Finite Group Databasebeta

← C3×#29523 #10648130

Subgroups — C3×#29523 #10648130

Order 768.

670 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2860C211yesmaximal, characteristic
2C3×C4×#125C211yesmaximal, characteristic, complemented
3C3×#2101C211yesmaximal, characteristic, complemented
4C3×#2092C211yesmaximal, characteristic
5C3×#2059C211yesmaximal, characteristic, complemented
6C3×#2025C211yesmaximal, characteristic
7C3×#1944C211yesmaximal, characteristic, complemented
8C3×#1943C211yesmaximal, characteristic, complemented
9C3×#1936C211yesmaximal, characteristic
10C3×#1936C211yesmaximal, characteristic
11C3×#1926C211yesmaximal, characteristic
12C3×#1925C211yesmaximal, characteristic
13C3×#1919C211yesmaximal, characteristic, complemented
14C3×#1918C211yesmaximal, characteristic, complemented
15C3×#1745C211yesmaximal, characteristic, complemented
16#29523C311yesmaximal, characteristic, direct product factor, Sylow
17C2×C3×#126—21no—
18C2×C3×#125C2211yescharacteristic
19C2×C3×#107C2211yescharacteristic
20C2×C3×#107C2211yescharacteristic
21C2×C3×#106C2211yescharacteristic
22C2×C3×#106C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C2×C3×#97C2211yescharacteristic
28C2×C3×#97C2211yescharacteristic
29C3×#390C2211yescharacteristic
30C3×#390C2211yescharacteristic
31C3×#390C2211yescharacteristic
32C3×#389C2211yescharacteristic
33C3×#389C2211yescharacteristic
34C3×#389C2211yescharacteristic
35C3×#389C2211yescharacteristic
36C3×#389C2211yescharacteristic
37C3×#388—21no—
38C3×#388—21no—
39C3×#388—21no—
40C3×#387—21no—
41C3×#387—21no—
42C3×#382C2211yescharacteristic
43C3×C4×C4⋊C4—21no—
44C3×C4×C4⋊C4—21no—
45C3×C4×C4⋊C4C2211yescharacteristic
46C3×C4×C4⋊C4C2211yescharacteristic
47C3×C43C2211yescharacteristic
48C3×#340C2211yescharacteristic
49C3×#340C2211yescharacteristic
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#336C2211yescharacteristic
57C3×#336C2211yescharacteristic
58C3×#336C2211yescharacteristic
59C3×#336C2211yescharacteristic
60#2860C2×C311yescharacteristic
61C4×#125C2×C311yescharacteristic, complemented
62#2101C2×C311yescharacteristic, complemented
63#2092C2×C311yescharacteristic
64#2059C2×C311yescharacteristic, complemented
65#2025C2×C311yescharacteristic
66#1944C2×C311yescharacteristic, complemented
67#1943C2×C311yescharacteristic, complemented
68#1936C2×C311yescharacteristic
69#1936C2×C311yescharacteristic
70#1926C2×C311yescharacteristic
71#1925C2×C311yescharacteristic
72#1919C2×C311yescharacteristic, complemented
73#1918C2×C311yescharacteristic, complemented
74#1745C2×C311yescharacteristic, complemented
75C22×C3×D4C2311yescharacteristic
76C22×C3×C8C2311yescharacteristic
77C22×C3×C8C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C3×#126—44no—
81C3×#125—22no—
82C3×#125—22no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C22⋊C4—21no—
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×C42—21no—
104C2×C3×C42—21no—
105C2×C3×C42D411yescharacteristic
106C2×C3×C42D411yescharacteristic
107C2×C3×C42D411yescharacteristic
108C2×C3×C42D411yescharacteristic
109C2×C3×C42C2311yescharacteristic
110C2×C3×C42C2311yescharacteristic
111C2×C3×C42C2311yescharacteristic
112C3×#107—22no—
113C3×#107—22no—
114C3×#106—22no—
115C3×#106—22no—
116C3×#102—22no—
117C3×#102—22no—
118C3×#102—22no—
119C3×#102—22no—
120C3×#97—22no—
121C3×#97—22no—
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—
131C2×#126—21no—
132C2×#125C22×C311yescharacteristic
133C2×#107C22×C311yescharacteristic
134C2×#107C22×C311yescharacteristic
135C2×#106C22×C311yescharacteristic
136C2×#106C22×C311yescharacteristic
137C2×#102C22×C311yescharacteristic
138C2×#102C22×C311yescharacteristic
139C2×#102C22×C311yescharacteristic
140C2×#102C22×C311yescharacteristic
