Finite Group Databasebeta

← C3×#25584 #10644191

Subgroups — C3×#25584 #10644191

Order 768.

1126 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#552C211yesmaximal, characteristic
2C2×C3×#491C211yesmaximal, characteristic, complemented
3C2×C3×#462C211yesmaximal, characteristic, complemented
4C3×#2542C211yesmaximal, characteristic
5C3×#2001C212yesmaximal, complemented
6C3×#2001C212yesmaximal, complemented
7C3×#1989C211yesmaximal, characteristic, complemented
8C3×#1955C211yesmaximal, characteristic
9C3×#1873C212yesmaximal, complemented
10C3×#1872C212yesmaximal, complemented
11C3×#1763C211yesmaximal, characteristic, complemented
12#25584C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×Q16C2212yes—
14C3×#552—24no—
15C3×#552—24no—
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×#126—21no—
18C2×C3×#125—21no—
19C2×C3×#125C2211yescharacteristic
20C2×C3×#123—21no—
21C2×C3×C22⋊Q8C2211yescharacteristic
22C2×C3×C22⋊Q8C2211yescharacteristic
23C2×C3×C22⋊Q8C2212yes—
24C2×C3×C22⋊Q8C2212yes—
25C2×C3×C4×Q8C2211yescharacteristic
26C2×C3×C4×D4C2211yescharacteristic
27C2×C3×#117—21no—
28C2×C3×#117C2211yescharacteristic
29C22×C3×C4⋊C4C2211yescharacteristic
30C3×#491—24no—
31C3×#462—24no—
32C2×C3×C4⋊C8C2211yescharacteristic
33C2×C3×#103C2212yes—
34C2×C3×#103C2212yes—
35C2×C3×#103C2212yes—
36C2×C3×#103C2212yes—
37C2×C3×C22⋊C8C2212yes—
38C2×C3×C22⋊C8C2212yes—
39C2×C3×#97C2211yescharacteristic
40C3×#399—21no—
41C3×#399—21no—
42C3×#390—21no—
43C3×#384—21no—
44C3×C4×C4⋊C4C2211yescharacteristic
45C3×#340C2212yes—
46C3×#340C2212yes—
47C3×#340C2212yes—
48C2×#552C2×C311yescharacteristic
49C2×#491C2×C311yescharacteristic, complemented
50C2×#462C2×C311yescharacteristic, complemented
51#2542C2×C311yescharacteristic
52#2001C2×C312yescomplemented
53#2001C2×C312yescomplemented
54#1989C2×C311yescharacteristic, complemented
55#1955C2×C311yescharacteristic
56#1873C2×C312yescomplemented
57#1872C2×C312yescomplemented
58#1763C2×C311yescharacteristic, complemented
59C22×C3×Q8C2312yes—
60C22×C3×Q8C2311yescharacteristic
61C22×C3×D4D411yescharacteristic
62C23×C3×C4—21no—
63C23×C3×C4C2311yescharacteristic
64C23×C3×C4C2311yescharacteristic
65C2×C3×Q16—24no—
66C2×C3×Q16—42no—
67C2×C3×Q16—42no—
68C22×C3×C8C2312yes—
69C22×C3×C8C2312yes—
70C3×C4⋊Q8—22no—
71C3×C4⋊Q8—22no—
72C3×#126—28no—
73C3×#125—28no—
74C3×#125—22no—
75C3×#125D414yes—
76C3×#123—28no—
77C3×C22⋊Q8—22no—
78C3×C22⋊Q8—22no—
79C3×C22⋊Q8—22no—
80C3×C22⋊Q8—22no—
81C3×C22⋊Q8—24no—
82C3×C22⋊Q8—24no—
83C3×C22⋊Q8—42no—
84C3×C22⋊Q8—42no—
85C3×C4×Q8—22no—
86C3×C4×Q8D414yes—
87C3×C4×D4—22no—
88C3×C4×D4—22no—
89C3×#117—28no—
90C3×#117—22no—
91C3×#117—22no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4—22no—
99C2×C3×C4⋊C4—22no—
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4—22no—
102C2×C3×C4⋊C4—22no—
