Finite Group Databasebeta

← #10849285

Subgroups — #10849285

Order 768.

1324 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5553C211yesmaximal, characteristic
2C2×#5009C211yesmaximal, characteristic, complemented
3C2×#4713C211yesmaximal, characteristic, complemented
4#73727C211yesmaximal, characteristic, complemented
5C3×#1762C211yesmaximal, characteristic, complemented
6#70177C212yesmaximal, complemented
7#70177C212yesmaximal, complemented
8#69838C211yesmaximal, characteristic, complemented
9#69487C211yesmaximal, characteristic, complemented
10#68895C212yesmaximal, complemented
11#68894C212yesmaximal, complemented
12#25655—31nomaximal, Sylow
13C22×#235—21no—
14C2×C3×C4×D4C2211yescharacteristic
15C2×#940C2212yes—
16C2×#930—21no—
17#5553—24no—
18#5553—24no—
19C2×#893C2212yes—
20C2×#884—22no—
21C2×#874C2211yescharacteristic
22C2×#870C2211yescharacteristic
23C2×C4×Dic6C2211yescharacteristic
24C2×C3×#102C2212yes—
25C2×C3×#97C2211yescharacteristic
26C3×C4×C4⋊C4C2211yescharacteristic
27C2×#832C2211yescharacteristic
28C2×#832C2211yescharacteristic
29C2×#821C2212yes—
30C2×#821C2212yes—
31#5009—24no—
32#4984C2212yes—
33#4984C2212yes—
34C2×#809C2212yes—
35C2×#809C2212yes—
36C2×#808C2211yescharacteristic
37C2×#807C2211yescharacteristic
38#4928—22no—
39C2×C4⋊(C3⋊C8)C2211yescharacteristic
40#4713—24no—
41#4661—21no—
42#4656—21no—
43C3×#340C2212yes—
44#4550C2212yes—
45#4550C2212yes—
46C2×#550—31no—
47C2×#490—31no—
48C2×#478—31no—
49#2584—31no—
50#2041—32no—
51#2041—32no—
52#1985—31no—
53#1982—31no—
54#1864—32no—
55#1862—32no—
56#1762D311yescharacteristic, complemented
57C22×C3×D4C2311yescharacteristic
58C23×C3×C4—21no—
59C22×#67—22no—
60C22×#67—22no—
61C2×#235—28no—
62C2×#235—48no—
63C22×D12—21no—
64C22×D12—21no—
65C22×D12C2312yes—
66C22×C4×D3—22no—
67C22×Dic6D411yescharacteristic
68C22×C3×C8C2312yes—
69C3×C4×D4—22no—
70C3×C4×D4D414yes—
71C2×C3×C4⋊C4C2311yescharacteristic
72C2×C3×C4⋊C4C2312yes—
73C2×C3×C22⋊C4—21no—
74C2×C3×C42—21no—
75C2×C3×C42C2311yescharacteristic
76#940—24no—
77#940—24no—
78#930—28no—
79C2×(C2×D3)⋊C4—22no—
80C2×C4⋊Dic3—22no—
81C2×C4⋊Dic3—22no—
82C2×C4⋊Dic3D411yescharacteristic
83C2×C4⋊Dic3C2311yescharacteristic
84C2×C4⋊Dic3C2311yescharacteristic
85C2×Dic3⋊C4—21no—
86C2×Dic3⋊C4—21no—
87C2×Dic3⋊C4D412yes—
88C2×C4×Dic3C2312yes—
89C22×C3⋊C8C2311yescharacteristic
90C22×C3⋊C8C2311yescharacteristic
91#893—24no—
92#893—24no—
93#884—216no—
94#874—22no—
95#874D414yes—
96#870—22no—
97#870—22no—
98C4×Dic6—22no—
99C4×Dic6—22no—
100C3×#102—24no—
101C3×#97—22no—
102C3×#95—21no—
103#832—22no—
104#832—22no—
105C3⋊#95—22no—
106C3⋊#95—22no—
107#821—24no—
108#821—24no—
109#809—24no—
110#809—24no—
