Finite Group Databasebeta

← C3×#25644 #10644251

Subgroups — C3×#25644 #10644251

Order 768.

1212 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#556C211yesmaximal, characteristic, complemented
2C2×C3×#496C211yesmaximal, characteristic
3C2×C3×#460C211yesmaximal, characteristic, complemented
4C3×#2548C211yesmaximal, characteristic, complemented
5C3×#2013C212yesmaximal, complemented
6C3×#2012C212yesmaximal, complemented
7C3×#1985C211yesmaximal, characteristic, complemented
8C3×#1983C211yesmaximal, characteristic, complemented
9C3×#1880C212yesmaximal, complemented
10C3×#1879C212yesmaximal, complemented
11C3×#1763C211yesmaximal, characteristic, complemented
12#25644C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2212yes—
14C22×C3×SD16C2212yes—
15C22×C3×D8C2212yes—
16C3×#556—24no—
17C3×#556—24no—
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×C3×#127C2211yescharacteristic
20C2×C3×#126—21no—
21C2×C3×#126—21no—
22C2×C3×#124—21no—
23C2×C3×#124C2211yescharacteristic
24C2×C3×#124C2211yescharacteristic
25C2×C3×C22⋊Q8—21no—
26C2×C3×C4×Q8C2211yescharacteristic
27C2×C3×C4×Q8C2211yescharacteristic
28C2×C3×C4×Q8C2211yescharacteristic
29C2×C3×C4×D4C2211yescharacteristic
30C2×C3×#117—21no—
31C3×#496—24no—
32C3×#460—24no—
33C2×C3×C4⋊C8C2211yescharacteristic
34C2×C3×#103C2212yes—
35C2×C3×#103C2212yes—
36C2×C3×#102C2212yes—
37C2×C3×#102C2212yes—
38C2×C3×#97C2211yescharacteristic
39C3×#394—21no—
40C3×#394—21no—
41C3×#390—21no—
42C3×#390C2212yes—
43C3×#390C2212yes—
44C3×C4×C4⋊C4C2211yescharacteristic
45C3×#340C2212yes—
46C3×#336C2212yes—
47C3×#336C2212yes—
48C2×#556C2×C311yescharacteristic, complemented
49C2×#496C2×C311yescharacteristic
50C2×#460C2×C311yescharacteristic, complemented
51#2548C2×C311yescharacteristic, complemented
52#2013C2×C312yescomplemented
53#2012C2×C312yescomplemented
54#1985C2×C311yescharacteristic, complemented
55#1983C2×C311yescharacteristic, complemented
56#1880C2×C312yescomplemented
57#1879C2×C312yescomplemented
58#1763C2×C311yescharacteristic, complemented
59C22×C3×Q8D411yescharacteristic
60C22×C3×Q8C2311yescharacteristic
61C22×C3×Q8C2311yescharacteristic
62C22×C3×D4—21no—
63C22×C3×D4—21no—
64C22×C3×D4C2312yes—
65C22×C3×D4C2311yescharacteristic
66C23×C3×C4—21no—
67C2×C3×SD16—24no—
68C2×C3×SD16—24no—
69C2×C3×SD16—24no—
70C2×C3×SD16—42no—
71C2×C3×SD16—42no—
72C2×C3×SD16—42no—
73C2×C3×D8—24no—
74C2×C3×D8—24no—
75C2×C3×D8—42no—
76C22×C3×C8C2312yes—
77C22×C3×C8C2312yes—
78C3×C4⋊Q8—22no—
79C3×C4⋊Q8—22no—
80C3×#127—22no—
81C3×#127—22no—
82C3×#126—28no—
83C3×#126—28no—
84C3×#124—22no—
85C3×#124—22no—
86C3×#124—22no—
87C3×#124—44no—
88C3×#124D414yes—
89C3×C22⋊Q8—28no—
90C3×C4×Q8—22no—
91C3×C4×Q8—22no—
92C3×C4×Q8—22no—
93C3×C4×Q8—22no—
94C3×C4×Q8—22no—
95C3×C4×Q8D414yes—
96C3×C4×D4—22no—
97C3×C4×D4—22no—
98C3×#117—28no—
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4—21no—
102C2×C3×C4⋊C4—21no—
103C2×C3×C4⋊C4D411yescharacteristic
104C2×C3×C4⋊C4D411yescharacteristic
105C2×C3×C4⋊C4D411yescharacteristic
106C2×C3×C4⋊C4C2312yes—
