Finite Group Databasebeta

← #10823076

Subgroups — #10823076

Order 768.

1454 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×AGL(1,ℤ/8)C211yesmaximal, characteristic
2#84074C211yesmaximal, characteristic, complemented
3#84072C211yesmaximal, characteristic, complemented
4C22×#833C211yesmaximal, characteristic, complemented
5C2×#5215C211yesmaximal, characteristic, complemented
6C2×#5124C211yesmaximal, characteristic, complemented
7#72093C218yesmaximal, complemented
8#71987C211yesmaximal, characteristic, complemented
9#23531—31nomaximal, Sylow
10C22×C3×#41C2211yescharacteristic
11C23×C3×D4C2211yescharacteristic
12C2×C3×AGL(1,ℤ/8)—22no—
13C2×C3×AGL(1,ℤ/8)—24no—
14C2×C3×AGL(1,ℤ/8)—24no—
15C2×C3×AGL(1,ℤ/8)C2214yes—
16C22×C3×SD16—21no—
17C22×C3×D8—21no—
18C22×C3×M16C2211yescharacteristic
19C2×#926C2211yescharacteristic
20C22×#208C2211yescharacteristic
21C22×#208C2211yescharacteristic
22C22×#208C2211yescharacteristic
23#5234—22no—
24#5234C2214yescomplemented
25#5233—22no—
26#5232C2214yescomplemented
27#5225—21no—
28C2×#833—24no—
29C2×#833C2214yescomplemented
30#5215—21no—
31#5215—22no—
32#5215—22no—
33C2×#832C2214yescomplemented
34#5207—22no—
35#5124—24no—
36C2×#822C2214yescomplemented
37#4554C2214yescomplemented
38C22×AGL(1,ℤ/8)D311yescharacteristic
39C2×#430—31no—
40#2896—31no—
41C2×#418—31no—
42C22×#100—31no—
43#2879—31no—
44#2051—38no—
45#1742—31no—
46C25×C3—21no—
47C2×C3×#41—21no—
48C2×C3×#41—21no—
49C2×C3×#41—22no—
50C2×C3×#41—22no—
51C2×C3×#41—22no—
52C2×C3×#41—22no—
53C2×C3×#41D412yes—
54C2×C3×#41C2312yes—
55C22×C3×Q8D411yescharacteristic
56C22×C3×D4—21no—
57C22×C3×D4—21no—
58C22×C3×D4—22no—
59C22×C3×D4—21no—
60C22×C3×D4—21no—
61C22×C3×D4—21no—
62C22×C3×D4—22no—
63C22×C3×D4—22no—
64C22×C3×D4—21no—
65C22×C3×D4D411yescharacteristic
66C22×C3×D4D414yes—
67C22×C3×D4C2312yes—
68C23×C3×C4—21no—
69C23×C3×C4C2311yescharacteristic
70C3×AGL(1,ℤ/8)—216no—
71C3×AGL(1,ℤ/8)—28no—
72C3×AGL(1,ℤ/8)—28no—
73C2×C3×SD16—28no—
74C2×C3×SD16—24no—
75C2×C3×D8—28no—
76C2×C3×D8—24no—
77C2×C3×M16—22no—
78C2×C3×M16—21no—
79C2×C3×M16D414yes—
80C2×C3×M16C2312yes—
81C22×C3×C8—21no—
82C2×#217—21no—
83#926—21no—
84#926—21no—
85#926—21no—
86#926—21no—
87C2×C4⋊Dic3C2312yes—
88C2×C4×Dic3—21no—
89C2×#208—22no—
90C2×#208—22no—
91C2×#208—21no—
92C2×#208—21no—
93C2×#208—22no—
94C2×#208—22no—
95C2×#208—22no—
96C2×#208—22no—
97C2×#208D412yes—
98C2×#208D412yes—
99C2×#208C2312yes—
100C2×#208C2312yes—
101C22×C3⋊C8—21no—
102C22×C3⋊C8—21no—
103C22×C3⋊C8C2312yes—
104#835—24no—
105#833—28no—
106#832—24no—
107#832—28no—
108#830—28no—
109#822—28no—
110C22×#41C2×D311yescharacteristic
111C23×D4C2×D311yescharacteristic
112C2×AGL(1,ℤ/8)—22no—
113C2×AGL(1,ℤ/8)—24no—
114C2×AGL(1,ℤ/8)—24no—
115C2×AGL(1,ℤ/8)C2×D314yes—
116C22×SD16—21no—
117C22×D8—21no—
118C22×M16—31no—
119C22×M16—31no—
120C22×M16—31no—
121C22×M16C2×D311yescharacteristic
122C2×#117—31no—
123#430—64no—
124C2×#108—34no—
125#419—61no—
126#418—62no—
127#418—62no—
128#418—61no—
