Finite Group Databasebeta

← #10824346

Subgroups — #10824346

Order 768.

1016 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4788C212yesmaximal, complemented
2#76477C211yesmaximal, characteristic, complemented
3#74319C211yesmaximal, characteristic
4C3×#2072C211yesmaximal, characteristic, complemented
5#70280C212yesmaximal, complemented
6#69688C211yesmaximal, characteristic
7#69570C211yesmaximal, characteristic, complemented
8#68900C212yesmaximal, complemented
9#68780C212yesmaximal, complemented
10#68629C211yesmaximal, characteristic
11#68602C211yesmaximal, characteristic, complemented
12#23678—31nomaximal, Sylow
13C2×Q8×Dic3C2211yescharacteristic
14C22×#215C2212yes—
15C2×#893—21no—
16C2×#893—21no—
17C2×#893C2212yes—
18C2×#874C2211yescharacteristic
19C2×C3×#103C2212yes—
20C3×#391C2211yescharacteristic
21C3×#384C2211yescharacteristic
22#5229C2211yescharacteristic
23C2×#821C2212yes—
24C2×#821C2212yes—
25C2×Dic3⋊C8C2212yes—
26C2×C8×Dic3C2212yes—
27#4988—21no—
28#4988—21no—
29#4988C2212yes—
30Dic3×C4⋊C4C2211yescharacteristic
31#4972C2211yescharacteristic
32C2×#809C2212yes—
33C2×#808C2212yes—
34C42×Dic3C2211yescharacteristic
35#4928C2211yescharacteristic
36#4788—28no—
37#4654—21no—
38#4653—21no—
39#4649—21no—
40#4648—21no—
41C3×#340C2212yes—
42C3×#336C2211yescharacteristic
43#4529C2212yes—
44#4524C2211yescharacteristic
45C2×C4×SD16—32no—
46C4×#127—31no—
47#2295—31no—
48#2084—31no—
49#2072D311yescharacteristic, complemented
50#2065—32no—
51#2055—32no—
52#2041—31no—
53#2014—31no—
54C4×C4⋊C8—31no—
55C4×#102—32no—
56C22×C3×Q8C2311yescharacteristic
57C22×D12—21no—
58C22×D12—21no—
59C22×D12C2312yes—
60C22×C4×D3—22no—
61C22×C3×C8C2312yes—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4C2312yes—
65C2×C3×C4⋊C4C2311yescharacteristic
66C2×C3×C42C2311yescharacteristic
67Q8×Dic3—22no—
68Q8×Dic3D414yescomplemented
69C2×#215—24no—
70C2×#215—24no—
71C2×#215—24no—
72C2×(C2×D3)⋊C4—22no—
73C2×(C2×D3)⋊C4—22no—
74C2×(C2×D3)⋊C4—22no—
75C2×C4⋊Dic3—21no—
76C2×C4⋊Dic3C2311yescharacteristic
77C2×C4⋊Dic3C2311yescharacteristic
78C2×Dic3⋊C4—21no—
79C2×Dic3⋊C4—21no—
80C2×C4×Dic3—21no—
81C2×C4×Dic3—21no—
82C2×C4×Dic3D411yescharacteristic
83C2×C4×Dic3D411yescharacteristic
84C2×C4×Dic3D411yescharacteristic
85C2×C4×Dic3D411yescharacteristic
86C2×C4×Dic3C2311yescharacteristic
87C2×C4×Dic3C2311yescharacteristic
88C22×C3⋊C8C2312yes—
89#893—24no—
90#893—24no—
91#893—44no—
92#893—44no—
93#874—24no—
94C3×#103—24no—
95C3×#95—21no—
96C3×#95—21no—
97C3⋊#95—21no—
98C3⋊#95—21no—
99C3⋊#95—21no—
100C3⋊#95—21no—
101C3⋊#95—21no—
102#821—24no—
103#821—24no—
104Dic3⋊C8—24no—
105C8×Dic3—24no—
106#809—24no—
