Finite Group Databasebeta

← #10831708

Subgroups — #10831708

Order 768.

1296 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×Q8×D12C211yesmaximal, characteristic, complemented
2C3×#2580C211yesmaximal, characteristic, complemented
3C2×#5008C211yesmaximal, characteristic, complemented
4C2×#5014C211yesmaximal, characteristic
5#71635C211yesmaximal, characteristic, complemented
6#69657C212yesmaximal
7#69638C212yesmaximal, complemented
8#69626C212yesmaximal, complemented
9#69605C212yesmaximal, complemented
10#69556C211yesmaximal, characteristic, complemented
11#69555C211yesmaximal, characteristic, complemented
12#24338—31nomaximal, Sylow
13C22×D3×Q8—21no—
14C2×C3×C4×Q8C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C22×#213C2211yescharacteristic
17Q8×D12—24no—
18Q8×D12—24no—
19C2×#897—22no—
20C2×#897—21no—
21C2×C4×D12C2212yes—
22C2×C4×D12C2211yescharacteristic
23C2×#869C2212yes—
24C2×#869C2211yescharacteristic
25C3×#399—21no—
26C3×#393—21no—
27C3×#390C2212yes—
28C3×#389C2212yes—
29C3×#385—22no—
30C3×#382C2211yescharacteristic
31C2×#832C2211yescharacteristic
32C2×#832C2212yes—
33#5014—24no—
34#5008—24no—
35C2×#809C2212yes—
36C2×#809C2211yescharacteristic
37C2×#808C2211yescharacteristic
38C2×#808C2212yes—
39C2×C4⋊(C3⋊C8)C2211yescharacteristic
40C2×#803C2211yescharacteristic
41#4529C2212yes—
42#4529C2212yes—
43#4529C2212yes—
44#4524C2212yes—
45#4524C2212yes—
46C2×D4×Q8—31no—
47C2×#476—31no—
48C2×#442—31no—
49#2580D311yescharacteristic, complemented
50#2058—32no—
51#2023—32no—
52#1993—31no—
53#1985—31no—
54#1879—32no—
55#1864—32no—
56#1804—31no—
57C22×C3×Q8D411yescharacteristic
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4—21no—
60C2×D3×Q8—28no—
61C2×D3×Q8—48no—
62C22×D12C2311yescharacteristic
63C22×C4×D3—22no—
64C22×C4×D3—21no—
65C22×Dic6—22no—
66C22×Dic6—21no—
67C3×C4×Q8—22no—
68C3×C4×Q8—22no—
69C3×C4×D4—22no—
70C3×C4×D4—22no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4D412yes—
73C2×C3×C4⋊C4D411yescharacteristic
74C2×C3×C4⋊C4C2311yescharacteristic
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C22⋊C4—22no—
77C2×C3×C22⋊C4—21no—
78C2×C3×C22⋊C4—22no—
79C2×C3×C42C2312yes—
80C2×C3×C42C2311yescharacteristic
81C2×#213—22no—
82C2×#213—22no—
83C2×#213Q814yescomplemented
84C2×(C2×D3)⋊C4—22no—
85C2×(C2×D3)⋊C4—21no—
86C2×C4⋊Dic3—22no—
87C2×C4⋊Dic3—21no—
88C2×C4⋊Dic3C2311yescharacteristic
89C2×C4⋊Dic3C2312yes—
90C22×C3⋊C8C2311yescharacteristic
91C22×C3⋊C8C2312yes—
92C22×C3⋊C8C2311yescharacteristic
93#897—28no—
94#897—216no—
95C4×D12—24no—
96C4×D12—22no—
97C4×D12—24no—
98C4×D12D414yes—
99#869—22no—
100#869—24no—
101#869—24no—
102#869D414yes—
103C3×#95—21no—
104C3×#95—21no—
105C3×#95—22no—
106#832—24no—
107#832—22no—
108#809—22no—
