Finite Group Databasebeta

← C3×#24184 #10642791

Subgroups — C3×#24184 #10642791

Order 768.

1462 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#550C211yesmaximal, characteristic, complemented
2C2×C3×#459C211yesmaximal, characteristic, complemented
3C2×C3×C4×D8C211yesmaximal, characteristic, complemented
4C3×#2538C211yesmaximal, characteristic, complemented
5C3×#2041C212yesmaximal, complemented
6C3×#2011C212yesmaximal, complemented
7C3×#1984C211yesmaximal, characteristic, complemented
8C3×#1957C211yesmaximal, characteristic, complemented
9C3×#1879C212yesmaximal, complemented
10C3×#1862C212yesmaximal, complemented
11C3×#1802C211yesmaximal, characteristic, complemented
12#24184C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41—21no—
14C22×C3×D8C2211yescharacteristic
15C22×C3×D8C2212yes—
16C3×#550—24no—
17C3×#550—24no—
18C2×C3×#127C2212yes—
19C2×C3×#127C2211yescharacteristic
20C2×C3×C22⋊Q8—21no—
21C2×C3×#121—21no—
22C2×C3×#121—22no—
23C2×C3×C4×Q8C2211yescharacteristic
24C2×C3×C4×D4C2212yes—
25C2×C3×C4×D4C2211yescharacteristic
26C2×C3×C4×D4C2211yescharacteristic
27C3×#459—24no—
28C3×C4×D8—24no—
29C2×C3×#107C2211yescharacteristic
30C2×C3×C4⋊C8C2211yescharacteristic
31C2×C3×#102C2212yes—
32C2×C3×#102C2211yescharacteristic
33C2×C3×#102C2212yes—
34C2×C3×#102C2211yescharacteristic
35C2×C3×#102C2212yes—
36C2×C3×#102C2212yes—
37C2×C3×C4×C8C2211yescharacteristic
38C3×#396—22no—
39C3×#390C2212yes—
40C3×#389C2212yes—
41C3×#384C2211yescharacteristic
42C3×#380—21no—
43C3×#340C2212yes—
44C3×#336C2212yes—
45C3×#336C2212yes—
46C2×#550C2×C311yescharacteristic, complemented
47C2×#459C2×C311yescharacteristic, complemented
48C2×C4×D8C2×C311yescharacteristic, complemented
49#2538C2×C311yescharacteristic, complemented
50#2041C2×C312yescomplemented
51#2011C2×C312yescomplemented
52#1984C2×C311yescharacteristic, complemented
53#1957C2×C311yescharacteristic, complemented
54#1879C2×C312yescomplemented
55#1862C2×C312yescomplemented
56#1802C2×C311yescharacteristic, complemented
57C2×C3×#41—28no—
58C2×C3×#41—44no—
59C2×C3×#41—44no—
60C22×C3×Q8D411yescharacteristic
61C22×C3×D4—22no—
62C22×C3×D4—21no—
63C22×C3×D4—22no—
64C22×C3×D4—21no—
65C22×C3×D4C2312yes—
66C22×C3×D4C2311yescharacteristic
67C22×C3×D4C2311yescharacteristic
68C23×C3×C4—22no—
69C23×C3×C4—21no—
70C23×C3×C4—21no—
71C2×C3×D8—22no—
72C2×C3×D8—22no—
73C2×C3×D8—24no—
74C2×C3×D8—24no—
75C2×C3×D8—22no—
76C2×C3×D8—42no—
77C22×C3×C8C2312yes—
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C3×#127—24no—
81C3×#127—24no—
82C3×#127—22no—
83C3×#127D414yes—
84C3×C22⋊Q8—44no—
85C3×#121—28no—
86C3×#121—216no—
87C3×C4×Q8—22no—
88C3×C4×Q8—22no—
89C3×C4×D4—24no—
90C3×C4×D4—24no—
91C3×C4×D4—22no—
92C3×C4×D4—22no—
93C3×C4×D4—22no—
