Finite Group Databasebeta

← #188424

Subgroups — #188424

Order 512.

1072 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34151C211yesmaximal, characteristic
2#21350C211yesmaximal, characteristic, complemented
3#20793C211yesmaximal, characteristic, complemented
4C2×#1495C211yesmaximal, characteristic
5C2×#1483C211yesmaximal, characteristic
6#8568C212yesmaximal, complemented
7#8567C212yesmaximal, complemented
8#8564C212yesmaximal, complemented
9#8563C212yesmaximal, complemented
10#7903C211yesmaximal, characteristic, complemented
11#7891C211yesmaximal, characteristic, complemented
12C2×#529C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#424—21no—
15C2×#424—21no—
16C2×#423C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18C2×C4×M16C2211yescharacteristic
19#2374—22no—
20#2373—22no—
21#2370C412yes—
22#2367C412yes—
23#2363C412yes—
24#2362C412yes—
25#2298—21no—
26#2295C2211yescharacteristic
27#2293—21no—
28#2293C2211yescharacteristic
29#2290—21no—
30C42×Q8C2211yescharacteristic
31C42×D4C2211yescharacteristic
32C4×#117—21no—
33C4×#117C2211yescharacteristic
34C4×#117C2211yescharacteristic
35#1940—22no—
36#1851—22no—
37#1843—21no—
38#1843—21no—
39#1842C2212yes—
40#1842C2212yes—
41C4×C4⋊C8C2212yes—
42C4×C4⋊C8C2212yes—
43#1495—24no—
44#1483—24no—
45C2×Q8⋊C8C2212yes—
46C2×Q8⋊C8C2212yes—
47C2×D4⋊C8C2212yes—
48C2×D4⋊C8C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51#1279C2212yes—
52#1279C2212yes—
53C22×M16—21no—
54C22×M16—21no—
55C23×C8—21no—
56#529—22no—
57#529—22no—
58#529D414yes—
59#529Q814yes—
60C2×#126—21no—
61C2×#125C2×C411yescharacteristic
62C2×#124C2×C411yescharacteristic
63C2×#123—21no—
64C2×C22⋊Q8C2×C411yescharacteristic
65C2×#121C2×C411yescharacteristic
66C2×C4×Q8C2×C412yes—
67C2×C4×Q8C2311yescharacteristic
68C2×C4×D4—21no—
69C2×C4×D4C2×C412yes—
70C2×C4×D4C2311yescharacteristic
71C2×#117—21no—
72C2×#117—21no—
73C2×#117C2×C412yes—
74C2×#117D411yescharacteristic
75C2×#117D411yescharacteristic
76C22×C42—21no—
77C22×C42C2311yescharacteristic
78#431—22no—
79#431—22no—
80#424—44no—
81#424—44no—
82#423—22no—
83#423—22no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8C2312yes—
96C2×C4⋊C8C2312yes—
97C2×C22⋊C8—21no—
98C4×M16—22no—
99C4×M16—22no—
100C2×#97—21no—
101C2×C4×C8C2312yes—
102C2×C4×C8C2312yes—
103#391C2×C412yes—
104#390—22no—
105#390C2×C412yes—
106#389—22no—
107#389C2×C412yes—
108#389C2×C412yes—
109#388—22no—
110#388—22no—
111#388—22no—
112#388—22no—
113#388—22no—
114#388—22no—
115#387—22no—
116#387—22no—
117#387—22no—
118#387—22no—
119#386—22no—
120#385—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4C2×C412yes—
127C4×C4⋊C4C2×C412yes—
128C4×C4⋊C4C2×C412yes—
129C4×C4⋊C4C2×C412yes—
130C4×C4⋊C4D412yes—
131C4×C4⋊C4C2311yescharacteristic
132C4×C4⋊C4C2311yescharacteristic
133C4×C22⋊C4—21no—
134C4×C22⋊C4—21no—
135C4×C22⋊C4—22no—
136C4×C22⋊C4—21no—
137C4×C22⋊C4—21no—
138C4×C22⋊C4—22no—
139C4×C22⋊C4—22no—
140C4×C22⋊C4C2×C412yes—
141C4×C22⋊C4D412yes—
142#376—21no—
143#376—21no—
144#376D411yescharacteristic
145#376D411yescharacteristic
146C43—21no—
147C43—21no—
148C43C2311yescharacteristic
