Finite Group Databasebeta

← C3×#23424 #10642031

Subgroups — C3×#23424 #10642031

Order 768.

2098 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×AGL(1,ℤ/8)C211yesmaximal, characteristic
2C23×C3×SD16C211yesmaximal, characteristic, complemented
3C2×C3×#536C211yesmaximal, characteristic, complemented
4C2×C3×#419C211yesmaximal, characteristic, complemented
5C2×C3×#416C211yesmaximal, characteristic, complemented
6C3×#2309C211yesmaximal, characteristic, complemented
7C3×#2013C214yesmaximal, complemented
8C3×#2012C214yesmaximal, complemented
9C3×#1729C211yesmaximal, characteristic, complemented
10#23424C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×#41C2211yescharacteristic
12C23×C3×Q8C2211yescharacteristic
13C23×C3×D4C2211yescharacteristic
14C2×C3×AGL(1,ℤ/8)—24no—
15C2×C3×AGL(1,ℤ/8)—24no—
16C2×C3×AGL(1,ℤ/8)—24no—
17C22×C3×SD16—24no—
18C22×C3×SD16—24no—
19C22×C3×SD16—24no—
20C22×C3×SD16C2214yes—
21C22×C3×SD16C2212yes—
22C22×C3×D8C2212yes—
23C22×C3×M16C2211yescharacteristic
24C23×C3×C8C2211yescharacteristic
25C3×#536—24no—
26C3×#536—24no—
27C2×C3×C4⋊Q8C2212yescomplemented
28C2×C3×#124C2212yescomplemented
29C2×C3×#123—22no—
30C2×C3×C22⋊Q8—22no—
31C2×C3×#117C2211yescharacteristic
32C2×C3×#117C2211yescharacteristic
33C3×#419—24no—
34C3×#416—24no—
35C2×C3×#103C2212yescomplemented
36C2×C3×#103C2214yescomplemented
37C2×C3×#102C2212yescomplemented
38C3×#390—22no—
39C3×#390C2214yes—
40C3×#379—22no—
41C3×#336C2214yescomplemented
42C22×AGL(1,ℤ/8)C2×C311yescharacteristic
43C23×SD16C2×C311yescharacteristic, complemented
44C2×#536C2×C311yescharacteristic, complemented
45C2×#419C2×C311yescharacteristic, complemented
46C2×#416C2×C311yescharacteristic, complemented
47#2309C2×C311yescharacteristic, complemented
48#2013C2×C314yescomplemented
49#2012C2×C314yescomplemented
50#1729C2×C311yescharacteristic, complemented
51C25×C3—21no—
52C2×C3×#41—22no—
53C2×C3×#41—22no—
54C2×C3×#41—22no—
55C2×C3×#41—22no—
56C2×C3×#41—22no—
57C2×C3×#41D414yes—
58C22×C3×Q8—21no—
59C22×C3×Q8—22no—
60C22×C3×Q8—22no—
61C22×C3×Q8—22no—
62C22×C3×Q8—22no—
63C22×C3×Q8—22no—
64C22×C3×Q8D414yes—
65C22×C3×Q8C2312yes—
66C22×C3×Q8C2311yescharacteristic
67C22×C3×D4—21no—
68C22×C3×D4—22no—
69C22×C3×D4—22no—
70C22×C3×D4—22no—
71C22×C3×D4—22no—
72C22×C3×D4—21no—
73C22×C3×D4—22no—
74C22×C3×D4D414yes—
75C22×C3×D4C2311yescharacteristic
76C22×C3×D4C2312yes—
77C23×C3×C4—21no—
78C23×C3×C4—21no—
79C23×C3×C4C2311yescharacteristic
80C3×AGL(1,ℤ/8)—232no—
81C2×C3×SD16—232no—
82C2×C3×SD16—28no—
83C2×C3×SD16—24no—
84C2×C3×SD16—28no—
85C2×C3×SD16—24no—
86C2×C3×SD16—28no—
87C2×C3×SD16—24no—
88C2×C3×D8—24no—
89C2×C3×D8—24no—
