Finite Group Databasebeta

← #196400

Subgroups — #196400

Order 512.

1532 conjugacy classes of subgroups.

The subgroup lattice has 619 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#431C211yesmaximal, characteristic
2#36625C211yesmaximal, characteristic
3C2×#1518C212yesmaximal, complemented
4#14682C211yesmaximal, characteristic
5#9112C214yesmaximal, complemented
6#9111C214yesmaximal
7#8026C212yesmaximal, complemented
8C2×#533C2211yescharacteristic
9C23×C42C2211yescharacteristic
10C2×#431—21no—
11C2×#431—21no—
12C2×#431—28no—
13C2×#431C2212yes—
14C2×#431C2212yes—
15C22×C4⋊C8—21no—
16C22×C22⋊C8C2212yes—
17C22×C4×C8—21no—
18#2857C411yescharacteristic, complemented
19#2857C411yescharacteristic, complemented
20#2857C411yescharacteristic, complemented
21#2857C411yescharacteristic, complemented
22#2855—21no—
23#2855C2212yes—
24#2851—21no—
25#2851—21no—
26#2848—21no—
27#2848C2212yes—
28C4×C22⋊Q8—21no—
29C4×C22⋊Q8—21no—
30C4×C22⋊Q8C412yescomplemented
31C4×C22⋊Q8C412yescomplemented
32#2296—21no—
33#2285C2211yescharacteristic
34C2×#382C2212yes—
35#2280—21no—
36C2×#380—21no—
37#1518—28no—
38C2×#328C2218yes—
39C4⋊#95—21no—
40#1434—21no—
41#1434C2212yes—
42C22⋊#95—21no—
43C22⋊#95C2212yes—
44#1294C2218yes—
45C24×C4D411yescharacteristic
46C24×C4D411yescharacteristic
47C24×C4C2311yescharacteristic
48C23×C8—22no—
49#533—24no—
50#533—24no—
51C2×C4⋊Q8C2×C411yescharacteristic
52C2×C4⋊Q8C2×C411yescharacteristic
53C2×#125—21no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8C2×C412yes—
58C2×C22⋊Q8C2×C412yes—
59C2×C22⋊Q8C2312yes—
60C2×C4×Q8—21no—
61C2×C4×Q8C2×C412yes—
62C2×#117—21no—
63C2×#117—21no—
64C22×C4⋊C4—22no—
65C22×C22⋊C4—22no—
66C22×C42—24no—
67C22×C42—24no—
68C22×C42C2×C411yescharacteristic
69C22×C42C2×C411yescharacteristic
70C22×C42C2×C411yescharacteristic
71C22×C42C2×C411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80#431—28no—
81#431—28no—
82#431—24no—
83#431—28no—
84#431D418yes—
85C2×C4⋊C8—28no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—28no—
90C2×C22⋊C8C2314yes—
91C2×C4×C8—28no—
92C2×C4×C8—22no—
93C2×C4×C8—22no—
94#391—21no—
95#391—21no—
96#391C2×C412yes—
97#391C2×C412yes—
98#389—21no—
99#389—21no—
100#389—21no—
101#389—21no—
102#389—21no—
103#389—21no—
104#389C2×C412yes—
105#389C2×C412yes—
106#387—22no—
107#387—22no—
108#387—22no—
109#387—22no—
110#385—22no—
111#385—22no—
112#385—22no—
113#385—22no—
114#385—22no—
115#385—22no—
116#384C2×C411yescharacteristic
117#384C2×C411yescharacteristic
118#382—21no—
119#382—24no—
120#382—24no—
121#381—22no—
122#380—21no—
123#380—21no—
124#380—21no—
125#380—21no—
126#380—21no—
127#380—21no—
128#380—21no—
129#380—44no—
130#380C2×C412yes—
131#380C2×C412yes—
132#380C2312yes—
133#379—22no—
134C4×C4⋊C4—22no—
135C4×C4⋊C4—22no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4C2×C412yes—
138C4×C22⋊C4—22no—
139C4×C22⋊C4—22no—
140C2×#95—22no—
141C2×#95—22no—
142C2×#95—22no—
143C2×#95—22no—
144C2×#95C2314yes—
145#328—216no—
146C25C2×D411yescharacteristic
147C22×Q8—21no—
148C22×Q8C22×C412yes—
149C23×C4—24no—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—24no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4C22⋊C411yescharacteristic
160C23×C4C22⋊C411yescharacteristic
161C23×C4C22⋊C411yescharacteristic
