Finite Group Databasebeta

← C3×#27641 #10646248

Subgroups — C3×#27641 #10646248

Order 768.

730 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2324C211yesmaximal, characteristic
2C3×#2314C211yesmaximal, characteristic, complemented
3C3×#2077C211yesmaximal, characteristic, complemented
4C3×#2077C211yesmaximal, characteristic, complemented
5C3×#2034C211yesmaximal, characteristic
6C3×#2033C211yesmaximal, characteristic
7C3×#1949C211yesmaximal, characteristic, complemented
8C3×#1948C211yesmaximal, characteristic, complemented
9C3×#1926C211yesmaximal, characteristic, complemented
10C3×#1926C211yesmaximal, characteristic, complemented
11C3×#1920C211yesmaximal, characteristic, complemented
12C3×#1918C211yesmaximal, characteristic, complemented
13C3×#1804C211yesmaximal, characteristic, complemented
14C3×#1802C211yesmaximal, characteristic, complemented
15C3×#1798C211yesmaximal, characteristic, complemented
16#27641C311yesmaximal, characteristic, direct product factor, Sylow
17C2×C3×#121C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C2×C3×#117C2211yescharacteristic
20C2×C3×#107C2211yescharacteristic
21C2×C3×#106C2211yescharacteristic
22C2×C3×#106C2211yescharacteristic
23C2×C3×#106C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C2×C3×#102C2211yescharacteristic
28C2×C3×#102C2211yescharacteristic
29C2×C3×#102C2211yescharacteristic
30C2×C3×C22⋊C8C2211yescharacteristic
31C2×C3×C22⋊C8C2211yescharacteristic
32C3×#390—21no—
33C3×#390C2211yescharacteristic
34C3×#389C2211yescharacteristic
35C3×#388—21no—
36C3×#388—21no—
37C3×#388—21no—
38C3×#387—21no—
39C3×#387—21no—
40C3×#386—21no—
41C3×#385—21no—
42C3×#384C2211yescharacteristic
43C3×#382C2211yescharacteristic
44C3×#380C2211yescharacteristic
45C3×#380C2211yescharacteristic
46C3×#380C2211yescharacteristic
47C3×C4×C4⋊C4C2211yescharacteristic
48C3×C4×C4⋊C4C2211yescharacteristic
49C3×C4×C4⋊C4C2211yescharacteristic
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#340C2211yescharacteristic
57C3×#340C2211yescharacteristic
58C3×#336C2211yescharacteristic
59C3×#336C2211yescharacteristic
60#2324C2×C311yescharacteristic
61#2314C2×C311yescharacteristic, complemented
62#2077C2×C311yescharacteristic, complemented
63#2077C2×C311yescharacteristic, complemented
64#2034C2×C311yescharacteristic
65#2033C2×C311yescharacteristic
66#1949C2×C311yescharacteristic, complemented
67#1948C2×C311yescharacteristic, complemented
68#1926C2×C311yescharacteristic, complemented
69#1926C2×C311yescharacteristic, complemented
70#1920C2×C311yescharacteristic, complemented
71#1918C2×C311yescharacteristic, complemented
72#1804C2×C311yescharacteristic, complemented
73#1802C2×C311yescharacteristic, complemented
74#1798C2×C311yescharacteristic, complemented
75C22×C3×D4—21no—
76C22×C3×D4C2311yescharacteristic
77C23×C3×C4—21no—
78C23×C3×C4C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C3×#121—22no—
84C3×#121—22no—
85C3×C4×D4—22no—
86C3×C4×D4—22no—
87C3×#117—22no—
88C3×#117—22no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4D411yescharacteristic
