Finite Group Databasebeta

← #10720440

Subgroups — #10720440

Order 768.

1646 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4943C211yesmaximal, characteristic
2C2×Dic3×C22⋊C4C212yesmaximal
3C3×#1439C211yesmaximal, characteristic
4#71564C212yesmaximal, complemented
5#71547C212yesmaximal
6#71546C211yesmaximal, characteristic
7#69407C212yesmaximal, complemented
8#69406C212yesmaximal, complemented
9#69405C212yesmaximal, complemented
10#14165—31nomaximal, Sylow
11C24×Dic3C2211yescharacteristic
12C22×C3×C4⋊C4C2211yescharacteristic
13C22×C3×C22⋊C4C2212yes—
14C22×C3×C22⋊C4C2211yescharacteristic
15C22×#217C2212yes—
16C22×#217C2211yescharacteristic
17C22×C4⋊Dic3C2211yescharacteristic
18C22×Dic3⋊C4C2212yes—
19C22×C4×Dic3C2212yes—
20C22×C4×Dic3C2212yes—
21C2×C3×#95C2212yes—
22C2×C3×#95C2211yescharacteristic
23C2×C3⋊#95C2212yes—
24C2×C3⋊#95C2212yes—
25C2×C3⋊#95C2212yes—
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2211yescharacteristic
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32#4943C418yescomplemented
33Dic3×C22⋊C4—28no—
34C2×#381—31no—
35C2×C4×C22⋊C4—32no—
36#1439—31no—
37#1439—32no—
38#1439D311yescharacteristic
39#1436—32no—
40#1436—32no—
41#1436—32no—
42#1436—32no—
43C25×C3C2311yescharacteristic
44C23×C3×C4C2311yescharacteristic
45C23×C3×C4C2311yescharacteristic
46C23×C3×C4C2312yes—
47C23×C3×C4C2312yes—
48C23×Dic3—41no—
49C23×Dic3—41no—
50C23×Dic3—41no—
51C23×Dic3—41no—
52C23×Dic3C2×C414yes—
53C23×Dic3D414yes—
54C23×Dic3Q814yes—
55C23×Dic3C2312yes—
56C23×Dic3C2312yes—
57C23×Dic3C2312yes—
58C23×Dic3C2312yes—
59C2×C3×C4⋊C4—24no—
60C2×C3×C4⋊C4—41no—
61C2×C3×C22⋊C4—24no—
62C2×C3×C22⋊C4—24no—
63C2×C3×C22⋊C4—24no—
64C2×C3×C22⋊C4—41no—
65C2×C3×C22⋊C4—41no—
66C2×C3×C22⋊C4C2×C414yes—
67C2×#217—24no—
68C2×#217—24no—
69C2×#217—24no—
70C2×#217—41no—
71C2×#217—41no—
72C2×#217C2×C414yes—
73C2×C4⋊Dic3—24no—
74C2×C4⋊Dic3—41no—
75C2×Dic3⋊C4—24no—
76C2×Dic3⋊C4—24no—
77C2×Dic3⋊C4—42no—
78C2×C4×Dic3—24no—
79C2×C4×Dic3—24no—
80C2×C4×Dic3—42no—
81C2×C4×Dic3—42no—
82C2×C4×Dic3—42no—
83C2×C4×Dic3—42no—
84C3×#95—24no—
85C3×#95—41no—
86C3⋊#95—24no—
87C3⋊#95—24no—
88C3⋊#95—24no—
89C3⋊#95—24no—
90C3⋊#95—24no—
91C3⋊#95—41no—
92C3⋊#95—42no—
93C3⋊#95C2×C418yes—
94C3⋊#95C2×C418yes—
95C24×C4—31no—
96C22×C4⋊C4—32no—
97C22×C4⋊C4—31no—
98C22×C4⋊C4C2×D311yescharacteristic
99C22×C22⋊C4—31no—
100C22×C22⋊C4—32no—
101C22×C22⋊C4C2×D312yes—
102C22×C22⋊C4C2×D311yescharacteristic
103C22×C42—32no—
104C22×C42—32no—
105#381—38no—
106C4×C22⋊C4—68no—
107C2×#95—32no—
108C2×#95—32no—
109C2×#95—32no—
110C2×#95—32no—
111C2×#95—32no—
112C2×#95—32no—
113C2×#95—32no—
114C2×#95—32no—
115C2×#95—31no—
116C2×#95C2×D312yes—
117C2×#95C2×D311yescharacteristic
118C24×C3—22no—
119C24×C3—22no—
120C24×C3—22no—
121C24×C3—22no—
122C24×C3—21no—
123C24×C3—21no—
124C24×C3—21no—
125C24×C3—21no—
126C24×C3C22×C412yes—
127C24×C3C2×D412yes—
128C24×C3C2×Q812yes—
