Finite Group Databasebeta

← C3×#26823 #10645430

Subgroups — C3×#26823 #10645430

Order 768.

796 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#488C211yesmaximal, characteristic
2C2×C3×C4×Q16C211yesmaximal, characteristic, complemented
3C3×#2453C211yesmaximal, characteristic, complemented
4C3×#2448C211yesmaximal, characteristic
5C3×#2067C211yesmaximal, characteristic
6C3×#2059C211yesmaximal, characteristic, complemented
7C3×#2056C211yesmaximal, characteristic, complemented
8C3×#2015C211yesmaximal, characteristic
9C3×#1968C211yesmaximal, characteristic
10C3×#1934C211yesmaximal, characteristic
11C3×#1933C211yesmaximal, characteristic, complemented
12C3×#1933C211yesmaximal, characteristic, complemented
13C3×#1880C211yesmaximal, characteristic
14C3×#1865C211yesmaximal, characteristic, complemented
15C3×C8×C4⋊C4C211yesmaximal, characteristic, complemented
16#26823C311yesmaximal, characteristic, direct product factor, Sylow
17C22×C3×Q16C2211yescharacteristic
18C22×C3×SD16C2211yescharacteristic
19C2×C3×#125C2211yescharacteristic
20C2×C3×#124C2211yescharacteristic
21C2×C3×C4×Q8C2211yescharacteristic
22C2×C3×C4×Q8C2211yescharacteristic
23C3×#488—24no—
24C3×C4×Q16—24no—
25C2×C3×#107C2211yescharacteristic
26C2×C3×#106C2211yescharacteristic
27C2×C3×C4⋊C8C2211yescharacteristic
28C2×C3×#103C2211yescharacteristic
29C2×C3×#103C2211yescharacteristic
30C2×C3×#103C2211yescharacteristic
31C2×C3×#103C2211yescharacteristic
32C2×C3×#103C2211yescharacteristic
33C2×C3×#102C2211yescharacteristic
34C2×C3×#102C2211yescharacteristic
35C2×C3×#102C2211yescharacteristic
36C2×C3×C4×C8C2211yescharacteristic
37C2×C3×C4×C8C2211yescharacteristic
38C2×C3×C4×C8C2211yescharacteristic
39C3×#400—21no—
40C3×#399—21no—
41C3×#398—21no—
42C3×#397—21no—
43C3×#391C2211yescharacteristic
44C3×#391C2211yescharacteristic
45C3×#391C2211yescharacteristic
46C3×#390C2211yescharacteristic
47C3×#390C2211yescharacteristic
48C3×#389—21no—
49C3×#389C2211yescharacteristic
50C3×#389C2211yescharacteristic
51C3×#389C2211yescharacteristic
52C3×#388—21no—
53C3×#388—21no—
54C3×#387—21no—
55C3×C4×C4⋊C4C2211yescharacteristic
56C3×#340C2211yescharacteristic
57C3×#340C2211yescharacteristic
58C3×#340C2211yescharacteristic
59C3×#340C2211yescharacteristic
60C3×#336C2211yescharacteristic
61C3×#336C2211yescharacteristic
62C2×#488C2×C311yescharacteristic
63C2×C4×Q16C2×C311yescharacteristic, complemented
64#2453C2×C311yescharacteristic, complemented
65#2448C2×C311yescharacteristic
66#2067C2×C311yescharacteristic
67#2059C2×C311yescharacteristic, complemented
68#2056C2×C311yescharacteristic, complemented
69#2015C2×C311yescharacteristic
70#1968C2×C311yescharacteristic
71#1934C2×C311yescharacteristic
72#1933C2×C311yescharacteristic, complemented
73#1933C2×C311yescharacteristic, complemented
74#1880C2×C311yescharacteristic
75#1865C2×C311yescharacteristic, complemented
76C8×C4⋊C4C2×C311yescharacteristic, complemented
77C22×C3×Q8C2311yescharacteristic
78C22×C3×Q8C2311yescharacteristic
79C22×C3×D4C2311yescharacteristic
80C2×C3×Q16—22no—
81C2×C3×Q16—22no—
82C2×C3×Q16—22no—
83C2×C3×SD16—22no—
84C2×C3×SD16—22no—
85C2×C3×SD16—41no—
86C22×C3×C8C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C22×C3×C8C2311yescharacteristic
90C3×#125—22no—
91C3×#125—41no—
92C3×#124—22no—
