Finite Group Databasebeta

← #189792

Subgroups — #189792

Order 512.

1174 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
1C4⋊C4×#41C211yesmaximal, characteristic
2#21374C211yesmaximal, characteristic
3#20721C211yesmaximal, characteristic
4#8710C212yesmaximal, complemented
5#8709C212yesmaximal, complemented
6#8707C212yesmaximal, complemented
7#8706C212yesmaximal, complemented
8#7904C211yesmaximal, characteristic, complemented
9#7900C211yesmaximal, characteristic, complemented
10#7896C211yesmaximal, characteristic, complemented
11#7892C211yesmaximal, characteristic, complemented
12C2×C4×#41—21no—
13C2×C4×#41C2211yescharacteristic
14C2×#432C2211yescharacteristic
15C2×#431C2211yescharacteristic
16C2×#431C2211yescharacteristic
17C2×C4×M16C2211yescharacteristic
18#2351—21no—
19Q8×C4⋊C4C412yes—
20#2349—21no—
21#2349—21no—
22#2349—21no—
23D4×C4⋊C4C412yes—
24D4×C4⋊C4C412yes—
25D4×C4⋊C4C412yes—
26#2296—21no—
27#2296—21no—
28#2296—21no—
29#2285C2211yescharacteristic
30#2285C2211yescharacteristic
31#2285C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35#1985—21no—
36#1955—21no—
37#1870—21no—
38#1870—21no—
39#1818—21no—
40#1797C2212yes—
41#1797C2212yes—
42#1780—21no—
43#1750—21no—
44#1750—21no—
45C4×C22⋊C8C2212yes—
46C4×C22⋊C8C2212yes—
47#1279C2212yes—
48#1279C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51#1279C2212yes—
52#1279C2212yes—
53#1279C2212yes—
54#1279C2212yes—
55C22×#41D411yescharacteristic
56C22×M16—21no—
57C4×#41—22no—
58C4×#41—22no—
59C4×#41—22no—
60C4×#41—22no—
61C4×#41—48no—
62C2×C4×Q8—22no—
63C2×C4×Q8C2×C411yescharacteristic
64C2×C4×Q8D411yescharacteristic
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4C2×C411yescharacteristic
69C2×C4×D4C2×C411yescharacteristic
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4D411yescharacteristic
72C2×C4×D4D411yescharacteristic
73C2×C4×D4D411yescharacteristic
74C2×#117—21no—
75C2×#117—21no—
76C2×#117—21no—
77C2×#117D411yescharacteristic
78C2×#117D411yescharacteristic
79C2×#117D411yescharacteristic
80C22×C4⋊C4C2×C412yes—
81C22×C4⋊C4C2×C412yes—
82C22×C4⋊C4C2×C412yes—
83C22×C42—21no—
84C22×C42—21no—
85C22×C42—21no—
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89#432—22no—
90#432—22no—
91#431—22no—
92#431—22no—
93#431—22no—
94#431—22no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8C2312yes—
97C2×C4⋊C8C2312yes—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105C4×M16—22no—
106C4×M16—22no—
107C2×#97—21no—
108C2×#97—21no—
109C2×C4×C8—21no—
110C2×C4×C8C2312yes—
111C2×C4×C8C2312yes—
112#389—22no—
113#389—22no—
114#389—22no—
115#389—22no—
116#389—22no—
117#389—22no—
118#385—22no—
119#385—22no—
120#385—22no—
121#385—22no—
122#385—22no—
123#385—22no—
124#384C2×C412yes—
125#384C2×C412yes—
126#384C2×C412yes—
127#382—21no—
128#382—21no—
129#382—21no—
130#380—21no—
131#380—21no—
132#380—21no—
133#380C2×C412yes—
134#380C2×C412yes—
135#380C2×C412yes—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4—21no—
140C4×C4⋊C4—21no—
141C4×C4⋊C4—21no—
142C4×C4⋊C4—21no—
143C4×C4⋊C4—21no—
144C4×C4⋊C4—21no—
145C4×C4⋊C4C2×C412yes—
146C4×C4⋊C4C2×C412yes—