141C2×#97C22×C311yescharacteristic
142C2×#97C22×C311yescharacteristic
143#390C22×C311yescharacteristic
144#390C22×C311yescharacteristic
145#390C22×C311yescharacteristic
146#389C22×C311yescharacteristic
147#389C22×C311yescharacteristic
148#389C22×C311yescharacteristic
149#389C22×C311yescharacteristic
150#389C22×C311yescharacteristic
151#388—21no—
152#388—21no—
153#388—21no—
154#387—21no—
155#387—21no—
156#382C22×C311yescharacteristic
157C4×C4⋊C4—21no—
158C4×C4⋊C4—21no—
159C4×C4⋊C4C22×C311yescharacteristic
160C4×C4⋊C4C22×C311yescharacteristic
161C43C22×C311yescharacteristic
162#340C22×C311yescharacteristic
163#340C22×C311yescharacteristic
164#340C22×C311yescharacteristic
165#340C22×C311yescharacteristic
166#340C22×C311yescharacteristic
167#340C22×C311yescharacteristic
168#340C22×C311yescharacteristic
169#340C22×C311yescharacteristic
170#336C22×C311yescharacteristic
171#336C22×C311yescharacteristic
172#336C22×C311yescharacteristic
173#336C22×C311yescharacteristic
174C24×C3—21no—
175C2×C3×D4—21no—
176C2×C3×D4—21no—
177C2×C3×D4—21no—
178C2×C3×D4—21no—
179C2×C3×D4—21no—
180C2×C3×D4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2411yescharacteristic
195C2×C3×C8—21no—
196C2×C3×C8—21no—
197C2×C3×C8—21no—
198C2×C3×C8—21no—
199C2×C3×C8—21no—
200C2×C3×C8—21no—
201C2×C3×C8—21no—
202C2×C3×C8—21no—
203C2×C3×C8—21no—
204C2×C3×C8—21no—
205C2×C3×C8#4112yes—
206C2×C3×C8#4112yes—
207C3×C4⋊C4—24no—
208C3×C4⋊C4—24no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—42no—
222C3×C4⋊C4—42no—
223C3×C4⋊C4—42no—
224C3×C4⋊C4—42no—
225C3×C4⋊C4—42no—
226C3×C4⋊C4#4112yes—
227C3×C4⋊C4#4112yes—
228C3×C22⋊C4—42no—
229C3×C22⋊C4—42no—
230C3×C22⋊C4—42no—
231C3×C22⋊C4—42no—
232C3×C22⋊C4—42no—
233C3×C22⋊C4—42no—
234C3×C42—22no—
235C3×C42—21no—
236C3×C42—21no—
237C3×C42—21no—
238C3×C42—21no—
239C3×C42—21no—
240C3×C42—21no—
241C3×C42—41no—
242C3×C42—41no—
243C3×C42—41no—
244C3×C42—42no—
245C3×C42—42no—
246C22×D4C23×C311yescharacteristic
247C22×C8C23×C311yescharacteristic
248C22×C8C23×C311yescharacteristic
249C22×C8C23×C311yescharacteristic
250C22×C8C23×C311yescharacteristic
251#126—44no—
252#125—22no—
253#125—22no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4C23×C311yescharacteristic
262C2×C4⋊C4C23×C311yescharacteristic
263C2×C4⋊C4C23×C311yescharacteristic
264C2×C4⋊C4C23×C311yescharacteristic
265C2×C4⋊C4C23×C311yescharacteristic
266C2×C4⋊C4C23×C311yescharacteristic
267C2×C4⋊C4C23×C311yescharacteristic
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42C3×D411yescharacteristic
277C2×C42C3×D411yescharacteristic
278C2×C42C3×D411yescharacteristic
279C2×C42C3×D411yescharacteristic
280C2×C42C23×C311yescharacteristic
281C2×C42C23×C311yescharacteristic
282C2×C42C23×C311yescharacteristic
283#107—22no—
284#107—22no—
285#106—22no—
286#106—22no—
287#102—22no—
288#102—22no—
289#102—22no—
290#102—22no—
291#97—22no—
292#97—22no—
293#95—21no—
294#95—21no—
295#95—21no—
296#95—21no—
297#95—21no—
298#95—21no—
299#95—21no—
300#95—21no—
301#95—21no—
302C23×C3—41no—
303C23×C3—41no—
304C23×C3—41no—
305C23×C3—41no—
306C23×C3—41no—
307C23×C3—41no—
308C23×C3—41no—
309C23×C3C22×D411yescharacteristic
310C3×D4—41no—
311C3×D4—41no—
312C3×D4—41no—
313C3×D4—41no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—21no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4AGL(1,ℤ/8)12yes—
353C2×C3×C4C2×#4111yescharacteristic
354C2×C3×C4C2×#4111yescharacteristic
355C2×C3×C4C2×#4111yescharacteristic
356C2×C3×C4C2×#4111yescharacteristic
357C2×C3×C42−1+411yescharacteristic
358C2×C3×C42−1+411yescharacteristic
359C3×C8—22no—
360C3×C8—22no—
361C3×C8—41no—
362C3×C8—41no—
363C24—21no—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4C2×C3×D411yescharacteristic
