Finite Group Databasebeta

← C3×#26776 #10645383

Subgroups — C3×#26776 #10645383

Order 768.

726 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2506C211yesmaximal, characteristic, complemented
2C3×#2367C211yesmaximal, characteristic
3C3×#2071C211yesmaximal, characteristic
4C3×#2067C211yesmaximal, characteristic
5C3×#2042C211yesmaximal, characteristic
6C3×#2042C211yesmaximal, characteristic
7C3×#1935C211yesmaximal, characteristic, complemented
8C3×#1928C211yesmaximal, characteristic, complemented
9C3×#1921C211yesmaximal, characteristic, complemented
10C3×#1920C211yesmaximal, characteristic, complemented
11C3×#1917C211yesmaximal, characteristic, complemented
12C3×#1865C211yesmaximal, characteristic, complemented
13C3×#1864C211yesmaximal, characteristic
14C3×#1803C211yesmaximal, characteristic, complemented
15C3×#1762C211yesmaximal, characteristic
16#26776C311yesmaximal, characteristic, direct product factor, Sylow
17C2×C3×#125—21no—
18C2×C3×#124C2211yescharacteristic
19C2×C3×C4×Q8C2211yescharacteristic
20C2×C3×C4×D4C2211yescharacteristic
21C2×C3×#107C2211yescharacteristic
22C2×C3×#106C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×#103C2211yescharacteristic
28C2×C3×#102C2211yescharacteristic
29C2×C3×#102C2211yescharacteristic
30C2×C3×#102C2211yescharacteristic
31C2×C3×#102C2211yescharacteristic
32C2×C3×#97C2211yescharacteristic
33C3×#400—21no—
34C3×#398—21no—
35C3×#391C2211yescharacteristic
36C3×#390C2211yescharacteristic
37C3×#389C2211yescharacteristic
38C3×#389C2211yescharacteristic
39C3×#389C2211yescharacteristic
40C3×#389C2211yescharacteristic
41C3×#388—21no—
42C3×#388—21no—
43C3×#387—21no—
44C3×#385—21no—
45C3×#384C2211yescharacteristic
46C3×C4×C4⋊C4C2211yescharacteristic
47C3×C4×C4⋊C4C2211yescharacteristic
48C3×C4×C4⋊C4C2211yescharacteristic
49C3×C4×C22⋊C4—21no—
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#336C2211yescharacteristic
57C3×#336C2211yescharacteristic
58C3×#336C2211yescharacteristic
59C3×#336C2211yescharacteristic
60#2506C2×C311yescharacteristic, complemented
61#2367C2×C311yescharacteristic
62#2071C2×C311yescharacteristic
63#2067C2×C311yescharacteristic
64#2042C2×C311yescharacteristic
65#2042C2×C311yescharacteristic
66#1935C2×C311yescharacteristic, complemented
67#1928C2×C311yescharacteristic, complemented
68#1921C2×C311yescharacteristic, complemented
69#1920C2×C311yescharacteristic, complemented
70#1917C2×C311yescharacteristic, complemented
71#1865C2×C311yescharacteristic, complemented
72#1864C2×C311yescharacteristic
73#1803C2×C311yescharacteristic, complemented
74#1762C2×C311yescharacteristic
75C22×C3×Q8C2311yescharacteristic
76C22×C3×D4C2311yescharacteristic
77C23×C3×C4—21no—
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C3×#125—44no—
83C3×#124—22no—
84C3×#124—22no—
85C3×C4×Q8—22no—
86C3×C4×Q8—41no—
87C3×C4×D4—22no—
88C3×C4×D4—41no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
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×C42—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×#103—22no—
119C3×#103—22no—
120C3×#103—22no—
121C3×#103—41no—
122C3×#102—22no—
123C3×#102—22no—
124C3×#102—22no—
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×#125—21no—
138C2×#124C22×C311yescharacteristic
139C2×C4×Q8C22×C311yescharacteristic
140C2×C4×D4C22×C311yescharacteristic
141C2×#107C22×C311yescharacteristic
142C2×#106C22×C311yescharacteristic
143C2×C4⋊C8C22×C311yescharacteristic
144C2×#103C22×C311yescharacteristic
145C2×#103C22×C311yescharacteristic
146C2×#103C22×C311yescharacteristic
147C2×#103C22×C311yescharacteristic
148C2×#102C22×C311yescharacteristic
149C2×#102C22×C311yescharacteristic
150C2×#102C22×C311yescharacteristic
151C2×#102C22×C311yescharacteristic