90C2×C3×D8—24no—
91C2×C3×M16—22no—
92C2×C3×M16—22no—
93C2×C3×M16D414yes—
94C22×C3×C8—22no—
95C22×C3×C8—22no—
96C22×C3×C8D414yes—
97C22×C3×C8C2312yes—
98C22×C3×C8C2312yes—
99C3×C4⋊Q8—24no—
100C3×C4⋊Q8—24no—
101C3×#124—24no—
102C3×#124—24no—
103C3×#123—216no—
104C3×C22⋊Q8—216no—
105C3×#117—22no—
106C3×#117—22no—
107C3×#117—22no—
108C3×#117—22no—
109C2×C3×C4⋊C4—24no—
110C2×C3×C4⋊C4D412yes—
111C2×C3×C4⋊C4C2312yes—
112C2×C3×C22⋊C4—24no—
113C2×C3×C22⋊C4—21no—
114C2×C3×C22⋊C4—28no—
115C2×C3×C22⋊C4D412yes—
116C2×C3×C42—21no—
117C2×C3×C42C2312yes—
118C3×#103—24no—
119C3×#103—28no—
120C3×#102—24no—
121C22×#41C22×C311yescharacteristic
122C23×Q8C22×C311yescharacteristic
123C23×D4C22×C311yescharacteristic
124C2×AGL(1,ℤ/8)—24no—
125C2×AGL(1,ℤ/8)—24no—
126C2×AGL(1,ℤ/8)—24no—
127C22×SD16—24no—
128C22×SD16—24no—
129C22×SD16—24no—
130C22×SD16C22×C314yes—
131C22×SD16C22×C312yes—
132C22×D8C22×C312yes—
133C22×M16C22×C311yescharacteristic
134C23×C8C22×C311yescharacteristic
135#536—24no—
136#536—24no—
137C2×C4⋊Q8C22×C312yescomplemented
138C2×#124C22×C312yescomplemented
139C2×#123—22no—
140C2×C22⋊Q8—22no—
141C2×#117C22×C311yescharacteristic
142C2×#117C22×C311yescharacteristic
143#419—24no—
144#416—24no—
145C2×#103C22×C312yescomplemented
146C2×#103C22×C314yescomplemented
147C2×#102C22×C312yescomplemented
148#390—22no—
149#390C22×C314yes—
150#379—22no—
151#336C22×C314yescomplemented
152C24×C3—22no—
153C24×C3—21no—
154C24×C3—24no—
155C24×C3—42no—
156C24×C3—42no—
157C24×C3—84no—
158C24×C3C2×D411yescharacteristic
159C3×#41—28no—
160C3×#41—28no—
161C3×#41—44no—
162C3×#41—44no—
163C2×C3×Q8—24no—
164C2×C3×Q8—22no—
165C2×C3×Q8—21no—
166C2×C3×Q8—28no—
167C2×C3×Q8—28no—
168C2×C3×Q8—22no—
169C2×C3×Q8—21no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C2×C3×Q8—21no—
173C2×C3×Q8—21no—
174C2×C3×Q8—22no—
175C2×C3×Q8—21no—
176C2×C3×Q8—44no—
177C2×C3×Q8—42no—
178C2×C3×Q8—44no—
179C2×C3×Q8—42no—
180C2×C3×Q8C2×D414yes—
181C2×C3×Q8C2×D412yes—
182C2×C3×D4—24no—
183C2×C3×D4—28no—
184C2×C3×D4—28no—
185C2×C3×D4—24no—
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4—21no—
189C2×C3×D4—22no—
190C2×C3×D4—22no—
191C2×C3×D4—22no—
192C2×C3×D4—22no—
193C2×C3×D4—21no—
194C2×C3×D4—21no—
195C2×C3×D4—22no—
196C2×C3×D4—42no—
197C2×C3×D4—42no—
198C2×C3×D4—42no—
199C2×C3×D4—44no—
200C2×C3×D4—44no—
201C2×C3×D4—42no—
202C2×C3×D4C2×D412yes—
203C2×C3×D4C2×D414yes—
204C22×C3×C4—24no—
205C22×C3×C4—22no—
206C22×C3×C4—21no—
207C22×C3×C4—24no—
208C22×C3×C4—22no—
209C22×C3×C4—42no—
210C22×C3×C4—42no—
211C22×C3×C4—42no—
212C22×C3×C4—42no—
213C22×C3×C4C2×D414yes—
214C22×C3×C4C2×D412yes—
215C22×C3×C4C2×D412yes—
216C22×C3×C4C2×D412yes—
