Finite Group Databasebeta

← C3×#24188 #10642795

Subgroups — C3×#24188 #10642795

Order 768.

1394 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#550C211yesmaximal, characteristic
2C2×C3×#460C211yesmaximal, characteristic, complemented
3C2×C3×C4×SD16C211yesmaximal, characteristic, complemented
4C3×#2582C211yesmaximal, characteristic, complemented
5C3×#2041C212yesmaximal, complemented
6C3×#2014C212yesmaximal, complemented
7C3×#1984C211yesmaximal, characteristic, complemented
8C3×#1957C211yesmaximal, characteristic, complemented
9C3×#1877C212yesmaximal, complemented
10C3×#1864C212yesmaximal, complemented
11C3×#1802C211yesmaximal, characteristic, complemented
12#24188C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41—21no—
14C22×C3×SD16C2211yescharacteristic
15C22×C3×SD16C2212yes—
16C3×#550—24no—
17C3×#550—24no—
18C2×C3×#127C2212yes—
19C2×C3×#127C2211yescharacteristic
20C2×C3×#121—21no—
21C2×C3×#121—22no—
22C2×C3×C4×Q8C2211yescharacteristic
23C2×C3×C4×Q8C2211yescharacteristic
24C2×C3×C4×D4C2212yes—
25C2×C3×C4×D4C2211yescharacteristic
26C3×#460—24no—
27C3×C4×SD16—24no—
28C2×C3×#106C2211yescharacteristic
29C2×C3×C4⋊C8C2211yescharacteristic
30C2×C3×#103C2211yescharacteristic
31C2×C3×#102C2212yes—
32C2×C3×#102C2211yescharacteristic
33C2×C3×#102C2212yes—
34C2×C3×#102C2212yes—
35C2×C3×#102C2212yes—
36C2×C3×C4×C8C2211yescharacteristic
37C3×#398—22no—
38C3×#391—21no—
39C3×#391C2212yes—
40C3×#389—21no—
41C3×#389C2212yes—
42C3×#384C2211yescharacteristic
43C3×#340C2212yes—
44C3×#336C2212yes—
45C3×#336C2212yes—
46C2×#550C2×C311yescharacteristic
47C2×#460C2×C311yescharacteristic, complemented
48C2×C4×SD16C2×C311yescharacteristic, complemented
49#2582C2×C311yescharacteristic, complemented
50#2041C2×C312yescomplemented
51#2014C2×C312yescomplemented
52#1984C2×C311yescharacteristic, complemented
53#1957C2×C311yescharacteristic, complemented
54#1877C2×C312yescomplemented
55#1864C2×C312yescomplemented
56#1802C2×C311yescharacteristic, complemented
57C2×C3×#41—28no—
58C2×C3×#41—44no—
59C2×C3×#41—44no—
60C22×C3×Q8D411yescharacteristic
61C22×C3×Q8C2311yescharacteristic
62C22×C3×D4—22no—
63C22×C3×D4—21no—
64C22×C3×D4—22no—
65C22×C3×D4—21no—
66C22×C3×D4C2312yes—
67C22×C3×D4C2311yescharacteristic
68C23×C3×C4—22no—
69C23×C3×C4—21no—
70C2×C3×SD16—22no—
71C2×C3×SD16—22no—
72C2×C3×SD16—24no—
73C2×C3×SD16—24no—
74C2×C3×SD16—22no—
75C2×C3×SD16—42no—
76C22×C3×C8C2312yes—
77C22×C3×C8C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C3×#127—24no—
80C3×#127—24no—
81C3×#127—22no—
82C3×#127D414yes—
83C3×#121—28no—
84C3×#121—216no—
85C3×C4×Q8—22no—
86C3×C4×Q8—22no—
87C3×C4×Q8—22no—
88C3×C4×Q8—22no—
89C3×C4×D4—24no—
90C3×C4×D4—24no—
91C3×C4×D4—22no—
92C3×C4×D4D414yes—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4—22no—
97C2×C3×C4⋊C4D411yescharacteristic
98C2×C3×C4⋊C4D412yes—
99C2×C3×C4⋊C4C2312yes—
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C4⋊C4C2311yescharacteristic
102C2×C3×C22⋊C4—22no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C42—21no—
105C2×C3×C42C2312yes—
106C2×C3×C42C2311yescharacteristic
107C3×#106—22no—
108C3×C4⋊C8—22no—
109C3×#103—22no—
110C3×#102—22no—
111C3×#102—24no—
112C3×#102—24no—
113C3×#102—42no—
114C3×#102—42no—
115C3×C4×C8—22no—
116C3×#95—22no—
117C3×#95—22no—
118C3×#95—22no—
119C22×#41—21no—
