Finite Group Databasebeta

← #10750412

Subgroups — #10750412

Order 768.

1746 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#869C211yesmaximal, characteristic, complemented
2C2×C3×#388C211yesmaximal, characteristic
3C2×#4953C212yesmaximal
4C2×#4947C212yesmaximal
5#71565C212yesmaximal, complemented
6#71546C212yesmaximal
7#71534C211yesmaximal, characteristic, complemented
8#69424C212yesmaximal, complemented
9#69422C212yesmaximal, complemented
10#16506—31nomaximal, Sylow
11C23×Dic6C2212yes—
12C22×C3×C4⋊C4C2211yescharacteristic
13C22×C3×C22⋊C4C2211yescharacteristic
14C22×C3×C22⋊C4C2211yescharacteristic
15C22×C3×C22⋊C4C2211yescharacteristic
16C22×C3×C42C2211yescharacteristic
17C22×#217C2214yes—
18C22×C4⋊Dic3C2212yes—
19C22×C4⋊Dic3C2212yes—
20C2×#869—24no—
21C2×#869—28no—
22C3×#388—24no—
23C2×C3×#95C2211yescharacteristic
24C2×C3×#95C2211yescharacteristic
25C2×C3⋊#95C2212yes—
26C2×C3⋊#95C2214yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2214yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2214yes—
31#4953—28no—
32#4947—28no—
33C22×C4⋊Q8—31no—
34C2×#397—32no—
35C2×#393—32no—
36C2×#388D311yescharacteristic
37#1446—32no—
38#1445—32no—
39#1441—32no—
40#1439—32no—
41#1434—31no—
42C25×C3C2311yescharacteristic
43C23×C3×C4C2311yescharacteristic
44C23×C3×C4C2311yescharacteristic
45C23×C3×C4C2311yescharacteristic
46C23×C3×C4C2311yescharacteristic
47C23×C3×C4C2311yescharacteristic
48C23×C3×C4C2311yescharacteristic
49C23×Dic3C2314yes—
50C23×Dic3C2314yes—
51C22×Dic6—24no—
52C22×Dic6—28no—
53C22×Dic6—24no—
54C22×Dic6—24no—
55C22×Dic6—44no—
56C2×C3×C4⋊C4—22no—
57C2×C3×C4⋊C4—24no—
58C2×C3×C22⋊C4—22no—
59C2×C3×C22⋊C4—22no—
60C2×C3×C22⋊C4—24no—
61C2×C3×C22⋊C4—22no—
62C2×C3×C22⋊C4—22no—
63C2×C3×C22⋊C4—24no—
64C2×C3×C22⋊C4D414yes—
65C2×C3×C42—24no—
66C2×C3×C42D414yes—
67C2×#217—28no—
68C2×#217—28no—
69C2×#217—44no—
70C2×C4⋊Dic3—24no—
71C2×C4⋊Dic3—24no—
72C2×C4⋊Dic3—24no—
73C2×C4⋊Dic3—28no—
74C2×C4⋊Dic3D418yes—
75#869—232no—
76C3×#95—22no—
77C3×#95—22no—
78C3⋊#95—28no—
79C3⋊#95—24no—
80C3⋊#95—24no—
81C3⋊#95—24no—
82C3⋊#95—28no—
83C3⋊#95—44no—
84C23×Q8—32no—
85C2×C4⋊Q8—68no—
86C2×C4⋊Q8—64no—
87C22×C4⋊C4—32no—
88C22×C4⋊C4—32no—
89C22×C4⋊C4C2×D311yescharacteristic
90C22×C22⋊C4—34no—
91C22×C22⋊C4C2×D311yescharacteristic
92C22×C22⋊C4C2×D311yescharacteristic
93C22×C22⋊C4C2×D311yescharacteristic
94C22×C42C2×D311yescharacteristic
95#397—68no—
96#393—68no—
97#388—24no—
98C2×#95—32no—
99C2×#95—34no—
100C2×#95—32no—
101C2×#95—34no—
102C2×#95—32no—
103C2×#95—34no—
104C2×#95C2×D311yescharacteristic
105C2×#95C2×D311yescharacteristic
106C24×C3—22no—
107C24×C3—21no—
108C24×C3—21no—
109C24×C3—22no—
110C24×C3—22no—
111C24×C3—22no—
112C24×C3—22no—
113C24×C3—22no—
114C24×C3C2×D412yes—
115C24×C3C2411yescharacteristic
116C22×C3×C4—22no—
117C22×C3×C4—22no—
118C22×C3×C4—22no—
119C22×C3×C4—22no—
120C22×C3×C4—22no—
121C22×C3×C4—22no—
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—22no—
127C22×C3×C4—21no—
128C22×C3×C4—21no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4C2×D412yes—
135C22×C3×C4C2×D412yes—
136C22×C3×C4C2×D414yes—
137C22×C3×C4C2×D412yes—
138C22×C3×C4C2×D412yes—
139C22×C3×C4C2×D412yes—
140C22×C3×C4#4112yes—
141C22×C3×C4#4114yes—
142C22×Dic3—24no—
143C22×Dic3—24no—
144C22×Dic3—24no—
145C22×Dic3—24no—
146C22×Dic3—24no—
