Finite Group Databasebeta

← #10720242

Subgroups — #10720242

Order 768.

1702 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4978C212yesmaximal, complemented
2C2×#4946C211yesmaximal, characteristic
3C2×Dic3×C22⋊C4C212yesmaximal
4C3×#1439C211yesmaximal, characteristic
5#71549C212yesmaximal
6#71546C211yesmaximal, characteristic
7#69407C212yesmaximal, complemented
8#69406C212yesmaximal, complemented
9#69403C212yesmaximal, complemented
10#14143—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⋊#95C2212yes—
29C2×C3⋊#95C2211yescharacteristic
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32#4978—28no—
33#4946C418yescomplemented
34Dic3×C22⋊C4—28no—
35C2×#387—32no—
36C2×#385—31no—
37C2×C4×C22⋊C4—32no—
38#1440—32no—
39#1439—31no—
40#1439D311yescharacteristic
41#1436—32no—
42#1436—32no—
43#1434—32no—
44C25×C3C2311yescharacteristic
45C23×C3×C4C2311yescharacteristic
46C23×C3×C4C2311yescharacteristic
47C23×C3×C4C2312yes—
48C23×C3×C4C2312yes—
49C23×Dic3—24no—
50C23×Dic3—24no—
51C23×Dic3—41no—
52C23×Dic3—41no—
53C23×Dic3C2×C414yes—
54C23×Dic3C2312yes—
55C23×Dic3C2312yes—
56C23×Dic3C2312yes—
57C23×Dic3C2312yes—
58C2×C3×C4⋊C4—24no—
59C2×C3×C4⋊C4—41no—
60C2×C3×C22⋊C4—24no—
61C2×C3×C22⋊C4—24no—
62C2×C3×C22⋊C4—24no—
63C2×C3×C22⋊C4—41no—
64C2×C3×C22⋊C4—41no—
65C2×C3×C22⋊C4C2×C414yes—
66C2×#217—22no—
67C2×#217—24no—
68C2×#217—22no—
69C2×#217—24no—
70C2×#217—24no—
71C2×#217C2×C414yes—
72C2×C4⋊Dic3—24no—
73C2×C4⋊Dic3—41no—
74C2×Dic3⋊C4—24no—
75C2×Dic3⋊C4—24no—
76C2×Dic3⋊C4C2×C418yes—
77C2×C4×Dic3—24no—
78C2×C4×Dic3—24no—
79C2×C4×Dic3—24no—
80C2×C4×Dic3—24no—
81C2×C4×Dic3—42no—
82C2×C4×Dic3—42no—
83C3×#95—24no—
84C3×#95—41no—
85C3⋊#95—24no—
86C3⋊#95—24no—
87C3⋊#95—24no—
88C3⋊#95—24no—
89C3⋊#95—24no—
90C3⋊#95—24no—
91C3⋊#95—41no—
92C3⋊#95C2×C418yes—
93C3⋊#95D418yes—
94C24×C4—31no—
95C22×C4⋊C4—32no—
96C22×C4⋊C4—31no—
97C22×C4⋊C4C2×D311yescharacteristic
98C22×C22⋊C4—31no—
99C22×C22⋊C4—32no—
100C22×C22⋊C4C2×D312yes—
101C22×C22⋊C4C2×D311yescharacteristic
102C22×C42—32no—
103C22×C42—32no—
104#387—68no—
105#385—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×C3#4112yes—
128C24×C3#4112yes—
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—21no—
143C22×C3×C4—22no—
144C22×C3×C4—21no—
145C22×C3×C4—22no—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4C22×C412yes—
151C22×C3×C4C22×C412yes—
152C22×C3×C4C2×D414yes—
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—216no—
177C22×Dic3—44no—
178C22×Dic3—44no—
179C22×Dic3—48no—
180C22×Dic3—44no—
181C22×Dic3C22×C414yes—
182C22×Dic3C22×C414yes—
183C22×Dic3C2×D414yes—
184C22×Dic3C2×D414yes—
185C22×Dic3#4114yes—
186C22×Dic3#4114yes—
187C3×C4⋊C4—44no—
188C3×C22⋊C4—216no—
189C3×C22⋊C4—44no—
190#217—216no—
191#217—28no—
192C4⋊Dic3—44no—
193Dic3⋊C4—216no—
194C4×Dic3—48no—
195C4×Dic3—48no—
196C25C22×D311yescharacteristic
197C23×C4—32no—
198C23×C4—32no—
199C23×C4—32no—
200C23×C4—32no—
201C23×C4—34no—
202C23×C4—64no—
203C23×C4—64no—
204C23×C4—121no—
205C23×C4—121no—
206C23×C4C22×D311yescharacteristic
207C23×C4C22×D311yescharacteristic
208C23×C4C22×D312yes—
209C23×C4C22×D312yes—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—38no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—64no—
214C2×C4⋊C4—64no—
215C2×C4⋊C4—64no—
