Finite Group Databasebeta

← C3×#29613 #10648220

Subgroups — C3×#29613 #10648220

Order 768.

1256 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#536C211yesmaximal, characteristic
2C3×#3095C211yesmaximal, characteristic, complemented
3C3×#3007C211yesmaximal, characteristic
4C3×#2434C211yesmaximal, characteristic, complemented
5C3×#2038C214yesmaximal, complemented
6C3×#2005C214yesmaximal, complemented
7C3×#1826C211yesmaximal, characteristic, complemented
8C3×#1789C211yesmaximal, characteristic
9C3×#1784C211yesmaximal, characteristic, complemented
10#29613C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×#41C2211yescharacteristic
12C23×C3×Q8C2211yescharacteristic
13C2×C3×#137C2212yes—
14C3×#536—22no—
15C3×#536—22no—
16C3×#536—22no—
17C3×#536C2214yes—
18C2×C3×C4⋊Q8—21no—
19C2×C3×C4⋊Q8C2212yescomplemented
20C2×C3×#124—21no—
21C2×C3×#123—22no—
22C2×C3×C22⋊Q8—21no—
23C2×C3×C22⋊Q8—21no—
24C2×C3×C22⋊Q8C2211yescharacteristic
25C2×C3×#117C2211yescharacteristic
26C3×#485—22no—
27C3×#484—21no—
28C3×#483—21no—
29C3×#452—22no—
30C3×#451—21no—
31C3×#450—21no—
32C3×#428C2212yescomplemented
33C2×C3×#104C2214yes—
34C3×#419C2212yescomplemented
35C3×#417C2212yes—
36C2×C3×#101C2214yescomplemented
37C3×#407C2211yescharacteristic
38C3×#407C2211yescharacteristic
39C3×#395—22no—
40C3×#391C2212yescomplemented
41C3×#385—22no—
42C3×C4×C22⋊C4C2211yescharacteristic
43C3×#339C2214yescomplemented
44C2×#536C2×C311yescharacteristic
45#3095C2×C311yescharacteristic, complemented
46#3007C2×C311yescharacteristic
47#2434C2×C311yescharacteristic, complemented
48#2038C2×C314yescomplemented
49#2005C2×C314yescomplemented
50#1826C2×C311yescharacteristic, complemented
51#1789C2×C311yescharacteristic
52#1784C2×C311yescharacteristic, complemented
53C2×C3×#41—21no—
54C2×C3×#41—21no—
55C2×C3×#41—44no—
56C2×C3×#41D412yes—
57C2×C3×#41C2312yes—
58C22×C3×Q8—21no—
59C22×C3×Q8—21no—
60C22×C3×Q8—21no—
61C22×C3×Q8—21no—
62C22×C3×Q8—44no—
63C22×C3×Q8D411yescharacteristic
64C22×C3×Q8D412yes—
65C22×C3×Q8C2311yescharacteristic
66C22×C3×Q8C2312yes—
67C22×C3×D4—21no—
68C22×C3×D4D411yescharacteristic
69C23×C3×C4—21no—
70C23×C3×C4D412yes—
71C23×C3×C4C2311yescharacteristic
72C3×#137—44no—
73C2×C3×Q16—22no—
74C2×C3×SD16—22no—
75C2×C3×M16C2312yes—
76C2×C3×M16C2312yes—
77C3×C4⋊Q8—24no—
78C3×C4⋊Q8—24no—
79C3×C4⋊Q8—44no—
80C3×#124—24no—
81C3×#124—24no—
82C3×#123—28no—
83C3×#123—28no—
84C3×C22⋊Q8—24no—
85C3×C22⋊Q8—24no—
86C3×C22⋊Q8—24no—
87C3×C22⋊Q8—24no—
88C3×C22⋊Q8—21no—
89C3×C22⋊Q8—21no—
90C3×C22⋊Q8—21no—
91C3×C22⋊Q8—21no—
92C3×#117—21no—
93C3×#117—21no—
94C3×#117D412yes—
95C3×#117C2312yes—
96C2×C3×C4⋊C4—22no—
97C2×C3×C4⋊C4—22no—
98C2×C3×C4⋊C4—24no—
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4D411yescharacteristic
101C2×C3×C4⋊C4D411yescharacteristic
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—21no—
108C2×C3×C42—21no—
