Finite Group Databasebeta

← #214717

Subgroups — #214717

Order 512.

949 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36031C211yesmaximal, characteristic
2#20634C211yesmaximal, characteristic
3#17351C211yesmaximal, characteristic
4#9469C212yesmaximal, complemented
5#9383C212yesmaximal, complemented
6#9200C212yesmaximal, complemented
7#9023C212yesmaximal, complemented
8#8761C212yesmaximal, complemented
9#8380C212yesmaximal, complemented
10C2×#554C2211yescharacteristic
11C2×#520C2211yescharacteristic
12#2972C2212yes—
13C2×#410—21no—
14#2628C411yescharacteristic, complemented
15#2627C411yescharacteristic, complemented
16#2627C2211yescharacteristic
17#2581C411yescharacteristic, complemented
18#2568C411yescharacteristic, complemented
19#2564C411yescharacteristic, complemented
20#2564C2211yescharacteristic
21#2487C411yescharacteristic, complemented
22#2487C2211yescharacteristic
23#2453—21no—
24#2450—21no—
25#2450—21no—
26#2442—21no—
27#2438—21no—
28#2438—21no—
29#2396C411yescharacteristic, complemented
30C2×#400C411yescharacteristic, complemented
31C2×#400C2211yescharacteristic
32#2349—21no—
33#2349C2211yescharacteristic
34#2339—21no—
35#2316C2212yes—
36#1824C2214yes—
37#1799—21no—
38#1799—21no—
39#1799C2212yes—
40#1799C2212yes—
41#1740—21no—
42#1520C2214yes—
43#1461—21no—
44#1459—21no—
45#1459C2212yes—
46#1458—21no—
47C2×#324—21no—
48C2×#324C2212yes—
49#1294C2214yes—
50#1289C2214yes—
51C22×#41D411yescharacteristic
52#554—44no—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#125—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8C2×C411yescharacteristic
58C2×C22⋊Q8C2311yescharacteristic
59#520—44no—
60C2×C4×Q8C2×C411yescharacteristic
61C2×C4×Q8C2×C411yescharacteristic
62C2×C4×Q8D411yescharacteristic
63C2×C4×D4—21no—
64C2×C4×D4—21no—
65C2×C4×D4C2×C411yescharacteristic
66C2×C4×D4D411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C2×C4×D4D411yescharacteristic
69C2×C4×D4C2311yescharacteristic
70C2×C4×D4C2311yescharacteristic
71C2×#117—22no—
72C2×#117C2×C411yescharacteristic
73C2×#117D411yescharacteristic
74C2×#117D411yescharacteristic
75C22×C4⋊C4C2×C411yescharacteristic
76C22×C4⋊C4C2×C411yescharacteristic
77C22×C4⋊C4D411yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C8×D4—44no—
81#431—22no—
82#410—44no—
83C2×#99—22no—
84C2×#99—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8C2314yes—
87#400—21no—
88#400—21no—
89#400—21no—
90#400—21no—
91#400C2×C411yescharacteristic
92#400C2×C411yescharacteristic
93#400C2×C411yescharacteristic
94#400C2×C411yescharacteristic
95#400C2×C411yescharacteristic
96#400C2×C411yescharacteristic
97#400C2×C411yescharacteristic
98#400C2×C411yescharacteristic
99#400C2×C411yescharacteristic
100#399C2×C411yescharacteristic
101#399C2×C411yescharacteristic
102#399C2×C411yescharacteristic
103#398—21no—
104#398—21no—
105#398—21no—
106#398—21no—
107#398—21no—
108#396—21no—
109#395—21no—
110#395C2×C411yescharacteristic
111#395C2×C411yescharacteristic
112#395C2×C411yescharacteristic
113#393—21no—
114#393C2×C411yescharacteristic
115#391—21no—
116#391—21no—
117#391—21no—
118#391—21no—
119#389—21no—
120#389—21no—
121#389—21no—
122#389—21no—
123#389—21no—
124#389—44no—
125#387—21no—
126#387—21no—
127#387—21no—
128#387—21no—
129#386—21no—
130#386—21no—
131#386—21no—
132#386—21no—
133#385—21no—
134#385—21no—