94C3×C4×D4D414yes—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4—22no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4D411yescharacteristic
99C2×C3×C4⋊C4D412yes—
100C2×C3×C4⋊C4C2312yes—
101C2×C3×C4⋊C4C2311yescharacteristic
102C2×C3×C4⋊C4C2311yescharacteristic
103C2×C3×C22⋊C4—22no—
104C2×C3×C22⋊C4—21no—
105C2×C3×C22⋊C4—22no—
106C2×C3×C22⋊C4—21no—
107C2×C3×C22⋊C4—22no—
108C2×C3×C42C2312yes—
109C2×C3×C42C2311yescharacteristic
110C3×#107—22no—
111C3×C4⋊C8—22no—
112C3×#102—22no—
113C3×#102—24no—
114C3×#102—24no—
115C3×#102—22no—
116C3×#102—42no—
117C3×#102—42no—
118C3×C4×C8—22no—
119C3×#95—22no—
120C22×#41—21no—
121C22×D8C22×C311yescharacteristic
122C22×D8C22×C312yes—
123#550—24no—
124#550—24no—
125C2×#127C22×C312yes—
126C2×#127C22×C311yescharacteristic
127C2×C22⋊Q8—21no—
128C2×#121—21no—
129C2×#121—22no—
130C2×C4×Q8C22×C311yescharacteristic
131C2×C4×D4C22×C312yes—
132C2×C4×D4C22×C311yescharacteristic
133C2×C4×D4C22×C311yescharacteristic
134#459—24no—
135C4×D8—24no—
136C2×#107C22×C311yescharacteristic
137C2×C4⋊C8C22×C311yescharacteristic
138C2×#102C22×C312yes—
139C2×#102C22×C312yes—
140C2×#102C22×C311yescharacteristic
141C2×#102C22×C311yescharacteristic
142C2×#102C22×C312yes—
143C2×#102C22×C312yes—
144C2×C4×C8C22×C311yescharacteristic
145#396—22no—
146#390C22×C312yes—
147#389C22×C312yes—
148#384C22×C311yescharacteristic
149#380—21no—
150#340C22×C312yes—
151#336C22×C312yes—
152#336C22×C312yes—
153C24×C3—21no—
154C24×C3—22no—
155C24×C3—21no—
156C3×#41—432no—
157C2×C3×Q8—22no—
158C2×C3×Q8—22no—
159C2×C3×Q8D814yescomplemented
160C2×C3×D4—28no—
161C2×C3×D4—28no—
162C2×C3×D4—24no—
163C2×C3×D4—24no—
164C2×C3×D4—21no—
165C2×C3×D4—21no—
166C2×C3×D4—22no—
167C2×C3×D4—21no—
168C2×C3×D4—22no—
169C2×C3×D4—21no—
170C2×C3×D4—22no—
171C2×C3×D4—21no—
172C2×C3×D4—21no—
173C2×C3×D4—21no—
174C2×C3×D4—22no—
175C2×C3×D4—21no—
176C2×C3×D4—21no—
177C2×C3×D4—22no—
178C2×C3×D4—21no—
179C2×C3×D4—44no—
180C2×C3×D4—44no—
181C2×C3×D4—42no—
182C2×C3×D4—42no—
183C2×C3×D4—81no—
184C2×C3×D4—81no—
185C2×C3×D4—82no—
186C2×C3×D4—82no—
187C2×C3×D4C2×D414yes—
188C2×C3×D4C2×D412yes—
189C2×C3×D4#4112yescomplemented
190C22×C3×C4—28no—
191C22×C3×C4—21no—
192C22×C3×C4—24no—
193C22×C3×C4—21no—
194C22×C3×C4—22no—
195C22×C3×C4—44no—
196C22×C3×C4—44no—
197C22×C3×C4—42no—
198C22×C3×C4—42no—
199C22×C3×C4—42no—
200C22×C3×C4—42no—
201C22×C3×C4—42no—
202C22×C3×C4C2×D412yes—
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D412yes—
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2411yescharacteristic
207C3×D8—28no—
208C3×D8—48no—
209C2×C3×C8—21no—
210C2×C3×C8—21no—
211C2×C3×C8—21no—
212C2×C3×C8—22no—
213C2×C3×C8—22no—
214C2×C3×C8—21no—
215C2×C3×C8—22no—
216C2×C3×C8—21no—
217C2×C3×C8#4112yescomplemented
218C3×C4⋊C4—24no—
219C3×C4⋊C4—22no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—22no—
