Finite Group Databasebeta

← C3×#24626 #10643233

Subgroups — C3×#24626 #10643233

Order 768.

1110 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#459C211yesmaximal, characteristic, complemented
2C2×C3×C4×D8C211yesmaximal, characteristic, complemented
3C3×#2541C211yesmaximal, characteristic, complemented
4C3×#2354C211yesmaximal, characteristic, complemented
5C3×#2054C212yesmaximal, complemented
6C3×#2011C212yesmaximal, complemented
7C3×#1982C212yesmaximal, complemented
8C3×#1957C211yesmaximal, characteristic, complemented
9C3×#1939C211yesmaximal, characteristic, complemented
10C3×#1925C212yesmaximal, complemented
11C3×#1802C211yesmaximal, characteristic, complemented
12#24626C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×D8C2211yescharacteristic
14C22×C3×D8C2212yes—
15C2×C3×#127C2211yescharacteristic
16C2×C3×#127C2212yes—
17C2×C3×#121—21no—
18C2×C3×#121—21no—
19C2×C3×C4×D4C2211yescharacteristic
20C2×C3×C4×D4C2211yescharacteristic
21C2×C3×#117—21no—
22C3×#459—24no—
23C3×C4×D8—24no—
24C2×C3×#107C2211yescharacteristic
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×C4⋊C8C2212yes—
27C2×C3×#102C2212yes—
28C2×C3×#102C2211yescharacteristic
29C2×C3×#102C2211yescharacteristic
30C2×C3×#102C2212yes—
31C2×C3×#102C2212yes—
32C2×C3×#102C2212yes—
33C2×C3×C4×C8C2211yescharacteristic
34C3×#392—22no—
35C3×#390C2212yes—
36C3×#390C2212yes—
37C3×#388—22no—
38C3×#384C2211yescharacteristic
39C3×#380—21no—
40C3×C4×C4⋊C4C2212yes—
41C3×C4×C4⋊C4C2211yescharacteristic
42C3×#340C2212yes—
43C3×#336C2212yes—
44C2×#459C2×C311yescharacteristic, complemented
45C2×C4×D8C2×C311yescharacteristic, complemented
46#2541C2×C311yescharacteristic, complemented
47#2354C2×C311yescharacteristic, complemented
48#2054C2×C312yescomplemented
49#2011C2×C312yescomplemented
50#1982C2×C312yescomplemented
51#1957C2×C311yescharacteristic, complemented
52#1939C2×C311yescharacteristic, complemented
53#1925C2×C312yescomplemented
54#1802C2×C311yescharacteristic, complemented
55C22×C3×D4—21no—
56C22×C3×D4—22no—
57C22×C3×D4—22no—
58C22×C3×D4—21no—
59C22×C3×D4C2312yes—
60C22×C3×D4C2311yescharacteristic
61C22×C3×D4C2311yescharacteristic
62C23×C3×C4—21no—
63C23×C3×C4—21no—
64C2×C3×D8—22no—
65C2×C3×D8—22no—
66C2×C3×D8—24no—
67C2×C3×D8—22no—
68C2×C3×D8—42no—
69C2×C3×D8—42no—
70C22×C3×C8C2312yes—
71C22×C3×C8C2311yescharacteristic
72C22×C3×C8C2311yescharacteristic
73C3×#127—24no—
74C3×#127—22no—
75C3×#127—22no—
76C3×#127—42no—
77C3×#121—44no—
78C3×#121—44no—
79C3×C4×D4—22no—
80C3×C4×D4—22no—
81C3×C4×D4—22no—
82C3×C4×D4D414yescomplemented
83C3×#117—44no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4D411yescharacteristic
87C2×C3×C4⋊C4D411yescharacteristic
88C2×C3×C4⋊C4C2312yes—
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—22no—
93C2×C3×C22⋊C4—22no—
94C2×C3×C22⋊C4—22no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—22no—
97C2×C3×C42—22no—
98C2×C3×C42C2312yes—
99C2×C3×C42C2311yescharacteristic
100C3×#107—22no—
101C3×C4⋊C8—22no—
102C3×C4⋊C8—42no—
103C3×#102—22no—
104C3×#102—24no—
105C3×#102—22no—
106C3×#102—24no—
107C3×#102—42no—
108C3×#102—42no—
109C3×C4×C8—22no—
110C3×#95—22no—
111C3×#95D412yes—
112C22×D8C22×C311yescharacteristic
113C22×D8C22×C312yes—
