Finite Group Databasebeta

← C3×#29451 #10648058

Subgroups — C3×#29451 #10648058

Order 768.

712 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2860C211yesmaximal, characteristic
2C3×#2295C211yesmaximal, characteristic, complemented
3C3×#2086C211yesmaximal, characteristic, complemented
4C3×#2067C211yesmaximal, characteristic
5C3×#2066C211yesmaximal, characteristic, complemented
6C3×#2055C211yesmaximal, characteristic
7C3×#1973C211yesmaximal, characteristic
8C3×#1943C211yesmaximal, characteristic, complemented
9C3×#1936C211yesmaximal, characteristic
10C3×#1935C211yesmaximal, characteristic, complemented
11C3×#1926C211yesmaximal, characteristic
12C3×#1921C211yesmaximal, characteristic, complemented
13C3×#1880C211yesmaximal, characteristic
14C3×#1863C211yesmaximal, characteristic, complemented
15C3×#1767C211yesmaximal, characteristic, complemented
16#29451C311yesmaximal, characteristic, direct product factor, Sylow
17C22×C3×SD16C2211yescharacteristic
18C2×C3×#126—21no—
19C2×C3×C4×Q8C2211yescharacteristic
20C2×C3×#107C2211yescharacteristic
21C2×C3×#107C2211yescharacteristic
22C2×C3×#106C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×#103C2211yescharacteristic
28C2×C3×#102C2211yescharacteristic
29C2×C3×#102C2211yescharacteristic
30C2×C3×#102C2211yescharacteristic
31C2×C3×#97C2211yescharacteristic
32C2×C3×#97C2211yescharacteristic
33C3×#391C2211yescharacteristic
34C3×#391C2211yescharacteristic
35C3×#390C2211yescharacteristic
36C3×#390C2211yescharacteristic
37C3×#390C2211yescharacteristic
38C3×#389C2211yescharacteristic
39C3×#389C2211yescharacteristic
40C3×#389C2211yescharacteristic
41C3×#389C2211yescharacteristic
42C3×#389C2211yescharacteristic
43C3×#388—21no—
44C3×#388—21no—
45C3×#388—21no—
46C3×#387—21no—
47C3×#387—21no—
48C3×C4×C4⋊C4—21no—
49C3×C4×C4⋊C4C2211yescharacteristic
50C3×#376—21no—
51C3×C43C2211yescharacteristic
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#340C2211yescharacteristic
57C3×#340C2211yescharacteristic
58C3×#336C2211yescharacteristic
59C3×#336C2211yescharacteristic
60#2860C2×C311yescharacteristic
61#2295C2×C311yescharacteristic, complemented
62#2086C2×C311yescharacteristic, complemented
63#2067C2×C311yescharacteristic
64#2066C2×C311yescharacteristic, complemented
65#2055C2×C311yescharacteristic
66#1973C2×C311yescharacteristic
67#1943C2×C311yescharacteristic, complemented
68#1936C2×C311yescharacteristic
69#1935C2×C311yescharacteristic, complemented
70#1926C2×C311yescharacteristic
71#1921C2×C311yescharacteristic, complemented
72#1880C2×C311yescharacteristic
73#1863C2×C311yescharacteristic, complemented
74#1767C2×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×#126—44no—
85C3×C4×Q8—22no—
86C3×C4×Q8—22no—
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⋊C4—21no—
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C4⋊C4C2311yescharacteristic
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×C22⋊C4—21no—
105C2×C3×C42—21no—
106C2×C3×C42—21no—
107C2×C3×C42D411yescharacteristic
108C2×C3×C42D411yescharacteristic