93C2×C3×C4⋊C4D411yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C22⋊C4—21no—
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×C22⋊C4D411yescharacteristic
109C2×C3×C22⋊C4D411yescharacteristic
110C2×C3×C42—21no—
111C2×C3×C42—21no—
112C2×C3×C42—21no—
113C2×C3×C42C2311yescharacteristic
114C2×C3×C42C2311yescharacteristic
115C3×#107—22no—
116C3×#106—22no—
117C3×#106—41no—
118C3×#106—41no—
119C3×C4⋊C8—22no—
120C3×C4⋊C8—22no—
121C3×#102—22no—
122C3×#102—22no—
123C3×#102—41no—
124C3×#102—41no—
125C3×C22⋊C8—41no—
126C3×C22⋊C8—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—
136C2×#121C22×C311yescharacteristic
137C2×C4×D4C22×C311yescharacteristic
138C2×#117C22×C311yescharacteristic
139C2×#107C22×C311yescharacteristic
140C2×#106C22×C311yescharacteristic
141C2×#106C22×C311yescharacteristic
142C2×#106C22×C311yescharacteristic
143C2×C4⋊C8C22×C311yescharacteristic
144C2×C4⋊C8C22×C311yescharacteristic
145C2×#102C22×C311yescharacteristic
146C2×#102C22×C311yescharacteristic
147C2×#102C22×C311yescharacteristic
148C2×#102C22×C311yescharacteristic
149C2×C22⋊C8C22×C311yescharacteristic
150C2×C22⋊C8C22×C311yescharacteristic
151#390—21no—
152#390C22×C311yescharacteristic
153#389C22×C311yescharacteristic
154#388—21no—
155#388—21no—
156#388—21no—
157#387—21no—
158#387—21no—
159#386—21no—
160#385—21no—
161#384C22×C311yescharacteristic
162#382C22×C311yescharacteristic
163#380C22×C311yescharacteristic
164#380C22×C311yescharacteristic
165#380C22×C311yescharacteristic
166C4×C4⋊C4C22×C311yescharacteristic
167C4×C4⋊C4C22×C311yescharacteristic
168C4×C4⋊C4C22×C311yescharacteristic
169#340C22×C311yescharacteristic
170#340C22×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—
180C24×C3C2×D411yescharacteristic
181C2×C3×D4—24no—
182C2×C3×D4—21no—
183C2×C3×D4—21no—
184C2×C3×D4—21no—
185C2×C3×D4—21no—
186C2×C3×D4—21no—
187C2×C3×D4—81no—
188C2×C3×D4—81no—
189C2×C3×D4#4112yes—
190C22×C3×C4—21no—
191C22×C3×C4—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×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—42no—
204C22×C3×C4—81no—
205C22×C3×C4—81no—
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2411yescharacteristic
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—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C3×C4⋊C4—22no—
225C3×C4⋊C4—22no—
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—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—81no—
240C3×C4⋊C4—81no—
241C3×C4⋊C4—81no—
242C3×C4⋊C4#4112yes—
243C3×C22⋊C4—24no—
244C3×C22⋊C4—22no—
245C3×C22⋊C4—22no—
246C3×C22⋊C4—42no—
247C3×C22⋊C4—81no—
248C3×C22⋊C4—81no—
249C3×C22⋊C4—81no—
250C3×C22⋊C4—81no—
251C3×C22⋊C4—81no—
252C3×C42—21no—
253C3×C42—21no—
254C3×C42—81no—
255C3×C42—81no—
256C3×C42—81no—
257C3×C42#4112yes—
258C3×C42#4112yes—
259C22×D4—21no—
260C22×D4C23×C311yescharacteristic
261C23×C4—21no—
262C23×C4C23×C311yescharacteristic
263C22×C8C23×C311yescharacteristic
264C22×C8C23×C311yescharacteristic
265C22×C8C23×C311yescharacteristic
266C22×C8C23×C311yescharacteristic
267#121—22no—
268#121—22no—
269C4×D4—22no—
270C4×D4—22no—
271#117—22no—
