Finite Group Databasebeta

← #10727911

Subgroups — #10727911

Order 768.

1418 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×C4⋊C4C211yesmaximal, characteristic
2C2×#4928C211yesmaximal, characteristic
3C2×#4928C212yesmaximal
4C2×#4933C211yesmaximal, characteristic
5C2×#4661C211yesmaximal, characteristic
6#71565C212yesmaximal
7#71565C212yesmaximal
8#71536C212yesmaximal
9#69423C211yesmaximal, characteristic
10#69419C211yesmaximal, characteristic
11#69410C211yesmaximal, characteristic
12#14818—31nomaximal, Sylow
13C22×C3×C4⋊C4C2212yes—
14C22×C3×C42C2212yes—
15C22×C3×C42C2211yescharacteristic
16C22×C4⋊Dic3C2211yescharacteristic
17C22×C4⋊Dic3C2212yes—
18C22×C4⋊Dic3C2211yescharacteristic
19C22×C4⋊Dic3C2212yes—
20C22×Dic3⋊C4C2211yescharacteristic
21C22×Dic3⋊C4C2212yes—
22C22×Dic3⋊C4C2211yescharacteristic
23C3×C4×C4⋊C4—24no—
24C2×C3×#95C2211yescharacteristic
25C2×C3×#95C2211yescharacteristic
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2211yescharacteristic
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2211yescharacteristic
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2211yescharacteristic
34C2×C3⋊#95C2211yescharacteristic
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37#4933—24no—
38#4928—24no—
39#4928—28no—
40#4661—24no—
41C2×#398—31no—
42C2×#389—32no—
43C2×#389—31no—
44C2×#387—31no—
45C2×C4×C4⋊C4D311yescharacteristic
46#1445—31no—
47#1444—31no—
48#1441—31no—
49#1441—32no—
50#1441—32no—
51#1436—32no—
52C23×C3×C4C2312yes—
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2312yes—
57C23×Dic3C2311yescharacteristic
58C23×Dic3C2311yescharacteristic
59C23×Dic3C2312yes—
60C23×Dic3C2311yescharacteristic
61C23×Dic3C2311yescharacteristic
62C23×Dic3C2312yes—
63C2×C3×C4⋊C4—24no—
64C2×C3×C4⋊C4—24no—
65C2×C3×C4⋊C4Q818yes—
66C2×C3×C42—22no—
67C2×C3×C42—22no—
68C2×C3×C42—24no—
69C2×C3×C42—24no—
70C2×C3×C42—42no—
71C2×C3×C42D414yes—
72C2×C4⋊Dic3—22no—
73C2×C4⋊Dic3—24no—
74C2×C4⋊Dic3—24no—
75C2×C4⋊Dic3—24no—
76C2×C4⋊Dic3—24no—
77C2×C4⋊Dic3—24no—
78C2×C4⋊Dic3—22no—
79C2×C4⋊Dic3—22no—
80C2×C4⋊Dic3—22no—
81C2×C4⋊Dic3—24no—
82C2×C4⋊Dic3—42no—
83C2×Dic3⋊C4—22no—
84C2×Dic3⋊C4—22no—
85C2×Dic3⋊C4—22no—
86C2×Dic3⋊C4—24no—
87C2×Dic3⋊C4—24no—
88C2×Dic3⋊C4—22no—
89C2×Dic3⋊C4—22no—
90C2×Dic3⋊C4—24no—
91C2×Dic3⋊C4D414yes—
92C3×#95—22no—
93C3×#95D414yes—
94C3⋊#95—24no—
95C3⋊#95—24no—
96C3⋊#95—22no—
97C3⋊#95—22no—
98C3⋊#95—22no—
99C3⋊#95—22no—
100C3⋊#95—24no—
101C3⋊#95—24no—
102C3⋊#95—24no—
103C3⋊#95—24no—
104C3⋊#95—42no—
105C22×C4⋊C4—31no—
106C22×C4⋊C4—31no—
107C22×C4⋊C4—31no—
108C22×C4⋊C4—32no—
109C22×C4⋊C4—32no—
110C22×C4⋊C4—31no—
111C22×C4⋊C4—32no—
112C22×C4⋊C4C2×D312yes—
113C22×C42C2×D312yes—
114C22×C42C2×D311yescharacteristic
115#398—64no—
116#389—68no—
117#389—64no—
118#387—64no—
119C4×C4⋊C4—24no—
120C2×#95—32no—
121C2×#95—31no—
122C2×#95—32no—
123C2×#95—32no—
124C2×#95—31no—
125C2×#95—31no—