109C2×C3×C42D411yescharacteristic
110C2×C3×C42D411yescharacteristic
111C2×C3×C42C2311yescharacteristic
112C2×C3×C42C2311yescharacteristic
113C2×C3×C42C2311yescharacteristic
114C3×#107—22no—
115C3×#107—22no—
116C3×#106—41no—
117C3×C4⋊C8—22no—
118C3×C4⋊C8—22no—
119C3×#103—22no—
120C3×#103—22no—
121C3×#103—41no—
122C3×#102—22no—
123C3×#102—22no—
124C3×#102—41no—
125C3×#97—41no—
126C3×#97—41no—
127C3×#95—21no—
128C3×#95—21no—
129C3×#95—21no—
130C3×#95—21no—
131C3×#95—21no—
132C3×#95—21no—
133C3×#95—21no—
134C3×#95—21no—
135C3×#95—21no—
136C3×#95—21no—
137C22×SD16C22×C311yescharacteristic
138C2×#126—21no—
139C2×C4×Q8C22×C311yescharacteristic
140C2×#107C22×C311yescharacteristic
141C2×#107C22×C311yescharacteristic
142C2×#106C22×C311yescharacteristic
143C2×C4⋊C8C22×C311yescharacteristic
144C2×C4⋊C8C22×C311yescharacteristic
145C2×#103C22×C311yescharacteristic
146C2×#103C22×C311yescharacteristic
147C2×#103C22×C311yescharacteristic
148C2×#102C22×C311yescharacteristic
149C2×#102C22×C311yescharacteristic
150C2×#102C22×C311yescharacteristic
151C2×#97C22×C311yescharacteristic
152C2×#97C22×C311yescharacteristic
153#391C22×C311yescharacteristic
154#391C22×C311yescharacteristic
155#390C22×C311yescharacteristic
156#390C22×C311yescharacteristic
157#390C22×C311yescharacteristic
158#389C22×C311yescharacteristic
159#389C22×C311yescharacteristic
160#389C22×C311yescharacteristic
161#389C22×C311yescharacteristic
162#389C22×C311yescharacteristic
163#388—21no—
164#388—21no—
165#388—21no—
166#387—21no—
167#387—21no—
168C4×C4⋊C4—21no—
169C4×C4⋊C4C22×C311yescharacteristic
170#376—21no—
171C43C22×C311yescharacteristic
172#340C22×C311yescharacteristic
173#340C22×C311yescharacteristic
174#340C22×C311yescharacteristic
175#340C22×C311yescharacteristic
176#340C22×C311yescharacteristic
177#340C22×C311yescharacteristic
178#336C22×C311yescharacteristic
179#336C22×C311yescharacteristic
180C24×C3—21no—
181C2×C3×Q8—21no—
182C2×C3×Q8—21no—
183C2×C3×Q8—21no—
184C2×C3×Q8—21no—
185C2×C3×Q8—21no—
186C2×C3×Q8#4112yes—
187C2×C3×D4—21no—
188C2×C3×D4—21no—
189C2×C3×D4—21no—
190C2×C3×D4—21no—
191C2×C3×D4—21no—
192C2×C3×D4—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×C4—21no—
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2411yescharacteristic
207C3×SD16—44no—
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#4112yes—
219C2×C3×C8#4112yes—
220C3×C4⋊C4—24no—
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—21no—
232C3×C4⋊C4—42no—
233C3×C4⋊C4—42no—
234C3×C4⋊C4—42no—
235C3×C4⋊C4—42no—
236C3×C4⋊C4—42no—
237C3×C4⋊C4#4112yes—
238C3×C22⋊C4—42no—
239C3×C22⋊C4—42no—
240C3×C22⋊C4—42no—
241C3×C22⋊C4—42no—
242C3×C22⋊C4—42no—
243C3×C22⋊C4—42no—
244C3×C42—24no—
245C3×C42—22no—
246C3×C42—22no—
247C3×C42—21no—
248C3×C42—21no—
249C3×C42—21no—
250C3×C42—21no—
251C3×C42—21no—
252C3×C42—21no—
253C3×C42—41no—
254C3×C42—41no—
255C3×C42—42no—
256C22×Q8C23×C311yescharacteristic
257C22×D4C23×C311yescharacteristic
258C2×SD16—22no—
259C2×SD16—22no—
260C2×SD16—41no—
261C22×C8C23×C311yescharacteristic
