Finite Group Databasebeta

← C3×#24625 #10643232

Subgroups — C3×#24625 #10643232

Order 768.

696 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3094C211yesmaximal, characteristic
2C2×C3×#482C211yesmaximal, characteristic
3C3×#2924C211yesmaximal, characteristic, complemented
4C3×#2911C211yesmaximal, characteristic, complemented
5C3×#2090C211yesmaximal, characteristic, complemented
6C3×#2089C211yesmaximal, characteristic, complemented
7C3×#2063C211yesmaximal, characteristic, complemented
8C3×#2063C211yesmaximal, characteristic, complemented
9C3×#1952C211yesmaximal, characteristic, complemented
10C3×#1951C211yesmaximal, characteristic, complemented
11C3×#1931C211yesmaximal, characteristic, complemented
12C3×#1931C211yesmaximal, characteristic, complemented
13C3×#1900C211yesmaximal, characteristic
14C3×#1855C211yesmaximal, characteristic, complemented
15C3×#1822C211yesmaximal, characteristic, complemented
16#24625C311yesmaximal, characteristic, direct product factor, Sylow
17C22×C3×M16C2211yescharacteristic
18C2×C3×#121C2211yescharacteristic
19C2×C3×#117C2211yescharacteristic
20C2×C3×#117C2211yescharacteristic
21C3×#482—21no—
22C3×#482—21no—
23C3×#482—21no—
24C3×#482—21no—
25C3×#482—21no—
26C3×#482—21no—
27C3×#481—21no—
28C3×#480—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×#418C2211yescharacteristic
50C3×#418C2211yescharacteristic
51C2×C3×#102C2211yescharacteristic
52C2×C3×#102C2211yescharacteristic
53C2×C3×#100C2211yescharacteristic
54C2×C3×#100C2211yescharacteristic
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#3094C2×C311yescharacteristic
69C2×#482C2×C311yescharacteristic
70#2924C2×C311yescharacteristic, complemented
71#2911C2×C311yescharacteristic, complemented
72#2090C2×C311yescharacteristic, complemented
73#2089C2×C311yescharacteristic, complemented
74#2063C2×C311yescharacteristic, complemented
75#2063C2×C311yescharacteristic, complemented
76#1952C2×C311yescharacteristic, complemented
77#1951C2×C311yescharacteristic, complemented
78#1931C2×C311yescharacteristic, complemented
79#1931C2×C311yescharacteristic, complemented
80#1900C2×C311yescharacteristic
81#1855C2×C311yescharacteristic, complemented
82#1822C2×C311yescharacteristic, complemented
83C22×C3×D4—21no—
84C22×C3×D4C2311yescharacteristic
85C23×C3×C4C2311yescharacteristic
86C2×C3×M16—21no—
87C2×C3×M16—22no—
88C2×C3×M16—22no—
89C2×C3×M16C2311yescharacteristic
90C2×C3×M16C2311yescharacteristic
91C2×C3×M16C2311yescharacteristic
92C2×C3×M16C2311yescharacteristic
93C2×C3×M16C2311yescharacteristic
94C2×C3×M16C2311yescharacteristic
95C22×C3×C8—21no—
96C22×C3×C8C2311yescharacteristic
97C22×C3×C8C2311yescharacteristic
98C3×#121—21no—
99C3×#121—21no—
100C3×#121—21no—
101C3×#121—21no—
102C3×#117—21no—
103C3×#117—21no—
104C3×#117—21no—
105C3×#117—21no—
106C3×#117—21no—
107C3×#117—21no—
108C3×#117D411yescharacteristic
109C3×#117D411yescharacteristic
110C3×#117C2311yescharacteristic
111C3×#117C2311yescharacteristic
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×#102—21no—
145C3×#102—21no—
146C3×#102—21no—
147C3×#102—21no—
148C3×#102—21no—
149C3×#102—21no—
150C3×#102—21no—
151C3×#102—21no—