126C2×#95—32no—
127C2×#95—32no—
128C2×#95—32no—
129C2×#95—31no—
130C2×#95—32no—
131C2×#95C2×D311yescharacteristic
132C2×#95C2×D311yescharacteristic
133C24×C3C2411yescharacteristic
134C22×C3×C4—22no—
135C22×C3×C4—21no—
136C22×C3×C4—21no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—21no—
140C22×C3×C4—21no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—21no—
144C22×C3×C4—21no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4C2×D412yes—
154C22×C3×C4C2×D412yes—
155C22×C3×C4C2×D412yes—
156C22×C3×C4C2×D412yes—
157C22×C3×C4C2×D412yes—
158C22×C3×C4C2×D414yes—
159C22×C3×C4C2×Q814yes—
160C22×C3×C4C2×Q814yes—
161C22×C3×C4C2×Q814yes—
162C22×C3×C4#4112yes—
163C22×C3×C4#4114yes—
164C22×C3×C4#4112yes—
165C22×Dic3—21no—
166C22×Dic3—21no—
167C22×Dic3—22no—
168C22×Dic3—22no—
169C22×Dic3—21no—
170C22×Dic3—22no—
171C22×Dic3—21no—
172C22×Dic3—21no—
173C22×Dic3—21no—
174C22×Dic3—22no—
175C22×Dic3—22no—
176C22×Dic3—22no—
177C22×Dic3—22no—
178C22×Dic3—22no—
179C22×Dic3—21no—
180C22×Dic3—21no—
181C22×Dic3—21no—
182C22×Dic3—21no—
183C22×Dic3—21no—
184C22×Dic3—22no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—22no—
188C22×Dic3—22no—
189C22×Dic3—21no—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—21no—
193C22×Dic3—22no—
194C22×Dic3—21no—
195C22×Dic3—22no—
196C22×Dic3—22no—
197C22×Dic3—22no—
198C22×Dic3C2×D412yes—
199C22×Dic3C2×D412yes—
200C22×Dic3#4114yes—
201C3×C4⋊C4—216no—
202C3×C42—28no—
203C3×C42—48no—
204C4⋊Dic3—216no—
205C4⋊Dic3—28no—
206C4⋊Dic3—48no—
207C4⋊Dic3—44no—
208Dic3⋊C4—28no—
209Dic3⋊C4—48no—
210Dic3⋊C4—44no—
211C23×C4—31no—
212C23×C4—31no—
213C23×C4—32no—
214C23×C4—32no—
215C23×C4—31no—
216C23×C4—31no—
217C23×C4C22×D312yes—
218C23×C4C22×D311yescharacteristic
219C23×C4C22×D312yes—
220C23×C4C22×D311yescharacteristic
221C23×C4C22×D311yescharacteristic
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—34no—
225C2×C4⋊C4—64no—
226C2×C4⋊C4—64no—
227C2×C4⋊C4—64no—
228C2×C4⋊C4—62no—
229C2×C4⋊C4—62no—
230C2×C4⋊C4—64no—
231C2×C4⋊C4—62no—
232C2×C4⋊C4—64no—
233C2×C4⋊C4—62no—
234C2×C4⋊C4—64no—
235C2×C4⋊C4—62no—
236C2×C4⋊C4—62no—
237C2×C4⋊C4—62no—
238C2×C4⋊C4—64no—
239C2×C4⋊C4—64no—
240C2×C4⋊C4—62no—
241C2×C4⋊C4—64no—
242C2×C4⋊C4—62no—
243C2×C4⋊C4—122no—
244C2×C4⋊C4Dic618yes—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—24no—
248C2×C42—24no—
249C2×C42—42no—
250C2×C42D1214yes—
251#95—22no—
252#95—64no—
253#95—64no—
254#95—64no—
255#95—62no—
256#95—62no—
257#95—62no—
258#95—62no—
259#95—64no—
260#95—64no—
261#95—64no—
262#95—122no—
263#95#6714yes—
264C23×C3C22×D411yescharacteristic
265C23×C3C22×D411yescharacteristic
266C23×C3C22×D411yescharacteristic
267C23×C3C22×Q812yes—
268C23×C3C2×#4111yescharacteristic
269C23×C3C2×#4112yes—
270C23×C3C2×#4111yescharacteristic
271C23×C3C2×#4112yes—
272C23×C3H(5,𝔽2)11yescharacteristic
273C23×C32−1+411yescharacteristic