107#808—24no—
108C22×SD16—32no—
109C2×#127—31no—
110C2×C4×Q8—31no—
111C2×C4×D4—32no—
112C2×C4×D4—61no—
113C2×C4×D4—61no—
114C4×SD16—68no—
115C2×#106—32no—
116C2×C4⋊C8—32no—
117C2×#103C2×D312yes—
118C2×#102—32no—
119C2×#102—32no—
120C2×#102—32no—
121C2×C4×C8—32no—
122#391—31no—
123#391C2×D311yescharacteristic
124#390—32no—
125#390—61no—
126#390—61no—
127#389—31no—
128#389—31no—
129#387—61no—
130#387—61no—
131#384C2×D311yescharacteristic
132C4×C4⋊C4—31no—
133C4×C4⋊C4—61no—
134#376—61no—
135C43—31no—
136#340—32no—
137#340C2×D312yes—
138#336—31no—
139#336C2×D311yescharacteristic
140C23×D3—22no—
141C2×C3×Q8—21no—
142C2×C3×Q8—22no—
143C2×C3×Q8—22no—
144C2×C3×Q8C2×D412yes—
145C22×C3×C4—21no—
146C22×C3×C4—21no—
147C22×C3×C4—22no—
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4C2411yescharacteristic
151C22×Dic3—21no—
152C22×Dic3—21no—
153C22×Dic3C2×D411yescharacteristic
154C22×Dic3C2×D411yescharacteristic
155C22×Dic3C2×D411yescharacteristic
156C22×Dic3C2×D411yescharacteristic
157C2×D12—22no—
158C2×D12—22no—
159C2×D12—22no—
160C2×D12—22no—
161C2×D12—24no—
162C2×D12—24no—
163C2×D12—22no—
164C2×D12—42no—
165C2×D12—42no—
166C2×D12—42no—
167C2×D12—42no—
168C2×D12#4114yes—
169C2×C4×D3—44no—
170C2×C4×D3—44no—
171C2×C4×D3—44no—
172C2×C3×C8—22no—
173C2×C3×C8—22no—
174C2×C3×C8—22no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—42no—
179C3×C4⋊C4—42no—
180C3×C42—22no—
181#215—216no—
182(C2×D3)⋊C4—44no—
183(C2×D3)⋊C4—44no—
184(C2×D3)⋊C4—44no—
185C4⋊Dic3—21no—
186C4⋊Dic3—24no—
187C4⋊Dic3—22no—
188C4⋊Dic3C2×D412yes—
189Dic3⋊C4—42no—
190Dic3⋊C4—42no—
191C4×Dic3—22no—
192C4×Dic3—22no—
193C4×Dic3—24no—
194C4×Dic3—22no—
195C4×Dic3—41no—
196C4×Dic3—42no—
197C4×Dic3SD1614yescomplemented
198C4×Dic3C2×D412yes—
199C4×Dic3#4112yes—
200C2×C3⋊C8—22no—
201C2×C3⋊C8—22no—
202C2×C3⋊C8#4114yes—
203C22×Q8C22×D311yescharacteristic
204C22×D4—32no—
205C22×D4—61no—
206C22×D4—61no—
207C23×C4—62no—
208C2×SD16—64no—
209C2×SD16—64no—
210C2×SD16—64no—
211C22×C8—32no—
212C22×C8C22×D312yes—
213#127—64no—
214C4×Q8—34no—
215C4×Q8—62no—
216C4×D4—64no—
217C4×D4—64no—
218C4×D4—124no—
219C4×D4—124no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—31no—
223C2×C4⋊C4—31no—
224C2×C4⋊C4—61no—
225C2×C4⋊C4—61no—
226C2×C4⋊C4—61no—
227C2×C4⋊C4C22×D312yes—
228C2×C4⋊C4C22×D311yescharacteristic
229C2×C22⋊C4—62no—
230C2×C22⋊C4—62no—
231C2×C22⋊C4—62no—
232C2×C42—31no—
233C2×C42—31no—
234C2×C42—31no—
235C2×C42—31no—
236C2×C42—31no—
237C2×C42—31no—
238C2×C42—61no—
239C2×C42—61no—
240C2×C42C22×D311yescharacteristic
241#106—64no—
242C4⋊C8—64no—
243#103—24no—
244#102—64no—
245#102—64no—
246#102—64no—
247C4×C8—64no—
248#95—21no—
