Finite Group Databasebeta

← #10823446

Subgroups — #10823446

Order 768.

1424 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5716C211yesmaximal, characteristic
2C2×#5164C212yesmaximal, complemented
3#76479C211yesmaximal, characteristic, complemented
4C3×#2014C211yesmaximal, characteristic, complemented
5#72054C212yesmaximal, complemented
6#71762C211yesmaximal, characteristic, complemented
7#70175C212yesmaximal, complemented
8#70152C212yesmaximal, complemented
9#69668C211yesmaximal, characteristic, complemented
10#68632C211yesmaximal, characteristic, complemented
11#68600C211yesmaximal, characteristic, complemented
12#23576—31nomaximal, Sylow
13C22×#237—21no—
14C22×C3×SD16C2212yes—
15C2×C3×#127C2211yescharacteristic
16C2×#946C2211yescharacteristic
17C2×Q8×Dic3C2211yescharacteristic
18C2×#938—21no—
19C2×#938C2212yes—
20C2×#936—21no—
21C2×D4×Dic3—21no—
22C2×D4×Dic3C2212yes—
23#5716—28no—
24C2×C42×D3C2211yescharacteristic
25C2×#869C2211yescharacteristic
26C2×C3×#102C2212yes—
27C3×#391C2211yescharacteristic
28C2×#835C2212yes—
29C2×#832C2212yes—
30C2×#832C2212yes—
31#5164—28no—
32C2×(C2×D3)⋊C8C2212yes—
33C2×#818C2212yes—
34C2×#815C2212yes—
35#4987C2211yescharacteristic
36Dic3×C4⋊C4C2211yescharacteristic
37#4928C2211yescharacteristic
38#4668—21no—
39#4666—21no—
40#4655—21no—
41#4649—21no—
42C3×#336C2211yescharacteristic
43#4551C2212yes—
44#4529C2212yes—
45#4524C2211yescharacteristic
46C2×#532—31no—
47C2×#440—32no—
48C4×C22⋊Q8—31no—
49#2057—31no—
50#2041—31no—
51#2033—32no—
52#2024—31no—
53#2014D311yescharacteristic, complemented
54#2000—32no—
55#1762—32no—
56#1749—31no—
57C22×C3×Q8C2311yescharacteristic
58C22×C3×D4—21no—
59C22×C3×D4—21no—
60C22×C3×D4C2312yes—
61C22×#67—22no—
62C23×Dic3—22no—
63C2×#237—28no—
64C2×#237—44no—
65C2×#237—44no—
66C22×C4×D3D411yescharacteristic
67C22×C4×D3D411yescharacteristic
68C22×C4×D3C2311yescharacteristic
69C22×Dic6—21no—
70C2×C3×SD16—24no—
71C2×C3×SD16—24no—
72C2×C3×SD16—24no—
73C22×C3×C8C2312yes—
74C3×#127—24no—
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C42C2311yescharacteristic
77C2×#217—22no—
78C2×#217—22no—
79#946—22no—
80#946—22no—
81Q8×Dic3—24no—
82#938—28no—
83#938—24no—
84#938—24no—
85#936—28no—
86D4×Dic3—28no—
87D4×Dic3—24no—
88D4×Dic3D418yes—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—21no—
91C2×C4⋊Dic3—21no—
92C2×C4⋊Dic3—21no—
93C2×C4⋊Dic3C2311yescharacteristic
94C2×C4⋊Dic3C2312yes—
95C2×C4⋊Dic3C2311yescharacteristic
96C2×Dic3⋊C4—21no—
97C2×Dic3⋊C4—21no—
98C2×C4×Dic3—21no—
99C2×C4×Dic3—21no—
100C2×C4×Dic3D411yescharacteristic
101C2×C4×Dic3D411yescharacteristic
102C2×C4×Dic3C2311yescharacteristic
103C22×C3⋊C8C2312yes—
104C42×D3—24no—
105#869—24no—
106C3×#102—24no—
107C3×#95—21no—
108C3×#95—21no—
109#835—24no—
110#832—24no—
111#832—24no—
112C3⋊#95—21no—
113C3⋊#95—21no—
114C3⋊#95—21no—
115(C2×D3)⋊C8—24no—
116#818—24no—
117#815—24no—
118C22×#41—61no—
119C22×SD16C2×D312yes—
120#532—68no—
121C2×C4⋊Q8—31no—
122C2×#127C2×D311yescharacteristic
123C2×#124—61no—
124C2×C22⋊Q8—31no—
