Finite Group Databasebeta

← #496547

Subgroups — #496547

Order 512.

1005 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35249C211yesmaximal, characteristic
2#20727C211yesmaximal, characteristic, complemented
3C2×#1801C211yesmaximal, characteristic, complemented
4#13302C212yesmaximal, complemented
5#13226C212yesmaximal, complemented
6#13092C214yesmaximal, complemented
7#12803C214yesmaximal, complemented
8C2×#532C411yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10#2920—21no—
11C2×#423—21no—
12C22×#101C2212yes—
13#2854—21no—
14#2854—21no—
15#2632—21no—
16#2632—21no—
17#2580—21no—
18#2580—21no—
19#2567C411yescharacteristic
20#2567C2211yescharacteristic
21#2548C411yescharacteristic
22#2548C2211yescharacteristic
23#2501C412yes—
24#2436C2212yes—
25#2406—21no—
26#2406—21no—
27#2326C412yes—
28#2326C2212yes—
29#2283C411yescharacteristic
30C2×#355C2218yes—
31#1821—21no—
32#1801—21no—
33#1801—21no—
34#1801—21no—
35#1801C2212yes—
36#1801C2212yes—
37#1801C2212yes—
38#1784C2212yes—
39C4×#101—21no—
40C4×#101C2212yes—
41C2×#343—21no—
42#1394C2218yes—
43C22×#41—21no—
44C23×Q8C2311yescharacteristic
45C22×M16—22no—
46#532—41no—
47#532—41no—
48#532—41no—
49#532—41no—
50C2×C4⋊Q8C2×C411yescharacteristic
51C2×#127C2×C411yescharacteristic
52C2×#124—21no—
53C2×#124C2×C412yes—
54C2×#124C2312yes—
55C2×#123—22no—
56C2×#123—22no—
57C2×#121—21no—
58C2×#121C2×C412yes—
59C2×C4×Q8—41no—
60C2×C4×Q8—41no—
61C2×C4×Q8—41no—
62C2×C4×Q8—41no—
63C2×C4×Q8C2×C411yescharacteristic
64C2×C4×Q8C2×C411yescharacteristic
65C2×C4×Q8D411yescharacteristic
66C2×C4×Q8D411yescharacteristic
67C2×C4×Q8D411yescharacteristic
68C2×C4×Q8D411yescharacteristic
69C2×C4×Q8C2311yescharacteristic
70C2×C4×Q8C2311yescharacteristic
71C2×C4×D4—21no—
72C2×C4×D4C2×C412yes—
73C22×C4⋊C4C2×C411yescharacteristic
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4D411yescharacteristic
76C22×C42C2×C411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79#432—41no—
80#432—41no—
81#431—41no—
82#431—41no—
83#423—22no—
84#423—22no—
85#423—22no—
86#423—41no—
87#423—41no—
88C2×C4⋊C8—41no—
89C2×C4⋊C8—41no—
90C2×#101—22no—
91C2×#101—22no—
92C2×#101—44no—
93C2×#101C2314yes—
94C2×#101C2314yes—
95#407—22no—
96C4×M16—22no—
97#399—22no—
98#399—22no—
99#395—21no—
100#395—21no—
101#395—21no—
102#395—21no—
103#394—21no—
104#394—21no—
105#394—21no—
106#394—21no—
107#393—22no—
108#393—22no—
109#391C2×C412yes—
110#391C2312yes—
111#390—21no—
112#390—21no—
113#390—21no—
114#390—21no—
115#390C2×C412yes—
116#390C2×C412yes—
117#389—21no—
118#389—21no—
119#389—21no—
120#389—21no—
121#388—22no—
122#388—22no—
123#388—22no—
124#388—22no—
125#386—22no—
126#386—22no—
127#385—22no—
128#385—22no—
129#385—22no—
130#385—22no—
131#384C2×C412yes—
132#384C2×C412yes—
133#383—21no—
134#382—21no—
135#381—21no—
136#380—21no—
137C4×C4⋊C4C2×C411yescharacteristic
138C4×C4⋊C4C2×C411yescharacteristic
139C4×C22⋊C4C2×C412yes—
140#376C2×C412yes—
141#355—48no—
142#343—22no—
143#343—22no—
144#343—22no—
145#343—22no—
146C2×#41—42no—
