Finite Group Databasebeta

← C3×#25493 #10644100

Subgroups — C3×#25493 #10644100

Order 768.

648 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3166C211yesmaximal, characteristic, complemented
2C3×#3077C211yesmaximal, characteristic
3C2×C3×#478C211yesmaximal, characteristic
4C3×#2962C211yesmaximal, characteristic, complemented
5C3×#2090C211yesmaximal, characteristic, complemented
6C3×#2089C211yesmaximal, characteristic, complemented
7C3×#2063C211yesmaximal, characteristic
8C3×#2063C211yesmaximal, characteristic
9C3×#1998C211yesmaximal, characteristic, complemented
10C3×#1952C211yesmaximal, characteristic, complemented
11C3×#1951C211yesmaximal, characteristic, complemented
12C3×#1950C211yesmaximal, characteristic, complemented
13C3×#1931C211yesmaximal, characteristic
14C3×#1931C211yesmaximal, characteristic
15C3×C4×#100C211yesmaximal, characteristic
16#25493C311yesmaximal, characteristic, direct product factor, Sylow
17C3×#533C2211yescharacteristic
18C2×C3×#125C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20C3×#501—21no—
21C3×#501—21no—
22C3×#499—21no—
23C3×#499—21no—
24C3×#478—21no—
25C3×#478—21no—
26C3×#478—21no—
27C3×#478—21no—
28C3×#478—21no—
29C3×#478—21no—
30C3×#476—21no—
31C3×#474—21no—
32C3×#446—21no—
33C3×#446—21no—
34C3×#446—21no—
35C3×C8×Q8—21no—
36C2×C3×#108C2211yescharacteristic
37C2×C3×#108C2211yescharacteristic
38C3×#428C2211yescharacteristic
39C3×#428C2211yescharacteristic
40C3×#428C2211yescharacteristic
41C3×#428C2211yescharacteristic
42C3×#428C2211yescharacteristic
43C3×#428C2211yescharacteristic
44C2×C3×#107C2211yescharacteristic
45C2×C3×#106C2211yescharacteristic
46C3×#424C2211yescharacteristic
47C3×#424C2211yescharacteristic
48C3×#423C2211yescharacteristic
49C2×C3×C4⋊C8C2211yescharacteristic
50C3×#418C2211yescharacteristic
51C3×#418C2211yescharacteristic
52C2×C3×#102C2211yescharacteristic
53C2×C3×#102C2211yescharacteristic
54C2×C3×#100C2211yescharacteristic
55C2×C3×#100C2211yescharacteristic
56C3×#405C2211yescharacteristic
57C3×#405C2211yescharacteristic
58C3×C4×M16C2211yescharacteristic
59C3×C4×M16C2211yescharacteristic
60C3×#343C2211yescharacteristic
61C3×#343C2211yescharacteristic
62C3×#343C2211yescharacteristic
63C3×#343C2211yescharacteristic
64C3×#343C2211yescharacteristic
65C3×#343C2211yescharacteristic
66C3×#341C2211yescharacteristic
67C3×#341C2211yescharacteristic
68#3166C2×C311yescharacteristic, complemented
69#3077C2×C311yescharacteristic
70C2×#478C2×C311yescharacteristic
71#2962C2×C311yescharacteristic, complemented
72#2090C2×C311yescharacteristic, complemented
73#2089C2×C311yescharacteristic, complemented
74#2063C2×C311yescharacteristic
75#2063C2×C311yescharacteristic
76#1998C2×C311yescharacteristic, complemented
77#1952C2×C311yescharacteristic, complemented
78#1951C2×C311yescharacteristic, complemented
