Finite Group Databasebeta

← C3×#26770 #10645377

Subgroups — C3×#26770 #10645377

Order 768.

798 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2509C211yesmaximal, characteristic, complemented
2C3×#2366C211yesmaximal, characteristic
3C3×#2073C211yesmaximal, characteristic, complemented
4C3×#2067C211yesmaximal, characteristic, complemented
5C3×#2041C211yesmaximal, characteristic
6C3×#2041C211yesmaximal, characteristic
7C3×#1936C211yesmaximal, characteristic, complemented
8C3×#1925C211yesmaximal, characteristic, complemented
9C3×#1920C211yesmaximal, characteristic, complemented
10C3×#1918C211yesmaximal, characteristic, complemented
11C3×#1917C211yesmaximal, characteristic, complemented
12C3×#1864C211yesmaximal, characteristic, complemented
13C3×#1864C211yesmaximal, characteristic, complemented
14C3×#1804C211yesmaximal, characteristic, complemented
15C3×#1762C211yesmaximal, characteristic, complemented
16#26770C311yesmaximal, characteristic, direct product factor, Sylow
17C2×C3×#127C2211yescharacteristic
18C2×C3×C4×D4—21no—
19C2×C3×C4×D4C2211yescharacteristic
20C2×C3×C4×D4C2211yescharacteristic
21C2×C3×#107C2211yescharacteristic
22C2×C3×#106C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C2×C3×#102C2211yescharacteristic
28C2×C3×#102C2211yescharacteristic
29C2×C3×#102C2211yescharacteristic
30C2×C3×#102C2211yescharacteristic
31C2×C3×#102C2211yescharacteristic
32C2×C3×#97C2211yescharacteristic
33C3×#400—21no—
34C3×#400—21no—
35C3×#390—21no—
36C3×#390C2211yescharacteristic
37C3×#390C2211yescharacteristic
38C3×#389—21no—
39C3×#389C2211yescharacteristic
40C3×#389C2211yescharacteristic
41C3×#389C2211yescharacteristic
42C3×#389C2211yescharacteristic
43C3×#387—21no—
44C3×#386—21no—
45C3×#386—21no—
46C3×#382C2211yescharacteristic
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×#336C2211yescharacteristic
57C3×#336C2211yescharacteristic
58C3×#336C2211yescharacteristic
59C3×#336C2211yescharacteristic
60#2509C2×C311yescharacteristic, complemented
61#2366C2×C311yescharacteristic
62#2073C2×C311yescharacteristic, complemented
63#2067C2×C311yescharacteristic, complemented
64#2041C2×C311yescharacteristic
65#2041C2×C311yescharacteristic
66#1936C2×C311yescharacteristic, complemented
67#1925C2×C311yescharacteristic, complemented
68#1920C2×C311yescharacteristic, complemented
69#1918C2×C311yescharacteristic, complemented
70#1917C2×C311yescharacteristic, complemented
71#1864C2×C311yescharacteristic, complemented
72#1864C2×C311yescharacteristic, complemented
73#1804C2×C311yescharacteristic, complemented
74#1762C2×C311yescharacteristic, complemented
75C22×C3×D4—21no—
76C22×C3×D4—21no—
77C22×C3×D4C2311yescharacteristic
78C22×C3×D4C2311yescharacteristic
79C23×C3×C4—21no—
80C23×C3×C4—21no—
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C3×#127—22no—
86C3×#127—22no—
87C3×C4×D4—22no—
88C3×C4×D4—22no—
89C3×C4×D4—41no—
90C3×C4×D4—41no—
91C3×C4×D4—44no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4D411yescharacteristic
97C2×C3×C4⋊C4D411yescharacteristic
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C4⋊C4C2311yescharacteristic
102C2×C3×C4⋊C4C2311yescharacteristic
103C2×C3×C4⋊C4C2311yescharacteristic
104C2×C3×C22⋊C4—21no—
105C2×C3×C22⋊C4—21no—
106C2×C3×C22⋊C4—21no—
