Finite Group Databasebeta

← C3×#24624 #10643231

Subgroups — C3×#24624 #10643231

Order 768.

670 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3095C211yesmaximal, characteristic
2C2×C3×#485C211yesmaximal, characteristic
3C3×#2924C211yesmaximal, characteristic
4C3×#2911C211yesmaximal, characteristic
5C3×#2090C211yesmaximal, characteristic, complemented
6C3×#2089C211yesmaximal, characteristic, complemented
7C3×#2064C211yesmaximal, characteristic, complemented
8C3×#2064C211yesmaximal, characteristic, complemented
9C3×#1952C211yesmaximal, characteristic, complemented
10C3×#1951C211yesmaximal, characteristic, complemented
11C3×#1930C211yesmaximal, characteristic, complemented
12C3×#1930C211yesmaximal, characteristic, complemented
13C3×#1901C211yesmaximal, characteristic
14C3×#1855C211yesmaximal, characteristic
15C3×#1822C211yesmaximal, characteristic
16#24624C311yesmaximal, characteristic, direct product factor, Sylow
17C22×C3×M16C2211yescharacteristic
18C2×C3×C22⋊Q8C2211yescharacteristic
19C2×C3×#117C2211yescharacteristic
20C2×C3×#117C2211yescharacteristic
21C3×#485—21no—
22C3×#485—21no—
23C3×#485—21no—
24C3×#485—21no—
25C3×#485—21no—
26C3×#485—21no—
27C3×#484—21no—
28C3×#483—21no—
29C3×#433—21no—
30C3×#433—21no—
31C3×#432—21no—
32C3×#431—21no—
33C2×C3×#108C2211yescharacteristic
34C2×C3×#108C2211yescharacteristic
35C3×#428—21no—
36C3×#428—21no—
37C3×#428C2211yescharacteristic
38C3×#428C2211yescharacteristic
39C3×#428C2211yescharacteristic
40C3×#428C2211yescharacteristic
41C3×#428C2211yescharacteristic
42C3×#428C2211yescharacteristic
43C3×#427—21no—
44C3×#427—21no—
45C2×C3×#107C2211yescharacteristic
46C2×C3×#106C2211yescharacteristic
47C3×#424C2211yescharacteristic
48C3×#424C2211yescharacteristic
49C3×#419C2211yescharacteristic
50C3×#419C2211yescharacteristic
51C2×C3×#103C2211yescharacteristic
52C2×C3×#103C2211yescharacteristic
53C2×C3×#101C2211yescharacteristic
54C2×C3×#101C2211yescharacteristic
55C3×#407C2211yescharacteristic
56C3×#407C2211yescharacteristic
57C2×C3×C22⋊C8C2211yescharacteristic
58C3×#405C2211yescharacteristic
59C3×#405C2211yescharacteristic
60C3×#343C2211yescharacteristic
61C3×#343C2211yescharacteristic
62C3×#343C2211yescharacteristic
63C3×#343C2211yescharacteristic
64C3×#343C2211yescharacteristic
65C3×#343C2211yescharacteristic
66C3×#341C2211yescharacteristic
67C3×#341C2211yescharacteristic
68#3095C2×C311yescharacteristic
69C2×#485C2×C311yescharacteristic
70#2924C2×C311yescharacteristic
71#2911C2×C311yescharacteristic
72#2090C2×C311yescharacteristic, complemented
73#2089C2×C311yescharacteristic, complemented
74#2064C2×C311yescharacteristic, complemented
75#2064C2×C311yescharacteristic, complemented
76#1952C2×C311yescharacteristic, complemented
77#1951C2×C311yescharacteristic, complemented
78#1930C2×C311yescharacteristic, complemented
79#1930C2×C311yescharacteristic, complemented
80#1901C2×C311yescharacteristic
81#1855C2×C311yescharacteristic