93C3×#124—41no—
94C3×C4×Q8—22no—
95C3×C4×Q8—22no—
96C3×C4×Q8—41no—
97C3×C4×Q8—41no—
98C2×C3×C4⋊C4—21no—
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4—21no—
102C2×C3×C4⋊C4—21no—
103C2×C3×C4⋊C4—21no—
104C2×C3×C4⋊C4—21no—
105C2×C3×C4⋊C4D411yescharacteristic
106C2×C3×C4⋊C4D411yescharacteristic
107C2×C3×C4⋊C4C2311yescharacteristic
108C2×C3×C4⋊C4C2311yescharacteristic
109C2×C3×C4⋊C4C2311yescharacteristic
110C2×C3×C4⋊C4C2311yescharacteristic
111C2×C3×C4⋊C4C2311yescharacteristic
112C2×C3×C22⋊C4—21no—
113C2×C3×C22⋊C4—21no—
114C2×C3×C22⋊C4—21no—
115C2×C3×C22⋊C4—21no—
116C2×C3×C22⋊C4—21no—
117C2×C3×C22⋊C4—21no—
118C2×C3×C42—21no—
119C2×C3×C42—21no—
120C2×C3×C42C2311yescharacteristic
121C2×C3×C42C2311yescharacteristic
122C2×C3×C42C2311yescharacteristic
123C3×#107—22no—
124C3×#106—41no—
125C3×C4⋊C8—22no—
126C3×#103—22no—
127C3×#103—22no—
128C3×#103—22no—
129C3×#103—22no—
130C3×#103D414yes—
131C3×#102—22no—
132C3×#102—22no—
133C3×#102—22no—
134C3×C4×C8—22no—
135C3×C4×C8—22no—
136C3×C4×C8—22no—
137C3×#95—21no—
138C3×#95—21no—
139C3×#95—21no—
140C3×#95—21no—
141C3×#95—21no—
142C3×#95—21no—
143C3×#95—21no—
144C3×#95—21no—
145C3×#95—21no—
146C3×#95D411yescharacteristic
147C3×#95D411yescharacteristic
148C22×Q16C22×C311yescharacteristic
149C22×SD16C22×C311yescharacteristic
150C2×#125C22×C311yescharacteristic
151C2×#124C22×C311yescharacteristic
152C2×C4×Q8C22×C311yescharacteristic
153C2×C4×Q8C22×C311yescharacteristic
154#488—24no—
155C4×Q16—24no—
156C2×#107C22×C311yescharacteristic
157C2×#106C22×C311yescharacteristic
158C2×C4⋊C8C22×C311yescharacteristic
159C2×#103C22×C311yescharacteristic
160C2×#103C22×C311yescharacteristic
161C2×#103C22×C311yescharacteristic
162C2×#103C22×C311yescharacteristic
163C2×#103C22×C311yescharacteristic
164C2×#102C22×C311yescharacteristic
165C2×#102C22×C311yescharacteristic
166C2×#102C22×C311yescharacteristic
167C2×C4×C8C22×C311yescharacteristic
168C2×C4×C8C22×C311yescharacteristic
169C2×C4×C8C22×C311yescharacteristic
170#400—21no—
171#399—21no—
172#398—21no—
173#397—21no—
174#391C22×C311yescharacteristic
175#391C22×C311yescharacteristic
176#391C22×C311yescharacteristic
177#390C22×C311yescharacteristic
178#390C22×C311yescharacteristic
179#389—21no—
180#389C22×C311yescharacteristic
181#389C22×C311yescharacteristic
182#389C22×C311yescharacteristic
183#388—21no—
184#388—21no—
185#387—21no—
186C4×C4⋊C4C22×C311yescharacteristic
187#340C22×C311yescharacteristic
188#340C22×C311yescharacteristic
189#340C22×C311yescharacteristic
190#340C22×C311yescharacteristic
191#336C22×C311yescharacteristic
192#336C22×C311yescharacteristic
193C24×C3—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8—21no—
204C2×C3×Q8C2×D412yes—
205C2×C3×Q8#4112yes—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C2×C3×D4—21no—
210C2×C3×D4—21no—
211C2×C3×D4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4—21no—
217C22×C3×C4—21no—
218C22×C3×C4—21no—
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4C2×D411yescharacteristic
224C22×C3×C4C2×D411yescharacteristic
225C22×C3×C4C2411yescharacteristic
226C3×Q16—28no—
227C3×SD16—44no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8C2×D412yes—
238C2×C3×C8#4112yes—
239C2×C3×C8#4112yes—
240C3×C4⋊C4—22no—
241C3×C4⋊C4—22no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—42no—