274C23×C32−1+412yes—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—24no—
278C2×C3×C4—22no—
279C2×C3×C4—24no—
280C2×C3×C4—22no—
281C2×C3×C4—24no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—24no—
285C2×C3×C4—24no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—24no—
289C2×C3×C4—24no—
290C2×C3×C4—22no—
291C2×C3×C4—24no—
292C2×C3×C4—24no—
293C2×C3×C4—22no—
294C2×C3×C4—24no—
295C2×C3×C4—24no—
296C2×C3×C4—24no—
297C2×C3×C4—24no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4#12114yes—
303C2×C3×C4#12114yes—
304C2×C3×C4C22⋊Q818yes—
305C2×C3×C4C4⋊Q814yes—
306C2×Dic3—22no—
307C2×Dic3—22no—
308C2×Dic3—24no—
309C2×Dic3—24no—
310C2×Dic3—24no—
311C2×Dic3—24no—
312C2×Dic3—22no—
313C2×Dic3—22no—
314C2×Dic3—42no—
315C2×Dic3—41no—
316C2×Dic3—41no—
317C2×Dic3—42no—
318C2×Dic3—42no—
319C2×Dic3—42no—
320C2×Dic3—41no—
321C2×Dic3—42no—
322C2×Dic3—41no—
323C2×Dic3—41no—
324C2×Dic3—41no—
325C2×Dic3—42no—
326C2×Dic3—41no—
327C2×Dic3—42no—
328C2×Dic3—41no—
329C2×Dic3—41no—
330C2×Dic3—41no—
331C2×Dic3—41no—
332C2×Dic3—41no—
333C2×Dic3—42no—
334C2×Dic3—42no—
335C2×Dic3—42no—
336C2×Dic3—42no—
337C2×Dic3—42no—
338C2×Dic3—42no—
339C2×Dic3—42no—
340C2×Dic3—42no—
341C2×Dic3—42no—
342C24C23×D311yescharacteristic
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—32no—
363C22×C4—32no—
364C22×C4—34no—
365C22×C4—61no—
366C22×C4—61no—
367C22×C4—62no—
368C22×C4—62no—
369C22×C4—61no—
370C22×C4—62no—
371C22×C4—62no—
372C22×C4—62no—
373C22×C4—61no—
374C22×C4—61no—
375C22×C4—61no—
376C22×C4—62no—
377C22×C4—61no—
378C22×C4—62no—
379C22×C4—62no—
380C22×C4—61no—
381C22×C4—61no—
382C22×C4—61no—
383C22×C4—62no—
384C22×C4—62no—
385C22×C4—62no—
386C22×C4—61no—
387C22×C4—62no—
388C22×C4—61no—
389C22×C4—62no—
390C22×C4—62no—
391C22×C4—62no—
392C22×C4—61no—
393C22×C4—62no—
394C22×C4—62no—
395C22×C4—62no—
396C22×C4—61no—
397C22×C4—62no—
398C22×C4C2×Dic614yes—
399C22×C4C2×Dic614yes—
400C22×C4C2×Dic614yes—
401C22×C4C2×D1212yes—
402C22×C4C2×D1212yes—
403C22×C4C2×D1212yes—
404C22×C4#23514yes—
405C22×C4#23512yes—
406C22×C4D3×D412yes—
407C22×C4#23912yes—
408C22×C4C2×#6714yes—
409C22×C4C2×#6712yes—
410C4⋊C4—216no—
411C4⋊C4—68no—
412C4⋊C4—616no—
413C4⋊C4—68no—
414C4⋊C4—128no—
415C4⋊C4—128no—
416C4⋊C4—124no—
417C4⋊C4—124no—
418C42—28no—
419C42—48no—
420C22×C3C2×#12111yescharacteristic
421C22×C3C2×#12111yescharacteristic
422C22×C3C2×C22⋊Q812yes—
423C22×C3#52511yescharacteristic
424C22×C3#52512yes—
425C22×C3#52912yes—
426C22×C3C2×C4⋊Q811yescharacteristic
427C22×C3#53211yescharacteristic
428C22×C3#53312yes—
429C22×C3#53611yescharacteristic
430C22×C3#53711yescharacteristic
431C22×C3#53912yes—
432C22×C3#54112yes—
433C22×C3D4211yescharacteristic
434C22×C3#54711yescharacteristic
435C22×C3#54712yes—
436C22×C3#54811yescharacteristic
437C22×C3D4×Q812yes—
438C22×C3#55212yes—
439C22×C3#55311yescharacteristic
440C22×C3#55412yes—
441C22×C3#55712yes—
442C22×C3#55712yes—