152C3×#100—22no—
153C3×#100—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×#121C22×C311yescharacteristic
168C2×#117C22×C311yescharacteristic
169C2×#117C22×C311yescharacteristic
170#482—21no—
171#482—21no—
172#482—21no—
173#482—21no—
174#482—21no—
175#482—21no—
176#481—21no—
177#480—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#418C22×C311yescharacteristic
199#418C22×C311yescharacteristic
200C2×#102C22×C311yescharacteristic
201C2×#102C22×C311yescharacteristic
202C2×#100C22×C311yescharacteristic
203C2×#100C22×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—21no—
218C24×C3#4111yescharacteristic
219C2×C3×D4—21no—
220C2×C3×D4—21no—
221C2×C3×D4—21no—
222C2×C3×D4—21no—
223C2×C3×D4—21no—
224C2×C3×D4—21no—
225C2×C3×D4—41no—
226C2×C3×D4—41no—
227C2×C3×D4—82no—
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×C4—21no—
235C22×C3×C4—21no—
236C22×C3×C4—21no—
237C22×C3×C4C2×D411yescharacteristic
238C22×C3×C4C2×D411yescharacteristic
239C22×C3×C4C2×D411yescharacteristic
240C22×C3×C4C2×D411yescharacteristic
241C22×C3×C4#4111yescharacteristic
242C22×C3×C4C2411yescharacteristic
243C3×M16—24no—
244C3×M16—21no—
245C3×M16—21no—
246C3×M16—21no—
247C3×M16—21no—
248C3×M16—21no—
249C3×M16—21no—
250C3×M16—21no—
251C3×M16—21no—
252C3×M16—21no—
253C3×M16—21no—
254C3×M16#4112yes—
255C3×M16#4112yes—
256C2×C3×C8—24no—
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×C8—21no—
265C2×C3×C8—21no—
266C2×C3×C8—21no—
267C2×C3×C8C2×D411yescharacteristic
268C2×C3×C8C2×D411yescharacteristic
269C2×C3×C8C2×D411yescharacteristic
270C2×C3×C8C2×D411yescharacteristic
271C3×C4⋊C4—21no—
272C3×C4⋊C4—21no—
273C3×C4⋊C4—21no—
274C3×C4⋊C4—21no—
275C3×C4⋊C4—21no—
276C3×C4⋊C4—21no—
277C3×C4⋊C4—21no—
278C3×C4⋊C4—21no—
279C3×C4⋊C4C2×D411yescharacteristic
280C3×C4⋊C4C2×D411yescharacteristic
281C3×C4⋊C4C2×D411yescharacteristic
282C3×C4⋊C4C2×D411yescharacteristic
283C3×C22⋊C4—21no—
284C3×C22⋊C4—21no—
285C3×C22⋊C4—21no—
286C3×C22⋊C4—21no—
287C3×C22⋊C4—41no—
288C3×C22⋊C4—41no—
289C3×C22⋊C4—41no—
290C3×C22⋊C4—41no—
291C3×C42—21no—
292C3×C42—21no—
293C3×C42—21no—
294C3×C42—21no—
295C3×C42—41no—
296C3×C42—41no—
297C22×D4—21no—
298C22×D4C23×C311yescharacteristic
299C23×C4C23×C311yescharacteristic
300C2×M16—21no—
301C2×M16—22no—
302C2×M16—22no—
303C2×M16C23×C311yescharacteristic
304C2×M16C23×C311yescharacteristic
305C2×M16C23×C311yescharacteristic
306C2×M16C23×C311yescharacteristic
307C2×M16C23×C311yescharacteristic
308C2×M16C23×C311yescharacteristic
309C22×C8—21no—
310C22×C8C23×C311yescharacteristic
311C22×C8C23×C311yescharacteristic
312#121—21no—
313#121—21no—
314#121—21no—
315#121—21no—
316#117—21no—
317#117—21no—
318#117—21no—
319#117—21no—
320#117—21no—
321#117—21no—
322#117C3×D411yescharacteristic
323#117C3×D411yescharacteristic
324#117C23×C311yescharacteristic
325#117C23×C311yescharacteristic
326C2×C4⋊C4C3×D411yescharacteristic
327C2×C4⋊C4C3×D411yescharacteristic
328C2×C4⋊C4C23×C311yescharacteristic
329C2×C4⋊C4C23×C311yescharacteristic
330C2×C4⋊C4C23×C311yescharacteristic
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C42—21no—
335C2×C42—21no—
336#108—22no—
337#108—22no—
338#107—21no—