107C2×C3×C4⋊C4C2311yescharacteristic
108C2×C3×C22⋊C4—21no—
109C2×C3×C22⋊C4—21no—
110C2×C3×C22⋊C4—21no—
111C2×C3×C22⋊C4—21no—
112C2×C3×C22⋊C4—21no—
113C2×C3×C22⋊C4—22no—
114C2×C3×C22⋊C4—22no—
115C2×C3×C22⋊C4—22no—
116C2×C3×C22⋊C4—22no—
117C2×C3×C42—21no—
118C2×C3×C42—21no—
119C2×C3×C42—21no—
120C2×C3×C42C2311yescharacteristic
121C2×C3×C42C2311yescharacteristic
122C2×C3×C42C2311yescharacteristic
123C3×C4⋊C8—22no—
124C3×#103—24no—
125C3×#103—42no—
126C3×#102—24no—
127C3×#102—42no—
128C3×#97—22no—
129C3×#95—21no—
130C22×SD16C22×C312yes—
131C22×SD16C22×C312yes—
132C22×D8C22×C312yes—
133#556—24no—
134#556—24no—
135C2×C4⋊Q8C22×C311yescharacteristic
136C2×#127C22×C311yescharacteristic
137C2×#126—21no—
138C2×#126—21no—
139C2×#124—21no—
140C2×#124C22×C311yescharacteristic
141C2×#124C22×C311yescharacteristic
142C2×C22⋊Q8—21no—
143C2×C4×Q8C22×C311yescharacteristic
144C2×C4×Q8C22×C311yescharacteristic
145C2×C4×Q8C22×C311yescharacteristic
146C2×C4×D4C22×C311yescharacteristic
147C2×#117—21no—
148#496—24no—
149#460—24no—
150C2×C4⋊C8C22×C311yescharacteristic
151C2×#103C22×C312yes—
152C2×#103C22×C312yes—
153C2×#102C22×C312yes—
154C2×#102C22×C312yes—
155C2×#97C22×C311yescharacteristic
156#394—21no—
157#394—21no—
158#390—21no—
159#390C22×C312yes—
160#390C22×C312yes—
161C4×C4⋊C4C22×C311yescharacteristic
162#340C22×C312yes—
163#336C22×C312yes—
164#336C22×C312yes—
165C24×C3—21no—
166C24×C3—22no—
167C2×C3×Q8—22no—
168C2×C3×Q8—22no—
169C2×C3×Q8—21no—
170C2×C3×Q8—21no—
171C2×C3×Q8—22no—
172C2×C3×Q8—22no—
173C2×C3×Q8—22no—
174C2×C3×Q8—22no—
175C2×C3×Q8—41no—
176C2×C3×Q8C2×D412yes—
177C2×C3×Q8C2×D412yes—
178C2×C3×D4—21no—
179C2×C3×D4—22no—
180C2×C3×D4—22no—
181C2×C3×D4—22no—
182C2×C3×D4—22no—
183C2×C3×D4—22no—
184C2×C3×D4—22no—
185C2×C3×D4—22no—
186C2×C3×D4—22no—
187C2×C3×D4—42no—
188C2×C3×D4—42no—
189C2×C3×D4—42no—
190C2×C3×D4—42no—
191C2×C3×D4—82no—
192C2×C3×D4C2×D412yes—
193C22×C3×C4—24no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—22no—
198C22×C3×C4—42no—
199C22×C3×C4—42no—
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2411yescharacteristic
207C3×SD16—48no—
208C3×SD16—48no—
209C3×D8—48no—
210C2×C3×C8—22no—
211C2×C3×C8—22no—
212C2×C3×C8—22no—
213C2×C3×C8—22no—
214C2×C3×C8—22no—
215C2×C3×C8#4114yes—
216C3×C4⋊C4—24no—
217C3×C4⋊C4—24no—
218C3×C4⋊C4—24no—
219C3×C4⋊C4—24no—
220C3×C4⋊C4—22no—
221C3×C4⋊C4—22no—
222C3×C4⋊C4—22no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—22no—
225C3×C4⋊C4—22no—
226C3×C4⋊C4C2×D412yes—
227C3×C22⋊C4—24no—
228C3×C22⋊C4—24no—
229C3×C22⋊C4—24no—
230C3×C22⋊C4—24no—
231C3×C22⋊C4—24no—
232C3×C22⋊C4—44no—
233C3×C22⋊C4—44no—
234C3×C22⋊C4—82no—
235C3×C22⋊C4—82no—
236C3×C42—24no—
237C3×C42—24no—
238C3×C42—21no—
239C3×C42—21no—
240C3×C42—21no—
241C3×C42—21no—
242C3×C42—42no—
243C3×C42C2×D412yes—
244C3×C42C2×D412yes—
245C22×Q8C3×D411yescharacteristic
246C22×Q8C23×C311yescharacteristic
247C22×Q8C23×C311yescharacteristic
248C22×D4—21no—
249C22×D4—21no—
250C22×D4C23×C312yes—
251C22×D4C23×C311yescharacteristic
