Finite Group Databasebeta

← #216346

Subgroups — #216346

Order 512.

899 conjugacy classes of subgroups.

The subgroup lattice has 589 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36638C211yesmaximal, characteristic
2#20808C211yesmaximal, characteristic, complemented
3#20808C211yesmaximal, characteristic, complemented
4#9866C212yesmaximal, complemented
5#9675C212yesmaximal, complemented
6#9074C212yesmaximal
7#9073C212yesmaximal, complemented
8#8568C212yesmaximal, complemented
9#8567C212yesmaximal, complemented
10C2×#431—21no—
11C2×#431C2212yes—
12#2866—21no—
13#2865—21no—
14#2864C411yescharacteristic
15#2863C411yescharacteristic
16#2861C411yescharacteristic
17#2860C411yescharacteristic
18#2855C411yescharacteristic
19#2854C411yescharacteristic
20#2853—21no—
21#2852—21no—
22#2851C411yescharacteristic
23#2850C411yescharacteristic
24C4×#125C2211yescharacteristic
25C4×#126—21no—
26C4×#124C2211yescharacteristic
27C4×C22⋊Q8C2211yescharacteristic
28C4×#123—21no—
29C4×#121C2211yescharacteristic
30#2298—21no—
31#2295C2211yescharacteristic
32#2293C2211yescharacteristic
33#2290—21no—
34C2×C43C2211yescharacteristic
35#1842—21no—
36#1842C2212yes—
37#1831—21no—
38#1831—21no—
39#1797—21no—
40#1797—21no—
41#1797—21no—
42#1797C2212yes—
43C4×C4⋊C8—21no—
44C4×C4⋊C8C2212yes—
45C4×C4⋊C8C2212yes—
46C4×C4⋊C8C2212yes—
47C2×Q8⋊C8C2214yes—
48C2×D4⋊C8C2214yes—
49#1279C2214yes—
50#1279C2214yes—
51C2×#126—21no—
52C2×#125C2×C411yescharacteristic
53C2×#124C2×C411yescharacteristic
54C2×#123—21no—
55C2×C22⋊Q8C2×C411yescharacteristic
56C2×#121C2×C411yescharacteristic
57C2×C4×Q8C2311yescharacteristic
58C2×C4×D4—21no—
59C2×C4×D4C2311yescharacteristic
60C2×#117—21no—
61C22×C42C2×C411yescharacteristic
62C22×C42C2×C411yescharacteristic
63C22×C42D411yescharacteristic
64C22×C42D411yescharacteristic
65C22×C42D411yescharacteristic
66C22×C42D411yescharacteristic
67C22×C42C2311yescharacteristic
68#431—44no—
69#431—44no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8C2314yes—
76C2×C22⋊C8—22no—
77C2×C22⋊C8—22no—
78C2×C4×C8—22no—
79C2×C4×C8C2314yes—
80#391C2×C411yescharacteristic
81#391C2×C411yescharacteristic
82#391C2×C411yescharacteristic
83#391C2×C411yescharacteristic
84#390C2×C411yescharacteristic
85#390C2×C411yescharacteristic
86#390C2×C411yescharacteristic
87#390C2×C411yescharacteristic
88#389C2×C411yescharacteristic
89#389C2×C411yescharacteristic
90#389C2×C411yescharacteristic
91#389C2×C411yescharacteristic
92#389C2×C411yescharacteristic
93#389C2×C411yescharacteristic
94#389C2×C411yescharacteristic
95#389C2×C411yescharacteristic
96#388—21no—
97#388—21no—
98#388—21no—
99#388—21no—
100#388—21no—
101#388—21no—
102#388—21no—
103#388—21no—
104#388—21no—
105#388—21no—
106#388—21no—
107#388—21no—
108#388—21no—
109#388—21no—
110#388—21no—
111#388—21no—
112#387—21no—
113#387—21no—
114#387—21no—
115#387—21no—
116#387—21no—
117#387—21no—
118#387—21no—
119#387—21no—
120#387—21no—
121#387—21no—
122#387—21no—
123#387—21no—
124#387—21no—
125#387—21no—
126#387—21no—
127#387—21no—
128#386—21no—
129#386—21no—
