Finite Group Databasebeta

← #194110

Subgroups — #194110

Order 512.

1416 conjugacy classes of subgroups.

The subgroup lattice has 599 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3246C211yesmaximal, characteristic
2#34206C211yesmaximal, characteristic
3#21400C211yesmaximal, characteristic, complemented
4#21312C211yesmaximal, characteristic, complemented
5C2×#1492C211yesmaximal, characteristic
6#8988C212yesmaximal, complemented
7#8985C212yesmaximal, complemented
8#8976C212yesmaximal, complemented
9#8973C212yesmaximal, complemented
10#8045C211yesmaximal, characteristic, complemented
11C2×#1279C211yesmaximal, characteristic, complemented
12C2×#533C2211yescharacteristic
13C2×#532C2211yescharacteristic
14C2×#520—21no—
15#3246—24no—
16#3246C418yescomplemented
17C2×#475—21no—
18C2×#476C2212yes—
19C2×#465—22no—
20C2×#460—21no—
21C2×#461C2212yes—
22C2×#423C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C22×C4×C8C2211yescharacteristic
25#2367—22no—
26#2366—21no—
27#2365—21no—
28#2354C2212yes—
29#2352—22no—
30D4×C4⋊C4C2212yes—
31#2325—22no—
32#2311—21no—
33C4×#121—22no—
34#2285C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36#1990C2211yescharacteristic
37#1990C2211yescharacteristic
38#1989—22no—
39#1988C2212yes—
40#1987C2212yes—
41#1956C2211yescharacteristic
42#1956C2211yescharacteristic
43#1892—22no—
44C4×#103—22no—
45#1736—21no—
46#1736—21no—
47#1492—24no—
48C2×D4⋊C8C2212yes—
49C2×D4⋊C8C2212yes—
50#1296C2212yes—
51#1296C2212yes—
52#1279C2212yes—
53#1279C2212yes—
54C22×#41—21no—
55C22×SD16—22no—
56C22×M16—21no—
57C23×C8—21no—
58C23×C8—21no—
59#533—22no—
60#533—22no—
61#533—22no—
62#533C2×C414yescomplemented
63#532—22no—
64#532—22no—
65#532C2×C414yes—
66#532D414yes—
67C2×C4⋊Q8D411yescharacteristic
68C2×C4⋊Q8C2311yescharacteristic
69C2×C4⋊Q8C2311yescharacteristic
70C2×#127D411yescharacteristic
71C2×#125—21no—
72C2×#124—21no—
73C2×C22⋊Q8—21no—
74C2×C22⋊Q8—21no—
75C2×#121—21no—
76C2×#121D412yes—
77#520—44no—
78#520—44no—
79C2×C4×Q8—22no—
80C2×C4×Q8—21no—
81C2×C4×Q8—21no—
82C2×C4×D4—24no—
83C2×C4×D4—22no—
84C2×C4×D4—22no—
85C2×C4×D4—21no—
86C2×C4×D4C2312yes—
87C2×#117—21no—
88C2×#117—22no—
89C2×#117—21no—
90C22×C4⋊C4—22no—
91C22×C4⋊C4D411yescharacteristic
92C22×C4⋊C4D411yescharacteristic
93C22×C42—21no—
94C22×C42C2311yescharacteristic
95#476—24no—
96#476C2×C418yescomplemented
97#475—28no—
98#465—216no—
99#461—24no—
100#461C2×C418yescomplemented
101#460—28no—
102C2×#106—22no—
103#423—22no—
104#423—22no—
105C2×C4⋊C8—21no—
106C2×C4⋊C8—22no—
107C2×C4⋊C8—21no—
108C2×C4⋊C8C2×C414yescomplemented
109C2×C4⋊C8C2312yes—
110C2×C4⋊C8C2312yes—
111C2×#103—22no—
112C2×#103—22no—
113C2×#103—22no—
114C2×#102—22no—
115C2×C4×C8—22no—
116C2×C4×C8—22no—
117C2×C4×C8—21no—
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120#391—22no—
121#390—22no—
122#390—22no—
123#389—22no—
124#388—24no—
125#388—24no—
126#387—24no—
127#386—24no—
128#386—24no—
129#385—24no—
130#384C2311yescharacteristic
131#384C2311yescharacteristic
132#382—21no—
133#380—21no—
134#380—22no—
135#380—22no—
136#380D412yes—
137C4×C4⋊C4—22no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4—21no—
140C4×C4⋊C4—21no—
141C4×C4⋊C4D412yes—
142C4×C4⋊C4D412yes—
143C4×C4⋊C4C2312yes—
144C4×C22⋊C4—24no—
145C2×#95—21no—
146#340—22no—
147#340—22no—
148#340—22no—
149#340—22no—
150#340—22no—
151#340—22no—
152D4⋊C8—24no—
153D4⋊C8—24no—
154C2×#41—24no—
155C2×#41—24no—
156C2×#41—42no—
157C2×#41—42no—
158C2×#41—42no—
