Finite Group Databasebeta

← C3×#23431 #10642038

Subgroups — C3×#23431 #10642038

Order 768.

2146 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#137C211yesmaximal, characteristic, complemented
2C22×C3×AGL(1,ℤ/8)C211yesmaximal, characteristic
3C22×C3×#124C211yesmaximal, characteristic, complemented
4C2×C3×#419C211yesmaximal, characteristic, complemented
5C2×C3×#418C211yesmaximal, characteristic, complemented
6C3×#2297C211yesmaximal, characteristic, complemented
7C3×#2015C214yesmaximal, complemented
8C3×#2012C214yesmaximal, complemented
9C3×#1728C211yesmaximal, characteristic, complemented
10#23431C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×#41C2211yescharacteristic
12C23×C3×Q8C2211yescharacteristic
13C23×C3×D4C2211yescharacteristic
14C2×C3×#137—24no—
15C2×C3×#137—24no—
16C2×C3×#137—24no—
17C2×C3×AGL(1,ℤ/8)—24no—
18C2×C3×AGL(1,ℤ/8)—24no—
19C2×C3×AGL(1,ℤ/8)—24no—
20C22×C3×Q16C2212yescomplemented
21C22×C3×SD16C2212yes—
22C22×C3×SD16C2212yescomplemented
23C22×C3×D8C2212yes—
24C22×C3×M16C2211yescharacteristic
25C22×C3×M16C2211yescharacteristic
26C2×C3×#124—24no—
27C2×C3×#124—24no—
28C2×C3×#124—22no—
29C2×C3×#124C2214yes—
30C2×C3×#117C2211yescharacteristic
31C22×C3×C22⋊C4—22no—
32C22×C3×C42C2211yescharacteristic
33C3×#419—24no—
34C3×#418—24no—
35C2×C3×#103C2214yescomplemented
36C2×C3×#102C2214yes—
37C3×#391C2212yescomplemented
38C3×#390C2212yescomplemented
39C3×#386—22no—
40C3×C4×C22⋊C4—22no—
41C3×#336C2214yescomplemented
42C22×#137C2×C311yescharacteristic, complemented
43C22×AGL(1,ℤ/8)C2×C311yescharacteristic
44C22×#124C2×C311yescharacteristic, complemented
45C2×#419C2×C311yescharacteristic, complemented
46C2×#418C2×C311yescharacteristic, complemented
47#2297C2×C311yescharacteristic, complemented
48#2015C2×C314yescomplemented
49#2012C2×C314yescomplemented
50#1728C2×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—22no—
59C22×C3×Q8—21no—
60C22×C3×Q8—22no—
61C22×C3×Q8—22no—
62C22×C3×Q8—22no—
63C22×C3×Q8—22no—
64C22×C3×Q8D414yes—
65C22×C3×Q8C2311yescharacteristic
66C22×C3×Q8C2312yes—
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×C4D412yes—
80C23×C3×C4C2311yescharacteristic
81C3×#137—232no—
82C3×AGL(1,ℤ/8)—232no—
83C2×C3×Q16—24no—
84C2×C3×Q16—24no—
85C2×C3×Q16—24no—
86C2×C3×SD16—24no—
87C2×C3×SD16—24no—
88C2×C3×SD16—24no—
89C2×C3×SD16—24no—
90C2×C3×SD16—24no—
91C2×C3×SD16—24no—
92C2×C3×D8—24no—
93C2×C3×D8—24no—
94C2×C3×D8—24no—
95C2×C3×M16—22no—
96C2×C3×M16—22no—
97C2×C3×M16—22no—
98C2×C3×M16—22no—
99C2×C3×M16D414yes—
100C2×C3×M16D414yes—
101C22×C3×C8C2312yes—
102C22×C3×C8C2312yes—
103C3×#124—216no—
104C3×#124—28no—
105C3×#124—28no—
106C3×#117—22no—
107C3×#117—22no—
108C2×C3×C4⋊C4C2312yes—
109C2×C3×C22⋊C4—216no—
110C2×C3×C22⋊C4—28no—
111C2×C3×C22⋊C4—21no—
112C2×C3×C22⋊C4—24no—
113C2×C3×C42—22no—
114C2×C3×C42—22no—
115C2×C3×C42—21no—
116C2×C3×C42D412yes—
117C2×C3×C42C2312yes—
118C3×#103—28no—
119C3×#102—28no—
120C3×#95—24no—
121C22×#41C22×C311yescharacteristic
122C23×Q8C22×C311yescharacteristic
123C23×D4C22×C311yescharacteristic
124C2×#137—24no—
125C2×#137—24no—
126C2×#137—24no—
127C2×AGL(1,ℤ/8)—24no—
128C2×AGL(1,ℤ/8)—24no—
129C2×AGL(1,ℤ/8)—24no—
130C22×Q16C22×C312yescomplemented