162C23×C4C22⋊C411yescharacteristic
163C23×C4C22⋊C411yescharacteristic
164C23×C4C22⋊C411yescharacteristic
165C23×C4C22⋊C411yescharacteristic
166C23×C4C22⋊C411yescharacteristic
167C23×C4SD1614yes—
168C23×C4Q1614yes—
169C23×C4C22×C411yescharacteristic
170C23×C4C22×C411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2411yescharacteristic
179C22×C8—28no—
180C22×C8—24no—
181C22×C8—44no—
182C22×C8—44no—
183C4⋊Q8—24no—
184C4⋊Q8—24no—
185#125—42no—
186#125—42no—
187C22⋊Q8—24no—
188C22⋊Q8—24no—
189C22⋊Q8—44no—
190C22⋊Q8—44no—
191C22⋊Q8—44no—
192C22⋊Q8—44no—
193C22⋊Q8—44no—
194C4×Q8—28no—
195C4×Q8—42no—
196C4×Q8—42no—
197#117—44no—
198#117—44no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4C22×C412yes—
216C2×C4⋊C4C22×C412yes—
217C2×C4⋊C4C22×C412yes—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C42—24no—
228C2×C42—24no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—24no—
232C2×C42—24no—
233C2×C42—24no—
234C2×C42—24no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42C22⋊C411yescharacteristic
246C2×C42C22⋊C411yescharacteristic
247C2×C42C22⋊C411yescharacteristic
248C2×C42C22⋊C411yescharacteristic
249C2×C42C22⋊C411yescharacteristic
250C2×C42C22⋊C411yescharacteristic
251C2×C42C22⋊C411yescharacteristic
252C2×C42C22⋊C411yescharacteristic
253C2×C42C22×C411yescharacteristic
254C2×C42C22×C411yescharacteristic
255C2×C42C22×C411yescharacteristic
256C2×C42C22×C411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D414yes—
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42#4114yes—
263C4⋊C8—28no—
264C4⋊C8—28no—
265C22⋊C8—28no—
266C22⋊C8—24no—
267C22⋊C8C2×D418yes—
268C4×C8—28no—
269C4×C8—28no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—22no—
280#95—24no—
281#95—24no—
282#95—44no—
283#95—44no—
284#95—44no—
285#95—44no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—41no—
291C24—41no—
292C24C2×C22⋊C411yescharacteristic
293C24C2×C22⋊C411yescharacteristic
294C24C2×SD1612yes—
295C24C2×Q1612yes—
296C24C22×D411yescharacteristic
297C2×Q8—24no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—42no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—216no—
321C22×C4—216no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
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—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—48no—
356C22×C4—416no—
357C22×C4—48no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4—44no—
362C22×C4—44no—
363C22×C4—44no—
364C22×C4—44no—
365C22×C4#10314yes—
366C22×C4#10314yes—
367C22×C4#10314yes—
368C22×C4#10314yes—
369C22×C4C2×C22⋊C411yescharacteristic
370C22×C4C2×C22⋊C411yescharacteristic
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C2×C22⋊C411yescharacteristic
373C22×C4C2×C22⋊C411yescharacteristic
374C22×C4C2×C22⋊C411yescharacteristic
375C22×C4C2×C22⋊C411yescharacteristic
376C22×C4C2×C22⋊C411yescharacteristic
377C22×C4C2×C22⋊C411yescharacteristic
378C22×C4C2×C22⋊C411yescharacteristic
379C22×C4#12114yes—
380C22×C4#12114yes—
381C22×C4C2×SD1612yes—
382C22×C4C2×SD1612yes—
383C22×C4C2×Q1612yes—
384C22×C4C2×Q1612yes—
385C22×C4#13514yes—
386C22×C4C23×C411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D412yes—
389C22×C4C2×#4112yes—
390C2×C8—28no—
391C2×C8—416no—
392C2×C8—44no—