129C24×C3C2411yescharacteristic
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—21no—
144C22×C3×C4—22no—
145C22×C3×C4—21no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—21no—
149C22×C3×C4—21no—
150C22×C3×C4C22×C412yes—
151C22×C3×C4C22×C412yes—
152C22×C3×C4#4114yes—
153C22×C3×C4#4114yes—
154C22×Dic3—22no—
155C22×Dic3—22no—
156C22×Dic3—22no—
157C22×Dic3—22no—
158C22×Dic3—22no—
159C22×Dic3—22no—
160C22×Dic3—22no—
161C22×Dic3—22no—
162C22×Dic3—22no—
163C22×Dic3—22no—
164C22×Dic3—22no—
165C22×Dic3—22no—
166C22×Dic3—22no—
167C22×Dic3—22no—
168C22×Dic3—22no—
169C22×Dic3—22no—
170C22×Dic3—22no—
171C22×Dic3—22no—
172C22×Dic3—22no—
173C22×Dic3—22no—
174C22×Dic3—22no—
175C22×Dic3—22no—
176C22×Dic3—44no—
177C22×Dic3—44no—
178C22×Dic3—44no—
179C22×Dic3—44no—
180C22×Dic3—44no—
181C22×Dic3—44no—
182C22×Dic3C4⋊C4116yescomplemented
183C22×Dic3C22×C414yes—
184C22×Dic3C22×C414yes—
185C22×Dic3C2×D414yes—
186C22×Dic3C2×Q814yes—
187C22×Dic3#4114yes—
188C22×Dic3#4114yes—
189C3×C4⋊C4—44no—
190C3×C22⋊C4—216no—
191C3×C22⋊C4—44no—
192#217—216no—
193#217—44no—
194C4⋊Dic3—44no—
195Dic3⋊C4—48no—
196C4×Dic3—48no—
197C4×Dic3—48no—
198C25C22×D311yescharacteristic
199C23×C4—32no—
200C23×C4—34no—
201C23×C4—34no—
202C23×C4—34no—
203C23×C4—32no—
204C23×C4—32no—
205C23×C4—32no—
206C23×C4—121no—
207C23×C4—121no—
208C23×C4—121no—
209C23×C4—121no—
210C23×C4C22×D311yescharacteristic
211C23×C4C22×D311yescharacteristic
212C23×C4C22×D312yes—
213C23×C4C22×D312yes—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—64no—
217C2×C4⋊C4—64no—
218C2×C4⋊C4—64no—
219C2×C4⋊C4—122no—
220C2×C4⋊C4—121no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—34no—
225C2×C22⋊C4—41no—
226C2×C22⋊C4—41no—
227C2×C22⋊C4—64no—
228C2×C22⋊C4—64no—
229C2×C22⋊C4—64no—
230C2×C22⋊C4—121no—
231C2×C22⋊C4—121no—
232C2×C22⋊C4C4×D314yes—
233C2×C42—64no—
234C2×C42—64no—
235C2×C42—122no—
236C2×C42—122no—
237C2×C42—122no—
238C2×C42—122no—
239#95—24no—
240#95—38no—
241#95—38no—
242#95—41no—
243#95—64no—
244#95—64no—
245#95—64no—
246#95—64no—
247#95—64no—
248#95—121no—
249#95—122no—
250C23×C3—24no—
251C23×C3—24no—
252C23×C3—22no—
253C23×C3—22no—
254C23×C3—24no—
255C23×C3—24no—
256C23×C3—24no—
257C23×C3—24no—
258C23×C3—22no—
259C23×C3—22no—
260C23×C3—22no—
261C23×C3—22no—
262C23×C3—42no—
263C23×C3—41no—
264C23×C3—41no—
265C23×C3—42no—
266C23×C3—42no—
267C23×C3—42no—
268C23×C3—42no—
269C23×C3—41no—
270C23×C3—42no—
271C23×C3—41no—
272C23×C3C2×C4⋊C414yes—
273C23×C3C23×C411yescharacteristic
274C23×C3C22×D411yescharacteristic
275C23×C3C22×Q811yescharacteristic
276C23×C3C2×#4112yes—
277C23×C3C2×#4112yes—
278C23×C3C2×#4112yes—
279C23×C3C2×#4112yes—
280C23×C3H(5,𝔽2)11yescharacteristic
281C23×C3H(5,𝔽2)11yescharacteristic
282C23×C3H(5,𝔽2)11yescharacteristic
283C23×C3H(5,𝔽2)11yescharacteristic
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—24no—
291C2×C3×C4—24no—
292C2×C3×C4—24no—
