Finite Group Databasebeta

← #201359

Subgroups — #201359

Order 512.

1232 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37453C211yesmaximal, characteristic
2#34359C211yesmaximal, characteristic
3#21753C211yesmaximal, characteristic
4#17578C211yesmaximal, characteristic
5C2×#1511C211yesmaximal, characteristic
6#9462C214yesmaximal, complemented
7#9460C214yesmaximal, complemented
8#7946C211yesmaximal, characteristic
9#7943C211yesmaximal, characteristic
10C22×C22⋊Q8C2211yescharacteristic
11C22×C22⋊Q8C2211yescharacteristic
12C2×#407C2211yescharacteristic
13C2×#407C2211yescharacteristic
14#2817—22no—
15#2627—22no—
16#2627—22no—
17#2621C414yescomplemented
18#2503C412yescomplemented
19#2503C2212yes—
20#2483—22no—
21#2434C412yescomplemented
22#2434C2212yes—
23#2378—22no—
24#2368—22no—
25#2344—22no—
26#2342—22no—
27#2332—22no—
28#2329—22no—
29#2313C2212yes—
30#2310C2212yes—
31#2280C2211yescharacteristic
32#2280C2211yescharacteristic
33C2×#381C2211yescharacteristic
34#1901—22no—
35#1901—22no—
36#1821C2212yes—
37#1784C2212yes—
38#1520—22no—
39#1511—22no—
40#1511C2214yes—
41#1511C2214yes—
42#1508—22no—
43C2×#328—22no—
44#1284C2214yes—
45#1284C2214yes—
46C23×Q8C2×C411yescharacteristic
47C23×Q8C2311yescharacteristic
48C24×C4C2311yescharacteristic
49C22×M16—21no—
50C22×M16—21no—
51C2×C4⋊Q8C2×C414yescomplemented
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8—21no—
68C2×C22⋊Q8—21no—
69C2×C22⋊Q8—21no—
70C2×C22⋊Q8C2×C412yes—
71C2×C22⋊Q8C2×C412yes—
72C2×C22⋊Q8C2×C412yes—
73C2×C22⋊Q8D412yes—
74C2×C22⋊Q8D412yes—
75C2×C22⋊Q8C2312yes—
76C2×#117—22no—
77C2×#117—21no—
78C2×#117C2×C412yes—
79C22×C4⋊C4—21no—
80C22×C4⋊C4C2×C411yescharacteristic
81C22×C4⋊C4D412yes—
82C22×C4⋊C4C2311yescharacteristic
83C22×C22⋊C4—21no—
84C22×C22⋊C4—21no—
85C22×C22⋊C4D412yes—
86C2×#101—24no—
87#407—21no—
88#407—21no—
89#407—21no—
90#407—21no—
91#407—21no—
92#407—21no—
93#407C2312yes—
94#407C2312yes—
95C2×C22⋊C8—21no—
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99#400—24no—
100#400—24no—
101#398—28no—
102#397—28no—
103#395—24no—
104#395—24no—
105#391—24no—
106#391—24no—
107#391C2×C414yescomplemented
108#391C2×C414yescomplemented
109#389—24no—
110#388—24no—
111#388—24no—
112#387—24no—
113#387—24no—
114#387—24no—
115#387—24no—
116#385—24no—
117#385—24no—
118#385—24no—
119#382C2×C414yescomplemented
120#381—21no—
121#381—22no—
122#381—21no—
123#381—24no—
124#381—21no—
125#381—24no—
126#381D412yes—
127#381C2312yes—
128#381C2312yes—
129#380—22no—
130#380—21no—
131#380—21no—
132#380—21no—
133#380D412yes—
134#379—21no—
135#379—21no—
136C4×C4⋊C4—24no—
137C4×C22⋊C4—24no—
138C4×C22⋊C4—22no—
139C4×C22⋊C4—21no—
140C4×C22⋊C4C2×C412yes—
141C4×C22⋊C4D412yes—
142C4×C22⋊C4C2312yes—
143C2×#95—21no—
144C2×#95D412yes—
145#343—24no—
146#328—28no—
147#328—28no—
148C25C2×D411yescharacteristic
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×Q8—21no—
156C22×Q8—21no—
157C22×Q8—21no—
158C22×Q8—82no—
159C22×Q8—82no—
160C22×Q8C22×C412yes—
161C22×Q8C22×C412yes—
