Finite Group Databasebeta

← C3×#2424 #84288

Subgroups — C3×#2424 #84288

Order 384.

922 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#121C211yesmaximal, characteristic, complemented
2C2×C3×#121C211yesmaximal, characteristic, complemented
3C2×C3×#120C211yesmaximal, characteristic, complemented
4C2×C3×#120C211yesmaximal, characteristic, complemented
5C2×C3×C4×D4C211yesmaximal, characteristic, complemented
6C22×C3×C22⋊C4C211yesmaximal, characteristic, complemented
7C3×#396C211yesmaximal, characteristic, complemented
8C3×#394C211yesmaximal, characteristic, complemented
9C3×#394C211yesmaximal, characteristic, complemented
10C3×#392C211yesmaximal, characteristic, complemented
11C3×#390C211yesmaximal, characteristic, complemented
12C3×#388C211yesmaximal, characteristic, complemented
13C3×#386C211yesmaximal, characteristic, complemented
14C3×#386C211yesmaximal, characteristic, complemented
15C3×#385C211yesmaximal, characteristic, complemented
16#2424C311yesmaximal, characteristic, direct product factor, Sylow
17C25×C3C2211yescharacteristic
18C22×C3×D4C2211yescharacteristic
19C22×C3×D4C2211yescharacteristic
20C22×C3×D4C2211yescharacteristic
21C22×C3×D4C2211yescharacteristic
22C22×C3×D4C2211yescharacteristic
23C22×C3×D4C2211yescharacteristic
24C22×C3×D4C2211yescharacteristic, complemented
25C23×C3×C4C2211yescharacteristic
26C23×C3×C4C2211yescharacteristic
27C23×C3×C4C2211yescharacteristic, complemented
28C3×#121—24no—
29C3×#121—24no—
30C3×#120—24no—
31C3×#120—24no—
32C3×C4×D4—24no—
33C2×C3×C4⋊C4C2211yescharacteristic, complemented
34C2×C3×C4⋊C4C2211yescharacteristic, complemented
35C2×C3×C4⋊C4C2211yescharacteristic, complemented
36C2×C3×C4⋊C4C2211yescharacteristic, complemented
37C2×C3×C22⋊C4—24no—
38C2×C3×C22⋊C4C2211yescharacteristic
39C2×C3×C22⋊C4C2211yescharacteristic
40C2×C3×C22⋊C4C2211yescharacteristic
41C2×C3×C22⋊C4C2211yescharacteristic
42C2×C3×C22⋊C4C2211yescharacteristic
43C2×C3×C22⋊C4C2211yescharacteristic
44C2×C3×C22⋊C4C2211yescharacteristic
45C2×C3×C22⋊C4C2211yescharacteristic
46C2×C3×C22⋊C4C2211yescharacteristic
47C2×C3×C22⋊C4C2211yescharacteristic
48C2×C3×C22⋊C4C2211yescharacteristic, complemented
49C2×C3×C22⋊C4C2211yescharacteristic, complemented
50C2×C3×C22⋊C4C2211yescharacteristic, complemented
51C2×C3×C22⋊C4C2211yescharacteristic, complemented
52C2×C3×C22⋊C4C2211yescharacteristic, complemented
53C2×C3×C42C2211yescharacteristic, complemented
54C3×#95C2211yescharacteristic, complemented
55C3×#95C2211yescharacteristic, complemented
56C3×#95C2211yescharacteristic, complemented
57C3×#95C2211yescharacteristic, complemented
58C2×#121C2×C311yescharacteristic, complemented
59C2×#121C2×C311yescharacteristic, complemented
60C2×#120C2×C311yescharacteristic, complemented