79#1950C2×C311yescharacteristic, complemented
80#1931C2×C311yescharacteristic
81#1931C2×C311yescharacteristic
82C4×#100C2×C311yescharacteristic
83C22×C3×D4C2311yescharacteristic
84C23×C3×C4—21no—
85C2×C3×M16C2311yescharacteristic
86C2×C3×M16C2311yescharacteristic
87C2×C3×M16C2311yescharacteristic
88C2×C3×M16C2311yescharacteristic
89C2×C3×M16C2311yescharacteristic
90C2×C3×M16C2311yescharacteristic
91C22×C3×C8C2311yescharacteristic
92C22×C3×C8C2311yescharacteristic
93C3×C4⋊Q8D411yescharacteristic
94C3×C4⋊Q8D411yescharacteristic
95C3×#125—21no—
96C3×#125—21no—
97C3×#125—21no—
98C3×#125—21no—
99C3×#125D411yescharacteristic
100C3×#125D411yescharacteristic
101C3×C22⋊Q8—21no—
102C3×C22⋊Q8—21no—
103C3×C4×Q8—21no—
104C3×C4×Q8—21no—
105C3×C4×D4—21no—
106C3×C4×D4—21no—
107C3×C4×D4—21no—
108C3×C4×D4—21no—
109C3×#117C2311yescharacteristic
110C3×#117C2311yescharacteristic
111C2×C3×C4⋊C4—21no—
112C2×C3×C4⋊C4—21no—
113C2×C3×C4⋊C4C2311yescharacteristic
114C2×C3×C4⋊C4C2311yescharacteristic
115C2×C3×C4⋊C4C2311yescharacteristic
116C2×C3×C22⋊C4—21no—
117C2×C3×C42C2311yescharacteristic
118C3×#108—22no—
119C3×#108—22no—
120C3×#107—21no—
121C3×#107—21no—
122C3×#107—21no—
123C3×#107—21no—
124C3×#107—21no—
125C3×#107—21no—
126C3×#107—21no—
127C3×#107—21no—
128C3×#106—21no—
129C3×#106—21no—
130C3×#106—21no—
131C3×#106—21no—
132C3×#106—21no—
133C3×#106—21no—
134C3×#106—21no—
135C3×#106—21no—
136C3×C4⋊C8—21no—
137C3×C4⋊C8—21no—
138C3×C4⋊C8—21no—
139C3×C4⋊C8—21no—
140C3×C4⋊C8—21no—
141C3×C4⋊C8—21no—
142C3×C4⋊C8D411yescharacteristic
143C3×C4⋊C8D411yescharacteristic
144C3×C4⋊C8D411yescharacteristic
145C3×C4⋊C8D411yescharacteristic
146C3×#102—21no—
147C3×#102—21no—
148C3×#102—21no—
149C3×#102—21no—
150C3×#102—21no—
151C3×#102—21no—
152C3×#102—21no—
153C3×#102—21no—
154C3×#100—22no—
155C3×#100Q814yescomplemented
156C3×#97—21no—
157C3×#97—21no—
158C3×#97—21no—
159C3×#97—21no—
160C3×C4×C8—21no—
161C3×C4×C8—21no—
162C3×C4×C8—21no—
163C3×C4×C8—21no—
164#533C22×C311yescharacteristic
165C2×#125C22×C311yescharacteristic
166C2×C4×D4C22×C311yescharacteristic
167#501—21no—
168#501—21no—
169#499—21no—
170#499—21no—
171#478—21no—
172#478—21no—
173#478—21no—
174#478—21no—
175#478—21no—
176#478—21no—
177#476—21no—
178#474—21no—
179#446—21no—
180#446—21no—
181#446—21no—
182C8×Q8—21no—
183C2×#108C22×C311yescharacteristic
184C2×#108C22×C311yescharacteristic
185#428C22×C311yescharacteristic
186#428C22×C311yescharacteristic
187#428C22×C311yescharacteristic
188#428C22×C311yescharacteristic
189#428C22×C311yescharacteristic
190#428C22×C311yescharacteristic
191C2×#107C22×C311yescharacteristic
192C2×#106C22×C311yescharacteristic
193#424C22×C311yescharacteristic
194#424C22×C311yescharacteristic
195#423C22×C311yescharacteristic
196C2×C4⋊C8C22×C311yescharacteristic
197#418C22×C311yescharacteristic
198#418C22×C311yescharacteristic
199C2×#102C22×C311yescharacteristic
