Finite Group Databasebeta

← #10848688

Subgroups — #10848688

Order 768.

1438 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5725C211yesmaximal, characteristic
2C2×C3×#487C211yesmaximal, characteristic, complemented
3#83269C211yesmaximal, characteristic, complemented
4C2×#5085C211yesmaximal, characteristic, complemented
5#71787C211yesmaximal, characteristic, complemented
6#70276C214yesmaximal, complemented
7#70272C214yesmaximal, complemented
8#69772C211yesmaximal, characteristic, complemented
9C4×(C2×D3)⋊C8C211yesmaximal, characteristic, complemented
10#25597—31nomaximal, Sylow
11C22×#239—21no—
12C2×C3×C4⋊Q8C2211yescharacteristic
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×#947—21no—
15C2×#946C2212yes—
16C22×#215C2214yes—
17#5725—24no—
18#5725—24no—
19C2×#895—21no—
20C2×#895C2212yes—
21C2×#893—21no—
22C2×#893C2212yes—
23C2×#874C2211yescharacteristic
24C2×C42×D3C2211yescharacteristic
25C3×#487—24no—
26C2×C3×#103C2214yes—
27C2×C3×C4×C8C2211yescharacteristic
28#5229C2212yes—
29C2×#821C2214yes—
30C2×(C2×D3)⋊C8C2212yes—
31#5085—24no—
32#4987—21no—
33#4987—21no—
34#4972—21no—
35#4972—21no—
36#4935C2211yescharacteristic
37C2×C4×C3⋊C8C2211yescharacteristic
38#4550C2212yes—
39#4529C2214yes—
40C2×#532—31no—
41C2×#492—31no—
42C2×#487D311yescharacteristic, complemented
43#2857—31no—
44#2000—34no—
45#1990—31no—
46#1957—31no—
47#1877—34no—
48C4×C22⋊C8—31no—
49C22×C3×Q8—21no—
50C22×C3×Q8C2312yes—
51C2×#239—28no—
52C2×#239—44no—
53C2×#239—44no—
54C22×D12—22no—
55C22×D12—21no—
56C22×D12—21no—
57C22×D12C2312yes—
58C22×C4×D3—22no—
59C22×C4×D3D411yescharacteristic
60C22×C4×D3D411yescharacteristic
61C22×C4×D3C2311yescharacteristic
62C22×C3×C8C2312yes—
63C3×C4⋊Q8—22no—
64C3×C4⋊Q8—22no—
65C3×C4⋊Q8—22no—
66C3×C4⋊Q8D414yes—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4C2312yes—
71C2×C3×C42C2311yescharacteristic
72#947—28no—
73#946—44no—
74C2×#215—28no—
75C2×#215—28no—
76C2×#215—44no—
77C2×(C2×D3)⋊C4—22no—
78C2×(C2×D3)⋊C4—22no—
79C2×(C2×D3)⋊C4—22no—
80C2×C4⋊Dic3—21no—
81C2×C4⋊Dic3—21no—
82C2×C4⋊Dic3C2312yes—
83C2×Dic3⋊C4—22no—
84C2×C4×Dic3D411yescharacteristic
85C2×C4×Dic3D411yescharacteristic
86C2×C4×Dic3C2311yescharacteristic
87C22×C3⋊C8C2312yes—
88#895—28no—
89#895—24no—
90#895—24no—
91#893—28no—
92#893—24no—
93#893—24no—
94#874—22no—
95#874D414yes—
96C42×D3—22no—
97C42×D3—22no—
98C3×#103—28no—
99C3×C4×C8—22no—
100C3⋊#95—22no—
101C3⋊#95—22no—
102#821—44no—
103(C2×D3)⋊C8—24no—
104C4×C3⋊C8—22no—
105C22×#41—61no—
106C22×SD16—34no—
107#532—64no—
108#532—64no—
109C2×C4⋊Q8C2×D311yescharacteristic
110C2×C4⋊Q8C2×D311yescharacteristic
111C2×#127—31no—
112C2×#124—61no—
113C2×C22⋊Q8—32no—
114C2×#121—32no—
115C2×#121—61no—
116C2×C4×D4—32no—
117C2×C4×D4—61no—
118C22×C42—31no—
