Finite Group Databasebeta

← C3×#33087 #10651694

Subgroups — C3×#33087 #10651694

Order 768.

728 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2644C211yesmaximal, characteristic
2C3×#2276C211yesmaximal, characteristic, complemented
3C3×#1926C211yesmaximal, characteristic
4C3×#1925C211yesmaximal, characteristic
5C3×#1925C211yesmaximal, characteristic
6C3×#1921C211yesmaximal, characteristic, complemented
7C3×#1919C211yesmaximal, characteristic, complemented
8C3×#1918C211yesmaximal, characteristic, complemented
9C3×#1880C211yesmaximal, characteristic
10C3×#1880C211yesmaximal, characteristic
11C3×#1880C211yesmaximal, characteristic
12C3×#1865C211yesmaximal, characteristic, complemented
13C3×#1863C211yesmaximal, characteristic, complemented
14C3×#1863C211yesmaximal, characteristic, complemented
15C3×#1840C211yesmaximal, characteristic, complemented
16#33087C311yesmaximal, characteristic, direct product factor, Sylow
17C22×C3×SD16C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×C4×Q8C2211yescharacteristic
20C2×C3×C4×Q8C2211yescharacteristic
21C2×C3×#107C2211yescharacteristic
22C2×C3×#107C2211yescharacteristic
23C2×C3×#106C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×#102C2211yescharacteristic
28C2×C3×#102C2211yescharacteristic
29C2×C3×#102C2211yescharacteristic
30C3×#397—21no—
31C3×#397—21no—
32C3×#397—21no—
33C3×#396—21no—
34C3×#390C2211yescharacteristic
35C3×#390C2211yescharacteristic
36C3×#390C2211yescharacteristic
37C3×#389C2211yescharacteristic
38C3×#389C2211yescharacteristic
39C3×#389C2211yescharacteristic
40C3×C4×C4⋊C4—21no—
41C3×C4×C4⋊C4—21no—
42C3×C4×C4⋊C4—21no—
43C3×C4×C4⋊C4C2211yescharacteristic
44C3×C4×C4⋊C4C2211yescharacteristic
45C3×C4×C4⋊C4C2211yescharacteristic
46C3×#376—21no—
47C3×#376C2211yescharacteristic
48C3×#340C2211yescharacteristic
49C3×#340C2211yescharacteristic
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54C3×#336C2211yescharacteristic
55C3×#336C2211yescharacteristic
56C3×#336C2211yescharacteristic
57C3×#336C2211yescharacteristic
58C3×#336C2211yescharacteristic
59C3×#336C2211yescharacteristic
60#2644C2×C311yescharacteristic
61#2276C2×C311yescharacteristic, complemented
62#1926C2×C311yescharacteristic
63#1925C2×C311yescharacteristic
64#1925C2×C311yescharacteristic
65#1921C2×C311yescharacteristic, complemented
66#1919C2×C311yescharacteristic, complemented
67#1918C2×C311yescharacteristic, complemented
68#1880C2×C311yescharacteristic
69#1880C2×C311yescharacteristic
70#1880C2×C311yescharacteristic
71#1865C2×C311yescharacteristic, complemented
72#1863C2×C311yescharacteristic, complemented
73#1863C2×C311yescharacteristic, complemented
74#1840C2×C311yescharacteristic, complemented
75C22×C3×Q8C2311yescharacteristic
76C22×C3×D4C2311yescharacteristic
77C2×C3×SD16—22no—
78C2×C3×SD16—22no—
79C2×C3×SD16—22no—
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C3×C4×Q8—22no—
85C3×C4×Q8—22no—
86C3×C4×Q8—22no—
87C3×C4×Q8—22no—