109#809—24no—
110#808—24no—
111#808—22no—
112C4⋊(C3⋊C8)—22no—
113#803—22no—
114C23×Q8—61no—
115C22×D8—31no—
116D4×Q8—64no—
117D4×Q8—64no—
118C2×C4⋊Q8—31no—
119C2×C4⋊Q8—32no—
120C2×C22⋊Q8—62no—
121C2×C22⋊Q8—61no—
122C2×C4×Q8C2×D311yescharacteristic
123C2×C4×D4—31no—
124C2×C4×D4—32no—
125C2×C4×D4C2×D311yescharacteristic
126#476—64no—
127#442—64no—
128C2×#106—31no—
129C2×#106—32no—
130C2×C4⋊C8—31no—
131C2×#102—32no—
132C2×#102—31no—
133C2×#102—32no—
134C2×#102—31no—
135C2×#97—31no—
136#399—21no—
137#393—21no—
138#390C2×D312yes—
139#389C2×D312yes—
140#385—22no—
141#382C2×D311yescharacteristic
142#340—32no—
143#340—32no—
144#340—32no—
145#336—32no—
146#336—32no—
147C24×C3—21no—
148C23×D3—21no—
149C2×C3×Q8—22no—
150C2×C3×Q8—24no—
151C2×C3×D4—21no—
152C2×C3×D4—21no—
153C2×C3×D4—21no—
154C2×C3×D4—21no—
155C2×C3×D4—21no—
156C2×C3×D4C2×Q812yescomplemented
157C22×C3×C4—22no—
158C22×C3×C4—21no—
159C22×C3×C4—42no—
160C22×C3×C4—82no—
161C22×C3×C4C2×D412yes—
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D412yes—
165C22×C3×C4C2411yescharacteristic
166C22×Dic3—21no—
167C22×Dic3—22no—
168D3×Q8—432no—
169C2×D12—21no—
170C2×D12—21no—
171C2×D12—21no—
172C2×D12C2×D412yes—
173C2×D12C2×Q812yes—
174C2×D12#4112yes—
175C2×C4×D3—28no—
176C2×C4×D3—24no—
177C2×C4×D3—44no—
178C2×C4×D3—44no—
179C2×C4×D3—44no—
180C2×Dic6—28no—
181C2×Dic6—24no—
182C2×Dic6—44no—
183C2×Dic6—44no—
184C2×Dic6—44no—
185C3×C4⋊C4—24no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—21no—
188C3×C4⋊C4—21no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—82no—
192C3×C22⋊C4—42no—
193C3×C22⋊C4—44no—
194C3×C22⋊C4—82no—
195C3×C42—22no—
196C3×C42—22no—
197C3×C42C2×D412yes—
198C3×C42C2×D412yes—
199#213—28no—
200(C2×D3)⋊C4—28no—
201(C2×D3)⋊C4—24no—
202C4⋊Dic3—28no—
203C4⋊Dic3—24no—
204C4⋊Dic3—22no—
205C4⋊Dic3—21no—
206C4⋊Dic3C2×D412yes—
207C4⋊Dic3C2×D414yes—
208C2×C3⋊C8—21no—
209C2×C3⋊C8—21no—
210C2×C3⋊C8—21no—
211C2×C3⋊C8—22no—
212C2×C3⋊C8—22no—
213C2×C3⋊C8—21no—
214C2×C3⋊C8—22no—
215C2×C3⋊C8—21no—
216C2×C3⋊C8C2×Q812yescomplemented
217C22×Q8—68no—
218C22×Q8—61no—
219C22×Q8—62no—
220C22×Q8—128no—
221C22×Q8#6711yescharacteristic
222C22×D4—31no—
223C22×D4C22×D311yescharacteristic
224C23×C4—21no—
225C23×C4—62no—
226C23×C4—61no—
227C2×D8—34no—
228C2×D8—62no—
229C2×D8—62no—
230C22×C8—31no—
231C22×C8—31no—
232C22×C8—32no—
233C4⋊Q8—34no—
234C4⋊Q8—64no—
235C4⋊Q8—64no—
236C4⋊Q8—62no—
237C22⋊Q8—68no—
238C22⋊Q8—616no—
239C4×Q8—22no—
240C4×Q8—22no—
241C4×D4—22no—
242C4×D4—22no—
243C4×D4—34no—
244C4×D4—64no—
245C4×D4—64no—
246C4×D4—62no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—31no—
249C2×C4⋊C4—32no—
250C2×C4⋊C4—62no—
251C2×C4⋊C4—61no—
252C2×C4⋊C4#6712yes—
253C2×C4⋊C4#6711yescharacteristic
