Finite Group Databasebeta

← #10723219

Subgroups — #10723219

Order 768.

1606 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4946C211yesmaximal, characteristic
2C2×#4942C212yesmaximal
3C2×#4933C212yesmaximal, complemented
4C3×#1437C211yesmaximal, characteristic
5#71549C212yesmaximal
6#71545C211yesmaximal, characteristic
7#69413C212yesmaximal, complemented
8#69409C212yesmaximal, complemented
9#69406C212yesmaximal, complemented
10#14409—31nomaximal, Sylow
11C24×Dic3C2211yescharacteristic
12C22×C3×C22⋊C4C2212yes—
13C22×C3×C22⋊C4C2211yescharacteristic
14C22×C3×C42C2211yescharacteristic
15C22×#217C2212yes—
16C22×#217C2211yescharacteristic
17C22×Dic3⋊C4C2212yes—
18C22×Dic3⋊C4C2212yes—
19C22×Dic3⋊C4C2212yes—
20C22×C4×Dic3C2211yescharacteristic
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#4946C418yescomplemented
33#4942—28no—
34#4933—28no—
35C2×#387—32no—
36C2×#385—31no—
37C2×#380—32no—
38#1442—32no—
39#1441—32no—
40#1440—32no—
41#1437—31no—
42#1437D311yescharacteristic
43#1436—32no—
44C25×C3C2311yescharacteristic
45C23×C3×C4C2312yes—
46C23×C3×C4C2311yescharacteristic
47C23×C3×C4C2311yescharacteristic
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×C22⋊C4—24no—
59C2×C3×C22⋊C4—24no—
60C2×C3×C22⋊C4—41no—
61C2×C3×C22⋊C4—41no—
62C2×C3×C22⋊C4—42no—
63C2×C3×C22⋊C4C2×C414yes—
64C2×C3×C42—24no—
65C2×C3×C42—41no—
66C2×#217—24no—
67C2×#217—24no—
68C2×#217—24no—
69C2×#217—42no—
70C2×#217C2×C414yes—
71C2×Dic3⋊C4—24no—
72C2×Dic3⋊C4—24no—
73C2×Dic3⋊C4—24no—
74C2×Dic3⋊C4—24no—
75C2×Dic3⋊C4—24no—
76C2×Dic3⋊C4—24no—
77C2×Dic3⋊C4—42no—
78C2×Dic3⋊C4—42no—
79C2×Dic3⋊C4C2×C418yes—
80C2×C4×Dic3—42no—
81C2×C4×Dic3—41no—
82C3×#95—42no—
83C3×#95—41no—
84C3⋊#95—24no—
85C3⋊#95—24no—
86C3⋊#95—24no—
87C3⋊#95—24no—
88C3⋊#95—42no—
89C3⋊#95—42no—
90C3⋊#95—41no—
91C3⋊#95C2×C418yes—
92C3⋊#95D418yes—
93C24×C4—31no—
94C22×C4⋊C4—32no—
95C22×C4⋊C4—32no—
96C22×C4⋊C4—32no—
97C22×C22⋊C4—31no—
98C22×C22⋊C4—32no—
99C22×C22⋊C4C2×D312yes—
100C22×C22⋊C4C2×D311yescharacteristic
101C22×C42—31no—
102C22×C42C2×D311yescharacteristic
103#387—68no—
104#385—38no—
105#380—68no—
106C2×#95—32no—
107C2×#95—32no—
108C2×#95—32no—
109C2×#95—32no—
110C2×#95—32no—
111C2×#95—32no—
112C2×#95—32no—
113C2×#95—31no—
114C2×#95—32no—
115C2×#95C2×D312yes—
116C2×#95C2×D311yescharacteristic
117C24×C3—22no—
118C24×C3—22no—
119C24×C3—22no—
120C24×C3—21no—
121C24×C3—21no—
122C24×C3—22no—
123C24×C3—22no—
124C24×C3C22×C412yes—
125C24×C3#4114yes—
126C24×C3C2411yescharacteristic
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—21no—
132C22×C3×C4—22no—
133C22×C3×C4—21no—
134C22×C3×C4—22no—
135C22×C3×C4—21no—
136C22×C3×C4—22no—
137C22×C3×C4—21no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4C22×C412yes—
149C22×C3×C4C22×C412yes—
150C22×C3×C4C2×D414yes—
151C22×Dic3—22no—
152C22×Dic3—22no—
153C22×Dic3—22no—
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—216no—
168C22×Dic3—22no—
169C22×Dic3—22no—
170C22×Dic3—22no—
171C22×Dic3—22no—
172C22×Dic3—22no—
173C22×Dic3—22no—
174C22×Dic3—22no—
175C22×Dic3—44no—
176C22×Dic3—44no—
177C22×Dic3—48no—
178C22×Dic3—44no—
179C22×Dic3C22×C414yes—
180C22×Dic3C22×C414yes—
181C22×Dic3C2×D414yes—
182C22×Dic3C2×D414yes—
183C22×Dic3#4114yes—
184C3×C22⋊C4—44no—
185C3×C22⋊C4—48no—
186C3×C42—44no—
187#217—28no—
188#217—48no—
189Dic3⋊C4—216no—
190Dic3⋊C4—48no—
191Dic3⋊C4—48no—
192C4×Dic3—44no—
193C25C22×D311yescharacteristic
194C23×C4—32no—
195C23×C4—32no—