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2×D412yes—
219C22×C3×C4C2×D412yes—
220C22×C3×C4#4112yes—
221C22×C3×C4C2411yescharacteristic
222C3×SD16—232no—
223C3×SD16—216no—
224C3×D8—216no—
225C3×M16—28no—
226C2×C3×C8—28no—
227C2×C3×C8—22no—
228C2×C3×C8—22no—
229C2×C3×C8—22no—
230C2×C3×C8—22no—
231C2×C3×C8C2×D414yes—
232C2×C3×C8C2×D414yes—
233C3×C4⋊C4—216no—
234C3×C4⋊C4—24no—
235C3×C4⋊C4—22no—
236C3×C4⋊C4—22no—
237C3×C22⋊C4—216no—
238C3×C22⋊C4—24no—
239C3×C22⋊C4—42no—
240C3×C22⋊C4—88no—
241C3×C42—22no—
242C3×C42—22no—
243C3×C42—42no—
244C25—21no—
245C2×#41—22no—
246C2×#41—22no—
247C2×#41—22no—
248C2×#41—22no—
249C2×#41—22no—
250C2×#41C3×D414yes—
251C22×Q8—21no—
252C22×Q8—22no—
253C22×Q8—22no—
254C22×Q8—22no—
255C22×Q8—22no—
256C22×Q8—22no—
257C22×Q8C3×D414yes—
258C22×Q8C23×C312yes—
259C22×Q8C23×C311yescharacteristic
260C22×D4—21no—
261C22×D4—22no—
262C22×D4—22no—
263C22×D4—22no—
264C22×D4—22no—
265C22×D4—21no—
266C22×D4—22no—
267C22×D4C3×D414yes—
268C22×D4C23×C311yescharacteristic
269C22×D4C23×C312yes—
270C23×C4—21no—
271C23×C4—21no—
272C23×C4C23×C311yescharacteristic
273AGL(1,ℤ/8)—232no—
274C2×SD16—232no—
275C2×SD16—28no—
276C2×SD16—24no—
277C2×SD16—28no—
278C2×SD16—24no—
279C2×SD16—28no—
280C2×SD16—24no—
281C2×D8—24no—
282C2×D8—24no—
283C2×D8—24no—
284C2×M16—22no—
285C2×M16—22no—
286C2×M16C3×D414yes—
287C22×C8—22no—
288C22×C8—22no—
289C22×C8C3×D414yes—
290C22×C8C23×C312yes—
291C22×C8C23×C312yes—
292C4⋊Q8—24no—
293C4⋊Q8—24no—
294#124—24no—
295#124—24no—
296#123—216no—
297C22⋊Q8—216no—
298#117—22no—
299#117—22no—
300#117—22no—
301#117—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4C3×D412yes—
304C2×C4⋊C4C23×C312yes—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—28no—
308C2×C22⋊C4C3×D412yes—
309C2×C42—21no—
310C2×C42C23×C312yes—
311#103—24no—
312#103—28no—
313#102—24no—
314C23×C3—24no—
315C23×C3—22no—
316C23×C3—24no—
317C23×C3—42no—
318C23×C3—41no—
319C23×C3—41no—
320C23×C3—42no—
321C23×C3—48no—
322C23×C3—44no—
323C23×C3—48no—
324C23×C3—88no—
325C23×C3—88no—
326C23×C3—88no—
327C23×C3#12012yes—
328C23×C3#12112yes—
329C23×C3#12712yes—
330C23×C3C22×D411yescharacteristic
331C3×Q8—24no—
332C3×Q8—22no—
333C3×Q8—24no—
334C3×Q8—22no—
335C3×Q8—42no—
336C3×Q8—41no—
337C3×Q8—42no—
338C3×Q8—41no—
339C3×Q8—48no—
340C3×D4—22no—
341C3×D4—24no—
342C3×D4—24no—
343C3×D4—22no—
344C3×D4—48no—
345C3×D4—41no—
346C3×D4—42no—
347C3×D4—42no—
348C3×D4—41no—
349C3×D4—48no—
350C2×C3×C4—24no—
351C2×C3×C4—24no—
352C2×C3×C4—22no—
353C2×C3×C4—28no—
354C2×C3×C4—41no—
355C2×C3×C4—42no—
356C2×C3×C4—41no—
357C2×C3×C4—42no—
358C2×C3×C4—48no—
359C2×C3×C4—48no—
360C2×C3×C4—42no—
361C2×C3×C4—44no—