130#386—21no—
131#386—21no—
132#385—21no—
133#385—21no—
134#385—21no—
135#385—21no—
136#383—21no—
137#382C2×C411yescharacteristic
138#382C2×C411yescharacteristic
139#381—21no—
140#380C2×C411yescharacteristic
141#380C2×C411yescharacteristic
142C4×C4⋊C4—21no—
143C4×C4⋊C4—21no—
144C4×C4⋊C4—21no—
145C4×C4⋊C4—21no—
146C4×C4⋊C4C2311yescharacteristic
147C4×C4⋊C4C2311yescharacteristic
148C4×C22⋊C4—21no—
149C4×C22⋊C4—21no—
150C4×C22⋊C4—21no—
151C4×C22⋊C4—21no—
152C4×C22⋊C4—21no—
153#376—21no—
154C43C2×C411yescharacteristic
155C43C2×C411yescharacteristic
156C43D411yescharacteristic
157C43D411yescharacteristic
158C43D411yescharacteristic
159C43D411yescharacteristic
160C43C2311yescharacteristic
161C43C2311yescharacteristic
162#336—22no—
163#336—22no—
164#336—22no—
165#336—22no—
166Q8⋊C8—44no—
167D4⋊C8—44no—
168C22×Q8C22×C411yescharacteristic
169C22×D4—21no—
170C22×D4C22×C411yescharacteristic
171C23×C4—21no—
172C23×C4C22⋊C411yescharacteristic
173C23×C4C22⋊C411yescharacteristic
174C23×C4C22⋊C411yescharacteristic
175C23×C4C22⋊C411yescharacteristic
176C23×C4C22×C411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C22×C8—24no—
180C22×C8—24no—
181#126—44no—
182#125—24no—
183#124—24no—
184#123—44no—
185C22⋊Q8—24no—
186#121—24no—
187C4×Q8—22no—
188C4×Q8—22no—
189C4×D4—22no—
190C4×D4—22no—
191C4×D4—44no—
192#117—42no—
193#117—42no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4C22×C411yescharacteristic
207C2×C4⋊C4C22×C411yescharacteristic
208C2×C4⋊C4C22×C411yescharacteristic
209C2×C4⋊C4C22×C411yescharacteristic
210C2×C4⋊C4C22×C411yescharacteristic
211C2×C4⋊C4C22×C411yescharacteristic
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C42—24no—
228C2×C42—24no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—42no—
237C2×C42—42no—
238C2×C42—42no—
239C2×C42C22⋊C411yescharacteristic
240C2×C42C22⋊C411yescharacteristic
241C2×C42C22⋊C411yescharacteristic
242C2×C42C22⋊C411yescharacteristic
243C2×C42C22⋊C411yescharacteristic
244C2×C42C22⋊C411yescharacteristic
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×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42C2411yescharacteristic
267C4⋊C8—24no—
268C4⋊C8—24no—
269C4⋊C8—44no—
270C4⋊C8—44no—
271C4⋊C8—44no—
272C4⋊C8—44no—
273C4⋊C8—44no—
274C22⋊C8—44no—
275C22⋊C8—44no—
276C4×C8—24no—
277C4×C8—24no—
278C4×C8—44no—
279#95—21no—
280#95—21no—
281#95—21no—
282#95—21no—
283#95—21no—
284#95—21no—
285#95—21no—
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—
303C24—21no—
304C24C2×C22⋊C411yescharacteristic
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—22no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—42no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4C2×C22⋊C411yescharacteristic
350C22×C4C2×C22⋊C411yescharacteristic
351C22×C4C2×C22⋊C411yescharacteristic
352C22×C4C2×C22⋊C411yescharacteristic
353C22×C4C2×C22⋊C411yescharacteristic
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4#13114yes—
359C22×C4C23×C411yescharacteristic
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C2×C8—44no—
363C2×C8—44no—
