Finite Group Databasebeta

← #10720447

Subgroups — #10720447

Order 768.

1690 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#879C211yesmaximal, characteristic
2C2×#4932C211yesmaximal, characteristic, complemented
3C2×#4649C211yesmaximal, characteristic, complemented
4C3×#1437C211yesmaximal, characteristic
5#71547C212yesmaximal
6#71545C212yesmaximal
7#71545C211yesmaximal, characteristic
8#71536C211yesmaximal, characteristic, complemented
9#69411C212yesmaximal, complemented
10#69406C212yesmaximal, complemented
11#69405C211yesmaximal, characteristic, complemented
12#14166—31nomaximal, Sylow
13C22×C3×C22⋊C4C2212yes—
14C22×C3×C22⋊C4C2211yescharacteristic
15C22×C3×C42C2211yescharacteristic
16C22×#217C2212yes—
17C22×#217C2211yescharacteristic
18C22×#217C2211yescharacteristic
19C22×C4⋊Dic3C2211yescharacteristic
20C22×Dic3⋊C4C2212yes—
21C22×Dic3⋊C4C2211yescharacteristic
22C22×Dic3⋊C4C2211yescharacteristic
23C22×C4×Dic3C2211yescharacteristic
24C22×C4×Dic3C2212yes—
25C22×C4×Dic3C2211yescharacteristic
26C2×#879—24no—
27C2×#879—24no—
28C2×#879C418yescomplemented
29C2×C3×#95C2211yescharacteristic
30C2×C3×#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2211yescharacteristic
38C2×C3⋊#95C2212yes—
39#4932—24no—
40#4649—24no—
41C22×#126—31no—
42C2×C4×C4⋊C4—31no—
43C2×#376—31no—
44#1441—32no—
45#1439—32no—
46#1437—32no—
47#1437—31no—
48#1437D311yescharacteristic
49#1436—31no—
50#1436—31no—
51#1436—32no—
52C25×C3C2311yescharacteristic
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2312yes—
55C23×C3×C4C2312yes—
56C23×C3×C4C2311yescharacteristic
57C23×Dic3C2311yescharacteristic
58C23×Dic3C2311yescharacteristic
59C23×Dic3C2312yes—
60C23×Dic3C2312yes—
61C23×Dic3C2311yescharacteristic
62C23×Dic3C2311yescharacteristic
63C2×C3×C22⋊C4—22no—
64C2×C3×C22⋊C4—22no—
65C2×C3×C22⋊C4—24no—
66C2×C3×C22⋊C4—24no—
67C2×C3×C22⋊C4—42no—
68C2×C3×C22⋊C4C2×C414yes—
69C2×C3×C42—22no—
70C2×C3×C42—24no—
71C2×#217—22no—
72C2×#217—22no—
73C2×#217—22no—
74C2×#217—22no—
75C2×#217—24no—
76C2×#217—24no—
77C2×#217—42no—
78C2×#217C2×C414yes—
79C2×#217C2×C414yes—
80C2×C4⋊Dic3—22no—
81C2×C4⋊Dic3—22no—
82C2×C4⋊Dic3C2×C414yes—
83C2×Dic3⋊C4—22no—
84C2×Dic3⋊C4—22no—
85C2×Dic3⋊C4—24no—
86C2×Dic3⋊C4—24no—
87C2×Dic3⋊C4—22no—
88C2×Dic3⋊C4—22no—
89C2×Dic3⋊C4—24no—
90C2×Dic3⋊C4C2×C414yes—
91C2×Dic3⋊C4C2×C414yes—
92C2×C4×Dic3—22no—
93C2×C4×Dic3—24no—
94C2×C4×Dic3—24no—
95C2×C4×Dic3—42no—
96C2×C4×Dic3—42no—
97C2×C4×Dic3C2×C414yes—
98C2×C4×Dic3D414yes—
99C2×C4×Dic3D414yes—
100#879—232no—
101C3×#95—22no—
102C3×#95—42no—
103C3⋊#95—22no—
104C3⋊#95—24no—
105C3⋊#95—24no—
106C3⋊#95—24no—
107C3⋊#95—24no—
108C3⋊#95—24no—
109C3⋊#95—24no—
110C3⋊#95—42no—
111C2×#126—38no—
112C2×#126—64no—
113C2×#126—64no—
114C22×C4⋊C4—31no—
115C22×C4⋊C4—31no—
116C22×C4⋊C4—31no—
117C22×C4⋊C4—32no—
118C22×C22⋊C4—31no—
119C22×C22⋊C4—31no—
120C22×C22⋊C4—32no—
121C22×C22⋊C4C2×D312yes—
122C22×C22⋊C4C2×D311yescharacteristic
123C22×C42—31no—
124C22×C42—31no—