88C3×C4×Q8—22no—
89C3×C4×Q8—22no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C4⋊C4C2311yescharacteristic
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4—21no—
105C2×C3×C22⋊C4—21no—
106C2×C3×C22⋊C4—21no—
107C2×C3×C22⋊C4—21no—
108C2×C3×C42—21no—
109C2×C3×C42—21no—
110C2×C3×C42—21no—
111C2×C3×C42—21no—
112C2×C3×C42—21no—
113C2×C3×C42—21no—
114C2×C3×C42C2311yescharacteristic
115C2×C3×C42C2311yescharacteristic
116C2×C3×C42C2311yescharacteristic
117C3×#107—22no—
118C3×#107—22no—
119C3×#106—22no—
120C3×#103—22no—
121C3×#103—22no—
122C3×#103—22no—
123C3×#102—22no—
124C3×#102—22no—
125C3×#102—22no—
126C3×#95—21no—
127C3×#95—21no—
128C3×#95—21no—
129C3×#95—21no—
130C3×#95—21no—
131C3×#95—21no—
132C3×#95D411yescharacteristic
133C3×#95D411yescharacteristic
134C3×#95D411yescharacteristic
135C3×#95D411yescharacteristic
136C22×SD16C22×C311yescharacteristic
137C2×C4×Q8C22×C311yescharacteristic
138C2×C4×Q8C22×C311yescharacteristic
139C2×C4×Q8C22×C311yescharacteristic
140C2×#107C22×C311yescharacteristic
141C2×#107C22×C311yescharacteristic
142C2×#106C22×C311yescharacteristic
143C2×#103C22×C311yescharacteristic
144C2×#103C22×C311yescharacteristic
145C2×#103C22×C311yescharacteristic
146C2×#102C22×C311yescharacteristic
147C2×#102C22×C311yescharacteristic
148C2×#102C22×C311yescharacteristic
149#397—21no—
150#397—21no—
151#397—21no—
152#396—21no—
153#390C22×C311yescharacteristic
154#390C22×C311yescharacteristic
155#390C22×C311yescharacteristic
156#389C22×C311yescharacteristic
157#389C22×C311yescharacteristic
158#389C22×C311yescharacteristic
159C4×C4⋊C4—21no—
160C4×C4⋊C4—21no—
161C4×C4⋊C4—21no—
162C4×C4⋊C4C22×C311yescharacteristic
163C4×C4⋊C4C22×C311yescharacteristic
164C4×C4⋊C4C22×C311yescharacteristic
165#376—21no—
166#376C22×C311yescharacteristic
167#340C22×C311yescharacteristic
168#340C22×C311yescharacteristic
169#340C22×C311yescharacteristic
170#340C22×C311yescharacteristic
171#340C22×C311yescharacteristic
172#340C22×C311yescharacteristic
173#336C22×C311yescharacteristic
174#336C22×C311yescharacteristic
175#336C22×C311yescharacteristic
176#336C22×C311yescharacteristic
177#336C22×C311yescharacteristic
178#336C22×C311yescharacteristic
179C24×C3—21no—
180C2×C3×Q8—21no—
181C2×C3×Q8—21no—
182C2×C3×Q8—21no—
183C2×C3×Q8#4112yes—
184C2×C3×Q8#4112yes—
185C2×C3×Q8#4112yes—
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4—21no—
189C2×C3×D4—21no—
190C2×C3×D4—21no—
191C2×C3×D4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2411yescharacteristic
206C3×SD16—44no—
207C2×C3×C8—21no—
208C2×C3×C8—21no—
209C2×C3×C8—21no—
210C2×C3×C8—21no—
211C2×C3×C8—21no—
212C2×C3×C8—21no—
213C2×C3×C8—21no—
214C2×C3×C8—21no—
215C2×C3×C8—21no—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C3×C4⋊C4—24no—
220C3×C4⋊C4—24no—
221C3×C4⋊C4—24no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—81no—
232C3×C4⋊C4—81no—
233C3×C4⋊C4—81no—
234C3×C4⋊C4#4112yes—
235C3×C4⋊C4#4112yes—
236C3×C4⋊C4#4112yes—