196C23×C4—32no—
197C23×C4—34no—
198C23×C4—32no—
199C23×C4—64no—
200C23×C4—64no—
201C23×C4—121no—
202C23×C4—121no—
203C23×C4C22×D312yes—
204C23×C4C22×D311yescharacteristic
205C23×C4C22×D311yescharacteristic
206C23×C4C22×D312yes—
207C2×C4⋊C4—38no—
208C2×C4⋊C4—64no—
209C2×C4⋊C4—64no—
210C2×C4⋊C4—64no—
211C2×C4⋊C4—64no—
212C2×C4⋊C4—64no—
213C2×C4⋊C4—64no—
214C2×C4⋊C4—122no—
215C2×C4⋊C4—122no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—34no—
219C2×C22⋊C4—41no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—64no—
223C2×C22⋊C4—64no—
224C2×C22⋊C4—64no—
225C2×C22⋊C4—122no—
226C2×C22⋊C4C4×D314yes—
227C2×C42—24no—
228C2×C42—41no—
229C2×C42—122no—
230C2×C42—121no—
231#95—38no—
232#95—38no—
233#95—42no—
234#95—41no—
235#95—64no—
236#95—64no—
237#95—64no—
238#95—64no—
239#95—121no—
240#95—122no—
241#95—122no—
242C23×C3—24no—
243C23×C3—22no—
244C23×C3—22no—
245C23×C3—24no—
246C23×C3—24no—
247C23×C3—24no—
248C23×C3—24no—
249C23×C3—24no—
250C23×C3—24no—
251C23×C3—24no—
252C23×C3—42no—
253C23×C3—42no—
254C23×C3—42no—
255C23×C3—42no—
256C23×C3—41no—
257C23×C3—42no—
258C23×C3—41no—
259C23×C3—41no—
260C23×C3—42no—
261C23×C3—41no—
262C23×C3#11714yes—
263C23×C3C23×C411yescharacteristic
264C23×C3C22×D412yes—
265C23×C3C2×#4112yes—
266C23×C3C2×#4112yes—
267C23×C3H(5,𝔽2)11yescharacteristic
268C23×C3H(5,𝔽2)11yescharacteristic
269C23×C3H(5,𝔽2)12yes—
270C23×C32−1+412yes—
271C23×C32−1+412yes—
272C2×C3×C4—24no—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—24no—
276C2×C3×C4—24no—
277C2×C3×C4—24no—
278C2×C3×C4—24no—
279C2×C3×C4—42no—
280C2×C3×C4—42no—
281C2×C3×C4—41no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4—41no—
285C2×C3×C4—41no—
286C2×C3×C4—42no—
287C2×C3×C4—42no—
288C2×C3×C4—41no—
289C2×C3×C4—42no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×Dic3—24no—
297C2×Dic3—24no—
298C2×Dic3—24no—
299C2×Dic3—24no—
300C2×Dic3—24no—
301C2×Dic3—24no—
302C2×Dic3—24no—
303C2×Dic3—24no—
304C2×Dic3—24no—
305C2×Dic3—24no—
306C2×Dic3—24no—
307C2×Dic3—24no—
308C2×Dic3—416no—
309C2×Dic3—42no—
310C2×Dic3—42no—
311C2×Dic3—42no—
312C2×Dic3—42no—
313C2×Dic3—42no—
314C2×Dic3—42no—
315C2×Dic3—42no—
316C2×Dic3—42no—
317C2×Dic3—42no—
318C2×Dic3—42no—
319C2×Dic3—42no—
320C2×Dic3—42no—
321C2×Dic3—42no—
322C2×Dic3—42no—
323C2×Dic3—42no—
324C2×Dic3C4×D418yes—
325C24—22no—
326C24—22no—
327C24—21no—
328C24—21no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24C2×C4×D312yes—
333C24#23514yes—
334C24C23×D311yescharacteristic
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—21no—
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—22no—
355C22×C4—22no—
356C22×C4—34no—
357C22×C4—34no—
358C22×C4—34no—
359C22×C4—34no—
360C22×C4—34no—
361C22×C4—616no—
362C22×C4—62no—
363C22×C4—62no—
364C22×C4—62no—
365C22×C4—62no—
366C22×C4—62no—
367C22×C4—62no—
368C22×C4—62no—
369C22×C4—62no—
370C22×C4—62no—
371C22×C4—62no—
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—124no—
386C22×C4—124no—
387C22×C4—124no—
388C22×C4—128no—
389C22×C4C2×C4×D312yes—
390C22×C4C2×C4×D312yes—
391C22×C4D3×D414yes—
392C4⋊C4—616no—
393C4⋊C4—128no—
394C4⋊C4—128no—
395C22⋊C4—44no—
396C22⋊C4—48no—
397C22⋊C4—68no—
398C22⋊C4—128no—
399C42—44no—
400C42—124no—
401C22×C3—24no—
402C22×C3—28no—
403C22×C3—44no—
404C22×C3—42no—
405C22×C3—42no—
406C22×C3—44no—
407C22×C3—44no—
408C22×C3—44no—
409C22×C3—44no—
410C22×C3C2×#11711yescharacteristic
411C22×C3C2×C4×D412yes—