119#492—64no—
120#487—24no—
121C2×#103C2×D314yes—
122C2×#102—34no—
123C2×C22⋊C8—32no—
124C2×C4×C8—31no—
125C2×C4×C8C2×D311yescharacteristic
126#391—32no—
127#389—61no—
128#389—61no—
129#384—31no—
130#380—61no—
131#380—61no—
132#340—34no—
133#336—32no—
134C23×D3—22no—
135C23×D3C2×D411yescharacteristic
136C2×C3×Q8—24no—
137C2×C3×Q8—22no—
138C2×C3×Q8—22no—
139C2×C3×Q8—24no—
140C2×C3×Q8—24no—
141C2×C3×Q8—42no—
142C2×C3×Q8—42no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4C2411yescharacteristic
148C22×Dic3—22no—
149C22×Dic3C2×D411yescharacteristic
150C22×Dic3C2×D411yescharacteristic
151C22×Dic3C2×D411yescharacteristic
152#239—432no—
153C2×D12—24no—
154C2×D12—24no—
155C2×D12—28no—
156C2×D12—24no—
157C2×D12—24no—
158C2×D12—22no—
159C2×D12—44no—
160C2×D12—44no—
161C2×D12—42no—
162C2×D12—42no—
163C2×D12—42no—
164C2×D12—42no—
165C2×D12C2×D414yes—
166C2×C4×D3—22no—
167C2×C4×D3—22no—
168C2×C4×D3—28no—
169C2×C4×D3—22no—
170C2×C4×D3—22no—
171C2×C4×D3—22no—
172C2×C4×D3—21no—
173C2×C4×D3—21no—
174C2×C4×D3—22no—
175C2×C4×D3—22no—
176C2×C4×D3—44no—
177C2×C4×D3—44no—
178C2×C4×D3SD1614yescomplemented
179C2×C3×C8—22no—
180C2×C3×C8—22no—
181C2×C3×C8—22no—
182C3×C4⋊C4—24no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—42no—
185C3×C4⋊C4—42no—
186C3×C4⋊C4C2×D414yes—
187C3×C42C2×D412yes—
188C3×C42C2×D412yes—
189#215—416no—
190(C2×D3)⋊C4—28no—
191(C2×D3)⋊C4—28no—
192(C2×D3)⋊C4—44no—
193C4⋊Dic3—24no—
194C4⋊Dic3—42no—
195C4⋊Dic3—42no—
196Dic3⋊C4—44no—
197C4×Dic3—22no—
198C4×Dic3—22no—
199C4×Dic3—21no—
200C4×Dic3—21no—
201C2×C3⋊C8—22no—
202C2×C3⋊C8—22no—
203C2×C3⋊C8#4114yescomplemented
204C2×#41—68no—
205C2×#41—124no—
206C2×#41—124no—
207C22×Q8—21no—
208C22×Q8C22×D312yes—
209C22×D4—32no—
210C22×D4—62no—
211C22×D4—61no—
212C22×D4—61no—
213C23×C4—31no—
214C23×C4—31no—
215C23×C4—31no—
216C23×C4—62no—
217C2×SD16—68no—
218C2×SD16—68no—
219C2×SD16—124no—
220C22×C8—32no—
221C22×C8C22×D312yes—
222C4⋊Q8—22no—
223C4⋊Q8—22no—
224C4⋊Q8—22no—
225C4⋊Q8#6714yes—
226#127—34no—
227#127—62no—
228#124—68no—
229C22⋊Q8—124no—
230#121—64no—
231#121—68no—
232#121—64no—
233C4×D4—64no—
234C4×D4—68no—
235C4×D4—64no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—32no—
240C2×C4⋊C4—62no—
241C2×C4⋊C4—61no—
242C2×C4⋊C4—61no—
243C2×C4⋊C4C22×D312yes—
244C2×C22⋊C4—62no—
245C2×C22⋊C4—62no—
246C2×C22⋊C4—62no—
247C2×C42—31no—
248C2×C42—31no—
249C2×C42—31no—
250C2×C42—62no—
251C2×C42—62no—
252C2×C42C22×D311yescharacteristic
253#103—28no—
254#102—124no—
255C22⋊C8—64no—
256C4×C8—22no—
257C4×C8—62no—
258#95—62no—
259#95—62no—
260C23×C3C22×D411yescharacteristic
261C22×D3—21no—
262C22×D3—21no—
263C22×D3—21no—
264C22×D3—21no—
265C22×D3—21no—
266C22×D3—21no—
267C22×D3—24no—
268C22×D3—42no—
269C22×D3—42no—
270C22×D3—42no—