159C2×#41—42no—
160C22×Q8—21no—
161C22×Q8—21no—
162C22×Q8—21no—
163C22×D4—22no—
164C22×D4—21no—
165C22×D4—21no—
166C22×D4C2×D412yes—
167C23×C4—21no—
168C23×C4—22no—
169C23×C4—24no—
170C23×C4—21no—
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C2×SD16—28no—
175C2×SD16—82no—
176C2×SD16—82no—
177C2×M16—42no—
178C2×M16—42no—
179C2×M16—42no—
180C22×C8—22no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—24no—
184C22×C8—22no—
185C22×C8—42no—
186C22×C8—42no—
187C22×C8—42no—
188C22×C8—81no—
189C22×C8—81no—
190C4⋊Q8—22no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8C22⋊C414yes—
198C4⋊Q8C22×C412yes—
199C4⋊Q8C22×C412yes—
200#127—22no—
201#127C22⋊C414yes—
202#125—24no—
203#125—42no—
204#124—24no—
205#124—24no—
206C22⋊Q8—24no—
207C22⋊Q8—24no—
208C22⋊Q8—42no—
209C22⋊Q8—42no—
210#121—24no—
211#121—24no—
212#121—24no—
213#121C22⋊C418yes—
214C4×Q8—24no—
215C4×Q8—24no—
216C4×Q8—48no—
217C4×Q8—42no—
218C4×Q8—42no—
219C4×D4—24no—
220C4×D4—24no—
221C4×D4—22no—
222C4×D4—22no—
223C4×D4—48no—
224C4×D4—416no—
225C4×D4—82no—
226C4×D4—82no—
227C4×D4C22×C414yes—
228C4×D4C2×D414yes—
229#117—24no—
230#117—44no—
231#117—42no—
232#117—42no—
233#117—42no—
234#117—44no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—41no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4SD1614yescomplemented
255C2×C4⋊C4C2×D412yes—
256C2×C4⋊C4C2×D412yes—
257C2×C4⋊C4C2×D412yes—
258C2×C4⋊C4C2×D412yes—
259C2×C4⋊C4C2×D412yes—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—41no—
267C2×C42—24no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—41no—
272C2×C42—81no—
273C2×C42—81no—
274C2×C42C22×C412yes—
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D412yes—
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42#4112yes—
280C2×C42#4112yes—
281C2×C42C2411yescharacteristic
282#106—28no—
283C4⋊C8—24no—
284C4⋊C8—22no—
285C4⋊C8—22no—
286C4⋊C8—22no—
287C4⋊C8—42no—
288C4⋊C8C22×C414yes—
289#103—28no—
290#103—44no—
291#103—44no—
292#102—28no—
293C4×C8—21no—
294C4×C8—21no—
295C4×C8—21no—
296C4×C8—21no—
297C4×C8—42no—
298#95—24no—
299#95—24no—
300#95—22no—
301#95—22no—
302#95—22no—
303#95—22no—
304C24—22no—
305C24#12011yescharacteristic
306#41—48no—
307#41—48no—
308#41—48no—
309#41—48no—
310C2×Q8—24no—
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—41no—
316C2×Q8—41no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—41no—
320C2×Q8—41no—
321C2×Q8—41no—
322C2×Q8—41no—
323C2×D4—28no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—24no—
330C2×D4—24no—
331C2×D4—44no—
332C2×D4—44no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—81no—
336C2×D4—81no—
337C2×D4C2×C22⋊C414yes—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—48no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—81no—
361C22×C4—81no—
362C22×C4—81no—
363C22×C4—81no—
364C22×C4—84no—
365C22×C4—84no—
366C22×C4C2×C22⋊C412yes—
367C22×C4C4×D412yes—
368C22×C4C4×D412yes—
369C22×C4#12011yescharacteristic
370C22×C4#12011yescharacteristic
371C22×C4#12011yescharacteristic
372C22×C4#12112yes—
373C22×C4#12112yes—
374C22×C4#12312yes—
375C22×C4C2×SD1612yes—
376C22×C4#13512yes—
377C22×C4AGL(1,ℤ/8)12yes—
378C22×C4#13712yes—
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D411yescharacteristic
382SD16—88no—
383M16—84no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—24no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—48no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—84no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—48no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—44no—