125C22×C42—32no—
126C22×C42C2×D311yescharacteristic
127C4×C4⋊C4—64no—
128#376—64no—
129C2×#95—32no—
130C2×#95—32no—
131C2×#95—32no—
132C2×#95—32no—
133C2×#95—32no—
134C2×#95—31no—
135C2×#95—32no—
136C2×#95—32no—
137C2×#95C2×D311yescharacteristic
138C2×#95C2×D312yes—
139C24×C3—21no—
140C24×C3—21no—
141C24×C3—22no—
142C24×C3—22no—
143C24×C3—22no—
144C24×C3—22no—
145C24×C3—22no—
146C24×C3—22no—
147C24×C3C22×C412yes—
148C24×C3C2411yescharacteristic
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4C22×C412yes—
168C22×C3×C4C22×C412yes—
169C22×C3×C4C2×D412yes—
170C22×C3×C4C2×D412yes—
171C22×C3×C4#4114yes—
172C22×C3×C4#4114yes—
173C22×Dic3—22no—
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—22no—
184C22×Dic3—22no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—22no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3—22no—
197C22×Dic3C22×C412yes—
198C22×Dic3C22×C412yes—
199C22×Dic3C22×C412yes—
200C22×Dic3C22×C412yes—
201C22×Dic3C2×D412yes—
202C22×Dic3C2×D412yes—
203C22×Dic3C2×D412yes—
204C22×Dic3C2×D412yes—
205C22×Dic3#4114yes—
206C22×Dic3#4114yes—
207C3×C22⋊C4—28no—
208C3×C22⋊C4—48no—
209C3×C42—28no—
210#217—28no—
211#217—28no—
212#217—48no—
213C4⋊Dic3—28no—
214Dic3⋊C4—216no—
215Dic3⋊C4—28no—
216Dic3⋊C4—28no—
217C4×Dic3—28no—
218C4×Dic3—48no—
219C4×Dic3C22⋊C4116yescomplemented
220C25C22×D311yescharacteristic
221C23×C4—31no—
222C23×C4—31no—
223C23×C4—31no—
224C23×C4—31no—
225C23×C4—32no—
226C23×C4—32no—
227C23×C4C22×D311yescharacteristic
228C23×C4C22×D312yes—
229C23×C4C22×D312yes—
230C23×C4C22×D311yescharacteristic
231#126—632no—
232C2×C4⋊C4—34no—
233C2×C4⋊C4—34no—
234C2×C4⋊C4—34no—
235C2×C4⋊C4—62no—
236C2×C4⋊C4—62no—
237C2×C4⋊C4—62no—
238C2×C4⋊C4—62no—
239C2×C4⋊C4—64no—
240C2×C4⋊C4—64no—
241C2×C4⋊C4—64no—
242C2×C4⋊C4—62no—
243C2×C4⋊C4—62no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—34no—
249C2×C22⋊C4—34no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—62no—
252C2×C22⋊C4—62no—
253C2×C22⋊C4—62no—
254C2×C22⋊C4—62no—
255C2×C22⋊C4—64no—
256C2×C22⋊C4—64no—
257C2×C22⋊C4—122no—
258C2×C22⋊C4C4×D314yes—
259C2×C42—22no—
260C2×C42—24no—
261C2×C42—34no—
262C2×C42—34no—
263C2×C42—34no—
264C2×C42—64no—
265C2×C42—62no—
266C2×C42—64no—
267C2×C42—122no—
268C2×C42—122no—
269#95—22no—
270#95—42no—
271#95—64no—
272#95—64no—
273#95—64no—
274#95—64no—
275#95—64no—
276#95—62no—
277#95—64no—
278#95—122no—
279C23×C3—22no—
280C23×C3—22no—
281C23×C3—22no—
282C23×C3—24no—
283C23×C3—24no—
284C23×C3—41no—
285C23×C3—42no—
286C23×C3—41no—
287C23×C3—42no—
288C23×C3—42no—
289C23×C3—41no—
290C23×C3—41no—
291C23×C3—42no—
292C23×C3—42no—
293C23×C3—42no—
294C23×C3—42no—
295C23×C3—42no—
296C23×C3—42no—
297C23×C3—42no—
298C23×C3—42no—
299C23×C3—42no—
300C23×C3C23×C411yescharacteristic
301C23×C3C22×D411yescharacteristic
302C23×C3C22×D411yescharacteristic
303C23×C3C2×#4112yes—
304C23×C3C2×#4112yes—
305C23×C3C2×#4112yes—
306C23×C3C2×#4112yes—
307C23×C3H(5,𝔽2)12yes—
308C23×C32−1+412yes—