114C2×#127C22×C311yescharacteristic
115C2×#127C22×C312yes—
116C2×#121—21no—
117C2×#121—21no—
118C2×C4×D4C22×C311yescharacteristic
119C2×C4×D4C22×C311yescharacteristic
120C2×#117—21no—
121#459—24no—
122C4×D8—24no—
123C2×#107C22×C311yescharacteristic
124C2×C4⋊C8C22×C311yescharacteristic
125C2×C4⋊C8C22×C312yes—
126C2×#102C22×C311yescharacteristic
127C2×#102C22×C311yescharacteristic
128C2×#102C22×C312yes—
129C2×#102C22×C312yes—
130C2×#102C22×C312yes—
131C2×#102C22×C312yes—
132C2×C4×C8C22×C311yescharacteristic
133#392—22no—
134#390C22×C312yes—
135#390C22×C312yes—
136#388—22no—
137#384C22×C311yescharacteristic
138#380—21no—
139C4×C4⋊C4C22×C312yes—
140C4×C4⋊C4C22×C311yescharacteristic
141#340C22×C312yes—
142#336C22×C312yes—
143C24×C3—21no—
144C24×C3—21no—
145C24×C3—22no—
146C2×C3×D4—21no—
147C2×C3×D4—22no—
148C2×C3×D4—21no—
149C2×C3×D4—21no—
150C2×C3×D4—21no—
151C2×C3×D4—21no—
152C2×C3×D4—21no—
153C2×C3×D4—22no—
154C2×C3×D4—22no—
155C2×C3×D4—22no—
156C2×C3×D4—21no—
157C2×C3×D4—21no—
158C2×C3×D4—21no—
159C2×C3×D4—21no—
160C2×C3×D4—22no—
161C2×C3×D4—22no—
162C2×C3×D4—44no—
163C2×C3×D4—44no—
164C2×C3×D4—44no—
165C2×C3×D4—42no—
166C2×C3×D4—42no—
167C2×C3×D4—44no—
168C2×C3×D4—42no—
169C2×C3×D4—42no—
170C2×C3×D4—81no—
171C2×C3×D4—82no—
172C2×C3×D4—81no—
173C2×C3×D4—82no—
174C2×C3×D4C2×D412yes—
175C2×C3×D4#4112yes—
176C22×C3×C4—21no—
177C22×C3×C4—24no—
178C22×C3×C4—22no—
179C22×C3×C4—21no—
180C22×C3×C4—42no—
181C22×C3×C4—42no—
182C22×C3×C4—42no—
183C22×C3×C4—42no—
184C22×C3×C4—42no—
185C22×C3×C4C2×D412yes—
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D412yes—
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2411yescharacteristic
190C3×D8—28no—
191C3×D8—48no—
192C2×C3×C8—21no—
193C2×C3×C8—21no—
194C2×C3×C8—21no—
195C2×C3×C8—22no—
196C2×C3×C8—21no—
197C2×C3×C8—22no—
198C2×C3×C8—21no—
199C2×C3×C8—22no—
200C2×C3×C8#4112yes—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—21no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—42no—
208C3×C4⋊C4—81no—
209C3×C4⋊C4D814yescomplemented
210C3×C4⋊C4C2×D412yes—
211C3×C22⋊C4—24no—
212C3×C22⋊C4—44no—
213C3×C22⋊C4—42no—
214C3×C22⋊C4—44no—
215C3×C22⋊C4—44no—
216C3×C22⋊C4—82no—
217C3×C42—22no—
218C3×C42—22no—
219C3×C42—44no—
220C3×C42C2×D412yes—
221C3×C42#4112yes—
222C22×D4—21no—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4—21no—
226C22×D4C23×C312yes—
227C22×D4C23×C311yescharacteristic
228C22×D4C23×C311yescharacteristic
229C23×C4—21no—
230C23×C4—21no—
231C2×D8—22no—
232C2×D8—22no—
233C2×D8—24no—
234C2×D8—22no—
235C2×D8—42no—
236C2×D8—42no—
237C22×C8C23×C312yes—
238C22×C8C23×C311yescharacteristic
239C22×C8C23×C311yescharacteristic
240#127—24no—
241#127—22no—
242#127—22no—
243#127—42no—
244#121—44no—
245#121—44no—
246C4×D4—22no—
247C4×D4—22no—
248C4×D4—22no—
249C4×D4C3×D414yescomplemented
250#117—44no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4C3×D411yescharacteristic
254C2×C4⋊C4C3×D411yescharacteristic
255C2×C4⋊C4C23×C311yescharacteristic
256C2×C4⋊C4C23×C312yes—
257C2×C4⋊C4C23×C311yescharacteristic
258C2×C22⋊C4—21no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C42—22no—
265C2×C42C23×C312yes—
266C2×C42C23×C311yescharacteristic
267#107—22no—