131C22×SD16C22×C312yes—
132C22×SD16C22×C312yescomplemented
133C22×D8C22×C312yes—
134C22×M16C22×C311yescharacteristic
135C22×M16C22×C311yescharacteristic
136C2×#124—24no—
137C2×#124—24no—
138C2×#124—22no—
139C2×#124C22×C314yes—
140C2×#117C22×C311yescharacteristic
141C22×C22⋊C4—22no—
142C22×C42C22×C311yescharacteristic
143#419—24no—
144#418—24no—
145C2×#103C22×C314yescomplemented
146C2×#102C22×C314yes—
147#391C22×C312yescomplemented
148#390C22×C312yescomplemented
149#386—22no—
150C4×C22⋊C4—22no—
151#336C22×C314yescomplemented
152C24×C3—24no—
153C24×C3—22no—
154C24×C3—21no—
155C24×C3—48no—
156C24×C3—42no—
157C24×C3—42no—
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—21no—
171C2×C3×Q8—22no—
172C2×C3×Q8—22no—
173C2×C3×Q8—21no—
174C2×C3×Q8—21no—
175C2×C3×Q8—22no—
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—24no—
185C2×C3×D4—28no—
186C2×C3×D4—22no—
187C2×C3×D4—21no—
188C2×C3×D4—21no—
189C2×C3×D4—22no—
190C2×C3×D4—22no—
191C2×C3×D4—21no—
192C2×C3×D4—21no—
193C2×C3×D4—22no—
194C2×C3×D4—22no—
195C2×C3×D4—21no—
196C2×C3×D4—42no—
197C2×C3×D4—42no—
198C2×C3×D4—44no—
199C2×C3×D4—44no—
200C2×C3×D4—42no—
201C2×C3×D4—42no—
202C2×C3×D4C2×D414yes—
203C2×C3×D4C2×D412yes—
204C22×C3×C4—24no—
205C22×C3×C4—24no—
206C22×C3×C4—21no—
207C22×C3×C4—22no—
208C22×C3×C4—24no—
209C22×C3×C4—22no—
210C22×C3×C4—44no—
211C22×C3×C4—42no—
212C22×C3×C4—42no—
213C22×C3×C4—42no—
214C22×C3×C4—42no—
215C22×C3×C4C2×D414yes—
216C22×C3×C4C2×D412yes—
217C22×C3×C4C2×D412yes—
218C22×C3×C4C2×D412yes—
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2×D412yes—
221C22×C3×C4C2×D412yes—
222C22×C3×C4#4112yes—
223C22×C3×C4C2411yescharacteristic
224C3×Q16—216no—
225C3×SD16—216no—
226C3×SD16—216no—
227C3×D8—216no—
228C3×M16—28no—
229C3×M16—28no—
230C2×C3×C8—22no—
231C2×C3×C8—22no—
232C2×C3×C8—22no—
233C2×C3×C8—22no—
234C2×C3×C8C2×D414yes—
235C2×C3×C8C2×D414yes—
236C3×C4⋊C4—22no—
237C3×C4⋊C4—22no—
238C3×C22⋊C4—232no—
239C3×C22⋊C4—48no—
240C3×C22⋊C4—42no—
241C3×C42—24no—
242C3×C42—22no—
243C3×C42—22no—
244C3×C42—42no—
245C25—21no—
246C2×#41—22no—
247C2×#41—22no—
248C2×#41—22no—
249C2×#41—22no—
250C2×#41—22no—
251C2×#41C3×D414yes—
252C22×Q8—22no—
253C22×Q8—21no—
254C22×Q8—22no—
255C22×Q8—22no—
256C22×Q8—22no—
257C22×Q8—22no—
258C22×Q8C3×D414yes—
259C22×Q8C23×C312yes—
260C22×Q8C23×C311yescharacteristic
261C22×D4—21no—
262C22×D4—22no—
263C22×D4—22no—
264C22×D4—22no—
265C22×D4—22no—
266C22×D4—21no—
267C22×D4—22no—
268C22×D4C3×D414yes—
269C22×D4C23×C312yes—
270C22×D4C23×C311yescharacteristic
271C23×C4—21no—
272C23×C4—21no—
273C23×C4C3×D412yes—
274C23×C4C23×C311yescharacteristic
275#137—232no—
276AGL(1,ℤ/8)—232no—
277C2×Q16—24no—
278C2×Q16—24no—
279C2×Q16—24no—
280C2×SD16—24no—
281C2×SD16—24no—
282C2×SD16—24no—
283C2×SD16—24no—
284C2×SD16—24no—
285C2×SD16—24no—
286C2×D8—24no—
287C2×D8—24no—
288C2×D8—24no—
289C2×M16—22no—
290C2×M16—22no—
291C2×M16—22no—
292C2×M16—22no—
293C2×M16C3×D414yes—
294C2×M16C3×D414yes—
295C22×C8C23×C312yes—
296C22×C8C23×C312yes—
297#124—216no—
298#124—28no—
299#124—28no—
300#117—22no—
301#117—22no—
302C2×C4⋊C4C23×C312yes—
303C2×C22⋊C4—216no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—24no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—21no—