125C2×#121—32no—
126C2×#121—61no—
127C2×C4×Q8—31no—
128C2×C4×D4—32no—
129C2×C4×D4—61no—
130C22×C42—31no—
131#440—68no—
132C2×#107—32no—
133C2×#103—32no—
134C2×#102—32no—
135C2×#102—32no—
136C2×#102C2×D312yes—
137C2×C22⋊C8—32no—
138C2×#97—32no—
139#391C2×D311yescharacteristic
140#389—31no—
141#387—61no—
142#385—61no—
143#380—31no—
144C4×C4⋊C4—31no—
145C4×C4⋊C4—61no—
146C4×C22⋊C4—61no—
147#340—32no—
148#340—32no—
149#336—31no—
150#336C2×D311yescharacteristic
151C24×C3—22no—
152C23×D3C2×D411yescharacteristic
153C2×C3×Q8—21no—
154C2×C3×Q8—22no—
155C2×C3×Q8—22no—
156C2×C3×Q8#4112yescomplemented
157C2×C3×D4—24no—
158C2×C3×D4—24no—
159C2×C3×D4—22no—
160C2×C3×D4—22no—
161C2×C3×D4—22no—
162C2×C3×D4—22no—
163C2×C3×D4—22no—
164C2×C3×D4—42no—
165C2×C3×D4—42no—
166C2×C3×D4—42no—
167C2×C3×D4—42no—
168C2×C3×D4C2×D414yes—
169C22×C3×C4—21no—
170C22×C3×C4—21no—
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2411yescharacteristic
174C2×#67—28no—
175C2×#67—44no—
176C2×#67—44no—
177C22×Dic3—21no—
178C22×Dic3—22no—
179C22×Dic3—28no—
180C22×Dic3—21no—
181C22×Dic3—44no—
182C22×Dic3—44no—
183C22×Dic3C2×D411yescharacteristic
184C22×Dic3C2×D411yescharacteristic
185C22×Dic3C2×D411yescharacteristic
186#237—432no—
187C2×C4×D3—22no—
188C2×C4×D3—22no—
189C2×C4×D3—22no—
190C2×C4×D3—22no—
191C2×C4×D3—22no—
192C2×C4×D3—22no—
193C2×C4×D3—21no—
194C2×C4×D3—21no—
195C2×C4×D3—22no—
196C2×C4×D3—41no—
197C2×Dic6—24no—
198C2×Dic6—42no—
199C2×Dic6—42no—
200C3×SD16—216no—
201C2×C3×C8—22no—
202C2×C3×C8—22no—
203C2×C3×C8#4114yescomplemented
204C3×C4⋊C4—22no—
205C3×C42—22no—
206#217—28no—
207#217—28no—
208(C2×D3)⋊C4—24no—
209(C2×D3)⋊C4—42no—
210C4⋊Dic3—24no—
211C4⋊Dic3—22no—
212C4⋊Dic3—21no—
213C4⋊Dic3—22no—
214C4⋊Dic3—42no—
215C4⋊Dic3C2×D414yes—
216C4⋊Dic3#4112yescomplemented
217Dic3⋊C4—24no—
218Dic3⋊C4—42no—
219C4×Dic3—22no—
220C4×Dic3—22no—
221C4×Dic3—42no—
222C4×Dic3—42no—
223C4×Dic3C2×D414yes—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—22no—
226C2×C3⋊C8—22no—
227C2×#41—68no—
228C2×#41—124no—
229C2×#41—124no—
230C22×Q8—61no—
231C22×Q8C22×D311yescharacteristic
232C22×D4—21no—
233C22×D4—21no—
234C22×D4—62no—
235C22×D4C22×D312yes—
236C23×C4—31no—
237C23×C4—31no—
238C23×C4—31no—
239C23×C4—62no—
240C2×SD16—24no—
241C2×SD16—24no—
242C2×SD16—24no—
243C22×C8—32no—
244C22×C8C22×D312yes—
245C4⋊Q8—64no—
246#127—24no—
247#124—68no—
248C22⋊Q8—62no—
249C22⋊Q8—62no—
250#121—64no—
251#121—68no—
252#121—64no—
253C4×Q8—64no—
254C4×D4—38no—
255C4×D4—64no—
256C4×D4—68no—
257C2×C4⋊C4—32no—
258C2×C4⋊C4—31no—
259C2×C4⋊C4—31no—
260C2×C4⋊C4—61no—
261C2×C4⋊C4—61no—
262C2×C4⋊C4—61no—
263C2×C4⋊C4—61no—
264C2×C4⋊C4C22×D311yescharacteristic
265C2×C22⋊C4—61no—
266C2×C22⋊C4—62no—
267C2×C22⋊C4—62no—
268C2×C22⋊C4—61no—
269C2×C42—31no—
270C2×C42—31no—
271C2×C42—31no—
272C2×C42—64no—
273C2×C42—61no—
274C2×C42—61no—
275C2×C42C22×D311yescharacteristic
276#107—64no—
277#103—64no—
278#102—24no—
279#102—64no—
280#102—64no—
281C22⋊C8—64no—
282#97—64no—
283#95—21no—
284#95—21no—
285#95—61no—