82#1822C2×C311yescharacteristic
83C22×C3×Q8C2311yescharacteristic
84C23×C3×C4C2311yescharacteristic
85C2×C3×M16—21no—
86C2×C3×M16—22no—
87C2×C3×M16—22no—
88C2×C3×M16C2311yescharacteristic
89C2×C3×M16C2311yescharacteristic
90C2×C3×M16C2311yescharacteristic
91C2×C3×M16C2311yescharacteristic
92C2×C3×M16C2311yescharacteristic
93C2×C3×M16C2311yescharacteristic
94C22×C3×C8—21no—
95C22×C3×C8C2311yescharacteristic
96C22×C3×C8C2311yescharacteristic
97C3×C22⋊Q8—21no—
98C3×C22⋊Q8—21no—
99C3×C22⋊Q8—21no—
100C3×C22⋊Q8—21no—
101C3×#117—21no—
102C3×#117—21no—
103C3×#117—21no—
104C3×#117—21no—
105C3×#117—21no—
106C3×#117—21no—
107C3×#117D411yescharacteristic
108C3×#117D411yescharacteristic
109C3×#117C2311yescharacteristic
110C3×#117C2311yescharacteristic
111C2×C3×C4⋊C4—21no—
112C2×C3×C4⋊C4D411yescharacteristic
113C2×C3×C4⋊C4D411yescharacteristic
114C2×C3×C4⋊C4C2311yescharacteristic
115C2×C3×C4⋊C4C2311yescharacteristic
116C2×C3×C4⋊C4C2311yescharacteristic
117C2×C3×C22⋊C4—21no—
118C2×C3×C22⋊C4—21no—
119C2×C3×C22⋊C4—21no—
120C2×C3×C42—21no—
121C2×C3×C42—21no—
122C3×#108—22no—
123C3×#108—22no—
124C3×#107—21no—
125C3×#107—21no—
126C3×#107—21no—
127C3×#107—21no—
128C3×#107—21no—
129C3×#107—21no—
130C3×#107—21no—
131C3×#107—21no—
132C3×#106—21no—
133C3×#106—21no—
134C3×#106—21no—
135C3×#106—21no—
136C3×#106—21no—
137C3×#106—21no—
138C3×#106—21no—
139C3×#106—21no—
140C3×C4⋊C8—21no—
141C3×C4⋊C8—21no—
142C3×C4⋊C8—21no—
143C3×C4⋊C8—21no—
144C3×#103—21no—
145C3×#103—21no—
146C3×#103—21no—
147C3×#103—21no—
148C3×#103—21no—
149C3×#103—21no—
150C3×#103—21no—
151C3×#103—21no—
152C3×#101—22no—
153C3×#101—22no—
154C3×C22⋊C8—21no—
155C3×C22⋊C8—21no—
156C3×C22⋊C8—21no—
157C3×C22⋊C8—21no—
158C3×C22⋊C8D411yescharacteristic
159C3×C22⋊C8D411yescharacteristic
160C3×C22⋊C8D411yescharacteristic
161C3×C22⋊C8D411yescharacteristic
162C3×#97—21no—
163C3×#97—21no—
164C3×C4×C8—21no—
165C3×C4×C8—21no—
166C22×M16C22×C311yescharacteristic
167C2×C22⋊Q8C22×C311yescharacteristic
168C2×#117C22×C311yescharacteristic
169C2×#117C22×C311yescharacteristic
170#485—21no—
171#485—21no—
172#485—21no—
173#485—21no—
174#485—21no—
175#485—21no—
176#484—21no—
177#483—21no—
178#433—21no—
179#433—21no—
180#432—21no—
181#431—21no—
182C2×#108C22×C311yescharacteristic
183C2×#108C22×C311yescharacteristic
184#428—21no—
185#428—21no—
186#428C22×C311yescharacteristic
187#428C22×C311yescharacteristic
188#428C22×C311yescharacteristic
189#428C22×C311yescharacteristic
190#428C22×C311yescharacteristic
191#428C22×C311yescharacteristic
192#427—21no—
193#427—21no—
194C2×#107C22×C311yescharacteristic
195C2×#106C22×C311yescharacteristic
196#424C22×C311yescharacteristic
197#424C22×C311yescharacteristic
198#419C22×C311yescharacteristic
199#419C22×C311yescharacteristic
200C2×#103C22×C311yescharacteristic
201C2×#103C22×C311yescharacteristic
202C2×#101C22×C311yescharacteristic
203C2×#101C22×C311yescharacteristic
204#407C22×C311yescharacteristic