61C2×#120C2×C311yescharacteristic, complemented
62C2×C4×D4C2×C311yescharacteristic, complemented
63C22×C22⋊C4C2×C311yescharacteristic, complemented
64#396C2×C311yescharacteristic, complemented
65#394C2×C311yescharacteristic, complemented
66#394C2×C311yescharacteristic, complemented
67#392C2×C311yescharacteristic, complemented
68#390C2×C311yescharacteristic, complemented
69#388C2×C311yescharacteristic, complemented
70#386C2×C311yescharacteristic, complemented
71#386C2×C311yescharacteristic, complemented
72#385C2×C311yescharacteristic, complemented
73C24×C3—22no—
74C24×C3—22no—
75C24×C3—22no—
76C24×C3—22no—
77C24×C3—22no—
78C24×C3—41no—
79C24×C3—41no—
80C24×C3C2311yescharacteristic
81C24×C3C2311yescharacteristic
82C24×C3C2311yescharacteristic
83C24×C3C2311yescharacteristic
84C24×C3C2311yescharacteristic
85C2×C3×D4—22no—
86C2×C3×D4—22no—
87C2×C3×D4—22no—
88C2×C3×D4—22no—
89C2×C3×D4—22no—
90C2×C3×D4—22no—
91C2×C3×D4—22no—
92C2×C3×D4—22no—
93C2×C3×D4—22no—
94C2×C3×D4—22no—
95C2×C3×D4—22no—
96C2×C3×D4—22no—
97C2×C3×D4—22no—
98C2×C3×D4—22no—
99C2×C3×D4—22no—
100C2×C3×D4—41no—
101C2×C3×D4—41no—
102C2×C3×D4—41no—
103C2×C3×D4—41no—
104C2×C3×D4—41no—
105C2×C3×D4D414yescomplemented
106C22×C3×C4—22no—
107C22×C3×C4—22no—
108C22×C3×C4—22no—
109C22×C3×C4—22no—
110C22×C3×C4—22no—
111C22×C3×C4—22no—
112C22×C3×C4—41no—
113C22×C3×C4—41no—
114C22×C3×C4—41no—
115C22×C3×C4C2311yescharacteristic
116C22×C3×C4C2311yescharacteristic
117C22×C3×C4C2311yescharacteristic
118C22×C3×C4C2311yescharacteristic
119C22×C3×C4C2311yescharacteristic
120C22×C3×C4C2311yescharacteristic
121C22×C3×C4C2311yescharacteristic
122C22×C3×C4C2311yescharacteristic
123C22×C3×C4C2311yescharacteristic
124C22×C3×C4C2311yescharacteristic
125C3×C4⋊C4—22no—
126C3×C4⋊C4—22no—
127C3×C4⋊C4—22no—
128C3×C4⋊C4—41no—
129C3×C22⋊C4—22no—
130C3×C22⋊C4—22no—
131C3×C22⋊C4—22no—
132C3×C22⋊C4—22no—
133C3×C22⋊C4—22no—
134C3×C22⋊C4—22no—
135C3×C22⋊C4—22no—
136C3×C22⋊C4—22no—
137C3×C22⋊C4—22no—
138C3×C22⋊C4—22no—
139C3×C22⋊C4—22no—
140C3×C22⋊C4—22no—
141C3×C22⋊C4—41no—
142C3×C22⋊C4D414yescomplemented
143C3×C22⋊C4D414yescomplemented
144C3×C42—22no—
145C25C22×C311yescharacteristic
146C22×D4C22×C311yescharacteristic
147C22×D4C22×C311yescharacteristic
148C22×D4C22×C311yescharacteristic
149C22×D4C22×C311yescharacteristic
150C22×D4C22×C311yescharacteristic
151C22×D4C22×C311yescharacteristic
152C22×D4C22×C311yescharacteristic, complemented
153C23×C4C22×C311yescharacteristic
154C23×C4C22×C311yescharacteristic
155C23×C4C22×C311yescharacteristic, complemented
156#121—24no—
157#121—24no—
158#120—24no—
159#120—24no—
160C4×D4—24no—
161C2×C4⋊C4C22×C311yescharacteristic, complemented
162C2×C4⋊C4C22×C311yescharacteristic, complemented
163C2×C4⋊C4C22×C311yescharacteristic, complemented