135#385—21no—
136#385—21no—
137#385—21no—
138#385—21no—
139#385—21no—
140#385—21no—
141#382—21no—
142#382C2×C411yescharacteristic
143#381—21no—
144#381—21no—
145#380—21no—
146#380—21no—
147#380—21no—
148#380—21no—
149#380—21no—
150#380—22no—
151#380C2×C411yescharacteristic
152#380C2314yes—
153#380C2311yescharacteristic
154C4×C4⋊C4—21no—
155C4×C4⋊C4C2×C411yescharacteristic
156C4×C22⋊C4—21no—
157C2×#95—21no—
158C2×#95—21no—
159C2×#95—21no—
160C2×#95—21no—
161C2×#95C2×C411yescharacteristic
162C2×#95C2311yescharacteristic
163#342—22no—
164#342—22no—
165#340—44no—
166#328—44no—
167#324—22no—
168#324—22no—
169#324—22no—
170#323—22no—
171C2×#41—42no—
172C2×#41—42no—
173C2×#41SD1614yes—
174C2×#41Q1614yes—
175C22×Q8—21no—
176C22×Q8C22⋊C411yescharacteristic
177C22×Q8C22×C411yescharacteristic
178C22×D4C22⋊C411yescharacteristic
179C22×D4C2×D411yescharacteristic
180C22×D4C2×D411yescharacteristic
181C23×C4—21no—
182C23×C4—21no—
183C23×C4—21no—
184C23×C4—21no—
185C23×C4C22⋊C411yescharacteristic
186C23×C4C22⋊C411yescharacteristic
187C23×C4C22⋊C411yescharacteristic
188C23×C4C22⋊C411yescharacteristic
189C23×C4C22×C411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2411yescharacteristic
194C22×C8—24no—
195C22×C8—42no—
196C4⋊Q8—42no—
197#125—44no—
198C22⋊Q8—24no—
199C22⋊Q8—44no—
200C22⋊Q8—44no—
201C22⋊Q8—42no—
202C4×Q8—42no—
203C4×Q8—42no—
204C4×Q8—42no—
205C4×D4—24no—
206C4×D4—24no—
207C4×D4—42no—
208C4×D4—42no—
209C4×D4—42no—
210C4×D4—42no—
211C4×D4—44no—
212C4×D4—44no—
213#117—42no—
214#117—42no—
215#117—42no—
216#117—44no—
217#117—44no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
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⋊C4C2×D411yescharacteristic
253C2×C4⋊C4C2×D411yescharacteristic
254C2×C4⋊C4C2×D411yescharacteristic
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4C22⋊C411yescharacteristic
265C2×C22⋊C4C22⋊C411yescharacteristic
266C2×C22⋊C4C22×C411yescharacteristic
267C2×C22⋊C4C2×D411yescharacteristic
268C2×C22⋊C4C2×D411yescharacteristic
269C2×C22⋊C4C2×D411yescharacteristic
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—42no—
273C2×C42C22⋊C411yescharacteristic
274C2×C42C22⋊C411yescharacteristic
275C2×C42C22⋊C411yescharacteristic
276C2×C42C22⋊C411yescharacteristic
277C2×C42C22×C411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C4⋊C8—42no—
280#99—44no—
281#99—44no—
282C22⋊C8—24no—
283C22⋊C8—24no—
284C22⋊C8—42no—
285C4×C8—42no—
286#95—21no—
287#95—21no—
288#95—21no—
289#95—21no—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—21no—
294#95—21no—
295#95—21no—
296#95—21no—
297#95—21no—
298#95—21no—
299#95—21no—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—21no—
304#95—21no—
305#95—21no—
306#95—21no—
307#95—21no—
308#95—21no—
309#95—21no—
310#95—21no—
311#95—24no—
312#95—42no—
313#95C22×C411yescharacteristic
314#95C22×C411yescharacteristic
315#95C22×C411yescharacteristic
316#95C22×C411yescharacteristic
317C24C2×C22⋊C411yescharacteristic
318C24C2×C22⋊C411yescharacteristic
319C24C22×D411yescharacteristic
320#41—48no—
321#41—48no—
322C2×Q8—22no—
323C2×Q8—41no—
324C2×Q8—41no—
325C2×Q8—42no—
326C2×Q8—81no—
327C2×Q8—81no—
328C2×Q8—81no—
329C2×Q8#10314yes—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4#10314yes—
337C2×D4C2×SD1612yes—
338C2×D4C2×SD1612yes—
339C2×D4C2×Q1612yes—
