Finite Group Databasebeta

← #10823169

Subgroups — #10823169

Order 768.

1570 conjugacy classes of subgroups.

The subgroup lattice has 556 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#84087C211yesmaximal, characteristic, complemented
3C2×#5233C211yesmaximal, characteristic, complemented
4#84044C211yesmaximal, characteristic, complemented
5C22×#832C211yesmaximal, characteristic, complemented
6C2×#5119C211yesmaximal, characteristic, complemented
7#72051C214yesmaximal, complemented
8#72040C214yesmaximal, complemented
9#71985C211yesmaximal, characteristic, complemented
10#23544—31nomaximal, Sylow
11C22×C3×#41C2211yescharacteristic
12C23×C3×D4C2211yescharacteristic
13C2×C3×AGL(1,ℤ/8)—24no—
14C2×C3×AGL(1,ℤ/8)—24no—
15C2×C3×AGL(1,ℤ/8)—24no—
16C22×C3×SD16C2212yes—
17C22×C3×D8C2212yes—
18C22×C3×M16C2211yescharacteristic
19C2×#926C2211yescharacteristic
20C2×#926C2211yescharacteristic
21C22×C4⋊Dic3C2211yescharacteristic
22C23×C3⋊C8C2211yescharacteristic
23#5247—22no—
24#5233—24no—
25#5229C2212yescomplemented
26C2×#835C2212yescomplemented
27#5221—22no—
28#5221C2214yes—
29#5221C2212yescomplemented
30#5220—22no—
31C2×#832—24no—
32C2×#832C2214yes—
33C2×#832C2212yescomplemented
34#5212—22no—
35#5119—24no—
36C2×Dic3⋊C8C2214yescomplemented
37#4551C2214yescomplemented
38C22×AGL(1,ℤ/8)D311yescharacteristic
39C2×#424—31no—
40C2×#416—31no—
41C22×#102—31no—
42#2311—31no—
43#2309—31no—
44#2055—34no—
45#2054—34no—
46#1736—31no—
47C25×C3—21no—
48C2×C3×#41—22no—
49C2×C3×#41—22no—
50C2×C3×#41—22no—
51C2×C3×#41—22no—
52C2×C3×#41—22no—
53C2×C3×#41D414yes—
54C22×C3×Q8C2311yescharacteristic
55C22×C3×D4—21no—
56C22×C3×D4—22no—
57C22×C3×D4—22no—
58C22×C3×D4—21no—
59C22×C3×D4—22no—
60C22×C3×D4—22no—
61C22×C3×D4—22no—
62C22×C3×D4D414yes—
63C22×C3×D4C2311yescharacteristic
64C22×C3×D4C2312yes—
65C23×C3×C4—21no—
66C23×C3×C4C2311yescharacteristic
67C23×Dic3—21no—
68C3×AGL(1,ℤ/8)—232no—
69C2×C3×SD16—24no—
70C2×C3×SD16—24no—
71C2×C3×SD16—24no—
72C2×C3×D8—24no—
73C2×C3×D8—24no—
74C2×C3×D8—24no—
75C2×C3×M16—22no—
76C2×C3×M16—22no—
77C2×C3×M16D414yes—
78C22×C3×C8C2312yes—
79C2×#217—21no—
80C2×#217—24no—
81C2×#217—24no—
82C2×#217—24no—
83C2×#217D412yes—
84#926—22no—
85#926—22no—
86#926—22no—
87#926—22no—
88C2×C4⋊Dic3—22no—
89C2×C4⋊Dic3—22no—
90C2×C4⋊Dic3—21no—
91C2×C4⋊Dic3D412yes—
92C2×C4⋊Dic3C2312yes—
93C2×C4⋊Dic3C2312yes—
94C2×C4×Dic3—21no—
95C2×C4×Dic3C2312yes—
96C22×C3⋊C8—22no—
97C22×C3⋊C8—22no—
98C22×C3⋊C8—22no—
99C22×C3⋊C8C2312yes—
100#835—24no—
101#832—28no—
102#832—24no—
103C3⋊#95—24no—
104Dic3⋊C8—28no—
105C22×#41C2×D311yescharacteristic
106C23×D4C2×D311yescharacteristic
107C2×AGL(1,ℤ/8)—24no—
108C2×AGL(1,ℤ/8)—24no—
109C2×AGL(1,ℤ/8)—24no—
110C22×SD16C2×D312yes—
111C22×D8C2×D312yes—
112C22×M16C2×D311yescharacteristic
113C23×C8—31no—
114C2×#117—31no—
115C2×#117—31no—
116C22×C4⋊C4—31no—
117#424—64no—
118C2×C4⋊C8—34no—
