3×#2042C2semidirect
7. C3×#2013C2semidirect
8. C3×#2012C2semidirect
9. C3×#1818C2semidirect
10. C3×#1788C2semidirect
11. C3×#1787C2semidirect
12. #29631C3direct (N × H)
46. C2×#536C2×C3semidirect
48. C2×#448C2×C3semidirect
49. #2624C2×C3semidirect
50. #2043C2×C3semidirect
51. #2042C2×C3semidirect
52. #2013C2×C3semidirect
53. #2012C2×C3semidirect
54. #1818C2×C3semidirect
55. #1788C2×C3semidirect
56. #1787C2×C3semidirect
566. C3#29631direct (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×#4492
C2×C3×#4482
C3×#26242
C3×#20432
C3×#20422
C3×#20132
C3×#20122
C3×#18182
C3×#17882
C3×#17872
#296313