237C3×C22⋊C4—81no—
238C3×C22⋊C4—81no—
239C3×C22⋊C4—81no—
240C3×C22⋊C4—81no—
241C3×C22⋊C4—81no—
242C3×C22⋊C4—81no—
243C3×C42—24no—
244C3×C42—24no—
245C3×C42—24no—
246C3×C42—21no—
247C3×C42—21no—
248C3×C42—21no—
249C3×C42—21no—
250C3×C42—21no—
251C3×C42—21no—
252C3×C42—81no—
253C3×C42—81no—
254C3×C42—81no—
255C22×Q8C23×C311yescharacteristic
256C22×D4C23×C311yescharacteristic
257C2×SD16—22no—
258C2×SD16—22no—
259C2×SD16—22no—
260C22×C8C23×C311yescharacteristic
261C22×C8C23×C311yescharacteristic
262C22×C8C23×C311yescharacteristic
263C22×C8C23×C311yescharacteristic
264C4×Q8—22no—
265C4×Q8—22no—
266C4×Q8—22no—
267C4×Q8—22no—
268C4×Q8—22no—
269C4×Q8—22no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4C23×C311yescharacteristic
277C2×C4⋊C4C23×C311yescharacteristic
278C2×C4⋊C4C23×C311yescharacteristic
279C2×C4⋊C4C23×C311yescharacteristic
280C2×C4⋊C4C23×C311yescharacteristic
281C2×C4⋊C4C23×C311yescharacteristic
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42C23×C311yescharacteristic
295C2×C42C23×C311yescharacteristic
296C2×C42C23×C311yescharacteristic
297#107—22no—
298#107—22no—
299#106—22no—
300#103—22no—
301#103—22no—
302#103—22no—
303#102—22no—
304#102—22no—
305#102—22no—
306#95—21no—
307#95—21no—
308#95—21no—
309#95—21no—
310#95—21no—
311#95—21no—
312#95C3×D411yescharacteristic
313#95C3×D411yescharacteristic
314#95C3×D411yescharacteristic
315#95C3×D411yescharacteristic
316C23×C3—41no—
317C23×C3—41no—
318C23×C3—41no—
319C23×C3—41no—
320C23×C3—41no—
321C23×C3—41no—
322C23×C3—41no—
323C23×C3C22×D411yescharacteristic
324C3×Q8—22no—
325C3×Q8—22no—
326C3×Q8—22no—
327C3×Q8#12614yes—
328C3×D4—41no—
329C3×D4—41no—
330C3×D4—41no—
331C3×D4—41no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4C2×#4111yescharacteristic
372C2×C3×C4C2×#4111yescharacteristic
373C2×C3×C4C2×#4111yescharacteristic
374C2×C3×C4C2×#4111yescharacteristic
375C2×C3×C4C2×#4111yescharacteristic
376C2×C3×C4C2×#4111yescharacteristic
377C3×C8—41no—
378C3×C8—41no—
379C3×C8—41no—
380C3×C8—41no—
381C24—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8C3×#4112yes—
386C2×Q8C3×#4112yes—
387C2×Q8C3×#4112yes—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4C2×C3×D411yescharacteristic
402C22×C4C2×C3×D411yescharacteristic
403C22×C4C2×C3×D411yescharacteristic
404C22×C4C2×C3×D411yescharacteristic
405C22×C4C2×C3×D411yescharacteristic
406C22×C4C2×C3×D411yescharacteristic
407C22×C4C24×C311yescharacteristic
408SD16—44no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—81no—
434C4⋊C4—81no—
435C4⋊C4—81no—
436C4⋊C4C3×#4112yes—
437C4⋊C4C3×#4112yes—
438C4⋊C4C3×#4112yes—
439C22⋊C4—81no—
440C22⋊C4—81no—
441C22⋊C4—81no—
442C22⋊C4—81no—
443C22⋊C4—81no—
444C22⋊C4—81no—
445C42—24no—
446C42—24no—
447C42—24no—
448C42—21no—
449C42—21no—
450C42—21no—
451C42—21no—
452C42—21no—
453C42—21no—
454C42—81no—
455C42—81no—
456C42—81no—
457C22×C3—81no—
458C22×C3—81no—
459C22×C3—81no—
460C22×C3—81no—
461C22×C3—81no—