129#417—34no—
130#416—62no—
131C2×#102—34no—
132C2×#100—34no—
133C2×#100—64no—
134#408—34no—
135#408—62no—
136#407—62no—
137#343—34no—
138C24×C3—21no—
139C24×C3—24no—
140C24×C3—22no—
141C24×C3—41no—
142C24×C3—41no—
143C24×C3—42no—
144C24×C3—84no—
145C24×C3#4111yescharacteristic
146C3×#41—24no—
147C3×#41—24no—
148C3×#41—22no—
149C3×#41—22no—
150C3×#41—22no—
151C3×#41—22no—
152C3×#41—42no—
153C3×#41—42no—
154C3×#41—42no—
155C3×#41—42no—
156C2×C3×Q8—21no—
157C2×C3×Q8—22no—
158C2×C3×Q8—22no—
159C2×C3×Q8C2×D412yes—
160C2×C3×D4—22no—
161C2×C3×D4—22no—
162C2×C3×D4—22no—
163C2×C3×D4—24no—
164C2×C3×D4—24no—
165C2×C3×D4—24no—
166C2×C3×D4—24no—
167C2×C3×D4—21no—
168C2×C3×D4—22no—
169C2×C3×D4—24no—
170C2×C3×D4—22no—
171C2×C3×D4—22no—
172C2×C3×D4—22no—
173C2×C3×D4—22no—
174C2×C3×D4—22no—
175C2×C3×D4—42no—
176C2×C3×D4—42no—
177C2×C3×D4—42no—
178C2×C3×D4—42no—
179C2×C3×D4—41no—
180C2×C3×D4—41no—
181C2×C3×D4—42no—
182C2×C3×D4—42no—
183C2×C3×D4—42no—
184C2×C3×D4C2×D412yes—
185C2×C3×D4C2×D414yes—
186C22×C3×C4—21no—
187C22×C3×C4—22no—
188C22×C3×C4—22no—
189C22×C3×C4—42no—
190C22×C3×C4—42no—
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D412yes—
194C22×C3×C4C2×D412yes—
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4#4112yes—
198C22×C3×C4#4111yescharacteristic
199C22×C3×C4#4112yes—
200C22×C3×C4C2411yescharacteristic
201C22×Dic3—22no—
202C3×SD16—216no—
203C3×D8—216no—
204C3×M16—24no—
205C3×M16—22no—
206C3×M16C2×D414yes—
207C2×C3×C8—24no—
208C2×C3×C8—22no—
209#217—41no—
210#217—41no—
211C4⋊Dic3—22no—
212C4⋊Dic3—22no—
213C4×Dic3—41no—
214C4×Dic3—41no—
215#208—24no—
216#208—24no—
217#208—22no—
218#208—22no—
219#208—22no—
220#208—22no—
221#208—42no—
222#208—42no—
223C2×C3⋊C8—22no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—22no—
226C2×C3⋊C8—22no—
227C2×C3⋊C8—42no—
228C2×C3⋊C8—42no—
229C2×C3⋊C8C2×D414yes—
230C25—21no—
231C2×#41—21no—
232C2×#41—21no—
233C2×#41—22no—
234C2×#41—22no—
235C2×#41—22no—
236C2×#41—22no—
237C2×#41#6712yes—
238C2×#41C22×D312yes—
239C22×Q8#6711yescharacteristic
240C22×D4—21no—
241C22×D4—21no—
242C22×D4—21no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—21no—
246C22×D4—21no—
247C22×D4—21no—
248C22×D4—22no—
249C22×D4#6711yescharacteristic
250C22×D4#6714yes—
251C22×D4C22×D312yes—
252C23×C4—21no—
253C23×C4C22×D311yescharacteristic
254AGL(1,ℤ/8)—216no—
255AGL(1,ℤ/8)—28no—
256AGL(1,ℤ/8)—28no—
257C2×SD16—28no—
258C2×SD16—24no—
259C2×D8—28no—
260C2×D8—24no—
261C2×M16—22no—
262C2×M16—21no—
263C2×M16—32no—
264C2×M16—32no—
265C2×M16—32no—
266C2×M16—32no—
267C2×M16—62no—
268C2×M16—62no—
269C2×M16—61no—
270C2×M16—61no—
271C2×M16—62no—
272C2×M16—62no—
273C2×M16—62no—
274C2×M16—62no—
275C2×M16#6714yes—
276C2×M16C22×D312yes—
277C22×C8—21no—
278C22×C8—32no—
279C22×C8—61no—
280C22×C8—61no—
281#117—61no—
282#117—61no—
283#117—61no—
284#117—61no—
285C2×C4⋊C4—32no—
286C2×C22⋊C4—61no—
287C2×C42—61no—
288#108—68no—
289#103—64no—
290#102—68no—
291#102—64no—
292#100—68no—
293C22⋊C8—68no—
294C23×C3—22no—