271C22×D3—42no—
272C22×D3—42no—
273C22×D3—42no—
274C22×D3C2×SD1612yes—
275C3×Q8—48no—
276C3×Q8—44no—
277C3×Q8—44no—
278C2×C3×C4—24no—
279C2×C3×C4—42no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4#12012yes—
285C2×C3×C4#12012yes—
286C2×C3×C4C2×SD1612yes—
287C2×C3×C4#13512yes—
288C2×C3×C4AGL(1,ℤ/8)12yes—
289C2×C3×C4#13712yes—
290C2×C3×C4C22×D411yescharacteristic
291C2×C3×C4C22×D411yescharacteristic
292C2×C3×C4C2×#4112yes—
293C2×C3×C4H(5,𝔽2)12yes—
294C2×Dic3—21no—
295C2×Dic3—21no—
296C2×Dic3—21no—
297C2×Dic3—21no—
298C2×Dic3—21no—
299C2×Dic3—21no—
300C2×Dic3—21no—
301C2×Dic3—21no—
302C2×Dic3—42no—
303C2×Dic3—42no—
304C2×Dic3—42no—
305C2×Dic3C2×SD1612yes—
306D12—28no—
307D12—416no—
308D12—44no—
309D12—48no—
310D12—48no—
311C4×D3—28no—
312C4×D3—416no—
313C4×D3—42no—
314C4×D3—42no—
315C4×D3—44no—
316C3×C8—42no—
317C3⋊C8—24no—
318C24—31no—
319C24—62no—
320#41—1232no—
321C2×Q8—24no—
322C2×Q8—22no—
323C2×Q8—24no—
324C2×Q8—24no—
325C2×Q8—22no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×D4—34no—
329C2×D4—68no—
330C2×D4—64no—
331C2×D4—64no—
332C2×D4—64no—
333C2×D4—64no—
334C2×D4—62no—
335C2×D4—124no—
336C2×D4—124no—
337C2×D4—122no—
338C2×D4—122no—
339C2×D4—122no—
340C2×D4—122no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—31no—
344C22×C4—34no—
345C22×C4—31no—
346C22×C4—31no—
347C22×C4—68no—
348C22×C4—62no—
349C22×C4—62no—
350C22×C4—62no—
351C22×C4—62no—
352C22×C4—62no—
353C22×C4—61no—
354C22×C4—62no—
355C22×C4—62no—
356C22×C4—61no—
357C22×C4—62no—
358C22×C4—124no—
359C22×C4—124no—
360C22×C4D3×D411yescharacteristic
361C22×C4D3×D411yescharacteristic
362C22×C4C23×D311yescharacteristic
363SD16—1216no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—34no—
368C2×C8—62no—
369C2×C8—62no—
370C4⋊C4—24no—
371C4⋊C4—22no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—64no—
375C4⋊C4—124no—
376C4⋊C4—122no—
377C4⋊C4—122no—
378C4⋊C4C2×#6714yes—
379C22⋊C4—68no—
380C22⋊C4—68no—
381C22⋊C4—124no—
382C42—62no—
383C42—62no—
384C42—61no—
385C42—61no—
386C42D3×D412yes—
387C42C2×#6712yes—
388C22×C3C2×#12011yescharacteristic
389C22×C3#53211yescharacteristic
390C22×C3#53511yescharacteristic
391C22×C3C22×SD1611yescharacteristic
392C22×C3C2×#13511yescharacteristic
393C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
394C22×C3C2×#13711yescharacteristic
395C2×D3—22no—
396C2×D3—22no—
397C2×D3—22no—
398C2×D3—41no—
399C2×D3—41no—
400C2×D3—41no—
401C2×D3—41no—
402C2×D3—44no—
403C2×D3—44no—
404C2×D3—44no—
405C2×D3—82no—
406C2×D3—82no—
407C2×D3—82no—
408C2×D3—82no—
409C3×C4—44no—
410C3×C4—82no—
411C3×C4#45014yes—
412C3×C4#45214yes—
413C3×C4#54612yes—
414C3×C4#55012yes—
415Dic3—22no—
416Dic3—41no—
417Dic3—41no—
418Dic3—82no—
419C23—32no—
420C23—61no—
421C23—61no—
422C23—61no—