111#808—22no—
112#807—22no—
113C4⋊(C3⋊C8)—22no—
114C22×#41—61no—
115#550—64no—
116#550—64no—
117C2×#127—32no—
118C2×#127—31no—
119C2×#125—31no—
120C2×#121—62no—
121C2×#121—61no—
122C2×C4×Q8—31no—
123C2×C4×D4—32no—
124C2×C4×D4C2×D311yescharacteristic
125#490—64no—
126#478—64no—
127C2×#107—31no—
128C2×#106—31no—
129C2×C4⋊C8—31no—
130C2×#102—31no—
131C2×#102—31no—
132C2×#102—32no—
133C2×#102—32no—
134C2×#102—32no—
135C2×#102—32no—
136C2×#102C2×D312yes—
137C2×#97C2×D311yescharacteristic
138#398—61no—
139#395—61no—
140#389—32no—
141#389—32no—
142#389—62no—
143C4×C4⋊C4C2×D311yescharacteristic
144#340C2×D312yes—
145#336—32no—
146#336—32no—
147C24×C3—21no—
148C23×D3—22no—
149C2×C3×D4—21no—
150C2×C3×D4—22no—
151C2×C3×D4—22no—
152C2×C3×D4C2×D412yes—
153C22×C3×C4—22no—
154C22×C3×C4—21no—
155C22×C3×C4—24no—
156C22×C3×C4—44no—
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2411yescharacteristic
160C2×#67—28no—
161C2×#67—28no—
162C2×#67—44no—
163C2×#67—44no—
164C2×#67—82no—
165C2×#67—82no—
166C22×Dic3—21no—
167C22×Dic3—21no—
168C22×Dic3C2×D412yes—
169C22×Dic3C2×D412yes—
170#235—432no—
171C2×D12—22no—
172C2×D12—22no—
173C2×D12—22no—
174C2×D12—22no—
175C2×D12—24no—
176C2×D12—22no—
177C2×D12—24no—
178C2×D12—44no—
179C2×D12—81no—
180C2×D12—81no—
181C2×D12C2×D414yes—
182C2×C4×D3—28no—
183C2×C4×D3—44no—
184C2×C4×D3—44no—
185C2×Dic6—22no—
186C2×Dic6—24no—
187C2×C3×C8—22no—
188C2×C3×C8—22no—
189C2×C3×C8—22no—
190C3×C4⋊C4—21no—
191C3×C4⋊C4—22no—
192C3×C4⋊C4C2×D412yes—
193C3×C4⋊C4#4114yescomplemented
194C3×C22⋊C4—24no—
195C3×C42—81no—
196C3×C42C2×D412yes—
197C3×C42C2×D412yes—
198(C2×D3)⋊C4—28no—
199C4⋊Dic3—22no—
200C4⋊Dic3—21no—
201C4⋊Dic3—21no—
202C4⋊Dic3—21no—
203C4⋊Dic3—21no—
204C4⋊Dic3—44no—
205C4⋊Dic3—82no—
206Dic3⋊C4—24no—
207Dic3⋊C4—42no—
208Dic3⋊C4—42no—
209C4×Dic3—22no—
210C4×Dic3—22no—
211C2×C3⋊C8—21no—
212C2×C3⋊C8—21no—
213C2×C3⋊C8—22no—
214C2×C3⋊C8—22no—
215C2×#41—68no—
216C2×#41—128no—
217C22×Q8—31no—
218C22×D4—32no—
219C22×D4—62no—
220C22×D4—62no—
221C22×D4—61no—
222C22×D4—61no—
223C22×D4C22×D311yescharacteristic
224C23×C4—21no—
225C23×C4—62no—
226C22×C8—31no—
227C22×C8—31no—
228C22×C8C22×D312yes—
229#127—34no—
230#127—64no—
231#127—64no—
232#127—62no—
233#125—62no—
234#125—62no—
235#121—616no—
236#121—68no—
237C4×Q8—62no—
238C4×Q8—62no—
239C4×D4—22no—
240C4×D4—64no—
241C4×D4—64no—
242C4×D4D1214yes—
243C2×C4⋊C4—32no—
244C2×C4⋊C4—31no—
245C2×C4⋊C4—31no—
246C2×C4⋊C4—31no—
247C2×C4⋊C4—62no—
248C2×C4⋊C4—61no—
249C2×C4⋊C4—61no—
250C2×C4⋊C4—62no—
251C2×C4⋊C4C22×D311yescharacteristic
252C2×C4⋊C4C22×D312yes—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—62no—
255C2×C42—21no—
256C2×C42—32no—
