Finite Group Databasebeta

← C3×#25614 #10644221

Subgroups — C3×#25614 #10644221

Order 768.

1216 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#542C211yesmaximal, characteristic, complemented
2C2×C3×#496C211yesmaximal, characteristic, complemented
3C2×C3×#486C211yesmaximal, characteristic, complemented
4C3×#2619C211yesmaximal, characteristic, complemented
5C3×#2000C212yesmaximal, complemented
6C3×#1999C212yesmaximal, complemented
7C3×#1993C211yesmaximal, characteristic, complemented
8C3×#1958C211yesmaximal, characteristic, complemented
9C3×#1880C212yesmaximal, complemented
10C3×#1879C212yesmaximal, complemented
11C3×#1750C211yesmaximal, characteristic, complemented
12#25614C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2212yes—
14C22×C3×D8C2212yes—
15C3×#542—24no—
16C3×#542—24no—
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#127C2211yescharacteristic
19C2×C3×#126—21no—
20C2×C3×#124—21no—
21C2×C3×#124C2211yescharacteristic
22C2×C3×#123—21no—
23C2×C3×C22⋊Q8—21no—
24C2×C3×C22⋊Q8C2211yescharacteristic
25C2×C3×#121C2211yescharacteristic
26C2×C3×#121C2212yes—
27C2×C3×C4×Q8C2211yescharacteristic
28C2×C3×C4×D4C2211yescharacteristic
29C2×C3×#117C2211yescharacteristic
30C3×#496—24no—
31C3×#486—24no—
32C2×C3×#103C2212yes—
33C2×C3×#102C2212yes—
34C2×C3×#102C2212yes—
35C2×C3×#102C2212yes—
36C2×C3×C22⋊C8C2212yes—
37C2×C3×#97C2211yescharacteristic
38C2×C3×C4×C8C2211yescharacteristic
39C3×#394—21no—
40C3×#394—21no—
41C3×#392—21no—
42C3×#392—21no—
43C3×#390C2212yes—
44C3×#382C2211yescharacteristic
45C3×#340C2212yes—
46C3×#340C2212yes—
47C3×#336C2212yes—
48C2×#542C2×C311yescharacteristic, complemented
49C2×#496C2×C311yescharacteristic, complemented
50C2×#486C2×C311yescharacteristic, complemented
51#2619C2×C311yescharacteristic, complemented
52#2000C2×C312yescomplemented
53#1999C2×C312yescomplemented
54#1993C2×C311yescharacteristic, complemented
55#1958C2×C311yescharacteristic, complemented
56#1880C2×C312yescomplemented
57#1879C2×C312yescomplemented
58#1750C2×C311yescharacteristic, complemented
59C22×C3×Q8—21no—
60C22×C3×Q8C2311yescharacteristic
61C22×C3×D4—21no—
62C22×C3×D4—22no—
63C22×C3×D4—21no—
64C22×C3×D4—21no—
65C22×C3×D4C2312yes—
66C22×C3×D4C2311yescharacteristic
67C23×C3×C4—21no—
68C23×C3×C4C2311yescharacteristic
69C2×C3×SD16—24no—
70C2×C3×SD16—24no—
71C2×C3×SD16—42no—
72C2×C3×D8—24no—
73C2×C3×D8—24no—
74C2×C3×D8—42no—
75C22×C3×C8C2312yes—
76C22×C3×C8C2312yes—
77C3×C4⋊Q8—22no—
78C3×C4⋊Q8D414yescomplemented
79C3×#127—22no—
80C3×#127—22no—
81C3×#126—28no—
82C3×#124—28no—
83C3×#124—22no—
84C3×#124D414yescomplemented
85C3×#123—28no—
86C3×C22⋊Q8—28no—
87C3×C22⋊Q8—22no—
88C3×C22⋊Q8—22no—
89C3×#121—22no—
90C3×#121—22no—
91C3×#121—42no—
92C3×#121—42no—
93C3×C4×Q8—22no—
94C3×C4×Q8—22no—
95C3×C4×D4—22no—
96C3×C4×D4—22no—
97C3×#117—22no—
98C3×#117—22no—
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4—21no—
102C2×C3×C4⋊C4D411yescharacteristic
103C2×C3×C4⋊C4D411yescharacteristic
104C2×C3×C4⋊C4C2312yes—