205#407C22×C311yescharacteristic
206C2×C22⋊C8C22×C311yescharacteristic
207#405C22×C311yescharacteristic
208#405C22×C311yescharacteristic
209#343C22×C311yescharacteristic
210#343C22×C311yescharacteristic
211#343C22×C311yescharacteristic
212#343C22×C311yescharacteristic
213#343C22×C311yescharacteristic
214#343C22×C311yescharacteristic
215#341C22×C311yescharacteristic
216#341C22×C311yescharacteristic
217C24×C3#4111yescharacteristic
218C2×C3×Q8—21no—
219C2×C3×Q8—21no—
220C2×C3×Q8—21no—
221C2×C3×Q8—21no—
222C2×C3×Q8—21no—
223C2×C3×Q8—21no—
224C22×C3×C4—21no—
225C22×C3×C4—21no—
226C22×C3×C4—21no—
227C22×C3×C4—21no—
228C22×C3×C4—21no—
229C22×C3×C4—21no—
230C22×C3×C4—21no—
231C22×C3×C4—21no—
232C22×C3×C4—21no—
233C22×C3×C4—21no—
234C22×C3×C4C2×D411yescharacteristic
235C22×C3×C4C2×D411yescharacteristic
236C22×C3×C4C2×D411yescharacteristic
237C22×C3×C4C2×D411yescharacteristic
238C22×C3×C4#4111yescharacteristic
239C22×C3×C4C2411yescharacteristic
240C3×M16—24no—
241C3×M16—21no—
242C3×M16—21no—
243C3×M16—21no—
244C3×M16—21no—
245C3×M16—21no—
246C3×M16—21no—
247C3×M16—21no—
248C3×M16—21no—
249C3×M16—21no—
250C3×M16—21no—
251C3×M16#4112yes—
252C3×M16#4112yes—
253C2×C3×C8—24no—
254C2×C3×C8—21no—
255C2×C3×C8—21no—
256C2×C3×C8—21no—
257C2×C3×C8—21no—
258C2×C3×C8—21no—
259C2×C3×C8—21no—
260C2×C3×C8—21no—
261C2×C3×C8—21no—
262C2×C3×C8—21no—
263C2×C3×C8—21no—
264C2×C3×C8C2×D411yescharacteristic
265C2×C3×C8C2×D411yescharacteristic
266C2×C3×C8C2×D411yescharacteristic
267C2×C3×C8C2×D411yescharacteristic
268C3×C4⋊C4—21no—
269C3×C4⋊C4—21no—
270C3×C4⋊C4—21no—
271C3×C4⋊C4—21no—
272C3×C4⋊C4—21no—
273C3×C4⋊C4—21no—
274C3×C4⋊C4—21no—
275C3×C4⋊C4—21no—
276C3×C4⋊C4—41no—
277C3×C4⋊C4—41no—
278C3×C4⋊C4C2×D411yescharacteristic
279C3×C4⋊C4C2×D411yescharacteristic
280C3×C4⋊C4C2×D411yescharacteristic
281C3×C4⋊C4C2×D411yescharacteristic
282C3×C22⋊C4—21no—
283C3×C22⋊C4—21no—
284C3×C22⋊C4—21no—
285C3×C22⋊C4—21no—
286C3×C22⋊C4—41no—
287C3×C22⋊C4—41no—
288C3×C22⋊C4—41no—
289C3×C22⋊C4—41no—
290C3×C42—21no—
291C3×C42—21no—
292C3×C42—21no—
293C3×C42—21no—
294C3×C42—41no—
295C3×C42—41no—
296C22×Q8C23×C311yescharacteristic
297C23×C4C23×C311yescharacteristic
298C2×M16—21no—
299C2×M16—22no—
300C2×M16—22no—
301C2×M16C23×C311yescharacteristic
302C2×M16C23×C311yescharacteristic
303C2×M16C23×C311yescharacteristic
304C2×M16C23×C311yescharacteristic
305C2×M16C23×C311yescharacteristic
306C2×M16C23×C311yescharacteristic
307C22×C8—21no—
308C22×C8C23×C311yescharacteristic
309C22×C8C23×C311yescharacteristic
310C22⋊Q8—21no—
311C22⋊Q8—21no—
312C22⋊Q8—21no—
313C22⋊Q8—21no—
314#117—21no—
315#117—21no—
316#117—21no—
317#117—21no—
318#117—21no—
319#117—21no—
320#117C3×D411yescharacteristic
321#117C3×D411yescharacteristic
322#117C23×C311yescharacteristic
323#117C23×C311yescharacteristic
324C2×C4⋊C4—21no—
325C2×C4⋊C4C3×D411yescharacteristic
326C2×C4⋊C4C3×D411yescharacteristic
327C2×C4⋊C4C23×C311yescharacteristic
328C2×C4⋊C4C23×C311yescharacteristic