216C2×C4⋊C4—121no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—34no—
221C2×C22⋊C4—41no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4—64no—
224C2×C22⋊C4—64no—
225C2×C22⋊C4—62no—
226C2×C22⋊C4—64no—
227C2×C22⋊C4—62no—
228C2×C22⋊C4C4×D314yes—
229C2×C42—64no—
230C2×C42—64no—
231C2×C42—64no—
232C2×C42—64no—
233C2×C42—122no—
234C2×C42—122no—
235#95—24no—
236#95—38no—
237#95—38no—
238#95—41no—
239#95—64no—
240#95—64no—
241#95—64no—
242#95—64no—
243#95—64no—
244#95—64no—
245#95—121no—
246C23×C3—24no—
247C23×C3—24no—
248C23×C3—22no—
249C23×C3—22no—
250C23×C3—24no—
251C23×C3—24no—
252C23×C3—24no—
253C23×C3—24no—
254C23×C3—22no—
255C23×C3—22no—
256C23×C3—22no—
257C23×C3—22no—
258C23×C3—41no—
259C23×C3—42no—
260C23×C3—42no—
261C23×C3—42no—
262C23×C3—41no—
263C23×C3—41no—
264C23×C3—42no—
265C23×C3—42no—
266C23×C3—42no—
267C23×C3—41no—
268C23×C3#11714yes—
269C23×C3C23×C411yescharacteristic
270C23×C3C22×D412yes—
271C23×C3C2×#4111yescharacteristic
272C23×C3C2×#4112yes—
273C23×C3C2×#4111yescharacteristic
274C23×C3C2×#4112yes—
275C23×C3C2×#4112yes—
276C23×C3H(5,𝔽2)11yescharacteristic
277C23×C3H(5,𝔽2)11yescharacteristic
278C23×C32−1+411yescharacteristic
279C23×C32−1+411yescharacteristic
280C2×C3×C4—24no—
281C2×C3×C4—24no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—24no—
285C2×C3×C4—24no—
286C2×C3×C4—24no—
287C2×C3×C4—24no—
288C2×C3×C4—24no—
289C2×C3×C4—24no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—41no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—42no—
302C2×C3×C4—41no—
303C2×C3×C4—42no—
304C2×Dic3—24no—
305C2×Dic3—24no—
306C2×Dic3—24no—
307C2×Dic3—24no—
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—416no—
320C2×Dic3—42no—
321C2×Dic3—42no—
322C2×Dic3—42no—
323C2×Dic3—42no—
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×Dic3C4×D418yes—
333C24—22no—
334C24—22no—
335C24—21no—
336C24—21no—
337C24—22no—
338C24—22no—
339C24—21no—
340C24—21no—
341C24C2×C4×D312yes—
342C24#23712yes—
343C24#23912yes—
344C24C23×D311yescharacteristic
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—34no—
366C22×C4—34no—
367C22×C4—34no—
368C22×C4—34no—
369C22×C4—34no—
370C22×C4—34no—
371C22×C4—616no—
372C22×C4—62no—
373C22×C4—62no—
374C22×C4—62no—
375C22×C4—62no—
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—124no—
395C22×C4—124no—
396C22×C4—124no—
397C22×C4—128no—
398C22×C4C2×C4×D312yes—
399C22×C4C2×C4×D312yes—
400C22×C4D3×D414yes—
401C22×C4#23714yes—
402C4⋊C4—44no—
403C4⋊C4—616no—
404C4⋊C4—124no—
405C22⋊C4—216no—
406C22⋊C4—44no—
407C22⋊C4—616no—
408C22⋊C4—68no—
409C42—128no—
410C42—128no—
411C22×C3—24no—
412C22×C3—28no—
413C22×C3—42no—
414C22×C3—44no—
415C22×C3—42no—
416C22×C3—42no—
417C22×C3—44no—
418C22×C3—42no—
419C22×C3—44no—
420C22×C3—42no—
421C22×C3—42no—
422C22×C3C2×#11711yescharacteristic
423C22×C3C2×C4×D412yes—
424C22×C3C4×#4112yes—
425C22×C3#51811yescharacteristic
426C22×C3#51911yescharacteristic
427C22×C3#52512yes—
428C22×C3#52512yes—
429C22×C3#52912yes—
430C22×C3#52912yes—
431C22×C3#53911yescharacteristic
432C22×C3#53911yescharacteristic
433C22×C3#53911yescharacteristic
434C22×C3#53911yescharacteristic
435C22×C3D4211yescharacteristic
436C22×C3#54612yes—
437C22×C3#54712yes—
438C22×C3#54812yes—
439C22×C3#55111yescharacteristic
440C22×C3#55111yescharacteristic
441C22×C3#55212yes—
