Finite Group Databasebeta

← #189151

Subgroups — #189151

Order 512.

1126 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2974C211yesmaximal, characteristic
2#21982C211yesmaximal, characteristic, complemented
3#20738C211yesmaximal, characteristic
4C2×#1802C211yesmaximal, characteristic
5C2×#1573C211yesmaximal, characteristic, complemented
6#8654C212yesmaximal, complemented
7#8652C212yesmaximal, complemented
8#8646C212yesmaximal, complemented
9#8644C212yesmaximal, complemented
10#8044C211yesmaximal, characteristic, complemented
11#8040C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13#2974—24no—
14#2974C418yescomplemented
15C2×#436C2212yes—
16C2×#435—22no—
17C2×C8×D4C2212yes—
18C2×#432C2211yescharacteristic
19C2×#432C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C22×#102—21no—
23C2×#407—21no—
24C22×C22⋊C8—21no—
25#2285C2211yescharacteristic
26C2×#384C2211yescharacteristic
27#1947—22no—
28#1852—21no—
29#1852—21no—
30#1845C2212yescomplemented
31#1834C2212yescomplemented
32#1824—21no—
33#1802—21no—
34#1802—21no—
35#1802C2211yescharacteristic
36#1802C2211yescharacteristic
37#1802C2211yescharacteristic
38#1802C2211yescharacteristic
39#1802C2212yes—
40#1802C2212yes—
41#1786—21no—
42#1762—21no—
43#1762—21no—
44C2×#340—21no—
45#1573—24no—
46C2×#335C2212yescomplemented
47C2×#334C2212yescomplemented
48#1296C2212yescomplemented
49#1296C2212yescomplemented
50#1296C2212yescomplemented
51#1296C2212yescomplemented
52C23×D4D411yescharacteristic
53C24×C4—21no—
54C22×M16—22no—
55C23×C8—22no—
56C23×C8—21no—
57C2×C4×D4—21no—
58C2×C4×D4—22no—
59C2×C4×D4—21no—
60C2×C4×D4—22no—
61C2×C4×D4C2×C414yes—
62C2×C4×D4D411yescharacteristic
63C2×C4×D4D411yescharacteristic
64C2×C4×D4Q814yes—
65C2×C4×D4C2311yescharacteristic
66C2×C4×D4C2311yescharacteristic
67C2×#117—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4D411yescharacteristic
72C22×C22⋊C4—21no—
73C22×C42C2311yescharacteristic
74#436—24no—
75#436C2×C418yescomplemented
76#435—216no—
77C8×D4—24no—
78C8×D4C2×C418yescomplemented
79#432—22no—
80#432—22no—
81#432—22no—
82#432C2×C414yes—
83#431C2×C414yes—
84#431D414yes—
85C2×#107—22no—
86C2×#107—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8D412yes—
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2312yes—
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2312yes—
95C2×#102—22no—
96C2×#102—22no—
97C2×#102—21no—
98C2×#102—21no—
99C2×#102—21no—
100C2×#102—21no—
101C2×#102—44no—
102#407—28no—
103C2×C22⋊C8—21no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—22no—
106C2×C22⋊C8—28no—
107C2×C22⋊C8—21no—
108C2×C22⋊C8D411yescharacteristic
109C2×C22⋊C8D411yescharacteristic
110C2×#97—21no—
111C2×#97—21no—
112C2×C4×C8C2312yes—
113#384—21no—
114#384—21no—
115#384—21no—
116#384C2311yescharacteristic
117#384C2312yes—
118#384C2311yescharacteristic
119#382—21no—
120#380—21no—
121#380—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124#340—22no—
125#340—22no—
126#340—22no—
127#340—22no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#335—24no—
133#334—24no—
134C25—21no—
135C22×D4—22no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—21no—
139C22×D4—41no—
140C22×D4—41no—
141C22×D4C4⋊C414yes—
142C22×D4C2×D411yescharacteristic