119#416—64no—
120C2×#103—32no—
121C2×#102—32no—
122C2×#102—34no—
123C2×#102—64no—
124#391—32no—
125#390—34no—
126#390—32no—
127#390—62no—
128#386—62no—
129#380—62no—
130#379—62no—
131#340—34no—
132C24×C3—22no—
133C24×C3—21no—
134C24×C3—24no—
135C24×C3—42no—
136C24×C3—42no—
137C24×C3—82no—
138C24×C3—82no—
139C24×C3C2×D411yescharacteristic
140C3×#41—28no—
141C3×#41—28no—
142C3×#41—44no—
143C3×#41—44no—
144C2×C3×Q8—21no—
145C2×C3×Q8—21no—
146C2×C3×Q8—21no—
147C2×C3×Q8—21no—
148C2×C3×Q8—21no—
149C2×C3×Q8C2×D412yes—
150C2×C3×D4—24no—
151C2×C3×D4—28no—
152C2×C3×D4—24no—
153C2×C3×D4—28no—
154C2×C3×D4—21no—
155C2×C3×D4—21no—
156C2×C3×D4—22no—
157C2×C3×D4—22no—
158C2×C3×D4—22no—
159C2×C3×D4—22no—
160C2×C3×D4—21no—
161C2×C3×D4—21no—
162C2×C3×D4—22no—
163C2×C3×D4—21no—
164C2×C3×D4—42no—
165C2×C3×D4—42no—
166C2×C3×D4—42no—
167C2×C3×D4—42no—
168C2×C3×D4—44no—
169C2×C3×D4—44no—
170C2×C3×D4C2×D412yes—
171C2×C3×D4C2×D414yes—
172C22×C3×C4—21no—
173C22×C3×C4—24no—
174C22×C3×C4—42no—
175C22×C3×C4—42no—
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2×D412yes—
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2×D412yes—
180C22×C3×C4#4112yes—
181C22×C3×C4#4112yes—
182C22×C3×C4#4112yes—
183C22×C3×C4C2411yescharacteristic
184C22×Dic3—22no—
185C22×Dic3—22no—
186C22×Dic3—42no—
187C22×Dic3—82no—
188C22×Dic3C2×D414yes—
189C3×SD16—216no—
190C3×D8—216no—
191C3×M16—28no—
192C2×C3×C8—22no—
193C2×C3×C8—22no—
194C2×C3×C8C2×D414yes—
195#217—24no—
196#217—42no—
197#217—84no—
198#217—84no—
199#217—84no—
200C4⋊Dic3—24no—
201C4⋊Dic3—22no—
202C4⋊Dic3—22no—
203C4⋊Dic3—22no—
204C4⋊Dic3—22no—
205C4⋊Dic3—42no—
206C4×Dic3—22no—
207C4×Dic3—22no—
208C4×Dic3—42no—
209C2×C3⋊C8—22no—
210C2×C3⋊C8—22no—
211C2×C3⋊C8—22no—
212C2×C3⋊C8—44no—
213C25—21no—
214C2×#41—22no—
215C2×#41—22no—
216C2×#41—22no—
217C2×#41—22no—
218C2×#41—22no—
219C2×#41#6714yes—
220C22×Q8C22×D311yescharacteristic
221C22×D4—21no—
222C22×D4—21no—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—22no—
227C22×D4—22no—
228C22×D4#6714yes—
229C22×D4C22×D311yescharacteristic
230C22×D4C22×D312yes—
231C23×C4—21no—
232C23×C4—61no—
233C23×C4C22×D311yescharacteristic
234AGL(1,ℤ/8)—232no—
235C2×SD16—24no—
236C2×SD16—24no—
237C2×SD16—24no—
238C2×D8—24no—
239C2×D8—24no—
240C2×D8—24no—
241C2×M16—22no—
242C2×M16—22no—
243C2×M16#6714yes—
244C22×C8—32no—
245C22×C8—62no—
246C22×C8—62no—
247C22×C8—62no—
248C22×C8C22×D312yes—
249#117—62no—
250#117—62no—
251#117—62no—
252#117—62no—
253C2×C4⋊C4—32no—
254C2×C4⋊C4—32no—
255C2×C4⋊C4—32no—
256C2×C4⋊C4—62no—
257C2×C4⋊C4—62no—
258C2×C4⋊C4—61no—
259C2×C22⋊C4—32no—
260C2×C22⋊C4—61no—
261C2×C22⋊C4—64no—
262C2×C22⋊C4—64no—
263C2×C22⋊C4—64no—
264C2×C42—32no—
265C2×C42—61no—
266C4⋊C8—68no—
267#103—64no—
268#102—68no—
269#102—64no—
270#95—64no—
271C23×C3—22no—
272C23×C3—22no—
273C23×C3—22no—
274C23×C3—24no—
275C23×C3—41no—
276C23×C3—41no—
