Finite Group Databasebeta

← #10725793

Subgroups — #10725793

Order 768.

1290 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4949C211yesmaximal, characteristic
2C3×#1440C211yesmaximal, characteristic
3#71549C212yesmaximal
4#71545C212yesmaximal
5#71545C211yesmaximal, characteristic
6#69417C212yesmaximal, complemented
7#69407C212yesmaximal, complemented
8#69406C212yesmaximal, complemented
9#69405C212yesmaximal, complemented
10#14618—31nomaximal, Sylow
11C22×C3×C22⋊C4C2211yescharacteristic
12C22×C3×C22⋊C4C2211yescharacteristic
13C22×C3×C22⋊C4C2211yescharacteristic
14C22×#217C2212yes—
15C22×#217C2212yes—
16C22×C4⋊Dic3C2211yescharacteristic
17C22×Dic3⋊C4C2212yes—
18C22×C4×Dic3C2211yescharacteristic
19C22×C4×Dic3C2212yes—
20C2×C3×#95C2212yes—
21C2×C3×#95C2212yes—
22C2×C3⋊#95C2212yes—
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⋊#95C2211yescharacteristic
32#4949C418yescomplemented
33C2×#399—31no—
34#1443—32no—
35#1440—32no—
36#1440D311yescharacteristic
37#1437—32no—
38#1437—31no—
39#1436—32no—
40#1436—32no—
41#1436—32no—
42C25×C3C2311yescharacteristic
43C23×C3×C4C2312yes—
44C23×C3×C4C2311yescharacteristic
45C23×C3×C4C2311yescharacteristic
46C23×C3×C4C2312yes—
47C23×Dic3C2312yes—
48C23×Dic3C2312yes—
49C23×Dic3C2312yes—
50C23×Dic3C2312yes—
51C2×C3×C22⋊C4—24no—
52C2×C3×C22⋊C4—41no—
53C2×C3×C22⋊C4—41no—
54C2×C3×C22⋊C4—41no—
55C2×C3×C22⋊C4—41no—
56C2×C3×C22⋊C4—42no—
57C2×C3×C22⋊C4C2×C414yes—
58C2×#217—24no—
59C2×#217—24no—
60C2×#217—42no—
61C2×#217—42no—
62C2×#217—42no—
63C2×#217C2×C418yes—
64C2×C4⋊Dic3—41no—
65C2×C4⋊Dic3—41no—
66C2×C4⋊Dic3C2×C414yes—
67C2×Dic3⋊C4—42no—
68C2×Dic3⋊C4—42no—
69C2×Dic3⋊C4C2×C418yes—
70C2×C4×Dic3—41no—
71C2×C4×Dic3—42no—
72C2×C4×Dic3—42no—
73C2×C4×Dic3—42no—
74C2×C4×Dic3—42no—
75C3×#95—24no—
76C3×#95—42no—
77C3⋊#95—24no—
78C3⋊#95—24no—
79C3⋊#95—24no—
80C3⋊#95—24no—
81C3⋊#95—24no—
82C3⋊#95—41no—
83C3⋊#95—42no—
84C3⋊#95—42no—
85C3⋊#95—42no—
86C3⋊#95C2×C414yes—
87C22×C4⋊C4—32no—
88C22×C4⋊C4—31no—
89C22×C22⋊C4—32no—
90C22×C22⋊C4—32no—
91C22×C22⋊C4C2×D311yescharacteristic
92C22×C22⋊C4C2×D311yescharacteristic
93C22×C22⋊C4C2×D311yescharacteristic
94C22×C42—32no—
95C22×C42—31no—
96#399—38no—
97C2×#95—31no—
98C2×#95—32no—
99C2×#95—32no—
100C2×#95—32no—
101C2×#95—32no—
102C2×#95—32no—
103C2×#95—32no—
104C2×#95—31no—
105C2×#95—32no—
106C2×#95—32no—
107C2×#95C2×D312yes—
108C2×#95C2×D312yes—
109C24×C3—22no—
110C24×C3—22no—
111C24×C3—22no—
112C24×C3—22no—
113C24×C3—22no—
114C24×C3—21no—
115C24×C3—21no—
116C24×C3—22no—
117C24×C3C22×C412yes—
118C24×C3C2411yescharacteristic
119C22×C3×C4—22no—
120C22×C3×C4—22no—
121C22×C3×C4—22no—
122C22×C3×C4—22no—
123C22×C3×C4—22no—
124C22×C3×C4—21no—
125C22×C3×C4—22no—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—21no—
129C22×C3×C4—22no—
130C22×C3×C4—21no—
131C22×C3×C4—21no—
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×C4C22×C412yes—
141C22×C3×C4C22×C412yes—
142C22×C3×C4#4114yes—
143C22×Dic3—22no—
144C22×Dic3—22no—
145C22×Dic3—22no—
146C22×Dic3—22no—
147C22×Dic3—22no—
148C22×Dic3—22no—
149C22×Dic3—22no—
150C22×Dic3—22no—
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×Dic3C22×C414yes—
168C22×Dic3C22×C414yes—
169C22×Dic3#4114yes—
170C22×Dic3#4114yes—
171C3×C22⋊C4—44no—
172C3×C22⋊C4—44no—
173C3×C22⋊C4—44no—
174#217—48no—
175#217—48no—
176C4⋊Dic3—44no—
177Dic3⋊C4—48no—
178C4×Dic3—44no—
179C4×Dic3—48no—