295C23×C3—24no—
296C23×C3—42no—
297C23×C3—42no—
298C23×C3—42no—
299C23×C3—42no—
300C23×C3—44no—
301C23×C3—44no—
302C23×C3—82no—
303C23×C3—84no—
304C23×C3—84no—
305C23×C3—84no—
306C23×C3—88no—
307C23×C3—88no—
308C23×C3#12111yescharacteristic
309C23×C3#12111yescharacteristic
310C23×C3#12312yes—
311C23×C3#12412yes—
312C23×C3C2×#4111yescharacteristic
313C3×Q8—22no—
314C3×Q8—22no—
315C3×Q8—42no—
316C3×D4—22no—
317C3×D4—24no—
318C3×D4—24no—
319C3×D4—22no—
320C3×D4—42no—
321C3×D4—42no—
322C3×D4—44no—
323C3×D4—42no—
324C3×D4—42no—
325C3×D4—44no—
326C3×D4—44no—
327C3×D4—42no—
328C2×C3×C4—22no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—44no—
333C2×C3×C4#12012yes—
334C2×C3×C4#12012yes—
335C2×C3×C4#12112yes—
336C2×C3×C4#12112yes—
337C2×C3×C4#12112yes—
338C2×C3×C4#12112yes—
339C2×C3×C4#12111yescharacteristic
340C2×C3×C4#12111yescharacteristic
341C2×C3×C4#12312yes—
342C2×C3×C4#12312yes—
343C2×C3×C4#12312yes—
344C2×C3×C4#12412yes—
345C2×C3×C4C22×D411yescharacteristic
346C2×C3×C4C22×D411yescharacteristic
347C2×C3×C4C22×D411yescharacteristic
348C2×C3×C4C22×D411yescharacteristic
349C2×C3×C4C2×#4111yescharacteristic
350C2×C3×C4C2×#4111yescharacteristic
351C2×Dic3—42no—
352C2×Dic3—82no—
353C3×C8—24no—
354C3⋊C8—24no—
355C3⋊C8—42no—
356C3⋊C8—42no—
357C24—24no—
358C24—21no—
359C24—22no—
360C24—42no—
361C24—41no—
362C24—41no—
363C24—84no—
364C24#23711yescharacteristic
365#41—24no—
366#41—24no—
367#41—22no—
368#41—22no—
369#41—22no—
370#41—22no—
371#41—42no—
372#41—42no—
373#41—42no—
374#41—42no—
375C2×Q8—21no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8C2×#6712yes—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—24no—
383C2×D4—24no—
384C2×D4—24no—
385C2×D4—21no—
386C2×D4—22no—
387C2×D4—24no—
388C2×D4—24no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—42no—
395C2×D4—42no—
396C2×D4—42no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—41no—
402C2×D4—41no—
403C2×D4C2×#6714yes—
404C2×D4C2×#6712yes—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—62no—
411C22×C4D3×D411yescharacteristic
412C22×C4D3×D411yescharacteristic
413C22×C4#23712yes—
414C22×C4#23711yescharacteristic
415C22×C4#23712yes—
416C22×C4C2×#6711yescharacteristic
417C22×C4C2×#6711yescharacteristic
418C22×C4C2×#6712yes—
419C22×C4C2×#6712yes—
420C22×C4C23×D311yescharacteristic
421SD16—216no—
422D8—216no—
423M16—24no—
424M16—22no—
425M16—64no—
426M16—64no—
427M16—62no—
428M16—62no—
429M16—62no—
430M16—62no—
431M16—122no—
432M16—122no—
433M16C2×#6714yes—
434C2×C8—24no—
435C2×C8—22no—
436C2×C8—34no—
437C2×C8—62no—
438C2×C8—62no—
439C2×C8—62no—
440C2×C8—62no—
441C2×C8—122no—
442C2×C8—122no—
443C4⋊C4—62no—
444C4⋊C4—62no—
445C22⋊C4—121no—
446C22⋊C4—121no—
447C42—121no—
448C42—121no—
449C22×C3—22no—
450C22×C3—22no—
451C22×C3—22no—
452C22×C3—42no—
453C22×C3—42no—
454C22×C3—42no—
455C22×C3—44no—
456C22×C3—44no—
457C22×C3—84no—
458C22×C3—84no—
459C22×C3—88no—
460C22×C3—88no—
461C22×C3—84no—
462C22×C3—84no—
463C22×C3#39412yes—
