Finite Group Databasebeta

← C3×#27033 #10645640

Subgroups — C3×#27033 #10645640

Order 768.

784 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2494C211yesmaximal, characteristic, complemented
2C3×#2361C211yesmaximal, characteristic
3C3×#2073C211yesmaximal, characteristic, complemented
4C3×#2067C211yesmaximal, characteristic, complemented
5C3×#2034C211yesmaximal, characteristic
6C3×#2033C211yesmaximal, characteristic
7C3×#1949C211yesmaximal, characteristic, complemented
8C3×#1942C211yesmaximal, characteristic, complemented
9C3×#1923C211yesmaximal, characteristic, complemented
10C3×#1921C211yesmaximal, characteristic, complemented
11C3×#1919C211yesmaximal, characteristic, complemented
12C3×#1862C211yesmaximal, characteristic, complemented
13C3×#1852C211yesmaximal, characteristic, complemented
14C3×#1824C211yesmaximal, characteristic, complemented
15C3×C4×#102C211yesmaximal, characteristic, complemented
16#27033C311yesmaximal, characteristic, direct product factor, Sylow
17C2×C3×#121C2211yescharacteristic
18C2×C3×C4×D4—21no—
19C2×C3×C4×D4C2211yescharacteristic
20C22×C3×C4⋊C4C2211yescharacteristic
21C2×C3×#107C2211yescharacteristic
22C2×C3×#107C2211yescharacteristic
23C2×C3×#107C2211yescharacteristic
24C2×C3×#106C2211yescharacteristic
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C2×C3×#102C2211yescharacteristic
28C2×C3×#102C2211yescharacteristic
29C2×C3×#102C2211yescharacteristic
30C2×C3×C22⋊C8C2211yescharacteristic
31C2×C3×C22⋊C8C2211yescharacteristic
32C2×C3×C4×C8C2211yescharacteristic
33C3×#401—21no—
34C3×#397—21no—
35C3×#390C2211yescharacteristic
36C3×#389C2211yescharacteristic
37C3×#389C2211yescharacteristic
38C3×#389C2211yescharacteristic
39C3×#388—21no—
40C3×#388—21no—
41C3×#387—21no—
42C3×#386—21no—
43C3×#385—21no—
44C3×#382C2211yescharacteristic
45C3×#380C2211yescharacteristic
46C3×#380C2211yescharacteristic
47C3×C4×C4⋊C4C2211yescharacteristic
48C3×C4×C4⋊C4C2211yescharacteristic
49C3×C4×C22⋊C4C2211yescharacteristic
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#340C2211yescharacteristic
57C3×#340C2211yescharacteristic
58C3×#336C2211yescharacteristic
59C3×#336C2211yescharacteristic
60#2494C2×C311yescharacteristic, complemented
61#2361C2×C311yescharacteristic
62#2073C2×C311yescharacteristic, complemented
63#2067C2×C311yescharacteristic, complemented
64#2034C2×C311yescharacteristic
65#2033C2×C311yescharacteristic
66#1949C2×C311yescharacteristic, complemented
67#1942C2×C311yescharacteristic, complemented
68#1923C2×C311yescharacteristic, complemented
69#1921C2×C311yescharacteristic, complemented
70#1919C2×C311yescharacteristic, complemented
71#1862C2×C311yescharacteristic, complemented
72#1852C2×C311yescharacteristic, complemented
73#1824C2×C311yescharacteristic, complemented
74C4×#102C2×C311yescharacteristic, complemented
75C22×C3×D4—21no—
76C22×C3×D4C2311yescharacteristic
77C23×C3×C4—21no—
78C23×C3×C4—21no—
79C23×C3×C4C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C3×#121—22no—
85C3×#121—22no—
86C3×C4×D4—22no—
87C3×C4×D4—41no—
88C3×C4×D4—44no—
89C2×C3×C4⋊C4—22no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—41no—
