Finite Group Databasebeta

← #194104

Subgroups — #194104

Order 512.

1512 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3208C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic, complemented
3#21260C211yesmaximal, characteristic
4C2×#1957C211yesmaximal, characteristic
5C2×#1493C211yesmaximal, characteristic, complemented
6#8979C214yesmaximal, complemented
7#8978C214yesmaximal, complemented
8#8043C211yesmaximal, characteristic, complemented
9#7890C211yesmaximal, characteristic, complemented
10C2×#533C2211yescharacteristic
11C22×#127C2211yescharacteristic
12#3208—24no—
13#3208C418yescomplemented
14C2×#492—21no—
15C2×#486—21no—
16C2×#460C2214yes—
17C2×#450—22no—
18C2×#431C2211yescharacteristic
19C2×#423C2211yescharacteristic
20C2×#418—21no—
21C2×#418—21no—
22C22×#102—22no—
23C22×C4×C8C2211yescharacteristic
24#2360—22no—
25#2360—22no—
26#2355—21no—
27#2355C2212yescomplemented
28Q8×C4⋊C4—21no—
29Q8×C4⋊C4C2212yescomplemented
30#2330—22no—
31#2317—21no—
32C4×C4⋊Q8—21no—
33C4×C4⋊Q8—21no—
34C2×#384C2211yescharacteristic
35#2274—21no—
36C4×#117C2211yescharacteristic
37#1982—21no—
38#1982—21no—
39#1982C2212yes—
40#1982C2212yes—
41#1957—22no—
42#1957C2214yes—
43#1493—24no—
44C2×Q8⋊C8C2214yescomplemented
45#1296C2214yescomplemented
46#1279C2212yescomplemented
47#1279C2212yescomplemented
48C23×D4—21no—
49C23×D4—21no—
50C23×D4—21no—
51C22×SD16—24no—
52C22×M16—21no—
53C23×C8—21no—
54#533—22no—
55#533—22no—
56#533C2×C414yes—
57#533D414yes—
58C2×C4⋊Q8D411yescharacteristic
59C2×C4⋊Q8D411yescharacteristic
60C2×#127—21no—
61C2×#127—22no—
62C2×#127—21no—
63C2×#127—21no—
64C2×#127—22no—
65C2×#127—22no—
66C2×#127C2×C414yes—
67C2×#127C2312yes—
68C2×#125—21no—
69C2×C22⋊Q8—21no—
70C2×C22⋊Q8D412yes—
71C2×C4×Q8—22no—
72C2×C4×Q8—22no—
73C2×C4×Q8—21no—
74C2×C4×Q8C2312yes—
75C2×#117—21no—
76C2×#117D411yescharacteristic
77C2×#117D411yescharacteristic
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4D412yes—
81C22×C42C2311yescharacteristic
82#492—28no—
83#486—28no—
84#460—28no—
85#460C2×C4116yescomplemented
86#450—216no—
87#431—22no—
88#431C2×C414yes—
89#423—22no—
90#423—22no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8C2312yes—
93C2×C4⋊C8C2312yes—
94#418—44no—
95#418—44no—
96C2×#102—24no—
97C2×#102—28no—
98C2×#102—22no—
99C2×#102—22no—
100C2×#102—22no—
101C2×#102—22no—
102C2×#102—48no—
103C2×C22⋊C8—21no—
104C2×C4×C8—21no—
105C2×C4×C8—22no—
106C2×C4×C8—22no—
107C2×C4×C8—21no—
108C2×C4×C8C2312yes—
109#391—22no—
110#391—22no—
111#389—22no—
112#389—22no—
113#389—22no—
114#389—24no—
115#389—22no—
116#389—24no—
117#387—24no—
118#387—24no—
119#385—24no—
120#384—22no—
121#384—21no—
122#384—21no—
123#384D412yes—
124#384C2312yes—
125#382—22no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4D411yescharacteristic
133C4×C4⋊C4D411yescharacteristic
134C4×C4⋊C4C2311yescharacteristic
135C4×C4⋊C4C2311yescharacteristic
136C4×C22⋊C4—24no—
137C4×C22⋊C4—21no—
138C4×C22⋊C4—21no—
139#376—21no—
140C43—21no—
141Q8⋊C8—28no—
142C25—41no—
143C22×Q8—21no—
144C22×Q8C2×D412yes—
145C22×D4—24no—
146C22×D4—24no—
147C22×D4—24no—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4—22no—
151C22×D4—44no—
152C22×D4—44no—
153C22×D4—41no—
154C22×D4—41no—
155C22×D4—41no—
156C22×D4—42no—
157C22×D4—42no—
158C22×D4—42no—
159C22×D4—42no—
160C22×D4—42no—
161C22×D4—42no—
162C22×D4—42no—
163C22×D4—82no—
164C22×D4—82no—