378C22×C4C2×C3×D411yescharacteristic
379C22×C4C2×C3×D411yescharacteristic
380C22×C4C2×C3×D411yescharacteristic
381C22×C4C2×C3×D411yescharacteristic
382C22×C4C2×C3×D411yescharacteristic
383C22×C4C24×C311yescharacteristic
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8C3×#4112yes—
395C2×C8C3×#4112yes—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4C3×#4112yes—
416C4⋊C4C3×#4112yes—
417C22⋊C4—42no—
418C22⋊C4—42no—
419C22⋊C4—42no—
420C22⋊C4—42no—
421C22⋊C4—42no—
422C22⋊C4—42no—
423C42—22no—
424C42—21no—
425C42—21no—
426C42—21no—
427C42—21no—
428C42—21no—
429C42—21no—
430C42—41no—
431C42—41no—
432C42—41no—
433C42—42no—
434C42—42no—
435C22×C3—81no—
436C22×C3—81no—
437C22×C3—81no—
438C22×C3—81no—
439C22×C3—81no—
440C22×C3—81no—
441C22×C3—81no—
442C22×C3#52511yescharacteristic
443C22×C3#53211yescharacteristic
444C22×C3#53611yescharacteristic
445C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
446C22×C3#57511yescharacteristic
447C22×C3#57511yescharacteristic
448C22×C3#57511yescharacteristic
449C3×C4—22no—
450C3×C4—41no—
451C3×C4—42no—
452C3×C4—41no—
453C3×C4—42no—
454C3×C4—41no—
455C3×C4—41no—
456C3×C4—41no—
457C3×C4—81no—
458C3×C4—81no—
459C3×C4—81no—
460C3×C4—81no—
461C3×C4—81no—
462C3×C4#55211yescharacteristic
463C3×C4#55211yescharacteristic
464C3×C4#55611yescharacteristic
465C3×C4#55611yescharacteristic
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23C22×C3×D411yescharacteristic
474D4—41no—
475D4—41no—
476D4—41no—
477D4—41no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4C3×AGL(1,ℤ/8)12yes—
517C2×C4C2×C3×#4111yescharacteristic
518C2×C4C2×C3×#4111yescharacteristic
519C2×C4C2×C3×#4111yescharacteristic
520C2×C4C2×C3×#4111yescharacteristic
521C2×C4C3×2−1+411yescharacteristic
522C2×C4C3×2−1+411yescharacteristic
523C8—22no—
524C8—22no—
525C8—41no—
526C8—41no—
527C2×C3—161no—
528C2×C3#260411yescharacteristic
529C2×C3#311111yescharacteristic
530C2×C3#311511yescharacteristic
531C2×C3#317111yescharacteristic
532C2×C3#318311yescharacteristic
533C2×C3#323411yescharacteristic
534C2×C3#324011yescharacteristic
535C22—81no—
536C22—81no—
537C22—81no—
538C22—81no—
539C22—81no—
540C22—81no—
541C22—81no—
542C22C3×#52511yescharacteristic
543C22C3×#53211yescharacteristic
544C22C3×#53611yescharacteristic
545C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
546C22C3×#57511yescharacteristic
547C22C3×#57511yescharacteristic
548C22C3×#57511yescharacteristic
549C4—22no—
550C4—41no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4—42no—
555C4—41no—
556C4—42no—
557C4—81no—
558C4—81no—
559C4—81no—
560C4—81no—
561C4—81no—
562C4C3×#55211yescharacteristic
563C4C3×#55211yescharacteristic
564C4C3×#55611yescharacteristic
565C4C3×#55611yescharacteristic
566C3#2952311yescharacteristic, direct product factor, Sylow
567C2—161no—
568C2C3×#260411yescharacteristic
569C2C3×#311111yescharacteristic
570C2C3×#311511yescharacteristic
571C2C3×#317111yescharacteristic
572C2C3×#318311yescharacteristic
573C2C3×#323411yescharacteristic
574C2C3×#324011yescharacteristic
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×C4×#125C2semidirect
3. C3×#2101C2semidirect
5. C3×#2059C2semidirect
7. C3×#1944C2semidirect
8. C3×#1943C2semidirect
13. C3×#1919C2semidirect
14. C3×#1918C2semidirect
15. C3×#1745C2semidirect
16. #29523C3direct (N × H)
61. C4×#125C2×C3semidirect
62. #2101C2×C3semidirect
64. #2059C2×C3semidirect
66. #1944C2×C3semidirect
67. #1943C2×C3semidirect
72. #1919C2×C3semidirect
73. #1918C2×C3semidirect
74. #1745C2×C3semidirect
566. C3#29523direct (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×C4×#1252
C3×#21012
C3×#20922
C3×#20592
C3×#20252
C3×#19442
C3×#19432
C3×#19362
C3×#19262
C3×#19252
C3×#19192
C3×#19182
C3×#17452
#295233