107C2×C3×C22⋊C4—21no—
108C2×C3×C22⋊C4—21no—
109C2×C3×C22⋊C4—21no—
110C2×C3×C42—21no—
111C2×C3×C42—21no—
112C2×C3×C42C2311yescharacteristic
113C2×C3×C42C2311yescharacteristic
114C2×C3×C42C2311yescharacteristic
115C3×#107—22no—
116C3×#106—22no—
117C3×C4⋊C8—41no—
118C3×#102—22no—
119C3×#102—22no—
120C3×#102—22no—
121C3×#102—22no—
122C3×#102—22no—
123C3×#102—22no—
124C3×#102—41no—
125C3×#102—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×#95D411yescharacteristic
136C3×#95D411yescharacteristic
137C2×#127C22×C311yescharacteristic
138C2×C4×D4—21no—
139C2×C4×D4C22×C311yescharacteristic
140C2×C4×D4C22×C311yescharacteristic
141C2×#107C22×C311yescharacteristic
142C2×#106C22×C311yescharacteristic
143C2×C4⋊C8C22×C311yescharacteristic
144C2×#102C22×C311yescharacteristic
145C2×#102C22×C311yescharacteristic
146C2×#102C22×C311yescharacteristic
147C2×#102C22×C311yescharacteristic
148C2×#102C22×C311yescharacteristic
149C2×#102C22×C311yescharacteristic
150C2×#102C22×C311yescharacteristic
151C2×#102C22×C311yescharacteristic
152C2×#97C22×C311yescharacteristic
153#400—21no—
154#400—21no—
155#390—21no—
156#390C22×C311yescharacteristic
157#390C22×C311yescharacteristic
158#389—21no—
159#389C22×C311yescharacteristic
160#389C22×C311yescharacteristic
161#389C22×C311yescharacteristic
162#389C22×C311yescharacteristic
163#387—21no—
164#386—21no—
165#386—21no—
166#382C22×C311yescharacteristic
167C4×C4⋊C4C22×C311yescharacteristic
168C4×C4⋊C4C22×C311yescharacteristic
169C4×C4⋊C4C22×C311yescharacteristic
170#340C22×C311yescharacteristic
171#340C22×C311yescharacteristic
172#340C22×C311yescharacteristic
173#340C22×C311yescharacteristic
174#340C22×C311yescharacteristic
175#340C22×C311yescharacteristic
176#336C22×C311yescharacteristic
177#336C22×C311yescharacteristic
178#336C22×C311yescharacteristic
179#336C22×C311yescharacteristic
180C24×C3—21no—
181C24×C3—21no—
182C2×C3×D4—24no—
183C2×C3×D4—24no—
184C2×C3×D4—21no—
185C2×C3×D4—21no—
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4—21no—
189C2×C3×D4—21no—
190C2×C3×D4—21no—
191C2×C3×D4—21no—
192C2×C3×D4—21no—
193C2×C3×D4—21no—
194C2×C3×D4—42no—
195C2×C3×D4—42no—
196C2×C3×D4—81no—
197C2×C3×D4—81no—
198C2×C3×D4#4112yes—
199C2×C3×D4#4112yes—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—42no—
207C22×C3×C4—42no—
208C22×C3×C4—42no—
209C22×C3×C4—42no—
210C22×C3×C4—81no—
211C22×C3×C4—81no—
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2411yescharacteristic
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—21no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C3×C4⋊C4—22no—
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—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—41no—
243C3×C4⋊C4—42no—
244C3×C4⋊C4—42no—
245C3×C4⋊C4—81no—
246C3×C4⋊C4—81no—
247C3×C4⋊C4#4112yes—
248C3×C4⋊C4#4112yes—
249C3×C22⋊C4—42no—
250C3×C22⋊C4—42no—
251C3×C22⋊C4—42no—
252C3×C22⋊C4—42no—
253C3×C22⋊C4—81no—
254C3×C22⋊C4—81no—
255C3×C42—21no—
256C3×C42—21no—
257C3×C42—21no—
258C3×C42—21no—
259C3×C42—21no—
260C3×C42—21no—
261C3×C42—42no—
262C3×C42—81no—
263C22×D4—21no—
264C22×D4—21no—
265C22×D4C23×C311yescharacteristic
266C22×D4C23×C311yescharacteristic
267C23×C4—21no—
268C23×C4—21no—
269C22×C8C23×C311yescharacteristic
270C22×C8C23×C311yescharacteristic
