Finite Group Databasebeta

← C3×#33090 #10651697

Subgroups — C3×#33090 #10651697

Order 768.

770 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2645C211yesmaximal, characteristic, complemented
2C3×#2275C211yesmaximal, characteristic
3C3×#1928C211yesmaximal, characteristic, complemented
4C3×#1928C211yesmaximal, characteristic, complemented
5C3×#1927C211yesmaximal, characteristic, complemented
6C3×#1921C211yesmaximal, characteristic, complemented
7C3×#1920C211yesmaximal, characteristic, complemented
8C3×#1918C211yesmaximal, characteristic, complemented
9C3×#1877C211yesmaximal, characteristic
10C3×#1877C211yesmaximal, characteristic
11C3×#1877C211yesmaximal, characteristic
12C3×#1864C211yesmaximal, characteristic
13C3×#1862C211yesmaximal, characteristic
14C3×#1862C211yesmaximal, characteristic
15C3×#1840C211yesmaximal, characteristic, complemented
16#33090C311yesmaximal, characteristic, direct product factor, Sylow
17C22×C3×SD16C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20C2×C3×C4×D4C2211yescharacteristic
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×#401—21no—
31C3×#400—21no—
32C3×#400—21no—
33C3×#400—21no—
34C3×#391C2211yescharacteristic
35C3×#391C2211yescharacteristic
36C3×#391C2211yescharacteristic
37C3×#389C2211yescharacteristic
38C3×#389C2211yescharacteristic
39C3×#389C2211yescharacteristic
40C3×C4×C4⋊C4C2211yescharacteristic
41C3×C4×C4⋊C4C2211yescharacteristic
42C3×C4×C4⋊C4C2211yescharacteristic
43C3×C4×C22⋊C4—21no—
44C3×C4×C22⋊C4—21no—
45C3×C4×C22⋊C4—21no—
46C3×#376C2211yescharacteristic
47C2×C3×#95—21no—
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#2645C2×C311yescharacteristic, complemented
61#2275C2×C311yescharacteristic
62#1928C2×C311yescharacteristic, complemented
63#1928C2×C311yescharacteristic, complemented
64#1927C2×C311yescharacteristic, complemented
65#1921C2×C311yescharacteristic, complemented
66#1920C2×C311yescharacteristic, complemented
67#1918C2×C311yescharacteristic, complemented
68#1877C2×C311yescharacteristic
69#1877C2×C311yescharacteristic
70#1877C2×C311yescharacteristic
71#1864C2×C311yescharacteristic
72#1862C2×C311yescharacteristic
73#1862C2×C311yescharacteristic
74#1840C2×C311yescharacteristic, complemented
75C22×C3×Q8C2311yescharacteristic
76C22×C3×D4C2311yescharacteristic
77C23×C3×C4—21no—
78C23×C3×C4—21no—
79C23×C3×C4—21no—
80C2×C3×SD16—22no—
81C2×C3×SD16—22no—
82C2×C3×SD16—22no—
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C3×C4×D4—22no—
88C3×C4×D4—22no—
89C3×C4×D4—22no—
90C3×C4×D4—22no—
91C3×C4×D4—22no—
92C3×C4×D4—22no—
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×C42—21no—
106C2×C3×C42—21no—
107C2×C3×C42—21no—
108C2×C3×C42C2311yescharacteristic
109C2×C3×C42C2311yescharacteristic
110C2×C3×C42C2311yescharacteristic
111C3×#107—22no—
112C3×#107—22no—
113C3×#106—22no—
114C3×#103—22no—
115C3×#103—22no—
116C3×#103—22no—
117C3×#102—22no—
118C3×#102—22no—
119C3×#102—22no—
120C3×#95—21no—
121C3×#95—21no—
122C3×#95—21no—
123C3×#95—21no—
124C3×#95—21no—
125C3×#95—21no—
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×D4C22×C311yescharacteristic
138C2×C4×D4C22×C311yescharacteristic
139C2×C4×D4C22×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#401—21no—
150#400—21no—
151#400—21no—
152#400—21no—
153#391C22×C311yescharacteristic
154#391C22×C311yescharacteristic
155#391C22×C311yescharacteristic