249#95—21no—
250#95—61no—
251#95—61no—
252#95—61no—
253#95—61no—
254#95—61no—
255C23×C3C22×D411yescharacteristic
256C22×D3—24no—
257C22×D3—42no—
258C22×D3—42no—
259C22×D3—42no—
260C22×D3—42no—
261C22×D3—42no—
262C22×D3—42no—
263C3×Q8—24no—
264C3×Q8—42no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—22no—
270C2×C3×C4—41no—
271C2×C3×C4—41no—
272C2×C3×C4—42no—
273C2×C3×C4—41no—
274C2×C3×C4—42no—
275C2×C3×C4—41no—
276C2×C3×C4—42no—
277C2×C3×C4—41no—
278C2×C3×C4C2×SD1612yes—
279C2×C3×C4#13512yes—
280C2×C3×C4C22×D411yescharacteristic
281C2×C3×C4C2×#4112yes—
282C2×C3×C4C2×#4112yes—
283C2×C3×C4C2×#4111yescharacteristic
284C2×Dic3—21no—
285C2×Dic3—21no—
286C2×Dic3—21no—
287C2×Dic3—21no—
288C2×Dic3—22no—
289C2×Dic3—21no—
290C2×Dic3—21no—
291C2×Dic3—21no—
292C2×Dic3—41no—
293C2×Dic3—41no—
294C2×Dic3—41no—
295C2×Dic3—41no—
296C2×Dic3—41no—
297C2×Dic3#12112yes—
298C2×Dic3#12112yes—
299C2×Dic3C2×SD1612yes—
300C2×Dic3C2×SD1612yes—
301C2×Dic3AGL(1,ℤ/8)12yes—
302D12—24no—
303D12—24no—
304D12—48no—
305D12—48no—
306D12—42no—
307D12—42no—
308C4×D3—88no—
309C3×C8—42no—
310C3⋊C8—24no—
311C24—62no—
312C2×Q8—21no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8D3×D412yes—
316C2×D4—34no—
317C2×D4—64no—
318C2×D4—64no—
319C2×D4—62no—
320C2×D4—62no—
321C2×D4—62no—
322C2×D4—62no—
323C2×D4—62no—
324C2×D4—122no—
325C2×D4—122no—
326C2×D4—122no—
327C2×D4—122no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—31no—
332C22×C4—31no—
333C22×C4—31no—
334C22×C4—31no—
335C22×C4—61no—
336C22×C4—61no—
337C22×C4—124no—
338C22×C4—124no—
339C22×C4—124no—
340C22×C4D3×D411yescharacteristic
341C22×C4D3×D411yescharacteristic
342C22×C4C23×D311yescharacteristic
343SD16—616no—
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—34no—
348C2×C8—62no—
349C2×C8—62no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—32no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—64no—
357C4⋊C4—61no—
358C4⋊C4—62no—
359C4⋊C4—122no—
360C4⋊C4—122no—
361C22⋊C4—124no—
362C22⋊C4—124no—
363C22⋊C4—124no—
364C42—22no—
365C42—34no—
366C42—32no—
367C42—32no—
368C42—62no—
369C42—64no—
370C42—62no—
371C42—62no—
372C42—121no—
373C42—122no—
374C22×C3C2×#12111yescharacteristic
375C22×C3#52511yescharacteristic
376C22×C3#53211yescharacteristic
377C22×C3C22×SD1611yescharacteristic
378C22×C3C2×#13511yescharacteristic
379C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
380C22×C3#57511yescharacteristic
381C2×D3—44no—
382C2×D3—44no—
383C2×D3—44no—
384C2×D3—82no—
385C2×D3—82no—
386C2×D3—82no—
387C2×D3—82no—
388C3×C4—22no—
389C3×C4—22no—
390C3×C4—42no—
391C3×C4—81no—
392C3×C4—82no—
393C3×C4#52512yes—
394C3×C4#52912yes—
395Dic3—22no—