340C2×D4C2×Q1612yes—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—44no—
365C22×C4—81no—
366C22×C4—81no—
367C22×C4—81no—
368C22×C4—81no—
369C22×C4—81no—
370C22×C4—81no—
371C22×C4—81no—
372C22×C4—81no—
373C22×C4—81no—
374C22×C4—81no—
375C22×C4—81no—
376C22×C4—81no—
377C22×C4—81no—
378C22×C4—81no—
379C22×C4#10314yes—
380C22×C4#10314yes—
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C22⋊C411yescharacteristic
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C2×SD1612yes—
392C22×C4C2×Q1612yes—
393C22×C4C23×C411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4H(5,𝔽2)14yes—
396C2×C8—44no—
397C2×C8—84no—
398C2×C8—162no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—84no—
420C4⋊C4—81no—
421C4⋊C4—81no—
422C4⋊C4—81no—
423C4⋊C4—81no—
424C4⋊C4—81no—
425C4⋊C4—81no—
426C4⋊C4—81no—
427C4⋊C4—81no—
428C4⋊C4—81no—
429C4⋊C4—84no—
430C4⋊C4—162no—
431C22⋊C4—22no—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—42no—
438C22⋊C4—42no—
439C22⋊C4—42no—
440C22⋊C4—42no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C22⋊C4—81no—
444C22⋊C4—81no—
445C22⋊C4—81no—
446C22⋊C4—81no—
447C42—22no—
448C42—22no—
449C42—41no—
450C42—41no—
451C42—41no—
452C42—41no—
453C42—42no—
454C42—42no—
455C42—84no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23C2×#10312yes—
467C23C2×#10312yes—
468C23C22×C22⋊C411yescharacteristic
469C23#53412yes—
470C23C22×SD1611yescharacteristic
471C23C22×Q1611yescharacteristic
472C23#57512yes—
473Q8—44no—
474Q8—82no—
475Q8—162no—
476D4—24no—
477D4—24no—
478D4—24no—
479D4—24no—
480D4—44no—
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—24no—
490C2×C4—24no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—44no—
498C2×C4—41no—
499C2×C4—44no—
500C2×C4—44no—
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—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—84no—
536C2×C4—84no—
537C2×C4—84no—
538C2×C4—161no—
539C2×C4—161no—
540C2×C4—162no—
541C2×C4—162no—
542C2×C4—162no—
543C2×C4—162no—
544C2×C4C2×#10312yes—
545C2×C4C2×#10312yes—
546C2×C4C2×#10312yes—
547C2×C4C2×#10312yes—
548C2×C4#51812yes—
549C2×C4#53412yes—
550C8—84no—
551C22—21no—
552C22—22no—
553C22—21no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—42no—
558C22—41no—
559C22—41no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—81no—
568C22#230811yescharacteristic
569C22C22×#10311yescharacteristic
570C22#289611yescharacteristic
571C22#318712yes—
572C22#318912yes—
573C4—22no—
574C4—22no—
575C4—22no—
576C4—22no—
577C4—22no—
578C4—22no—
579C4—41no—
580C4—41no—
581C4—41no—
582C4—41no—
583C4—41no—
584C4—41no—
585C4—81no—
586C4—81no—
587C4—81no—
588C4—81no—
589C4—81no—
590C4—81no—
591C4—81no—
592C4—81no—
593C4—164no—
594C2—21no—
595C2—22no—
596C2—21no—
597C2—22no—
598C2—42no—
599C2—42no—
600C2#2175911yescharacteristic
601C2#2177711yescharacteristic
602C2#2178511yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9469C2semidirect
5. #9383C2semidirect
6. #9200C2semidirect
7. #9023C2semidirect
8. #8761C2semidirect
9. #8380C2semidirect
14. #2628C4semidirect
15. #2627C4semidirect
17. #2581C4semidirect
18. #2568C4semidirect
19. #2564C4semidirect
21. #2487C4semidirect
29. #2396C4semidirect
30. C2×#400C4semidirect

Maximal subgroups

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

subgroupindex
#360312
#206342
#173512
#94692
#93832
#92002
#90232
#87612
#83802