224C3×C4⋊C4—22no—
225C3×C4⋊C4—42no—
226C3×C4⋊C4—44no—
227C3×C4⋊C4—81no—
228C3×C4⋊C4C2×D412yes—
229C3×C22⋊C4—28no—
230C3×C22⋊C4—24no—
231C3×C22⋊C4—42no—
232C3×C22⋊C4—44no—
233C3×C22⋊C4—82no—
234C3×C42—22no—
235C3×C42—22no—
236C3×C42C2×D412yes—
237C3×C42C2×D412yes—
238C2×#41—28no—
239C2×#41—44no—
240C2×#41—44no—
241C22×Q8C3×D411yescharacteristic
242C22×D4—22no—
243C22×D4—21no—
244C22×D4—22no—
245C22×D4—21no—
246C22×D4C23×C312yes—
247C22×D4C23×C311yescharacteristic
248C22×D4C23×C311yescharacteristic
249C23×C4—22no—
250C23×C4—21no—
251C23×C4—21no—
252C2×D8—22no—
253C2×D8—22no—
254C2×D8—24no—
255C2×D8—24no—
256C2×D8—22no—
257C2×D8—42no—
258C22×C8C23×C312yes—
259C22×C8C23×C311yescharacteristic
260C22×C8C23×C311yescharacteristic
261#127—24no—
262#127—24no—
263#127—22no—
264#127C3×D414yes—
265C22⋊Q8—44no—
266#121—28no—
267#121—216no—
268C4×Q8—22no—
269C4×Q8—22no—
270C4×D4—24no—
271C4×D4—24no—
272C4×D4—22no—
273C4×D4—22no—
274C4×D4—22no—
275C4×D4C3×D414yes—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4C3×D411yescharacteristic
280C2×C4⋊C4C3×D412yes—
281C2×C4⋊C4C23×C311yescharacteristic
282C2×C4⋊C4C23×C312yes—
283C2×C4⋊C4C23×C311yescharacteristic
284C2×C22⋊C4—22no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C42C23×C312yes—
290C2×C42C23×C311yescharacteristic
291#107—22no—
292C4⋊C8—22no—
293#102—22no—
294#102—24no—
295#102—24no—
296#102—22no—
297#102—42no—
298#102—42no—
299C4×C8—22no—
300#95—22no—
301C23×C3—24no—
302C23×C3—22no—
303C23×C3—22no—
304C23×C3—41no—
305C23×C3—41no—
306C23×C3—42no—
307C23×C3—42no—
308C23×C3—41no—
309C23×C3—41no—
310C23×C3—41no—
311C23×C3—42no—
312C23×C3—41no—
313C23×C3—42no—
314C23×C3—41no—
315C23×C3—41no—
316C23×C3—41no—
317C23×C3—42no—
318C23×C3—41no—
319C23×C3—42no—
320C23×C3—41no—
321C23×C3—41no—
322C23×C3C22×D411yescharacteristic
323C3×Q8—28no—
324C3×D4—22no—
325C3×D4—22no—
326C3×D4—22no—
327C3×D4—22no—
328C3×D4—24no—
329C3×D4—24no—
330C3×D4—416no—
331C3×D4—42no—
332C3×D4—41no—
333C3×D4—42no—
334C3×D4—41no—
335C3×D4—41no—
336C3×D4—41no—
337C3×D4—48no—
338C3×D4—84no—
339C3×D4—88no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—416no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—48no—
352C2×C3×C4—41no—
353C2×C3×C4—42no—
354C2×C3×C4—42no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—42no—
358C2×C3×C4—84no—
359C2×C3×C4#12012yes—
360C2×C3×C4#12012yes—
361C2×C3×C4C2×D814yes—
362C2×C3×C4C2×D812yes—
363C2×C3×C4AGL(1,ℤ/8)12yes—
364C2×C3×C4C22×D411yescharacteristic
365C2×C3×C4C22×D411yescharacteristic
366C2×C3×C4C2×#4111yescharacteristic
367C2×C3×C4C2×#4111yescharacteristic
368C2×C3×C4H(5,𝔽2)11yescharacteristic
369C2×C3×C42−1+411yescharacteristic
370C3×C8—22no—
371C3×C8—41no—
372C3×C8—42no—
373C24—21no—
374C24—22no—
375C24—21no—
376#41—432no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8C3×D814yescomplemented
