Finite Group Databasebeta

← C3×#25678 #10644285

Subgroups — C3×#25678 #10644285

Order 768.

1114 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#529C211yesmaximal, characteristic
2C2×C3×#501C211yesmaximal, characteristic, complemented
3C2×C3×#489C211yesmaximal, characteristic, complemented
4C3×#2857C211yesmaximal, characteristic, complemented
5C3×#2036C212yesmaximal, complemented
6C3×#2034C212yesmaximal
7C3×#1992C211yesmaximal, characteristic, complemented
8C3×#1959C211yesmaximal, characteristic, complemented
9C3×#1865C212yesmaximal, complemented
10C3×#1862C212yesmaximal, complemented
11C3×#1749C211yesmaximal, characteristic, complemented
12#25678C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C3×#529—24no—
15C3×#529—24no—
16C2×C3×#126—21no—
17C2×C3×#125C2211yescharacteristic
18C2×C3×#124C2211yescharacteristic
19C2×C3×#123—21no—
20C2×C3×C22⋊Q8—21no—
21C2×C3×C22⋊Q8C2211yescharacteristic
22C2×C3×#121C2211yescharacteristic
23C2×C3×C4×Q8C2211yescharacteristic
24C2×C3×C4×D4—21no—
25C2×C3×C4×D4C2211yescharacteristic
26C2×C3×#117—21no—
27C22×C3×C42C2211yescharacteristic
28C3×#501—24no—
29C3×#489—24no—
30C2×C3×#107C2212yes—
31C2×C3×#106C2212yes—
32C2×C3×#103C2212yes—
33C2×C3×#103C2212yes—
34C2×C3×#102C2212yes—
35C2×C3×#102C2212yes—
36C2×C3×C22⋊C8C2212yes—
37C2×C3×#97C2211yescharacteristic
38C2×C3×#97C2211yescharacteristic
39C3×#391—21no—
40C3×#389—21no—
41C3×#389C2212yes—
42C3×#384C2211yescharacteristic
43C3×#380—21no—
44C3×#380C2212yes—
45C3×#340C2212yes—
46C3×#340C2212yes—
47C3×#336C2212yes—
48C2×#529C2×C311yescharacteristic
49C2×#501C2×C311yescharacteristic, complemented
50C2×#489C2×C311yescharacteristic, complemented
51#2857C2×C311yescharacteristic, complemented
52#2036C2×C312yescomplemented
53#2034C2×C312yes—
54#1992C2×C311yescharacteristic, complemented
55#1959C2×C311yescharacteristic, complemented
56#1865C2×C312yescomplemented
57#1862C2×C312yescomplemented
58#1749C2×C311yescharacteristic, complemented
59C22×C3×Q8—21no—
60C22×C3×Q8C2311yescharacteristic
61C22×C3×D4—21no—
62C22×C3×D4C2311yescharacteristic
63C23×C3×C4—21no—
64C23×C3×C4D411yescharacteristic
65C23×C3×C4D411yescharacteristic
66C23×C3×C4C2311yescharacteristic
67C22×C3×C8C2312yes—
68C22×C3×C8C2312yes—
69C3×C4⋊Q8—22no—
70C3×C4⋊Q8—22no—
71C3×#126—28no—
72C3×#125—22no—
73C3×#125D414yes—
74C3×#124—22no—
75C3×#124D414yes—
76C3×#123—28no—
77C3×C22⋊Q8—22no—
78C3×C22⋊Q8—22no—
79C3×C22⋊Q8—44no—
80C3×#121—22no—
81C3×#121—22no—
82C3×C4×Q8—22no—
83C3×C4×Q8—22no—
84C3×C4×D4—28no—
85C3×C4×D4—22no—
86C3×C4×D4—22no—
87C3×#117—28no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4—22no—
96C2×C3×C4⋊C4C2312yes—
97C2×C3×C4⋊C4C2312yes—
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4—21no—
105C2×C3×C22⋊C4—22no—
106C2×C3×C42—21no—
107C2×C3×C42—22no—
108C2×C3×C42—22no—