180C25C22×D311yescharacteristic
181C23×C4—32no—
182C23×C4—32no—
183C23×C4—32no—
184C23×C4—32no—
185C23×C4C22×D312yes—
186C23×C4C22×D311yescharacteristic
187C23×C4C22×D311yescharacteristic
188C23×C4C22×D312yes—
189C2×C4⋊C4—38no—
190C2×C4⋊C4—34no—
191C2×C4⋊C4—122no—
192C2×C4⋊C4—122no—
193C2×C4⋊C4—121no—
194C2×C4⋊C4—121no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—38no—
197C2×C22⋊C4—41no—
198C2×C22⋊C4—41no—
199C2×C22⋊C4—41no—
200C2×C22⋊C4—42no—
201C2×C22⋊C4—41no—
202C2×C22⋊C4—64no—
203C2×C22⋊C4—64no—
204C2×C22⋊C4—122no—
205C2×C22⋊C4—122no—
206C2×C22⋊C4—122no—
207C2×C22⋊C4C4×D314yes—
208C2×C42—122no—
209C2×C42—121no—
210C2×C42—122no—
211C2×C42—122no—
212C2×C42—122no—
213#95—24no—
214#95—34no—
215#95—42no—
216#95—64no—
217#95—64no—
218#95—64no—
219#95—64no—
220#95—64no—
221#95—121no—
222#95—122no—
223#95—122no—
224#95—122no—
225C23×C3—22no—
226C23×C3—24no—
227C23×C3—24no—
228C23×C3—22no—
229C23×C3—22no—
230C23×C3—42no—
231C23×C3—42no—
232C23×C3—42no—
233C23×C3—42no—
234C23×C3—42no—
235C23×C3—42no—
236C23×C3—42no—
237C23×C3—41no—
238C23×C3—42no—
239C23×C3—42no—
240C23×C3—42no—
241C23×C3—41no—
242C23×C3—42no—
243C23×C3—41no—
244C23×C3—41no—
245C23×C3—42no—
246C23×C3C23×C411yescharacteristic
247C23×C3C2×#4112yes—
248C23×C3C2×#4112yes—
249C23×C3C2×#4112yes—
250C23×C3H(5,𝔽2)11yescharacteristic
251C23×C3H(5,𝔽2)12yes—
252C23×C3H(5,𝔽2)11yescharacteristic
253C23×C3H(5,𝔽2)12yes—
254C23×C32−1+412yes—
255C2×C3×C4—24no—
256C2×C3×C4—24no—
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—24no—
260C2×C3×C4—24no—
261C2×C3×C4—24no—
262C2×C3×C4—42no—
263C2×C3×C4—42no—
264C2×C3×C4—41no—
265C2×C3×C4—42no—
266C2×C3×C4—41no—
267C2×C3×C4—42no—
268C2×C3×C4—41no—
269C2×C3×C4—42no—
270C2×C3×C4—41no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4—42no—
274C2×C3×C4—42no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×Dic3—24no—
280C2×Dic3—24no—
281C2×Dic3—24no—
282C2×Dic3—24no—
283C2×Dic3—24no—
284C2×Dic3—24no—
285C2×Dic3—24no—
286C2×Dic3—24no—
287C2×Dic3—24no—
288C2×Dic3—24no—
289C2×Dic3—24no—
290C2×Dic3—24no—
291C2×Dic3—42no—
292C2×Dic3—42no—
293C2×Dic3—42no—
294C2×Dic3—42no—
295C2×Dic3—42no—
296C2×Dic3—42no—
297C2×Dic3—42no—
298C2×Dic3—42no—
299C2×Dic3—42no—
300C2×Dic3—42no—
301C2×Dic3—42no—
302C2×Dic3—42no—
303C2×Dic3—42no—
304C2×Dic3—42no—
305C2×Dic3—42no—
306C2×Dic3—42no—
307C24—22no—
308C24—21no—
309C24—21no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24C2×C4×D312yes—
316C24C23×D311yescharacteristic
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—34no—
339C22×C4—34no—
340C22×C4—34no—
341C22×C4—34no—
342C22×C4—62no—
343C22×C4—62no—
344C22×C4—62no—
345C22×C4—62no—
346C22×C4—62no—
347C22×C4—62no—
348C22×C4—62no—
349C22×C4—62no—
350C22×C4—62no—
351C22×C4—62no—
352C22×C4—62no—
353C22×C4—62no—
354C22×C4—62no—
355C22×C4—62no—
356C22×C4—62no—
357C22×C4—62no—
358C22×C4—62no—
359C22×C4—62no—
360C22×C4—62no—
361C22×C4—62no—
362C22×C4—62no—
363C22×C4—62no—
364C22×C4—62no—
365C22×C4—62no—
366C22×C4C2×C4×D312yes—
367C22×C4C2×C4×D312yes—
368C22×C4#23714yes—
369C4⋊C4—128no—
370C4⋊C4—124no—
371C22⋊C4—44no—
372C22⋊C4—44no—
373C22⋊C4—44no—
374C22⋊C4—128no—
375C22⋊C4—128no—
376C42—124no—
377C42—128no—
378C22×C3—42no—
379C22×C3—44no—
380C22×C3—42no—
381C22×C3—44no—
382C22×C3—42no—
383C22×C3—82no—
384C22×C3—82no—
385C22×C3—82no—
386C22×C3—82no—
387C22×C3C4×#4111yescharacteristic