109C2×C3×C42C2312yes—
110C3×#107—22no—
111C3×#106—22no—
112C3×#104—44no—
113C3×#103—22no—
114C3×#103—22no—
115C3×#103—22no—
116C3×#102—22no—
117C3×#101—44no—
118C3×C22⋊C8—21no—
119C3×C22⋊C8—21no—
120C3×C22⋊C8—21no—
121C3×C22⋊C8—21no—
122C3×#95—21no—
123C3×#95—24no—
124C22×#41C22×C311yescharacteristic
125C23×Q8C22×C311yescharacteristic
126C2×#137C22×C312yes—
127#536—22no—
128#536—22no—
129#536—22no—
130#536C22×C314yes—
131C2×C4⋊Q8—21no—
132C2×C4⋊Q8C22×C312yescomplemented
133C2×#124—21no—
134C2×#123—22no—
135C2×C22⋊Q8—21no—
136C2×C22⋊Q8—21no—
137C2×C22⋊Q8C22×C311yescharacteristic
138C2×#117C22×C311yescharacteristic
139#485—22no—
140#484—21no—
141#483—21no—
142#452—22no—
143#451—21no—
144#450—21no—
145#428C22×C312yescomplemented
146C2×#104C22×C314yes—
147#419C22×C312yescomplemented
148#417C22×C312yes—
149C2×#101C22×C314yescomplemented
150#407C22×C311yescharacteristic
151#407C22×C311yescharacteristic
152#395—22no—
153#391C22×C312yescomplemented
154#385—22no—
155C4×C22⋊C4C22×C311yescharacteristic
156#339C22×C314yescomplemented
157C24×C3—21no—
158C24×C3C2×D411yescharacteristic
159C3×#41—24no—
160C3×#41—24no—
161C3×#41—42no—
162C3×#41—42no—
163C3×#41—44no—
164C3×#41—44no—
165C2×C3×Q8—22no—
166C2×C3×Q8—22no—
167C2×C3×Q8—21no—
168C2×C3×Q8—22no—
169C2×C3×Q8—22no—
170C2×C3×Q8—24no—
171C2×C3×Q8—24no—
172C2×C3×Q8—21no—
173C2×C3×Q8—21no—
174C2×C3×Q8—24no—
175C2×C3×Q8—44no—
176C2×C3×Q8—42no—
177C2×C3×Q8—44no—
178C2×C3×Q8—44no—
179C2×C3×Q8—44no—
180C2×C3×Q8—44no—
181C2×C3×Q8C2×D412yes—
182C2×C3×D4—22no—
183C2×C3×D4—22no—
184C2×C3×D4—21no—
185C2×C3×D4—42no—
186C2×C3×D4—44no—
187C2×C3×D4C2×D412yes—
188C22×C3×C4—24no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—22no—
192C22×C3×C4—21no—
193C22×C3×C4—22no—
194C22×C3×C4—22no—
195C22×C3×C4—22no—
196C22×C3×C4—22no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—44no—
200C22×C3×C4—44no—
201C22×C3×C4C2×D414yes—
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2411yescharacteristic
208C3×Q16—44no—
209C3×SD16—44no—
210C3×M16—24no—
211C3×M16—24no—
212C2×C3×C8—22no—
213C2×C3×C8—22no—
214C3×C4⋊C4—28no—
215C3×C4⋊C4—28no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—48no—
221C3×C4⋊C4—41no—
222C3×C4⋊C4—41no—
223C3×C4⋊C4C2×D412yes—
224C3×C4⋊C4C2×D412yes—
225C3×C22⋊C4—24no—
226C3×C22⋊C4—28no—
227C3×C22⋊C4—28no—
228C3×C22⋊C4—22no—
229C3×C22⋊C4—22no—
230C3×C22⋊C4—41no—
231C3×C22⋊C4—41no—
232C3×C42—22no—
233C3×C42—22no—
234C3×C42—24no—
235C2×#41—21no—
236C2×#41—21no—
237C2×#41—44no—
238C2×#41C3×D412yes—
239C2×#41C23×C312yes—
240C22×Q8—21no—
241C22×Q8—21no—
242C22×Q8—21no—
243C22×Q8—21no—
244C22×Q8—44no—
245C22×Q8C3×D411yescharacteristic
246C22×Q8C3×D412yes—
247C22×Q8C23×C311yescharacteristic
248C22×Q8C23×C312yes—
249C22×D4—21no—
250C22×D4C3×D411yescharacteristic
251C23×C4—21no—
252C23×C4C3×D412yes—
253C23×C4C23×C311yescharacteristic
254#137—44no—