105C2×C3×C4⋊C4C2311yescharacteristic
106C2×C3×C4⋊C4C2311yescharacteristic
107C2×C3×C22⋊C4—21no—
108C2×C3×C22⋊C4—21no—
109C2×C3×C22⋊C4—21no—
110C2×C3×C22⋊C4—21no—
111C2×C3×C22⋊C4—21no—
112C2×C3×C22⋊C4—22no—
113C2×C3×C22⋊C4—22no—
114C2×C3×C22⋊C4—22no—
115C2×C3×C22⋊C4D411yescharacteristic
116C2×C3×C22⋊C4D411yescharacteristic
117C2×C3×C42—21no—
118C2×C3×C42C2311yescharacteristic
119C2×C3×C42C2311yescharacteristic
120C3×#103—42no—
121C3×#102—24no—
122C3×#102—24no—
123C3×#102—42no—
124C3×C22⋊C8—24no—
125C3×#97—22no—
126C3×C4×C8—22no—
127C3×#95—21no—
128C3×#95—21no—
129C22×SD16C22×C312yes—
130C22×D8C22×C312yes—
131#542—24no—
132#542—24no—
133C2×C4⋊Q8C22×C311yescharacteristic
134C2×#127C22×C311yescharacteristic
135C2×#126—21no—
136C2×#124—21no—
137C2×#124C22×C311yescharacteristic
138C2×#123—21no—
139C2×C22⋊Q8—21no—
140C2×C22⋊Q8C22×C311yescharacteristic
141C2×#121C22×C311yescharacteristic
142C2×#121C22×C312yes—
143C2×C4×Q8C22×C311yescharacteristic
144C2×C4×D4C22×C311yescharacteristic
145C2×#117C22×C311yescharacteristic
146#496—24no—
147#486—24no—
148C2×#103C22×C312yes—
149C2×#102C22×C312yes—
150C2×#102C22×C312yes—
151C2×#102C22×C312yes—
152C2×C22⋊C8C22×C312yes—
153C2×#97C22×C311yescharacteristic
154C2×C4×C8C22×C311yescharacteristic
155#394—21no—
156#394—21no—
157#392—21no—
158#392—21no—
159#390C22×C312yes—
160#382C22×C311yescharacteristic
161#340C22×C312yes—
162#340C22×C312yes—
163#336C22×C312yes—
164C24×C3—21no—
165C24×C3—22no—
166C24×C3C2×D411yescharacteristic
167C2×C3×Q8—24no—
168C2×C3×Q8—21no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8—82no—
172C2×C3×Q8C2×D412yes—
173C2×C3×D4—24no—
174C2×C3×D4—21no—
175C2×C3×D4—22no—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—22no—
181C2×C3×D4—22no—
182C2×C3×D4—22no—
183C2×C3×D4—44no—
184C2×C3×D4—42no—
185C2×C3×D4—42no—
186C2×C3×D4—82no—
187C2×C3×D4—82no—
188C2×C3×D4—82no—
189C2×C3×D4—82no—
190C2×C3×D4—82no—
191C2×C3×D4C2×D412yes—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—24no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—22no—
199C22×C3×C4—22no—
200C22×C3×C4—22no—
201C22×C3×C4—82no—
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2411yescharacteristic
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—22no—
220C3×C4⋊C4—22no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—22no—
224C3×C4⋊C4—22no—
225C3×C4⋊C4C2×D412yes—
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—22no—
233C3×C22⋊C4—22no—
234C3×C22⋊C4—44no—
235C3×C22⋊C4—82no—
236C3×C22⋊C4—82no—
237C3×C42—24no—
238C3×C42—21no—
239C3×C42—21no—
240C3×C42C2×D412yes—
241C3×C42C2×D412yes—
242C22×Q8—21no—
243C22×Q8C23×C311yescharacteristic
244C22×D4—21no—
245C22×D4—22no—
246C22×D4—21no—
247C22×D4—21no—
248C22×D4C23×C312yes—
249C22×D4C23×C311yescharacteristic
250C23×C4—21no—
251C23×C4C23×C311yescharacteristic
252C2×SD16—24no—
253C2×SD16—24no—
254C2×SD16—42no—
255C2×D8—24no—
256C2×D8—24no—
257C2×D8—42no—
258C22×C8C23×C312yes—
259C22×C8C23×C312yes—
260C4⋊Q8—22no—
