Finite Group Databasebeta

← C3×#29459 #10648066

Subgroups — C3×#29459 #10648066

Order 768.

660 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2644C211yesmaximal, characteristic
2C3×#2320C211yesmaximal, characteristic, complemented
3C3×#2087C211yesmaximal, characteristic, complemented
4C3×#2067C211yesmaximal, characteristic
5C3×#2066C211yesmaximal, characteristic, complemented
6C3×#2055C211yesmaximal, characteristic
7C3×#1973C211yesmaximal, characteristic
8C3×#1943C211yesmaximal, characteristic, complemented
9C3×#1934C211yesmaximal, characteristic
10C3×#1933C211yesmaximal, characteristic, complemented
11C3×#1926C211yesmaximal, characteristic
12C3×#1918C211yesmaximal, characteristic, complemented
13C3×#1880C211yesmaximal, characteristic
14C3×#1863C211yesmaximal, characteristic, complemented
15C3×#1768C211yesmaximal, characteristic, complemented
16#29459C311yesmaximal, characteristic, direct product factor, Sylow
17C22×C3×SD16C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×#107C2211yescharacteristic
20C2×C3×#107C2211yescharacteristic
21C2×C3×#106C2211yescharacteristic
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×#102C2211yescharacteristic
28C2×C3×#102C2211yescharacteristic
29C2×C3×#102C2211yescharacteristic
30C2×C3×#97C2211yescharacteristic
31C2×C3×C4×C8C2211yescharacteristic
32C3×#397—21no—
33C3×#397—21no—
34C3×#397—21no—
35C3×#396—21no—
36C3×#391C2211yescharacteristic
37C3×#391C2211yescharacteristic
38C3×#390C2211yescharacteristic
39C3×#390C2211yescharacteristic
40C3×#390C2211yescharacteristic
41C3×#389C2211yescharacteristic
42C3×#389C2211yescharacteristic
43C3×#389C2211yescharacteristic
44C3×#389C2211yescharacteristic
45C3×#389C2211yescharacteristic
46C3×#387—21no—
47C3×#387—21no—
48C3×#383—21no—
49C3×#382—21no—
50C3×C4×C4⋊C4C2211yescharacteristic
51C3×#376C2211yescharacteristic
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#340C2211yescharacteristic
57C3×#340C2211yescharacteristic
58C3×#336C2211yescharacteristic
59C3×#336C2211yescharacteristic
60#2644C2×C311yescharacteristic
61#2320C2×C311yescharacteristic, complemented
62#2087C2×C311yescharacteristic, complemented
63#2067C2×C311yescharacteristic
64#2066C2×C311yescharacteristic, complemented
65#2055C2×C311yescharacteristic
66#1973C2×C311yescharacteristic
67#1943C2×C311yescharacteristic, complemented
68#1934C2×C311yescharacteristic
69#1933C2×C311yescharacteristic, complemented
70#1926C2×C311yescharacteristic
71#1918C2×C311yescharacteristic, complemented
72#1880C2×C311yescharacteristic
73#1863C2×C311yescharacteristic, complemented
74#1768C2×C311yescharacteristic, complemented
75C22×C3×Q8C2311yescharacteristic
76C22×C3×D4C2311yescharacteristic
77C2×C3×SD16—22no—
78C2×C3×SD16—22no—
79C2×C3×SD16—41no—
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C3×C4×Q8—22no—
85C3×C4×Q8—22no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C22⋊C4—21no—
99C2×C3×C22⋊C4—21no—
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C42—21no—
105C2×C3×C42—21no—
106C2×C3×C42C2311yescharacteristic
107C2×C3×C42C2311yescharacteristic
108C2×C3×C42C2311yescharacteristic
109C3×#107—22no—
110C3×#107—22no—
111C3×#106—41no—
112C3×C4⋊C8—22no—
113C3×C4⋊C8—22no—
114C3×#103—22no—
115C3×#103—22no—
116C3×#103—41no—
117C3×#102—22no—
118C3×#102—22no—
119C3×#102—41no—
120C3×#97—41no—
121C3×C4×C8—41no—
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×Q8C22×C311yescharacteristic
138C2×#107C22×C311yescharacteristic
139C2×#107C22×C311yescharacteristic
140C2×#106C22×C311yescharacteristic
141C2×C4⋊C8C22×C311yescharacteristic
142C2×C4⋊C8C22×C311yescharacteristic