200C2×#102C22×C311yescharacteristic
201C2×#100C22×C311yescharacteristic
202C2×#100C22×C311yescharacteristic
203#405C22×C311yescharacteristic
204#405C22×C311yescharacteristic
205C4×M16C22×C311yescharacteristic
206C4×M16C22×C311yescharacteristic
207#343C22×C311yescharacteristic
208#343C22×C311yescharacteristic
209#343C22×C311yescharacteristic
210#343C22×C311yescharacteristic
211#343C22×C311yescharacteristic
212#343C22×C311yescharacteristic
213#341C22×C311yescharacteristic
214#341C22×C311yescharacteristic
215C24×C3—21no—
216C2×C3×Q8—21no—
217C2×C3×Q8—21no—
218C2×C3×D4—21no—
219C2×C3×D4—21no—
220C2×C3×D4—21no—
221C2×C3×D4—21no—
222C2×C3×D4—21no—
223C2×C3×D4C2×Q812yes—
224C22×C3×C4—21no—
225C22×C3×C4—21no—
226C22×C3×C4—21no—
227C22×C3×C4—41no—
228C22×C3×C4—41no—
229C22×C3×C4—82no—
230C22×C3×C4#4111yescharacteristic
231C22×C3×C4#4111yescharacteristic
232C22×C3×C4C2411yescharacteristic
233C3×M16—21no—
234C3×M16—21no—
235C3×M16—21no—
236C3×M16—21no—
237C3×M16—21no—
238C3×M16—21no—
239C3×M16—21no—
240C3×M16—21no—
241C3×M16—21no—
242C3×M16C2×Q812yes—
243C3×M16C2×Q812yes—
244C3×M16#4112yes—
245C2×C3×C8—21no—
246C2×C3×C8—21no—
247C2×C3×C8—21no—
248C2×C3×C8—21no—
249C2×C3×C8—21no—
250C2×C3×C8—21no—
251C2×C3×C8—21no—
252C2×C3×C8—21no—
253C2×C3×C8—21no—
254C2×C3×C8—21no—
255C2×C3×C8C2×D411yescharacteristic
256C2×C3×C8C2×D411yescharacteristic
257C2×C3×C8C2×D411yescharacteristic
258C2×C3×C8C2×D411yescharacteristic
259C3×C4⋊C4—21no—
260C3×C4⋊C4—21no—
261C3×C4⋊C4—21no—
262C3×C4⋊C4—21no—
263C3×C4⋊C4—21no—
264C3×C4⋊C4—21no—
265C3×C4⋊C4—21no—
266C3×C4⋊C4—21no—
267C3×C4⋊C4—21no—
268C3×C4⋊C4—21no—
269C3×C4⋊C4—21no—
270C3×C4⋊C4—21no—
271C3×C4⋊C4—41no—
272C3×C4⋊C4—41no—
273C3×C4⋊C4—41no—
274C3×C4⋊C4—41no—
275C3×C4⋊C4C2×D411yescharacteristic
276C3×C4⋊C4C2×D411yescharacteristic
277C3×C4⋊C4C2×D411yescharacteristic
278C3×C4⋊C4C2×D411yescharacteristic
279C3×C22⋊C4—21no—
280C3×C22⋊C4—21no—
281C3×C22⋊C4—41no—
282C3×C22⋊C4—41no—
283C3×C42—21no—
284C3×C42—21no—
285C3×C42C2×D411yescharacteristic
286C3×C42C2×D411yescharacteristic
287C3×C42C2×D411yescharacteristic
288C3×C42C2×D411yescharacteristic
289C22×D4C23×C311yescharacteristic
290C23×C4—21no—
291C2×M16C23×C311yescharacteristic
292C2×M16C23×C311yescharacteristic
293C2×M16C23×C311yescharacteristic
294C2×M16C23×C311yescharacteristic
295C2×M16C23×C311yescharacteristic
296C2×M16C23×C311yescharacteristic
297C22×C8C23×C311yescharacteristic
298C22×C8C23×C311yescharacteristic
299C4⋊Q8C3×D411yescharacteristic
300C4⋊Q8C3×D411yescharacteristic
301#125—21no—
302#125—21no—
303#125—21no—
304#125—21no—
305#125C3×D411yescharacteristic
306#125C3×D411yescharacteristic
307C22⋊Q8—21no—
308C22⋊Q8—21no—
309C4×Q8—21no—
310C4×Q8—21no—
311C4×D4—21no—
312C4×D4—21no—
313C4×D4—21no—
314C4×D4—21no—
315#117C23×C311yescharacteristic
316#117C23×C311yescharacteristic