152C2×#97C22×C311yescharacteristic
153#400—21no—
154#398—21no—
155#391C22×C311yescharacteristic
156#390C22×C311yescharacteristic
157#389C22×C311yescharacteristic
158#389C22×C311yescharacteristic
159#389C22×C311yescharacteristic
160#389C22×C311yescharacteristic
161#388—21no—
162#388—21no—
163#387—21no—
164#385—21no—
165#384C22×C311yescharacteristic
166C4×C4⋊C4C22×C311yescharacteristic
167C4×C4⋊C4C22×C311yescharacteristic
168C4×C4⋊C4C22×C311yescharacteristic
169C4×C22⋊C4—21no—
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—
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#4112yes—
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—42no—
201C22×C3×C4—42no—
202C22×C3×C4—81no—
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2411yescharacteristic
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—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C3×C4⋊C4—22no—
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—21no—
233C3×C4⋊C4—41no—
234C3×C4⋊C4—42no—
235C3×C4⋊C4—42no—
236C3×C4⋊C4—42no—
237C3×C4⋊C4—42no—
238C3×C4⋊C4—42no—
239C3×C4⋊C4—42no—
240C3×C4⋊C4—42no—
241C3×C4⋊C4#4112yes—
242C3×C4⋊C4#4112yes—
243C3×C22⋊C4—24no—
244C3×C22⋊C4—24no—
245C3×C22⋊C4—42no—
246C3×C22⋊C4—42no—
247C3×C22⋊C4—81no—
248C3×C42—21no—
249C3×C42—21no—
250C3×C42—21no—
251C3×C42—21no—
252C3×C42—21no—
253C3×C42—21no—
254C3×C42—42no—
255C3×C42—42no—
256C3×C42—81no—
257C22×Q8C23×C311yescharacteristic
258C22×D4C23×C311yescharacteristic
259C23×C4—21no—
260C22×C8C23×C311yescharacteristic
261C22×C8C23×C311yescharacteristic
262C22×C8C23×C311yescharacteristic
263C22×C8C23×C311yescharacteristic
264#125—44no—
265#124—22no—
266#124—22no—
267C4×Q8—22no—
268C4×Q8—41no—
269C4×D4—22no—
270C4×D4—41no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4C3×D411yescharacteristic
279C2×C4⋊C4C3×D411yescharacteristic
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×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42C23×C311yescharacteristic
295C2×C42C23×C311yescharacteristic
296C2×C42C23×C311yescharacteristic
297#107—22no—
298#106—22no—
299C4⋊C8—41no—
300#103—22no—
301#103—22no—
302#103—22no—
303#103—41no—
304#102—22no—
305#102—22no—
306#102—22no—
307#102—41no—
308#97—41no—
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#95C3×D411yescharacteristic
318#95C3×D411yescharacteristic
319C23×C3—22no—
320C23×C3—41no—
321C23×C3—41no—
322C23×C3—41no—
323C23×C3—41no—
324C23×C3—41no—
325C23×C3—41no—
326C23×C3C22×D411yescharacteristic
327C3×Q8—22no—
328C3×Q8—22no—
329C3×Q8—41no—
330C3×Q8—41no—
331C3×D4—22no—
332C3×D4—22no—
333C3×D4—41no—
334C3×D4—41no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—22no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
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—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
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—84no—
374C2×C3×C4AGL(1,ℤ/8)12yes—
375C2×C3×C4C2×#4111yescharacteristic
376C2×C3×C4C2×#4111yescharacteristic
377C2×C3×C4C2×#4111yescharacteristic
378C2×C3×C4C2×#4111yescharacteristic
379C2×C3×C4H(5,𝔽2)11yescharacteristic
380C2×C3×C42−1+411yescharacteristic
381C3×C8—41no—
382C3×C8—41no—
383C3×C8—41no—
384C3×C8—41no—
385C24—21no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8C3×#4112yes—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4C3×#4112yes—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—81no—
408C22×C4C2×C3×D411yescharacteristic
409C22×C4C2×C3×D411yescharacteristic
410C22×C4C2×C3×D411yescharacteristic
411C22×C4C2×C3×D411yescharacteristic
412C22×C4C2×C3×D411yescharacteristic
413C22×C4C2×C3×D411yescharacteristic
414C22×C4C24×C311yescharacteristic