120C22×SD16C22×C311yescharacteristic
121C22×SD16C22×C312yes—
122#550—24no—
123#550—24no—
124C2×#127C22×C312yes—
125C2×#127C22×C311yescharacteristic
126C2×#121—21no—
127C2×#121—22no—
128C2×C4×Q8C22×C311yescharacteristic
129C2×C4×Q8C22×C311yescharacteristic
130C2×C4×D4C22×C312yes—
131C2×C4×D4C22×C311yescharacteristic
132#460—24no—
133C4×SD16—24no—
134C2×#106C22×C311yescharacteristic
135C2×C4⋊C8C22×C311yescharacteristic
136C2×#103C22×C311yescharacteristic
137C2×#102C22×C312yes—
138C2×#102C22×C312yes—
139C2×#102C22×C311yescharacteristic
140C2×#102C22×C312yes—
141C2×#102C22×C312yes—
142C2×C4×C8C22×C311yescharacteristic
143#398—22no—
144#391—21no—
145#391C22×C312yes—
146#389—21no—
147#389C22×C312yes—
148#384C22×C311yescharacteristic
149#340C22×C312yes—
150#336C22×C312yes—
151#336C22×C312yes—
152C24×C3—22no—
153C24×C3—21no—
154C3×#41—432no—
155C2×C3×Q8—22no—
156C2×C3×Q8—22no—
157C2×C3×Q8—21no—
158C2×C3×Q8—21no—
159C2×C3×Q8—21no—
160C2×C3×Q8—21no—
161C2×C3×Q8—21no—
162C2×C3×Q8SD1614yescomplemented
163C2×C3×Q8#4112yescomplemented
164C2×C3×D4—28no—
165C2×C3×D4—28no—
166C2×C3×D4—24no—
167C2×C3×D4—24no—
168C2×C3×D4—22no—
169C2×C3×D4—21no—
170C2×C3×D4—22no—
171C2×C3×D4—21no—
172C2×C3×D4—22no—
173C2×C3×D4—21no—
174C2×C3×D4—21no—
175C2×C3×D4—22no—
176C2×C3×D4—21no—
177C2×C3×D4—22no—
178C2×C3×D4—44no—
179C2×C3×D4—44no—
180C2×C3×D4—42no—
181C2×C3×D4—42no—
182C2×C3×D4—81no—
183C2×C3×D4—81no—
184C2×C3×D4—82no—
185C2×C3×D4—82no—
186C2×C3×D4C2×D414yes—
187C2×C3×D4C2×D412yes—
188C22×C3×C4—28no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—24no—
192C22×C3×C4—22no—
193C22×C3×C4—21no—
194C22×C3×C4—44no—
195C22×C3×C4—44no—
196C22×C3×C4—42no—
197C22×C3×C4—42no—
198C22×C3×C4C2×D412yes—
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D412yes—
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2411yescharacteristic
203C3×SD16—28no—
204C3×SD16—48no—
205C2×C3×C8—21no—
206C2×C3×C8—21no—
207C2×C3×C8—21no—
208C2×C3×C8—22no—
209C2×C3×C8—22no—
210C2×C3×C8—21no—
211C2×C3×C8—22no—
212C2×C3×C8—21no—
213C2×C3×C8#4112yescomplemented
214C3×C4⋊C4—24no—
215C3×C4⋊C4—22no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—22no—
220C3×C4⋊C4—22no—
221C3×C4⋊C4—42no—
222C3×C4⋊C4—42no—
223C3×C4⋊C4—81no—
224C3×C4⋊C4—82no—
225C3×C4⋊C4C2×D412yes—
226C3×C22⋊C4—28no—
227C3×C22⋊C4—24no—
228C3×C42—22no—
229C3×C42—22no—
230C3×C42—42no—
231C3×C42C2×D412yes—
232C3×C42C2×D412yes—
233C2×#41—28no—
234C2×#41—44no—
235C2×#41—44no—
236C22×Q8C3×D411yescharacteristic
237C22×Q8C23×C311yescharacteristic
238C22×D4—22no—
239C22×D4—21no—
240C22×D4—22no—
241C22×D4—21no—
242C22×D4C23×C312yes—
243C22×D4C23×C311yescharacteristic
244C23×C4—22no—
245C23×C4—21no—
246C2×SD16—22no—
247C2×SD16—22no—
248C2×SD16—24no—
249C2×SD16—24no—
250C2×SD16—22no—
251C2×SD16—42no—
252C22×C8C23×C312yes—
253C22×C8C23×C311yescharacteristic
254C22×C8C23×C311yescharacteristic
255#127—24no—
256#127—24no—
257#127—22no—
258#127C3×D414yes—
259#121—28no—
260#121—216no—
261C4×Q8—22no—
262C4×Q8—22no—
263C4×Q8—22no—
264C4×Q8—22no—
265C4×D4—24no—
266C4×D4—24no—
267C4×D4—22no—
268C4×D4C3×D414yes—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4C3×D411yescharacteristic
274C2×C4⋊C4C3×D412yes—
275C2×C4⋊C4C23×C311yescharacteristic
276C2×C4⋊C4C23×C312yes—
277C2×C4⋊C4C23×C311yescharacteristic
278C2×C22⋊C4—22no—