262C22×C8C23×C311yescharacteristic
263C22×C8C23×C311yescharacteristic
264C22×C8C23×C311yescharacteristic
265#126—44no—
266C4×Q8—22no—
267C4×Q8—22no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4C23×C311yescharacteristic
275C2×C4⋊C4C23×C311yescharacteristic
276C2×C4⋊C4C23×C311yescharacteristic
277C2×C4⋊C4C23×C311yescharacteristic
278C2×C4⋊C4C23×C311yescharacteristic
279C2×C4⋊C4C23×C311yescharacteristic
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42C3×D411yescharacteristic
289C2×C42C3×D411yescharacteristic
290C2×C42C3×D411yescharacteristic
291C2×C42C3×D411yescharacteristic
292C2×C42C23×C311yescharacteristic
293C2×C42C23×C311yescharacteristic
294C2×C42C23×C311yescharacteristic
295#107—22no—
296#107—22no—
297#106—41no—
298C4⋊C8—22no—
299C4⋊C8—22no—
300#103—22no—
301#103—22no—
302#103—41no—
303#102—22no—
304#102—22no—
305#102—41no—
306#97—41no—
307#97—41no—
308#95—21no—
309#95—21no—
310#95—21no—
311#95—21no—
312#95—21no—
313#95—21no—
314#95—21no—
315#95—21no—
316#95—21no—
317#95—21no—
318C23×C3—41no—
319C23×C3—41no—
320C23×C3—41no—
321C23×C3—41no—
322C23×C3—41no—
323C23×C3—41no—
324C23×C3—41no—
325C23×C3C22×D411yescharacteristic
326C3×Q8—22no—
327C3×Q8—22no—
328C3×Q8—41no—
329C3×Q8—41no—
330C3×D4—41no—
331C3×D4—41no—
332C3×D4—41no—
333C3×D4—41no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
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—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×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4AGL(1,ℤ/8)12yes—
372C2×C3×C4#13712yes—
373C2×C3×C4C2×#4111yescharacteristic
374C2×C3×C4C2×#4111yescharacteristic
375C2×C3×C4C2×#4111yescharacteristic
376C2×C3×C4C2×#4111yescharacteristic
377C2×C3×C4H(5,𝔽2)11yescharacteristic
378C2×C3×C42−1+411yescharacteristic
379C3×C8—22no—
380C3×C8—22no—
381C3×C8—41no—
382C3×C8—41no—
383C24—21no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8C3×#4112yes—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4C2×C3×D411yescharacteristic
404C22×C4C2×C3×D411yescharacteristic
405C22×C4C2×C3×D411yescharacteristic
406C22×C4C2×C3×D411yescharacteristic
407C22×C4C2×C3×D411yescharacteristic
408C22×C4C2×C3×D411yescharacteristic
409C22×C4C24×C311yescharacteristic
410SD16—44no—
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—
421C2×C8C3×#4112yes—
422C2×C8C3×#4112yes—
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—21no—
434C4⋊C4—21no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4C3×#4112yes—
441C22⋊C4—42no—
442C22⋊C4—42no—
443C22⋊C4—42no—
444C22⋊C4—42no—
445C22⋊C4—42no—
446C22⋊C4—42no—
447C42—24no—
448C42—22no—
449C42—22no—
450C42—21no—
451C42—21no—
452C42—21no—
453C42—21no—
454C42—21no—
455C42—21no—
456C42—41no—
457C42—41no—
458C42—42no—
459C22×C3—81no—
460C22×C3—81no—
461C22×C3—81no—
462C22×C3—81no—
463C22×C3—81no—
464C22×C3—81no—
465C22×C3—81no—
466C22×C3#52511yescharacteristic
467C22×C3#53211yescharacteristic
468C22×C3#53711yescharacteristic
469C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
470C22×C3C2×#13711yescharacteristic
