Finite Group Databasebeta

← #214335

Subgroups — #214335

Order 512.

1014 conjugacy classes of subgroups.

The subgroup lattice has 613 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35248C211yesmaximal, characteristic
2#21820C211yesmaximal, characteristic
3#13786C211yesmaximal, characteristic
4#9145C212yesmaximal
5#9144C212yesmaximal
6#9116C212yesmaximal
7#9112C212yesmaximal
8#8714C212yesmaximal
9#8333C212yesmaximal
10C2×#533C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C2×#432C2211yescharacteristic
13C2×#431C2211yescharacteristic
14#2855C2212yes—
15#2855C2212yes—
16#2634—21no—
17#2634—21no—
18#2581C412yescomplemented
19#2581C412yescomplemented
20#2568C411yescharacteristic, complemented
21#2568C411yescharacteristic, complemented
22#2546C411yescharacteristic, complemented
23#2546C411yescharacteristic, complemented
24#2503—21no—
25#2484—21no—
26#2484—21no—
27#2434—21no—
28#2327—21no—
29#2327—21no—
30C2×#391—21no—
31#2283C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×#376C2211yescharacteristic
34#1985—21no—
35#1985—21no—
36#1984—21no—
37#1984—21no—
38#1817C2212yes—
39#1817C2212yes—
40C2×#328C2212yes—
41C2×#328C2212yes—
42C2×#328C2212yes—
43C2×#328C2212yes—
44#1436—21no—
45#1436—21no—
46C4⋊#95—21no—
47#1434C2212yes—
48#1434C2212yes—
49#1294C2212yes—
50#1294C2212yes—
51#1294C2212yes—
52#1294C2212yes—
53C23×Q8D411yescharacteristic
54#533—41no—
55#533—41no—
56#533—41no—
57#533—41no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8C2×C411yescharacteristic
61C2×C4⋊Q8C2×C411yescharacteristic
62C2×#125—21no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8C2312yes—
65C2×C4×Q8—21no—
66C2×C4×Q8—41no—
67C2×C4×Q8—41no—
68C2×C4×Q8—41no—
69C2×C4×Q8—41no—
70C2×C4×Q8C2×C412yes—
71C2×C4×Q8C2×C411yescharacteristic
72C2×C4×Q8C2×C411yescharacteristic
73C2×C4×Q8C2×C411yescharacteristic
74C2×C4×Q8C2×C411yescharacteristic
75C2×C4×Q8D411yescharacteristic
76C2×C4×Q8D411yescharacteristic
77C2×C4×Q8D411yescharacteristic
78C2×C4×Q8D411yescharacteristic
79C2×#117—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4D411yescharacteristic
82C22×C4⋊C4D411yescharacteristic
83C22×C4⋊C4D411yescharacteristic
84C22×C42—21no—
85C22×C42—21no—
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89#432—41no—
90#432—41no—
91#431—41no—
92#431—41no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C22⋊C8C2312yes—
96C2×C22⋊C8C2312yes—
97C2×#97—21no—
98C2×C4×C8—21no—
99#400—21no—
100#400—21no—
101#400—21no—
102#400—21no—
103#400—22no—
104#400—22no—
105#398—21no—
106#398—21no—
107#398—21no—
108#398—21no—
109#395—22no—
110#395—22no—
111#391—28no—
112#391—21no—
113#391—21no—
114#391—21no—
115#391—21no—
116#391C2×C412yes—
117#391C2×C412yes—
118#391C2×C412yes—
119#391C2×C412yes—
120#389C2×C412yes—
121#389C2×C412yes—
122#389C2×C412yes—
123#389C2×C412yes—
124#387—22no—
125#387—22no—
126#387—22no—
127#387—22no—
128#387—22no—
129#387—22no—
130#387—22no—
131#387—22no—
132#385—22no—