162C22×Q8C2×D412yes—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—22no—
166C23×C4—82no—
167C23×C4C22×C411yescharacteristic
168C23×C4C22×C411yescharacteristic
169C23×C4C2×D412yes—
170C23×C4C2×D412yes—
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4#4111yescharacteristic
179C23×C4#4111yescharacteristic
180C23×C4#4111yescharacteristic
181C23×C4#4111yescharacteristic
182C23×C4C2411yescharacteristic
183C2×M16—22no—
184C2×M16—22no—
185C2×M16—41no—
186C2×M16—41no—
187C2×M16—82no—
188C2×M16—82no—
189C22×C8—41no—
190C22×C8—41no—
191C4⋊Q8—84no—
192C22⋊Q8—22no—
193C22⋊Q8—22no—
194C22⋊Q8—22no—
195C22⋊Q8—22no—
196C22⋊Q8—22no—
197C22⋊Q8—22no—
198C22⋊Q8—22no—
199C22⋊Q8—22no—
200C22⋊Q8—22no—
201C22⋊Q8—22no—
202C22⋊Q8—22no—
203C22⋊Q8—22no—
204C22⋊Q8—22no—
205C22⋊Q8—22no—
206C22⋊Q8—22no—
207C22⋊Q8—22no—
208C22⋊Q8—22no—
209C22⋊Q8—22no—
210C22⋊Q8—22no—
211C22⋊Q8—22no—
212C22⋊Q8—22no—
213C22⋊Q8—22no—
214C22⋊Q8—22no—
215C22⋊Q8—22no—
216C22⋊Q8—41no—
217C22⋊Q8—41no—
218C22⋊Q8—41no—
219C22⋊Q8—41no—
220C22⋊Q8—41no—
221C22⋊Q8—41no—
222C22⋊Q8—41no—
223C22⋊Q8—41no—
224#117—82no—
225#117—84no—
226#117—82no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4C22⋊C414yes—
252C2×C4⋊C4C22⋊C414yes—
253C2×C4⋊C4C22×C412yes—
254C2×C4⋊C4C22×C412yes—
255C2×C4⋊C4C2×D412yes—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—41no—
271C2×C22⋊C4—41no—
272C2×C22⋊C4—41no—
273C2×C22⋊C4—41no—
274C2×C22⋊C4—82no—
275C2×C22⋊C4C22⋊C414yes—
276C2×C22⋊C4C22⋊C414yes—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—22no—
280C2×C42C22×C414yes—
281#101—84no—
282C22⋊C8—22no—
283C22⋊C8—22no—
284C22⋊C8—22no—
285C22⋊C8—22no—
286C22⋊C8—41no—
287C22⋊C8—41no—
288C22⋊C8—41no—
289C22⋊C8—41no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—24no—
294#95—22no—
295#95—28no—
296#95—28no—
297#95—28no—
298#95—22no—
299#95—22no—
300#95—22no—
301#95—22no—
302#95—22no—
303#95C2×D414yes—
304#95C2×D414yes—
305C24—42no—
306C24—42no—
307C24—42no—
308C24C2×C22⋊C411yescharacteristic
309C24C2×C22⋊C411yescharacteristic
310C24#12011yescharacteristic
311C24#12011yescharacteristic
312C24#12311yescharacteristic
313C24#12311yescharacteristic
314C24C22×D411yescharacteristic
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8—22no—
321C2×Q8—44no—
322C2×Q8—44no—
323C2×Q8—44no—
324C2×Q8—41no—
325C2×Q8—41no—
326C2×Q8—82no—
327C2×Q8—81no—
328C2×Q8—82no—
329C2×Q8—81no—
330C2×Q8—82no—
331C2×Q8—82no—
332C2×Q8—82no—
333C2×Q8—82no—
334C2×Q8—82no—
335C2×Q8—82no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—44no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—44no—
353C22×C4—44no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—44no—
359C22×C4—81no—
360C22×C4—81no—
361C22×C4—81no—
362C22×C4—81no—
363C22×C4—81no—
364C22×C4—81no—
365C22×C4—81no—
366C22×C4—81no—
367C22×C4—82no—
368C22×C4—82no—
369C22×C4—82no—
370C22×C4C2×C22⋊C414yes—
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C2×C22⋊C411yescharacteristic
373C22×C4C2×C22⋊C414yes—
374C22×C4C4×D411yescharacteristic
375C22×C4C4×D411yescharacteristic
376C22×C4C4×D411yescharacteristic