156#389C22×C311yescharacteristic
157#389C22×C311yescharacteristic
158#389C22×C311yescharacteristic
159C4×C4⋊C4C22×C311yescharacteristic
160C4×C4⋊C4C22×C311yescharacteristic
161C4×C4⋊C4C22×C311yescharacteristic
162C4×C22⋊C4—21no—
163C4×C22⋊C4—21no—
164C4×C22⋊C4—21no—
165#376C22×C311yescharacteristic
166C2×#95—21no—
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—21no—
184C2×C3×Q8—21no—
185C2×C3×Q8—21no—
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4—21no—
189C2×C3×D4#4112yes—
190C2×C3×D4#4112yes—
191C2×C3×D4#4112yes—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—24no—
196C22×C3×C4—24no—
197C22×C3×C4—24no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—81no—
203C22×C3×C4—81no—
204C22×C3×C4—81no—
205C22×C3×C4—81no—
206C22×C3×C4—81no—
207C22×C3×C4—81no—
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2411yescharacteristic
215C3×SD16—44no—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—81no—
238C3×C4⋊C4—81no—
239C3×C4⋊C4—81no—
240C3×C4⋊C4#4112yes—
241C3×C4⋊C4#4112yes—
242C3×C4⋊C4#4112yes—
243C3×C22⋊C4—24no—
244C3×C22⋊C4—24no—
245C3×C22⋊C4—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
257C23×C4—21no—
258C23×C4—21no—
259C23×C4—21no—
260C2×SD16—22no—
261C2×SD16—22no—
262C2×SD16—22no—
263C22×C8C23×C311yescharacteristic
264C22×C8C23×C311yescharacteristic
265C22×C8C23×C311yescharacteristic
266C22×C8C23×C311yescharacteristic
267C4×D4—22no—
268C4×D4—22no—
269C4×D4—22no—
270C4×D4—22no—
271C4×D4—22no—
272C4×D4—22no—
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×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42C23×C311yescharacteristic
289C2×C42C23×C311yescharacteristic
290C2×C42C23×C311yescharacteristic
291#107—22no—
292#107—22no—
293#106—22no—
294#103—22no—
295#103—22no—
296#103—22no—
297#102—22no—
298#102—22no—
299#102—22no—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—21no—
304#95—21no—
305#95—21no—
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—22no—
317C23×C3—22no—
318C23×C3—22no—
319C23×C3—41no—
320C23×C3—41no—
321C23×C3—41no—
322C23×C3—41no—
323C23×C3C22×D411yescharacteristic
324C3×Q8—41no—
325C3×Q8—41no—
326C3×Q8—41no—
327C3×Q8—41no—
328C3×D4—22no—
329C3×D4—22no—
330C3×D4—22no—
331C3×D4#12614yes—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
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—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
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×C4—84no—
372C2×C3×C4—84no—
373C2×C3×C4—84no—
374C2×C3×C4C2×#4111yescharacteristic
375C2×C3×C4C2×#4111yescharacteristic
376C2×C3×C4C2×#4111yescharacteristic
377C2×C3×C4C2×#4111yescharacteristic
378C2×C3×C4C2×#4111yescharacteristic
379C2×C3×C4C2×#4111yescharacteristic
380C3×C8—41no—
381C3×C8—41no—
382C3×C8—41no—
383C3×C8—41no—
384C24—21no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4C3×#4112yes—
395C2×D4C3×#4112yes—
396C2×D4C3×#4112yes—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—81no—
408C22×C4—81no—
409C22×C4—81no—
410C22×C4—81no—
411C22×C4—81no—
412C22×C4—81no—
413C22×C4C2×C3×D411yescharacteristic
414C22×C4C2×C3×D411yescharacteristic
415C22×C4C2×C3×D411yescharacteristic
416C22×C4C2×C3×D411yescharacteristic
417C22×C4C2×C3×D411yescharacteristic
418C22×C4C2×C3×D411yescharacteristic
419C22×C4C24×C311yescharacteristic
420SD16—44no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—81no—
443C4⋊C4—81no—
