Finite Group Databasebeta

← #10732920

Subgroups — #10732920

Order 768.

2294 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×#936C211yesmaximal, characteristic
2C22×#935C211yesmaximal, characteristic
3C2×#4950C212yesmaximal, complemented
4C2×#4656C212yesmaximal, complemented
5C3×#1438C211yesmaximal, characteristic
6#71549C212yesmaximal, complemented
7#71546C214yesmaximal, complemented
8#69405C212yesmaximal, complemented
9#15262—31nomaximal, Sylow
10C23×C3×D4C2211yescharacteristic
11C24×Dic3C2211yescharacteristic
12C23×Dic6C2211yescharacteristic
13C22×C3×C22⋊C4C2212yes—
14C22×C3×C22⋊C4C2212yes—
15C22×#217C2212yes—
16C22×#217C2212yes—
17C22×#217C2211yescharacteristic
18C22×#217C2212yes—
19C2×#936—24no—
20C2×#936—28no—
21C2×#935—24no—
22C2×#935—28no—
23C22×C4⋊Dic3C2214yes—
24C22×Dic3⋊C4C2212yes—
25C22×C4×Dic3C2211yescharacteristic
26C2×C3×#95C2212yes—
27C2×C3⋊#95C2214yes—
28C2×C3⋊#95C2214yes—
29C2×C3⋊#95C2214yes—
30#4950—28no—
31#4656—28no—
32C22×#124—31no—
33C22×#123—31no—
34C2×#395—32no—
35C2×#385—32no—
36#1440—32no—
37#1439—34no—
38#1438D311yescharacteristic
39#1436—32no—
40C25×C3C2311yescharacteristic
41C25×C3C2311yescharacteristic
42C22×C3×D4—24no—
43C22×C3×D4—24no—
44C22×C3×D4—42no—
45C22×C3×D4D414yes—
46C23×C3×C4C2314yes—
47C23×C3×C4C2311yescharacteristic
48C23×Dic3—22no—
49C23×Dic3—24no—
50C23×Dic3—24no—
51C23×Dic3—42no—
52C23×Dic3C2312yes—
53C23×Dic3C2312yes—
54C23×Dic3C2312yes—
55C23×Dic3C2312yes—
56C22×Dic6—22no—
57C22×Dic6—24no—
58C22×Dic6—24no—
59C22×Dic6—42no—
60C2×C3×C22⋊C4—24no—
61C2×C3×C22⋊C4—28no—
62C2×C3×C22⋊C4—28no—
63C2×C3×C22⋊C4—42no—
64C2×#217—22no—
65C2×#217—24no—
66C2×#217—24no—
67C2×#217—24no—
68C2×#217—24no—
69C2×#217—24no—
70C2×#217—24no—
71C2×#217—24no—
72C2×#217—24no—
73C2×#217—24no—
74C2×#217D418yes—
75#936—232no—
76#935—232no—
77C2×C4⋊Dic3—28no—
78C2×C4⋊Dic3—28no—
79C2×C4⋊Dic3—44no—
80C2×Dic3⋊C4—24no—
81C2×Dic3⋊C4—24no—
82C2×Dic3⋊C4D418yes—
83C2×C4×Dic3—22no—
84C2×C4×Dic3—24no—
85C3×#95—24no—
86C3⋊#95—28no—
87C3⋊#95—28no—
88C3⋊#95—28no—
89C23×Q8—31no—
90C23×D4C2×D311yescharacteristic
91C24×C4—31no—
92C2×#124—68no—
93C2×#124—64no—
94C2×#123—68no—
95C2×#123—64no—
96C22×C4⋊C4—32no—
97C22×C4⋊C4—34no—
98C22×C22⋊C4—32no—
99C22×C22⋊C4—32no—
100C22×C22⋊C4—31no—
101C22×C22⋊C4—32no—
102C22×C22⋊C4C2×D312yes—
103C22×C22⋊C4C2×D312yes—
104C22×C42—31no—
105#395—68no—
106#385—68no—
107C2×#95—34no—
108C2×#95—34no—
109C2×#95—34no—
110C2×#95C2×D312yes—
111C24×C3—22no—
112C24×C3—24no—
113C24×C3—22no—
114C24×C3—24no—
115C24×C3—22no—
116C24×C3—22no—
117C24×C3—24no—
118C24×C3—22no—
119C24×C3—24no—
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—22no—
129C22×C3×C4—22no—
130C22×C3×C4—24no—
131C22×C3×C4—24no—
132C22×C3×C4—24no—
133C22×C3×C4—24no—
134C22×C3×C4—24no—
135C22×C3×C4—24no—
136C22×C3×C4C2×D412yes—
137C22×C3×C4C2×D414yes—
138C22×C3×C4#4118yes—
139C22×Dic3—28no—
140C22×Dic3—24no—
141C22×Dic3—24no—
142C22×Dic3—22no—
143C22×Dic3—22no—
144C22×Dic3—22no—
145C22×Dic3—22no—
146C22×Dic3—22no—
147C22×Dic3—22no—
148C22×Dic3—22no—
149C22×Dic3—22no—
150C22×Dic3—24no—
151C22×Dic3—24no—
152C22×Dic3—24no—
153C22×Dic3—24no—
154C22×Dic3—24no—
155C22×Dic3—24no—
156C22×Dic3—416no—