252C3×C4⋊C4—42no—
253C3×C4⋊C4—42no—
254C3×C4⋊C4—42no—
255C3×C4⋊C4—42no—
256C3×C4⋊C4—42no—
257C3×C4⋊C4—81no—
258C3×C4⋊C4C2×D412yes—
259C3×C22⋊C4—42no—
260C3×C22⋊C4—42no—
261C3×C22⋊C4—42no—
262C3×C22⋊C4—42no—
263C3×C22⋊C4—81no—
264C3×C22⋊C4—81no—
265C3×C42—21no—
266C3×C42—21no—
267C3×C42—21no—
268C3×C42—21no—
269C3×C42—21no—
270C3×C42—21no—
271C3×C42—42no—
272C3×C42—42no—
273C22×Q8C23×C311yescharacteristic
274C22×Q8C23×C311yescharacteristic
275C22×D4C23×C311yescharacteristic
276C2×Q16—22no—
277C2×Q16—22no—
278C2×Q16—22no—
279C2×SD16—22no—
280C2×SD16—22no—
281C2×SD16—41no—
282C22×C8C23×C311yescharacteristic
283C22×C8C23×C311yescharacteristic
284C22×C8C23×C311yescharacteristic
285C22×C8C23×C311yescharacteristic
286#125—22no—
287#125—41no—
288#124—22no—
289#124—41no—
290C4×Q8—22no—
291C4×Q8—22no—
292C4×Q8—41no—
293C4×Q8—41no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4C3×D411yescharacteristic
302C2×C4⋊C4C3×D411yescharacteristic
303C2×C4⋊C4C23×C311yescharacteristic
304C2×C4⋊C4C23×C311yescharacteristic
305C2×C4⋊C4C23×C311yescharacteristic
306C2×C4⋊C4C23×C311yescharacteristic
307C2×C4⋊C4C23×C311yescharacteristic
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C42—21no—
315C2×C42—21no—
316C2×C42C23×C311yescharacteristic
317C2×C42C23×C311yescharacteristic
318C2×C42C23×C311yescharacteristic
319#107—22no—
320#106—41no—
321C4⋊C8—22no—
322#103—22no—
323#103—22no—
324#103—22no—
325#103—22no—
326#103C3×D414yes—
327#102—22no—
328#102—22no—
329#102—22no—
330C4×C8—22no—
331C4×C8—22no—
332C4×C8—22no—
333#95—21no—
334#95—21no—
335#95—21no—
336#95—21no—
337#95—21no—
338#95—21no—
339#95—21no—
340#95—21no—
341#95—21no—
342#95C3×D411yescharacteristic
343#95C3×D411yescharacteristic
344C23×C3—41no—
345C23×C3—41no—
346C23×C3—41no—
347C23×C3—41no—
348C23×C3—41no—
349C23×C3—41no—
350C23×C3—41no—
351C23×C3C22×D411yescharacteristic
352C3×Q8—22no—
353C3×Q8—22no—
354C3×Q8—22no—
355C3×Q8—22no—
356C3×Q8—41no—
357C3×Q8—41no—
358C3×Q8—41no—
359C3×Q8—41no—
360C3×D4—41no—
361C3×D4—41no—
362C3×D4—41no—
363C3×D4—41no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—22no—
373C2×C3×C4—22no—
374C2×C3×C4—22no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4#13512yes—
402C2×C3×C4#13512yes—
403C2×C3×C4C22×D411yescharacteristic
404C2×C3×C4C2×#4111yescharacteristic
405C2×C3×C4C2×#4111yescharacteristic
406C2×C3×C4C2×#4111yescharacteristic
407C2×C3×C4H(5,𝔽2)11yescharacteristic
408C2×C3×C42−1+411yescharacteristic
409C3×C8—22no—
410C3×C8—22no—
411C3×C8—22no—
412C3×C8—41no—
413C24—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8C2×C3×D412yes—
425C2×Q8C3×#4112yes—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4C2×C3×D411yescharacteristic
440C22×C4C2×C3×D411yescharacteristic
441C22×C4C2×C3×D411yescharacteristic
442C22×C4C2×C3×D411yescharacteristic
443C22×C4C2×C3×D411yescharacteristic
444C22×C4C2×C3×D411yescharacteristic
445C22×C4C24×C311yescharacteristic
446Q16—28no—
447SD16—44no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8C2×C3×D412yes—
458C2×C8C3×#4112yes—
459C2×C8C3×#4112yes—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—81no—
478C4⋊C4C2×C3×D412yes—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C22⋊C4—81no—
484C22⋊C4—81no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—42no—