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4C23×C311yescharacteristic
320C2×C4⋊C4C23×C311yescharacteristic
321C2×C4⋊C4C23×C311yescharacteristic
322C2×C22⋊C4—21no—
323C2×C42C23×C311yescharacteristic
324#108—22no—
325#108—22no—
326#107—21no—
327#107—21no—
328#107—21no—
329#107—21no—
330#107—21no—
331#107—21no—
332#107—21no—
333#107—21no—
334#106—21no—
335#106—21no—
336#106—21no—
337#106—21no—
338#106—21no—
339#106—21no—
340#106—21no—
341#106—21no—
342C4⋊C8—21no—
343C4⋊C8—21no—
344C4⋊C8—21no—
345C4⋊C8—21no—
346C4⋊C8—21no—
347C4⋊C8—21no—
348C4⋊C8C3×D411yescharacteristic
349C4⋊C8C3×D411yescharacteristic
350C4⋊C8C3×D411yescharacteristic
351C4⋊C8C3×D411yescharacteristic
352#102—21no—
353#102—21no—
354#102—21no—
355#102—21no—
356#102—21no—
357#102—21no—
358#102—21no—
359#102—21no—
360#100—22no—
361#100C3×Q814yescomplemented
362#97—21no—
363#97—21no—
364#97—21no—
365#97—21no—
366C4×C8—21no—
367C4×C8—21no—
368C4×C8—21no—
369C4×C8—21no—
370C23×C3—22no—
371C23×C3—41no—
372C23×C3—41no—
373C23×C3—81no—
374C23×C3—81no—
375C23×C3C2×#4111yescharacteristic
376C3×Q8—42no—
377C3×Q8—81no—
378C3×D4—22no—
379C3×D4—22no—
380C3×D4—41no—
381C3×D4—41no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—82no—
392C2×C3×C4—82no—
393C2×C3×C4—81no—
394C2×C3×C4—81no—
395C2×C3×C4—81no—
396C2×C3×C4#12111yescharacteristic
397C2×C3×C4#12111yescharacteristic
398C2×C3×C4#12111yescharacteristic
399C2×C3×C4#12111yescharacteristic
400C2×C3×C4C22×D411yescharacteristic
401C2×C3×C4C22×D411yescharacteristic
402C2×C3×C4C22×Q811yescharacteristic
403C2×C3×C4C2×#4111yescharacteristic
404C2×C3×C42−1+411yescharacteristic
405C2×C3×C42−1+411yescharacteristic
406C3×C8—21no—
407C3×C8—21no—
408C3×C8—21no—
409C3×C8—21no—
410C3×C8—22no—
411C3×C8—22no—
412C3×C8—22no—
413C3×C8—41no—
414C3×C8—41no—
415C3×C8—41no—
416C24—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4C2×C3×Q812yes—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—82no—
431C22×C4C3×#4111yescharacteristic
432C22×C4C3×#4111yescharacteristic
433C22×C4C24×C311yescharacteristic
434M16—21no—
435M16—21no—
436M16—21no—
437M16—21no—
438M16—21no—
439M16—21no—
440M16—21no—
441M16—21no—
442M16—21no—
443M16C2×C3×Q812yes—
444M16C2×C3×Q812yes—
445M16C3×#4112yes—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8C2×C3×D411yescharacteristic
457C2×C8C2×C3×D411yescharacteristic
458C2×C8C2×C3×D411yescharacteristic
459C2×C8C2×C3×D411yescharacteristic
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—41no—
473C4⋊C4—41no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C4⋊C4C2×C3×D411yescharacteristic
477C4⋊C4C2×C3×D411yescharacteristic
478C4⋊C4C2×C3×D411yescharacteristic
479C4⋊C4C2×C3×D411yescharacteristic
480C22⋊C4—21no—
481C22⋊C4—21no—
482C22⋊C4—41no—
483C22⋊C4—41no—
484C42—21no—
485C42—21no—
486C42C2×C3×D411yescharacteristic
487C42C2×C3×D411yescharacteristic