271C22×C8C23×C311yescharacteristic
272C22×C8C23×C311yescharacteristic
273#127—22no—
274#127—22no—
275C4×D4—22no—
276C4×D4—22no—
277C4×D4—41no—
278C4×D4—41no—
279C4×D4—44no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4C3×D411yescharacteristic
285C2×C4⋊C4C3×D411yescharacteristic
286C2×C4⋊C4C23×C311yescharacteristic
287C2×C4⋊C4C23×C311yescharacteristic
288C2×C4⋊C4C23×C311yescharacteristic
289C2×C4⋊C4C23×C311yescharacteristic
290C2×C4⋊C4C23×C311yescharacteristic
291C2×C4⋊C4C23×C311yescharacteristic
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42C23×C311yescharacteristic
301C2×C42C23×C311yescharacteristic
302C2×C42C23×C311yescharacteristic
303#107—22no—
304#106—22no—
305C4⋊C8—41no—
306#102—22no—
307#102—22no—
308#102—22no—
309#102—22no—
310#102—22no—
311#102—22no—
312#102—41no—
313#102—41no—
314#97—41no—
315#95—21no—
316#95—21no—
317#95—21no—
318#95—21no—
319#95—21no—
320#95—21no—
321#95—21no—
322#95—21no—
323#95C3×D411yescharacteristic
324#95C3×D411yescharacteristic
325C23×C3—22no—
326C23×C3—22no—
327C23×C3—41no—
328C23×C3—41no—
329C23×C3—41no—
330C23×C3—41no—
331C23×C3—41no—
332C23×C3—41no—
333C23×C3—41no—
334C23×C3—41no—
335C23×C3—41no—
336C23×C3—41no—
337C23×C3—41no—
338C23×C3—41no—
339C23×C3C22×D411yescharacteristic
340C3×D4—22no—
341C3×D4—22no—
342C3×D4—22no—
343C3×D4—22no—
344C3×D4—48no—
345C3×D4—41no—
346C3×D4—41no—
347C3×D4—41no—
348C3×D4—41no—
349C3×D4—84no—
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—22no—
358C2×C3×C4—22no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
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—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—84no—
386C2×C3×C4—84no—
387C2×C3×C4AGL(1,ℤ/8)12yes—
388C2×C3×C4C2×#4111yescharacteristic
389C2×C3×C4C2×#4111yescharacteristic
390C2×C3×C4C2×#4111yescharacteristic
391C2×C3×C4C2×#4111yescharacteristic
392C2×C3×C4H(5,𝔽2)11yescharacteristic
393C2×C3×C42−1+411yescharacteristic
394C3×C8—41no—
395C3×C8—41no—
396C3×C8—41no—
397C3×C8—41no—
398C24—21no—
399C24—21no—
400C2×D4—24no—
401C2×D4—24no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—81no—
415C2×D4—81no—
416C2×D4C3×#4112yes—
417C2×D4C3×#4112yes—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—81no—
429C22×C4—81no—
430C22×C4C2×C3×D411yescharacteristic
431C22×C4C2×C3×D411yescharacteristic
432C22×C4C2×C3×D411yescharacteristic
433C22×C4C2×C3×D411yescharacteristic
434C22×C4C2×C3×D411yescharacteristic
435C22×C4C2×C3×D411yescharacteristic
436C22×C4C24×C311yescharacteristic
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C4⋊C4—22no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—41no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—81no—
464C4⋊C4—81no—
465C4⋊C4C3×#4112yes—
466C4⋊C4C3×#4112yes—
467C22⋊C4—42no—
468C22⋊C4—42no—
469C22⋊C4—42no—
470C22⋊C4—42no—
471C22⋊C4—81no—
472C22⋊C4—81no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—42no—
480C42—81no—
481C22×C3—42no—
482C22×C3—42no—
483C22×C3—42no—
484C22×C3—42no—
485C22×C3—42no—
486C22×C3—42no—
487C22×C3—81no—
488C22×C3—81no—
489C22×C3—81no—
490C22×C3—81no—
491C22×C3—81no—
492C22×C3—81no—
493C22×C3—81no—
494C22×C3—81no—
495C22×C3#52511yescharacteristic
496C22×C3#54611yescharacteristic
497C22×C3#54811yescharacteristic