147C4×C4⋊C4C2×C412yes—
148C4×C4⋊C4C2311yescharacteristic
149C4×C4⋊C4C2311yescharacteristic
150C4×C4⋊C4C2311yescharacteristic
151C4×C4⋊C4C2311yescharacteristic
152C2×#95—21no—
153C2×#95—21no—
154C2×#95—21no—
155#336—22no—
156#336—22no—
157#336—22no—
158#336—22no—
159C2×#41—22no—
160C2×#41—22no—
161C2×#41—24no—
162C2×#41—24no—
163C22×Q8C22⋊C411yescharacteristic
164C22×D4C22⋊C411yescharacteristic
165C22×D4C22⋊C411yescharacteristic
166C22×D4C22⋊C411yescharacteristic
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4C22×C411yescharacteristic
171C23×C4C22×C411yescharacteristic
172C23×C4C22×C411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C2×M16—44no—
180C2×M16—42no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C4×Q8—22no—
186C4×Q8—22no—
187C4×Q8—22no—
188C4×Q8—22no—
189C4×Q8—48no—
190C4×D4—22no—
191C4×D4—22no—
192C4×D4—22no—
193C4×D4—22no—
194C4×D4—22no—
195C4×D4—22no—
196C4×D4—22no—
197C4×D4—22no—
198C4×D4—22no—
199C4×D4—22no—
200C4×D4—22no—
201C4×D4—22no—
202C4×D4—48no—
203C4×D4—48no—
204C4×D4—48no—
205#117—22no—
206#117—22no—
207#117—22no—
208#117—22no—
209#117—22no—
210#117—22no—
211#117—44no—
212#117—44no—
213#117—44no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—22no—
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—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4C22⋊C411yescharacteristic
230C2×C4⋊C4C22⋊C411yescharacteristic
231C2×C4⋊C4C22⋊C411yescharacteristic
232C2×C4⋊C4C22⋊C411yescharacteristic
233C2×C4⋊C4C22⋊C411yescharacteristic
234C2×C4⋊C4C22⋊C411yescharacteristic
235C2×C4⋊C4C22×C412yes—
236C2×C4⋊C4C22×C412yes—
237C2×C4⋊C4C22×C412yes—
238C2×C4⋊C4C22×C412yes—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4C22⋊C411yescharacteristic
243C2×C22⋊C4C22⋊C411yescharacteristic
244C2×C22⋊C4C22⋊C411yescharacteristic
245C2×C22⋊C4C22⋊C411yescharacteristic
246C2×C22⋊C4C22⋊C411yescharacteristic
247C2×C22⋊C4C22⋊C411yescharacteristic
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—44no—
264C2×C42—44no—
265C2×C42—44no—
266C2×C42C22×C411yescharacteristic
267C2×C42C22×C411yescharacteristic
268C2×C42C22×C411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42#4112yes—
276C2×C42#4112yes—
277C2×C42#4112yes—
278C2×C42#4112yes—
279C2×C42C2411yescharacteristic
280C4⋊C8—22no—
281C4⋊C8—22no—
282C4⋊C8—22no—
283C4⋊C8—22no—
284C4⋊C8—42no—
285C22⋊C8—44no—
286C22⋊C8—44no—
287C22⋊C8—42no—
288C22⋊C8—42no—
289C22⋊C8—42no—
290C22⋊C8—44no—
291C22⋊C8—44no—
292#97—42no—
293#97—42no—
294C4×C8—22no—
295C4×C8—22no—
296C4×C8—22no—
297C4×C8—22no—
298C4×C8—42no—
299#95—22no—
300#95—22no—
301#95—22no—
302#95—22no—
303#95—22no—
304#95—22no—
305#95—21no—
306#95—21no—
307#95—21no—
308#95—21no—
309C24C2×C22⋊C411yescharacteristic
310C24C2×C22⋊C411yescharacteristic
311C24C2×C22⋊C411yescharacteristic
312#41—216no—
313#41—216no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—22no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—44no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4—82no—
362C22×C4—82no—
363C22×C4—82no—
364C22×C4C2×C22⋊C411yescharacteristic
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4C2×C22⋊C411yescharacteristic