443C3×C4—24no—
444C3×C4—24no—
445C3×C4—24no—
446C3×C4—28no—
447C3×C4—44no—
448Dic3—42no—
449Dic3—44no—
450Dic3—42no—
451Dic3—81no—
452Dic3—81no—
453Dic3—82no—
454C23C22×Dic612yes—
455C23C22×D1211yescharacteristic
456C23C2×#23511yescharacteristic
457C23C2×#23512yes—
458C23C2×D3×D411yescharacteristic
459C23#100411yescharacteristic
460C23C2×#23911yescharacteristic
461C23#100711yescharacteristic
462C23D3×#4112yes—
463C23#101012yes—
464C23C22×#6711yescharacteristic
465C2×C4—24no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—62no—
493C2×C4—62no—
494C2×C4—62no—
495C2×C4—64no—
496C2×C4—62no—
497C2×C4—64no—
498C2×C4—64no—
499C2×C4—64no—
500C2×C4—121no—
501C2×C4—122no—
502C2×C4—122no—
503C2×C4—121no—
504C2×C4—121no—
505C2×C4—121no—
506C2×C4—121no—
507C2×C4—122no—
508C2×C4—121no—
509C2×C4—122no—
510C2×C4—122no—
511C2×C4—121no—
512C2×C4—122no—
513C2×C4—122no—
514C2×C4—122no—
515C2×C4—121no—
516C2×C4—122no—
517C2×C4—121no—
518C2×C4—121no—
519C2×C4—121no—
520C2×C4—122no—
521C2×C4—122no—
522C2×C4—121no—
523C2×C4—122no—
524C2×C4—122no—
525C2×C4—122no—
526C2×C4—122no—
527C2×C4—122no—
528C2×C4#86914yes—
529C2×C4#89514yes—
530C2×C4#92418yes—
531C2×C4#93014yes—
532C2×C3#239211yescharacteristic
533C2×C3#241311yescharacteristic
534C2×C3#241611yescharacteristic
535C2×C3#244912yes—
536C2×C3#245312yes—
537C2×C3#246212yes—
538C2×C3#254611yescharacteristic
539C2×C3#258411yescharacteristic
540C2×C3#258512yes—
541C2×C3#266411yescharacteristic
542C2×C3#285711yescharacteristic
543C22C2×#86911yescharacteristic
544C22#547012yes—
545C22#547711yescharacteristic
546C22C2×#89511yescharacteristic
547C22#550711yescharacteristic
548C22#551011yescharacteristic
549C22#551411yescharacteristic
550C22#552411yescharacteristic
551C22#552611yescharacteristic
552C22#553112yes—
553C22D4×Dic612yes—
554C22#553912yes—
555C22#554712yes—
556C22#556412yes—
557C22#556912yes—
558C22#557012yes—
559C22#558012yes—
560C22#558312yes—
561C22C2×#92412yes—
562C22C2×#93011yescharacteristic
563C22D4×#6711yescharacteristic
564C22#581711yescharacteristic
565C22#582412yes—
566C4—24no—
567C4—24no—
568C4—24no—
569C4—28no—
570C4—44no—
571C4—122no—
572C4—122no—
573C4—124no—
574C4—242no—
575C4—241no—
576C4—241no—
577C3#1481811yescharacteristic, complemented, Sylow
578C2#7367812yes—
579C2#7371111yescharacteristic
580C2#7372711yescharacteristic
581C2#7372912yes—
582C2#7374711yescharacteristic
583C2#7377611yescharacteristic
584C2#8207611yescharacteristic
585C2#8219311yescharacteristic
586C2#8224611yescharacteristic
587C2#8236012yes—
588C2#8247112yes—
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC2×C3×C4×C4⋊C4C2
hypercenterC2×C4×C4⋊C4D3
hypocenterC3#14818
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
577. C3#14818semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×C4⋊C42
C2×#49282
C2×#49332
C2×#46612
#715652
#715362
#694232
#694192
#694102
#148183