293C2×C3×C4—24no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—41no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4—41no—
308C2×Dic3—24no—
309C2×Dic3—24no—
310C2×Dic3—24no—
311C2×Dic3—24no—
312C2×Dic3—24no—
313C2×Dic3—24no—
314C2×Dic3—24no—
315C2×Dic3—24no—
316C2×Dic3—24no—
317C2×Dic3—24no—
318C2×Dic3—24no—
319C2×Dic3—24no—
320C2×Dic3—24no—
321C2×Dic3—24no—
322C2×Dic3—24no—
323C2×Dic3—416no—
324C2×Dic3—42no—
325C2×Dic3—42no—
326C2×Dic3—42no—
327C2×Dic3—42no—
328C2×Dic3—42no—
329C2×Dic3—42no—
330C2×Dic3—42no—
331C2×Dic3—42no—
332C2×Dic3—42no—
333C2×Dic3—42no—
334C2×Dic3—42no—
335C2×Dic3—42no—
336C2×Dic3C2×C4⋊C418yes—
337C24—22no—
338C24—22no—
339C24—21no—
340C24—21no—
341C24—22no—
342C24—22no—
343C24—21no—
344C24—21no—
345C24C2×C4×D312yes—
346C24D3×D412yes—
347C24D3×Q812yes—
348C24C23×D311yescharacteristic
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—34no—
370C22×C4—34no—
371C22×C4—34no—
372C22×C4—316no—
373C22×C4—34no—
374C22×C4—34no—
375C22×C4—34no—
376C22×C4—62no—
377C22×C4—62no—
378C22×C4—62no—
379C22×C4—62no—
380C22×C4—62no—
381C22×C4—62no—
382C22×C4—62no—
383C22×C4—62no—
384C22×C4—62no—
385C22×C4—62no—
386C22×C4—62no—
387C22×C4—62no—
388C22×C4—62no—
389C22×C4—62no—
390C22×C4—62no—
391C22×C4—62no—
392C22×C4—62no—
393C22×C4—62no—
394C22×C4—62no—
395C22×C4—62no—
396C22×C4—62no—
397C22×C4—62no—
398C22×C4—124no—
399C22×C4—124no—
400C22×C4—124no—
401C22×C4—124no—
402C22×C4—124no—
403C22×C4—124no—
404C22×C4C2×C4×D312yes—
405C22×C4C2×C4×D312yes—
406C22×C4#23714yes—
407C22×C4#23714yes—
408C4⋊C4—44no—
409C4⋊C4—128no—
410C4⋊C4—124no—
411C22⋊C4—216no—
412C22⋊C4—44no—
413C22⋊C4—616no—
414C22⋊C4—124no—
415C42—128no—
416C42—128no—
417C22×C3—24no—
418C22×C3—28no—
419C22×C3—42no—
420C22×C3—44no—
421C22×C3—42no—
422C22×C3—42no—
423C22×C3—42no—
424C22×C3—44no—
425C22×C3—42no—
426C22×C3—42no—
427C22×C3—44no—
428C22×C3C22×C4⋊C411yescharacteristic
429C22×C3C4×#4112yes—
430C22×C3C4×#4112yes—
431C22×C3#51811yescharacteristic
432C22×C3#51811yescharacteristic
433C22×C3#52512yes—
434C22×C3#52512yes—
435C22×C3#53312yes—
436C22×C3#53312yes—
437C22×C3#53411yescharacteristic
438C22×C3#53411yescharacteristic
439C22×C3#54311yescharacteristic
440C22×C3#54311yescharacteristic
441C22×C3#55112yes—
442C22×C3#55111yescharacteristic
443C22×C3#55111yescharacteristic
444C22×C3#55312yes—
445C22×C3#55312yes—
446C22×C3#55312yes—
447C22×C3#55312yes—
448C22×C3#55911yescharacteristic
449C22×C3#55912yes—
450C22×C3#55911yescharacteristic
451C3×C4—44no—
452C3×C4—44no—
453C3×C4—42no—
454C3×C4—42no—
455Dic3—28no—
456Dic3—44no—
457Dic3—44no—
458Dic3—44no—
459C23—22no—
460C23—22no—
461C23—24no—
462C23—22no—
463C23—22no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—22no—
468C23—22no—
469C23—24no—
470C23—24no—
471C23—42no—
472C23—41no—
473C23—41no—
474C23—42no—
475C23—42no—
476C23—41no—
477C23—41no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23D3×C4⋊C414yes—