157C22×Dic3—44no—
158C22×Dic3—44no—
159C22×Dic3C2×D414yes—
160C22×Dic3C2×D414yes—
161C22×Dic3C2×D414yes—
162C22×Dic3C2×D414yes—
163C2×Dic6—28no—
164C2×Dic6—416no—
165C2×Dic6—44no—
166C2×Dic6—44no—
167C3×C22⋊C4—216no—
168C3×C22⋊C4—48no—
169#217—28no—
170#217—216no—
171#217—216no—
172#217—216no—
173C4⋊Dic3—416no—
174Dic3⋊C4—216no—
175C4×Dic3—28no—
176C25C22×D311yescharacteristic
177C25C22×D311yescharacteristic
178C22×Q8—64no—
179C22×Q8—62no—
180C22×Q8—64no—
181C22×Q8—122no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—42no—
185C22×D4D1214yes—
186C23×C4—32no—
187C23×C4—32no—
188C23×C4—32no—
189C23×C4—32no—
190C23×C4—64no—
191C23×C4—62no—
192C23×C4—64no—
193C23×C4—122no—
194C23×C4C22×D314yes—
195C23×C4C22×D311yescharacteristic
196#124—632no—
197#123—632no—
198C2×C4⋊C4—38no—
199C2×C4⋊C4—64no—
200C2×C4⋊C4—64no—
201C2×C4⋊C4—68no—
202C2×C4⋊C4—68no—
203C2×C4⋊C4—124no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—28no—
206C2×C22⋊C4—28no—
207C2×C22⋊C4—38no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—64no—
210C2×C22⋊C4—64no—
211C2×C22⋊C4—64no—
212C2×C22⋊C4—64no—
213C2×C22⋊C4—64no—
214C2×C22⋊C4—62no—
215C2×C22⋊C4—64no—
216C2×C22⋊C4—64no—
217C2×C22⋊C4—64no—
218C2×C22⋊C4—64no—
219C2×C42—64no—
220C2×C42—62no—
221#95—24no—
222#95—68no—
223#95—68no—
224#95—68no—
225C23×C3—22no—
226C23×C3—24no—
227C23×C3—22no—
228C23×C3—24no—
229C23×C3—28no—
230C23×C3—28no—
231C23×C3—28no—
232C23×C3—22no—
233C23×C3—22no—
234C23×C3—22no—
235C23×C3—28no—
236C23×C3—42no—
237C23×C3—42no—
238C23×C3—44no—
239C23×C3—44no—
240C23×C3—42no—
241C23×C3—42no—
242C23×C3—42no—
243C23×C3—42no—
244C23×C3—42no—
245C23×C3—42no—
246C23×C3—44no—
247C23×C3—44no—
248C23×C3—44no—
249C23×C3—44no—
250C23×C3—42no—
251C23×C3—42no—
252C23×C3#12014yes—
253C23×C3C22×D411yescharacteristic
254C23×C3C22×D412yes—
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—22no—
262C2×C3×C4—28no—
263C2×C3×C4—28no—
264C2×C3×C4—28no—
265C2×C3×C4—28no—
266C2×C3×C4—44no—
267C2×C3×C4—44no—
268C2×C3×C4—44no—
269C2×C3×C4—44no—
270C2×C3×C4#12014yes—
271C2×Dic3—24no—
272C2×Dic3—24no—
273C2×Dic3—24no—
274C2×Dic3—24no—
275C2×Dic3—24no—
276C2×Dic3—24no—
277C2×Dic3—28no—
278C2×Dic3—28no—
279C2×Dic3—28no—
280C2×Dic3—416no—
281C2×Dic3—42no—
282C2×Dic3—42no—
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—
292Dic6—416no—
293C24—22no—
294C24—24no—
295C24—22no—
296C24—24no—
297C24—22no—
298C24—22no—
299C24—24no—
300C24—22no—
301C24—24no—
302C24C2×D1212yes—
303C24C2×D1212yes—
304C24D3×D414yes—
305C24C23×D311yescharacteristic
306C2×Q8—68no—
307C2×Q8—1216no—
308C2×Q8—124no—
309C2×Q8—124no—
310C2×D4—28no—
311C2×D4—28no—
312C2×D4—44no—
313C2×D4—416no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—34no—
323C22×C4—34no—
324C22×C4—34no—
325C22×C4—34no—
326C22×C4—68no—
327C22×C4—62no—
328C22×C4—62no—
329C22×C4—62no—
330C22×C4—62no—
331C22×C4—64no—
332C22×C4—64no—
333C22×C4—62no—
334C22×C4—62no—
335C22×C4—62no—
336C22×C4—62no—
337C22×C4—64no—
338C22×C4—64no—
339C22×C4—64no—
340C22×C4—64no—
341C22×C4—64no—
342C22×C4—64no—
343C22×C4—1216no—
344C22×C4—124no—
345C22×C4—124no—
346C22×C4C2×D1212yes—
347C22×C4D3×D414yes—
348C22×C4#23718yes—
349C4⋊C4—616no—
350C4⋊C4—1216no—