471C22×C3#57511yescharacteristic
472C22×C3#57511yescharacteristic
473C3×C4—22no—
474C3×C4—22no—
475C3×C4—42no—
476C3×C4—41no—
477C3×C4—42no—
478C3×C4—41no—
479C3×C4—41no—
480C3×C4—41no—
481C3×C4—81no—
482C3×C4—81no—
483C3×C4—81no—
484C3×C4—81no—
485C3×C4—81no—
486C3×C4#55111yescharacteristic
487C3×C4#55211yescharacteristic
488C3×C4#55311yescharacteristic
489C3×C4#55611yescharacteristic
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23C22×C3×D411yescharacteristic
498Q8—22no—
499Q8—22no—
500Q8—41no—
501Q8—41no—
502D4—41no—
503D4—41no—
504D4—41no—
505D4—41no—
506C2×C4—21no—
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—22no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
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×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4C3×AGL(1,ℤ/8)12yes—
544C2×C4C3×#13712yes—
545C2×C4C2×C3×#4111yescharacteristic
546C2×C4C2×C3×#4111yescharacteristic
547C2×C4C2×C3×#4111yescharacteristic
548C2×C4C2×C3×#4111yescharacteristic
549C2×C4C3×H(5,𝔽2)11yescharacteristic
550C2×C4C3×2−1+411yescharacteristic
551C8—22no—
552C8—22no—
553C8—41no—
554C8—41no—
555C2×C3—161no—
556C2×C3#260111yescharacteristic
557C2×C3#311511yescharacteristic
558C2×C3#311711yescharacteristic
559C2×C3#317211yescharacteristic
560C2×C3#318011yescharacteristic
561C2×C3#325411yescharacteristic
562C2×C3#325611yescharacteristic
563C22—81no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22—81no—
570C22C3×#52511yescharacteristic
571C22C3×#53211yescharacteristic
572C22C3×#53711yescharacteristic
573C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
574C22C2×C3×#13711yescharacteristic
575C22C3×#57511yescharacteristic
576C22C3×#57511yescharacteristic
577C4—22no—
578C4—22no—
579C4—41no—
580C4—41no—
581C4—41no—
582C4—42no—
583C4—41no—
584C4—42no—
585C4—81no—
586C4—81no—
587C4—81no—
588C4—81no—
589C4—81no—
590C4C3×#55111yescharacteristic
591C4C3×#55211yescharacteristic
592C4C3×#55311yescharacteristic
593C4C3×#55611yescharacteristic
594C3#2945111yescharacteristic, direct product factor, Sylow
595C2—161no—
596C2C3×#260111yescharacteristic
597C2C3×#311511yescharacteristic
598C2C3×#311711yescharacteristic
599C2C3×#317211yescharacteristic
600C2C3×#318011yescharacteristic
601C2C3×#325411yescharacteristic
602C2C3×#325611yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×C43 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#2295C2semidirect
3. C3×#2086C2semidirect
5. C3×#2066C2semidirect
8. C3×#1943C2semidirect
10. C3×#1935C2semidirect
12. C3×#1921C2semidirect
14. C3×#1863C2semidirect
15. C3×#1767C2semidirect
16. #29451C3direct (N × H)
61. #2295C2×C3semidirect
62. #2086C2×C3semidirect
64. #2066C2×C3semidirect
67. #1943C2×C3semidirect
69. #1935C2×C3semidirect
71. #1921C2×C3semidirect
73. #1863C2×C3semidirect
74. #1767C2×C3semidirect
594. C3#29451direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#28602
C3×#22952
C3×#20862
C3×#20672
C3×#20662
C3×#20552
C3×#19732
C3×#19432
C3×#19362
C3×#19352
C3×#19262
C3×#19212
C3×#18802
C3×#18632
C3×#17672
#294513