Finite Group Databasebeta

← #194882

Subgroups — #194882

Order 512.

1712 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2564C211yesmaximal, characteristic
2#34181C211yesmaximal, characteristic
3#21759C212yesmaximal, complemented
4C2×#1459C211yesmaximal, characteristic
5#9023C212yesmaximal, complemented
6#9021C212yesmaximal, complemented
7#9015C212yesmaximal, complemented
8#9013C212yesmaximal, complemented
9C2×#1294C212yesmaximal, complemented
10C22×C22⋊Q8C2211yescharacteristic
11C2×#518—21no—
12C22×C4×D4C2211yescharacteristic
13C22×C22⋊C8C2212yes—
14#2564—24no—
15#2564C418yescomplemented
16C2×#400C2211yescharacteristic
17C2×#399C2211yescharacteristic
18C2×#395C2211yescharacteristic
19C2×#393C2211yescharacteristic
20#2349—24no—
21D4×C4⋊C4—24no—
22#2316C2212yes—
23#2315—21no—
24#2315C2212yes—
25C2×#387—21no—
26C2×#386—21no—
27C2×#385—21no—
28#2285—21no—
29C2×#380—21no—
30C2×#380—21no—
31C2×#380C2212yes—
32C22×#95C2211yescharacteristic
33#1892—24no—
34#1872—24no—
35#1824C2214yes—
36#1787C2214yes—
37#1459—24no—
38C2×#324C2212yes—
39C2×#323C2212yes—
40#1294C2214yes—
41#1294C2214yes—
42C23×Q8C2311yescharacteristic
43C23×D4D411yescharacteristic
44C24×C4—21no—
45C24×C4D411yescharacteristic
46C24×C4C2311yescharacteristic
47C23×C8—22no—
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8—24no—
52C2×C22⋊Q8C2×C414yescomplemented
53#518—44no—
54#518—44no—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4—24no—
58C2×C4×D4—28no—
59C2×C4×D4C2×C414yes—
60C2×C4×D4D414yes—
61C2×#117—22no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4D414yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C22⋊C4—22no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4D411yescharacteristic
75C22×C22⋊C4C2311yescharacteristic
76C22×C42D411yescharacteristic
77C2×#103—28no—
78C2×C22⋊C8—24no—
79C2×C22⋊C8—24no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8C2314yes—
82#400—22no—
83#400C2×C414yescomplemented
84#399—22no—
85#399C2×C414yescomplemented
86#395—22no—
87#395C2×C414yescomplemented
88#393—22no—
89#393C2×C414yescomplemented
90#389—28no—
91#387—28no—
92#386—28no—
93#385—28no—
94#385—28no—
95#385—28no—
96#384—22no—
97#382—22no—
98#380—22no—
99#380—22no—
100#380—22no—
101#380—22no—
102#380—22no—
103#380—22no—
104#380—28no—
105#380D414yes—
106#380C2314yes—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95C2×C414yescomplemented
118C2×#95C2311yescharacteristic
119C2×#95C2311yescharacteristic
120#340—28no—
121#324—24no—
122#323—24no—
123C25C2×D411yescharacteristic
124C25C2×D411yescharacteristic
125C22×Q8—21no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—82no—
129C22×Q8C22×C412yes—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4C22⋊C414yes—
135C22×D4SD1614yes—
136C22×D4Q1614yes—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—24no—
145C23×C4—24no—
146C23×C4—24no—
147C23×C4—44no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4—82no—
151C23×C4—82no—
152C23×C4—82no—
153C23×C4C22⋊C414yes—
154C23×C4C22×C412yes—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D412yes—
160C23×C4C2×D412yes—
161C23×C4C2×D412yes—
162C23×C4#4114yes—
163C23×C4C2411yescharacteristic
164C22×C8—24no—
165C22×C8—44no—
166C22×C8—84no—
167C22⋊Q8—28no—
168C22⋊Q8—28no—
169C22⋊Q8—44no—
170C22⋊Q8—44no—
171C4×D4—28no—
172C4×D4—28no—
173C4×D4—28no—
174C4×D4—28no—
175C4×D4—432no—
176#117—44no—
177#117—44no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—84no—
195C2×C4⋊C4—84no—
196C2×C4⋊C4—82no—
197C2×C4⋊C4C22×C412yes—
198C2×C4⋊C4C22×C412yes—
199C2×C4⋊C4C2×D412yes—
200C2×C4⋊C4C2×D418yes—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—82no—
219C2×C22⋊C4C22⋊C414yes—
220C2×C22⋊C4C22×C412yes—
221C2×C22⋊C4C2×D412yes—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—22no—
225C2×C42—24no—
226C2×C42C22⋊C414yes—
227#103—88no—
228C22⋊C8—24no—
229C22⋊C8—24no—
230C22⋊C8—44no—
231#95—24no—
232#95—24no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95—24no—
237#95—24no—
238#95—24no—
239#95—21no—
240#95—21no—
241#95—24no—
242#95C22×C412yes—
243#95C22×C412yes—
244C24—21no—
245C24—21no—
246C24—21no—
247C24—21no—
248C24—42no—
249C24—42no—
250C24—42no—
251C24—42no—
252C24C2×C22⋊C412yes—
253C24C2×C22⋊C412yes—
254C24#12012yes—
255C24#12312yes—
256C24C2×SD1612yes—
257C24C2×SD1612yes—
258C24C2×Q1612yes—
259C24C2×Q1612yes—
260C24AGL(1,ℤ/8)12yes—
261C24#13712yes—
262C24C22×D411yescharacteristic
263C2×Q8—24no—
264C2×Q8—48no—
265C2×Q8—42no—