164C2×C4⋊C4C22×C311yescharacteristic, complemented
165C2×C22⋊C4—24no—
166C2×C22⋊C4C22×C311yescharacteristic
167C2×C22⋊C4C22×C311yescharacteristic
168C2×C22⋊C4C22×C311yescharacteristic
169C2×C22⋊C4C22×C311yescharacteristic
170C2×C22⋊C4C22×C311yescharacteristic
171C2×C22⋊C4C22×C311yescharacteristic
172C2×C22⋊C4C22×C311yescharacteristic
173C2×C22⋊C4C22×C311yescharacteristic
174C2×C22⋊C4C22×C311yescharacteristic
175C2×C22⋊C4C22×C311yescharacteristic
176C2×C22⋊C4C22×C311yescharacteristic, complemented
177C2×C22⋊C4C22×C311yescharacteristic, complemented
178C2×C22⋊C4C22×C311yescharacteristic, complemented
179C2×C22⋊C4C22×C311yescharacteristic, complemented
180C2×C22⋊C4C22×C311yescharacteristic, complemented
181C2×C42C22×C311yescharacteristic, complemented
182#95C22×C311yescharacteristic, complemented
183#95C22×C311yescharacteristic, complemented
184#95C22×C311yescharacteristic, complemented
185#95C22×C311yescharacteristic, complemented
186C23×C3—21no—
187C23×C3—21no—
188C23×C3—21no—
189C23×C3—21no—
190C23×C3—28no—
191C23×C3—21no—
192C23×C3—21no—
193C23×C3—21no—
194C23×C3—21no—
195C23×C3—21no—
196C23×C3—21no—
197C23×C3—21no—
198C23×C3—21no—
199C23×C3—21no—
200C23×C3—21no—
201C23×C3—21no—
202C23×C3—21no—
203C23×C3—21no—
204C23×C3—21no—
205C23×C3—21no—
206C23×C3—21no—
207C23×C3—21no—
208C23×C3—21no—
209C23×C3—21no—
210C23×C3—21no—
211C23×C3—21no—
212C23×C3—21no—
213C23×C3—21no—
214C23×C3—21no—
215C23×C3—21no—
216C23×C3—44no—
217C23×C3—44no—
218C23×C3—44no—
219C23×C3—44no—
220C23×C3—44no—
221C23×C3—44no—
222C23×C3C2×D412yes—
223C23×C3C2×D412yes—
224C23×C3C2×D412yes—
225C23×C3C2×D412yes—
226C23×C3#4112yes—
227C23×C3#4112yes—
228C23×C3C2411yescharacteristic
229C3×D4—28no—
230C3×D4—44no—
231C3×D4—44no—
232C3×D4—44no—
233C3×D4—44no—
234C3×D4—44no—
235C3×D4—44no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—44no—
262C2×C3×C4—44no—
263C2×C3×C4—44no—
264C2×C3×C4C2×D412yes—
265C2×C3×C4C2×D412yes—
266C2×C3×C4C2×D412yes—
267C2×C3×C4C2×D412yes—
268C2×C3×C4C2×D412yes—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—41no—
275C24—41no—
276C24C23×C311yescharacteristic
277C24C23×C311yescharacteristic
278C24C23×C311yescharacteristic
279C24C23×C311yescharacteristic
280C24C23×C311yescharacteristic
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4C3×D414yescomplemented
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4C23×C311yescharacteristic
312C22×C4C23×C311yescharacteristic
313C22×C4C23×C311yescharacteristic
314C22×C4C23×C311yescharacteristic
315C22×C4C23×C311yescharacteristic
316C22×C4C23×C311yescharacteristic
317C22×C4C23×C311yescharacteristic
318C22×C4C23×C311yescharacteristic
319C22×C4C23×C311yescharacteristic
320C22×C4C23×C311yescharacteristic