444C4⋊C4—81no—
445C4⋊C4C3×#4112yes—
446C4⋊C4C3×#4112yes—
447C4⋊C4C3×#4112yes—
448C22⋊C4—24no—
449C22⋊C4—24no—
450C22⋊C4—24no—
451C42—21no—
452C42—21no—
453C42—21no—
454C42—21no—
455C42—21no—
456C42—21no—
457C42—81no—
458C42—81no—
459C42—81no—
460C22×C3—24no—
461C22×C3—42no—
462C22×C3—42no—
463C22×C3—42no—
464C22×C3—42no—
465C22×C3—42no—
466C22×C3—42no—
467C22×C3#52511yescharacteristic
468C22×C3#52511yescharacteristic
469C22×C3#52511yescharacteristic
470C22×C3#57611yescharacteristic
471C22×C3#57711yescharacteristic
472C22×C3#57711yescharacteristic
473C22×C3#57811yescharacteristic
474C3×C4—41no—
475C3×C4—41no—
476C3×C4—42no—
477C3×C4—42no—
478C3×C4—41no—
479C3×C4—41no—
480C3×C4—41no—
481C3×C4—42no—
482C3×C4—41no—
483C3×C4—81no—
484C3×C4—81no—
485C3×C4—81no—
486C3×C4—81no—
487C3×C4C2×#12611yescharacteristic
488C3×C4#52911yescharacteristic
489C3×C4#52911yescharacteristic
490C3×C4#52911yescharacteristic
491C23—22no—
492C23—22no—
493C23—22no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23C22×C3×D411yescharacteristic
499Q8—41no—
500Q8—41no—
501Q8—41no—
502Q8—41no—
503D4—22no—
504D4—22no—
505D4—22no—
506D4C3×#12614yes—
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×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—84no—
547C2×C4—84no—
548C2×C4—84no—
549C2×C4C2×C3×#4111yescharacteristic
550C2×C4C2×C3×#4111yescharacteristic
551C2×C4C2×C3×#4111yescharacteristic
552C2×C4C2×C3×#4111yescharacteristic
553C2×C4C2×C3×#4111yescharacteristic
554C2×C4C2×C3×#4111yescharacteristic
555C8—41no—
556C8—41no—
557C8—41no—
558C8—41no—
559C2×C3—44no—
560C2×C3#285211yescharacteristic
561C2×C3#311811yescharacteristic
562C2×C3#311811yescharacteristic
563C2×C3#312011yescharacteristic
564C2×C3#312011yescharacteristic
565C2×C3#312311yescharacteristic
566C2×C3#312411yescharacteristic
567C22—24no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22C3×#52511yescharacteristic
575C22C3×#52511yescharacteristic
576C22C3×#52511yescharacteristic
577C22C3×#57611yescharacteristic
578C22C3×#57711yescharacteristic
579C22C3×#57711yescharacteristic
580C22C3×#57811yescharacteristic
581C4—41no—
582C4—41no—
583C4—42no—
584C4—42no—
585C4—41no—
586C4—41no—
587C4—41no—
588C4—42no—
589C4—41no—
590C4—81no—
591C4—81no—
592C4—81no—
593C4—81no—
594C4C2×C3×#12611yescharacteristic
595C4C3×#52911yescharacteristic
596C4C3×#52911yescharacteristic
597C4C3×#52911yescharacteristic
598C3#3309011yescharacteristic, direct product factor, Sylow
599C2—44no—
600C2C3×#285211yescharacteristic
601C2C3×#311811yescharacteristic
602C2C3×#311811yescharacteristic
603C2C3×#312011yescharacteristic
604C2C3×#312011yescharacteristic
605C2C3×#312311yescharacteristic
606C2C3×#312411yescharacteristic
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
1. C3×#2645C2semidirect
3. C3×#1928C2semidirect
4. C3×#1928C2semidirect
5. C3×#1927C2semidirect
6. C3×#1921C2semidirect
7. C3×#1920C2semidirect
8. C3×#1918C2semidirect
15. C3×#1840C2semidirect
16. #33090C3direct (N × H)
60. #2645C2×C3semidirect
62. #1928C2×C3semidirect
63. #1928C2×C3semidirect
64. #1927C2×C3semidirect
65. #1921C2×C3semidirect
66. #1920C2×C3semidirect
67. #1918C2×C3semidirect
74. #1840C2×C3semidirect
598. C3#33090direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#26452
C3×#22752
C3×#19282
C3×#19272
C3×#19212
C3×#19202
C3×#19182
C3×#18772
C3×#18642
C3×#18622
C3×#18402
#330903