277C23×C3—42no—
278C23×C3—42no—
279C23×C3—42no—
280C23×C3—44no—
281C23×C3—44no—
282C23×C3—42no—
283C23×C3—48no—
284C23×C3—84no—
285C23×C3—84no—
286C23×C3—88no—
287C23×C3—88no—
288C23×C3#12112yes—
289C23×C3#12112yes—
290C23×C3#12112yes—
291C23×C3C22×D411yescharacteristic
292C3×Q8—22no—
293C3×Q8—22no—
294C3×Q8—41no—
295C3×Q8—41no—
296C3×D4—22no—
297C3×D4—24no—
298C3×D4—24no—
299C3×D4—22no—
300C3×D4—48no—
301C3×D4—42no—
302C3×D4—41no—
303C3×D4—42no—
304C3×D4—41no—
305C3×D4—48no—
306C2×C3×C4—22no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4—48no—
310C2×C3×C4#12014yes—
311C2×C3×C4#12112yes—
312C2×C3×C4#12112yes—
313C2×C3×C4#12112yes—
314C2×C3×C4#12314yes—
315C2×C3×C4#12314yes—
316C2×C3×C4#12414yes—
317C2×C3×C4C22×D411yescharacteristic
318C2×C3×C4C22×D411yescharacteristic
319C2×C3×C4C22×D411yescharacteristic
320C2×C3×C4C2×#4111yescharacteristic
321C2×C3×C4C2×#4111yescharacteristic
322C2×C3×C4C2×#4111yescharacteristic
323C2×Dic3—24no—
324C2×Dic3—24no—
325C2×Dic3—24no—
326C2×Dic3—42no—
327C2×Dic3—42no—
328C2×Dic3—42no—
329C2×Dic3—44no—
330C2×Dic3—42no—
331C2×Dic3—84no—
332C3×C8—24no—
333C3⋊C8—42no—
334C24—21no—
335C24—22no—
336C24—24no—
337C24—42no—
338C24—42no—
339C24—82no—
340C24—82no—
341C24D3×D411yescharacteristic
342#41—28no—
343#41—28no—
344#41—44no—
345#41—44no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8C2×#6712yes—
352C2×D4—24no—
353C2×D4—28no—
354C2×D4—24no—
355C2×D4—28no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—21no—
359C2×D4—21no—
360C2×D4—22no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—44no—
369C2×D4—44no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4C2×#6712yes—
373C2×D4C2×#6714yes—
374C22×C4—21no—
375C22×C4—24no—
376C22×C4—34no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—62no—
380C22×C4—62no—
381C22×C4—122no—
382C22×C4—242no—
383C22×C4D3×D411yescharacteristic
384C22×C4#23712yes—
385C22×C4#23712yes—
386C22×C4#23712yes—
387C22×C4C2×#6712yes—
388C22×C4C2×#6712yes—
389C22×C4C2×#6712yes—
390C22×C4C23×D311yescharacteristic
391SD16—216no—
392D8—216no—
393M16—28no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—62no—
397C2×C8—62no—
398C2×C8—62no—
399C2×C8—124no—
400C2×C8C2×#6714yes—
401C4⋊C4—64no—
402C4⋊C4—62no—
403C4⋊C4—62no—
404C4⋊C4—62no—
405C4⋊C4—62no—
406C4⋊C4—122no—
407C22⋊C4—64no—
408C22⋊C4—122no—
409C22⋊C4—244no—
410C22⋊C4—244no—
411C22⋊C4—244no—
412C42—62no—
413C42—62no—
414C42—122no—
415C22×C3—22no—
416C22×C3—22no—
417C22×C3—24no—
418C22×C3—24no—
419C22×C3—42no—
420C22×C3—44no—
421C22×C3—44no—
422C22×C3—44no—
423C22×C3—44no—
424C22×C3—816no—
425C22×C3—81no—
426C22×C3—81no—
427C22×C3—84no—
428C22×C3—82no—
429C22×C3—84no—
430C22×C3#39414yes—
431C22×C3C2×#12111yescharacteristic
432C22×C3C2×#12111yescharacteristic
433C22×C3C2×#12111yescharacteristic
434C22×C3#57612yes—
435C22×C3#57712yes—
436C3×C4—42no—
437C3×C4#39414yes—