488C42C2×C3×D411yescharacteristic
489C42C2×C3×D411yescharacteristic
490C22×C3—21no—
491C22×C3—21no—
492C22×C3—42no—
493C22×C3—82no—
494C22×C3—82no—
495C22×C3—81no—
496C22×C3—81no—
497C22×C3C2×#12111yescharacteristic
498C22×C3#53311yescharacteristic
499C22×C3#54111yescharacteristic
500C3×C4—21no—
501C3×C4—21no—
502C3×C4—21no—
503C3×C4—21no—
504C3×C4—41no—
505C3×C4—41no—
506C3×C4—41no—
507C3×C4—41no—
508C3×C4—81no—
509C3×C4—81no—
510C3×C4—81no—
511C3×C4#54711yescharacteristic
512C3×C4#54811yescharacteristic
513C3×C4D4×Q811yescharacteristic
514C3×C4D4×Q811yescharacteristic
515C23—22no—
516C23—41no—
517C23—41no—
518C23—81no—
519C23—81no—
520C23C2×C3×#4111yescharacteristic
521Q8—42no—
522Q8—81no—
523D4—22no—
524D4—22no—
525D4—41no—
526D4—41no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4C3×#12111yescharacteristic
542C2×C4C3×#12111yescharacteristic
543C2×C4C3×#12111yescharacteristic
544C2×C4C3×#12111yescharacteristic
545C2×C4C22×C3×D411yescharacteristic
546C2×C4C22×C3×D411yescharacteristic
547C2×C4C22×C3×Q811yescharacteristic
548C2×C4C2×C3×#4111yescharacteristic
549C2×C4C3×2−1+411yescharacteristic
550C2×C4C3×2−1+411yescharacteristic
551C8—21no—
552C8—21no—
553C8—21no—
554C8—21no—
555C8—22no—
556C8—22no—
557C8—22no—
558C8—41no—
559C8—41no—
560C8—41no—
561C2×C3—21no—
562C2×C3—21no—
563C2×C3—82no—
564C2×C3#242911yescharacteristic
565C2×C3#306711yescharacteristic
566C2×C3#306811yescharacteristic
567C22—21no—
568C22—21no—
569C22—42no—
570C22—82no—
571C22—82no—
572C22—81no—
573C22—81no—
574C22C2×C3×#12111yescharacteristic
575C22C3×#53311yescharacteristic
576C22C3×#54111yescharacteristic
577C4—21no—
578C4—21no—
579C4—21no—
580C4—21no—
581C4—41no—
582C4—41no—
583C4—41no—
584C4—41no—
585C4—81no—
586C4—81no—
587C4—81no—
588C4C3×#54711yescharacteristic
589C4C3×#54811yescharacteristic
590C4C3×D4×Q811yescharacteristic
591C4C3×D4×Q811yescharacteristic
592C3#2549311yescharacteristic, direct product factor, Sylow
593C2—21no—
594C2—21no—
595C2—82no—
596C2C3×#242911yescharacteristic
597C2C3×#306711yescharacteristic
598C2C3×#306811yescharacteristic
subgroupG/H
center, socleC22×C3C2×#121
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3166C2semidirect
4. C3×#2962C2semidirect
5. C3×#2090C2semidirect
6. C3×#2089C2semidirect
9. C3×#1998C2semidirect
10. C3×#1952C2semidirect
11. C3×#1951C2semidirect
12. C3×#1950C2semidirect
16. #25493C3direct (N × H)
68. #3166C2×C3semidirect
71. #2962C2×C3semidirect
72. #2090C2×C3semidirect
73. #2089C2×C3semidirect
76. #1998C2×C3semidirect
77. #1952C2×C3semidirect
78. #1951C2×C3semidirect
79. #1950C2×C3semidirect
155. C3×#100Q8semidirect
361. #100C3×Q8semidirect
592. C3#25493direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#31662
C3×#30772
C2×C3×#4782
C3×#29622
C3×#20902
C3×#20892
C3×#20632
C3×#19982
C3×#19522
C3×#19512
C3×#19502
C3×#19312
C3×C4×#1002
#254933