339#107—21no—
340#107—21no—
341#107—21no—
342#107—21no—
343#107—21no—
344#107—21no—
345#107—21no—
346#106—21no—
347#106—21no—
348#106—21no—
349#106—21no—
350#106—21no—
351#106—21no—
352#106—21no—
353#106—21no—
354C4⋊C8—21no—
355C4⋊C8—21no—
356C4⋊C8—21no—
357C4⋊C8—21no—
358#102—21no—
359#102—21no—
360#102—21no—
361#102—21no—
362#102—21no—
363#102—21no—
364#102—21no—
365#102—21no—
366#100—22no—
367#100—22no—
368C22⋊C8—21no—
369C22⋊C8—21no—
370C22⋊C8—21no—
371C22⋊C8—21no—
372C22⋊C8C3×D411yescharacteristic
373C22⋊C8C3×D411yescharacteristic
374C22⋊C8C3×D411yescharacteristic
375C22⋊C8C3×D411yescharacteristic
376#97—21no—
377#97—21no—
378C4×C8—21no—
379C4×C8—21no—
380C23×C3—21no—
381C23×C3—21no—
382C23×C3—41no—
383C23×C3—41no—
384C23×C3—41no—
385C23×C3—41no—
386C23×C3—41no—
387C23×C3—81no—
388C23×C3—81no—
389C23×C3#12111yescharacteristic
390C23×C3#12111yescharacteristic
391C23×C3C2×#4111yescharacteristic
392C3×D4—41no—
393C3×D4—41no—
394C3×D4—41no—
395C3×D4—41no—
396C3×D4—82no—
397C3×D4—82no—
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—21no—
405C2×C3×C4—21no—
406C2×C3×C4—21no—
407C2×C3×C4—21no—
408C2×C3×C4—21no—
409C2×C3×C4—21no—
410C2×C3×C4—21no—
411C2×C3×C4—21no—
412C2×C3×C4—42no—
413C2×C3×C4—41no—
414C2×C3×C4—41no—
415C2×C3×C4—41no—
416C2×C3×C4—81no—
417C2×C3×C4—81no—
418C2×C3×C4—81no—
419C2×C3×C4—81no—
420C2×C3×C4#12111yescharacteristic
421C2×C3×C4#12111yescharacteristic
422C2×C3×C4C22×D411yescharacteristic
423C2×C3×C4C22×D411yescharacteristic
424C2×C3×C4C2×#4111yescharacteristic
425C2×C3×C4C2×#4111yescharacteristic
426C2×C3×C42−1+411yescharacteristic
427C2×C3×C42−1+411yescharacteristic
428C3×C8—22no—
429C3×C8—22no—
430C3×C8—21no—
431C3×C8—21no—
432C3×C8—21no—
433C3×C8—21no—
434C3×C8—41no—
435C3×C8—41no—
436C3×C8—41no—
437C3×C8—41no—
438C24—21no—
439C24C3×#4111yescharacteristic
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—41no—
447C2×D4—41no—
448C2×D4—82no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4C2×C3×D411yescharacteristic
459C22×C4C2×C3×D411yescharacteristic
460C22×C4C2×C3×D411yescharacteristic
461C22×C4C2×C3×D411yescharacteristic
462C22×C4C3×#4111yescharacteristic
463C22×C4C24×C311yescharacteristic
464M16—24no—
465M16—21no—
466M16—21no—
467M16—21no—
468M16—21no—
469M16—21no—
470M16—21no—
471M16—21no—
472M16—21no—
473M16—21no—
474M16—21no—
475M16C3×#4112yes—
476M16C3×#4112yes—
477C2×C8—24no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8C2×C3×D411yescharacteristic
489C2×C8C2×C3×D411yescharacteristic
490C2×C8C2×C3×D411yescharacteristic
491C2×C8C2×C3×D411yescharacteristic
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4C2×C3×D411yescharacteristic
501C4⋊C4C2×C3×D411yescharacteristic
502C4⋊C4C2×C3×D411yescharacteristic
503C4⋊C4C2×C3×D411yescharacteristic
504C22⋊C4—21no—
505C22⋊C4—21no—
506C22⋊C4—21no—
507C22⋊C4—21no—
508C22⋊C4—41no—
509C22⋊C4—41no—
510C22⋊C4—41no—
511C22⋊C4—41no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—41no—
517C42—41no—
518C22×C3—21no—
519C22×C3—21no—
520C22×C3—21no—
521C22×C3—21no—
522C22×C3—21no—
523C22×C3—21no—
524C22×C3—21no—
525C22×C3—21no—
526C22×C3—81no—
527C22×C3—81no—
528C22×C3—81no—