252C23×C4—21no—
253C2×SD16—24no—
254C2×SD16—24no—
255C2×SD16—24no—
256C2×SD16—42no—
257C2×SD16—42no—
258C2×SD16—42no—
259C2×D8—24no—
260C2×D8—24no—
261C2×D8—42no—
262C22×C8C23×C312yes—
263C22×C8C23×C312yes—
264C4⋊Q8—22no—
265C4⋊Q8—22no—
266#127—22no—
267#127—22no—
268#126—28no—
269#126—28no—
270#124—22no—
271#124—22no—
272#124—22no—
273#124—44no—
274#124C3×D414yes—
275C22⋊Q8—28no—
276C4×Q8—22no—
277C4×Q8—22no—
278C4×Q8—22no—
279C4×Q8—22no—
280C4×Q8—22no—
281C4×Q8C3×D414yes—
282C4×D4—22no—
283C4×D4—22no—
284#117—28no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4C3×D411yescharacteristic
290C2×C4⋊C4C3×D411yescharacteristic
291C2×C4⋊C4C3×D411yescharacteristic
292C2×C4⋊C4C23×C312yes—
293C2×C4⋊C4C23×C311yescharacteristic
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42C23×C311yescharacteristic
307C2×C42C23×C311yescharacteristic
308C2×C42C23×C311yescharacteristic
309C4⋊C8—22no—
310#103—24no—
311#103—42no—
312#102—24no—
313#102—42no—
314#97—22no—
315#95—21no—
316C23×C3—22no—
317C23×C3—42no—
318C23×C3—41no—
319C23×C3—41no—
320C23×C3—42no—
321C23×C3—42no—
322C23×C3—42no—
323C23×C3—42no—
324C23×C3—42no—
325C23×C3—42no—
326C23×C3—42no—
327C23×C3—42no—
328C23×C3C22×D411yescharacteristic
329C3×Q8—22no—
330C3×Q8—22no—
331C3×Q8—24no—
332C3×Q8—44no—
333C3×Q8—41no—
334C3×Q8—41no—
335C3×Q8—42no—
336C3×D4—24no—
337C3×D4—41no—
338C3×D4—42no—
339C3×D4—41no—
340C3×D4—42no—
341C3×D4—42no—
342C3×D4—42no—
343C3×D4—84no—
344C3×D4—84no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—22no—
350C2×C3×C4—21no—
351C2×C3×C4—22no—
352C2×C3×C4—21no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—48no—
361C2×C3×C4—42no—
362C2×C3×C4—42no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—42no—
366C2×C3×C4—42no—
367C2×C3×C4—42no—
368C2×C3×C4#12012yes—
369C2×C3×C4#12012yes—
370C2×C3×C4AGL(1,ℤ/8)12yes—
371C2×C3×C4C22×D411yescharacteristic
372C2×C3×C4C22×D411yescharacteristic
373C2×C3×C4C2×#4112yes—
374C2×C3×C4H(5,𝔽2)12yes—
375C3×C8—24no—
376C3×C8—42no—
377C24—21no—
378C24—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—41no—
388C2×Q8C2×C3×D412yes—
389C2×Q8C2×C3×D412yes—
390C2×D4—21no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—82no—
404C2×D4C2×C3×D412yes—
405C22×C4—24no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4C2×C3×D411yescharacteristic
413C22×C4C2×C3×D411yescharacteristic
414C22×C4C2×C3×D411yescharacteristic
415C22×C4C2×C3×D411yescharacteristic
416C22×C4C2×C3×D411yescharacteristic
417C22×C4C2×C3×D411yescharacteristic
418C22×C4C24×C311yescharacteristic
419SD16—48no—
420SD16—48no—
421D8—48no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8C3×#4114yes—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—21no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4C2×C3×D412yes—
439C22⋊C4—24no—
440C22⋊C4—24no—
441C22⋊C4—24no—
442C22⋊C4—24no—
443C22⋊C4—24no—
444C22⋊C4—44no—
445C22⋊C4—44no—
446C22⋊C4—82no—
447C22⋊C4—82no—
448C42—24no—
449C42—24no—
450C42—21no—
451C42—21no—
452C42—21no—
453C42—21no—
454C42—42no—
455C42C2×C3×D412yes—