143C22×D4C2×D411yescharacteristic
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—24no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—24no—
151C23×C4—41no—
152C23×C4—82no—
153C23×C4—82no—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4#4111yescharacteristic
157C2×M16—28no—
158C2×M16—82no—
159C2×M16—82no—
160C22×C8—22no—
161C22×C8—28no—
162C22×C8—22no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—42no—
166C22×C8—42no—
167C22×C8—42no—
168C22×C8—82no—
169C22×C8—82no—
170C22×C8C2×D412yes—
171C22×C8C2×D412yes—
172C4×D4—24no—
173C4×D4—24no—
174C4×D4—22no—
175C4×D4—22no—
176C4×D4—24no—
177C4×D4—24no—
178C4×D4—21no—
179C4×D4—21no—
180C4×D4—21no—
181C4×D4—21no—
182C4×D4C4⋊C414yes—
183C4×D4C4⋊C414yes—
184C4×D4C22×C412yes—
185C4×D4C22×C412yes—
186C4×D4C2×Q812yes—
187C4×D4C2×Q812yes—
188#117—82no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—82no—
206C2×C4⋊C4—82no—
207C2×C4⋊C4C4⋊C414yes—
208C2×C4⋊C4C2×D411yescharacteristic
209C2×C4⋊C4C2×D411yescharacteristic
210C2×C22⋊C4—24no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—41no—
217C2×C42—41no—
218C2×C42C22×C412yes—
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D412yes—
221C2×C42C2×D411yescharacteristic
222C2×C42C2×Q812yes—
223C2×C42#4111yescharacteristic
224C2×C42#4112yes—
225C2×C42C2411yescharacteristic
226#107—82no—
227#107—82no—
228C4⋊C8—22no—
229C4⋊C8—21no—
230C4⋊C8—22no—
231C4⋊C8—21no—
232C4⋊C8—22no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—42no—
236C4⋊C8C22⋊C418yes—
237C4⋊C8C22×C414yes—
238#102—42no—
239#102—42no—
240#102—44no—
241#102—42no—
242#102—44no—
243#102—42no—
244C22⋊C8—24no—
245C22⋊C8—28no—
246C22⋊C8—28no—
247C22⋊C8—42no—
248C22⋊C8C22⋊C414yes—
249C22⋊C8C22⋊C414yes—
250#97—24no—
251#97—42no—
252C4×C8C22×C414yes—
253C4×C8C2×D414yes—
254#95—41no—
255#95—41no—
256C24—21no—
257C24—21no—
258C24—24no—
259C24—42no—
260C24—42no—
261C24—42no—
262C24—42no—
263C24—81no—
264C24—81no—
265C24#12111yescharacteristic
266C2×D4—28no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—24no—
276C2×D4—44no—
277C2×D4—44no—
278C2×D4—44no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4C2×C4⋊C412yes—
282C2×D4C2×C4⋊C412yes—
283C2×D4AGL(1,ℤ/8)12yes—
284C2×D4#13712yes—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—24no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—88no—
310C22×C4—88no—
311C22×C4—81no—
312C22×C4—81no—
313C22×C4—81no—
314C22×C4—81no—
315C22×C4—81no—
316C22×C4—81no—
317C22×C4—82no—
318C22×C4—82no—
319C22×C4—82no—
320C22×C4C2×C22⋊C412yes—
321C22×C4C2×C4⋊C412yes—
322C22×C4#11712yes—
323C22×C4#12112yes—
324C22×C4#12111yescharacteristic
325C22×C4#12111yescharacteristic
326C22×C4#12111yescharacteristic
327C22×C4C22⋊Q812yes—
328C22×C4C22⋊Q812yes—
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C2×#4111yescharacteristic
332M16—88no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—24no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—42no—
349C2×C8—88no—
350C2×C8—84no—
351C2×C8C2×C22⋊C414yes—
352C2×C8C2×C22⋊C414yes—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—24no—
356C4⋊C4—42no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—82no—
367C4⋊C4—81no—
368C4⋊C4—81no—
369C4⋊C4—82no—
370C4⋊C4—82no—
371C4⋊C4—82no—
372C4⋊C4C2×C4⋊C412yes—