362C2×C3×C4#12014yes—
363C2×C3×C4#12014yes—
364C2×C3×C4#12012yes—
365C2×C3×C4#12114yes—
366C2×C3×C4#12114yes—
367C2×C3×C4#12112yes—
368C2×C3×C4#12712yes—
369C2×C3×C4C22×D411yescharacteristic
370C2×C3×C4C22×D411yescharacteristic
371C2×C3×C4C22×D411yescharacteristic
372C2×C3×C4C22×D411yescharacteristic
373C2×C3×C4C22×D411yescharacteristic
374C2×C3×C4C2×#4111yescharacteristic
375C3×C8—24no—
376C3×C8—24no—
377C24—21no—
378C24—22no—
379C24—24no—
380C24—42no—
381C24—42no—
382C24—84no—
383C24C2×C3×D411yescharacteristic
384#41—28no—
385#41—28no—
386#41—44no—
387#41—44no—
388C2×Q8—24no—
389C2×Q8—22no—
390C2×Q8—21no—
391C2×Q8—28no—
392C2×Q8—28no—
393C2×Q8—22no—
394C2×Q8—21no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—22no—
400C2×Q8—21no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—42no—
404C2×Q8—42no—
405C2×Q8C2×C3×D414yes—
406C2×Q8C2×C3×D412yes—
407C2×D4—24no—
408C2×D4—28no—
409C2×D4—28no—
410C2×D4—24no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—22no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—42no—
427C2×D4C2×C3×D412yes—
428C2×D4C2×C3×D414yes—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4C2×C3×D414yes—
439C22×C4C2×C3×D412yes—
440C22×C4C2×C3×D412yes—
441C22×C4C2×C3×D412yes—
442C22×C4C2×C3×D411yescharacteristic
443C22×C4C2×C3×D412yes—
444C22×C4C2×C3×D412yes—
445C22×C4C3×#4112yes—
446C22×C4C24×C311yescharacteristic
447SD16—232no—
448SD16—216no—
449D8—216no—
450M16—28no—
451C2×C8—28no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8C2×C3×D414yes—
457C2×C8C2×C3×D414yes—
458C4⋊C4—216no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C22⋊C4—216no—
463C22⋊C4—24no—
464C22⋊C4—42no—
465C22⋊C4—88no—
466C42—22no—
467C42—22no—
468C42—42no—
469C22×C3—24no—
470C22×C3—24no—
471C22×C3—24no—
472C22×C3—42no—
473C22×C3—44no—
474C22×C3—44no—
475C22×C3—48no—
476C22×C3—816no—
477C22×C3—82no—
478C22×C3—84no—
479C22×C3—84no—
480C22×C3—82no—
481C22×C3#39214yes—
482C22×C3C2×#12011yescharacteristic
483C22×C3C2×#12111yescharacteristic
484C22×C3C2×#12711yescharacteristic
485C22×C3#57714yes—
486C3×C4—42no—
487C3×C4—44no—
488C3×C4—44no—
489C3×C4—82no—
490C3×C4#39214yes—
491C3×C4C2×#12011yescharacteristic
492C3×C4C2×#12011yescharacteristic
493C3×C4C2×#12111yescharacteristic
494C3×C4C2×#12111yescharacteristic
495C23—24no—
496C23—22no—
497C23—24no—
498C23—48no—
499C23—44no—
500C23—48no—
501C23—42no—
502C23—41no—
503C23—41no—
504C23—42no—
505C23—88no—
506C23—88no—
507C23—88no—
508C23C3×#12012yes—
509C23C3×#12112yes—
510C23C3×#12712yes—
511C23C22×C3×D411yescharacteristic
512Q8—24no—
513Q8—22no—
514Q8—24no—
515Q8—22no—
516Q8—48no—
517Q8—42no—
518Q8—41no—
519Q8—42no—