423C23—61no—
424C23—61no—
425C23—61no—
426C23—64no—
427C23—122no—
428C23—122no—
429C23—122no—
430C23—122no—
431C23—122no—
432C23—122no—
433C23C2×D3×D411yescharacteristic
434Q8—48no—
435Q8—44no—
436Q8—44no—
437D4—68no—
438D4—1216no—
439D4—128no—
440D4—128no—
441D4—124no—
442C2×C4—24no—
443C2×C4—32no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—61no—
450C2×C4—61no—
451C2×C4—68no—
452C2×C4—61no—
453C2×C4—61no—
454C2×C4—61no—
455C2×C4—61no—
456C2×C4—61no—
457C2×C4—61no—
458C2×C4—1216no—
459C2×C4—122no—
460C2×C4—124no—
461C2×C4—122no—
462C2×C4—122no—
463C2×C4—122no—
464C2×C4—122no—
465C2×C4D3×SD1612yes—
466C2×C4#91412yes—
467C2×C4#91812yes—
468C2×C4#91912yes—
469C2×C4#93712yes—
470C2×C4#93712yes—
471C2×C4C2×D3×D411yescharacteristic
472C2×C4D3×#4112yes—
473C2×C4#100912yes—
474C2×C4C22×#6711yescharacteristic
475C8—42no—
476C8—64no—
477C2×C3#243111yescharacteristic
478C2×C3C2×#45011yescharacteristic
479C2×C3C2×#45211yescharacteristic
480C2×C3#316711yescharacteristic
481C2×C3#318211yescharacteristic
482C2×C3#320611yescharacteristic
483C2×C3#320711yescharacteristic
484D3—42no—
485D3—84no—
486C22—62no—
487C22—62no—
488C22—62no—
489C22—121no—
490C22—121no—
491C22—121no—
492C22—121no—
493C22—124no—
494C22—124no—
495C22—124no—
496C22—242no—
497C22—242no—
498C22—242no—
499C22—242no—
500C22#566811yescharacteristic
501C22#567411yescharacteristic
502C22C2×D3×SD1611yescharacteristic
503C22C2×#91411yescharacteristic
504C22C2×#91811yescharacteristic
505C22C2×#91911yescharacteristic
506C22C2×#93711yescharacteristic
507C4—44no—
508C4—62no—
509C4—82no—
510C4—121no—
511C4—121no—
512C4—242no—
513C4#516914yes—
514C4#518714yes—
515C4#567612yes—
516C4#583412yes—
517C3#2559711yescharacteristic, complemented, Sylow
518C2—122no—
519C2—244no—
520C2#8159211yescharacteristic
521C2#8159511yescharacteristic
522C2#8160211yescharacteristic
523C2#8160311yescharacteristic
524C2#8295811yescharacteristic
525C2C2×#516911yescharacteristic
526C2C2×#518711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#487C2
socleC23×C3C22×D4
prime radicalC2×#5725C2
hypercenterC2×#487D3
hypocenterC3#25597
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#487

Complements

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

NH ≅ G/Ntype
2. C2×C3×#487C2semidirect
3. #83269C2semidirect
4. C2×#5085C2semidirect
5. #71787C2semidirect
6. #70276C2semidirect
7. #70272C2semidirect
8. #69772C2semidirect
9. C4×(C2×D3)⋊C8C2semidirect
42. C2×#487D3semidirect
178. C2×C4×D3SD16semidirect
203. C2×C3⋊C8#41semidirect
517. C3#25597semidirect

Maximal subgroups

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

subgroupindex
C2×#57252
C2×C3×#4872
#832692
C2×#50852
#717872
#702762
#702722
#697722
C4×(C2×D3)⋊C82
#255973