254C2×C4⋊C4C22×D311yescharacteristic
255C2×C4⋊C4C22×D312yes—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—62no—
260C2×C22⋊C4—61no—
261C2×C42C22×D312yes—
262C2×C42C22×D311yescharacteristic
263#106—62no—
264#106—64no—
265C4⋊C8—62no—
266#102—64no—
267#102—62no—
268#102—62no—
269#102—64no—
270#97—62no—
271#95—21no—
272#95—21no—
273#95—22no—
274C23×C3—22no—
275C23×C3—41no—
276C23×C3—41no—
277C23×C3—42no—
278C23×C3—42no—
279C23×C3C22×D411yescharacteristic
280C22×D3—22no—
281C22×D3—22no—
282C22×D3—22no—
283C22×D3—42no—
284C22×D3—42no—
285C3×Q8—28no—
286C3×D4—22no—
287C3×D4—22no—
288C3×D4—41no—
289C3×D4—41no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—41no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—84no—
303C2×C3×C4#12012yes—
304C2×C3×C4#12012yes—
305C2×C3×C4AGL(1,ℤ/8)14yes—
306C2×C3×C4C22×D411yescharacteristic
307C2×C3×C4C22×D411yescharacteristic
308C2×C3×C4C22×Q811yescharacteristic
309C2×C3×C4C2×#4111yescharacteristic
310C2×C3×C42−1+411yescharacteristic
311C2×C3×C42−1+411yescharacteristic
312C2×Dic3—22no—
313C2×Dic3—24no—
314C2×Dic3—42no—
315C2×Dic3—42no—
316C2×Dic3—42no—
317D12—22no—
318D12—22no—
319D12—22no—
320D12C22⋊Q814yes—
321C4×D3—48no—
322C4×D3—416no—
323Dic6—48no—
324Dic6—416no—
325C3⋊C8—22no—
326C3⋊C8—41no—
327C3⋊C8—42no—
328C24—21no—
329C24—61no—
330C2×Q8—22no—
331C2×Q8—24no—
332C2×Q8—68no—
333C2×Q8—64no—
334C2×Q8—124no—
335C2×Q8—124no—
336C2×Q8—1232no—
337C2×Q8—124no—
338C2×D4—21no—
339C2×D4—21no—
340C2×D4—21no—
341C2×D4—21no—
342C2×D4—21no—
343C2×D4—32no—
344C2×D4—32no—
345C2×D4—32no—
346C2×D4—61no—
347C2×D4—61no—
348C2×D4—61no—
349C2×D4D3×Q812yescomplemented
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—42no—
353C22×C4—68no—
354C22×C4—62no—
355C22×C4—61no—
356C22×C4—64no—
357C22×C4—82no—
358C22×C4—124no—
359C22×C4—124no—
360C22×C4—124no—
361C22×C4C2×#6712yes—
362C22×C4C2×#6711yescharacteristic
363C22×C4C2×#6711yescharacteristic
364C22×C4C2×#6712yes—
365C22×C4C23×D311yescharacteristic
366D8—68no—
367C2×C8—32no—
368C2×C8—61no—
369C2×C8—61no—
370C2×C8—61no—
371C2×C8—62no—
372C2×C8—62no—
373C2×C8—61no—
374C2×C8—62no—
375C2×C8—61no—
376C4⋊C4—24no—
377C4⋊C4—22no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—32no—
383C4⋊C4—34no—
384C4⋊C4—68no—
385C4⋊C4—64no—
386C4⋊C4—61no—
387C4⋊C4—62no—
388C4⋊C4—82no—
389C22⋊C4—42no—
390C22⋊C4—44no—
391C22⋊C4—68no—
392C22⋊C4—64no—
393C22⋊C4—82no—
394C42—22no—
395C42—22no—
396C42D3×D412yes—
397C42D3×D412yes—
398C22×C3—44no—
399C22×C3—42no—
400C22×C3—82no—
401C22×C3—81no—
402C22×C3—81no—
403C22×C3C2×#12011yescharacteristic
404C22×C3#54811yescharacteristic
405C22×C3D4×Q811yescharacteristic
406C22×C3C2×AGL(1,ℤ/8)12yes—
407C22×C3#57712yes—
408C2×D3—24no—
409C2×D3—44no—
410C2×D3—44no—
