Finite Group Databasebeta

← #196289

Subgroups — #196289

Order 512.

1263 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60248C211yesmaximal, characteristic, complemented
2C4×#536C211yesmaximal, characteristic
3#17450C212yesmaximal
4#17365C211yesmaximal, characteristic, complemented
5#9106C212yesmaximal, complemented
6#9104C212yesmaximal, complemented
7#9102C212yesmaximal, complemented
8#9100C212yesmaximal, complemented
9#7856C212yesmaximal, complemented
10C2×#536C411yescharacteristic
11C2×C4×#41C2211yescharacteristic
12C22×C4×Q8C2211yescharacteristic
13#2984—24no—
14#2983—22no—
15#2983C2212yes—
16#2982C2212yes—
17Q8×M16—24no—
18#2920—22no—
19#2920C2212yes—
20#2920C2212yes—
21#2327C2211yescharacteristic
22#2327C2211yescharacteristic
23#2326C2211yescharacteristic
24#2326C2211yescharacteristic
25#2323—22no—
26#2322—22no—
27#2317C411yescharacteristic
28#2311C411yescharacteristic
29#2310C411yescharacteristic
30C4×C4⋊Q8C411yescharacteristic
31C4×C4⋊Q8C411yescharacteristic
32C4×#124C411yescharacteristic
33C4×#124C411yescharacteristic
34C4×C22⋊Q8—22no—
35C4×#123—22no—
36C4×#117C2211yescharacteristic
37#1489—24no—
38#1488—24no—
39#1479C2214yes—
40#1478C2214yes—
41#1467—22no—
42#1463—22no—
43#1461C2212yes—
44#1461C2212yes—
45#1274C2214yes—
46#1274C2214yes—
47C22×#41C2×C411yescharacteristic
48C23×Q8C2×C411yescharacteristic
49#536—24no—
50#536—24no—
51C2×C4⋊Q8C2×C411yescharacteristic
52C2×C4⋊Q8C2×C411yescharacteristic
53C2×#124C2×C411yescharacteristic
54C2×#124C2×C411yescharacteristic
55C2×#123—22no—
56C2×C22⋊Q8—22no—
57C4×#41—22no—
58C4×#41—22no—
59C4×#41—22no—
60C4×#41—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8C2×C411yescharacteristic
66C2×C4×Q8C2×C411yescharacteristic
67C2×C4×Q8C2×C411yescharacteristic
68C2×C4×Q8D412yes—
69C2×C4×Q8D412yes—
70C2×C4×Q8C2311yescharacteristic
71C2×C4×Q8C2311yescharacteristic
72C2×C4×Q8C2311yescharacteristic
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4C2×C411yescharacteristic
76C2×C4×D4C2311yescharacteristic
77C2×#117—21no—
78C2×#117C2×C411yescharacteristic
79C2×#117C2×C411yescharacteristic
80C2×#117C2×C411yescharacteristic
81C22×C4⋊C4C2×C411yescharacteristic
82C22×C4⋊C4D412yes—
83C22×C42—21no—
84C22×C42D412yes—
85C22×C42C2311yescharacteristic
86#446—216no—
87#446—28no—
88C8×Q8—28no—
89#432—24no—
90#432—24no—
91#432—22no—
92#432—22no—
93#432—22no—
94#432—22no—
95#431—24no—
96#431—24no—
97#431—22no—
98#431D414yes—
99#431C2314yes—
100#431C2314yes—
101#423—24no—
102#423—22no—
103#423—22no—
104#407—22no—
105#407—22no—
106C4×M16—24no—
107C4×M16—22no—
108C4×M16D414yes—
109#391—22no—
110#391C2×C411yescharacteristic
111#391C2×C411yescharacteristic
112#391C2×C411yescharacteristic
113#391C2×C411yescharacteristic
114#391C2×C411yescharacteristic
115#391C2×C411yescharacteristic
116#390C2×C411yescharacteristic
117#390C2×C411yescharacteristic
118#389—22no—
119#389—22no—
120#389—22no—
121#388—22no—
122#388—22no—
123#388—22no—
124#388—22no—
125#387—22no—
126#387—22no—
127#387—22no—
128#387—22no—
129#386—22no—
130#385—22no—
131#385—22no—
132#385—22no—
133#384C2×C411yescharacteristic
134#384C2×C411yescharacteristic
135#382C2×C411yescharacteristic
136#382C2×C411yescharacteristic
137#381—22no—
138#380—22no—
139C4×C4⋊C4—22no—
140C4×C4⋊C4—22no—
141C4×C4⋊C4—21no—
142C4×C4⋊C4—21no—
143C4×C22⋊C4—21no—
144C4×C22⋊C4—21no—
145C4×C22⋊C4—22no—
146C4×C22⋊C4—22no—