255C2×Q16—22no—
256C2×SD16—22no—
257C2×M16C23×C312yes—
258C2×M16C23×C312yes—
259C4⋊Q8—24no—
260C4⋊Q8—24no—
261C4⋊Q8—44no—
262#124—24no—
263#124—24no—
264#123—28no—
265#123—28no—
266C22⋊Q8—24no—
267C22⋊Q8—24no—
268C22⋊Q8—24no—
269C22⋊Q8—24no—
270C22⋊Q8—21no—
271C22⋊Q8—21no—
272C22⋊Q8—21no—
273C22⋊Q8—21no—
274#117—21no—
275#117—21no—
276#117C3×D412yes—
277#117C23×C312yes—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4C3×D411yescharacteristic
283C2×C4⋊C4C3×D411yescharacteristic
284C2×C4⋊C4C23×C311yescharacteristic
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C42—21no—
291C2×C42C23×C312yes—
292#107—22no—
293#106—22no—
294#104—44no—
295#103—22no—
296#103—22no—
297#103—22no—
298#102—22no—
299#101—44no—
300C22⋊C8—21no—
301C22⋊C8—21no—
302C22⋊C8—21no—
303C22⋊C8—21no—
304#95—21no—
305#95—24no—
306C23×C3—22no—
307C23×C3—22no—
308C23×C3—22no—
309C23×C3—41no—
310C23×C3—42no—
311C23×C3—42no—
312C23×C3—42no—
313C23×C3#12011yescharacteristic
314C23×C3#12011yescharacteristic
315C23×C3C22×D411yescharacteristic
316C3×Q8—22no—
317C3×Q8—22no—
318C3×Q8—44no—
319C3×Q8—44no—
320C3×Q8—42no—
321C3×Q8—44no—
322C3×Q8—44no—
323C3×Q8—42no—
324C3×Q8—42no—
325C3×D4—22no—
326C3×D4—22no—
327C3×D4—42no—
328C3×D4—44no—
329C3×D4—44no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—24no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—24no—
339C2×C3×C4—22no—
340C2×C3×C4—24no—
341C2×C3×C4—24no—
342C2×C3×C4—44no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4—42no—
346C2×C3×C4—44no—
347C2×C3×C4—44no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4—44no—
351C2×C3×C4—42no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—42no—
355C2×C3×C4#12011yescharacteristic
356C2×C3×C4#12011yescharacteristic
357C2×C3×C4C22×D411yescharacteristic
358C2×C3×C4C22×D411yescharacteristic
359C2×C3×C4H(5,𝔽2)14yes—
360C3×C8—42no—
361C3×C8—42no—
362C24—21no—
363C24C2×C3×D411yescharacteristic
364#41—24no—
365#41—24no—
366#41—42no—
367#41—42no—
368#41—44no—
369#41—44no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—21no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—24no—
376C2×Q8—24no—
377C2×Q8—21no—
378C2×Q8—21no—
379C2×Q8—24no—
380C2×Q8—44no—
381C2×Q8—42no—
382C2×Q8—44no—
383C2×Q8—44no—
384C2×Q8—44no—
385C2×Q8—44no—
386C2×Q8C2×C3×D412yes—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—21no—
390C2×D4—42no—
391C2×D4—44no—
392C2×D4C2×C3×D412yes—
393C22×C4—24no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4C2×C3×D414yes—
407C22×C4C2×C3×D411yescharacteristic
408C22×C4C2×C3×D411yescharacteristic
409C22×C4C2×C3×D411yescharacteristic
410C22×C4C2×C3×D411yescharacteristic
411C22×C4C2×C3×D411yescharacteristic
412C22×C4C24×C311yescharacteristic
413Q16—44no—
414SD16—44no—
415M16—24no—
416M16—24no—
417C2×C8—22no—
418C2×C8—22no—
419C4⋊C4—28no—