257C2×C42C22×D311yescharacteristic
258#107—62no—
259#106—62no—
260C4⋊C8—62no—
261#102—24no—
262#102—64no—
263#102—64no—
264#102—62no—
265#102—62no—
266#102—64no—
267#102—64no—
268#97—22no—
269#95—21no—
270#95—62no—
271#95—62no—
272C23×C3—22no—
273C23×C3—42no—
274C23×C3—42no—
275C23×C3—41no—
276C23×C3—41no—
277C23×C3C22×D411yescharacteristic
278C22×D3—24no—
279C22×D3—42no—
280C22×D3—42no—
281C22×D3—42no—
282C22×D3—42no—
283C22×D3—42no—
284C22×D3—42no—
285C3×D4—22no—
286C3×D4—22no—
287C3×D4—42no—
288C2×C3×C4—22no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—22no—
292C2×C3×C4—24no—
293C2×C3×C4—48no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4#12012yes—
298C2×C3×C4#12012yes—
299C2×C3×C4C22×D411yescharacteristic
300C2×C3×C4C22×D411yescharacteristic
301C2×C3×C4C2×#4112yes—
302C2×C3×C42−1+412yes—
303#67—416no—
304#67—88no—
305C2×Dic3—22no—
306C2×Dic3—22no—
307C2×Dic3—22no—
308C2×Dic3—22no—
309C2×Dic3—22no—
310C2×Dic3—41no—
311C2×Dic3—41no—
312C2×Dic3—42no—
313C2×Dic3—41no—
314C2×Dic3—41no—
315C2×Dic3AGL(1,ℤ/8)14yes—
316D12—24no—
317D12—24no—
318D12—48no—
319D12—42no—
320D12—42no—
321D12—84no—
322C4×D3—416no—
323Dic6—28no—
324C3×C8—42no—
325C3⋊C8—41no—
326C3⋊C8—41no—
327C24—21no—
328C24—62no—
329#41—1232no—
330C2×Q8—64no—
331C2×Q8—62no—
332C2×D4—22no—
333C2×D4—21no—
334C2×D4—22no—
335C2×D4—34no—
336C2×D4—68no—
337C2×D4—68no—
338C2×D4—64no—
339C2×D4—64no—
340C2×D4—62no—
341C2×D4—62no—
342C2×D4—62no—
343C2×D4—62no—
344C2×D4—62no—
345C2×D4—124no—
346C2×D4—124no—
347C2×D4—124no—
348C2×D4—241no—
349C2×D4—241no—
350C2×D4—242no—
351C2×D4—242no—
352C2×D4C2×D1212yes—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—24no—
356C22×C4—32no—
357C22×C4—32no—
358C22×C4—44no—
359C22×C4—68no—
360C22×C4—61no—
361C22×C4—61no—
362C22×C4—124no—
363C22×C4—124no—
364C22×C4D3×D411yescharacteristic
365C22×C4D3×D411yescharacteristic
366C22×C4C23×D311yescharacteristic
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—61no—
371C2×C8—61no—
372C2×C8—62no—
373C2×C8—62no—
374C4⋊C4—21no—
375C4⋊C4—22no—
376C4⋊C4—64no—
377C4⋊C4—62no—
378C4⋊C4—61no—
379C4⋊C4—61no—
380C4⋊C4—61no—
381C4⋊C4—61no—
382C4⋊C4—124no—
383C4⋊C4—122no—
384C4⋊C4—122no—
385C4⋊C4—242no—
386C4⋊C4C2×D1212yes—
387C4⋊C4#23514yescomplemented
388C22⋊C4—24no—
389C22⋊C4—68no—
390C42—62no—
391C42—62no—
392C42—81no—
393C42C2×D1212yes—
394C42D3×D412yes—
395C22×C3—42no—
396C22×C3—42no—
397C22×C3—42no—
398C22×C3—82no—
399C22×C3—82no—
400C22×C3C2×#12011yescharacteristic
401C22×C3#54812yes—
402C22×C3C2×AGL(1,ℤ/8)12yes—
403C22×C3#57611yescharacteristic
404C22×C3#57711yescharacteristic
405C2×D3—44no—
406C2×D3—44no—
407C2×D3—44no—
408C2×D3—82no—