380C2×D4—28no—
381C2×D4—28no—
382C2×D4—24no—
383C2×D4—24no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—22no—
387C2×D4—21no—
388C2×D4—22no—
389C2×D4—21no—
390C2×D4—22no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—22no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—22no—
398C2×D4—21no—
399C2×D4—44no—
400C2×D4—44no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—81no—
404C2×D4—81no—
405C2×D4—82no—
406C2×D4—82no—
407C2×D4C2×C3×D414yes—
408C2×D4C2×C3×D412yes—
409C2×D4C3×#4112yescomplemented
410C22×C4—28no—
411C22×C4—21no—
412C22×C4—24no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4C2×C3×D412yes—
423C22×C4C2×C3×D411yescharacteristic
424C22×C4C2×C3×D412yes—
425C22×C4C2×C3×D411yescharacteristic
426C22×C4C24×C311yescharacteristic
427D8—28no—
428D8—48no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—21no—
435C2×C8—22no—
436C2×C8—21no—
437C2×C8C3×#4112yescomplemented
438C4⋊C4—24no—
439C4⋊C4—22no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—42no—
446C4⋊C4—44no—
447C4⋊C4—81no—
448C4⋊C4C2×C3×D412yes—
449C22⋊C4—28no—
450C22⋊C4—24no—
451C22⋊C4—42no—
452C22⋊C4—44no—
453C22⋊C4—82no—
454C42—22no—
455C42—22no—
456C42C2×C3×D412yes—
457C42C2×C3×D412yes—
458C22×C3—42no—
459C22×C3—42no—
460C22×C3—42no—
461C22×C3—42no—
462C22×C3—44no—
463C22×C3—44no—
464C22×C3—42no—
465C22×C3—44no—
466C22×C3—42no—
467C22×C3—81no—
468C22×C3—82no—
469C22×C3—82no—
470C22×C3—81no—
471C22×C3—81no—
472C22×C3—82no—
473C22×C3—81no—
474C22×C3—82no—
475C22×C3—81no—
476C22×C3—81no—
477C22×C3—81no—
478C22×C3—81no—
479C22×C3C2×#12011yescharacteristic
480C22×C3#54811yescharacteristic
481C22×C3#55011yescharacteristic
482C22×C3C22×D811yescharacteristic
483C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
484C22×C3#57611yescharacteristic
485C22×C3#57711yescharacteristic
486C3×C4—24no—
487C3×C4—22no—
488C3×C4—42no—
489C3×C4—42no—
490C3×C4—82no—
491C3×C4—81no—
492C3×C4#44714yes—
493C3×C4#52511yescharacteristic
494C3×C4#53711yescharacteristic
495C3×C4#54611yescharacteristic
496C3×C4#54711yescharacteristic
497C23—24no—
498C23—22no—
499C23—22no—
500C23—41no—
501C23—42no—
502C23—41no—
503C23—42no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—42no—
508C23—41no—
509C23—42no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—42no—
515C23—42no—
516C23—41no—
517C23—41no—
518C23C22×C3×D411yescharacteristic
519Q8—28no—
520D4—22no—
521D4—22no—
522D4—22no—
523D4—22no—
524D4—24no—
525D4—24no—
526D4—416no—
527D4—48no—
528D4—42no—
529D4—41no—
530D4—42no—
531D4—41no—
532D4—41no—
533D4—41no—
534D4—84no—
535D4—88no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—416no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—48no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—84no—