266C2×Q8—82no—
267C2×Q8—82no—
268C2×Q8—84no—
269C2×D4—28no—
270C2×D4—28no—
271C2×D4—28no—
272C2×D4—28no—
273C2×D4—28no—
274C2×D4—28no—
275C2×D4#103116yescomplemented
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—24no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—28no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—44no—
297C22×C4—48no—
298C22×C4—42no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—48no—
316C22×C4—48no—
317C22×C4—48no—
318C22×C4—82no—
319C22×C4—82no—
320C22×C4—82no—
321C22×C4—84no—
322C22×C4—84no—
323C22×C4—84no—
324C22×C4—84no—
325C22×C4—84no—
326C22×C4—84no—
327C22×C4—84no—
328C22×C4—84no—
329C22×C4—88no—
330C22×C4—88no—
331C22×C4C2×C22⋊C412yes—
332C22×C4C2×C22⋊C412yes—
333C22×C4C2×C22⋊C412yes—
334C22×C4C2×C22⋊C412yes—
335C22×C4C4×D414yes—
336C22×C4C4×D414yes—
337C22×C4#12011yescharacteristic
338C22×C4#12011yescharacteristic
339C22×C4#12114yes—
340C22×C4#12114yes—
341C22×C4#12312yes—
342C22×C4C2×SD1612yes—
343C22×C4C2×Q1612yes—
344C22×C4C23×C411yescharacteristic
345C22×C4C22×D412yes—
346C22×C4C22×D411yescharacteristic
347C22×C4C2×#4112yes—
348C2×C8—48no—
349C2×C8—44no—
350C2×C8—88no—
351C4⋊C4—24no—
352C4⋊C4—24no—
353C4⋊C4—24no—
354C4⋊C4—48no—
355C4⋊C4—48no—
356C4⋊C4—48no—
357C4⋊C4—48no—
358C4⋊C4—42no—
359C4⋊C4—44no—
360C4⋊C4—44no—
361C4⋊C4—84no—
362C4⋊C4—84no—
363C4⋊C4—84no—
364C22⋊C4—28no—
365C22⋊C4—24no—
366C22⋊C4—24no—
367C22⋊C4—48no—
368C22⋊C4—48no—
369C22⋊C4—44no—
370C22⋊C4—44no—
371C22⋊C4—48no—
372C22⋊C4—48no—
373C22⋊C4—84no—
374C42—24no—
375C42—24no—
376C42—48no—
377C23—22no—
378C23—24no—
379C23—22no—
380C23—24no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—22no—
385C23—22no—
386C23—22no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—44no—
391C23—44no—
392C23—42no—
393C23—44no—
394C23—44no—
395C23—42no—
396C23—42no—
397C23—44no—
398C23—44no—
399C23—42no—
400C23—42no—
401C23—81no—
402C23—82no—
403C23—82no—
404C23—81no—
405C23#38614yes—
406C23C2×#10314yes—
407C23C2×#10314yes—
408C23#41714yes—
409C23#45014yes—
410C23#45114yes—
411C23#48314yes—
412C23#48414yes—
413C23C22×C22⋊C411yescharacteristic
414C23C2×#12011yescharacteristic
415C23C2×#12311yescharacteristic
416C23C22×SD1611yescharacteristic
417C23C22×Q1611yescharacteristic
418C23C2×AGL(1,ℤ/8)11yescharacteristic
419C23C2×#13711yescharacteristic
420Q8—48no—
421Q8—84no—
422D4—232no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—24no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—44no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—48no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—48no—
461C2×C4—48no—
462C2×C4—48no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—84no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—816no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—168no—
476C2×C4—168no—
477C2×C4#38612yes—
478C2×C4#38612yes—
479C2×C4C2×#10314yes—
480C2×C4C2×C4×D412yes—
481C2×C4C2×#12112yes—
482C8—84no—
483C22—22no—
484C22—28no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—28no—
489C22—21no—
490C22—21no—
491C22—24no—
492C22—21no—
493C22—21no—
494C22—24no—
495C22—24no—
496C22—44no—
497C22—44no—
498C22—44no—
499C22—44no—
500C22—44no—
501C22—44no—
502C22—44no—
503C22—42no—
504C22—42no—
505C22—82no—
506C22—82no—
507C22—82no—
508C22#187218yes—
509C22C2×#38611yescharacteristic
510C22C22×#10311yescharacteristic
511C22C2×#41711yescharacteristic
512C22C2×#45011yescharacteristic
513C22C2×#45111yescharacteristic
514C22#301812yes—
515C22#302612yes—
516C22C2×#48311yescharacteristic
517C22C2×#48411yescharacteristic
518C4—24no—
519C4—22no—
520C4—22no—
521C4—22no—
522C4—22no—
523C4—44no—
524C4—42no—
525C4—42no—
526C4—82no—
527C4—88no—
528C4—82no—
529C4—82no—
530C2—21no—
531C2—21no—
532C2—21no—
533C2—21no—
534C2—28no—
535C2—44no—
536C2#1838112yes—
537C2#1838312yes—
538C2C2×#187211yescharacteristic
539C2#2095211yescharacteristic
540C2#2096011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21759C2semidirect
5. #9023C2semidirect
6. #9021C2semidirect
7. #9015C2semidirect
8. #9013C2semidirect
9. C2×#1294C2semidirect
15. #2564C4semidirect
52. C2×C22⋊Q8C2×C4semidirect
83. #400C2×C4semidirect
85. #399C2×C4semidirect
87. #395C2×C4semidirect
89. #393C2×C4semidirect
117. C2×#95C2×C4semidirect
275. C2×D4#103semidirect

Maximal subgroups

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

subgroupindex
C2×#25642
#341812
#217592
C2×#14592
#90232
#90212
#90152
#90132
C2×#12942