388C22×C3C4×#4112yes—
389C22×C3#51811yescharacteristic
390C22×C3#51812yes—
391C22×C3#51911yescharacteristic
392C22×C3#53912yes—
393C22×C3#53912yes—
394C22×C3#53912yes—
395C22×C3#54212yes—
396C22×C3#54212yes—
397C22×C3#54212yes—
398C22×C3#55112yes—
399C22×C3#55111yescharacteristic
400C22×C3#55312yes—
401C22×C3#55312yes—
402C22×C3#55312yes—
403C22×C3#55911yescharacteristic
404C22×C3#55912yes—
405C22×C3#56011yescharacteristic
406C22×C3#56011yescharacteristic
407C22×C3#56012yes—
408C3×C4—42no—
409C3×C4—42no—
410C3×C4—44no—
411C3×C4—82no—
412Dic3—44no—
413Dic3—44no—
414Dic3—44no—
415Dic3—44no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—24no—
420C23—24no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—41no—
425C23—42no—
426C23—41no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—41no—
432C23—41no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23C22×C4×D311yescharacteristic
438C23C2×#23712yes—
439C23#100411yescharacteristic
440C23#100412yes—
441C23#100411yescharacteristic
442C23D3×#4112yes—
443C23D3×#4112yes—
444C23#100912yes—
445C23#101012yes—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—64no—
471C2×C4—64no—
472C2×C4—64no—
473C2×C4—64no—
474C2×C4—64no—
475C2×C4—64no—
476C2×C4—64no—
477C2×C4—64no—
478C2×C4—64no—
479C2×C4—64no—
480C2×C4—64no—
481C2×C4—64no—
482C2×C4—122no—
483C2×C4—122no—
484C2×C4—122no—
485C2×C4—122no—
486C2×C4—122no—
487C2×C4—122no—
488C2×C4—122no—
489C2×C4—122no—
490C2×C4—122no—
491C2×C4—122no—
492C2×C4—122no—
493C2×C4—122no—
494C2×C4—122no—
495C2×C4—122no—
496C2×C4—122no—
497C2×C4—122no—
498C2×C3—82no—
499C2×C3#233211yescharacteristic
500C2×C3#233212yes—
501C2×C3#237411yescharacteristic
502C2×C3#237412yes—
503C2×C3#238011yescharacteristic
504C2×C3#279612yes—
505C2×C3#279612yes—
506C2×C3#279612yes—
507C2×C3#280112yes—
508C22—42no—
509C22—42no—
510C22—42no—
511C22—44no—
512C22—44no—
513C22—82no—
514C22—82no—
515C22—82no—
516C22—82no—
517C22#548611yescharacteristic
518C22#551611yescharacteristic
519C22#552211yescharacteristic
520C22C4×#23712yes—
521C22#554512yes—
522C22#558711yescharacteristic
523C22#559511yescharacteristic
524C22#560112yes—
525C22#560312yes—
526C22#561412yes—
527C22#562012yes—
528C22#566012yes—
529C22#566112yes—
530C22#566212yes—
531C22#566411yescharacteristic
532C22#566512yes—
533C22#566611yescharacteristic
534C22#566912yes—
535C22#568112yes—
536C22#570512yes—
537C22#570712yes—
538C4—44no—
539C4—42no—
540C4—42no—
541C4—82no—
542C4—124no—
543C4—124no—
544C4—124no—
545C4—124no—
546C3#1461811yescharacteristic, complemented, Sylow
547C2—82no—
548C2#7405711yescharacteristic
549C2#7468111yescharacteristic
550C2#7468511yescharacteristic
551C2#7546412yes—
552C2#7548712yes—
553C2#7707412yes—
554C2#7707812yes—
555C2#7772112yes—
556C2#7772212yes—
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC3×#1440C2
socle, cosocleC24×C3C24
prime radicalC25×C3C23
hypercenter#1440D3
hypocenterC3#14618
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1440

Complements

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

NH ≅ G/Ntype
6. #69417C2semidirect
7. #69407C2semidirect
8. #69406C2semidirect
9. #69405C2semidirect
32. #4949C4semidirect
546. C3#14618semidirect

Maximal subgroups

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

subgroupindex
C2×#49492
C3×#14402
#715492
#715452
#694172
#694072
#694062
#694052
#146183