268C4⋊C8—22no—
269C4⋊C8—42no—
270#102—22no—
271#102—24no—
272#102—22no—
273#102—24no—
274#102—42no—
275#102—42no—
276C4×C8—22no—
277#95—22no—
278#95C3×D412yes—
279C23×C3—22no—
280C23×C3—22no—
281C23×C3—41no—
282C23×C3—41no—
283C23×C3—41no—
284C23×C3—41no—
285C23×C3—41no—
286C23×C3—41no—
287C23×C3—41no—
288C23×C3—41no—
289C23×C3—41no—
290C23×C3—41no—
291C23×C3—41no—
292C23×C3—41no—
293C23×C3—42no—
294C23×C3—42no—
295C23×C3—42no—
296C23×C3—42no—
297C23×C3—42no—
298C23×C3—42no—
299C23×C3—42no—
300C23×C3C22×D411yescharacteristic
301C3×D4—22no—
302C3×D4—22no—
303C3×D4—22no—
304C3×D4—22no—
305C3×D4—42no—
306C3×D4—41no—
307C3×D4—42no—
308C3×D4—41no—
309C3×D4—42no—
310C3×D4—41no—
311C3×D4—41no—
312C3×D4—42no—
313C3×D4—88no—
314C3×D4—84no—
315C3×D4—88no—
316C3×D4—84no—
317C2×C3×C4—22no—
318C2×C3×C4—21no—
319C2×C3×C4—22no—
320C2×C3×C4—21no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—41no—
326C2×C3×C4—41no—
327C2×C3×C4—48no—
328C2×C3×C4—41no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—42no—
334C2×C3×C4—84no—
335C2×C3×C4#12112yes—
336C2×C3×C4#12112yes—
337C2×C3×C4C2×D814yes—
338C2×C3×C4C2×D812yes—
339C2×C3×C4AGL(1,ℤ/8)12yes—
340C2×C3×C4C22×D411yescharacteristic
341C2×C3×C4C2×#4111yescharacteristic
342C2×C3×C4C2×#4111yescharacteristic
343C2×C3×C4C2×#4111yescharacteristic
344C2×C3×C4H(5,𝔽2)11yescharacteristic
345C2×C3×C4H(5,𝔽2)11yescharacteristic
346C3×C8—22no—
347C3×C8—41no—
348C3×C8—42no—
349C24—21no—
350C24—21no—
351C24—22no—
352C2×D4—21no—
353C2×D4—22no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—44no—
369C2×D4—44no—
370C2×D4—44no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—44no—
374C2×D4—42no—
375C2×D4—42no—
376C2×D4—81no—
377C2×D4—82no—
378C2×D4—81no—
379C2×D4—82no—
380C2×D4C2×C3×D412yes—
381C2×D4C3×#4112yes—
382C22×C4—21no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4C2×C3×D412yes—
392C22×C4C2×C3×D411yescharacteristic
393C22×C4C2×C3×D412yes—
394C22×C4C2×C3×D411yescharacteristic
395C22×C4C24×C311yescharacteristic
396D8—28no—
397D8—48no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—22no—
402C2×C8—21no—
403C2×C8—22no—
404C2×C8—21no—
405C2×C8—22no—
406C2×C8C3×#4112yes—
407C4⋊C4—22no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—22no—
411C4⋊C4—21no—
412C4⋊C4—22no—
413C4⋊C4—42no—
414C4⋊C4—81no—
415C4⋊C4C3×D814yescomplemented
416C4⋊C4C2×C3×D412yes—
417C22⋊C4—24no—
418C22⋊C4—44no—
419C22⋊C4—42no—
420C22⋊C4—44no—
421C22⋊C4—44no—
422C22⋊C4—82no—
423C42—22no—
424C42—22no—
425C42—44no—
426C42C2×C3×D412yes—
427C42C3×#4112yes—
428C22×C3—42no—
429C22×C3—42no—
430C22×C3—42no—
431C22×C3—42no—
432C22×C3—42no—
433C22×C3—42no—
434C22×C3—81no—
435C22×C3—81no—
436C22×C3—81no—
437C22×C3—81no—
438C22×C3—81no—
439C22×C3—81no—
440C22×C3—81no—
441C22×C3—81no—
442C22×C3—82no—
443C22×C3—82no—
444C22×C3—82no—
445C22×C3—82no—
446C22×C3—82no—
447C22×C3—82no—
448C22×C3—82no—
449C22×C3C2×#12111yescharacteristic
450C22×C3#54611yescharacteristic
451C22×C3#55011yescharacteristic
452C22×C3C22×D811yescharacteristic
453C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
454C22×C3#57611yescharacteristic
455C22×C3#57711yescharacteristic
456C3×C4—24no—
457C3×C4—22no—
458C3×C4—42no—
459C3×C4—42no—
460C3×C4—82no—