464C22×C3#39412yes—
465C22×C3C2×#12111yescharacteristic
466C22×C3C2×#12311yescharacteristic
467C22×C3C2×#12411yescharacteristic
468C3×C4—42no—
469C3×C4#39412yes—
470C3×C4#39412yes—
471C3×C4C2×#12011yescharacteristic
472C3×C4C2×#12111yescharacteristic
473C3×C4C2×#12111yescharacteristic
474C3×C4C2×#12311yescharacteristic
475Dic3—82no—
476C23—22no—
477C23—24no—
478C23—42no—
479C23—42no—
480C23—44no—
481C23—44no—
482C23—42no—
483C23—42no—
484C23—82no—
485C23—84no—
486C23—84no—
487C23—84no—
488C23—88no—
489C23—88no—
490C23#93512yes—
491C23#93612yes—
492C23#93911yescharacteristic
493C23#93911yescharacteristic
494C23C2×#23711yescharacteristic
495Q8—22no—
496Q8—22no—
497Q8—42no—
498D4—22no—
499D4—24no—
500D4—22no—
501D4—24no—
502D4—44no—
503D4—42no—
504D4—42no—
505D4—44no—
506D4—44no—
507D4—42no—
508D4—42no—
509D4—42no—
510C2×C4—22no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—122no—
516C2×C4—242no—
517C2×C4#93512yes—
518C2×C4#93512yes—
519C2×C4#93512yes—
520C2×C4#93612yes—
521C2×C4#93812yes—
522C2×C4#93812yes—
523C2×C4#93911yescharacteristic
524C2×C4#93911yescharacteristic
525C2×C4#93912yes—
526C2×C4#93912yes—
527C2×C4#95312yes—
528C2×C4#95312yes—
529C2×C4C2×D3×D411yescharacteristic
530C2×C4C2×#23711yescharacteristic
531C2×C4C2×#23711yescharacteristic
532C2×C4C22×#6711yescharacteristic
533C2×C4C22×#6711yescharacteristic
534C2×C4C22×#6711yescharacteristic
535C8—24no—
536C8—64no—
537C8—122no—
538C8—122no—
539C2×C3—22no—
540C2×C3—21no—
541C2×C3—21no—
542C2×C3—22no—
543C2×C3—82no—
544C2×C3—84no—
545C2×C3C2×#39411yescharacteristic
546C2×C3#306812yes—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—44no—
554C22—44no—
555C22—84no—
556C22—88no—
557C22—84no—
558C22—88no—
559C22—84no—
560C22—84no—
561C22#522212yes—
562C22#522212yes—
563C22C2×#93511yescharacteristic
564C22C2×#93611yescharacteristic
565C22C2×#93911yescharacteristic
566C4—42no—
567C4—242no—
568C4#522212yes—
569C4#522212yes—
570C4C2×#93511yescharacteristic
571C4C2×#93811yescharacteristic
572C4C2×#93911yescharacteristic
573C4C2×#95311yescharacteristic
574C3#2353111yescharacteristic, complemented, Sylow
575C2—22no—
576C2—22no—
577C2—21no—
578C2—21no—
579C2—82no—
580C2—84no—
581C2#8394312yes—
582C2C2×#522211yescharacteristic
subgroupG/H
centerC22C2×#939
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC22×C3×AGL(1,ℤ/8)C2
socleC22×C3C2×#121
hypercenterC22×AGL(1,ℤ/8)D3
hypocenterC3#23531
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
2. #84074C2semidirect
3. #84072C2semidirect
4. C22×#833C2semidirect
5. C2×#5215C2semidirect
6. C2×#5124C2semidirect
7. #72093C2semidirect
8. #71987C2semidirect
24. #5234C22semidirect
26. #5232C22semidirect
29. C2×#833C22semidirect
33. C2×#832C22semidirect
36. C2×#822C22semidirect
37. #4554C22semidirect
574. C3#23531semidirect

Maximal subgroups

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

subgroupindex
C22×C3×AGL(1,ℤ/8)2
#840742
#840722
C22×#8332
C2×#52152
C2×#51242
#720932
#719872
#235313