261C4⋊Q8C3×D414yescomplemented
262#127—22no—
263#127—22no—
264#126—28no—
265#124—28no—
266#124—22no—
267#124C3×D414yescomplemented
268#123—28no—
269C22⋊Q8—28no—
270C22⋊Q8—22no—
271C22⋊Q8—22no—
272#121—22no—
273#121—22no—
274#121—42no—
275#121—42no—
276C4×Q8—22no—
277C4×Q8—22no—
278C4×D4—22no—
279C4×D4—22no—
280#117—22no—
281#117—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4C3×D411yescharacteristic
286C2×C4⋊C4C3×D411yescharacteristic
287C2×C4⋊C4C23×C312yes—
288C2×C4⋊C4C23×C311yescharacteristic
289C2×C4⋊C4C23×C311yescharacteristic
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4C3×D411yescharacteristic
299C2×C22⋊C4C3×D411yescharacteristic
300C2×C42—21no—
301C2×C42C23×C311yescharacteristic
302C2×C42C23×C311yescharacteristic
303#103—42no—
304#102—24no—
305#102—24no—
306#102—42no—
307C22⋊C8—24no—
308#97—22no—
309C4×C8—22no—
310#95—21no—
311#95—21no—
312C23×C3—21no—
313C23×C3—22no—
314C23×C3—21no—
315C23×C3—21no—
316C23×C3—22no—
317C23×C3—22no—
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×C3—42no—
329C23×C3C22×D411yescharacteristic
330C3×Q8—24no—
331C3×Q8—42no—
332C3×Q8—84no—
333C3×D4—24no—
334C3×D4—42no—
335C3×D4—42no—
336C3×D4—42no—
337C3×D4—42no—
338C3×D4—42no—
339C3×D4—84no—
340C3×D4—84no—
341C3×D4—84no—
342C3×D4—88no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—21no—
346C2×C3×C4—22no—
347C2×C3×C4—21no—
348C2×C3×C4—22no—
349C2×C3×C4—21no—
350C2×C3×C4—22no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—42no—
358C2×C3×C4—42no—
359C2×C3×C4—42no—
360C2×C3×C4—42no—
361C2×C3×C4—42no—
362C2×C3×C4—42no—
363C2×C3×C4—42no—
364C2×C3×C4—84no—
365C2×C3×C4#12012yes—
366C2×C3×C4#12012yes—
367C2×C3×C4C22×D411yescharacteristic
368C2×C3×C4C22×D411yescharacteristic
369C2×C3×C4C2×#4112yes—
370C2×C3×C4H(5,𝔽2)12yes—
371C3×C8—24no—
372C3×C8—42no—
373C24—21no—
374C24—22no—
375C24C2×C3×D411yescharacteristic
376C2×Q8—24no—
377C2×Q8—21no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—82no—
381C2×Q8C2×C3×D412yes—
382C2×D4—24no—
383C2×D4—21no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—44no—
393C2×D4—42no—
394C2×D4—42no—
395C2×D4—82no—
396C2×D4—82no—
397C2×D4—82no—
398C2×D4—82no—
399C2×D4—82no—
400C2×D4C2×C3×D412yes—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—24no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—82no—
411C22×C4C2×C3×D411yescharacteristic
412C22×C4C2×C3×D411yescharacteristic
413C22×C4C2×C3×D411yescharacteristic
414C22×C4C2×C3×D411yescharacteristic
415C22×C4C2×C3×D411yescharacteristic
416C22×C4C24×C311yescharacteristic
417SD16—48no—
418D8—48no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8C3×#4114yes—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4C2×C3×D412yes—
435C4⋊C4C2×C3×D412yes—
436C22⋊C4—24no—
437C22⋊C4—24no—
438C22⋊C4—24no—
439C22⋊C4—24no—
440C22⋊C4—24no—
441C22⋊C4—22no—
442C22⋊C4—22no—
443C22⋊C4—44no—
444C22⋊C4—82no—
445C22⋊C4—82no—
446C42—24no—
447C42—21no—
448C42—21no—
449C42C2×C3×D412yes—
450C42C2×C3×D412yes—
451C22×C3—42no—