310C2×C42C3×D412yes—
311C2×C42C23×C312yes—
312#103—28no—
313#102—28no—
314#95—24no—
315C23×C3—24no—
316C23×C3—24no—
317C23×C3—22no—
318C23×C3—416no—
319C23×C3—41no—
320C23×C3—42no—
321C23×C3—41no—
322C23×C3—42no—
323C23×C3—44no—
324C23×C3—48no—
325C23×C3—48no—
326C23×C3—88no—
327C23×C3—88no—
328C23×C3#12012yes—
329C23×C3#12112yes—
330C23×C3#12712yes—
331C23×C3C22×D411yescharacteristic
332C3×Q8—22no—
333C3×Q8—24no—
334C3×Q8—22no—
335C3×Q8—24no—
336C3×Q8—42no—
337C3×Q8—42no—
338C3×Q8—41no—
339C3×Q8—41no—
340C3×Q8—48no—
341C3×D4—22no—
342C3×D4—24no—
343C3×D4—22no—
344C3×D4—24no—
345C3×D4—48no—
346C3×D4—42no—
347C3×D4—42no—
348C3×D4—41no—
349C3×D4—41no—
350C3×D4—48no—
351C2×C3×C4—24no—
352C2×C3×C4—24no—
353C2×C3×C4—22no—
354C2×C3×C4—24no—
355C2×C3×C4—24no—
356C2×C3×C4—48no—
357C2×C3×C4—41no—
358C2×C3×C4—42no—
359C2×C3×C4—41no—
360C2×C3×C4—42no—
361C2×C3×C4—48no—
362C2×C3×C4—48no—
363C2×C3×C4—42no—
364C2×C3×C4—44no—
365C2×C3×C4#12014yes—
366C2×C3×C4#12014yes—
367C2×C3×C4#12012yes—
368C2×C3×C4#12114yes—
369C2×C3×C4#12114yes—
370C2×C3×C4#12112yes—
371C2×C3×C4#12712yes—
372C2×C3×C4C22×D411yescharacteristic
373C2×C3×C4C22×D411yescharacteristic
374C2×C3×C4C22×D411yescharacteristic
375C2×C3×C4C22×D411yescharacteristic
376C2×C3×C4C22×D411yescharacteristic
377C2×C3×C4C2×#4111yescharacteristic
378C3×C8—24no—
379C3×C8—24no—
380C24—24no—
381C24—21no—
382C24—22no—
383C24—48no—
384C24—42no—
385C24—42no—
386C24C2×C3×D411yescharacteristic
387#41—28no—
388#41—28no—
389#41—44no—
390#41—44no—
391C2×Q8—24no—
392C2×Q8—22no—
393C2×Q8—21no—
394C2×Q8—28no—
395C2×Q8—28no—
396C2×Q8—22no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—22no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8—42no—
407C2×Q8—42no—
408C2×Q8C2×C3×D414yes—
409C2×Q8C2×C3×D412yes—
410C2×D4—24no—
411C2×D4—28no—
412C2×D4—24no—
413C2×D4—28no—
414C2×D4—22no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—21no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—44no—
427C2×D4—42no—
428C2×D4—44no—
429C2×D4—42no—
430C2×D4C2×C3×D414yes—
431C2×D4C2×C3×D412yes—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—44no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4C2×C3×D414yes—
444C22×C4C2×C3×D412yes—
445C22×C4C2×C3×D412yes—
446C22×C4C2×C3×D412yes—
447C22×C4C2×C3×D411yescharacteristic
448C22×C4C2×C3×D412yes—
449C22×C4C2×C3×D412yes—
450C22×C4C3×#4112yes—
451C22×C4C24×C311yescharacteristic
452Q16—216no—
453SD16—216no—
454SD16—216no—
455D8—216no—
456M16—28no—
457M16—28no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8C2×C3×D414yes—
463C2×C8C2×C3×D414yes—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C22⋊C4—232no—
467C22⋊C4—48no—
468C22⋊C4—42no—
469C42—24no—
470C42—22no—
471C42—22no—
472C42—42no—
473C22×C3—24no—
474C22×C3—24no—
475C22×C3—24no—
476C22×C3—42no—
477C22×C3—44no—
478C22×C3—48no—
479C22×C3—44no—
480C22×C3—816no—
481C22×C3—82no—
482C22×C3—84no—
483C22×C3—82no—
484C22×C3—84no—
485C22×C3#39214yes—
486C22×C3C2×#12011yescharacteristic
487C22×C3C2×#12111yescharacteristic
488C22×C3C2×#12711yescharacteristic
489C22×C3#57514yes—
490C3×C4—42no—
491C3×C4—44no—