492C42—42no—
493C22×C3—81no—
494C22×C3—81no—
495C22×C3—81no—
496C22×C3—81no—
497C22×C3—81no—
498C22×C3—81no—
499C22×C3—81no—
500C22×C3#53211yescharacteristic
501C22×C3D4211yescharacteristic
502C22×C3#54811yescharacteristic
503C22×C3C2×#13511yescharacteristic
504C22×C3C2×#13511yescharacteristic
505C22×C3#57711yescharacteristic
506C22×C3#57811yescharacteristic
507C3×C4—22no—
508C3×C4—22no—
509C3×C4—41no—
510C3×C4—42no—
511C3×C4—42no—
512C3×C4—41no—
513C3×C4—41no—
514C3×C4—42no—
515C3×C4—41no—
516C3×C4—81no—
517C3×C4—81no—
518C3×C4—81no—
519C3×C4—81no—
520C3×C4#52511yescharacteristic
521C3×C4#54211yescharacteristic
522C3×C4#54711yescharacteristic
523C3×C4#55111yescharacteristic
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23C22×C3×D411yescharacteristic
532Q8—22no—
533Q8—22no—
534Q8—22no—
535Q8—22no—
536Q8—41no—
537Q8—41no—
538Q8—41no—
539Q8—41no—
540D4—41no—
541D4—41no—
542D4—41no—
543D4—41no—
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—22no—
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—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4C3×#13512yes—
582C2×C4C3×#13512yes—
583C2×C4C22×C3×D411yescharacteristic
584C2×C4C2×C3×#4111yescharacteristic
585C2×C4C2×C3×#4111yescharacteristic
586C2×C4C2×C3×#4111yescharacteristic
587C2×C4C3×H(5,𝔽2)11yescharacteristic
588C2×C4C3×2−1+411yescharacteristic
589C8—22no—
590C8—22no—
591C8—22no—
592C8—41no—
593C2×C3—161no—
594C2×C3#246111yescharacteristic
595C2×C3#316811yescharacteristic
596C2×C3#317811yescharacteristic
597C2×C3#328411yescharacteristic
598C2×C3#328511yescharacteristic
599C2×C3#336411yescharacteristic
600C2×C3#336511yescharacteristic
601C22—81no—
602C22—81no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22—81no—
608C22C3×#53211yescharacteristic
609C22C3×D4211yescharacteristic
610C22C3×#54811yescharacteristic
611C22C2×C3×#13511yescharacteristic
612C22C2×C3×#13511yescharacteristic
613C22C3×#57711yescharacteristic
614C22C3×#57811yescharacteristic
615C4—22no—
616C4—22no—
617C4—41no—
618C4—42no—
619C4—42no—
620C4—41no—
621C4—41no—
622C4—42no—
623C4—41no—
624C4—81no—
625C4—81no—
626C4—81no—
627C4—81no—
628C4C3×#52511yescharacteristic
629C4C3×#54211yescharacteristic
630C4C3×#54711yescharacteristic
631C4C3×#55111yescharacteristic
632C3#2682311yescharacteristic, direct product factor, Sylow
633C2—161no—
634C2C3×#246111yescharacteristic
635C2C3×#316811yescharacteristic
636C2C3×#317811yescharacteristic
637C2C3×#328411yescharacteristic
638C2C3×#328511yescharacteristic
639C2C3×#336411yescharacteristic
640C2C3×#336511yescharacteristic
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
2. C2×C3×C4×Q16C2semidirect
3. C3×#2453C2semidirect
6. C3×#2059C2semidirect
7. C3×#2056C2semidirect
11. C3×#1933C2semidirect
12. C3×#1933C2semidirect
14. C3×#1865C2semidirect
15. C3×C8×C4⋊C4C2semidirect
16. #26823C3direct (N × H)
63. C2×C4×Q16C2×C3semidirect
64. #2453C2×C3semidirect
67. #2059C2×C3semidirect
68. #2056C2×C3semidirect
72. #1933C2×C3semidirect
73. #1933C2×C3semidirect
75. #1865C2×C3semidirect
76. C8×C4⋊C4C2×C3semidirect
632. C3#26823direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4882
C2×C3×C4×Q162
C3×#24532
C3×#24482
C3×#20672
C3×#20592
C3×#20562
C3×#20152
C3×#19682
C3×#19342
C3×#19332
C3×#18802
C3×#18652
C3×C8×C4⋊C42
#268233