482C23C22×C4×D311yescharacteristic
483C23C2×D3×D411yescharacteristic
484C23C2×#23712yes—
485C23C2×#23712yes—
486C23#100411yescharacteristic
487C23#100411yescharacteristic
488C23C2×D3×Q811yescharacteristic
489C23D3×#4112yes—
490C23D3×#4112yes—
491C23#100911yescharacteristic
492C23#100911yescharacteristic
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—38no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—64no—
519C2×C4—64no—
520C2×C4—64no—
521C2×C4—64no—
522C2×C4—64no—
523C2×C4—64no—
524C2×C4—64no—
525C2×C4—64no—
526C2×C4—64no—
527C2×C4—64no—
528C2×C4—64no—
529C2×C4—64no—
530C2×C4—64no—
531C2×C4—64no—
532C2×C4—64no—
533C2×C4—1216no—
534C2×C4—122no—
535C2×C4—122no—
536C2×C4—122no—
537C2×C4—122no—
538C2×C4—122no—
539C2×C4—122no—
540C2×C4—122no—
541C2×C4—122no—
542C2×C4—122no—
543C2×C4—122no—
544C2×C4—122no—
545C2×C4—122no—
546C2×C3—44no—
547C2×C3#229612yes—
548C2×C3#231511yescharacteristic
549C2×C3#237411yescharacteristic
550C2×C3#237411yescharacteristic
551C2×C3#237412yes—
552C2×C3#260011yescharacteristic
553C2×C3#260012yes—
554C2×C3#260011yescharacteristic
555C2×C3#260711yescharacteristic
556C2×C3#260711yescharacteristic
557C2×C3#260812yes—
558C22—24no—
559C22—28no—
560C22—42no—
561C22—44no—
562C22—42no—
563C22—42no—
564C22—44no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—44no—
569C22#548611yescharacteristic
570C22C2×D3×C4⋊C411yescharacteristic
571C22#552111yescharacteristic
572C22C4×#23712yes—
573C22C4×#23712yes—
574C22#558612yes—
575C22#558711yescharacteristic
576C22#560112yes—
577C22#560511yescharacteristic
578C22#560511yescharacteristic
579C22#561412yes—
580C22#561812yes—
581C22#562212yes—
582C22#562212yes—
583C22#563411yescharacteristic
584C22#563411yescharacteristic
585C22#565612yes—
586C22#565812yes—
587C22#566411yescharacteristic
588C22#567911yescharacteristic
589C22#568112yes—
590C22#570912yes—
591C22#571311yescharacteristic
592C4—42no—
593C4—44no—
594C4—44no—
595C4—42no—
596C4—68no—
597C4—124no—
598C4—124no—
599C4—124no—
600C3#1416511yescharacteristic, complemented, Sylow
601C2—44no—
602C2#7400511yescharacteristic
603C2#7437812yes—
604C2#7468111yescharacteristic
605C2#7548812yes—
606C2#7630811yescharacteristic
607C2#7669311yescharacteristic
608C2#7680711yescharacteristic
609C2#7696312yes—
610C2#7732912yes—
611C2#7753411yescharacteristic
612C2#7804611yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC3×#1439C2
socle, cosocleC24×C3C24
prime radicalC25×C3C23
hypercenter#1439D3
hypocenterC3#14165
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
4. #71564C2semidirect
7. #69407C2semidirect
8. #69406C2semidirect
9. #69405C2semidirect
32. #4943C4semidirect
182. C22×Dic3C4⋊C4semidirect
600. C3#14165semidirect

Maximal subgroups

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

subgroupindex
C2×#49432
C2×Dic3×C22⋊C42
C3×#14392
#715642
#715472
#715462
#694072
#694062
#694052
#141653