147C2×#41—42no—
148C2×#41—84no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—21no—
152C22×Q8—41no—
153C22×Q8—41no—
154C22×Q8—42no—
155C22×Q8C22×C411yescharacteristic
156C22×Q8C2×D411yescharacteristic
157C22×Q8C2×D411yescharacteristic
158C22×Q8C2411yescharacteristic
159C22×D4—22no—
160C22×D4—21no—
161C22×D4—21no—
162C22×D4C22×C412yes—
163C23×C4—22no—
164C23×C4C22⋊C411yescharacteristic
165C23×C4C22⋊C411yescharacteristic
166C23×C4C22⋊C411yescharacteristic
167C23×C4C22⋊C411yescharacteristic
168C23×C4C22×C411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C2×M16—24no—
172C2×M16—24no—
173C2×M16—42no—
174C2×M16—44no—
175C2×M16—44no—
176C22×C8—42no—
177C4⋊Q8—41no—
178C4⋊Q8—41no—
179#127—41no—
180#127—41no—
181#124—24no—
182#124—24no—
183#124—42no—
184#124—42no—
185#124—44no—
186#123—84no—
187#123—84no—
188#121—42no—
189#121—42no—
190#121—41no—
191#121—41no—
192#121—41no—
193#121—41no—
194C4×Q8—41no—
195C4×Q8—41no—
196C4×Q8—41no—
197C4×Q8—41no—
198C4×Q8—41no—
199C4×Q8—41no—
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×D4—42no—
211C4×D4—42no—
212C4×D4—42no—
213C4×D4—42no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4C22⋊C411yescharacteristic
228C2×C4⋊C4C22⋊C411yescharacteristic
229C2×C4⋊C4C22⋊C411yescharacteristic
230C2×C4⋊C4C22⋊C411yescharacteristic
231C2×C4⋊C4C22⋊C411yescharacteristic
232C2×C4⋊C4C22⋊C411yescharacteristic
233C2×C4⋊C4C22×C412yes—
234C2×C4⋊C4C22×C411yescharacteristic
235C2×C4⋊C4C22×C411yescharacteristic
236C2×C4⋊C4C2×D411yescharacteristic
237C2×C4⋊C4C2×D411yescharacteristic
238C2×C4⋊C4C2×D411yescharacteristic
239C2×C4⋊C4C2×D411yescharacteristic
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42C22⋊C411yescharacteristic
264C2×C42C22⋊C411yescharacteristic
265C2×C42C22⋊C411yescharacteristic
266C2×C42C22⋊C411yescharacteristic
267C2×C42C22⋊C411yescharacteristic
268C2×C42C22⋊C411yescharacteristic
269C2×C42C22×C412yes—
270C2×C42C22×C411yescharacteristic
271C2×C42C22×C411yescharacteristic
272C2×C42C22×C412yes—
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C4⋊C8—42no—
278C4⋊C8—42no—
279C4⋊C8—42no—
280C4⋊C8—42no—
281C4⋊C8—42no—
282C4⋊C8—42no—
283#101—24no—
284#101—28no—
285#101—24no—
286#101—44no—
287#101—44no—
288C22⋊C8—42no—
289C22⋊C8—42no—
290#97—42no—
291C4×C8—42no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—21no—
296#95—21no—
297#95—22no—
298#95—22no—
299#95—22no—
300#95—22no—
301C24—22no—
302C24C2×C22⋊C411yescharacteristic
303#41—88no—
304#41—88no—
305C2×Q8—22no—
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—41no—
312C2×Q8—41no—
313C2×Q8—41no—
314C2×Q8—41no—
315C2×Q8—41no—
316C2×Q8—41no—
317C2×Q8—44no—
318C2×Q8—44no—
319C2×Q8—44no—
320C2×Q8—44no—
321C2×Q8—42no—
322C2×Q8—82no—
323C2×Q8H(5,𝔽2)12yes—
324C2×Q82−1+412yes—
325C2×D4—24no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4—44no—
332C2×D4—44no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—82no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—42no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—44no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—82no—