377C22×C4C4×D411yescharacteristic
378C22×C4C4×D411yescharacteristic
379C22×C4C4×D411yescharacteristic
380C22×C4C4×D411yescharacteristic
381C22×C4C4×D411yescharacteristic
382C22×C4#12011yescharacteristic
383C22×C4#12011yescharacteristic
384C22×C4#12111yescharacteristic
385C22×C4#12111yescharacteristic
386C22×C4#12111yescharacteristic
387C22×C4#12111yescharacteristic
388C22×C4#12111yescharacteristic
389C22×C4#12111yescharacteristic
390C22×C4#12111yescharacteristic
391C22×C4#12111yescharacteristic
392C22×C4#12311yescharacteristic
393C22×C4#12311yescharacteristic
394C22×C4C23×C411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C2×#4111yescharacteristic
399C22×C4C2×#4111yescharacteristic
400M16—44no—
401M16—44no—
402M16—82no—
403M16—82no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—82no—
407C2×C8—82no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—82no—
425C4⋊C4—88no—
426C4⋊C4—84no—
427C4⋊C4—82no—
428C4⋊C4—84no—
429C22⋊C4—24no—
430C22⋊C4—24no—
431C22⋊C4—24no—
432C22⋊C4—24no—
433C22⋊C4—44no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—42no—
439C22⋊C4—42no—
440C22⋊C4—42no—
441C22⋊C4—42no—
442C22⋊C4—82no—
443C22⋊C4—82no—
444C22⋊C4—82no—
445C42—44no—
446C42—84no—
447C42—82no—
448C42—84no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—88no—
471C23#38611yescharacteristic
472C23#38611yescharacteristic
473C23#38611yescharacteristic
474C23#38611yescharacteristic
475C23C22×C22⋊C411yescharacteristic
476C23C2×#12011yescharacteristic
477C23C2×#12311yescharacteristic
478Q8—82no—
479Q8—82no—
480Q8—82no—
481Q8—82no—
482Q8—82no—
483Q8—82no—
484Q8—82no—
485Q8—82no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—28no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—164no—
541C2×C4#38611yescharacteristic
542C2×C4#38611yescharacteristic
543C2×C4#38611yescharacteristic
544C2×C4#38611yescharacteristic
545C2×C4C2×C4×D411yescharacteristic
546C2×C4C2×C4×D411yescharacteristic
547C2×C4C2×#12111yescharacteristic
548C2×C4C2×#12111yescharacteristic
549C8—82no—
550C8—82no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22—21no—
569C22—44no—
570C22—44no—
571C22—44no—
572C22—82no—
573C22—82no—
574C22—82no—
575C22—82no—
576C22#191212yes—
577C22C2×#38611yescharacteristic
578C22#288312yes—
579C22#302212yes—
580C4—21no—
581C4—21no—
582C4—21no—
583C4—21no—
584C4—21no—
585C4—21no—
586C4—21no—
587C4—21no—
588C4—44no—
589C4—44no—
590C4—84no—
591C4—84no—
592C4—84no—
593C4—82no—
594C4—82no—
595C2—22no—
596C2—22no—
597C2—22no—
598C2—41no—
599C2—41no—
600C2—41no—
601C2—41no—
602C2#2095511yescharacteristic
603C2#2098411yescharacteristic
604C2C2×#191211yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9462C2semidirect
7. #9460C2semidirect
17. #2621C4semidirect
18. #2503C4semidirect
21. #2434C4semidirect
51. C2×C4⋊Q8C2×C4semidirect
107. #391C2×C4semidirect
108. #391C2×C4semidirect
119. #382C2×C4semidirect

Maximal subgroups

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

subgroupindex
#374532
#343592
#217532
#175782
C2×#15112
#94622
#94602
#79462
#79432