103C2×C3×C4⋊C4—21no—
104C2×C3×C4⋊C4D411yescharacteristic
105C2×C3×C4⋊C4C2311yescharacteristic
106C2×C3×C4⋊C4C2311yescharacteristic
107C2×C3×C4⋊C4C2311yescharacteristic
108C2×C3×C4⋊C4C2312yes—
109C2×C3×C22⋊C4—21no—
110C2×C3×C22⋊C4—21no—
111C2×C3×C22⋊C4—21no—
112C2×C3×C22⋊C4—22no—
113C2×C3×C22⋊C4—22no—
114C2×C3×C22⋊C4D411yescharacteristic
115C2×C3×C22⋊C4D411yescharacteristic
116C2×C3×C42—21no—
117C2×C3×C42—21no—
118C2×C3×C42—21no—
119C2×C3×C42C2311yescharacteristic
120C3×C4⋊C8—22no—
121C3×#103—24no—
122C3×#103—24no—
123C3×#103—24no—
124C3×#103—24no—
125C3×C22⋊C8—42no—
126C3×C22⋊C8—42no—
127C3×#97—22no—
128C3×#95—21no—
129C22×Q16C22×C312yes—
130#552—24no—
131#552—24no—
132C2×C4⋊Q8C22×C311yescharacteristic
133C2×#126—21no—
134C2×#125—21no—
135C2×#125C22×C311yescharacteristic
136C2×#123—21no—
137C2×C22⋊Q8C22×C311yescharacteristic
138C2×C22⋊Q8C22×C311yescharacteristic
139C2×C22⋊Q8C22×C312yes—
140C2×C22⋊Q8C22×C312yes—
141C2×C4×Q8C22×C311yescharacteristic
142C2×C4×D4C22×C311yescharacteristic
143C2×#117—21no—
144C2×#117C22×C311yescharacteristic
145C22×C4⋊C4C22×C311yescharacteristic
146#491—24no—
147#462—24no—
148C2×C4⋊C8C22×C311yescharacteristic
149C2×#103C22×C312yes—
150C2×#103C22×C312yes—
151C2×#103C22×C312yes—
152C2×#103C22×C312yes—
153C2×C22⋊C8C22×C312yes—
154C2×C22⋊C8C22×C312yes—
155C2×#97C22×C311yescharacteristic
156#399—21no—
157#399—21no—
158#390—21no—
159#384—21no—
160C4×C4⋊C4C22×C311yescharacteristic
161#340C22×C312yes—
162#340C22×C312yes—
163#340C22×C312yes—
164C24×C3C2×D411yescharacteristic
165C24×C3C2×D411yescharacteristic
166C2×C3×Q8—21no—
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—22no—
174C2×C3×Q8—22no—
175C2×C3×Q8C2×D412yes—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4—41no—
179C22×C3×C4—24no—
180C22×C3×C4—22no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—22no—
190C22×C3×C4—22no—
191C22×C3×C4—22no—
192C22×C3×C4—22no—
193C22×C3×C4—22no—
194C22×C3×C4—42no—
195C22×C3×C4—42no—
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2411yescharacteristic
201C3×Q16—48no—
202C2×C3×C8—22no—
203C2×C3×C8—22no—
204C2×C3×C8—22no—
205C2×C3×C8—22no—
206C2×C3×C8—22no—
207C2×C3×C8—22no—
208C3×C4⋊C4—24no—
209C3×C4⋊C4—24no—
210C3×C4⋊C4—24no—
211C3×C4⋊C4—24no—
212C3×C4⋊C4—24no—
213C3×C4⋊C4—24no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—22no—
219C3×C4⋊C4—44no—
220C3×C4⋊C4—44no—
221C3×C4⋊C4—42no—
222C3×C4⋊C4—42no—
223C3×C4⋊C4C2×D412yes—
224C3×C4⋊C4C2×D412yes—
225C3×C4⋊C4C2×D412yes—
226C3×C4⋊C4#4114yes—
227C3×C22⋊C4—24no—
228C3×C22⋊C4—24no—
229C3×C22⋊C4—24no—
230C3×C22⋊C4—22no—
231C3×C22⋊C4—22no—
232C3×C22⋊C4—44no—
233C3×C22⋊C4—44no—
234C3×C42—24no—
235C3×C42—24no—
236C3×C42—42no—
237C3×C42C2×D412yes—
238C3×C42C2×D412yes—
239C22×Q8C23×C312yes—
240C22×Q8C23×C311yescharacteristic
241C22×D4C3×D411yescharacteristic
242C23×C4—21no—