520Q8—41no—
521D4—22no—
522D4—24no—
523D4—24no—
524D4—22no—
525D4—48no—
526D4—48no—
527D4—41no—
528D4—42no—
529D4—42no—
530D4—41no—
531C2×C4—28no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—48no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—48no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4C3×#12012yes—
544C2×C4C3×#12014yes—
545C2×C4C3×#12014yes—
546C2×C4C3×#12112yes—
547C2×C4C3×#12114yes—
548C2×C4C3×#12114yes—
549C2×C4C3×#12712yes—
550C2×C4C22×C3×D411yescharacteristic
551C2×C4C22×C3×D411yescharacteristic
552C2×C4C22×C3×D411yescharacteristic
553C2×C4C22×C3×D411yescharacteristic
554C2×C4C22×C3×D411yescharacteristic
555C2×C4C2×C3×#4111yescharacteristic
556C8—24no—
557C8—24no—
558C2×C3—24no—
559C2×C3—82no—
560C2×C3—84no—
561C2×C3C2×#39211yescharacteristic
562C2×C3#301012yes—
563C2×C3#304212yes—
564C2×C3#315412yes—
565C22—24no—
566C22—24no—
567C22—24no—
568C22—42no—
569C22—44no—
570C22—44no—
571C22—48no—
572C22—816no—
573C22—84no—
574C22—82no—
575C22—84no—
576C22—82no—
577C22C3×#39214yes—
578C22C2×C3×#12011yescharacteristic
579C22C2×C3×#12111yescharacteristic
580C22C2×C3×#12711yescharacteristic
581C22C3×#57714yes—
582C4—42no—
583C4—44no—
584C4—44no—
585C4—82no—
586C4C3×#39214yes—
587C4C2×C3×#12011yescharacteristic
588C4C2×C3×#12011yescharacteristic
589C4C2×C3×#12111yescharacteristic
590C4C2×C3×#12111yescharacteristic
591C3#2342411yescharacteristic, direct product factor, Sylow
592C2—24no—
593C2—84no—
594C2—82no—
595C2C2×C3×#39211yescharacteristic
596C2C3×#301012yes—
597C2C3×#304212yes—
598C2C3×#315412yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×AGL(1,ℤ/8)C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×C3×SD16C2semidirect
3. C2×C3×#536C2semidirect
4. C2×C3×#419C2semidirect
5. C2×C3×#416C2semidirect
6. C3×#2309C2semidirect
7. C3×#2013C2semidirect
8. C3×#2012C2semidirect
9. C3×#1729C2semidirect
10. #23424C3direct (N × H)
27. C2×C3×C4⋊Q8C22semidirect
28. C2×C3×#124C22semidirect
35. C2×C3×#103C22semidirect
36. C2×C3×#103C22semidirect
37. C2×C3×#102C22semidirect
41. C3×#336C22semidirect
43. C23×SD16C2×C3semidirect
44. C2×#536C2×C3semidirect
45. C2×#419C2×C3semidirect
46. C2×#416C2×C3semidirect
47. #2309C2×C3semidirect
48. #2013C2×C3semidirect
49. #2012C2×C3semidirect
50. #1729C2×C3semidirect
137. C2×C4⋊Q8C22×C3semidirect
138. C2×#124C22×C3semidirect
145. C2×#103C22×C3semidirect
146. C2×#103C22×C3semidirect
147. C2×#102C22×C3semidirect
151. #336C22×C3semidirect
591. C3#23424direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×AGL(1,ℤ/8)2
C23×C3×SD162
C2×C3×#5362
C2×C3×#4192
C2×C3×#4162
C3×#23092
C3×#20132
C3×#20122
C3×#17292
#234243