409C2×D3—82no—
410C2×D3—82no—
411C2×D3—82no—
412C3×C4—22no—
413C3×C4—22no—
414C3×C4—44no—
415C3×C4—42no—
416C3×C4#53211yescharacteristic
417C3×C4#53611yescharacteristic
418C3×C4#54712yes—
419Dic3—24no—
420Dic3—42no—
421Dic3—81no—
422Dic3—81no—
423C23—22no—
424C23—42no—
425C23—41no—
426C23—41no—
427C23—42no—
428C23—64no—
429C23—122no—
430C23—122no—
431C23—122no—
432C23—122no—
433C23—122no—
434C23—122no—
435C23C2×D3×D411yescharacteristic
436Q8—68no—
437D4—22no—
438D4—22no—
439D4—42no—
440D4—64no—
441D4—64no—
442D4—128no—
443D4—1216no—
444D4—122no—
445D4—122no—
446D4—244no—
447D4—248no—
448C2×C4—22no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—34no—
454C2×C4—48no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—62no—
459C2×C4—62no—
460C2×C4—62no—
461C2×C4—62no—
462C2×C4—62no—
463C2×C4—1216no—
464C2×C4—121no—
465C2×C4—121no—
466C2×C4—122no—
467C2×C4—121no—
468C2×C4—121no—
469C2×C4#88212yes—
470C2×C4#88212yes—
471C2×C4C22×D1211yescharacteristic
472C2×C4C2×#23512yes—
473C2×C4C2×D3×D411yescharacteristic
474C2×C4#101012yes—
475C8—42no—
476C8—121no—
477C8—121no—
478C2×C3—82no—
479C2×C3#243211yescharacteristic
480C2×C3#300311yescharacteristic
481C2×C3#301211yescharacteristic
482C2×C3#336612yes—
483C2×C3#336812yes—
484D3—84no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—82no—
489C22—82no—
490C22—124no—
491C22—124no—
492C22—124no—
493C22—242no—
494C22—242no—
495C22—242no—
496C22—242no—
497C22C2×#88211yescharacteristic
498C22#555212yes—
499C22D3×AGL(1,ℤ/8)12yes—
500C22#577411yescharacteristic
501C22#577811yescharacteristic
502C4—22no—
503C4—22no—
504C4—42no—
505C4—44no—
506C4—64no—
507C4—122no—
508C4—241no—
509C4—241no—
510C4#547711yescharacteristic
511C4#552611yescharacteristic
512C4#555112yes—
513C3#2565511yescharacteristic, complemented, Sylow
514C2—82no—
515C2—244no—
516C2#7348911yescharacteristic
517C2#7888611yescharacteristic
518C2#7892611yescharacteristic
519C2#7975212yes—
520C2#7985112yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1762C2
socleC23×C3C22×D4
prime radicalC2×#5553C2
hypercenter#1762D3
hypocenterC3#25655
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
2. C2×#5009C2semidirect
3. C2×#4713C2semidirect
4. #73727C2semidirect
5. C3×#1762C2semidirect
6. #70177C2semidirect
7. #70177C2semidirect
8. #69838C2semidirect
9. #69487C2semidirect
10. #68895C2semidirect
11. #68894C2semidirect
56. #1762D3semidirect
193. C3×C4⋊C4#41semidirect
387. C4⋊C4#235semidirect
513. C3#25655semidirect

Maximal subgroups

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

subgroupindex
C2×#55532
C2×#50092
C2×#47132
#737272
C3×#17622
#701772
#698382
#694872
#688952
#688942
#256553