329C2×C4⋊C4C23×C311yescharacteristic
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C42—21no—
334C2×C42—21no—
335#108—22no—
336#108—22no—
337#107—21no—
338#107—21no—
339#107—21no—
340#107—21no—
341#107—21no—
342#107—21no—
343#107—21no—
344#107—21no—
345#106—21no—
346#106—21no—
347#106—21no—
348#106—21no—
349#106—21no—
350#106—21no—
351#106—21no—
352#106—21no—
353C4⋊C8—21no—
354C4⋊C8—21no—
355C4⋊C8—21no—
356C4⋊C8—21no—
357#103—21no—
358#103—21no—
359#103—21no—
360#103—21no—
361#103—21no—
362#103—21no—
363#103—21no—
364#103—21no—
365#101—22no—
366#101—22no—
367C22⋊C8—21no—
368C22⋊C8—21no—
369C22⋊C8—21no—
370C22⋊C8—21no—
371C22⋊C8C3×D411yescharacteristic
372C22⋊C8C3×D411yescharacteristic
373C22⋊C8C3×D411yescharacteristic
374C22⋊C8C3×D411yescharacteristic
375#97—21no—
376#97—21no—
377C4×C8—21no—
378C4×C8—21no—
379C23×C3—21no—
380C23×C3—21no—
381C23×C3—41no—
382C23×C3—41no—
383C23×C3#12111yescharacteristic
384C23×C3#12111yescharacteristic
385C23×C3C2×#4111yescharacteristic
386C3×Q8—41no—
387C3×Q8—41no—
388C3×Q8—41no—
389C3×Q8—41no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—21no—
397C2×C3×C4—21no—
398C2×C3×C4—21no—
399C2×C3×C4—21no—
400C2×C3×C4—21no—
401C2×C3×C4—21no—
402C2×C3×C4—21no—
403C2×C3×C4—21no—
404C2×C3×C4—42no—
405C2×C3×C4—41no—
406C2×C3×C4—41no—
407C2×C3×C4—41no—
408C2×C3×C4—41no—
409C2×C3×C4—41no—
410C2×C3×C4—41no—
411C2×C3×C4—81no—
412C2×C3×C4—81no—
413C2×C3×C4—81no—
414C2×C3×C4—81no—
415C2×C3×C4#12111yescharacteristic
416C2×C3×C4#12111yescharacteristic
417C2×C3×C4C22×D411yescharacteristic
418C2×C3×C4C22×D411yescharacteristic
419C2×C3×C4C2×#4111yescharacteristic
420C2×C3×C4C2×#4111yescharacteristic
421C2×C3×C42−1+411yescharacteristic
422C2×C3×C42−1+411yescharacteristic
423C3×C8—22no—
424C3×C8—22no—
425C3×C8—21no—
426C3×C8—21no—
427C3×C8—21no—
428C3×C8—21no—
429C3×C8—41no—
430C3×C8—41no—
431C3×C8—41no—
432C3×C8—41no—
433C24C3×#4111yescharacteristic
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4C2×C3×D411yescharacteristic
451C22×C4C2×C3×D411yescharacteristic
452C22×C4C2×C3×D411yescharacteristic
453C22×C4C2×C3×D411yescharacteristic
454C22×C4C3×#4111yescharacteristic
455C22×C4C24×C311yescharacteristic
456M16—24no—
457M16—21no—
458M16—21no—
459M16—21no—
460M16—21no—
461M16—21no—
462M16—21no—
463M16—21no—
464M16—21no—
465M16—21no—
466M16—21no—
467M16C3×#4112yes—
468M16C3×#4112yes—
469C2×C8—24no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8C2×C3×D411yescharacteristic
481C2×C8C2×C3×D411yescharacteristic
482C2×C8C2×C3×D411yescharacteristic
483C2×C8C2×C3×D411yescharacteristic
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—41no—
493C4⋊C4—41no—
494C4⋊C4C2×C3×D411yescharacteristic
495C4⋊C4C2×C3×D411yescharacteristic
496C4⋊C4C2×C3×D411yescharacteristic
497C4⋊C4C2×C3×D411yescharacteristic
498C22⋊C4—21no—
499C22⋊C4—21no—
500C22⋊C4—21no—
501C22⋊C4—21no—
502C22⋊C4—41no—
503C22⋊C4—41no—
504C22⋊C4—41no—
505C22⋊C4—41no—