442C22×C3#55312yes—
443C22×C3#55612yes—
444C22×C3#55911yescharacteristic
445C3×C4—42no—
446C3×C4—42no—
447C3×C4—44no—
448C3×C4—44no—
449Dic3—28no—
450Dic3—44no—
451Dic3—44no—
452Dic3—44no—
453C23—22no—
454C23—22no—
455C23—24no—
456C23—22no—
457C23—22no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—22no—
462C23—22no—
463C23—24no—
464C23—24no—
465C23—42no—
466C23—41no—
467C23—42no—
468C23—41no—
469C23—42no—
470C23—41no—
471C23—41no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23#89214yes—
476C23C22×C4×D311yescharacteristic
477C23C2×D3×D412yes—
478C23C2×#23711yescharacteristic
479C23C2×#23712yes—
480C23#100411yescharacteristic
481C23#100411yescharacteristic
482C23C2×#23911yescharacteristic
483C23D3×#4112yes—
484C23D3×#4112yes—
485C23#101011yescharacteristic
486C23#101011yescharacteristic
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—38no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—64no—
513C2×C4—64no—
514C2×C4—64no—
515C2×C4—64no—
516C2×C4—64no—
517C2×C4—64no—
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—1216no—
528C2×C4—122no—
529C2×C4—122no—
530C2×C4—122no—
531C2×C4—122no—
532C2×C4—122no—
533C2×C4—122no—
534C2×C4—122no—
535C2×C4—122no—
536C2×C4—122no—
537C2×C4—122no—
538C2×C4—122no—
539C2×C4—122no—
540C2×C3—44no—
541C2×C3#229012yes—
542C2×C3#231311yescharacteristic
543C2×C3#235811yescharacteristic
544C2×C3#236212yes—
545C2×C3#237411yescharacteristic
546C2×C3#245511yescharacteristic
547C2×C3#246012yes—
548C2×C3#246311yescharacteristic
549C2×C3#246612yes—
550C2×C3#251711yescharacteristic
551C2×C3#252311yescharacteristic
552C22—28no—
553C22—24no—
554C22—42no—
555C22—44no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—44no—
561C22—44no—
562C22—42no—
563C22#548611yescharacteristic
564C22C2×#89211yescharacteristic
565C22#551611yescharacteristic
566C22C4×#23712yes—
567C22C4×D3×D412yes—
568C22#558612yes—
569C22#559011yescharacteristic
570C22#560112yes—
571C22#560211yescharacteristic
572C22#560211yescharacteristic
573C22#561112yes—
574C22#561212yes—
575C22#562412yes—
576C22#562712yes—
577C22#564511yescharacteristic
578C22#564511yescharacteristic
579C22#564912yes—
580C22#565012yes—
581C22#565911yescharacteristic
582C22#567712yes—
583C22#567911yescharacteristic
584C22#570212yes—
585C22#571311yescharacteristic
586C4—42no—
587C4—44no—
588C4—42no—
589C4—44no—
590C4—68no—
591C4—124no—
592C4—124no—
593C4—124no—
594C3#1414311yescharacteristic, complemented, Sylow
595C2—44no—
596C2#7386711yescharacteristic
597C2#7437712yes—
598C2#7488011yescharacteristic
599C2#7547812yes—
600C2#7630811yescharacteristic
601C2#7667411yescharacteristic
602C2#7678711yescharacteristic
603C2#7691312yes—
604C2#7729712yes—
605C2#7750211yescharacteristic
606C2#7803211yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC3×#1439C2
socle, cosocleC24×C3C24
prime radicalC25×C3C23
hypercenter#1439D3
hypocenterC3#14143
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
1. C2×#4978C2semidirect
7. #69407C2semidirect
8. #69406C2semidirect
9. #69403C2semidirect
33. #4946C4semidirect
594. C3#14143semidirect

Maximal subgroups

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

subgroupindex
C2×#49782
C2×#49462
C2×Dic3×C22⋊C42
C3×#14392
#715492
#715462
#694072
#694062
#694032
#141433