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×C4⋊C4C2311yescharacteristic
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×C42C2311yescharacteristic
113C2×C3×C42C2311yescharacteristic
114C3×#107—22no—
115C3×#107—22no—
116C3×#107—41no—
117C3×#106—22no—
118C3×C4⋊C8—22no—
119C3×#102—22no—
120C3×#102—22no—
121C3×#102—22no—
122C3×#102—41no—
123C3×C22⋊C8—22no—
124C3×C22⋊C8—41no—
125C3×C4×C8—41no—
126C3×#95—21no—
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×#95D411yescharacteristic
136C3×#95D411yescharacteristic
137C2×#121C22×C311yescharacteristic
138C2×C4×D4—21no—
139C2×C4×D4C22×C311yescharacteristic
140C22×C4⋊C4C22×C311yescharacteristic
141C2×#107C22×C311yescharacteristic
142C2×#107C22×C311yescharacteristic
143C2×#107C22×C311yescharacteristic
144C2×#106C22×C311yescharacteristic
145C2×C4⋊C8C22×C311yescharacteristic
146C2×#102C22×C311yescharacteristic
147C2×#102C22×C311yescharacteristic
148C2×#102C22×C311yescharacteristic
149C2×#102C22×C311yescharacteristic
150C2×C22⋊C8C22×C311yescharacteristic
151C2×C22⋊C8C22×C311yescharacteristic
152C2×C4×C8C22×C311yescharacteristic
153#401—21no—
154#397—21no—
155#390C22×C311yescharacteristic
156#389C22×C311yescharacteristic
157#389C22×C311yescharacteristic
158#389C22×C311yescharacteristic
159#388—21no—
160#388—21no—
161#387—21no—
162#386—21no—
163#385—21no—
164#382C22×C311yescharacteristic
165#380C22×C311yescharacteristic
166#380C22×C311yescharacteristic
167C4×C4⋊C4C22×C311yescharacteristic
168C4×C4⋊C4C22×C311yescharacteristic
169C4×C22⋊C4C22×C311yescharacteristic
170#340C22×C311yescharacteristic
171#340C22×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—
181C24×C3C2×D411yescharacteristic
182C2×C3×D4—24no—
183C2×C3×D4—21no—
184C2×C3×D4—21no—
185C2×C3×D4—21no—
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4—42no—
189C2×C3×D4—42no—
190C2×C3×D4#4112yes—
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—21no—
204C22×C3×C4—42no—
205C22×C3×C4—42no—
206C22×C3×C4—42no—
207C22×C3×C4—42no—
208C22×C3×C4—81no—
209C22×C3×C4—81no—
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2411yescharacteristic
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#4112yes—
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—21no—
240C3×C4⋊C4—42no—
241C3×C4⋊C4—42no—
242C3×C4⋊C4—42no—
243C3×C4⋊C4—81no—
244C3×C4⋊C4#4112yes—
245C3×C4⋊C4#4112yes—
246C3×C22⋊C4—24no—
247C3×C22⋊C4—22no—
248C3×C22⋊C4—22no—
249C3×C22⋊C4—42no—
250C3×C22⋊C4—42no—
251C3×C22⋊C4—42no—
252C3×C22⋊C4—81no—
253C3×C22⋊C4—81no—
254C3×C42—21no—
255C3×C42—21no—
256C3×C42—21no—
257C3×C42—21no—
258C3×C42—42no—
259C3×C42—81no—
260C22×D4—21no—
261C22×D4C23×C311yescharacteristic
262C23×C4—21no—
263C23×C4—21no—
264C23×C4C23×C311yescharacteristic
265C22×C8C23×C311yescharacteristic
266C22×C8C23×C311yescharacteristic
267C22×C8C23×C311yescharacteristic
268C22×C8C23×C311yescharacteristic
269#121—22no—
270#121—22no—
271C4×D4—22no—
272C4×D4—41no—
273C4×D4—44no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—41no—
280C2×C4⋊C4C23×C311yescharacteristic
281C2×C4⋊C4C23×C311yescharacteristic
282C2×C4⋊C4C23×C311yescharacteristic
283C2×C4⋊C4C23×C311yescharacteristic
