Finite Group Databasebeta

← #10742710

Subgroups — #10742710

Order 768.

2354 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#123C211yesmaximal, characteristic
2C22×#935C211yesmaximal, characteristic
3C2×#5210C212yesmaximal, complemented
4C2×#4946C212yesmaximal, complemented
5#72131C211yesmaximal, characteristic
6#71568C214yesmaximal, complemented
7#71549C212yesmaximal, complemented
8#71548C212yesmaximal, complemented
9#15949—31nomaximal, Sylow
10C23×C3×D4C2211yescharacteristic
11C24×C3×C4C2211yescharacteristic
12C24×Dic3C2211yescharacteristic
13C2×C3×#123—24no—
14C2×C3×#123—28no—
15C22×C3×C4⋊C4C2212yes—
16C22×C3×C22⋊C4C2211yescharacteristic
17C22×C3×C22⋊C4C2212yes—
18C22×#217C2212yes—
19C22×#217C2211yescharacteristic
20C22×#217C2212yes—
21C22×#217C2212yes—
22C2×#935—24no—
23C2×#935—28no—
24C22×Dic3⋊C4C2212yes—
25C22×Dic3⋊C4C2214yes—
26#5210—28no—
27C2×C3⋊#95C2214yes—
28C2×C3⋊#95C2214yes—
29C2×C3⋊#95C2214yes—
30C2×C3⋊#95C2212yes—
31#4946—28no—
32C22×#123—31no—
33C22×#123D311yescharacteristic
34C2×#385—32no—
35C2×#385—32no—
36#1444—34no—
37#1440—32no—
38#1440—32no—
39#1438—31no—
40C25×C3C2311yescharacteristic
41C25×C3C2311yescharacteristic
42C22×C3×D4—24no—
43C22×C3×D4—24no—
44C22×C3×D4—42no—
45C22×C3×D4D414yes—
46C23×C3×C4—22no—
47C23×C3×C4—24no—
48C23×C3×C4—24no—
49C23×C3×C4—42no—
50C23×C3×C4C2312yes—
51C23×C3×C4C2311yescharacteristic
52C23×C3×C4C2312yes—
53C23×Dic3—22no—
54C23×Dic3—24no—
55C23×Dic3—24no—
56C23×Dic3—42no—
57C23×Dic3C2312yes—
58C23×Dic3C2312yes—
59C23×Dic3C2314yes—
60C3×#123—232no—
61C2×C3×C4⋊C4—24no—
62C2×C3×C4⋊C4—24no—
63C2×C3×C4⋊C4—24no—
64C2×C3×C22⋊C4—22no—
65C2×C3×C22⋊C4—24no—
66C2×C3×C22⋊C4—24no—
67C2×C3×C22⋊C4—24no—
68C2×C3×C22⋊C4D418yes—
69C2×#217—24no—
70C2×#217—24no—
71C2×#217—24no—
72C2×#217—22no—
73C2×#217—24no—
74C2×#217—24no—
75C2×#217—28no—
76C2×#217—28no—
77C2×#217D418yes—
78#935—232no—
79C2×Dic3⋊C4—24no—
80C2×Dic3⋊C4—24no—
81C2×Dic3⋊C4—24no—
82C2×Dic3⋊C4—28no—
83C2×Dic3⋊C4—28no—
84C2×Dic3⋊C4—44no—
85C3⋊#95—28no—
86C3⋊#95—28no—
87C3⋊#95—28no—
88C3⋊#95—42no—
89C23×D4C2×D311yescharacteristic
90C24×C4—31no—
91C24×C4C2×D311yescharacteristic
92C2×#123—24no—
93C2×#123—28no—
94C2×#123—68no—
95C2×#123—64no—
96C22×C4⋊C4—32no—
97C22×C4⋊C4—34no—
98C22×C4⋊C4C2×D312yes—
99C22×C22⋊C4—32no—
100C22×C22⋊C4—31no—
101C22×C22⋊C4—32no—
102C22×C22⋊C4—32no—
103C22×C22⋊C4C2×D311yescharacteristic
104C22×C22⋊C4C2×D312yes—
105#385—68no—
106#385—68no—
107C2×#95—34no—
108C2×#95—34no—
109C2×#95—34no—
110C2×#95—32no—
111C24×C3—22no—
112C24×C3—22no—
113C24×C3—24no—
114C24×C3—24no—
115C24×C3—22no—
116C24×C3—22no—
117C24×C3—24no—
118C24×C3—24no—
119C24×C3C2×D414yes—
120C24×C3C2×D412yes—
121C24×C3C2×D412yes—
122C24×C3C2×D414yes—
123C24×C3C2411yescharacteristic
124C2×C3×D4—28no—
125C2×C3×D4—28no—
126C2×C3×D4—44no—
127C2×C3×D4—416no—
128C22×C3×C4—28no—
129C22×C3×C4—24no—
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—24no—
137C22×C3×C4—22no—
138C22×C3×C4—24no—
139C22×C3×C4—24no—
140C22×C3×C4—416no—
141C22×C3×C4—44no—
142C22×C3×C4—44no—
143C22×C3×C4C2×D414yes—
144C22×C3×C4C2×D414yes—
145C22×C3×C4C2×D412yes—
146C22×Dic3—28no—
147C22×Dic3—24no—
148C22×Dic3—24no—
149C22×Dic3—24no—
150C22×Dic3—22no—
151C22×Dic3—22no—
152C22×Dic3—22no—
153C22×Dic3—22no—