393C2×C8—44no—
394C4⋊C4—24no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—44no—
400C4⋊C4—44no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—44no—
406C4⋊C4—42no—
407C4⋊C4—44no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—44no—
411C4⋊C4—88no—
412C22⋊C4—44no—
413C22⋊C4—44no—
414C22⋊C4—44no—
415C22⋊C4—44no—
416C22⋊C4—44no—
417C22⋊C4—44no—
418C22⋊C4—88no—
419C42—22no—
420C42—22no—
421C42—21no—
422C42—21no—
423C42—21no—
424C42—21no—
425C42—22no—
426C42—22no—
427C42—22no—
428C42—22no—
429C42—22no—
430C42—22no—
431C42—22no—
432C42—22no—
433C42—44no—
434C42—44no—
435C42C4×D414yes—
436C42C4×D414yes—
437C42#12012yes—
438C42#12012yes—
439C42#12312yes—
440C42#12312yes—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—24no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—21no—
459C23—42no—
460C23—42no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23#41012yes—
466C23C2×#10312yes—
467C23C2×#10312yes—
468C23#42112yes—
469C23C22×C22⋊C411yescharacteristic
470C23C2×#12112yes—
471C23C22×SD1611yescharacteristic
472C23C22×Q1611yescharacteristic
473C23C2×#13512yes—
474Q8—84no—
475Q8—84no—
476C2×C4—24no—
477C2×C4—28no—
478C2×C4—24no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—28no—
498C2×C4—28no—
499C2×C4—28no—
500C2×C4—28no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—84no—
532C2×C4—84no—
533C2×C4—84no—
534C2×C4—84no—
535C2×C4—84no—
536C2×C4—84no—
537C2×C4—84no—
538C2×C4#38612yes—
539C2×C4#38612yes—
540C2×C4#38612yes—
541C2×C4#38612yes—
542C2×C4#41012yes—
543C2×C4C2×#10312yes—
544C2×C4C2×#10312yes—
545C2×C4C2×#10312yes—
546C2×C4C2×#10312yes—
547C2×C4#41612yes—
548C2×C4#41612yes—
549C2×C4#42112yes—
550C2×C4#46518yes—
551C2×C4#46718yes—
552C2×C4C2×C4×D412yes—
553C2×C4C2×#12011yescharacteristic
554C2×C4C2×#12311yescharacteristic
555C8—48no—
556C22—21no—
557C22—21no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—21no—
565C22—21no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—48no—
574C22—48no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22#151814yescomplemented
580C22C2×#38611yescharacteristic
581C22C2×#41011yescharacteristic
582C22C2×#41011yescharacteristic
583C22C22×#10311yescharacteristic
584C22C2×#41611yescharacteristic
585C22C2×#42111yescharacteristic
586C22C2×#42111yescharacteristic
587C22C2×#46512yes—
588C22C2×#46712yes—
589C4—22no—
590C4—22no—
591C4—22no—
592C4—22no—
593C4—22no—
594C4—21no—
595C4—21no—
596C4—22no—
597C4—22no—
598C4—41no—
599C4—41no—
600C4—82no—
601C4—82no—
602C4—82no—
603C4—82no—
604C4—164no—
605C4#189214yes—
606C4#189214yes—
607C2—22no—
608C2—22no—
609C2—24no—
610C2—42no—
611C2C2×#151811yescharacteristic
612C2#1757111yescharacteristic
613C2#1757311yescharacteristic
614C2#1757511yescharacteristic
615C2#2090211yescharacteristic
616C2C2×#189211yescharacteristic
617C2#2098311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1518C2semidirect
5. #9112C2semidirect
7. #8026C2semidirect
18. #2857C4semidirect
19. #2857C4semidirect
20. #2857C4semidirect
21. #2857C4semidirect
30. C4×C22⋊Q8C4semidirect
31. C4×C22⋊Q8C4semidirect
579. C22#1518semidirect

Maximal subgroups

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

subgroupindex
C22×#4312
#366252
C2×#15182
#146822
#91122
#91112
#80262