456C42C2×C3×D412yes—
457C22×C3—42no—
458C22×C3—42no—
459C22×C3—42no—
460C22×C3—82no—
461C22×C3—82no—
462C22×C3—82no—
463C22×C3—82no—
464C22×C3—82no—
465C22×C3—82no—
466C22×C3—82no—
467C22×C3—82no—
468C22×C3—82no—
469C22×C3C2×#12011yescharacteristic
470C22×C3#55012yes—
471C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
472C22×C3#57511yescharacteristic
473C22×C3#57712yes—
474C3×C4—22no—
475C3×C4—22no—
476C3×C4—22no—
477C3×C4—42no—
478C3×C4—42no—
479C3×C4—41no—
480C3×C4—42no—
481C3×C4—41no—
482C3×C4—41no—
483C3×C4—82no—
484C3×C4#53211yescharacteristic
485C3×C4#53511yescharacteristic
486C3×C4#54612yes—
487C23—22no—
488C23—41no—
489C23—41no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23C22×C3×D411yescharacteristic
500Q8—22no—
501Q8—22no—
502Q8—24no—
503Q8—44no—
504Q8—41no—
505Q8—41no—
506Q8—42no—
507D4—24no—
508D4—41no—
509D4—42no—
510D4—41no—
511D4—42no—
512D4—42no—
513D4—42no—
514D4—84no—
515D4—84no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—48no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4C3×#12012yes—
540C2×C4C3×#12012yes—
541C2×C4C3×AGL(1,ℤ/8)12yes—
542C2×C4C22×C3×D411yescharacteristic
543C2×C4C22×C3×D411yescharacteristic
544C2×C4C2×C3×#4112yes—
545C2×C4C3×H(5,𝔽2)12yes—
546C8—24no—
547C8—42no—
548C2×C3—82no—
549C2×C3—162no—
550C2×C3#243111yescharacteristic
551C2×C3#300611yescharacteristic
552C2×C3#301011yescharacteristic
553C2×C3#339712yes—
554C2×C3#340112yes—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—82no—
559C22—82no—
560C22—82no—
561C22—82no—
562C22—82no—
563C22—82no—
564C22—82no—
565C22—82no—
566C22—82no—
567C22C2×C3×#12011yescharacteristic
568C22C3×#55012yes—
569C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
570C22C3×#57511yescharacteristic
571C22C3×#57712yes—
572C4—22no—
573C4—22no—
574C4—22no—
575C4—42no—
576C4—42no—
577C4—41no—
578C4—42no—
579C4—41no—
580C4—41no—
581C4—82no—
582C4C3×#53211yescharacteristic
583C4C3×#53511yescharacteristic
584C4C3×#54612yes—
585C3#2564411yescharacteristic, direct product factor, Sylow
586C2—82no—
587C2—162no—
588C2C3×#243111yescharacteristic
589C2C3×#300611yescharacteristic
590C2C3×#301011yescharacteristic
591C2C3×#339712yes—
592C2C3×#340112yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#496C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#556C2semidirect
3. C2×C3×#460C2semidirect
4. C3×#2548C2semidirect
5. C3×#2013C2semidirect
6. C3×#2012C2semidirect
7. C3×#1985C2semidirect
8. C3×#1983C2semidirect
9. C3×#1880C2semidirect
10. C3×#1879C2semidirect
11. C3×#1763C2semidirect
12. #25644C3direct (N × H)
48. C2×#556C2×C3semidirect
50. C2×#460C2×C3semidirect
51. #2548C2×C3semidirect
52. #2013C2×C3semidirect
53. #2012C2×C3semidirect
54. #1985C2×C3semidirect
55. #1983C2×C3semidirect
56. #1880C2×C3semidirect
57. #1879C2×C3semidirect
58. #1763C2×C3semidirect
585. C3#25644direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5562
C2×C3×#4962
C2×C3×#4602
C3×#25482
C3×#20132
C3×#20122
C3×#19852
C3×#19832
C3×#18802
C3×#18792
C3×#17632
#256443