286#95—61no—
287#95—61no—
288C23×C3—24no—
289C23×C3—42no—
290C23×C3—42no—
291C23×C3—42no—
292C23×C3—42no—
293C23×C3—42no—
294C23×C3—42no—
295C23×C3C22×D411yescharacteristic
296C22×D3—21no—
297C22×D3—21no—
298C22×D3—21no—
299C22×D3—21no—
300C22×D3—21no—
301C22×D3—21no—
302C22×D3AGL(1,ℤ/8)12yes—
303C3×Q8—24no—
304C3×Q8—42no—
305C3×D4—24no—
306C3×D4—24no—
307C3×D4—48no—
308C3×D4—48no—
309C3×D4—42no—
310C3×D4—42no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—22no—
316C2×C3×C4—41no—
317C2×C3×C4—41no—
318C2×C3×C4—41no—
319C2×C3×C4—41no—
320C2×C3×C4—41no—
321C2×C3×C4AGL(1,ℤ/8)12yes—
322C2×C3×C4#13712yes—
323C2×C3×C4C22×D412yes—
324C2×C3×C4C2×#4111yescharacteristic
325C2×C3×C4C2×#4112yes—
326C2×C3×C4C2×#4111yescharacteristic
327#67—416no—
328C2×Dic3—21no—
329C2×Dic3—21no—
330C2×Dic3—21no—
331C2×Dic3—21no—
332C2×Dic3—21no—
333C2×Dic3—24no—
334C2×Dic3—21no—
335C2×Dic3—22no—
336C2×Dic3—21no—
337C2×Dic3—416no—
338C2×Dic3—41no—
339C2×Dic3—42no—
340C2×Dic3—42no—
341C2×Dic3—41no—
342C2×Dic3—41no—
343C2×Dic3—41no—
344C2×Dic3—41no—
345C2×Dic3#12012yes—
346C2×Dic3#12012yes—
347C4×D3—28no—
348C4×D3—44no—
349C4×D3—42no—
350C4×D3—42no—
351Dic6—48no—
352C3×C8—24no—
353C3⋊C8—42no—
354C24—22no—
355C24—31no—
356#41—1232no—
357C2×Q8—21no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—64no—
361C2×Q8—122no—
362C2×Q8—122no—
363C2×Q8#23712yescomplemented
364C2×D4—24no—
365C2×D4—24no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—68no—
376C2×D4—124no—
377C2×D4—124no—
378C2×D4D3×D414yes—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—31no—
382C22×C4—31no—
383C22×C4—31no—
384C22×C4—62no—
385C22×C4—62no—
386C22×C4—62no—
387C22×C4—68no—
388C22×C4—62no—
389C22×C4—61no—
390C22×C4—62no—
391C22×C4—62no—
392C22×C4—62no—
393C22×C4—62no—
394C22×C4—61no—
395C22×C4—61no—
396C22×C4—61no—
397C22×C4—121no—
398C22×C4—124no—
399C22×C4—124no—
400C22×C4D3×D411yescharacteristic
401C22×C4D3×D411yescharacteristic
402C22×C4C23×D311yescharacteristic
403SD16—216no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—62no—
407C2×C8—62no—
408C2×C8—62no—
409C2×C8#23714yescomplemented
410C4⋊C4—22no—
411C4⋊C4—34no—
412C4⋊C4—32no—
413C4⋊C4—64no—
414C4⋊C4—64no—
415C4⋊C4—62no—
416C4⋊C4—61no—
417C4⋊C4—62no—
418C4⋊C4—122no—
419C4⋊C4—122no—
420C22⋊C4—64no—
421C22⋊C4—68no—
422C22⋊C4—68no—
423C22⋊C4—122no—
424C42—22no—
425C42—34no—
426C42—62no—
427C42—62no—
428C42—122no—
429C42—122no—
430C22×C3—44no—
431C22×C3—44no—
432C22×C3—44no—
433C22×C3—82no—
434C22×C3—82no—
435C22×C3—82no—
436C22×C3—82no—
437C22×C3C2×#12011yescharacteristic
438C22×C3#52511yescharacteristic
439C22×C3#53211yescharacteristic
440C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
441C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
442C22×C3C2×#13711yescharacteristic
443C22×C3#57511yescharacteristic
444C2×D3—22no—
445C2×D3—22no—
446C2×D3—22no—
447C2×D3—41no—
448C2×D3—41no—
449C2×D3—41no—
450C2×D3—41no—
451C3×C4—22no—
452C3×C4—22no—
453C3×C4—42no—
454C3×C4—81no—
455C3×C4#52512yes—
456C3×C4#53212yes—