243C23×C4C23×C311yescharacteristic
244C23×C4C23×C311yescharacteristic
245C2×Q16—24no—
246C2×Q16—42no—
247C2×Q16—42no—
248C22×C8C23×C312yes—
249C22×C8C23×C312yes—
250C4⋊Q8—22no—
251C4⋊Q8—22no—
252#126—28no—
253#125—28no—
254#125—22no—
255#125C3×D414yes—
256#123—28no—
257C22⋊Q8—22no—
258C22⋊Q8—22no—
259C22⋊Q8—22no—
260C22⋊Q8—22no—
261C22⋊Q8—24no—
262C22⋊Q8—24no—
263C22⋊Q8—42no—
264C22⋊Q8—42no—
265C4×Q8—22no—
266C4×Q8C3×D414yes—
267C4×D4—22no—
268C4×D4—22no—
269#117—28no—
270#117—22no—
271#117—22no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4C3×D411yescharacteristic
285C2×C4⋊C4C23×C311yescharacteristic
286C2×C4⋊C4C23×C311yescharacteristic
287C2×C4⋊C4C23×C311yescharacteristic
288C2×C4⋊C4C23×C312yes—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4C3×D411yescharacteristic
295C2×C22⋊C4C3×D411yescharacteristic
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42C23×C311yescharacteristic
300C4⋊C8—22no—
301#103—24no—
302#103—24no—
303#103—24no—
304#103—24no—
305C22⋊C8—42no—
306C22⋊C8—42no—
307#97—22no—
308#95—21no—
309C23×C3—21no—
310C23×C3—22no—
311C23×C3—21no—
312C23×C3—22no—
313C23×C3—21no—
314C23×C3—22no—
315C23×C3—21no—
316C23×C3—21no—
317C23×C3—22no—
318C23×C3#13712yes—
319C23×C3C22×D411yescharacteristic
320C3×Q8—22no—
321C3×Q8—22no—
322C3×Q8—42no—
323C3×Q8—42no—
324C3×Q8—42no—
325C3×Q8—42no—
326C3×Q8—42no—
327C3×D4—44no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—22no—
331C2×C3×C4—21no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—21no—
336C2×C3×C4—22no—
337C2×C3×C4—24no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—48no—
341C2×C3×C4—42no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—42no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—42no—
354C2×C3×C4—42no—
355C2×C3×C4—42no—
356C2×C3×C4—42no—
357C2×C3×C4—42no—
358C2×C3×C4#12012yes—
359C2×C3×C4#12012yes—
360C2×C3×C4C22×D411yescharacteristic
361C2×C3×C4C22×D411yescharacteristic
362C2×C3×C4C2×#4112yes—
363C2×C3×C4H(5,𝔽2)12yes—
364C3×C8—42no—
365C3×C8—42no—
366C24C2×C3×D411yescharacteristic
367C24C2×C3×D411yescharacteristic
368C2×Q8—21no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8C2×C3×D412yes—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—41no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4C2×C3×D411yescharacteristic
399C22×C4C2×C3×D411yescharacteristic
400C22×C4C2×C3×D411yescharacteristic
401C22×C4C2×C3×D411yescharacteristic
402C22×C4C24×C311yescharacteristic
403Q16—48no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—22no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—22no—
421C4⋊C4—44no—
422C4⋊C4—44no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4C2×C3×D412yes—
426C4⋊C4C2×C3×D412yes—
427C4⋊C4C2×C3×D412yes—
428C4⋊C4C3×#4114yes—
429C22⋊C4—24no—
430C22⋊C4—24no—