370C22×C4C2×C22⋊C411yescharacteristic
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C2×C22⋊C411yescharacteristic
373C22×C4#11712yes—
374C22×C4#11712yes—
375C22×C4#12312yes—
376C22×C4#12312yes—
377C22×C4#12312yes—
378C22×C4#12312yes—
379C22×C4AGL(1,ℤ/8)14yes—
380C22×C4#13714yes—
381C22×C4C23×C411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384M16—48no—
385C2×C8—24no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—44no—
411C4⋊C4—44no—
412C4⋊C4—44no—
413C4⋊C4—44no—
414C4⋊C4—44no—
415C4⋊C4—44no—
416C4⋊C4—82no—
417C4⋊C4—82no—
418C4⋊C4—82no—
419C4⋊C4#13114yes—
420C22⋊C4—22no—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C22⋊C4—22no—
424C22⋊C4—22no—
425C22⋊C4—22no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C42—21no—
430C42—21no—
431C42—21no—
432C42—21no—
433C42—21no—
434C42—21no—
435C42—21no—
436C42—21no—
437C42—21no—
438C42—21no—
439C42—21no—
440C42—21no—
441C42—44no—
442C42—44no—
443C42—42no—
444C42—42no—
445C42—42no—
446C42—42no—
447C42—44no—
448C42#11712yes—
449C42#11712yes—
450C42#12312yes—
451C42#12312yes—
452C42#12312yes—
453C42#12312yes—
454C42C2×#4111yescharacteristic
455C42C2×#4111yescharacteristic
456C42C2×#4111yescharacteristic
457C42C2×#4111yescharacteristic
458C23—22no—
459C23—22no—
460C23—22no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—21no—
469C23—21no—
470C23#38112yes—
471C23#41812yes—
472C23#41912yes—
473C23C22×C22⋊C411yescharacteristic
474Q8—28no—
475D4—28no—
476D4—28no—
477D4—28no—
478C2×C4—22no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—88no—
529C2×C4—88no—
530C2×C4—88no—
531C2×C4#38112yes—
532C2×C4#38112yes—
533C2×C4#38112yes—
534C2×C4#41812yes—
535C2×C4#41912yes—
536C2×C4C2×#11711yescharacteristic
537C2×C4C2×#11711yescharacteristic
538C2×C4C2×#12311yescharacteristic
539C2×C4C2×#12311yescharacteristic
540C2×C4C2×#12311yescharacteristic
541C2×C4C2×#12311yescharacteristic
542C2×C4C2×#13112yes—
543C2×C4#56812yes—
544C2×C4C2×AGL(1,ℤ/8)12yes—
545C2×C4C2×#13712yes—
546C8—44no—
547C8—82no—
548C8—82no—
549C8—82no—
550C22—24no—
551C22—24no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22C2×#38111yescharacteristic
566C22#288012yes—
567C22C2×#41811yescharacteristic
568C22C2×#41911yescharacteristic
569C22#290112yes—
570C4—22no—
571C4—22no—
572C4—22no—
573C4—22no—
574C4—21no—
575C4—21no—
576C4—22no—
577C4—44no—
578C4—41no—
579C4—41no—
580C4—41no—
581C4—41no—
582C4—82no—
583C4—82no—
584C4—82no—
585C4#292211yescharacteristic
586C4#292311yescharacteristic
587C4#292512yes—
588C4#309212yes—
589C4#309212yes—
590C2—42no—
591C2—42no—
592C2—42no—
593C2#1751112yes—
594C2#1751412yes—
595C2#2068411yescharacteristic
596C2#2070011yescharacteristic
597C2#2070611yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8710C2semidirect
5. #8709C2semidirect
6. #8707C2semidirect
7. #8706C2semidirect
8. #7904C2semidirect
9. #7900C2semidirect
10. #7896C2semidirect
11. #7892C2semidirect

Maximal subgroups

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

subgroupindex
C4⋊C4×#412
#213742
#207212
#87102
#87092
#87072
#87062
#79042
#79002
#78962
#78922