147#376C2311yescharacteristic
148#376C2311yescharacteristic
149C43C2×C411yescharacteristic
150C43C2×C411yescharacteristic
151Q8⋊C8—28no—
152D4⋊C8—28no—
153#324—24no—
154#324—24no—
155C2×#41—22no—
156C2×#41—22no—
157C2×#41—22no—
158C2×#41—22no—
159C2×#41—42no—
160C22×Q8—22no—
161C22×Q8—22no—
162C22×Q8—22no—
163C22×Q8—22no—
164C22×Q8—42no—
165C22×Q8C22⋊C412yes—
166C22×Q8C22×C411yescharacteristic
167C22×Q8C22×C411yescharacteristic
168C22×Q8C22×C411yescharacteristic
169C22×D4—21no—
170C22×D4C22×C411yescharacteristic
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4C22⋊C412yes—
175C23×C4C22⋊C412yes—
176C23×C4C22×C411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C2×M16—24no—
180C2×M16—42no—
181C4⋊Q8—22no—
182C4⋊Q8—22no—
183C4⋊Q8—22no—
184C4⋊Q8M1614yescomplemented
185#124—22no—
186#124—22no—
187#124—22no—
188#124M1614yescomplemented
189#123—216no—
190C22⋊Q8—216no—
191C4×Q8—22no—
192C4×Q8—22no—
193C4×Q8—22no—
194C4×Q8—22no—
195C4×Q8—24no—
196C4×Q8—24no—
197C4×Q8—24no—
198C4×Q8—24no—
199C4×Q8—22no—
200C4×Q8—22no—
201C4×Q8—22no—
202C4×Q8—22no—
203C4×Q8—22no—
204C4×Q8—22no—
205C4×Q8—22no—
206C4×Q8—22no—
207C4×D4—22no—
208C4×D4—22no—
209C4×D4—22no—
210C4×D4—22no—
211C4×D4—41no—
212C4×D4—41no—
213C4×D4—42no—
214C4×D4—42no—
215C4×D4—41no—
216C4×D4—41no—
217#117—22no—
218#117—22no—
219#117—22no—
220#117—22no—
221#117—42no—
222#117—42no—
223#117—42no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4C22⋊C412yes—
237C2×C4⋊C4C22⋊C412yes—
238C2×C4⋊C4C22⋊C412yes—
239C2×C4⋊C4C22×C411yescharacteristic
240C2×C4⋊C4C22×C411yescharacteristic
241C2×C4⋊C4C22×C411yescharacteristic
242C2×C4⋊C4C22×C411yescharacteristic
243C2×C4⋊C4C2×D412yes—
244C2×C4⋊C4C2×D412yes—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—42no—
266C2×C42—42no—
267C2×C42C22⋊C412yes—
268C2×C42C22⋊C412yes—
269C2×C42C22×C411yescharacteristic
270C2×C42C22×C411yescharacteristic
271C2×C42C22×C411yescharacteristic
272C2×C42C22×C411yescharacteristic
273C2×C42C22×C411yescharacteristic
274C2×C42C2×D412yes—
275C2×C42C2×D412yes—
276C2×C42C2×D412yes—
277C2×C42C2×D412yes—
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42#4112yes—
281C2×C42#4112yes—
282C2×C42C2411yescharacteristic
283C4⋊C8—28no—
284C4⋊C8—28no—
285C4⋊C8—24no—
286C4⋊C8—24no—
287C4⋊C8—24no—
288C4⋊C8—42no—
289C22⋊C8—24no—
290C22⋊C8—24no—
291C22⋊C8—24no—
292C22⋊C8—42no—
293#97—28no—
294#97—24no—
295#97—42no—
296C4×C8—28no—
297C4×C8—24no—
298C4×C8C2×D418yes—
299#95—22no—
300#95—22no—
301#95—22no—
302#95—22no—
303#95—22no—
304#95—22no—
305#95—22no—
306#95—22no—
307#95—21no—
308#95—21no—
309#95—21no—
310#95—21no—
311C24—21no—
312C24C2×C22⋊C411yescharacteristic
313#41—44no—
314#41—44no—
315#41—44no—
316#41—44no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—21no—
320C2×Q8—21no—
321C2×Q8—21no—
322C2×Q8—22no—
323C2×Q8—22no—
324C2×Q8—24no—
325C2×Q8—24no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—42no—
329C2×Q8—44no—
330C2×Q8—44no—
331C2×Q8—44no—
332C2×Q8—44no—
333C2×Q8C2×M1612yes—
334C2×Q8C2×M1612yes—
335C2×Q8C2×M1612yes—
336C2×D4—24no—
337C2×D4—22no—
338C2×D4—21no—
339C2×D4—22no—
340C2×D4—42no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4C2×M1612yes—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—44no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—81no—
369C22×C4—82no—
370C22×C4—82no—
371C22×C4C2×C22⋊C412yes—
372C22×C4C2×C22⋊C411yescharacteristic