165C23×C4—22no—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C2×SD16—216no—
170C2×SD16—84no—
171C2×SD16—84no—
172C2×M16—42no—
173C2×M16—42no—
174C2×M16—42no—
175C22×C8—24no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—42no—
180C22×C8—42no—
181C4⋊Q8—22no—
182C4⋊Q8—22no—
183C4⋊Q8C22⋊C414yes—
184C4⋊Q8C22⋊C414yes—
185#127—24no—
186#127—24no—
187#127—24no—
188#127—22no—
189#127—22no—
190#127—22no—
191#127—42no—
192#127—42no—
193#127—42no—
194#127C22×C414yes—
195#125—24no—
196#125—24no—
197C22⋊Q8—24no—
198C22⋊Q8—24no—
199C22⋊Q8—24no—
200C22⋊Q8C22⋊C418yes—
201C4×Q8—24no—
202C4×Q8—24no—
203C4×Q8—22no—
204C4×Q8—22no—
205C4×Q8—84no—
206C4×Q8—84no—
207C4×Q8C22×C414yes—
208C4×Q8C2×D414yes—
209#117—24no—
210#117—24no—
211#117—22no—
212#117—22no—
213#117—22no—
214#117SD1614yescomplemented
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—82no—
231C2×C4⋊C4C2×D411yescharacteristic
232C2×C4⋊C4C2×D411yescharacteristic
233C2×C4⋊C4C2×D412yes—
234C2×C4⋊C4C2×D411yescharacteristic
235C2×C4⋊C4C2×D411yescharacteristic
236C2×C4⋊C4C2×D414yes—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C42—21no—
241C2×C42—22no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—24no—
248C2×C42—24no—
249C2×C42C22×C412yes—
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D412yes—
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42#4112yes—
255C2×C42#4112yes—
256C2×C42C2411yescharacteristic
257C4⋊C8—22no—
258C4⋊C8—22no—
259C4⋊C8—22no—
260C4⋊C8—42no—
261C4⋊C8C22×C414yes—
262#102—216no—
263#102—44no—
264#102—44no—
265#102—416no—
266#102—44no—
267#102—44no—
268C22⋊C8—24no—
269C4×C8—24no—
270C4×C8—22no—
271C4×C8—22no—
272C4×C8—42no—
273#95—24no—
274#95—24no—
275#95—24no—
276#95—24no—
277#95—21no—
278#95—21no—
279#95—21no—
280#95—21no—
281C24—42no—
282C24—44no—
283C24—82no—
284C24—82no—
285C24—82no—
286C24—82no—
287C24—82no—
288C24—82no—
289C24#12011yescharacteristic
290C2×Q8—22no—
291C2×Q8—24no—
292C2×Q8—24no—
293C2×Q8—24no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8C2×C22⋊C414yes—
297C2×D4—24no—
298C2×D4—24no—
299C2×D4—24no—
300C2×D4—44no—
301C2×D4—44no—
302C2×D4—416no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4—44no—
311C2×D4—44no—
312C2×D4—44no—
313C2×D4—44no—
314C2×D4—44no—
315C2×D4—48no—
316C2×D4—48no—
317C2×D4—88no—
318C2×D4—88no—
319C2×D4—82no—
320C2×D4—82no—
321C2×D4—88no—
322C2×D4—82no—
323C2×D4—82no—
324C2×D4—82no—
325C2×D4—82no—
326C2×D4—84no—
327C2×D4—84no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—24no—
338C22×C4—21no—
339C22×C4—44no—
340C22×C4—82no—
341C22×C4—82no—
342C22×C4C2×C22⋊C412yes—
343C22×C4C4×D412yes—
344C22×C4C4×D412yes—
345C22×C4#12011yescharacteristic
346C22×C4#12011yescharacteristic
347C22×C4#12011yescharacteristic
348C22×C4#12112yes—
349C22×C4#12112yes—
350C22×C4#12312yes—
351C22×C4C2×SD1612yes—
352C22×C4#13512yes—
353C22×C4AGL(1,ℤ/8)12yes—
354C22×C4#13712yes—
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D411yescharacteristic
358SD16—816no—
359M16—84no—
360C2×C8—24no—
361C2×C8—24no—
362C2×C8—42no—
363C2×C8—48no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—22no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—28no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—44no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—44no—
387C4⋊C4—44no—
388C4⋊C4—44no—
389C4⋊C4—82no—
390C4⋊C4—84no—
391C4⋊C4—84no—