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C4⋊C4—22no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—41no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4C3×#4112yes—
447C4⋊C4C3×#4112yes—
448C22⋊C4—24no—
449C22⋊C4—24no—
450C22⋊C4—42no—
451C22⋊C4—42no—
452C22⋊C4—81no—
453C42—21no—
454C42—21no—
455C42—21no—
456C42—21no—
457C42—21no—
458C42—21no—
459C42—42no—
460C42—42no—
461C42—81no—
462C22×C3—42no—
463C22×C3—42no—
464C22×C3—42no—
465C22×C3—81no—
466C22×C3—81no—
467C22×C3—81no—
468C22×C3—81no—
469C22×C3#52511yescharacteristic
470C22×C3#54611yescharacteristic
471C22×C3#54811yescharacteristic
472C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
473C22×C3#57511yescharacteristic
474C22×C3#57711yescharacteristic
475C22×C3#57811yescharacteristic
476C3×C4—22no—
477C3×C4—41no—
478C3×C4—42no—
479C3×C4—41no—
480C3×C4—42no—
481C3×C4—41no—
482C3×C4—41no—
483C3×C4—41no—
484C3×C4—42no—
485C3×C4—81no—
486C3×C4—81no—
487C3×C4—81no—
488C3×C4—81no—
489C3×C4#52911yescharacteristic
490C3×C4#53911yescharacteristic
491C3×C4#55211yescharacteristic
492C3×C4#55311yescharacteristic
493C23—22no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23C22×C3×D411yescharacteristic
501Q8—22no—
502Q8—22no—
503Q8—41no—
504Q8—41no—
505D4—22no—
506D4—22no—
507D4—41no—
508D4—41no—
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—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—22no—
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×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—84no—
548C2×C4C3×AGL(1,ℤ/8)12yes—
549C2×C4C2×C3×#4111yescharacteristic
550C2×C4C2×C3×#4111yescharacteristic
551C2×C4C2×C3×#4111yescharacteristic
552C2×C4C2×C3×#4111yescharacteristic
553C2×C4C3×H(5,𝔽2)11yescharacteristic
554C2×C4C3×2−1+411yescharacteristic
555C8—41no—
556C8—41no—
557C8—41no—
558C8—41no—
559C2×C3—82no—
560C2×C3#246011yescharacteristic
561C2×C3#311511yescharacteristic
562C2×C3#312011yescharacteristic
563C2×C3#330811yescharacteristic
564C2×C3#332311yescharacteristic
565C2×C3#337311yescharacteristic
566C2×C3#337711yescharacteristic
567C22—42no—
568C22—42no—
569C22—42no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22C3×#52511yescharacteristic
575C22C3×#54611yescharacteristic
576C22C3×#54811yescharacteristic
577C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
578C22C3×#57511yescharacteristic
579C22C3×#57711yescharacteristic
580C22C3×#57811yescharacteristic
581C4—22no—
582C4—42no—
583C4—41no—
584C4—41no—
585C4—41no—
586C4—42no—
587C4—41no—
588C4—42no—
589C4—41no—
590C4—81no—
591C4—81no—
592C4—81no—
593C4—81no—
594C4C3×#52911yescharacteristic
595C4C3×#53911yescharacteristic
596C4C3×#55211yescharacteristic
597C4C3×#55311yescharacteristic
598C3#2677611yescharacteristic, direct product factor, Sylow
599C2—82no—
600C2C3×#246011yescharacteristic
601C2C3×#311511yescharacteristic
602C2C3×#312011yescharacteristic
603C2C3×#330811yescharacteristic
604C2C3×#332311yescharacteristic
605C2C3×#337311yescharacteristic
606C2C3×#337711yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
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×#2506C2semidirect
7. C3×#1935C2semidirect
8. C3×#1928C2semidirect
9. C3×#1921C2semidirect
10. C3×#1920C2semidirect
11. C3×#1917C2semidirect
12. C3×#1865C2semidirect
14. C3×#1803C2semidirect
16. #26776C3direct (N × H)
60. #2506C2×C3semidirect
66. #1935C2×C3semidirect
67. #1928C2×C3semidirect
68. #1921C2×C3semidirect
69. #1920C2×C3semidirect
70. #1917C2×C3semidirect
71. #1865C2×C3semidirect
73. #1803C2×C3semidirect
598. C3#26776direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#25062
C3×#23672
C3×#20712
C3×#20672
C3×#20422
C3×#19352
C3×#19282
C3×#19212
C3×#19202
C3×#19172
C3×#18652
C3×#18642
C3×#18032
C3×#17622
#267763