555C2×C4C3×#12012yes—
556C2×C4C3×#12012yes—
557C2×C4C2×C3×D814yes—
558C2×C4C2×C3×D812yes—
559C2×C4C3×AGL(1,ℤ/8)12yes—
560C2×C4C22×C3×D411yescharacteristic
561C2×C4C22×C3×D411yescharacteristic
562C2×C4C2×C3×#4111yescharacteristic
563C2×C4C2×C3×#4111yescharacteristic
564C2×C4C3×H(5,𝔽2)11yescharacteristic
565C2×C4C3×2−1+411yescharacteristic
566C8—22no—
567C8—41no—
568C8—42no—
569C2×C3—82no—
570C2×C3—82no—
571C2×C3—84no—
572C2×C3#240711yescharacteristic
573C2×C3C2×#44711yescharacteristic
574C2×C3#301211yescharacteristic
575C2×C3#335811yescharacteristic
576C2×C3#336611yescharacteristic
577C2×C3#339111yescharacteristic
578C2×C3#339711yescharacteristic
579C22—42no—
580C22—42no—
581C22—44no—
582C22—44no—
583C22—42no—
584C22—44no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—81no—
589C22—82no—
590C22—81no—
591C22—82no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22—81no—
596C22—81no—
597C22—82no—
598C22—82no—
599C22—81no—
600C22C2×C3×#12011yescharacteristic
601C22C3×#54811yescharacteristic
602C22C3×#55011yescharacteristic
603C22C22×C3×D811yescharacteristic
604C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
605C22C3×#57611yescharacteristic
606C22C3×#57711yescharacteristic
607C4—24no—
608C4—22no—
609C4—42no—
610C4—42no—
611C4—82no—
612C4—81no—
613C4C3×#44714yes—
614C4C3×#52511yescharacteristic
615C4C3×#53711yescharacteristic
616C4C3×#54611yescharacteristic
617C4C3×#54711yescharacteristic
618C3#2418411yescharacteristic, direct product factor, Sylow
619C2—82no—
620C2—84no—
621C2—82no—
622C2C3×#240711yescharacteristic
623C2C2×C3×#44711yescharacteristic
624C2C3×#301211yescharacteristic
625C2C3×#335811yescharacteristic
626C2C3×#336611yescharacteristic
627C2C3×#339111yescharacteristic
628C2C3×#339711yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
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
1. C2×C3×#550C2semidirect
2. C2×C3×#459C2semidirect
3. C2×C3×C4×D8C2semidirect
4. C3×#2538C2semidirect
5. C3×#2041C2semidirect
6. C3×#2011C2semidirect
7. C3×#1984C2semidirect
8. C3×#1957C2semidirect
9. C3×#1879C2semidirect
10. C3×#1862C2semidirect
11. C3×#1802C2semidirect
12. #24184C3direct (N × H)
46. C2×#550C2×C3semidirect
47. C2×#459C2×C3semidirect
48. C2×C4×D8C2×C3semidirect
49. #2538C2×C3semidirect
50. #2041C2×C3semidirect
51. #2011C2×C3semidirect
52. #1984C2×C3semidirect
53. #1957C2×C3semidirect
54. #1879C2×C3semidirect
55. #1862C2×C3semidirect
56. #1802C2×C3semidirect
159. C2×C3×Q8D8semidirect
189. C2×C3×D4#41semidirect
217. C2×C3×C8#41semidirect
379. C2×Q8C3×D8semidirect
409. C2×D4C3×#41semidirect
437. C2×C8C3×#41semidirect
618. C3#24184direct (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×#4592
C2×C3×C4×D82
C3×#25382
C3×#20412
C3×#20112
C3×#19842
C3×#19572
C3×#18792
C3×#18622
C3×#18022
#241843