284C2×C4⋊C4C23×C311yescharacteristic
285C2×C4⋊C4C23×C311yescharacteristic
286C2×C4⋊C4C23×C311yescharacteristic
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4C3×D411yescharacteristic
294C2×C22⋊C4C3×D411yescharacteristic
295C2×C42—21no—
296C2×C42—21no—
297C2×C42C23×C311yescharacteristic
298C2×C42C23×C311yescharacteristic
299#107—22no—
300#107—22no—
301#107—41no—
302#106—22no—
303C4⋊C8—22no—
304#102—22no—
305#102—22no—
306#102—22no—
307#102—41no—
308C22⋊C8—22no—
309C22⋊C8—41no—
310C4×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—
320#95C3×D411yescharacteristic
321#95C3×D411yescharacteristic
322C23×C3—21no—
323C23×C3—21no—
324C23×C3—21no—
325C23×C3—21no—
326C23×C3—21no—
327C23×C3—21no—
328C23×C3—22no—
329C23×C3—41no—
330C23×C3—41no—
331C23×C3—41no—
332C23×C3—41no—
333C23×C3—41no—
334C23×C3—41no—
335C23×C3#13512yes—
336C23×C3C22×D411yescharacteristic
337C3×D4—22no—
338C3×D4—22no—
339C3×D4—48no—
340C3×D4—41no—
341C3×D4—41no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—22no—
346C2×C3×C4—21no—
347C2×C3×C4—22no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
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—41no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—84no—
382C2×C3×C4—84no—
383C2×C3×C4AGL(1,ℤ/8)12yes—
384C2×C3×C4C2×#4111yescharacteristic
385C2×C3×C4C2×#4111yescharacteristic
386C2×C3×C4C2×#4111yescharacteristic
387C2×C3×C4C2×#4111yescharacteristic
388C2×C3×C4H(5,𝔽2)11yescharacteristic
389C2×C3×C42−1+411yescharacteristic
390C3×C8—22no—
391C3×C8—41no—
392C3×C8—41no—
393C3×C8—41no—
394C24—21no—
395C24C2×C3×D411yescharacteristic
396C2×D4—24no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4C3×#4112yes—
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—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—81no—
423C22×C4—81no—
424C22×C4C2×C3×D411yescharacteristic
425C22×C4C2×C3×D411yescharacteristic
426C22×C4C2×C3×D411yescharacteristic
427C22×C4C2×C3×D411yescharacteristic
428C22×C4C2×C3×D411yescharacteristic
429C22×C4C24×C311yescharacteristic
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8C3×#4112yes—
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—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—81no—
458C4⋊C4C3×#4112yes—
459C4⋊C4C3×#4112yes—
460C22⋊C4—24no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—42no—
464C22⋊C4—42no—
465C22⋊C4—42no—
466C22⋊C4—81no—
467C22⋊C4—81no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—42no—
473C42—81no—
474C22×C3—22no—
475C22×C3—22no—
476C22×C3—22no—
477C22×C3—41no—
478C22×C3—41no—
479C22×C3—41no—
480C22×C3—41no—
481C22×C3—42no—
482C22×C3—42no—
483C22×C3—42no—
484C22×C3—81no—
485C22×C3—81no—
486C22×C3—81no—
487C22×C3—81no—
488C22×C3#52511yescharacteristic
489C22×C3#54611yescharacteristic
490C22×C3#54711yescharacteristic
491C22×C3C2×#13511yescharacteristic
492C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
493C22×C3#57611yescharacteristic
494C22×C3#57811yescharacteristic
495C3×C4—22no—
496C3×C4—41no—
497C3×C4—41no—
498C3×C4—42no—
499C3×C4—41no—
500C3×C4—41no—
501C3×C4—42no—
502C3×C4—81no—
503C3×C4—81no—
504C3×C4—81no—