279C2×C22⋊C4—21no—
280C2×C42—21no—
281C2×C42C23×C312yes—
282C2×C42C23×C311yescharacteristic
283#106—22no—
284C4⋊C8—22no—
285#103—22no—
286#102—22no—
287#102—24no—
288#102—24no—
289#102—42no—
290#102—42no—
291C4×C8—22no—
292#95—22no—
293#95—22no—
294#95—22no—
295C23×C3—24no—
296C23×C3—22no—
297C23×C3—41no—
298C23×C3—42no—
299C23×C3—42no—
300C23×C3—41no—
301C23×C3—42no—
302C23×C3—41no—
303C23×C3—42no—
304C23×C3—41no—
305C23×C3—42no—
306C23×C3—41no—
307C23×C3—42no—
308C23×C3—41no—
309C23×C3C22×D411yescharacteristic
310C3×Q8—28no—
311C3×Q8—22no—
312C3×Q8—22no—
313C3×Q8—41no—
314C3×Q8—41no—
315C3×D4—22no—
316C3×D4—22no—
317C3×D4—24no—
318C3×D4—24no—
319C3×D4—416no—
320C3×D4—42no—
321C3×D4—42no—
322C3×D4—41no—
323C3×D4—41no—
324C3×D4—48no—
325C3×D4—84no—
326C3×D4—88no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—416no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—48no—
342C2×C3×C4—41no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4—42no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4#12012yes—
349C2×C3×C4#12012yes—
350C2×C3×C4C2×SD1614yes—
351C2×C3×C4C2×SD1612yes—
352C2×C3×C4AGL(1,ℤ/8)12yes—
353C2×C3×C4C22×D411yescharacteristic
354C2×C3×C4C22×D411yescharacteristic
355C2×C3×C4C2×#4111yescharacteristic
356C2×C3×C4C2×#4111yescharacteristic
357C2×C3×C4H(5,𝔽2)11yescharacteristic
358C2×C3×C42−1+411yescharacteristic
359C3×C8—22no—
360C3×C8—41no—
361C3×C8—42no—
362C24—22no—
363C24—21no—
364#41—432no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—21no—
368C2×Q8—21no—
369C2×Q8—21no—
370C2×Q8—21no—
371C2×Q8—21no—
372C2×Q8C3×SD1614yescomplemented
373C2×Q8C3×#4112yescomplemented
374C2×D4—28no—
375C2×D4—28no—
376C2×D4—24no—
377C2×D4—24no—
378C2×D4—22no—
379C2×D4—21no—
380C2×D4—22no—
381C2×D4—21no—
382C2×D4—22no—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—22no—
386C2×D4—21no—
387C2×D4—22no—
388C2×D4—44no—
389C2×D4—44no—
390C2×D4—42no—
391C2×D4—42no—
392C2×D4—81no—
393C2×D4—81no—
394C2×D4—82no—
395C2×D4—82no—
396C2×D4C2×C3×D414yes—
397C2×D4C2×C3×D412yes—
398C22×C4—28no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4C2×C3×D412yes—
409C22×C4C2×C3×D411yescharacteristic
410C22×C4C2×C3×D412yes—
411C22×C4C2×C3×D411yescharacteristic
412C22×C4C24×C311yescharacteristic
413SD16—28no—
414SD16—48no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—21no—
421C2×C8—22no—
422C2×C8—21no—
423C2×C8C3×#4112yescomplemented
424C4⋊C4—24no—
425C4⋊C4—22no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—81no—
434C4⋊C4—82no—
435C4⋊C4C2×C3×D412yes—
436C22⋊C4—28no—
437C22⋊C4—24no—
438C42—22no—
439C42—22no—
440C42—42no—
441C42C2×C3×D412yes—
442C42C2×C3×D412yes—
443C22×C3—42no—
444C22×C3—44no—
445C22×C3—44no—
446C22×C3—42no—
447C22×C3—44no—
448C22×C3—42no—
449C22×C3—81no—
450C22×C3—82no—
451C22×C3—82no—
452C22×C3—81no—
453C22×C3—82no—
454C22×C3—81no—
455C22×C3—82no—
456C22×C3—81no—
457C22×C3C2×#12011yescharacteristic
458C22×C3#54811yescharacteristic
459C22×C3#55011yescharacteristic
460C22×C3C22×SD1611yescharacteristic
461C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
462C22×C3#57611yescharacteristic
463C22×C3#57711yescharacteristic
464C3×C4—24no—
465C3×C4—22no—
466C3×C4—42no—
467C3×C4—42no—
468C3×C4—42no—
469C3×C4—82no—
470C3×C4—81no—
471C3×C4#45014yes—
472C3×C4#52511yescharacteristic