366C22×C4—82no—
367C22×C4#9911yescharacteristic
368C22×C4#9911yescharacteristic
369C22×C4#9911yescharacteristic
370C22×C4#9911yescharacteristic
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×C4C2×C22⋊C411yescharacteristic
380C22×C4C2×C22⋊C411yescharacteristic
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4C23×C411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385M16—44no—
386M16—44no—
387M16—44no—
388M16—44no—
389M16—84no—
390C2×C8—44no—
391C2×C8—44no—
392C2×C8—84no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—82no—
411C4⋊C4—82no—
412C4⋊C4—82no—
413C4⋊C4—82no—
414C22⋊C4—44no—
415C22⋊C4—44no—
416C22⋊C4—44no—
417C22⋊C4—42no—
418C22⋊C4—42no—
419C22⋊C4—44no—
420C22⋊C4—42no—
421C22⋊C4—42no—
422C22⋊C4—44no—
423C22⋊C4—82no—
424C22⋊C4—82no—
425C22⋊C4—82no—
426C22⋊C4—82no—
427C42—21no—
428C42—21no—
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—21no—
442C42—21no—
443C42—21no—
444C42—21no—
445C42—21no—
446C42—21no—
447C42—21no—
448C42—21no—
449C42—21no—
450C42—21no—
451C42—24no—
452C42—42no—
453C23—21no—
454C23—21no—
455C23—22no—
456C23—41no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23C2×#9911yescharacteristic
465C23C2×#9911yescharacteristic
466C23C22×C22⋊C411yescharacteristic
467Q8—42no—
468Q8—42no—
469Q8—44no—
470Q8—44no—
471Q8—42no—
472Q8—42no—
473Q8—84no—
474D4—84no—
475D4—82no—
476D4—84no—
477D4—84no—
478D4—84no—
479D4—82no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
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—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—88no—
536C2×C4C2×#9911yescharacteristic
537C2×C4C2×#9911yescharacteristic
538C2×C4C2×#9911yescharacteristic
539C2×C4C2×#9911yescharacteristic
540C2×C4#41012yes—
541C2×C4#52012yes—
542C2×C4#53511yescharacteristic
543C2×C4#53611yescharacteristic
544C2×C4#53712yes—
545C8—84no—
546C8—84no—
547C22—21no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—42no—
555C22—42no—
556C22—82no—
557C22—82no—
558C22—82no—
559C22—82no—
560C22—82no—
561C22—82no—
562C22—82no—
563C22#213412yes—
564C22#231111yescharacteristic
565C22C22×#9911yescharacteristic
566C22C2×#41011yescharacteristic
567C4—21no—
568C4—21no—
569C4—21no—
570C4—21no—
571C4—22no—
572C4—21no—
573C4—22no—
574C4—22no—
575C4—41no—
576C4—41no—
577C4—41no—
578C4—41no—
579C4—41no—
580C4—41no—
581C4—41no—
582C4—41no—
583C4—82no—
584C4—82no—
585C4—82no—
586C4#213312yes—
587C2—21no—
588C2—21no—
589C2—22no—
590C2—41no—
591C2—162no—
592C2#2187411yescharacteristic
593C2C2×#213311yescharacteristic
594C2C2×#213411yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
prime radicalC2×#532C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20727C2semidirect
3. C2×#1801C2semidirect
4. #13302C2semidirect
5. #13226C2semidirect
6. #13092C2semidirect
7. #12803C2semidirect

Maximal subgroups

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

subgroupindex
#352492
#207272
C2×#18012
#133022
#132262
#130922
#128032