529C22×C3—81no—
530C22×C3—81no—
531C22×C3—161no—
532C22×C3—161no—
533C22×C3C2×#12111yescharacteristic
534C22×C3#55211yescharacteristic
535C22×C3#55211yescharacteristic
536C3×C4—21no—
537C3×C4—21no—
538C3×C4—41no—
539C3×C4—41no—
540C3×C4—41no—
541C3×C4—41no—
542C3×C4—81no—
543C3×C4—81no—
544C3×C4—81no—
545C3×C4#52511yescharacteristic
546C3×C4#53611yescharacteristic
547C3×C4#54711yescharacteristic
548C3×C4#54711yescharacteristic
549C23—21no—
550C23—21no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—81no—
557C23—81no—
558C23C3×#12111yescharacteristic
559C23C3×#12111yescharacteristic
560C23C2×C3×#4111yescharacteristic
561D4—41no—
562D4—41no—
563D4—41no—
564D4—41no—
565D4—82no—
566D4—82no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—81no—
586C2×C4—81no—
587C2×C4—81no—
588C2×C4—81no—
589C2×C4C3×#12111yescharacteristic
590C2×C4C3×#12111yescharacteristic
591C2×C4C22×C3×D411yescharacteristic
592C2×C4C22×C3×D411yescharacteristic
593C2×C4C2×C3×#4111yescharacteristic
594C2×C4C2×C3×#4111yescharacteristic
595C2×C4C3×2−1+411yescharacteristic
596C2×C4C3×2−1+411yescharacteristic
597C8—22no—
598C8—22no—
599C8—21no—
600C8—21no—
601C8—21no—
602C8—21no—
603C8—41no—
604C8—41no—
605C8—41no—
606C8—41no—
607C2×C3—21no—
608C2×C3—21no—
609C2×C3—41no—
610C2×C3—41no—
611C2×C3—161no—
612C2×C3#241311yescharacteristic
613C2×C3#306711yescharacteristic
614C2×C3#306811yescharacteristic
615C22—21no—
616C22—21no—
617C22—21no—
618C22—21no—
619C22—21no—
620C22—21no—
621C22—21no—
622C22—21no—
623C22—81no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22—161no—
629C22—161no—
630C22C2×C3×#12111yescharacteristic
631C22C3×#55211yescharacteristic
632C22C3×#55211yescharacteristic
633C4—21no—
634C4—21no—
635C4—41no—
636C4—41no—
637C4—41no—
638C4—41no—
639C4—81no—
640C4—81no—
641C4—81no—
642C4C3×#52511yescharacteristic
643C4C3×#53611yescharacteristic
644C4C3×#54711yescharacteristic
645C4C3×#54711yescharacteristic
646C3#2462511yescharacteristic, direct product factor, Sylow
647C2—21no—
648C2—21no—
649C2—41no—
650C2—41no—
651C2—161no—
652C2C3×#241311yescharacteristic
653C2C3×#306711yescharacteristic
654C2C3×#306811yescharacteristic
subgroupG/H
center, socleC22×C3C2×#121
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#121C22
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
3. C3×#2924C2semidirect
4. C3×#2911C2semidirect
5. C3×#2090C2semidirect
6. C3×#2089C2semidirect
7. C3×#2063C2semidirect
8. C3×#2063C2semidirect
9. C3×#1952C2semidirect
10. C3×#1951C2semidirect
11. C3×#1931C2semidirect
12. C3×#1931C2semidirect
14. C3×#1855C2semidirect
15. C3×#1822C2semidirect
16. #24625C3direct (N × H)
70. #2924C2×C3semidirect
71. #2911C2×C3semidirect
72. #2090C2×C3semidirect
73. #2089C2×C3semidirect
74. #2063C2×C3semidirect
75. #2063C2×C3semidirect
76. #1952C2×C3semidirect
77. #1951C2×C3semidirect
78. #1931C2×C3semidirect
79. #1931C2×C3semidirect
81. #1855C2×C3semidirect
82. #1822C2×C3semidirect
646. C3#24625direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#30942
C2×C3×#4822
C3×#29242
C3×#29112
C3×#20902
C3×#20892
C3×#20632
C3×#19522
C3×#19512
C3×#19312
C3×#19002
C3×#18552
C3×#18222
#246253