143C2×#103C22×C311yescharacteristic
144C2×#103C22×C311yescharacteristic
145C2×#103C22×C311yescharacteristic
146C2×#102C22×C311yescharacteristic
147C2×#102C22×C311yescharacteristic
148C2×#102C22×C311yescharacteristic
149C2×#97C22×C311yescharacteristic
150C2×C4×C8C22×C311yescharacteristic
151#397—21no—
152#397—21no—
153#397—21no—
154#396—21no—
155#391C22×C311yescharacteristic
156#391C22×C311yescharacteristic
157#390C22×C311yescharacteristic
158#390C22×C311yescharacteristic
159#390C22×C311yescharacteristic
160#389C22×C311yescharacteristic
161#389C22×C311yescharacteristic
162#389C22×C311yescharacteristic
163#389C22×C311yescharacteristic
164#389C22×C311yescharacteristic
165#387—21no—
166#387—21no—
167#383—21no—
168#382—21no—
169C4×C4⋊C4C22×C311yescharacteristic
170#376C22×C311yescharacteristic
171#340C22×C311yescharacteristic
172#340C22×C311yescharacteristic
173#340C22×C311yescharacteristic
174#340C22×C311yescharacteristic
175#340C22×C311yescharacteristic
176#340C22×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#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#4112yes—
218C2×C3×C8#4112yes—
219C3×C4⋊C4—24no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
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—81no—
235C3×C4⋊C4—81no—
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—21no—
245C3×C42—21no—
246C3×C42—21no—
247C3×C42—21no—
248C3×C42—21no—
249C3×C42—21no—
250C3×C42—81no—
251C22×Q8C23×C311yescharacteristic
252C22×D4C23×C311yescharacteristic
253C2×SD16—22no—
254C2×SD16—22no—
255C2×SD16—41no—
256C22×C8C23×C311yescharacteristic
257C22×C8C23×C311yescharacteristic
258C22×C8C23×C311yescharacteristic
259C22×C8C23×C311yescharacteristic
260C4×Q8—22no—
261C4×Q8—22no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4C23×C311yescharacteristic
269C2×C4⋊C4C23×C311yescharacteristic
270C2×C4⋊C4C23×C311yescharacteristic
271C2×C4⋊C4C23×C311yescharacteristic
272C2×C4⋊C4C23×C311yescharacteristic
273C2×C4⋊C4C23×C311yescharacteristic
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42C23×C311yescharacteristic
283C2×C42C23×C311yescharacteristic
284C2×C42C23×C311yescharacteristic
285#107—22no—
286#107—22no—
287#106—41no—
288C4⋊C8—22no—
289C4⋊C8—22no—
290#103—22no—
291#103—22no—
292#103—41no—
293#102—22no—
294#102—22no—
295#102—41no—
296#97—41no—
297C4×C8—41no—
298#95—21no—
299#95—21no—
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#95C3×D411yescharacteristic
309#95C3×D411yescharacteristic
310#95C3×D411yescharacteristic
311#95C3×D411yescharacteristic
312C23×C3—41no—
313C23×C3—41no—
314C23×C3—41no—
315C23×C3—41no—
316C23×C3—41no—
317C23×C3—41no—
318C23×C3—41no—
319C23×C3C22×D411yescharacteristic
320C3×Q8—22no—
321C3×Q8—22no—
322C3×Q8—41no—
323C3×Q8—41no—
324C3×D4—41no—
325C3×D4—41no—
326C3×D4—41no—
327C3×D4—41no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
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—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
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—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
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×C4C2×#4111yescharacteristic
368C2×C3×C4C2×#4111yescharacteristic
369C2×C3×C4C2×#4111yescharacteristic
370C2×C3×C4C2×#4111yescharacteristic
371C2×C3×C4H(5,𝔽2)11yescharacteristic
372C2×C3×C42−1+411yescharacteristic
373C3×C8—22no—
374C3×C8—22no—
375C3×C8—41no—
376C3×C8—41no—
377C24—21no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8C3×#4112yes—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4C2×C3×D411yescharacteristic
398C22×C4C2×C3×D411yescharacteristic
399C22×C4C2×C3×D411yescharacteristic