412C22×C3#51811yescharacteristic
413C22×C3#51911yescharacteristic
414C22×C3#52012yes—
415C22×C3#53612yes—
416C22×C3#53712yes—
417C22×C3#53912yes—
418C22×C3#53912yes—
419C22×C3#54112yes—
420C22×C3#54212yes—
421C22×C3D4211yescharacteristic
422C22×C3#54612yes—
423C22×C3#54612yes—
424C22×C3#54812yes—
425C22×C3#55111yescharacteristic
426C22×C3#55212yes—
427C22×C3#55312yes—
428C22×C3#56011yescharacteristic
429C22×C3#56212yes—
430C22×C3#56311yescharacteristic
431C3×C4—42no—
432C3×C4—42no—
433C3×C4—44no—
434C3×C4—82no—
435Dic3—28no—
436Dic3—44no—
437Dic3—44no—
438Dic3—82no—
439C23—22no—
440C23—24no—
441C23—22no—
442C23—24no—
443C23—24no—
444C23—24no—
445C23—24no—
446C23—24no—
447C23—24no—
448C23—24no—
449C23—42no—
450C23—41no—
451C23—42no—
452C23—41no—
453C23—42no—
454C23—41no—
455C23—41no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23#87214yes—
460C23C22×C4×D311yescharacteristic
461C23C2×#23512yes—
462C23C2×D3×D412yes—
463C23#100411yescharacteristic
464C23#100412yes—
465C23#100411yescharacteristic
466C23D3×#4112yes—
467C23#101012yes—
468C23#101012yes—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—38no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—64no—
495C2×C4—64no—
496C2×C4—64no—
497C2×C4—64no—
498C2×C4—64no—
499C2×C4—64no—
500C2×C4—64no—
501C2×C4—64no—
502C2×C4—64no—
503C2×C4—64no—
504C2×C4—64no—
505C2×C4—64no—
506C2×C4—1216no—
507C2×C4—122no—
508C2×C4—122no—
509C2×C4—122no—
510C2×C4—122no—
511C2×C4—122no—
512C2×C4—122no—
513C2×C4—122no—
514C2×C4—122no—
515C2×C4—122no—
516C2×C4—122no—
517C2×C4—122no—
518C2×C4—122no—
519C2×C4—122no—
520C2×C4—122no—
521C2×C4—122no—
522C2×C3—44no—
523C2×C3#231311yescharacteristic
524C2×C3#232312yes—
525C2×C3#235212yes—
526C2×C3#235811yescharacteristic
527C2×C3#237611yescharacteristic
528C2×C3#268112yes—
529C2×C3#269012yes—
530C2×C3#269112yes—
531C2×C3#274712yes—
532C22—24no—
533C22—28no—
534C22—42no—
535C22—44no—
536C22—44no—
537C22—42no—
538C22—44no—
539C22—44no—
540C22—44no—
541C22C2×#87211yescharacteristic
542C22#548611yescharacteristic
543C22#549012yes—
544C22#551611yescharacteristic
545C22#553112yes—
546C22#554012yes—
547C22C4×D3×D412yes—
548C22#554612yes—
549C22#555212yes—
550C22#556412yes—
551C22#556412yes—
552C22#559011yescharacteristic
553C22#560812yes—
554C22#561212yes—
555C22#562112yes—
556C22#564812yes—
557C22#565911yescharacteristic
558C22#566912yes—
559C22#567211yescharacteristic
560C22#570112yes—
561C22#572011yescharacteristic
562C4—42no—
563C4—44no—
564C4—42no—
565C4—68no—
566C4—82no—
567C4—124no—
568C4—124no—
569C4—242no—
570C3#1440911yescharacteristic, complemented, Sylow
571C2—44no—
572C2#7344511yescharacteristic
573C2#7346112yes—
574C2#7479412yes—
575C2#7488011yescharacteristic
576C2#7546012yes—
577C2#7567212yes—
578C2#7573012yes—
579C2#7630911yescharacteristic
580C2#7638112yes—
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC3×#1437C2
socle, cosocleC24×C3C24
prime radicalC25×C3C23
hypercenter#1437D3
hypocenterC3#14409
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1437

Complements

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

NH ≅ G/Ntype
3. C2×#4933C2semidirect
7. #69413C2semidirect
8. #69409C2semidirect
9. #69406C2semidirect
32. #4946C4semidirect
570. C3#14409semidirect

Maximal subgroups

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

subgroupindex
C2×#49462
C2×#49422
C2×#49332
C3×#14372
#715492
#715452
#694132
#694092
#694062
#144093