373C22×C4C2×C22⋊C411yescharacteristic
374C22×C4C2×C22⋊C412yes—
375C22×C4C2×C22⋊C412yes—
376C22×C4C2×C22⋊C412yes—
377C22×C4C2×C22⋊C411yescharacteristic
378C22×C4C4×D412yes—
379C22×C4C4×D412yes—
380C22×C4#12011yescharacteristic
381C22×C4#12011yescharacteristic
382C22×C4#12112yes—
383C22×C4#12112yes—
384C22×C4#12311yescharacteristic
385C22×C4#12311yescharacteristic
386C22×C4C23×C411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D411yescharacteristic
389M16—48no—
390M16—84no—
391C2×C8—28no—
392C2×C8—44no—
393C4⋊C4—24no—
394C4⋊C4—28no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—28no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—42no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—42no—
409C4⋊C4—82no—
410C4⋊C4—82no—
411C22⋊C4—24no—
412C22⋊C4—28no—
413C22⋊C4—28no—
414C22⋊C4—42no—
415C22⋊C4—41no—
416C22⋊C4—41no—
417C22⋊C4—41no—
418C22⋊C4—41no—
419C22⋊C4—42no—
420C22⋊C4—82no—
421C22⋊C4—82no—
422C42—22no—
423C42—22no—
424C42—22no—
425C42—22no—
426C42—21no—
427C42—21no—
428C42—22no—
429C42—22no—
430C42—22no—
431C42—22no—
432C42—22no—
433C42—22no—
434C42—42no—
435C42—42no—
436C42—42no—
437C42—42no—
438C42—41no—
439C42—41no—
440C42—81no—
441C42—81no—
442C42C4×D412yes—
443C42C4×D412yes—
444C42#12012yes—
445C42#12112yes—
446C42#12112yes—
447C42#12312yes—
448C42C2×M1612yes—
449C42C2×M1612yes—
450C42C22×D412yes—
451C42C2×#4112yes—
452C23—21no—
453C23—21no—
454C23—22no—
455C23—41no—
456C23—41no—
457C23—42no—
458C23—42no—
459C23—41no—
460C23—41no—
461C23#38611yescharacteristic
462C23#38611yescharacteristic
463C23C22×C22⋊C411yescharacteristic
464Q8—24no—
465Q8—24no—
466Q8—24no—
467Q8—48no—
468Q8—42no—
469Q8—42no—
470Q8—42no—
471D4—24no—
472D4—44no—
473D4—44no—
474D4—42no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—48no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—44no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—82no—
524C2×C4—84no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4#38611yescharacteristic
528C2×C4#38612yes—
529C2×C4#38612yes—
530C2×C4#38611yescharacteristic
531C2×C4#40714yes—
532C2×C4C2×C4×D412yes—
533C2×C4C2×#12011yescharacteristic
534C2×C4C2×#12112yes—
535C2×C4C2×#12311yescharacteristic
536C2×C4C22×M1611yescharacteristic
537C2×C4C22×M1611yescharacteristic
538C2×C4#56812yes—
539C8—48no—
540C8—84no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—21no—
549C22—42no—
550C22—42no—
551C22—44no—
552C22—42no—
553C22—82no—
554C22—81no—
555C22—81no—
556C22C2×#38611yescharacteristic
557C22C2×#40711yescharacteristic
558C22#287911yescharacteristic
559C4—22no—
560C4—22no—
561C4—21no—
562C4—21no—
563C4—21no—
564C4—22no—
565C4—24no—
566C4—42no—
567C4—42no—
568C4—42no—
569C4—42no—
570C4—42no—
571C4—42no—
572C4—81no—
573C4—81no—
574C4—81no—
575C4—81no—
576C4#191212yes—
577C4D4×M1612yes—
578C4D4×M1612yes—
579C2—21no—
580C2—21no—
581C2—41no—
582C2—41no—
583C2—82no—
584C2#2089811yescharacteristic
585C2#2100911yescharacteristic
586C2C2×#191211yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #60248C2semidirect
4. #17365C2semidirect
5. #9106C2semidirect
6. #9104C2semidirect
7. #9102C2semidirect
8. #9100C2semidirect
9. #7856C2semidirect
184. C4⋊Q8M16semidirect
188. #124M16semidirect

Maximal subgroups

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

subgroupindex
#602482
C4×#5362
#174502
#173652
#91062
#91042
#91022
#91002
#78562