400C22×C4C2×C3×D411yescharacteristic
401C22×C4C2×C3×D411yescharacteristic
402C22×C4C2×C3×D411yescharacteristic
403C22×C4C24×C311yescharacteristic
404SD16—44no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8C3×#4112yes—
416C2×C8C3×#4112yes—
417C4⋊C4—24no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—81no—
430C4⋊C4—81no—
431C4⋊C4—81no—
432C4⋊C4—81no—
433C4⋊C4—81no—
434C4⋊C4C3×#4112yes—
435C22⋊C4—81no—
436C22⋊C4—81no—
437C22⋊C4—81no—
438C22⋊C4—81no—
439C22⋊C4—81no—
440C22⋊C4—81no—
441C42—24no—
442C42—21no—
443C42—21no—
444C42—21no—
445C42—21no—
446C42—21no—
447C42—21no—
448C42—81no—
449C22×C3—81no—
450C22×C3—81no—
451C22×C3—81no—
452C22×C3—81no—
453C22×C3—81no—
454C22×C3—81no—
455C22×C3—81no—
456C22×C3#52511yescharacteristic
457C22×C3#53211yescharacteristic
458C22×C3#53711yescharacteristic
459C22×C3#57611yescharacteristic
460C22×C3#57711yescharacteristic
461C22×C3#57711yescharacteristic
462C22×C3#57811yescharacteristic
463C3×C4—42no—
464C3×C4—41no—
465C3×C4—42no—
466C3×C4—41no—
467C3×C4—41no—
468C3×C4—41no—
469C3×C4—41no—
470C3×C4—41no—
471C3×C4—81no—
472C3×C4—81no—
473C3×C4—81no—
474C3×C4—81no—
475C3×C4—81no—
476C3×C4#55111yescharacteristic
477C3×C4#55211yescharacteristic
478C3×C4#55311yescharacteristic
479C3×C4#55611yescharacteristic
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23C22×C3×D411yescharacteristic
488Q8—22no—
489Q8—22no—
490Q8—41no—
491Q8—41no—
492D4—41no—
493D4—41no—
494D4—41no—
495D4—41no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4C2×C3×#4111yescharacteristic
536C2×C4C2×C3×#4111yescharacteristic
537C2×C4C2×C3×#4111yescharacteristic
538C2×C4C2×C3×#4111yescharacteristic
539C2×C4C3×H(5,𝔽2)11yescharacteristic
540C2×C4C3×2−1+411yescharacteristic
541C8—22no—
542C8—22no—
543C8—41no—
544C8—41no—
545C2×C3—161no—
546C2×C3#260111yescharacteristic
547C2×C3#311911yescharacteristic
548C2×C3#312111yescharacteristic
549C2×C3#317811yescharacteristic
550C2×C3#317911yescharacteristic
551C2×C3#326011yescharacteristic
552C2×C3#326211yescharacteristic
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22—81no—
560C22C3×#52511yescharacteristic
561C22C3×#53211yescharacteristic
562C22C3×#53711yescharacteristic
563C22C3×#57611yescharacteristic
564C22C3×#57711yescharacteristic
565C22C3×#57711yescharacteristic
566C22C3×#57811yescharacteristic
567C4—41no—
568C4—41no—
569C4—41no—
570C4—41no—
571C4—42no—
572C4—41no—
573C4—42no—
574C4—41no—
575C4—81no—
576C4—81no—
577C4—81no—
578C4—81no—
579C4—81no—
580C4C3×#55111yescharacteristic
581C4C3×#55211yescharacteristic
582C4C3×#55311yescharacteristic
583C4C3×#55611yescharacteristic
584C3#2945911yescharacteristic, direct product factor, Sylow
585C2—161no—
586C2C3×#260111yescharacteristic
587C2C3×#311911yescharacteristic
588C2C3×#312111yescharacteristic
589C2C3×#317811yescharacteristic
590C2C3×#317911yescharacteristic
591C2C3×#326011yescharacteristic
592C2C3×#326211yescharacteristic
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×#2320C2semidirect
3. C3×#2087C2semidirect
5. C3×#2066C2semidirect
8. C3×#1943C2semidirect
10. C3×#1933C2semidirect
12. C3×#1918C2semidirect
14. C3×#1863C2semidirect
15. C3×#1768C2semidirect
16. #29459C3direct (N × H)
61. #2320C2×C3semidirect
62. #2087C2×C3semidirect
64. #2066C2×C3semidirect
67. #1943C2×C3semidirect
69. #1933C2×C3semidirect
71. #1918C2×C3semidirect
73. #1863C2×C3semidirect
74. #1768C2×C3semidirect
584. C3#29459direct (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×#23202
C3×#20872
C3×#20672
C3×#20662
C3×#20552
C3×#19732
C3×#19432
C3×#19342
C3×#19332
C3×#19262
C3×#19182
C3×#18802
C3×#18632
C3×#17682
#294593