420C4⋊C4—28no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—48no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4C2×C3×D412yes—
429C4⋊C4C2×C3×D412yes—
430C22⋊C4—28no—
431C22⋊C4—28no—
432C22⋊C4—24no—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C42—22no—
438C42—22no—
439C42—24no—
440C22×C3—21no—
441C22×C3—21no—
442C22×C3—21no—
443C22×C3—21no—
444C22×C3—21no—
445C22×C3—21no—
446C22×C3—21no—
447C22×C3—44no—
448C22×C3—42no—
449C22×C3—42no—
450C22×C3—42no—
451C22×C3—84no—
452C22×C3#45612yes—
453C22×C3C2×#12011yescharacteristic
454C22×C3#53412yes—
455C3×C4—21no—
456C3×C4—21no—
457C3×C4—44no—
458C3×C4—44no—
459C3×C4—42no—
460C3×C4—42no—
461C3×C4—42no—
462C3×C4—81no—
463C3×C4—81no—
464C3×C4#53412yes—
465C3×C4#53512yes—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—41no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23C3×#12011yescharacteristic
474C23C3×#12011yescharacteristic
475C23C22×C3×D411yescharacteristic
476Q8—22no—
477Q8—22no—
478Q8—44no—
479Q8—44no—
480Q8—42no—
481Q8—44no—
482Q8—44no—
483Q8—42no—
484Q8—42no—
485D4—22no—
486D4—22no—
487D4—42no—
488D4—44no—
489D4—44no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4C3×#12011yescharacteristic
516C2×C4C3×#12011yescharacteristic
517C2×C4C22×C3×D411yescharacteristic
518C2×C4C22×C3×D411yescharacteristic
519C2×C4C3×H(5,𝔽2)14yes—
520C8—42no—
521C8—42no—
522C2×C3—22no—
523C2×C3—22no—
524C2×C3—41no—
525C2×C3—82no—
526C2×C3#261311yescharacteristic
527C2×C3C2×#45611yescharacteristic
528C2×C3#302011yescharacteristic
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—44no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—84no—
541C22C3×#45612yes—
542C22C2×C3×#12011yescharacteristic
543C22C3×#53412yes—
544C4—21no—
545C4—21no—
546C4—44no—
547C4—42no—
548C4—42no—
549C4—44no—
550C4—42no—
551C4—81no—
552C4—81no—
553C4C3×#53412yes—
554C4C3×#53512yes—
555C3#2961311yescharacteristic, direct product factor, Sylow
556C2—22no—
557C2—22no—
558C2—41no—
559C2—82no—
560C2C3×#261311yescharacteristic
561C2C2×C3×#45611yescharacteristic
562C2C3×#302011yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#3095C2semidirect
4. C3×#2434C2semidirect
5. C3×#2038C2semidirect
6. C3×#2005C2semidirect
7. C3×#1826C2semidirect
9. C3×#1784C2semidirect
10. #29613C3direct (N × H)
19. C2×C3×C4⋊Q8C22semidirect
32. C3×#428C22semidirect
34. C3×#419C22semidirect
36. C2×C3×#101C22semidirect
40. C3×#391C22semidirect
43. C3×#339C22semidirect
45. #3095C2×C3semidirect
47. #2434C2×C3semidirect
48. #2038C2×C3semidirect
49. #2005C2×C3semidirect
50. #1826C2×C3semidirect
52. #1784C2×C3semidirect
132. C2×C4⋊Q8C22×C3semidirect
145. #428C22×C3semidirect
147. #419C22×C3semidirect
149. C2×#101C22×C3semidirect
153. #391C22×C3semidirect
156. #339C22×C3semidirect
555. C3#29613direct (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
C3×#30952
C3×#30072
C3×#24342
C3×#20382
C3×#20052
C3×#18262
C3×#17892
C3×#17842
#296133