109C2×C3×C42D411yescharacteristic
110C2×C3×C42D411yescharacteristic
111C2×C3×C42C2311yescharacteristic
112C2×C3×C42C2311yescharacteristic
113C3×#107—24no—
114C3×#106—24no—
115C3×#103—24no—
116C3×#103—24no—
117C3×#102—24no—
118C3×#102—24no—
119C3×C22⋊C8—24no—
120C3×#97—22no—
121C3×#97—22no—
122C3×#95—22no—
123C3×#95—22no—
124C2×C4⋊Q8C22×C311yescharacteristic
125#529—24no—
126#529—24no—
127C2×#126—21no—
128C2×#125C22×C311yescharacteristic
129C2×#124C22×C311yescharacteristic
130C2×#123—21no—
131C2×C22⋊Q8—21no—
132C2×C22⋊Q8C22×C311yescharacteristic
133C2×#121C22×C311yescharacteristic
134C2×C4×Q8C22×C311yescharacteristic
135C2×C4×D4—21no—
136C2×C4×D4C22×C311yescharacteristic
137C2×#117—21no—
138C22×C42C22×C311yescharacteristic
139#501—24no—
140#489—24no—
141C2×#107C22×C312yes—
142C2×#106C22×C312yes—
143C2×#103C22×C312yes—
144C2×#103C22×C312yes—
145C2×#102C22×C312yes—
146C2×#102C22×C312yes—
147C2×C22⋊C8C22×C312yes—
148C2×#97C22×C311yescharacteristic
149C2×#97C22×C311yescharacteristic
150#391—21no—
151#389—21no—
152#389C22×C312yes—
153#384C22×C311yescharacteristic
154#380—21no—
155#380C22×C312yes—
156#340C22×C312yes—
157#340C22×C312yes—
158#336C22×C312yes—
159C24×C3—21no—
160C24×C3C2×D411yescharacteristic
161C2×C3×Q8—21no—
162C2×C3×Q8—22no—
163C2×C3×Q8—22no—
164C2×C3×Q8—42no—
165C2×C3×Q8—42no—
166C2×C3×Q8—42no—
167C2×C3×Q8C2×D412yes—
168C2×C3×D4—24no—
169C2×C3×D4—21no—
170C2×C3×D4—22no—
171C2×C3×D4—22no—
172C2×C3×D4—42no—
173C2×C3×D4—42no—
174C2×C3×D4C2×D412yes—
175C22×C3×C4—24no—
176C22×C3×C4—22no—
177C22×C3×C4—21no—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—22no—
189C22×C3×C4—41no—
190C22×C3×C4—41no—
191C22×C3×C4—42no—
192C22×C3×C4—42no—
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2411yescharacteristic
199C2×C3×C8—22no—
200C2×C3×C8—22no—
201C2×C3×C8—22no—
202C2×C3×C8—22no—
203C2×C3×C8—22no—
204C2×C3×C8#4114yes—
205C3×C4⋊C4—24no—
206C3×C4⋊C4—24no—
207C3×C4⋊C4—24no—
208C3×C4⋊C4—24no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—22no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4—42no—
216C3×C4⋊C4—42no—
217C3×C4⋊C4—42no—
218C3×C4⋊C4—44no—
219C3×C4⋊C4C2×D412yes—
220C3×C4⋊C4C2×D412yes—
221C3×C22⋊C4—24no—
222C3×C22⋊C4—24no—
223C3×C22⋊C4—24no—
224C3×C22⋊C4—24no—
225C3×C22⋊C4—24no—
226C3×C22⋊C4—44no—
227C3×C42—24no—
228C3×C42—22no—
229C3×C42—22no—
230C3×C42—21no—
231C3×C42—21no—
232C3×C42C2×D412yes—
233C3×C42C2×D412yes—
234C22×Q8—21no—
235C22×Q8C23×C311yescharacteristic
236C22×D4—21no—
237C22×D4C23×C311yescharacteristic
238C23×C4—21no—
239C23×C4C3×D411yescharacteristic
240C23×C4C3×D411yescharacteristic
241C23×C4C23×C311yescharacteristic
242C22×C8C23×C312yes—
243C22×C8C23×C312yes—
244C4⋊Q8—22no—
245C4⋊Q8—22no—
246#126—28no—
247#125—22no—
248#125C3×D414yes—
249#124—22no—
250#124C3×D414yes—
251#123—28no—