392C4⋊C4C2×C22⋊C414yes—
393C4⋊C4C2×SD1612yes—
394C4⋊C4C2×SD1612yes—
395C22⋊C4—28no—
396C22⋊C4—24no—
397C22⋊C4—42no—
398C42—24no—
399C42—24no—
400C42—24no—
401C42—42no—
402C42—42no—
403C42—42no—
404C42—42no—
405C42—42no—
406C42—84no—
407C42—84no—
408C42C2×C22⋊C412yes—
409C42C4×D412yes—
410C42C4×D412yes—
411C42#12112yes—
412C42#12112yes—
413C42#12312yes—
414C42C23×C411yescharacteristic
415C42C22×D411yescharacteristic
416C42C2×#4111yescharacteristic
417C42C2×#4111yescharacteristic
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—44no—
423C23—88no—
424C23—88no—
425C23—88no—
426C23—82no—
427C23—82no—
428C23—82no—
429C23—82no—
430C23—82no—
431C23—82no—
432C23—164no—
433C23—164no—
434C23—164no—
435C23—164no—
436C23#38612yes—
437C23#44812yes—
438C23#44912yes—
439C23C2×#12011yescharacteristic
440Q8—28no—
441Q8—44no—
442Q8—48no—
443D4—48no—
444D4—48no—
445D4—44no—
446D4—44no—
447D4—84no—
448D4—88no—
449D4—84no—
450D4—88no—
451D4—88no—
452D4—82no—
453D4—82no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—44no—
477C2×C4—41no—
478C2×C4—44no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—44no—
482C2×C4—41no—
483C2×C4—164no—
484C2×C4#38612yes—
485C2×C4#38612yes—
486C2×C4#38612yes—
487C2×C4C4×SD1614yes—
488C2×C4#44014yes—
489C2×C4#44812yes—
490C2×C4#44912yes—
491C2×C4#45312yes—
492C2×C4#45412yes—
493C2×C4#46014yes—
494C2×C4#46414yes—
495C2×C4C22×C22⋊C411yescharacteristic
496C2×C4C2×C4×D411yescharacteristic
497C2×C4C2×C4×D411yescharacteristic
498C2×C4C2×#12111yescharacteristic
499C2×C4C2×#12111yescharacteristic
500C2×C4C2×#12311yescharacteristic
501C2×C4C22×SD1611yescharacteristic
502C2×C4C2×#13511yescharacteristic
503C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
504C2×C4C2×#13711yescharacteristic
505C2×C4#57612yes—
506C2×C4#57712yes—
507C8—44no—
508C8—44no—
509C8—82no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—22no—
516C22—84no—
517C22—84no—
518C22—84no—
519C22—1616no—
520C22—162no—
521C22—162no—
522C22—162no—
523C22—162no—
524C22#188014yes—
525C22C2×#38611yescharacteristic
526C22C2×#44811yescharacteristic
527C22C2×#44911yescharacteristic
528C22#300811yescharacteristic
529C22#301011yescharacteristic
530C22C2×#45311yescharacteristic
531C22C2×#45411yescharacteristic
532C4—22no—
533C4—21no—
534C4—21no—
535C4—44no—
536C4—42no—
537C4—44no—
538C4—42no—
539C4—81no—
540C4—84no—
541C4—81no—
542C4#188014yes—
543C4#188314yes—
544C4#289712yes—
545C4C2×C4×SD1611yescharacteristic
546C4C2×#44011yescharacteristic
547C4C2×#46011yescharacteristic
548C4C2×#46411yescharacteristic
549C4#309612yes—
550C2—24no—
551C2—164no—
552C2#1862211yescharacteristic
553C2#1862411yescharacteristic
554C2#1864211yescharacteristic
555C2#1864411yescharacteristic
556C2#2091411yescharacteristic
557C2C2×#188011yescharacteristic
558C2C2×#188311yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#127C22
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. #34214C2semidirect
5. C2×#1493C2semidirect
6. #8979C2semidirect
7. #8978C2semidirect
8. #8043C2semidirect
9. #7890C2semidirect
13. #3208C4semidirect
27. #2355C22semidirect
29. Q8×C4⋊C4C22semidirect
44. C2×Q8⋊C8C22semidirect
45. #1296C22semidirect
46. #1279C22semidirect
47. #1279C22semidirect
85. #460C2×C4semidirect
214. #117SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#32082
#342142
#212602
C2×#19572
C2×#14932
#89792
#89782
#80432
#78902