147C22×Dic3—24no—
148C22×Dic3—24no—
149C22×Dic3—24no—
150C22×Dic3—24no—
151C22×Dic3—24no—
152C22×Dic3—24no—
153C22×Dic3—24no—
154C22×Dic3—24no—
155C22×Dic3C2×D418yes—
156C2×Dic6—216no—
157C2×Dic6—416no—
158C2×Dic6—416no—
159C2×Dic6—48no—
160C2×Dic6—48no—
161C3×C4⋊C4—44no—
162C3×C22⋊C4—28no—
163C3×C22⋊C4—44no—
164C3×C22⋊C4—44no—
165C3×C42—28no—
166#217—416no—
167C4⋊Dic3—216no—
168C4⋊Dic3—216no—
169C25C22×D311yescharacteristic
170C22×Q8—68no—
171C22×Q8—64no—
172C22×Q8—64no—
173C22×Q8—64no—
174C22×Q8—124no—
175C23×C4—34no—
176C23×C4—34no—
177C23×C4C22×D311yescharacteristic
178C23×C4C22×D311yescharacteristic
179C23×C4C22×D311yescharacteristic
180C23×C4C22×D311yescharacteristic
181C23×C4C22×D311yescharacteristic
182C23×C4C22×D311yescharacteristic
183C4⋊Q8—632no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—38no—
187C2×C4⋊C4—68no—
188C2×C4⋊C4—64no—
189C2×C4⋊C4—64no—
190C2×C4⋊C4—64no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—68no—
198C2×C22⋊C4—68no—
199C2×C22⋊C4—124no—
200C2×C22⋊C4D1214yes—
201C2×C42—24no—
202C2×C42#6714yes—
203#95—22no—
204#95—22no—
205#95—68no—
206#95—68no—
207#95—64no—
208#95—64no—
209#95—64no—
210#95—124no—
211C23×C3—22no—
212C23×C3—24no—
213C23×C3—24no—
214C23×C3—24no—
215C23×C3—41no—
216C23×C3—41no—
217C23×C3—41no—
218C23×C3—44no—
219C23×C3—41no—
220C23×C3—41no—
221C23×C3—41no—
222C23×C3—42no—
223C23×C3—44no—
224C23×C3—42no—
225C23×C3—44no—
226C23×C3—42no—
227C23×C3—44no—
228C23×C3C22×D411yescharacteristic
229C23×C3C22×D412yes—
230C23×C3C22×D411yescharacteristic
231C23×C3C2×#4111yescharacteristic
232C23×C3C2×#4112yes—
233C23×C3H(5,𝔽2)12yes—
234C23×C32−1+412yes—
235C23×C32−1+412yes—
236C23×C32−1+412yes—
237C2×C3×C4—22no—
238C2×C3×C4—22no—
239C2×C3×C4—22no—
240C2×C3×C4—28no—
241C2×C3×C4—22no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—24no—
245C2×C3×C4—24no—
246C2×C3×C4—28no—
247C2×C3×C4—22no—
248C2×C3×C4—24no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—42no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
258C2×C3×C4—44no—
259C2×C3×C4—44no—
260C2×C3×C4#12014yes—
261C2×C3×C4#12114yes—
262C2×C3×C4#12414yes—
263C2×Dic3—28no—
264C2×Dic3—28no—
265C2×Dic3—28no—
266C2×Dic3—44no—
267C2×Dic3—44no—
268C2×Dic3—44no—
269C2×Dic3—44no—
270C2×Dic3—44no—
271C2×Dic3—44no—
272C2×Dic3—44no—
273C2×Dic3—44no—
274C2×Dic3—44no—
275C2×Dic3—44no—
276C2×Dic3—44no—
277Dic6—432no—
278C24—21no—
279C24—22no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—21no—
286C24C2×D1212yes—
287C24C23×D311yescharacteristic
288C2×Q8—616no—
289C2×Q8—1216no—
290C2×Q8—1216no—
291C2×Q8—128no—
292C2×Q8—128no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—38no—
312C22×C4—64no—
313C22×C4—64no—
314C22×C4—64no—
315C22×C4—64no—
316C22×C4—64no—
317C22×C4—64no—
318C22×C4—64no—
319C22×C4—64no—
320C22×C4—64no—
321C22×C4—64no—
322C22×C4—64no—
323C22×C4—64no—
324C22×C4—64no—
325C22×C4C2×D1212yes—
326C22×C4C2×D1212yes—
327C22×C4#23512yes—
328C22×C4D3×D414yes—
329C22×C4#23714yes—
330C22×C4C2×#6712yes—
331C22×C4C2×#6712yes—
332C22×C4C2×#6712yes—
333C4⋊C4—44no—
334C4⋊C4—616no—
335C4⋊C4—616no—
336C22⋊C4—28no—
337C22⋊C4—44no—
338C22⋊C4—44no—
339C22⋊C4—1216no—
340C42—28no—
341C22×C3—42no—
342C22×C3—42no—
343C22×C3—48no—
344C22×C3—42no—
345C22×C3—81no—
346C22×C3—81no—
347C22×C3—81no—
348C22×C3—81no—
349C22×C3—84no—