149C43C2311yescharacteristic
150#336—22no—
151#336—22no—
152Q8⋊C8—24no—
153Q8⋊C8—24no—
154D4⋊C8—24no—
155D4⋊C8—24no—
156C22×Q8C22×C411yescharacteristic
157C22×D4—21no—
158C22×D4C22×C411yescharacteristic
159C23×C4—21no—
160C23×C4—22no—
161C23×C4C22×C411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C2×M16—44no—
165C2×M16—42no—
166C2×M16—42no—
167C2×M16—42no—
168C2×M16—42no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—42no—
174C22×C8—42no—
175C22×C8—42no—
176#126—24no—
177#126—24no—
178#125—22no—
179#125C4⋊C414yes—
180#124—22no—
181#124C4⋊C414yes—
182#123—24no—
183#123—24no—
184C22⋊Q8—22no—
185C22⋊Q8C4⋊C414yes—
186#121—22no—
187#121C4⋊C414yes—
188C4×Q8—21no—
189C4×Q8—21no—
190C4×Q8—24no—
191C4×Q8—42no—
192C4×Q8C2×D412yes—
193C4×Q8C2×Q812yes—
194C4×D4—24no—
195C4×D4—24no—
196C4×D4—24no—
197C4×D4—21no—
198C4×D4—21no—
199C4×D4—42no—
200C4×D4C2×D412yes—
201C4×D4C2×Q812yes—
202#117—24no—
203#117—24no—
204#117—22no—
205#117—22no—
206#117—22no—
207#117—22no—
208#117—24no—
209#117—42no—
210#117—42no—
211#117—42no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4C22⋊C412yes—
220C2×C4⋊C4C22⋊C412yes—
221C2×C4⋊C4C22×C412yes—
222C2×C4⋊C4C22×C411yescharacteristic
223C2×C4⋊C4C22×C411yescharacteristic
224C2×C4⋊C4C22×C412yes—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4C22⋊C412yes—
235C2×C22⋊C4C22⋊C412yes—
236C2×C42—21no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42C22×C412yes—
247C2×C42C22×C412yes—
248C2×C42C22×C411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D412yes—
252C2×C42C2×D412yes—
253C2×C42C2×D412yes—
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2×Q812yes—
259C2×C42#4112yes—
260C2×C42#4112yes—
261C2×C42C2411yescharacteristic
262C4⋊C8—22no—
263C4⋊C8—22no—
264C4⋊C8—22no—
265C4⋊C8—22no—
266C4⋊C8—44no—
267C4⋊C8—44no—
268C4⋊C8—44no—
269C4⋊C8—44no—
270C4⋊C8—42no—
271C4⋊C8—42no—
272C4⋊C8—42no—
273C4⋊C8—44no—
274C4⋊C8—44no—
275C22⋊C8—42no—
276#97—42no—
277C4×C8—22no—
278C4×C8—22no—
279C4×C8—22no—
280C4×C8—22no—
281#95—22no—
282#95—22no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—21no—
289#95—21no—
290#95C22⋊C412yes—
291#95C22⋊C412yes—
292#95C22⋊C412yes—
293#95C22⋊C412yes—
294C24—21no—
295C24C2×C22⋊C411yescharacteristic
296C2×Q8—21no—
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8C2×C4⋊C412yes—
300C2×D4—24no—
301C2×D4—21no—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—82no—
305C2×D4C2×C4⋊C412yes—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—42no—
320C22×C4—44no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—44no—
326C22×C4—82no—
327C22×C4C2×C22⋊C412yes—
328C22×C4C2×C22⋊C412yes—
329C22×C4C2×C22⋊C411yescharacteristic
330C22×C4C2×C22⋊C411yescharacteristic
331C22×C4C2×C22⋊C411yescharacteristic
332C22×C4C2×C22⋊C412yes—
333C22×C4C2×C22⋊C412yes—
334C22×C4C2×C4⋊C412yes—
335C22×C4#11712yes—
336C22×C4#12112yes—
337C22×C4#12112yes—
338C22×C4C22⋊Q812yes—
339C22×C4C22⋊Q812yes—
340C22×C4C23×C411yescharacteristic
341C22×C4C22×D411yescharacteristic
342C22×C4C22×D411yescharacteristic
343M16—48no—
344M16—84no—
345C2×C8—24no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—84no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—44no—