364C2×C8—44no—
365C2×C8—44no—
366C2×C8—44no—
367C2×C8—44no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C22⋊C4—42no—
388C22⋊C4—42no—
389C22⋊C4—42no—
390C22⋊C4—42no—
391C22⋊C4—42no—
392C22⋊C4—42no—
393C22⋊C4—42no—
394C22⋊C4—41no—
395C22⋊C4—41no—
396C22⋊C4—42no—
397C22⋊C4—42no—
398C22⋊C4—42no—
399C22⋊C4—42no—
400C22⋊C4—42no—
401C22⋊C4—42no—
402C22⋊C4—42no—
403C42—22no—
404C42—21no—
405C42—21no—
406C42—22no—
407C42—22no—
408C42—22no—
409C42—22no—
410C42—21no—
411C42—21no—
412C42—22no—
413C42—22no—
414C42—22no—
415C42—22no—
416C42—22no—
417C42—22no—
418C42—22no—
419C42—22no—
420C42—22no—
421C42—22no—
422C42—22no—
423C42—22no—
424C42—41no—
425C42—41no—
426C42—41no—
427C42—41no—
428C42—41no—
429C42—41no—
430C42—42no—
431C42—41no—
432C42—41no—
433C42—42no—
434C42#13114yes—
435C42AGL(1,ℤ/8)12yes—
436C42#13712yes—
437C42H(5,𝔽2)12yes—
438C422−1+412yes—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23#40812yes—
453C23C22×C22⋊C411yescharacteristic
454Q8—44no—
455D4—44no—
456D4—84no—
457C2×C4—28no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—28no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
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—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
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—82no—
516C2×C4—82no—
517C2×C4#40812yes—
518C2×C4#40812yes—
519C2×C4#40812yes—
520C2×C4#41712yes—
521C2×C4#41712yes—
522C2×C4#42112yes—
523C2×C4#42112yes—
524C2×C4#52012yes—
525C2×C4#53511yescharacteristic
526C2×C4#53611yescharacteristic
527C2×C4#53712yes—
528C2×C4C2×#13112yes—
529C2×C4C2×#13112yes—
530C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
531C2×C4C2×#13711yescharacteristic
532C2×C4#57512yes—
533C8—84no—
534C8—84no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22—41no—
542C22—81no—
543C22—81no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22—81no—
548C22—81no—
549C22#231111yescharacteristic
550C22C2×#40811yescharacteristic
551C22C2×#40811yescharacteristic
552C22C2×#41711yescharacteristic
553C22#289611yescharacteristic
554C22C2×#42111yescharacteristic
555C22C2×#42111yescharacteristic
556C4—22no—
557C4—22no—
558C4—22no—
559C4—22no—
560C4—22no—
561C4—22no—
562C4—22no—
563C4—21no—
564C4—21no—
565C4—41no—
566C4—41no—
567C4—41no—
568C4—41no—
569C4—81no—
570C4—81no—
571C4—81no—
572C4—81no—
573C4—81no—
574C4—81no—
575C4—81no—
576C4#299414yes—
577C4#325112yes—
578C4#325212yes—
579C2—42no—
580C2—161no—
581C2#1746511yescharacteristic
582C2#1746611yescharacteristic
583C2#1747711yescharacteristic
584C2#1748111yescharacteristic
585C2#2184811yescharacteristic
586C2#2189011yescharacteristic
587C2#2190011yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC2×#121C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20808C2semidirect
3. #20808C2semidirect
4. #9866C2semidirect
5. #9675C2semidirect
7. #9073C2semidirect
8. #8568C2semidirect
9. #8567C2semidirect

Maximal subgroups

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

subgroupindex
#366382
#208082
#98662
#96752
#90742
#90732
#85682
#85672