396Dic3—22no—
397Dic3—41no—
398Dic3—42no—
399Dic3—81no—
400Dic3#46014yes—
401C23—64no—
402C23—122no—
403C23—122no—
404C23—122no—
405C23—122no—
406C23—122no—
407C23—122no—
408C23C2×D3×D411yescharacteristic
409Q8—24no—
410Q8—42no—
411D4—64no—
412D4—64no—
413D4—128no—
414D4—128no—
415D4—122no—
416D4—122no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—32no—
423C2×C4—32no—
424C2×C4—32no—
425C2×C4—32no—
426C2×C4—32no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—61no—
436C2×C4—61no—
437C2×C4—61no—
438C2×C4—61no—
439C2×C4—61no—
440C2×C4—61no—
441C2×C4—61no—
442C2×C4—62no—
443C2×C4—121no—
444C2×C4—121no—
445C2×C4—121no—
446C2×C4—121no—
447C2×C4—121no—
448C2×C4—248no—
449C2×C4D3×SD1612yes—
450C2×C4#91912yes—
451C2×C4C2×D3×D411yescharacteristic
452C2×C4D3×#4112yes—
453C2×C4D3×#4112yes—
454C2×C4D3×#4111yescharacteristic
455C8—42no—
456C8—64no—
457C2×C3#239211yescharacteristic
458C2×C3C2×#46011yescharacteristic
459C2×C3#303511yescharacteristic
460C2×C3#310111yescharacteristic
461C2×C3#310511yescharacteristic
462C2×C3#316711yescharacteristic
463C2×C3#317111yescharacteristic
464D3—84no—
465C22—124no—
466C22—124no—
467C22—124no—
468C22—242no—
469C22—242no—
470C22—242no—
471C22—242no—
472C22D3×#12111yescharacteristic
473C22#565611yescharacteristic
474C22#566811yescharacteristic
475C22C2×D3×SD1611yescharacteristic
476C22C2×#91911yescharacteristic
477C22D3×AGL(1,ℤ/8)11yescharacteristic
478C22#577711yescharacteristic
479C4—22no—
480C4—22no—
481C4—34no—
482C4—42no—
483C4—62no—
484C4—62no—
485C4—81no—
486C4—82no—
487C4—121no—
488C4—122no—
489C4—241no—
490C4#565612yes—
491C4#570612yes—
492C3#2367811yescharacteristic, complemented, Sylow
493C2—244no—
494C2#7745711yescharacteristic
495C2D3×#46011yescharacteristic
496C2#8047511yescharacteristic
497C2#8141211yescharacteristic
498C2#8145311yescharacteristic
499C2#8159211yescharacteristic
500C2#8166411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2072C2
socleC23×C3C22×D4
prime radicalC2×#874C22
hypercenter#2072D3
hypocenterC3#23678
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2072

Complements

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

NH ≅ G/Ntype
1. C2×#4788C2semidirect
2. #76477C2semidirect
4. C3×#2072C2semidirect
5. #70280C2semidirect
7. #69570C2semidirect
8. #68900C2semidirect
9. #68780C2semidirect
11. #68602C2semidirect
49. #2072D3semidirect
68. Q8×Dic3D4semidirect
197. C4×Dic3SD16semidirect
492. C3#23678semidirect

Maximal subgroups

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

subgroupindex
C2×#47882
#764772
#743192
C3×#20722
#702802
#696882
#695702
#689002
#687802
#686292
#686022
#236783