492C3×C4—44no—
493C3×C4—82no—
494C3×C4#39214yes—
495C3×C4C2×#12011yescharacteristic
496C3×C4C2×#12011yescharacteristic
497C3×C4C2×#12111yescharacteristic
498C3×C4C2×#12111yescharacteristic
499C23—24no—
500C23—24no—
501C23—22no—
502C23—416no—
503C23—44no—
504C23—48no—
505C23—48no—
506C23—41no—
507C23—42no—
508C23—41no—
509C23—42no—
510C23—88no—
511C23—88no—
512C23C3×#12012yes—
513C23C3×#12112yes—
514C23C3×#12712yes—
515C23C22×C3×D411yescharacteristic
516Q8—22no—
517Q8—24no—
518Q8—22no—
519Q8—24no—
520Q8—48no—
521Q8—42no—
522Q8—42no—
523Q8—41no—
524Q8—41no—
525D4—22no—
526D4—24no—
527D4—22no—
528D4—24no—
529D4—48no—
530D4—48no—
531D4—42no—
532D4—42no—
533D4—41no—
534D4—41no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—48no—
541C2×C4—48no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—48no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4C3×#12012yes—
550C2×C4C3×#12014yes—
551C2×C4C3×#12014yes—
552C2×C4C3×#12112yes—
553C2×C4C3×#12114yes—
554C2×C4C3×#12114yes—
555C2×C4C3×#12712yes—
556C2×C4C22×C3×D411yescharacteristic
557C2×C4C22×C3×D411yescharacteristic
558C2×C4C22×C3×D411yescharacteristic
559C2×C4C22×C3×D411yescharacteristic
560C2×C4C22×C3×D411yescharacteristic
561C2×C4C2×C3×#4111yescharacteristic
562C8—24no—
563C8—24no—
564C2×C3—24no—
565C2×C3—82no—
566C2×C3—84no—
567C2×C3C2×#39211yescharacteristic
568C2×C3#300512yes—
569C2×C3#304012yes—
570C2×C3#315312yes—
571C22—24no—
572C22—24no—
573C22—24no—
574C22—42no—
575C22—44no—
576C22—48no—
577C22—44no—
578C22—816no—
579C22—84no—
580C22—82no—
581C22—82no—
582C22—84no—
583C22C3×#39214yes—
584C22C2×C3×#12011yescharacteristic
585C22C2×C3×#12111yescharacteristic
586C22C2×C3×#12711yescharacteristic
587C22C3×#57514yes—
588C4—42no—
589C4—44no—
590C4—44no—
591C4—82no—
592C4C3×#39214yes—
593C4C2×C3×#12011yescharacteristic
594C4C2×C3×#12011yescharacteristic
595C4C2×C3×#12111yescharacteristic
596C4C2×C3×#12111yescharacteristic
597C3#2343111yescharacteristic, direct product factor, Sylow
598C2—24no—
599C2—84no—
600C2—82no—
601C2C2×C3×#39211yescharacteristic
602C2C3×#300512yes—
603C2C3×#304012yes—
604C2C3×#315312yes—
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 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C3×#137C2semidirect
3. C22×C3×#124C2semidirect
4. C2×C3×#419C2semidirect
5. C2×C3×#418C2semidirect
6. C3×#2297C2semidirect
7. C3×#2015C2semidirect
8. C3×#2012C2semidirect
9. C3×#1728C2semidirect
10. #23431C3direct (N × H)
20. C22×C3×Q16C22semidirect
22. C22×C3×SD16C22semidirect
35. C2×C3×#103C22semidirect
37. C3×#391C22semidirect
38. C3×#390C22semidirect
41. C3×#336C22semidirect
42. C22×#137C2×C3semidirect
44. C22×#124C2×C3semidirect
45. C2×#419C2×C3semidirect
46. C2×#418C2×C3semidirect
47. #2297C2×C3semidirect
48. #2015C2×C3semidirect
49. #2012C2×C3semidirect
50. #1728C2×C3semidirect
130. C22×Q16C22×C3semidirect
132. C22×SD16C22×C3semidirect
145. C2×#103C22×C3semidirect
147. #391C22×C3semidirect
148. #390C22×C3semidirect
151. #336C22×C3semidirect
597. C3#23431direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1372
C22×C3×AGL(1,ℤ/8)2
C22×C3×#1242
C2×C3×#4192
C2×C3×#4182
C3×#22972
C3×#20152
C3×#20122
C3×#17282
#234313