452C22×C3—41no—
453C22×C3—42no—
454C22×C3—42no—
455C22×C3—42no—
456C22×C3—41no—
457C22×C3—41no—
458C22×C3—42no—
459C22×C3—82no—
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×C3C2×#12011yescharacteristic
469C22×C3#53211yescharacteristic
470C22×C3#53511yescharacteristic
471C22×C3#57612yes—
472C22×C3#57712yes—
473C3×C4—22no—
474C3×C4—22no—
475C3×C4—41no—
476C3×C4—42no—
477C3×C4—41no—
478C3×C4—42no—
479C3×C4—42no—
480C3×C4—41no—
481C3×C4—82no—
482C3×C4#54612yes—
483C3×C4#55012yes—
484C23—22no—
485C23—22no—
486C23—21no—
487C23—22no—
488C23—21no—
489C23—21no—
490C23—42no—
491C23—42no—
492C23—41no—
493C23—41no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23C22×C3×D411yescharacteristic
502Q8—24no—
503Q8—42no—
504Q8—84no—
505D4—24no—
506D4—42no—
507D4—42no—
508D4—42no—
509D4—42no—
510D4—42no—
511D4—84no—
512D4—84no—
513D4—84no—
514D4—88no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—84no—
537C2×C4C3×#12012yes—
538C2×C4C3×#12012yes—
539C2×C4C22×C3×D411yescharacteristic
540C2×C4C22×C3×D411yescharacteristic
541C2×C4C2×C3×#4112yes—
542C2×C4C3×H(5,𝔽2)12yes—
543C8—24no—
544C8—42no—
545C2×C3—81no—
546C2×C3—82no—
547C2×C3—162no—
548C2×C3#243111yescharacteristic
549C2×C3#300811yescharacteristic
550C2×C3#301011yescharacteristic
551C2×C3#317912yes—
552C2×C3#322412yes—
553C22—42no—
554C22—41no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—41no—
559C22—41no—
560C22—42no—
561C22—82no—
562C22—82no—
563C22—82no—
564C22—82no—
565C22—82no—
566C22—82no—
567C22—82no—
568C22—82no—
569C22—82no—
570C22C2×C3×#12011yescharacteristic
571C22C3×#53211yescharacteristic
572C22C3×#53511yescharacteristic
573C22C3×#57612yes—
574C22C3×#57712yes—
575C4—22no—
576C4—22no—
577C4—41no—
578C4—42no—
579C4—41no—
580C4—42no—
581C4—42no—
582C4—41no—
583C4—82no—
584C4C3×#54612yes—
585C4C3×#55012yes—
586C3#2561411yescharacteristic, direct product factor, Sylow
587C2—81no—
588C2—82no—
589C2—162no—
590C2C3×#243111yescharacteristic
591C2C3×#300811yescharacteristic
592C2C3×#301011yescharacteristic
593C2C3×#317912yes—
594C2C3×#322412yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#542C2semidirect
2. C2×C3×#496C2semidirect
3. C2×C3×#486C2semidirect
4. C3×#2619C2semidirect
5. C3×#2000C2semidirect
6. C3×#1999C2semidirect
7. C3×#1993C2semidirect
8. C3×#1958C2semidirect
9. C3×#1880C2semidirect
10. C3×#1879C2semidirect
11. C3×#1750C2semidirect
12. #25614C3direct (N × H)
48. C2×#542C2×C3semidirect
49. C2×#496C2×C3semidirect
50. C2×#486C2×C3semidirect
51. #2619C2×C3semidirect
52. #2000C2×C3semidirect
53. #1999C2×C3semidirect
54. #1993C2×C3semidirect
55. #1958C2×C3semidirect
56. #1880C2×C3semidirect
57. #1879C2×C3semidirect
58. #1750C2×C3semidirect
78. C3×C4⋊Q8D4semidirect
84. C3×#124D4semidirect
261. C4⋊Q8C3×D4semidirect
267. #124C3×D4semidirect
586. C3#25614direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5422
C2×C3×#4962
C2×C3×#4862
C3×#26192
C3×#20002
C3×#19992
C3×#19932
C3×#19582
C3×#18802
C3×#18792
C3×#17502
#256143