Finite Group Databasebeta

← #188425

Subgroups — #188425

Order 512.

1024 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
1#34151C211yesmaximal, characteristic
2#21350C211yesmaximal, characteristic, complemented
3#20755C211yesmaximal, characteristic, complemented
4C2×#1495C211yesmaximal, characteristic
5C2×#1483C211yesmaximal, characteristic
6#8570C212yesmaximal, complemented
7#8569C212yesmaximal, complemented
8#8566C212yesmaximal, complemented
9#8565C212yesmaximal, complemented
10#7903C211yesmaximal, characteristic, complemented
11#7891C211yesmaximal, characteristic, complemented
12C2×#529C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#423C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C2×C4×M16C2211yescharacteristic
17#2374—22no—
18#2373—22no—
19#2370C412yes—
20#2367C412yes—
21#2363C412yes—
22#2362C412yes—
23#2298—21no—
24#2295C2211yescharacteristic
25#2293—21no—
26#2293C2211yescharacteristic
27#2290—21no—
28C42×Q8C2211yescharacteristic
29C42×D4C2211yescharacteristic
30C4×#117—21no—
31C4×#117C2211yescharacteristic
32C4×#117C2211yescharacteristic
33#1940—22no—
34#1851—22no—
35#1832—21no—
36#1832—21no—
37#1831C2212yes—
38#1831C2212yes—
39C4×C4⋊C8C2212yes—
40C4×C4⋊C8C2212yes—
41#1729—21no—
42#1729—21no—
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#423—22no—
81#423—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8C2312yes—
90C2×C4⋊C8C2312yes—
91C2×C22⋊C8—21no—
92C4×M16—22no—
93C4×M16—22no—
94C2×#97—21no—
95C2×C4×C8C2312yes—
96C2×C4×C8C2312yes—
97#391C2×C412yes—
98#390—22no—
99#390C2×C412yes—
100#389—22no—
101#389C2×C412yes—
102#389C2×C412yes—
103#388—22no—
104#388—22no—
105#388—22no—
106#388—22no—
107#388—22no—
108#388—22no—
109#387—22no—
110#387—22no—
111#387—22no—
112#387—22no—
113#386—22no—
114#385—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4C2×C412yes—
121C4×C4⋊C4C2×C412yes—
122C4×C4⋊C4C2×C412yes—
123C4×C4⋊C4C2×C412yes—
124C4×C4⋊C4D412yes—
125C4×C4⋊C4C2311yescharacteristic
126C4×C4⋊C4C2311yescharacteristic
127C4×C22⋊C4—21no—
128C4×C22⋊C4—21no—
129C4×C22⋊C4—22no—
130C4×C22⋊C4—21no—
131C4×C22⋊C4—21no—
132C4×C22⋊C4—22no—
133C4×C22⋊C4—22no—
134C4×C22⋊C4C2×C412yes—
135C4×C22⋊C4D412yes—
136#376—21no—
137#376—21no—
138#376D411yescharacteristic
139#376D411yescharacteristic
140C43—21no—
141C43—21no—
142C43C2311yescharacteristic
143C43C2311yescharacteristic
144#336—22no—
145#336—22no—
146#336—22no—
147#336—22no—
148#336—22no—
149#336—22no—
150Q8⋊C8—24no—
151Q8⋊C8—24no—
152D4⋊C8—24no—
153D4⋊C8—24no—
154C22×Q8C22×C411yescharacteristic
155C22×D4—21no—
156C22×D4C22×C411yescharacteristic
157C23×C4—21no—
158C23×C4—22no—
159C23×C4C22×C411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C2×M16—44no—
163C2×M16—42no—
164C2×M16—42no—
165C2×M16—42no—
166C2×M16—42no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—42no—
172C22×C8—42no—
173C22×C8—42no—
174#126—24no—
175#126—24no—
176#125—22no—
177#125C4⋊C414yes—
178#124—22no—
179#124C4⋊C414yes—
180#123—24no—
181#123—24no—
182C22⋊Q8—22no—
183C22⋊Q8C4⋊C414yes—
184#121—22no—
185#121C4⋊C414yes—
186C4×Q8—21no—
187C4×Q8—21no—
188C4×Q8—24no—
189C4×Q8—42no—
190C4×Q8C2×D412yes—
191C4×Q8C2×Q812yes—
192C4×D4—24no—
193C4×D4—24no—
194C4×D4—24no—
195C4×D4—21no—
196C4×D4—21no—
197C4×D4—42no—
198C4×D4C2×D412yes—
199C4×D4C2×Q812yes—
200#117—24no—
201#117—24no—
202#117—24no—
203#117—41no—
204#117—41no—
205#117—41no—
206#117—41no—
207#117—42no—
208#117—42no—
209#117—42no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4C22⋊C412yes—
218C2×C4⋊C4C22⋊C412yes—
219C2×C4⋊C4C22×C412yes—
220C2×C4⋊C4C22×C411yescharacteristic
221C2×C4⋊C4C22×C411yescharacteristic
222C2×C4⋊C4C22×C412yes—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4C22⋊C412yes—
233C2×C22⋊C4C22⋊C412yes—
234C2×C42—21no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—42no—
243C2×C42—42no—
244C2×C42C22×C412yes—
245C2×C42C22×C412yes—
246C2×C42C22×C411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C2×C42C2×D411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D412yes—
252C2×C42C2×D412yes—
253C2×C42C2×D412yes—
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2×Q812yes—