506C42—21no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42—41no—
511C42—41no—
512C22×C3—21no—
513C22×C3—21no—
514C22×C3—21no—
515C22×C3—21no—
516C22×C3—21no—
517C22×C3—21no—
518C22×C3—21no—
519C22×C3—21no—
520C22×C3—81no—
521C22×C3—81no—
522C22×C3C2×#12111yescharacteristic
523C22×C3#55211yescharacteristic
524C22×C3#55211yescharacteristic
525C3×C4—21no—
526C3×C4—21no—
527C3×C4—41no—
528C3×C4—41no—
529C3×C4—41no—
530C3×C4—41no—
531C3×C4—81no—
532C3×C4—81no—
533C3×C4—81no—
534C3×C4—81no—
535C3×C4#52511yescharacteristic
536C3×C4#53611yescharacteristic
537C3×C4#54711yescharacteristic
538C3×C4#54711yescharacteristic
539C23—21no—
540C23—21no—
541C23—41no—
542C23—41no—
543C23C3×#12111yescharacteristic
544C23C3×#12111yescharacteristic
545C23C2×C3×#4111yescharacteristic
546Q8—41no—
547Q8—41no—
548Q8—41no—
549Q8—41no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—81no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4C3×#12111yescharacteristic
576C2×C4C3×#12111yescharacteristic
577C2×C4C22×C3×D411yescharacteristic
578C2×C4C22×C3×D411yescharacteristic
579C2×C4C2×C3×#4111yescharacteristic
580C2×C4C2×C3×#4111yescharacteristic
581C2×C4C3×2−1+411yescharacteristic
582C2×C4C3×2−1+411yescharacteristic
583C8—22no—
584C8—22no—
585C8—21no—
586C8—21no—
587C8—21no—
588C8—21no—
589C8—41no—
590C8—41no—
591C8—41no—
592C8—41no—
593C2×C3—21no—
594C2×C3—21no—
595C2×C3—41no—
596C2×C3—41no—
597C2×C3#241311yescharacteristic
598C2×C3#306711yescharacteristic
599C2×C3#306911yescharacteristic
600C22—21no—
601C22—21no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—21no—
606C22—21no—
607C22—21no—
608C22—81no—
609C22—81no—
610C22C2×C3×#12111yescharacteristic
611C22C3×#55211yescharacteristic
612C22C3×#55211yescharacteristic
613C4—21no—
614C4—21no—
615C4—41no—
616C4—41no—
617C4—41no—
618C4—41no—
619C4—81no—
620C4—81no—
621C4—81no—
622C4—81no—
623C4C3×#52511yescharacteristic
624C4C3×#53611yescharacteristic
625C4C3×#54711yescharacteristic
626C4C3×#54711yescharacteristic
627C3#2462411yescharacteristic, direct product factor, Sylow
628C2—21no—
629C2—21no—
630C2—41no—
631C2—41no—
632C2C3×#241311yescharacteristic
633C2C3×#306711yescharacteristic
634C2C3×#306911yescharacteristic
subgroupG/H
center, socleC22×C3C2×#121
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
5. C3×#2090C2semidirect
6. C3×#2089C2semidirect
7. C3×#2064C2semidirect
8. C3×#2064C2semidirect
9. C3×#1952C2semidirect
10. C3×#1951C2semidirect
11. C3×#1930C2semidirect
12. C3×#1930C2semidirect
16. #24624C3direct (N × H)
72. #2090C2×C3semidirect
73. #2089C2×C3semidirect
74. #2064C2×C3semidirect
75. #2064C2×C3semidirect
76. #1952C2×C3semidirect
77. #1951C2×C3semidirect
78. #1930C2×C3semidirect
79. #1930C2×C3semidirect
627. C3#24624direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#30952
C2×C3×#4852
C3×#29242
C3×#29112
C3×#20902
C3×#20892
C3×#20642
C3×#19522
C3×#19512
C3×#19302
C3×#19012
C3×#18552
C3×#18222
#246243