462C22×C3—81no—
463C22×C3—81no—
464C22×C3#52511yescharacteristic
465C22×C3#52511yescharacteristic
466C22×C3#52511yescharacteristic
467C22×C3#57611yescharacteristic
468C22×C3#57711yescharacteristic
469C22×C3#57711yescharacteristic
470C22×C3#57811yescharacteristic
471C3×C4—24no—
472C3×C4—41no—
473C3×C4—41no—
474C3×C4—42no—
475C3×C4—42no—
476C3×C4—41no—
477C3×C4—41no—
478C3×C4—41no—
479C3×C4—42no—
480C3×C4—41no—
481C3×C4—81no—
482C3×C4—81no—
483C3×C4—81no—
484C3×C4C2×#12611yescharacteristic
485C3×C4#52911yescharacteristic
486C3×C4#52911yescharacteristic
487C3×C4#52911yescharacteristic
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23C22×C3×D411yescharacteristic
496Q8—22no—
497Q8—22no—
498Q8—22no—
499Q8C3×#12614yes—
500D4—41no—
501D4—41no—
502D4—41no—
503D4—41no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4C2×C3×#4111yescharacteristic
544C2×C4C2×C3×#4111yescharacteristic
545C2×C4C2×C3×#4111yescharacteristic
546C2×C4C2×C3×#4111yescharacteristic
547C2×C4C2×C3×#4111yescharacteristic
548C2×C4C2×C3×#4111yescharacteristic
549C8—41no—
550C8—41no—
551C8—41no—
552C8—41no—
553C2×C3—161no—
554C2×C3#285211yescharacteristic
555C2×C3#311911yescharacteristic
556C2×C3#311911yescharacteristic
557C2×C3#312111yescharacteristic
558C2×C3#312111yescharacteristic
559C2×C3#312411yescharacteristic
560C2×C3#312611yescharacteristic
561C22—81no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22C3×#52511yescharacteristic
569C22C3×#52511yescharacteristic
570C22C3×#52511yescharacteristic
571C22C3×#57611yescharacteristic
572C22C3×#57711yescharacteristic
573C22C3×#57711yescharacteristic
574C22C3×#57811yescharacteristic
575C4—24no—
576C4—41no—
577C4—41no—
578C4—42no—
579C4—42no—
580C4—41no—
581C4—41no—
582C4—41no—
583C4—42no—
584C4—41no—
585C4—81no—
586C4—81no—
587C4—81no—
588C4C2×C3×#12611yescharacteristic
589C4C3×#52911yescharacteristic
590C4C3×#52911yescharacteristic
591C4C3×#52911yescharacteristic
592C3#3308711yescharacteristic, direct product factor, Sylow
593C2—161no—
594C2C3×#285211yescharacteristic
595C2C3×#311911yescharacteristic
596C2C3×#311911yescharacteristic
597C2C3×#312111yescharacteristic
598C2C3×#312111yescharacteristic
599C2C3×#312411yescharacteristic
600C2C3×#312611yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×#376 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#2276C2semidirect
6. C3×#1921C2semidirect
7. C3×#1919C2semidirect
8. C3×#1918C2semidirect
12. C3×#1865C2semidirect
13. C3×#1863C2semidirect
14. C3×#1863C2semidirect
15. C3×#1840C2semidirect
16. #33087C3direct (N × H)
61. #2276C2×C3semidirect
65. #1921C2×C3semidirect
66. #1919C2×C3semidirect
67. #1918C2×C3semidirect
71. #1865C2×C3semidirect
72. #1863C2×C3semidirect
73. #1863C2×C3semidirect
74. #1840C2×C3semidirect
592. C3#33087direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#26442
C3×#22762
C3×#19262
C3×#19252
C3×#19212
C3×#19192
C3×#19182
C3×#18802
C3×#18652
C3×#18632
C3×#18402
#330873