252C22⋊Q8—22no—
253C22⋊Q8—22no—
254C22⋊Q8—44no—
255#121—22no—
256#121—22no—
257C4×Q8—22no—
258C4×Q8—22no—
259C4×D4—28no—
260C4×D4—22no—
261C4×D4—22no—
262#117—28no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4C23×C312yes—
272C2×C4⋊C4C23×C312yes—
273C2×C4⋊C4C23×C311yescharacteristic
274C2×C4⋊C4C23×C311yescharacteristic
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—22no—
281C2×C42—21no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42C3×D411yescharacteristic
285C2×C42C3×D411yescharacteristic
286C2×C42C23×C311yescharacteristic
287C2×C42C23×C311yescharacteristic
288#107—24no—
289#106—24no—
290#103—24no—
291#103—24no—
292#102—24no—
293#102—24no—
294C22⋊C8—24no—
295#97—22no—
296#97—22no—
297#95—22no—
298#95—22no—
299C23×C3—21no—
300C23×C3—21no—
301C23×C3—21no—
302C23×C3—21no—
303C23×C3—21no—
304C23×C3—21no—
305C23×C3—22no—
306C23×C3—41no—
307C23×C3—41no—
308C23×C3—41no—
309C23×C3—41no—
310C23×C3—41no—
311C23×C3—41no—
312C23×C3#13712yes—
313C23×C3C22×D411yescharacteristic
314C3×Q8—24no—
315C3×Q8—42no—
316C3×Q8—84no—
317C3×D4—24no—
318C3×D4—48no—
319C3×D4—42no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—44no—
336C2×C3×C4—44no—
337C2×C3×C4—48no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—42no—
345C2×C3×C4—42no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4—42no—
351C2×C3×C4—42no—
352C2×C3×C4#12012yes—
353C2×C3×C4#12012yes—
354C2×C3×C4AGL(1,ℤ/8)12yes—
355C2×C3×C4C22×D411yescharacteristic
356C2×C3×C4C22×D411yescharacteristic
357C2×C3×C4C2×#4112yes—
358C2×C3×C42−1+412yes—
359C3×C8—24no—
360C3×C8—42no—
361C24—21no—
362C24C2×C3×D411yescharacteristic
363C2×Q8—21no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—42no—
367C2×Q8—42no—
368C2×Q8—42no—
369C2×Q8C2×C3×D412yes—
370C2×D4—24no—
371C2×D4—21no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—42no—
375C2×D4—42no—
376C2×D4C2×C3×D412yes—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4C2×C3×D411yescharacteristic
396C22×C4C2×C3×D411yescharacteristic
397C22×C4C2×C3×D411yescharacteristic
398C22×C4C2×C3×D411yescharacteristic
399C22×C4C2×C3×D411yescharacteristic
400C22×C4C24×C311yescharacteristic
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8C3×#4114yes—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—44no—
421C4⋊C4C2×C3×D412yes—
422C4⋊C4C2×C3×D412yes—
423C22⋊C4—24no—
424C22⋊C4—24no—
425C22⋊C4—24no—
426C22⋊C4—24no—
427C22⋊C4—24no—
428C22⋊C4—44no—
429C42—24no—
430C42—22no—
431C42—22no—
432C42—21no—
433C42—21no—
434C42C2×C3×D412yes—
435C42C2×C3×D412yes—
436C22×C3—22no—
437C22×C3—22no—
438C22×C3—22no—
439C22×C3—41no—
440C22×C3—41no—
441C22×C3—41no—
442C22×C3—41no—
443C22×C3—42no—
444C22×C3—42no—
445C22×C3—42no—
446C22×C3—81no—
447C22×C3—81no—
448C22×C3—81no—
449C22×C3—81no—
450C22×C3C2×#12011yescharacteristic
451C22×C3#53211yescharacteristic