351C22⋊C4—216no—
352C22⋊C4—48no—
353C22⋊C4—68no—
354C22⋊C4—616no—
355C22⋊C4—616no—
356C22⋊C4—616no—
357C42—68no—
358C22×C3—24no—
359C22×C3—28no—
360C22×C3—48no—
361C22×C3—44no—
362C22×C3—48no—
363C22×C3—44no—
364C22×C3—42no—
365C22×C3—44no—
366C22×C3—44no—
367C22×C3—44no—
368C22×C3—84no—
369C22×C3—84no—
370C22×C3C2×#12011yescharacteristic
371C22×C3C2×#12011yescharacteristic
372C22×C3#52512yes—
373C22×C3#53212yes—
374C22×C3#53411yescharacteristic
375C22×C3#53612yes—
376C22×C3#53612yes—
377C22×C3#54214yes—
378C22×C3D4214yes—
379C22×C3#54714yes—
380C22×C3#54714yes—
381C22×C3#54814yes—
382C22×C3#55112yes—
383C22×C3#56312yes—
384C3×C4—24no—
385C3×C4—48no—
386Dic3—44no—
387Dic3—44no—
388Dic3—44no—
389Dic3—44no—
390C23—24no—
391C23—22no—
392C23—22no—
393C23—28no—
394C23—24no—
395C23—28no—
396C23—28no—
397C23—22no—
398C23—22no—
399C23—22no—
400C23—28no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—44no—
406C23—44no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—44no—
414C23—44no—
415C23—44no—
416C23—44no—
417C23#88214yes—
418C23C22×D1211yescharacteristic
419C23C2×D3×D412yes—
420C23C2×D3×D412yes—
421C23C2×#23714yes—
422C23#100412yes—
423C23#101014yes—
424Q8—1216no—
425D4—416no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—28no—
429C2×C4—28no—
430C2×C4—28no—
431C2×C4—28no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—64no—
437C2×C4—64no—
438C2×C4—64no—
439C2×C4—64no—
440C2×C4—64no—
441C2×C4—64no—
442C2×C4—68no—
443C2×C4—68no—
444C2×C4—68no—
445C2×C4—1216no—
446C2×C4—122no—
447C2×C4—122no—
448C2×C4—122no—
449C2×C4—122no—
450C2×C4—122no—
451C2×C4—124no—
452C2×C4—124no—
453C2×C4—122no—
454C2×C4—124no—
455C2×C4—124no—
456C2×C4—124no—
457C2×C4#88214yes—
458C2×C3—44no—
459C2×C3—82no—
460C2×C3#240311yescharacteristic
461C2×C3#240612yes—
462C2×C3#243212yes—
463C2×C3#246114yes—
464C2×C3#267414yes—
465C2×C3#271712yes—
466C22—24no—
467C22—28no—
468C22—44no—
469C22—48no—
470C22—44no—
471C22—48no—
472C22—44no—
473C22—42no—
474C22—44no—
475C22—44no—
476C22—84no—
477C22—84no—
478C22C2×#88211yescharacteristic
479C22C2×#88211yescharacteristic
480C22#549311yescharacteristic
481C22#555514yes—
482C22#558612yes—
483C22#558712yes—
484C22#559014yes—
485C22#559814yes—
486C22#560314yes—
487C22#561114yes—
488C22#564812yes—
489C22#565112yes—
490C22#567512yes—
491C22#571612yes—
492C4—24no—
493C4—48no—
494C4—124no—
495C4—124no—
496C4—124no—
497C4—124no—
498C3#1526211yescharacteristic, complemented, Sylow
499C2—44no—
500C2—82no—
501C2#7385211yescharacteristic
502C2#7468712yes—
503C2#7482512yes—
504C2#7644512yes—
505C2#7668014yes—
506C2#7700214yes—
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC3×#1438C2
prime radicalC23×C3×D4C22
hypercenter#1438D3
hypocenterC3#15262
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1438

Complements

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

NH ≅ G/Ntype
3. C2×#4950C2semidirect
4. C2×#4656C2semidirect
6. #71549C2semidirect
7. #71546C2semidirect
8. #69405C2semidirect
498. C3#15262semidirect

Maximal subgroups

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

subgroupindex
C22×#9362
C22×#9352
C2×#49502
C2×#46562
C3×#14382
#715492
#715462
#694052
#152623