309C2×C3×C4—24no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—24no—
313C2×C3×C4—22no—
314C2×C3×C4—24no—
315C2×C3×C4—24no—
316C2×C3×C4—24no—
317C2×C3×C4—24no—
318C2×C3×C4—24no—
319C2×C3×C4—24no—
320C2×C3×C4—24no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—41no—
325C2×C3×C4—41no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4C2×C22⋊C414yes—
333C2×Dic3—22no—
334C2×Dic3—22no—
335C2×Dic3—24no—
336C2×Dic3—24no—
337C2×Dic3—24no—
338C2×Dic3—22no—
339C2×Dic3—22no—
340C2×Dic3—24no—
341C2×Dic3—22no—
342C2×Dic3—22no—
343C2×Dic3—22no—
344C2×Dic3—22no—
345C2×Dic3—24no—
346C2×Dic3—24no—
347C2×Dic3—24no—
348C2×Dic3—24no—
349C2×Dic3—24no—
350C2×Dic3—24no—
351C2×Dic3—24no—
352C2×Dic3—41no—
353C2×Dic3—42no—
354C2×Dic3—41no—
355C2×Dic3—42no—
356C2×Dic3—42no—
357C2×Dic3—42no—
358C2×Dic3—42no—
359C2×Dic3—42no—
360C2×Dic3—42no—
361C2×Dic3—42no—
362C2×Dic3—42no—
363C2×Dic3—42no—
364C2×Dic3—42no—
365C2×Dic3C2×C22⋊C414yes—
366C2×Dic3C2×C22⋊C414yes—
367C24—22no—
368C24—21no—
369C24—21no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24C2×C4×D312yes—
376C24C23×D311yescharacteristic
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—32no—
396C22×C4—32no—
397C22×C4—32no—
398C22×C4—32no—
399C22×C4—32no—
400C22×C4—32no—
401C22×C4—32no—
402C22×C4—32no—
403C22×C4—34no—
404C22×C4—34no—
405C22×C4—62no—
406C22×C4—62no—
407C22×C4—62no—
408C22×C4—62no—
409C22×C4—62no—
410C22×C4—62no—
411C22×C4—62no—
412C22×C4—62no—
413C22×C4—62no—
414C22×C4—61no—
415C22×C4—61no—
416C22×C4—61no—
417C22×C4—61no—
418C22×C4—62no—
419C22×C4—62no—
420C22×C4—62no—
421C22×C4—62no—
422C22×C4—62no—
423C22×C4—62no—
424C22×C4—62no—
425C22×C4—62no—
426C22×C4—62no—
427C22×C4—62no—
428C22×C4—62no—
429C22×C4C2×C4×D312yes—
430C22×C4C2×C4×D312yes—
431C22×C4#23514yes—
432C22×C4D3×D412yes—
433C22×C4D3×D412yes—
434C22×C4#23714yes—
435C4⋊C4—68no—
436C4⋊C4—68no—
437C4⋊C4—616no—
438C4⋊C4—68no—
439C22⋊C4—28no—
440C22⋊C4—48no—
441C22⋊C4—68no—
442C22⋊C4—68no—
443C22⋊C4—128no—
444C42—28no—
445C42—316no—
446C42—68no—
447C42—128no—
448C22×C3—42no—
449C22×C3—44no—
450C22×C3—44no—
451C22×C3—42no—
452C22×C3—42no—
453C22×C3—82no—
454C22×C3—82no—
455C22×C3—82no—
456C22×C3—82no—
457C22×C3C22×C22⋊C411yescharacteristic
458C22×C3C4×#4111yescharacteristic
459C22×C3C4×#4111yescharacteristic
460C22×C3C4×#4112yes—
461C22×C3#52012yes—
462C22×C3#52511yescharacteristic
463C22×C3#52512yes—
464C22×C3#52511yescharacteristic
465C22×C3#53212yes—
466C22×C3#53212yes—
467C22×C3#53511yescharacteristic
468C22×C3#53611yescharacteristic
469C22×C3#53712yes—
470C22×C3#55112yes—
471C22×C3#55112yes—
472C22×C3#55212yes—
473C22×C3#55212yes—
474C22×C3#55312yes—
475C22×C3#55312yes—
476C22×C3#55612yes—
477C22×C3#55612yes—
478C3×C4—24no—
479C3×C4—42no—
480C3×C4—44no—
481C3×C4—44no—
482Dic3—24no—
483Dic3—24no—
484Dic3—44no—
485Dic3—44no—
486Dic3—42no—
487Dic3—42no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—24no—
492C23—24no—
493C23—42no—
494C23—41no—