452C22×C3#53611yescharacteristic
453C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
454C22×C3C2×#13711yescharacteristic
455C22×C3#57511yescharacteristic
456C22×C3#57511yescharacteristic
457C3×C4—22no—
458C3×C4—22no—
459C3×C4—22no—
460C3×C4—41no—
461C3×C4—42no—
462C3×C4—42no—
463C3×C4—41no—
464C3×C4—42no—
465C3×C4—82no—
466C3×C4—82no—
467C3×C4#54712yes—
468C3×C4#54812yes—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—22no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23C3×#13712yes—
483C23C22×C3×D411yescharacteristic
484Q8—24no—
485Q8—42no—
486Q8—84no—
487D4—24no—
488D4—48no—
489D4—42no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—48no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4C3×#12012yes—
523C2×C4C3×#12012yes—
524C2×C4C3×AGL(1,ℤ/8)12yes—
525C2×C4C22×C3×D411yescharacteristic
526C2×C4C22×C3×D411yescharacteristic
527C2×C4C2×C3×#4112yes—
528C2×C4C3×2−1+412yes—
529C8—24no—
530C8—42no—
531C2×C3—42no—
532C2×C3—82no—
533C2×C3#243211yescharacteristic
534C2×C3#300611yescharacteristic
535C2×C3#300711yescharacteristic
536C2×C3#317111yescharacteristic
537C2×C3#318111yescharacteristic
538C2×C3#323911yescharacteristic
539C2×C3#324011yescharacteristic
540C22—22no—
541C22—22no—
542C22—22no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—41no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22C2×C3×#12011yescharacteristic
555C22C3×#53211yescharacteristic
556C22C3×#53611yescharacteristic
557C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
558C22C2×C3×#13711yescharacteristic
559C22C3×#57511yescharacteristic
560C22C3×#57511yescharacteristic
561C4—22no—
562C4—22no—
563C4—22no—
564C4—41no—
565C4—42no—
566C4—42no—
567C4—41no—
568C4—42no—
569C4—82no—
570C4—82no—
571C4C3×#54712yes—
572C4C3×#54812yes—
573C3#2567811yescharacteristic, direct product factor, Sylow
574C2—42no—
575C2—82no—
576C2C3×#243211yescharacteristic
577C2C3×#300611yescharacteristic
578C2C3×#300711yescharacteristic
579C2C3×#317111yescharacteristic
580C2C3×#318111yescharacteristic
581C2C3×#323911yescharacteristic
582C2C3×#324011yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#501C2semidirect
3. C2×C3×#489C2semidirect
4. C3×#2857C2semidirect
5. C3×#2036C2semidirect
7. C3×#1992C2semidirect
8. C3×#1959C2semidirect
9. C3×#1865C2semidirect
10. C3×#1862C2semidirect
11. C3×#1749C2semidirect
12. #25678C3direct (N × H)
49. C2×#501C2×C3semidirect
50. C2×#489C2×C3semidirect
51. #2857C2×C3semidirect
52. #2036C2×C3semidirect
54. #1992C2×C3semidirect
55. #1959C2×C3semidirect
56. #1865C2×C3semidirect
57. #1862C2×C3semidirect
58. #1749C2×C3semidirect
573. C3#25678direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5292
C2×C3×#5012
C2×C3×#4892
C3×#28572
C3×#20362
C3×#20342
C3×#19922
C3×#19592
C3×#18652
C3×#18622
C3×#17492
#256783