350C22×C3C2×#12011yescharacteristic
351C22×C3C2×#12111yescharacteristic
352C22×C3C2×#12411yescharacteristic
353C22×C3#53511yescharacteristic
354C22×C3#53614yes—
355C22×C3#53612yes—
356C22×C3#53611yescharacteristic
357C22×C3#53611yescharacteristic
358C22×C3#53611yescharacteristic
359C22×C3#54112yes—
360C22×C3#54214yes—
361C22×C3D4212yes—
362C22×C3#54612yes—
363C22×C3#54712yes—
364C22×C3#54814yes—
365C22×C3#55612yes—
366C22×C3#56314yes—
367C3×C4—24no—
368C3×C4—24no—
369C3×C4—24no—
370C3×C4—42no—
371C3×C4—81no—
372C3×C4—81no—
373Dic3—48no—
374Dic3—84no—
375C23—24no—
376C23—22no—
377C23—24no—
378C23—24no—
379C23—44no—
380C23—44no—
381C23—44no—
382C23—42no—
383C23—41no—
384C23—42no—
385C23—41no—
386C23—42no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—44no—
392C23C22×D1211yescharacteristic
393C23C2×#23511yescharacteristic
394C23C2×D3×D412yes—
395C23C2×#23712yes—
396C23#100912yes—
397C23#101012yes—
398C23#101012yes—
399C23#101012yes—
400C23C22×#6711yescharacteristic
401Q8—1232no—
402C2×C4—28no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—28no—
412C2×C4—22no—
413C2×C4—24no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—44no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—44no—
425C2×C4—68no—
426C2×C4—68no—
427C2×C4—68no—
428C2×C4—124no—
429C2×C4—124no—
430C2×C4—124no—
431C2×C4—124no—
432C2×C4—124no—
433C2×C4—124no—
434C2×C4—124no—
435C2×C4—124no—
436C2×C4—124no—
437C2×C4—124no—
438C2×C4—124no—
439C2×C4#93014yes—
440C2×C4#93614yes—
441C2×C4#93714yes—
442C2×C3—82no—
443C2×C3#240811yescharacteristic
444C2×C3#243611yescharacteristic
445C2×C3#254711yescharacteristic
446C2×C3#261411yescharacteristic
447C2×C3#262111yescharacteristic
448C2×C3#264011yescharacteristic
449C2×C3#267414yes—
450C2×C3#271614yes—
451C2×C3#282111yescharacteristic
452C22—42no—
453C22—48no—
454C22—42no—
455C22—42no—
456C22—81no—
457C22—81no—
458C22—84no—
459C22—81no—
460C22—81no—
461C22#551812yes—
462C22#552612yes—
463C22#554212yes—
464C22D4×D1212yes—
465C22#555012yes—
466C22#555512yes—
467C22#559814yes—
468C22#564814yes—
469C22#568214yes—
470C22#571214yes—
471C22C2×#93011yescharacteristic
472C22C2×#93611yescharacteristic
473C22C2×#93711yescharacteristic
474C22#582511yescharacteristic
475C22#582511yescharacteristic
476C22#582511yescharacteristic
477C22#583011yescharacteristic
478C4—24no—
479C4—24no—
480C4—24no—
481C4—42no—
482C4—81no—
483C4—81no—
484C4—128no—
485C4—244no—
486C3#1650611yescharacteristic, complemented, Sylow
487C2—82no—
488C2#7566614yes—
489C2#7569514yes—
490C2#8226211yescharacteristic
491C2#8238211yescharacteristic
492C2#8240311yescharacteristic
493C2#8265411yescharacteristic
494C2#8286411yescharacteristic
495C2#8297811yescharacteristic
496C2#8310311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC2×C3×#388C2
prime radicalC25×C3C23
hypercenterC2×#388D3
hypocenterC3#16506
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#388

Complements

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

NH ≅ G/Ntype
1. C22×#869C2semidirect
5. #71565C2semidirect
7. #71534C2semidirect
8. #69424C2semidirect
9. #69422C2semidirect
486. C3#16506semidirect

Maximal subgroups

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

subgroupindex
C22×#8692
C2×C3×#3882
C2×#49532
C2×#49472
#715652
#715462
#715342
#694242
#694222
#165063