495C23—41no—
496C23—42no—
497C23—42no—
498C23—41no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—41no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23C22×C4×D311yescharacteristic
510C23C2×#23512yes—
511C23C2×D3×D411yescharacteristic
512C23C2×D3×D411yescharacteristic
513C23C2×#23712yes—
514C23#100412yes—
515C23D3×#4112yes—
516C23D3×#4112yes—
517C23#101012yes—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—34no—
531C2×C4—34no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—62no—
544C2×C4—62no—
545C2×C4—62no—
546C2×C4—62no—
547C2×C4—64no—
548C2×C4—64no—
549C2×C4—64no—
550C2×C4—64no—
551C2×C4—64no—
552C2×C4—64no—
553C2×C4—64no—
554C2×C4—62no—
555C2×C4—62no—
556C2×C4—62no—
557C2×C4—62no—
558C2×C4—64no—
559C2×C4—64no—
560C2×C4—64no—
561C2×C4—64no—
562C2×C4—121no—
563C2×C4—122no—
564C2×C4—122no—
565C2×C4—122no—
566C2×C4—122no—
567C2×C4—122no—
568C2×C4—122no—
569C2×C4—122no—
570C2×C4—122no—
571C2×C4—121no—
572C2×C4—122no—
573C2×C4—122no—
574C2×C4—122no—
575C2×C4D3×C22⋊C414yes—
576C2×C3—82no—
577C2×C3#229711yescharacteristic
578C2×C3#229711yescharacteristic
579C2×C3#231111yescharacteristic
580C2×C3#237312yes—
581C2×C3#237312yes—
582C2×C3#260112yes—
583C2×C3#260112yes—
584C2×C3#260312yes—
585C2×C3#260412yes—
586C22—44no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—44no—
591C22—82no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22C4×#23511yescharacteristic
596C22C2×D3×C22⋊C411yescharacteristic
597C22#549512yes—
598C22#552211yescharacteristic
599C22#552912yes—
600C22C4×#23712yes—
601C22#554012yes—
602C22#554212yes—
603C22#554612yes—
604C22#554712yes—
605C22#554812yes—
606C22#558611yescharacteristic
607C22#559612yes—
608C22#560812yes—
609C22#561412yes—
610C22#565012yes—
611C22#565611yescharacteristic
612C22#567511yescharacteristic
613C22#567912yes—
614C22#570212yes—
615C22#571511yescharacteristic
616C4—24no—
617C4—42no—
618C4—44no—
619C4—44no—
620C4—64no—
621C4—64no—
622C4—124no—
623C4—122no—
624C4—124no—
625C4—122no—
626C3#1416611yescharacteristic, complemented, Sylow
627C2—82no—
628C2#7347712yes—
629C2#7356311yescharacteristic
630C2#7465811yescharacteristic
631C2#7477712yes—
632C2#7548512yes—
633C2#7564812yes—
634C2#7589712yes—
635C2#7629411yescharacteristic
636C2#7637112yes—
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC3×#1437C2
socle, cosocleC24×C3C24
prime radicalC25×C3C23
hypercenter#1437D3
hypocenterC3#14166
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1437

Complements

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

NH ≅ G/Ntype
2. C2×#4932C2semidirect
3. C2×#4649C2semidirect
8. #71536C2semidirect
9. #69411C2semidirect
10. #69406C2semidirect
11. #69405C2semidirect
28. C2×#879C4semidirect
219. C4×Dic3C22⋊C4semidirect
626. C3#14166semidirect

Maximal subgroups

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

subgroupindex
C22×#8792
C2×#49322
C2×#46492
C3×#14372
#715472
#715452
#715362
#694112
#694062
#694052
#141663