473C3×C4#53711yescharacteristic
474C3×C4#54611yescharacteristic
475C3×C4#54711yescharacteristic
476C23—24no—
477C23—22no—
478C23—42no—
479C23—41no—
480C23—42no—
481C23—41no—
482C23—42no—
483C23—41no—
484C23—42no—
485C23—41no—
486C23—41no—
487C23—42no—
488C23—42no—
489C23—41no—
490C23C22×C3×D411yescharacteristic
491Q8—28no—
492Q8—22no—
493Q8—22no—
494Q8—41no—
495Q8—41no—
496D4—22no—
497D4—22no—
498D4—24no—
499D4—24no—
500D4—416no—
501D4—48no—
502D4—42no—
503D4—42no—
504D4—41no—
505D4—41no—
506D4—84no—
507D4—88no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—416no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—48no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4C3×#12012yes—
530C2×C4C3×#12012yes—
531C2×C4C2×C3×SD1614yes—
532C2×C4C2×C3×SD1612yes—
533C2×C4C3×AGL(1,ℤ/8)12yes—
534C2×C4C22×C3×D411yescharacteristic
535C2×C4C22×C3×D411yescharacteristic
536C2×C4C2×C3×#4111yescharacteristic
537C2×C4C2×C3×#4111yescharacteristic
538C2×C4C3×H(5,𝔽2)11yescharacteristic
539C2×C4C3×2−1+411yescharacteristic
540C8—22no—
541C8—41no—
542C8—42no—
543C2×C3—82no—
544C2×C3—84no—
545C2×C3#240711yescharacteristic
546C2×C3C2×#45011yescharacteristic
547C2×C3#301211yescharacteristic
548C2×C3#336211yescharacteristic
549C2×C3#336811yescharacteristic
550C2×C3#339211yescharacteristic
551C2×C3#339911yescharacteristic
552C22—42no—
553C22—44no—
554C22—44no—
555C22—42no—
556C22—44no—
557C22—42no—
558C22—82no—
559C22—81no—
560C22—82no—
561C22—81no—
562C22—81no—
563C22—82no—
564C22—82no—
565C22—81no—
566C22C2×C3×#12011yescharacteristic
567C22C3×#54811yescharacteristic
568C22C3×#55011yescharacteristic
569C22C22×C3×SD1611yescharacteristic
570C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
571C22C3×#57611yescharacteristic
572C22C3×#57711yescharacteristic
573C4—24no—
574C4—22no—
575C4—42no—
576C4—42no—
577C4—42no—
578C4—82no—
579C4—81no—
580C4C3×#45014yes—
581C4C3×#52511yescharacteristic
582C4C3×#53711yescharacteristic
583C4C3×#54611yescharacteristic
584C4C3×#54711yescharacteristic
585C3#2418811yescharacteristic, direct product factor, Sylow
586C2—82no—
587C2—84no—
588C2C3×#240711yescharacteristic
589C2C2×C3×#45011yescharacteristic
590C2C3×#301211yescharacteristic
591C2C3×#336211yescharacteristic
592C2C3×#336811yescharacteristic
593C2C3×#339211yescharacteristic
594C2C3×#339911yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#550C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#460C2semidirect
3. C2×C3×C4×SD16C2semidirect
4. C3×#2582C2semidirect
5. C3×#2041C2semidirect
6. C3×#2014C2semidirect
7. C3×#1984C2semidirect
8. C3×#1957C2semidirect
9. C3×#1877C2semidirect
10. C3×#1864C2semidirect
11. C3×#1802C2semidirect
12. #24188C3direct (N × H)
47. C2×#460C2×C3semidirect
48. C2×C4×SD16C2×C3semidirect
49. #2582C2×C3semidirect
50. #2041C2×C3semidirect
51. #2014C2×C3semidirect
52. #1984C2×C3semidirect
53. #1957C2×C3semidirect
54. #1877C2×C3semidirect
55. #1864C2×C3semidirect
56. #1802C2×C3semidirect
162. C2×C3×Q8SD16semidirect
163. C2×C3×Q8#41semidirect
213. C2×C3×C8#41semidirect
372. C2×Q8C3×SD16semidirect
373. C2×Q8C3×#41semidirect
423. C2×C8C3×#41semidirect
585. C3#24188direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5502
C2×C3×#4602
C2×C3×C4×SD162
C3×#25822
C3×#20412
C3×#20142
C3×#19842
C3×#19572
C3×#18772
C3×#18642
C3×#18022
#241883