461C3×C4—81no—
462C3×C4#45914yes—
463C3×C4#52511yescharacteristic
464C3×C4#53511yescharacteristic
465C3×C4#55311yescharacteristic
466C3×C4#55511yescharacteristic
467C23—22no—
468C23—22no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23C22×C3×D411yescharacteristic
489D4—22no—
490D4—22no—
491D4—22no—
492D4—22no—
493D4—42no—
494D4—41no—
495D4—42no—
496D4—41no—
497D4—42no—
498D4—41no—
499D4—41no—
500D4—42no—
501D4—88no—
502D4—84no—
503D4—88no—
504D4—84no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—48no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—84no—
523C2×C4C3×#12112yes—
524C2×C4C3×#12112yes—
525C2×C4C2×C3×D814yes—
526C2×C4C2×C3×D812yes—
527C2×C4C3×AGL(1,ℤ/8)12yes—
528C2×C4C22×C3×D411yescharacteristic
529C2×C4C2×C3×#4111yescharacteristic
530C2×C4C2×C3×#4111yescharacteristic
531C2×C4C2×C3×#4111yescharacteristic
532C2×C4C3×H(5,𝔽2)11yescharacteristic
533C2×C4C3×H(5,𝔽2)11yescharacteristic
534C8—22no—
535C8—42no—
536C8—41no—
537C2×C3—82no—
538C2×C3—82no—
539C2×C3—162no—
540C2×C3#241411yescharacteristic
541C2×C3C2×#45911yescharacteristic
542C2×C3#304511yescharacteristic
543C2×C3#329411yescharacteristic
544C2×C3#332511yescharacteristic
545C2×C3#339111yescharacteristic
546C2×C3#339711yescharacteristic
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22—81no—
560C22—81no—
561C22—82no—
562C22—82no—
563C22—82no—
564C22—82no—
565C22—82no—
566C22—82no—
567C22—82no—
568C22C2×C3×#12111yescharacteristic
569C22C3×#54611yescharacteristic
570C22C3×#55011yescharacteristic
571C22C22×C3×D811yescharacteristic
572C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
573C22C3×#57611yescharacteristic
574C22C3×#57711yescharacteristic
575C4—24no—
576C4—22no—
577C4—42no—
578C4—42no—
579C4—82no—
580C4—81no—
581C4C3×#45914yes—
582C4C3×#52511yescharacteristic
583C4C3×#53511yescharacteristic
584C4C3×#55311yescharacteristic
585C4C3×#55511yescharacteristic
586C3#2462611yescharacteristic, direct product factor, Sylow
587C2—82no—
588C2—82no—
589C2—162no—
590C2C3×#241411yescharacteristic
591C2C2×C3×#45911yescharacteristic
592C2C3×#304511yescharacteristic
593C2C3×#329411yescharacteristic
594C2C3×#332511yescharacteristic
595C2C3×#339111yescharacteristic
596C2C3×#339711yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×C4×C4⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#459C2semidirect
2. C2×C3×C4×D8C2semidirect
3. C3×#2541C2semidirect
4. C3×#2354C2semidirect
5. C3×#2054C2semidirect
6. C3×#2011C2semidirect
7. C3×#1982C2semidirect
8. C3×#1957C2semidirect
9. C3×#1939C2semidirect
10. C3×#1925C2semidirect
11. C3×#1802C2semidirect
12. #24626C3direct (N × H)
44. C2×#459C2×C3semidirect
45. C2×C4×D8C2×C3semidirect
46. #2541C2×C3semidirect
47. #2354C2×C3semidirect
48. #2054C2×C3semidirect
49. #2011C2×C3semidirect
50. #1982C2×C3semidirect
51. #1957C2×C3semidirect
52. #1939C2×C3semidirect
53. #1925C2×C3semidirect
54. #1802C2×C3semidirect
82. C3×C4×D4D4semidirect
209. C3×C4⋊C4D8semidirect
249. C4×D4C3×D4semidirect
415. C4⋊C4C3×D8semidirect
586. C3#24626direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4592
C2×C3×C4×D82
C3×#25412
C3×#23542
C3×#20542
C3×#20112
C3×#19822
C3×#19572
C3×#19392
C3×#19252
C3×#18022
#246263