431C22⋊C4—24no—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C42—24no—
437C42—24no—
438C42—42no—
439C42C2×C3×D412yes—
440C42C2×C3×D412yes—
441C22×C3—22no—
442C22×C3—24no—
443C22×C3—41no—
444C22×C3—41no—
445C22×C3—42no—
446C22×C3—41no—
447C22×C3—41no—
448C22×C3—41no—
449C22×C3—42no—
450C22×C3—42no—
451C22×C3C2×#12011yescharacteristic
452C22×C3#54612yes—
453C22×C3C2×#13711yescharacteristic
454C22×C3#57511yescharacteristic
455C22×C3#57812yes—
456C3×C4—22no—
457C3×C4—22no—
458C3×C4—41no—
459C3×C4—41no—
460C3×C4—42no—
461C3×C4—42no—
462C3×C4—42no—
463C3×C4—42no—
464C3×C4—44no—
465C3×C4—82no—
466C3×C4#53211yescharacteristic
467C3×C4#53511yescharacteristic
468C3×C4#55012yes—
469C23—21no—
470C23—22no—
471C23—21no—
472C23—22no—
473C23—21no—
474C23—21no—
475C23—22no—
476C23—21no—
477C23—22no—
478C23C3×#13712yes—
479C23C22×C3×D411yescharacteristic
480Q8—22no—
481Q8—22no—
482Q8—42no—
483Q8—42no—
484Q8—42no—
485Q8—42no—
486Q8—42no—
487D4—44no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—48no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4C3×#12012yes—
519C2×C4C3×#12012yes—
520C2×C4C22×C3×D411yescharacteristic
521C2×C4C22×C3×D411yescharacteristic
522C2×C4C2×C3×#4112yes—
523C2×C4C3×H(5,𝔽2)12yes—
524C8—42no—
525C8—42no—
526C2×C3—42no—
527C2×C3—81no—
528C2×C3#243111yescharacteristic
529C2×C3#300711yescharacteristic
530C2×C3#301111yescharacteristic
531C2×C3#331212yes—
532C2×C3#331512yes—
533C22—22no—
534C22—24no—
535C22—41no—
536C22—41no—
537C22—42no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22—42no—
542C22—42no—
543C22C2×C3×#12011yescharacteristic
544C22C3×#54612yes—
545C22C2×C3×#13711yescharacteristic
546C22C3×#57511yescharacteristic
547C22C3×#57812yes—
548C4—22no—
549C4—22no—
550C4—41no—
551C4—41no—
552C4—42no—
553C4—42no—
554C4—42no—
555C4—42no—
556C4—44no—
557C4—82no—
558C4C3×#53211yescharacteristic
559C4C3×#53511yescharacteristic
560C4C3×#55012yes—
561C3#2558411yescharacteristic, direct product factor, Sylow
562C2—42no—
563C2—81no—
564C2C3×#243111yescharacteristic
565C2C3×#300711yescharacteristic
566C2C3×#301111yescharacteristic
567C2C3×#331212yes—
568C2C3×#331512yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×D4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#491C2semidirect
3. C2×C3×#462C2semidirect
5. C3×#2001C2semidirect
6. C3×#2001C2semidirect
7. C3×#1989C2semidirect
9. C3×#1873C2semidirect
10. C3×#1872C2semidirect
11. C3×#1763C2semidirect
12. #25584C3direct (N × H)
49. C2×#491C2×C3semidirect
50. C2×#462C2×C3semidirect
52. #2001C2×C3semidirect
53. #2001C2×C3semidirect
54. #1989C2×C3semidirect
56. #1873C2×C3semidirect
57. #1872C2×C3semidirect
58. #1763C2×C3semidirect
561. C3#25584direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5522
C2×C3×#4912
C2×C3×#4622
C3×#25422
C3×#20012
C3×#19892
C3×#19552
C3×#18732
C3×#18722
C3×#17632
#255843