272#117—22no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4C3×D411yescharacteristic
277C2×C4⋊C4C3×D411yescharacteristic
278C2×C4⋊C4C23×C311yescharacteristic
279C2×C4⋊C4C23×C311yescharacteristic
280C2×C4⋊C4C23×C311yescharacteristic
281C2×C4⋊C4C23×C311yescharacteristic
282C2×C4⋊C4C23×C311yescharacteristic
283C2×C4⋊C4C23×C311yescharacteristic
284C2×C4⋊C4C23×C311yescharacteristic
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4C3×D411yescharacteristic
293C2×C22⋊C4C3×D411yescharacteristic
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42C23×C311yescharacteristic
298C2×C42C23×C311yescharacteristic
299#107—22no—
300#106—22no—
301#106—41no—
302#106—41no—
303C4⋊C8—22no—
304C4⋊C8—22no—
305#102—22no—
306#102—22no—
307#102—41no—
308#102—41no—
309C22⋊C8—41no—
310C22⋊C8—41no—
311#95—21no—
312#95—21no—
313#95—21no—
314#95—21no—
315#95—21no—
316#95—21no—
317#95—21no—
318#95—21no—
319#95—21no—
320C23×C3—21no—
321C23×C3—22no—
322C23×C3—21no—
323C23×C3—21no—
324C23×C3—21no—
325C23×C3—21no—
326C23×C3—21no—
327C23×C3—21no—
328C23×C3—41no—
329C23×C3—41no—
330C23×C3—41no—
331C23×C3—41no—
332C23×C3—41no—
333C23×C3—41no—
334C23×C3C22×D411yescharacteristic
335C3×D4—22no—
336C3×D4—22no—
337C3×D4—41no—
338C3×D4—41no—
339C3×D4—84no—
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—22no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
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—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—84no—
379C2×C3×C4#12312yes—
380C2×C3×C4#12312yes—
381C2×C3×C4C2×#4111yescharacteristic
382C2×C3×C4C2×#4111yescharacteristic
383C2×C3×C4C2×#4111yescharacteristic
384C2×C3×C4C2×#4111yescharacteristic
385C2×C3×C4H(5,𝔽2)11yescharacteristic
386C2×C3×C42−1+411yescharacteristic
387C3×C8—41no—
388C3×C8—41no—
389C3×C8—41no—
390C3×C8—41no—
391C24—21no—
392C24C2×C3×D411yescharacteristic
393C2×D4—24no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—81no—
400C2×D4—81no—
401C2×D4C3×#4112yes—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—42no—
416C22×C4—81no—
417C22×C4—81no—
418C22×C4C2×C3×D411yescharacteristic
419C22×C4C2×C3×D411yescharacteristic
420C22×C4C2×C3×D411yescharacteristic
421C22×C4C2×C3×D411yescharacteristic
422C22×C4C2×C3×D411yescharacteristic
423C22×C4C24×C311yescharacteristic
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—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—81no—
452C4⋊C4—81no—
453C4⋊C4—81no—
454C4⋊C4C3×#4112yes—
455C22⋊C4—24no—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—42no—
459C22⋊C4—81no—
460C22⋊C4—81no—
461C22⋊C4—81no—
462C22⋊C4—81no—
463C22⋊C4—81no—
464C42—21no—
465C42—21no—
466C42—81no—
467C42—81no—
468C42—81no—
469C42C3×#4112yes—
470C42C3×#4112yes—
471C22×C3—41no—
472C22×C3—41no—
473C22×C3—41no—
474C22×C3—42no—
475C22×C3—41no—
476C22×C3—42no—
477C22×C3—41no—
478C22×C3—42no—
479C22×C3—41no—
480C22×C3—41no—
481C22×C3—81no—
482C22×C3—81no—
483C22×C3—81no—
484C22×C3—81no—
485C22×C3C2×#12311yescharacteristic
486C22×C3#52511yescharacteristic
487C22×C3#53711yescharacteristic
488C22×C3#57611yescharacteristic
489C22×C3#57711yescharacteristic
490C22×C3#57711yescharacteristic