424C4⋊C4—44no—
425C4⋊C4—42no—
426C4⋊C4C2×C22⋊C414yes—
427C4⋊C4C2×SD1614yes—
428C22⋊C4—24no—
429C22⋊C4—24no—
430C22⋊C4—28no—
431C22⋊C4—48no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C22⋊C4—84no—
435C22⋊C4—84no—
436C42—48no—
437C42—42no—
438C42—42no—
439C42—44no—
440C42—44no—
441C42—82no—
442C42C2×C22⋊C412yes—
443C42C4×D412yes—
444C42C4×D412yes—
445C42#12112yes—
446C42#12112yes—
447C42#12312yes—
448C42C23×C411yescharacteristic
449C42C22×D411yescharacteristic
450C42C2×#4111yescharacteristic
451C42C2×#4111yescharacteristic
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—24no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23#38612yes—
464C23#45012yes—
465C23#45212yes—
466C23C2×#12011yescharacteristic
467Q8—44no—
468Q8—48no—
469Q8—42no—
470Q8—42no—
471Q8—82no—
472Q8—81no—
473Q8—81no—
474D4—24no—
475D4—24no—
476D4—416no—
477D4—42no—
478D4—42no—
479D4—48no—
480D4—84no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—84no—
515C2×C4—84no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—168no—
521C2×C4#38612yes—
522C2×C4#38612yes—
523C2×C4#38612yes—
524C2×C4C4×SD1614yes—
525C2×C4#44014yes—
526C2×C4#45012yes—
527C2×C4#45212yes—
528C2×C4#46114yes—
529C2×C4#46314yes—
530C2×C4C22×C22⋊C411yescharacteristic
531C2×C4C2×C4×D411yescharacteristic
532C2×C4C2×C4×D411yescharacteristic
533C2×C4C2×#12111yescharacteristic
534C2×C4C2×#12111yescharacteristic
535C2×C4C2×#12311yescharacteristic
536C2×C4C22×SD1611yescharacteristic
537C2×C4C2×#13511yescharacteristic
538C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
539C2×C4C2×#13711yescharacteristic
540C2×C4#57611yescharacteristic
541C2×C4#57711yescharacteristic
542C2×C4#57711yescharacteristic
543C2×C4#57811yescharacteristic
544C8—42no—
545C8—42no—
546C8—44no—
547C8—82no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—44no—
555C22—44no—
556C22—44no—
557C22—82no—
558C22—82no—
559C22—82no—
560C22—82no—
561C22#187714yes—
562C22C2×#38611yescharacteristic
563C22C2×#45011yescharacteristic
564C22C2×#45211yescharacteristic
565C22#301211yescharacteristic
566C22#301311yescharacteristic
567C22#301811yescharacteristic
568C22#301811yescharacteristic
569C4—22no—
570C4—21no—
571C4—21no—
572C4—22no—
573C4—44no—
574C4—44no—
575C4—82no—
576C4—82no—
577C4—82no—
578C4—82no—
579C4—82no—
580C4#187714yes—
581C4#289711yescharacteristic
582C4#289811yescharacteristic
583C4C2×C4×SD1611yescharacteristic
584C4C2×#44011yescharacteristic
585C4C2×#46111yescharacteristic
586C4C2×#46311yescharacteristic
587C4#309811yescharacteristic
588C4#309911yescharacteristic
589C2—24no—
590C2—84no—
591C2#1866311yescharacteristic
592C2#1866911yescharacteristic
593C2#1867511yescharacteristic
594C2#1868111yescharacteristic
595C2#2091511yescharacteristic
596C2C2×#187711yescharacteristic
597C2#2094211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
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
3. #21400C2semidirect
4. #21312C2semidirect
6. #8988C2semidirect
7. #8985C2semidirect
8. #8976C2semidirect
9. #8973C2semidirect
10. #8045C2semidirect
11. C2×#1279C2semidirect
16. #3246C4semidirect
62. #533C2×C4semidirect
96. #476C2×C4semidirect
100. #461C2×C4semidirect
108. C2×C4⋊C8C2×C4semidirect
254. C2×C4⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#32462
#342062
#214002
#213122
C2×#14922
#89882
#89852
#89762
#89732
#80452
C2×#12792