438C3×C4C2×#12011yescharacteristic
439C3×C4C2×#12311yescharacteristic
440C3×C4C2×#12311yescharacteristic
441C3×C4C2×#12411yescharacteristic
442Dic3—44no—
443Dic3—82no—
444Dic3—82no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—24no—
449C23—48no—
450C23—42no—
451C23—44no—
452C23—44no—
453C23—42no—
454C23—41no—
455C23—42no—
456C23—41no—
457C23—42no—
458C23—84no—
459C23—84no—
460C23—88no—
461C23—88no—
462C23#93812yes—
463C23#93912yes—
464C23#93912yes—
465C23C2×D3×D411yescharacteristic
466Q8—22no—
467Q8—22no—
468Q8—41no—
469Q8—41no—
470D4—24no—
471D4—22no—
472D4—22no—
473D4—24no—
474D4—48no—
475D4—48no—
476D4—42no—
477D4—41no—
478D4—42no—
479D4—41no—
480C2×C4—22no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—48no—
484C2×C4—64no—
485C2×C4—64no—
486C2×C4—64no—
487C2×C4—122no—
488C2×C4—124no—
489C2×C4—122no—
490C2×C4—122no—
491C2×C4—122no—
492C2×C4—244no—
493C2×C4#93514yes—
494C2×C4#93514yes—
495C2×C4#93614yes—
496C2×C4#93812yes—
497C2×C4#93912yes—
498C2×C4#93912yes—
499C2×C4#95314yes—
500C2×C4C2×#23711yescharacteristic
501C2×C4C2×#23711yescharacteristic
502C2×C4C2×#23711yescharacteristic
503C2×C4C22×#6711yescharacteristic
504C2×C4C22×#6711yescharacteristic
505C2×C4C22×#6711yescharacteristic
506C8—24no—
507C8—122no—
508C2×C3—24no—
509C2×C3—82no—
510C2×C3—84no—
511C2×C3C2×#39411yescharacteristic
512C2×C3#304111yescharacteristic
513C2×C3#304211yescharacteristic
514C2×C3#304512yes—
515C2×C3#306212yes—
516C22—22no—
517C22—22no—
518C22—24no—
519C22—24no—
520C22—42no—
521C22—44no—
522C22—44no—
523C22—44no—
524C22—44no—
525C22—816no—
526C22—81no—
527C22—81no—
528C22—84no—
529C22—84no—
530C22—82no—
531C22#522214yes—
532C22#577412yes—
533C22#577512yes—
534C22C2×#93811yescharacteristic
535C22C2×#93911yescharacteristic
536C22C2×#93911yescharacteristic
537C4—42no—
538C4—124no—
539C4—242no—
540C4—242no—
541C4#522214yes—
542C4C2×#93511yescharacteristic
543C4C2×#93511yescharacteristic
544C4C2×#93611yescharacteristic
545C4C2×#95311yescharacteristic
546C3#2354411yescharacteristic, complemented, Sylow
547C2—24no—
548C2—84no—
549C2—82no—
550C2#8393411yescharacteristic
551C2#8393511yescharacteristic
552C2#8393612yes—
553C2#8395112yes—
554C2C2×#522211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC22×C3×AGL(1,ℤ/8)C2
socleC23×C3C22×D4
hypercenterC22×AGL(1,ℤ/8)D3
hypocenterC3#23544
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
2. #84087C2semidirect
3. C2×#5233C2semidirect
4. #84044C2semidirect
5. C22×#832C2semidirect
6. C2×#5119C2semidirect
7. #72051C2semidirect
8. #72040C2semidirect
9. #71985C2semidirect
25. #5229C22semidirect
26. C2×#835C22semidirect
29. #5221C22semidirect
33. C2×#832C22semidirect
36. C2×Dic3⋊C8C22semidirect
37. #4551C22semidirect
546. C3#23544semidirect

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
#840872
C2×#52332
#840442
C22×#8322
C2×#51192
#720512
#720402
#719852
#235443