491C22×C3#57711yescharacteristic
492C3×C4—22no—
493C3×C4—22no—
494C3×C4—41no—
495C3×C4—41no—
496C3×C4—42no—
497C3×C4—41no—
498C3×C4—81no—
499C3×C4—81no—
500C3×C4—81no—
501C3×C4—81no—
502C3×C4—81no—
503C3×C4—81no—
504C3×C4#55211yescharacteristic
505C3×C4#55311yescharacteristic
506C3×C4#55511yescharacteristic
507C3×C4#55611yescharacteristic
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—22no—
514C23—21no—
515C23—21no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23C22×C3×D411yescharacteristic
523D4—22no—
524D4—22no—
525D4—41no—
526D4—41no—
527D4—84no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—84no—
567C2×C4C3×#12312yes—
568C2×C4C3×#12312yes—
569C2×C4C2×C3×#4111yescharacteristic
570C2×C4C2×C3×#4111yescharacteristic
571C2×C4C2×C3×#4111yescharacteristic
572C2×C4C2×C3×#4111yescharacteristic
573C2×C4C3×H(5,𝔽2)11yescharacteristic
574C2×C4C3×2−1+411yescharacteristic
575C8—41no—
576C8—41no—
577C8—41no—
578C8—41no—
579C2×C3—81no—
580C2×C3—82no—
581C2×C3#248511yescharacteristic
582C2×C3#309611yescharacteristic
583C2×C3#309911yescharacteristic
584C2×C3#311811yescharacteristic
585C2×C3#312511yescharacteristic
586C2×C3#325711yescharacteristic
587C2×C3#326011yescharacteristic
588C22—41no—
589C22—42no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—42no—
596C22—41no—
597C22—42no—
598C22—81no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22C2×C3×#12311yescharacteristic
603C22C3×#52511yescharacteristic
604C22C3×#53711yescharacteristic
605C22C3×#57611yescharacteristic
606C22C3×#57711yescharacteristic
607C22C3×#57711yescharacteristic
608C22C3×#57711yescharacteristic
609C4—22no—
610C4—22no—
611C4—41no—
612C4—41no—
613C4—41no—
614C4—42no—
615C4—81no—
616C4—81no—
617C4—81no—
618C4—81no—
619C4—81no—
620C4—81no—
621C4C3×#55211yescharacteristic
622C4C3×#55311yescharacteristic
623C4C3×#55511yescharacteristic
624C4C3×#55611yescharacteristic
625C3#2764111yescharacteristic, direct product factor, Sylow
626C2—81no—
627C2—82no—
628C2C3×#248511yescharacteristic
629C2C3×#309611yescharacteristic
630C2C3×#309911yescharacteristic
631C2C3×#311811yescharacteristic
632C2C3×#312511yescharacteristic
633C2C3×#325711yescharacteristic
634C2C3×#326011yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#2314C2semidirect
3. C3×#2077C2semidirect
4. C3×#2077C2semidirect
7. C3×#1949C2semidirect
8. C3×#1948C2semidirect
9. C3×#1926C2semidirect
10. C3×#1926C2semidirect
11. C3×#1920C2semidirect
12. C3×#1918C2semidirect
13. C3×#1804C2semidirect
14. C3×#1802C2semidirect
15. C3×#1798C2semidirect
16. #27641C3direct (N × H)
61. #2314C2×C3semidirect
62. #2077C2×C3semidirect
63. #2077C2×C3semidirect
66. #1949C2×C3semidirect
67. #1948C2×C3semidirect
68. #1926C2×C3semidirect
69. #1926C2×C3semidirect
70. #1920C2×C3semidirect
71. #1918C2×C3semidirect
72. #1804C2×C3semidirect
73. #1802C2×C3semidirect
74. #1798C2×C3semidirect
625. C3#27641direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#23242
C3×#23142
C3×#20772
C3×#20342
C3×#20332
C3×#19492
C3×#19482
C3×#19262
C3×#19202
C3×#19182
C3×#18042
C3×#18022
C3×#17982
#276413