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—41no—
325C22⋊C4—22no—
326C22⋊C4—22no—
327C22⋊C4—22no—
328C22⋊C4—22no—
329C22⋊C4—22no—
330C22⋊C4—22no—
331C22⋊C4—22no—
332C22⋊C4—22no—
333C22⋊C4—22no—
334C22⋊C4—22no—
335C22⋊C4—22no—
336C22⋊C4—22no—
337C22⋊C4—41no—
338C22⋊C4C3×D414yescomplemented
339C22⋊C4C3×D414yescomplemented
340C42—22no—
341C22×C3—22no—
342C22×C3—22no—
343C22×C3—22no—
344C22×C3—22no—
345C22×C3—22no—
346C22×C3—22no—
347C22×C3—22no—
348C22×C3—22no—
349C22×C3—22no—
350C22×C3—22no—
351C22×C3—22no—
352C22×C3—22no—
353C22×C3—22no—
354C22×C3—22no—
355C22×C3—22no—
356C22×C3—416no—
357C22×C3—41no—
358C22×C3—41no—
359C22×C3—41no—
360C22×C3—41no—
361C22×C3—41no—
362C22×C3—41no—
363C22×C3—41no—
364C22×C3—41no—
365C22×C3—41no—
366C22×C3—41no—
367C22×C3—41no—
368C22×C3—41no—
369C22×C3—41no—
370C22×C3—41no—
371C22×C3—41no—
372C22×C3—41no—
373C22×C3—41no—
374C22×C3—41no—
375C22×C3—41no—
376C22×C3#12114yescomplemented
377C22×C3C22×D411yescharacteristic
378C22×C3C22×D411yescharacteristic
379C22×C3C22×D411yescharacteristic
380C22×C3C2×#4111yescharacteristic
381C22×C3C2×#4111yescharacteristic
382C22×C3H(5,𝔽2)11yescharacteristic
383C22×C3H(5,𝔽2)11yescharacteristic
384C3×C4—22no—
385C3×C4—22no—
386C3×C4—22no—
387C3×C4—22no—
388C3×C4—22no—
389C3×C4—41no—
390C3×C4—41no—
391C3×C4—41no—
392C3×C4—41no—
393C3×C4—41no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—28no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—44no—
425C23—44no—
426C23—44no—
427C23—44no—
428C23—44no—
429C23—44no—
430C23C2×C3×D412yes—
431C23C2×C3×D412yes—
432C23C2×C3×D412yes—
433C23C2×C3×D412yes—
434C23C3×#4112yes—
435C23C3×#4112yes—
436C23C24×C311yescharacteristic
437D4—28no—
438D4—44no—
439D4—44no—
440D4—44no—
441D4—44no—
442D4—44no—
443D4—44no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4C2×C3×D412yes—
473C2×C4C2×C3×D412yes—
474C2×C4C2×C3×D412yes—
475C2×C4C2×C3×D412yes—
476C2×C4C2×C3×D412yes—
477C2×C3—24no—
478C2×C3—42no—
479C2×C3—42no—
480C2×C3—42no—
481C2×C3—81no—
482C2×C3C2×#12111yescharacteristic
483C2×C3#52511yescharacteristic
484C2×C3#53811yescharacteristic
485C2×C3D4211yescharacteristic
486C2×C3D4211yescharacteristic
487C2×C3#54611yescharacteristic
488C2×C3#54611yescharacteristic
489C22—22no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—22no—
504C22—416no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22—41no—
516C22—41no—
517C22—41no—
518C22—41no—
519C22—41no—
520C22—41no—
521C22—41no—
522C22—41no—
523C22—41no—
524C22C3×#12114yescomplemented
525C22C22×C3×D411yescharacteristic
526C22C22×C3×D411yescharacteristic
527C22C22×C3×D411yescharacteristic
528C22C2×C3×#4111yescharacteristic
529C22C2×C3×#4111yescharacteristic
530C22C3×H(5,𝔽2)11yescharacteristic
531C22C3×H(5,𝔽2)11yescharacteristic