505C3×C4—81no—
506C3×C4—81no—
507C3×C4#52911yescharacteristic
508C3×C4#53911yescharacteristic
509C3×C4#55311yescharacteristic
510C3×C4#55611yescharacteristic
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—22no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23C3×#13512yes—
525C23C22×C3×D411yescharacteristic
526D4—22no—
527D4—22no—
528D4—48no—
529D4—41no—
530D4—41no—
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—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
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—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—84no—
571C2×C4—84no—
572C2×C4C3×AGL(1,ℤ/8)12yes—
573C2×C4C2×C3×#4111yescharacteristic
574C2×C4C2×C3×#4111yescharacteristic
575C2×C4C2×C3×#4111yescharacteristic
576C2×C4C2×C3×#4111yescharacteristic
577C2×C4C3×H(5,𝔽2)11yescharacteristic
578C2×C4C3×2−1+411yescharacteristic
579C8—22no—
580C8—41no—
581C8—41no—
582C8—41no—
583C2×C3—42no—
584C2×C3—82no—
585C2×C3#246611yescharacteristic
586C2×C3#310511yescharacteristic
587C2×C3#312411yescharacteristic
588C2×C3#330011yescharacteristic
589C2×C3#331811yescharacteristic
590C2×C3#333911yescharacteristic
591C2×C3#335311yescharacteristic
592C22—22no—
593C22—22no—
594C22—22no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—81no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22C3×#52511yescharacteristic
607C22C3×#54611yescharacteristic
608C22C3×#54711yescharacteristic
609C22C2×C3×#13511yescharacteristic
610C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
611C22C3×#57611yescharacteristic
612C22C3×#57811yescharacteristic
613C4—22no—
614C4—41no—
615C4—41no—
616C4—42no—
617C4—41no—
618C4—41no—
619C4—42no—
620C4—81no—
621C4—81no—
622C4—81no—
623C4—81no—
624C4—81no—
625C4C3×#52911yescharacteristic
626C4C3×#53911yescharacteristic
627C4C3×#55311yescharacteristic
628C4C3×#55611yescharacteristic
629C3#2703311yescharacteristic, direct product factor, Sylow
630C2—42no—
631C2—82no—
632C2C3×#246611yescharacteristic
633C2C3×#310511yescharacteristic
634C2C3×#312411yescharacteristic
635C2C3×#330011yescharacteristic
636C2C3×#331811yescharacteristic
637C2C3×#333911yescharacteristic
638C2C3×#335311yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×C4×C22⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#2494C2semidirect
3. C3×#2073C2semidirect
4. C3×#2067C2semidirect
7. C3×#1949C2semidirect
8. C3×#1942C2semidirect
9. C3×#1923C2semidirect
10. C3×#1921C2semidirect
11. C3×#1919C2semidirect
12. C3×#1862C2semidirect
13. C3×#1852C2semidirect
14. C3×#1824C2semidirect
15. C3×C4×#102C2semidirect
16. #27033C3direct (N × H)
60. #2494C2×C3semidirect
62. #2073C2×C3semidirect
63. #2067C2×C3semidirect
66. #1949C2×C3semidirect
67. #1942C2×C3semidirect
68. #1923C2×C3semidirect
69. #1921C2×C3semidirect
70. #1919C2×C3semidirect
71. #1862C2×C3semidirect
72. #1852C2×C3semidirect
73. #1824C2×C3semidirect
74. C4×#102C2×C3semidirect
629. C3#27033direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#24942
C3×#23612
C3×#20732
C3×#20672
C3×#20342
C3×#20332
C3×#19492
C3×#19422
C3×#19232
C3×#19212
C3×#19192
C3×#18622
C3×#18522
C3×#18242
C3×C4×#1022
#270333