411C2×D3—42no—
412C2×D3—42no—
413C3×C4—22no—
414C3×C4—24no—
415C3×C4—22no—
416C3×C4—42no—
417C3×C4—82no—
418C3×C4—81no—
419C3×C4C2×C22⋊Q811yescharacteristic
420C3×C4#53611yescharacteristic
421C3×C4#54811yescharacteristic
422C3×C4D4×Q811yescharacteristic
423Dic3—44no—
424Dic3—42no—
425C23—22no—
426C23—41no—
427C23—42no—
428C23—41no—
429C23—42no—
430C23—62no—
431C23—62no—
432C23—62no—
433C23—122no—
434C23—122no—
435C23C22×#6711yescharacteristic
436Q8—28no—
437Q8—128no—
438Q8—1216no—
439D4—22no—
440D4—22no—
441D4—34no—
442D4—41no—
443D4—41no—
444D4—62no—
445D4—62no—
446D4—62no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—64no—
460C2×C4—62no—
461C2×C4—84no—
462C2×C4—128no—
463C2×C4—1216no—
464C2×C4—122no—
465C2×C4—122no—
466C2×C4—122no—
467C2×C4#93712yes—
468C2×C4#93712yes—
469C2×C4#94914yes—
470C2×C4C2×D3×D411yescharacteristic
471C2×C4C2×D3×D411yescharacteristic
472C2×C4C2×D3×Q811yescharacteristic
473C2×C4#100711yescharacteristic
474C2×C4D3×#4111yescharacteristic
475C2×C4#101011yescharacteristic
476C8—62no—
477C8—122no—
478C8—121no—
479C2×C3—82no—
480C2×C3#240911yescharacteristic
481C2×C3#300311yescharacteristic
482C2×C3#301011yescharacteristic
483C2×C3#337312yes—
484C2×C3#338412yes—
485D3—44no—
486C22—42no—
487C22—44no—
488C22—64no—
489C22—82no—
490C22—81no—
491C22—81no—
492C22—124no—
493C22—124no—
494C22—122no—
495C22—122no—
496C22C2×#93711yescharacteristic
497C22Q8×#6711yescharacteristic
498C22C2×#94912yes—
499C22#583111yescharacteristic
500C22#583612yes—
501C4—22no—
502C4—22no—
503C4—24no—
504C4—42no—
505C4—82no—
506C4—81no—
507C4—124no—
508C4—122no—
509C4D3×C22⋊Q811yescharacteristic
510C4#562811yescharacteristic
511C4#563211yescharacteristic
512C4#563311yescharacteristic
513C3#2433811yescharacteristic, complemented, Sylow
514C2—82no—
515C2—124no—
516C2#8275111yescharacteristic
517C2#8292711yescharacteristic
518C2#8339011yescharacteristic
519C2#8363012yes—
520C2#8371512yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2580C2
socleC23×C3C22×D4
prime radicalC22×#213C22
hypercenter#2580D3
hypocenterC3#24338
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2580

Complements

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

NH ≅ G/Ntype
1. C2×Q8×D12C2semidirect
2. C3×#2580C2semidirect
3. C2×#5008C2semidirect
5. #71635C2semidirect
7. #69638C2semidirect
8. #69626C2semidirect
9. #69605C2semidirect
10. #69556C2semidirect
11. #69555C2semidirect
49. #2580D3semidirect
83. C2×#213Q8semidirect
156. C2×C3×D4C2×Q8semidirect
216. C2×C3⋊C8C2×Q8semidirect
349. C2×D4D3×Q8semidirect
513. C3#24338semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×D122
C3×#25802
C2×#50082
C2×#50142
#716352
#696572
#696382
#696262
#696052
#695562
#695552
#243383