371C4⋊C4—44no—
372C4⋊C4—82no—
373C4⋊C4C2×C4⋊C412yes—
374C4⋊C4C2×C4⋊C412yes—
375C4⋊C4#13114yes—
376C22⋊C4—24no—
377C22⋊C4—22no—
378C22⋊C4—22no—
379C22⋊C4—24no—
380C22⋊C4—24no—
381C22⋊C4—24no—
382C22⋊C4—24no—
383C22⋊C4—24no—
384C22⋊C4—44no—
385C22⋊C4—44no—
386C22⋊C4—44no—
387C22⋊C4—82no—
388C42—22no—
389C42—22no—
390C42—22no—
391C42—22no—
392C42—22no—
393C42—22no—
394C42—21no—
395C42—21no—
396C42—21no—
397C42—21no—
398C42—42no—
399C42—42no—
400C42—44no—
401C42—44no—
402C42—41no—
403C42—41no—
404C42—41no—
405C42—41no—
406C42—44no—
407C42—82no—
408C42—82no—
409C42C2×C4⋊C412yes—
410C42#11712yes—
411C42#12112yes—
412C42#12112yes—
413C42C22⋊Q812yes—
414C42C22⋊Q812yes—
415C42AGL(1,ℤ/8)12yes—
416C42AGL(1,ℤ/8)12yes—
417C42#13712yes—
418C42#13712yes—
419C42C22×D411yescharacteristic
420C42C22×Q811yescharacteristic
421C42C2×#4111yescharacteristic
422C42C2×#4111yescharacteristic
423C23—22no—
424C23—21no—
425C23—21no—
426C23—22no—
427C23—22no—
428C23—42no—
429C23—41no—
430C23—41no—
431C23—42no—
432C23#38012yes—
433C23C22×C22⋊C411yescharacteristic
434Q8—24no—
435Q8—41no—
436Q8—41no—
437D4—24no—
438D4—41no—
439D4—41no—
440D4—84no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—88no—
483C2×C4#38012yes—
484C2×C4#38012yes—
485C2×C4#38012yes—
486C2×C4#41714yes—
487C2×C4#41714yes—
488C2×C4C22×C4⋊C411yescharacteristic
489C2×C4C2×#11711yescharacteristic
490C2×C4C2×#12111yescharacteristic
491C2×C4C2×#12111yescharacteristic
492C2×C4C2×C22⋊Q811yescharacteristic
493C2×C4C2×C22⋊Q811yescharacteristic
494C2×C4C2×#13112yes—
495C2×C4#56812yes—
496C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
497C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
498C2×C4C2×#13711yescharacteristic
499C2×C4C2×#13711yescharacteristic
500C8—44no—
501C8—82no—
502C8—82no—
503C8—82no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—42no—
508C22—42no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22—82no—
513C22—82no—
514C22C2×#38011yescharacteristic
515C22#288012yes—
516C22C2×#41711yescharacteristic
517C22C2×#41711yescharacteristic
518C22#290112yes—
519C4—24no—
520C4—24no—
521C4—22no—
522C4—22no—
523C4—21no—
524C4—21no—
525C4—22no—
526C4—42no—
527C4—44no—
528C4—42no—
529C4—42no—
530C4—42no—
531C4—42no—
532C4—82no—
533C4#290612yes—
534C4#292211yescharacteristic
535C4#292311yescharacteristic
536C4#303811yescharacteristic
537C4#303911yescharacteristic
538C4#307911yescharacteristic
539C4#307911yescharacteristic
540C2—42no—
541C2—82no—
542C2#1751212yes—
543C2#1751512yes—
544C2#2063011yescharacteristic
545C2#2064811yescharacteristic
546C2#2065511yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC2×#121C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21350C2semidirect
3. #20793C2semidirect
6. #8568C2semidirect
7. #8567C2semidirect
8. #8564C2semidirect
9. #8563C2semidirect
10. #7903C2semidirect
11. #7891C2semidirect

Maximal subgroups

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

subgroupindex
#341512
#213502
#207932
C2×#14952
C2×#14832
#85682
#85672
#85642
#85632
#79032
#78912