532C4—22no—
533C4—22no—
534C4—22no—
535C4—22no—
536C4—22no—
537C4—41no—
538C4—41no—
539C4—41no—
540C4—41no—
541C4—41no—
542C3#242411yescharacteristic, direct product factor, Sylow
543C2—24no—
544C2—42no—
545C2—42no—
546C2—42no—
547C2—81no—
548C2C2×C3×#12111yescharacteristic
549C2C3×#52511yescharacteristic
550C2C3×#53811yescharacteristic
551C2C3×D4211yescharacteristic
552C2C3×D4211yescharacteristic
553C2C3×#54611yescharacteristic
554C2C3×#54611yescharacteristic
subgroupG/H
center, socleC23×C3C24
commutator, Frattini, cosocleC23C24×C3
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C3 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#121C2semidirect
2. C2×C3×#121C2semidirect
3. C2×C3×#120C2semidirect
4. C2×C3×#120C2semidirect
5. C2×C3×C4×D4C2semidirect
6. C22×C3×C22⋊C4C2semidirect
7. C3×#396C2semidirect
8. C3×#394C2semidirect
9. C3×#394C2semidirect
10. C3×#392C2semidirect
11. C3×#390C2semidirect
12. C3×#388C2semidirect
13. C3×#386C2semidirect
14. C3×#386C2semidirect
15. C3×#385C2semidirect
16. #2424C3direct (N × H)
24. C22×C3×D4C22semidirect
27. C23×C3×C4C22semidirect
33. C2×C3×C4⋊C4C22semidirect
34. C2×C3×C4⋊C4C22semidirect
35. C2×C3×C4⋊C4C22semidirect
36. C2×C3×C4⋊C4C22semidirect
48. C2×C3×C22⋊C4C22semidirect
49. C2×C3×C22⋊C4C22semidirect
50. C2×C3×C22⋊C4C22semidirect
51. C2×C3×C22⋊C4C22semidirect
52. C2×C3×C22⋊C4C22semidirect
53. C2×C3×C42C22semidirect
54. C3×#95C22semidirect
55. C3×#95C22semidirect
56. C3×#95C22semidirect
57. C3×#95C22semidirect
58. C2×#121C2×C3semidirect
59. C2×#121C2×C3semidirect
60. C2×#120C2×C3semidirect
61. C2×#120C2×C3semidirect
62. C2×C4×D4C2×C3semidirect
63. C22×C22⋊C4C2×C3semidirect
64. #396C2×C3semidirect
65. #394C2×C3semidirect
66. #394C2×C3semidirect
67. #392C2×C3semidirect
68. #390C2×C3semidirect
69. #388C2×C3semidirect
70. #386C2×C3semidirect
71. #386C2×C3semidirect
72. #385C2×C3semidirect
105. C2×C3×D4D4semidirect
142. C3×C22⋊C4D4semidirect
143. C3×C22⋊C4D4semidirect
152. C22×D4C22×C3semidirect
155. C23×C4C22×C3semidirect
161. C2×C4⋊C4C22×C3semidirect
162. C2×C4⋊C4C22×C3semidirect
163. C2×C4⋊C4C22×C3semidirect
164. C2×C4⋊C4C22×C3semidirect
176. C2×C22⋊C4C22×C3semidirect
177. C2×C22⋊C4C22×C3semidirect
178. C2×C22⋊C4C22×C3semidirect
179. C2×C22⋊C4C22×C3semidirect
180. C2×C22⋊C4C22×C3semidirect
181. C2×C42C22×C3semidirect
182. #95C22×C3semidirect
183. #95C22×C3semidirect
184. #95C22×C3semidirect
185. #95C22×C3semidirect
301. C2×D4C3×D4semidirect
338. C22⋊C4C3×D4semidirect
339. C22⋊C4C3×D4semidirect
376. C22×C3#121semidirect
524. C22C3×#121semidirect
542. C3#2424direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#1212
C2×C3×#1202
C2×C3×C4×D42
C22×C3×C22⋊C42
C3×#3962
C3×#3942
C3×#3922
C3×#3902
C3×#3882
C3×#3862
C3×#3852
#24243