133#385—22no—
134#383—21no—
135#382—21no—
136#382—21no—
137#382C2×C412yes—
138#381—22no—
139#381—22no—
140#381—21no—
141#380C2312yes—
142C4×C4⋊C4—41no—
143C4×C4⋊C4—41no—
144C4×C4⋊C4C2×C411yescharacteristic
145C4×C4⋊C4C2×C411yescharacteristic
146C4×C22⋊C4—21no—
147#376—21no—
148#376—41no—
149#376—41no—
150C2×#95—22no—
151C2×#95—22no—
152C2×#95—21no—
153C2×#95—22no—
154C2×#95—22no—
155C2×#95C2312yes—
156C2×#95C2312yes—
157#336—22no—
158#336—22no—
159#336—22no—
160#336—22no—
161#328—42no—
162#328—42no—
163#328—42no—
164#328—42no—
165C22×Q8—24no—
166C22×Q8—21no—
167C22×Q8—41no—
168C22×Q8—41no—
169C22×Q8—42no—
170C22×Q8C22⋊C411yescharacteristic
171C22×Q8C22⋊C411yescharacteristic
172C22×Q8C22⋊C411yescharacteristic
173C22×Q8C22⋊C411yescharacteristic
174C22×Q8C22×C412yes—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2411yescharacteristic
184C22×C8—22no—
185C22×C8—22no—
186C4⋊Q8—41no—
187C4⋊Q8—41no—
188C4⋊Q8—41no—
189C4⋊Q8—41no—
190C4⋊Q8—44no—
191C4⋊Q8—44no—
192#125—42no—
193#125—42no—
194C22⋊Q8—22no—
195C22⋊Q8—22no—
196C22⋊Q8—22no—
197C22⋊Q8—22no—
198C22⋊Q8—42no—
199C22⋊Q8—42no—
200C4×Q8—41no—
201C4×Q8—41no—
202C4×Q8—41no—
203C4×Q8—41no—
204C4×Q8—41no—
205C4×Q8—41no—
206C4×Q8—41no—
207C4×Q8—41no—
208C4×Q8—41no—
209C4×Q8—41no—
210C4×Q8—41no—
211C4×Q8—41no—
212C4×Q8—41no—
213C4×Q8—41no—
214C4×Q8—41no—
215C4×Q8—41no—
216C4×Q8—42no—
217C4×Q8—42no—
218C4×Q8—42no—
219C4×Q8—42no—
220#117—42no—
221#117—42no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4C22⋊C411yescharacteristic
241C2×C4⋊C4C22⋊C411yescharacteristic
242C2×C4⋊C4C22⋊C411yescharacteristic
243C2×C4⋊C4C22⋊C411yescharacteristic
244C2×C4⋊C4C22⋊C411yescharacteristic
245C2×C4⋊C4C22⋊C411yescharacteristic
246C2×C4⋊C4C22⋊C411yescharacteristic
247C2×C4⋊C4C22⋊C411yescharacteristic
248C2×C4⋊C4C22⋊C411yescharacteristic
249C2×C4⋊C4C22⋊C411yescharacteristic
250C2×C4⋊C4C22⋊C411yescharacteristic
251C2×C4⋊C4C22⋊C411yescharacteristic
252C2×C4⋊C4C22×C412yes—
253C2×C4⋊C4C22×C412yes—
254C2×C4⋊C4C22×C412yes—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—24no—
270C2×C42—22no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—42no—
274C2×C42—42no—
275C2×C42—42no—
276C2×C42—82no—
277C2×C42C22×C411yescharacteristic
278C2×C42C22×C411yescharacteristic
279C2×C42C22×C411yescharacteristic
280C2×C42C22×C411yescharacteristic
281C2×C42C22×C411yescharacteristic
282C2×C42C22×C411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D411yescharacteristic
289C4⋊C8—42no—
290C4⋊C8—42no—
291C22⋊C8—22no—
292C22⋊C8—22no—
293C22⋊C8—22no—
294C22⋊C8—22no—
295#97—42no—
296C4×C8—42no—
297#95—28no—
298#95—28no—
299#95—22no—
300#95—22no—
301#95—21no—
302#95—21no—
303#95—22no—
304#95—22no—
305#95—22no—
306#95—22no—
307#95—22no—
308#95—22no—
309#95—22no—
310#95—22no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95—22no—
315#95—22no—
316#95—22no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—42no—
321#95—82no—
322#95—82no—