154C22×Dic3—24no—
155C22×Dic3—24no—
156C22×Dic3—24no—
157C22×Dic3—24no—
158C22×Dic3—24no—
159C22×Dic3—24no—
160C22×Dic3—24no—
161C22×Dic3—416no—
162C22×Dic3—44no—
163C22×Dic3—44no—
164C22×Dic3C2×D414yes—
165C22×Dic3C2×D414yes—
166C22×Dic3#4118yes—
167C3×C4⋊C4—216no—
168C3×C22⋊C4—28no—
169C3×C22⋊C4—216no—
170#217—28no—
171#217—216no—
172#217—216no—
173#217—216no—
174Dic3⋊C4—216no—
175Dic3⋊C4—416no—
176C25C22×D311yescharacteristic
177C25C22×D311yescharacteristic
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—42no—
181C22×D4#6714yes—
182C23×C4—22no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—32no—
186C23×C4—34no—
187C23×C4—32no—
188C23×C4—42no—
189C23×C4—64no—
190C23×C4—62no—
191C23×C4—64no—
192C23×C4—122no—
193C23×C4C22×D312yes—
194C23×C4C22×D312yes—
195C23×C4C22×D311yescharacteristic
196#123—232no—
197#123—632no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—64no—
202C2×C4⋊C4—64no—
203C2×C4⋊C4—64no—
204C2×C4⋊C4—68no—
205C2×C4⋊C4—68no—
206C2×C4⋊C4—124no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—38no—
212C2×C22⋊C4—64no—
213C2×C22⋊C4—68no—
214C2×C22⋊C4—68no—
215C2×C22⋊C4—64no—
216C2×C22⋊C4—64no—
217C2×C22⋊C4—62no—
218C2×C22⋊C4—64no—
219C2×C22⋊C4—64no—
220C2×C22⋊C4#6718yes—
221#95—68no—
222#95—68no—
223#95—68no—
224#95—122no—
225C23×C3—22no—
226C23×C3—22no—
227C23×C3—24no—
228C23×C3—24no—
229C23×C3—22no—
230C23×C3—22no—
231C23×C3—24no—
232C23×C3—24no—
233C23×C3—28no—
234C23×C3—28no—
235C23×C3—28no—
236C23×C3—28no—
237C23×C3—28no—
238C23×C3—28no—
239C23×C3—42no—
240C23×C3—42no—
241C23×C3—42no—
242C23×C3—42no—
243C23×C3—44no—
244C23×C3—44no—
245C23×C3—44no—
246C23×C3—44no—
247C23×C3—42no—
248C23×C3—42no—
249C23×C3—42no—
250C23×C3—42no—
251C23×C3#12014yes—
252C23×C3#12014yes—
253C23×C3C22×D412yes—
254C23×C3C22×D411yescharacteristic
255C23×C3C22×D412yes—
256C23×C3C2×#4114yes—
257C23×C3H(5,𝔽2)12yes—
258C23×C32−1+414yes—
259C3×D4—416no—
260C2×C3×C4—22no—
261C2×C3×C4—24no—
262C2×C3×C4—24no—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—28no—
266C2×C3×C4—28no—
267C2×C3×C4—416no—
268C2×C3×C4—42no—
269C2×C3×C4—42no—
270C2×C3×C4—42no—
271C2×C3×C4—42no—
272C2×C3×C4—44no—
273C2×C3×C4—44no—
274C2×C3×C4—44no—
275C2×Dic3—24no—
276C2×Dic3—24no—
277C2×Dic3—28no—
278C2×Dic3—28no—
279C2×Dic3—28no—
280C2×Dic3—28no—
281C2×Dic3—28no—
282C2×Dic3—416no—
283C2×Dic3—42no—
284C2×Dic3—42no—
285C2×Dic3—42no—
286C2×Dic3—42no—
287C2×Dic3—44no—
288C2×Dic3—44no—
289C2×Dic3—44no—
290C2×Dic3—44no—
291C2×Dic3—44no—
292C2×Dic3—44no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—24no—
297C24—24no—
298C24—24no—
299C24—24no—
300C24—22no—
301C24D3×D414yes—
302C24C2×#6712yes—
303C24C2×#6712yes—
304C24C2×#6714yes—
305C24C23×D311yescharacteristic
306C2×D4—28no—
307C2×D4—28no—
308C2×D4—44no—
309C2×D4—416no—
310C22×C4—28no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—34no—
323C22×C4—34no—
324C22×C4—38no—
325C22×C4—416no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—68no—
329C22×C4—64no—
330C22×C4—62no—
331C22×C4—62no—
332C22×C4—64no—
333C22×C4—64no—
334C22×C4—64no—
335C22×C4—64no—
336C22×C4—64no—
337C22×C4—64no—
338C22×C4—64no—
339C22×C4—64no—
340C22×C4—64no—
341C22×C4—62no—
342C22×C4—62no—
343C22×C4—1216no—
344C22×C4—124no—
345C22×C4—124no—