257C2×C42#4112yes—
258C2×C42#4112yes—
259C2×C42C2411yescharacteristic
260C4⋊C8—22no—
261C4⋊C8—22no—
262C4⋊C8—22no—
263C4⋊C8—22no—
264C4⋊C8—44no—
265C4⋊C8—44no—
266C4⋊C8—42no—
267C4⋊C8—42no—
268C4⋊C8—42no—
269C22⋊C8—42no—
270#97—42no—
271C4×C8—22no—
272C4×C8—22no—
273C4×C8—22no—
274C4×C8—22no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—22no—
280#95—22no—
281#95—22no—
282#95—21no—
283#95—21no—
284#95C22⋊C412yes—
285#95C22⋊C412yes—
286#95C22⋊C412yes—
287#95C22⋊C412yes—
288C24—21no—
289C24C2×C22⋊C411yescharacteristic
290C2×Q8—21no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8C2×C4⋊C412yes—
294C2×D4—24no—
295C2×D4—21no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—82no—
299C2×D4C2×C4⋊C412yes—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—42no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—82no—
321C22×C4C2×C22⋊C412yes—
322C22×C4C2×C22⋊C412yes—
323C22×C4C2×C22⋊C412yes—
324C22×C4C2×C22⋊C412yes—
325C22×C4C2×C22⋊C411yescharacteristic
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×C4⋊C412yes—
329C22×C4#11712yes—
330C22×C4#12112yes—
331C22×C4#12112yes—
332C22×C4C22⋊Q812yes—
333C22×C4C22⋊Q812yes—
334C22×C4C23×C411yescharacteristic
335C22×C4C22×D411yescharacteristic
336C22×C4C22×D411yescharacteristic
337M16—48no—
338M16—84no—
339C2×C8—24no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—84no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—44no—
365C4⋊C4—44no—
366C4⋊C4—82no—
367C4⋊C4C2×C4⋊C412yes—
368C4⋊C4C2×C4⋊C412yes—
369C4⋊C4#13114yes—
370C22⋊C4—24no—
371C22⋊C4—22no—
372C22⋊C4—22no—
373C22⋊C4—24no—
374C22⋊C4—24no—
375C22⋊C4—24no—
376C22⋊C4—42no—
377C22⋊C4—42no—
378C22⋊C4—44no—
379C22⋊C4—44no—
380C22⋊C4—44no—
381C22⋊C4—82no—
382C42—22no—
383C42—22no—
384C42—22no—
385C42—22no—
386C42—22no—
387C42—22no—
388C42—21no—
389C42—21no—
390C42—21no—
391C42—21no—
392C42—21no—
393C42—21no—
394C42—21no—
395C42—21no—
396C42—42no—
397C42—42no—
398C42—44no—
399C42—44no—
400C42—44no—
401C42—41no—
402C42—41no—
403C42—41no—
404C42—41no—
405C42—82no—
406C42—82no—
407C42C2×C4⋊C412yes—
408C42#11712yes—
409C42#12112yes—
410C42#12112yes—
411C42C22⋊Q812yes—
412C42C22⋊Q812yes—
413C42C22×D411yescharacteristic
414C42C22×Q811yescharacteristic
415C42C2×#4111yescharacteristic
416C42C2×#4111yescharacteristic
417C23—22no—
418C23—21no—
419C23—21no—
420C23—22no—
421C23—22no—
422C23—42no—
423C23—41no—
424C23—41no—
425C23—42no—
426C23#38012yes—
427C23C22×C22⋊C411yescharacteristic
428Q8—24no—
429Q8—41no—
430Q8—41no—
431D4—24no—
432D4—41no—
433D4—41no—
434D4—84no—
435C2×C4—22no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—88no—
479C2×C4#38012yes—
480C2×C4#38012yes—
481C2×C4#38012yes—
482C2×C4C22×C4⋊C411yescharacteristic
483C2×C4C2×#11711yescharacteristic
484C2×C4C2×#12111yescharacteristic
485C2×C4C2×#12111yescharacteristic
486C2×C4C2×C22⋊Q811yescharacteristic
487C2×C4C2×C22⋊Q811yescharacteristic
488C2×C4C2×#13112yes—
489C2×C4#56812yes—
490C2×C4#57511yescharacteristic
491C2×C4#57511yescharacteristic
492C2×C4#57511yescharacteristic
493C2×C4#57511yescharacteristic
494C8—44no—
495C8—82no—
496C8—82no—
497C8—82no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—82no—
507C22—82no—
508C22C2×#38011yescharacteristic
509C22#288012yes—
510C22#289611yescharacteristic
511C22#289611yescharacteristic
512C22#290112yes—
513C4—22no—
514C4—22no—
515C4—21no—
516C4—21no—
517C4—22no—
518C4—42no—
519C4—42no—
520C4—44no—
521C4—42no—
522C4—42no—
523C4—42no—
524C4—42no—
525C4—42no—
526C4—82no—
527C4#290612yes—
528C4#292211yescharacteristic
529C4#292311yescharacteristic
530C4#304011yescharacteristic
531C4#304011yescharacteristic
532C4#308011yescharacteristic
533C4#308011yescharacteristic
534C2—42no—
535C2—82no—
536C2#1751312yes—
537C2#1751612yes—
538C2#2063011yescharacteristic
539C2#2064911yescharacteristic
540C2#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. #20755C2semidirect
6. #8570C2semidirect
7. #8569C2semidirect
8. #8566C2semidirect
9. #8565C2semidirect
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
#207552
C2×#14952
C2×#14832
#85702
#85692
#85662
#85652
#79032
#78912