457Dic3—22no—
458Dic3—22no—
459Dic3—42no—
460Dic3—41no—
461Dic3—44no—
462Dic3—81no—
463C23—24no—
464C23—32no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—61no—
472C23—61no—
473C23—61no—
474C23—61no—
475C23—61no—
476C23—61no—
477C23C2×D3×D411yescharacteristic
478Q8—24no—
479Q8—42no—
480Q8—128no—
481D4—24no—
482D4—24no—
483D4—48no—
484D4—48no—
485D4—42no—
486D4—42no—
487D4—1216no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—32no—
494C2×C4—32no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—68no—
501C2×C4—61no—
502C2×C4—61no—
503C2×C4—61no—
504C2×C4—61no—
505C2×C4—61no—
506C2×C4—61no—
507C2×C4—64no—
508C2×C4—62no—
509C2×C4—61no—
510C2×C4—124no—
511C2×C4—1216no—
512C2×C4—121no—
513C2×C4—122no—
514C2×C4—122no—
515C2×C4—121no—
516C2×C4—122no—
517C2×C4—122no—
518C2×C4—121no—
519C2×C4—121no—
520C2×C4—121no—
521C2×C4#91412yes—
522C2×C4#91512yes—
523C2×C4C2×D3×D412yes—
524C2×C4C2×#23711yescharacteristic
525C2×C4C2×#23712yes—
526C2×C4D3×#4111yescharacteristic
527C8—24no—
528C8—122no—
529C2×C3—84no—
530C2×C3#238811yescharacteristic
531C2×C3#300311yescharacteristic
532C2×C3#300711yescharacteristic
533C2×C3#311211yescharacteristic
534C2×C3#311511yescharacteristic
535C2×C3#318011yescharacteristic
536C2×C3#318211yescharacteristic
537D3—42no—
538C22—44no—
539C22—44no—
540C22—44no—
541C22—62no—
542C22—62no—
543C22—62no—
544C22—82no—
545C22—82no—
546C22—82no—
547C22—82no—
548C22—121no—
549C22—121no—
550C22—121no—
551C22—121no—
552C22D3×#12011yescharacteristic
553C22#560611yescharacteristic
554C22#571611yescharacteristic
555C22C2×#91411yescharacteristic
556C22C2×#91511yescharacteristic
557C22D3×AGL(1,ℤ/8)11yescharacteristic
558C22#577311yescharacteristic
559C4—22no—
560C4—22no—
561C4—42no—
562C4—62no—
563C4—62no—
564C4—81no—
565C4—124no—
566C4—121no—
567C4—122no—
568C4—241no—
569C4#558612yes—
570C4#559612yes—
571C3#2357611yescharacteristic, complemented, Sylow
572C2—84no—
573C2—122no—
574C2#7666511yescharacteristic
575C2#8014611yescharacteristic
576C2#8014711yescharacteristic
577C2#8036211yescharacteristic
578C2#8036311yescharacteristic
579C2#8175411yescharacteristic
580C2#8183811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2014C2
socleC23×C3C22×D4
prime radicalC2×#5716C2
hypercenter#2014D3
hypocenterC3#23576
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2014

Complements

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

NH ≅ G/Ntype
2. C2×#5164C2semidirect
3. #76479C2semidirect
4. C3×#2014C2semidirect
5. #72054C2semidirect
6. #71762C2semidirect
7. #70175C2semidirect
8. #70152C2semidirect
9. #69668C2semidirect
10. #68632C2semidirect
11. #68600C2semidirect
53. #2014D3semidirect
156. C2×C3×Q8#41semidirect
203. C2×C3×C8#41semidirect
216. C4⋊Dic3#41semidirect
363. C2×Q8#237semidirect
409. C2×C8#237semidirect
571. C3#23576semidirect

Maximal subgroups

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

subgroupindex
C2×#57162
C2×#51642
#764792
C3×#20142
#720542
#717622
#701752
#701522
#696682
#686322
#686002
#235763