498C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
499C22×C3#57511yescharacteristic
500C22×C3#57611yescharacteristic
501C22×C3#57711yescharacteristic
502C3×C4—22no—
503C3×C4—42no—
504C3×C4—41no—
505C3×C4—41no—
506C3×C4—41no—
507C3×C4—41no—
508C3×C4—41no—
509C3×C4—42no—
510C3×C4—81no—
511C3×C4—81no—
512C3×C4—81no—
513C3×C4—81no—
514C3×C4#52911yescharacteristic
515C3×C4#53911yescharacteristic
516C3×C4#55211yescharacteristic
517C3×C4#55311yescharacteristic
518C23—22no—
519C23—22no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23C22×C3×D411yescharacteristic
533D4—22no—
534D4—22no—
535D4—22no—
536D4—22no—
537D4—48no—
538D4—41no—
539D4—41no—
540D4—41no—
541D4—41no—
542D4—84no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—22no—
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—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—84no—
579C2×C4—84no—
580C2×C4C3×AGL(1,ℤ/8)12yes—
581C2×C4C2×C3×#4111yescharacteristic
582C2×C4C2×C3×#4111yescharacteristic
583C2×C4C2×C3×#4111yescharacteristic
584C2×C4C2×C3×#4111yescharacteristic
585C2×C4C3×H(5,𝔽2)11yescharacteristic
586C2×C4C3×2−1+411yescharacteristic
587C8—41no—
588C8—41no—
589C8—41no—
590C8—41no—
591C2×C3—82no—
592C2×C3—82no—
593C2×C3#246011yescharacteristic
594C2×C3#311411yescharacteristic
595C2×C3#311811yescharacteristic
596C2×C3#330911yescharacteristic
597C2×C3#332411yescharacteristic
598C2×C3#336611yescharacteristic
599C2×C3#337111yescharacteristic
600C22—42no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22C3×#52511yescharacteristic
615C22C3×#54611yescharacteristic
616C22C3×#54811yescharacteristic
617C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
618C22C3×#57511yescharacteristic
619C22C3×#57611yescharacteristic
620C22C3×#57711yescharacteristic
621C4—22no—
622C4—41no—
623C4—41no—
624C4—41no—
625C4—41no—
626C4—42no—
627C4—42no—
628C4—41no—
629C4—81no—
630C4—81no—
631C4—81no—
632C4—81no—
633C4C3×#52911yescharacteristic
634C4C3×#53911yescharacteristic
635C4C3×#55211yescharacteristic
636C4C3×#55311yescharacteristic
637C3#2677011yescharacteristic, direct product factor, Sylow
638C2—82no—
639C2—82no—
640C2C3×#246011yescharacteristic
641C2C3×#311411yescharacteristic
642C2C3×#311811yescharacteristic
643C2C3×#330911yescharacteristic
644C2C3×#332411yescharacteristic
645C2C3×#336611yescharacteristic
646C2C3×#337111yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×C4×C4⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#2509C2semidirect
3. C3×#2073C2semidirect
4. C3×#2067C2semidirect
7. C3×#1936C2semidirect
8. C3×#1925C2semidirect
9. C3×#1920C2semidirect
10. C3×#1918C2semidirect
11. C3×#1917C2semidirect
12. C3×#1864C2semidirect
13. C3×#1864C2semidirect
14. C3×#1804C2semidirect
15. C3×#1762C2semidirect
16. #26770C3direct (N × H)
60. #2509C2×C3semidirect
62. #2073C2×C3semidirect
63. #2067C2×C3semidirect
66. #1936C2×C3semidirect
67. #1925C2×C3semidirect
68. #1920C2×C3semidirect
69. #1918C2×C3semidirect
70. #1917C2×C3semidirect
71. #1864C2×C3semidirect
72. #1864C2×C3semidirect
73. #1804C2×C3semidirect
74. #1762C2×C3semidirect
637. C3#26770direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#25092
C3×#23662
C3×#20732
C3×#20672
C3×#20412
C3×#19362
C3×#19252
C3×#19202
C3×#19182
C3×#19172
C3×#18642
C3×#18042
C3×#17622
#267703