373C4⋊C4C2×C4⋊C412yes—
374C22⋊C4—22no—
375C22⋊C4—22no—
376C22⋊C4—22no—
377C22⋊C4—22no—
378C22⋊C4—24no—
379C22⋊C4—24no—
380C22⋊C4—82no—
381C42—82no—
382C42C2×C22⋊C412yes—
383C42C2×C4⋊C412yes—
384C42#11712yes—
385C42#12112yes—
386C42C22⋊Q812yes—
387C42C22⋊Q812yes—
388C42C23×C411yescharacteristic
389C42C22×D411yescharacteristic
390C42C22×Q811yescharacteristic
391C42C2×#4111yescharacteristic
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—22no—
397C23—22no—
398C23—22no—
399C23—22no—
400C23—48no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23—48no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—84no—
413C23—84no—
414C23—84no—
415C23—84no—
416C23—84no—
417C23#38012yes—
418C23#47212yes—
419C23#47312yes—
420C23C2×#12111yescharacteristic
421D4—22no—
422D4—22no—
423D4—22no—
424D4—22no—
425D4—22no—
426D4—48no—
427D4—41no—
428D4—41no—
429D4#42814yes—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—84no—
457C2×C4—84no—
458C2×C4—84no—
459C2×C4—82no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—82no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—82no—
468C2×C4—164no—
469C2×C4#38012yes—
470C2×C4#38012yes—
471C2×C4#38012yes—
472C2×C4C22×C22⋊C411yescharacteristic
473C2×C4C22×C4⋊C411yescharacteristic
474C2×C4C2×#11711yescharacteristic
475C2×C4C2×#12111yescharacteristic
476C2×C4C2×C22⋊Q811yescharacteristic
477C2×C4C2×C22⋊Q811yescharacteristic
478C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
479C2×C4C2×#13711yescharacteristic
480C2×C4#57511yescharacteristic
481C2×C4#57511yescharacteristic
482C2×C4#57612yes—
483C2×C4#57712yes—
484C8—24no—
485C8—24no—
486C8—44no—
487C8—81no—
488C8—82no—
489C8—81no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—24no—
497C22—42no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—816no—
507C22—81no—
508C22—81no—
509C22—81no—
510C22—81no—
511C22#181614yes—
512C22C2×#38011yescharacteristic
513C22#304111yescharacteristic
514C22#304211yescharacteristic
515C22#305511yescharacteristic
516C22#305611yescharacteristic
517C22C2×#47211yescharacteristic
518C22C2×#47311yescharacteristic
519C4—22no—
520C4—22no—
521C4—21no—
522C4—21no—
523C4—22no—
524C4—42no—
525C4—42no—
526C4—161no—
527C4—162no—
528C4—161no—
529C4#289712yes—
530C4C2×#42811yescharacteristic
531C4#291111yescharacteristic
532C4#305511yescharacteristic
533C4#305611yescharacteristic
534C4#308112yes—
535C2—24no—
536C2—44no—
537C2—82no—
538C2#1945811yescharacteristic
539C2#1945911yescharacteristic
540C2#1946411yescharacteristic
541C2#1946511yescharacteristic
542C2#2065011yescharacteristic
543C2#2066111yescharacteristic
544C2C2×#181611yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21982C2semidirect
5. C2×#1573C2semidirect
6. #8654C2semidirect
7. #8652C2semidirect
8. #8646C2semidirect
9. #8644C2semidirect
10. #8044C2semidirect
11. #8040C2semidirect
14. #2974C4semidirect
30. #1845C22semidirect
31. #1834C22semidirect
46. C2×#335C22semidirect
47. C2×#334C22semidirect
48. #1296C22semidirect
49. #1296C22semidirect
50. #1296C22semidirect
51. #1296C22semidirect
75. #436C2×C4semidirect
78. C8×D4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29742
#219822
#207382
C2×#18022
C2×#15732
#86542
#86522
#86462
#86442
#80442
#80402