346C22×C4C2×#6714yes—
347C22×C4C2×#6714yes—
348C22×C4C2×#6712yes—
349C4⋊C4—216no—
350C4⋊C4—616no—
351C4⋊C4—1216no—
352C22⋊C4—28no—
353C22⋊C4—216no—
354C22⋊C4—68no—
355C22⋊C4—616no—
356C22⋊C4—616no—
357C22⋊C4—616no—
358C22×C3—24no—
359C22×C3—24no—
360C22×C3—28no—
361C22×C3—28no—
362C22×C3—44no—
363C22×C3—44no—
364C22×C3—48no—
365C22×C3—48no—
366C22×C3—48no—
367C22×C3—48no—
368C22×C3—44no—
369C22×C3—44no—
370C22×C3C2×#12011yescharacteristic
371C22×C3C2×#12011yescharacteristic
372C22×C3#52512yes—
373C22×C3#52512yes—
374C22×C3#53411yescharacteristic
375C22×C3#53612yes—
376C22×C3#53612yes—
377C22×C3#53914yes—
378C22×C3D4214yes—
379C22×C3#54714yes—
380C22×C3#54714yes—
381C22×C3#54714yes—
382C22×C3#55912yes—
383C22×C3#56312yes—
384C3×C4—44no—
385C3×C4—42no—
386C3×C4—44no—
387Dic3—44no—
388Dic3—44no—
389Dic3—48no—
390C23—24no—
391C23—28no—
392C23—24no—
393C23—22no—
394C23—28no—
395C23—28no—
396C23—22no—
397C23—22no—
398C23—28no—
399C23—28no—
400C23—28no—
401C23—24no—
402C23—24no—
403C23—22no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—44no—
408C23—42no—
409C23—44no—
410C23—44no—
411C23—42no—
412C23—42no—
413C23—44no—
414C23—42no—
415C23—42no—
416C23#93714yes—
417C23#95314yes—
418C23C2×D3×D412yes—
419C23#100412yes—
420C23D3×#4114yes—
421C23#101014yes—
422C23C22×#6711yescharacteristic
423C23C22×#6712yes—
424D4—416no—
425C2×C4—22no—
426C2×C4—28no—
427C2×C4—22no—
428C2×C4—28no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—22no—
432C2×C4—416no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—68no—
441C2×C4—68no—
442C2×C4—68no—
443C2×C4—64no—
444C2×C4—64no—
445C2×C4—68no—
446C2×C4—68no—
447C2×C4—1216no—
448C2×C4—122no—
449C2×C4—124no—
450C2×C4—122no—
451C2×C4—124no—
452C2×C4—124no—
453C2×C4—124no—
454C2×C4—122no—
455C2×C4—124no—
456C2×C4—124no—
457C2×C4—122no—
458C2×C3—44no—
459C2×C3—44no—
460C2×C3#240311yescharacteristic
461C2×C3#240612yes—
462C2×C3#240612yes—
463C2×C3#245514yes—
464C2×C3#268014yes—
465C2×C3#272412yes—
466C22—24no—
467C22—24no—
468C22—28no—
469C22—28no—
470C22—44no—
471C22—44no—
472C22—48no—
473C22—48no—
474C22—44no—
475C22—44no—
476C22—48no—
477C22—48no—
478C22#564812yes—
479C22#565112yes—
480C22#565514yes—
481C22#565612yes—
482C22#566014yes—
483C22#566412yes—
484C22C2×#93711yescharacteristic
485C22D4×#6714yes—
486C22#580411yescharacteristic
487C22#582414yes—
488C22#582414yes—
489C22#582512yes—
490C22#582812yes—
491C22C2×#95311yescharacteristic
492C4—44no—
493C4—42no—
494C4—44no—
495C4—124no—
496C4—124no—
497C4—128no—
498C3#1594911yescharacteristic, complemented, Sylow
499C2—44no—
500C2—44no—
501C2#8284112yes—
502C2#8287114yes—
503C2#8288812yes—
504C2#8289814yes—
505C2#8405211yescharacteristic
506C2#8409012yes—
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC22×C3×#123C2
prime radicalC23×C3×D4C22
hypercenterC22×#123D3
hypocenterC3#15949
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×#123

Complements

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

NH ≅ G/Ntype
3. C2×#5210C2semidirect
4. C2×#4946C2semidirect
6. #71568C2semidirect
7. #71549C2semidirect
8. #71548C2semidirect
498. C3#15949semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#1232
C22×#9352
C2×#52102
C2×#49462
#721312
#715682
#715492
#715482
#159493