323C24C22×D411yescharacteristic
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—41no—
332C2×Q8—41no—
333C2×Q8—41no—
334C2×Q8—41no—
335C2×Q8—41no—
336C2×Q8—41no—
337C2×Q8—48no—
338C2×Q8—48no—
339C2×Q8—42no—
340C2×Q8—42no—
341C2×Q8—44no—
342C2×Q8—42no—
343C2×Q8#10414yes—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4C23×C411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4H(5,𝔽2)11yescharacteristic
402C22×C4H(5,𝔽2)11yescharacteristic
403C22×C4H(5,𝔽2)11yescharacteristic
404C22×C4H(5,𝔽2)11yescharacteristic
405C2×C8—44no—
406C2×C8—44no—
407C2×C8—42no—
408C2×C8—42no—
409C4⋊C4—22no—
410C4⋊C4—22no—
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—42no—
424C4⋊C4—42no—
425C4⋊C4—48no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—44no—
429C4⋊C4—42no—
430C4⋊C4—44no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—44no—
434C4⋊C4—44no—
435C4⋊C4—44no—
436C22⋊C4—42no—
437C22⋊C4—42no—
438C22⋊C4—42no—
439C22⋊C4—42no—
440C42—21no—
441C42—21no—
442C42—21no—
443C42—21no—
444C42—21no—
445C42—21no—
446C42—21no—
447C42—21no—
448C42—21no—
449C42—21no—
450C42—21no—
451C42—21no—
452C42—21no—
453C42—21no—
454C42—21no—
455C42—21no—
456C42—21no—
457C42—21no—
458C42—21no—
459C42—21no—
460C42—21no—
461C42—21no—
462C42—21no—
463C42—21no—
464C42—48no—
465C42—44no—
466C42—42no—
467C42—42no—
468C42—84no—
469C23—22no—
470C23—21no—
471C23—21no—
472C23C22×C22⋊C411yescharacteristic
473C23#53411yescharacteristic
474C23#53411yescharacteristic
475C23#57712yes—
476C23#57812yes—
477Q8—44no—
478Q8—44no—
479Q8—44no—
480Q8—44no—
481Q8—82no—
482Q8—82no—
483Q8—84no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4C2×#10412yes—
557C2×C4C2×#10412yes—
558C2×C4C2×#10412yes—
559C2×C4#42112yes—
560C2×C4#51811yescharacteristic
561C2×C4#51811yescharacteristic
562C2×C4#53411yescharacteristic
563C2×C4#53411yescharacteristic
564C8—82no—
565C8—82no—
566C22—21no—
567C22—21no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—21no—
573C22—21no—
574C22#230811yescharacteristic
575C22#288412yes—
576C22#289812yes—
577C22C22×#10411yescharacteristic
578C22C2×#42111yescharacteristic
579C22#320011yescharacteristic
580C22#320112yes—
581C22#320211yescharacteristic
582C4—22no—
583C4—22no—
584C4—22no—
585C4—22no—
586C4—21no—
587C4—21no—
588C4—21no—
589C4—21no—
590C4—41no—
591C4—41no—
592C4—41no—
593C4—41no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—41no—
598C4—84no—
599C4—84no—
600C4—82no—
601C4—82no—
602C4—82no—
603C4—82no—
604C2—21no—
605C2—22no—
606C2—21no—
607C2#1754912yes—
608C2#1760112yes—
609C2#2175111yescharacteristic
610C2#2177111yescharacteristic
611C2#2177311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
18. #2581C4semidirect
19. #2581C4semidirect
20. #2568C4semidirect
21. #2568C4semidirect
22. #2546C4semidirect
23. #2546C4semidirect

Maximal subgroups

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

subgroupindex
#352482
#218202
#137862
#91452
#91442
#91162
#91122
#87142
#83332