Finite Group Databasebeta

← #207708

Subgroups — #207708

Order 512.

929 conjugacy classes of subgroups.

The subgroup lattice has 587 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2865C211yesmaximal, characteristic
2#20778C211yesmaximal, characteristic, complemented
3C4×#431C211yesmaximal, characteristic, complemented
4#9880C212yesmaximal, complemented
5#9651C212yesmaximal
6#9082C212yesmaximal, complemented
7#9080C212yesmaximal
8#8155C212yesmaximal
9#8154C212yesmaximal, complemented
10C2×#431C2212yes—
11#2865—21no—
12#2865—21no—
13#2865—21no—
14#2865—21no—
15#2865—21no—
16#2865—21no—
17#2865—21no—
18#2865—21no—
19C2×#388C411yescharacteristic
20C2×#388C411yescharacteristic
21C2×#388C411yescharacteristic
22C2×#388C411yescharacteristic
23C2×#388C2211yescharacteristic
24C2×#388C2211yescharacteristic
25C2×#388C2211yescharacteristic
26C2×#387C411yescharacteristic
27C2×#387C411yescharacteristic
28C2×#387C411yescharacteristic
29C2×#387C411yescharacteristic
30C2×#387C2211yescharacteristic
31C2×#387C2211yescharacteristic
32C2×#387C2211yescharacteristic
33C2×C43C2211yescharacteristic
34#1839—21no—
35#1831—21no—
36#1817C2212yes—
37#1797C2212yes—
38#1767—21no—
39C4×C4⋊C8—21no—
40C4×C4⋊C8—21no—
41#1749C2212yes—
42C4×C22⋊C8C2212yes—
43C4×C22⋊C8C2212yes—
44#1745—21no—
45#1744—21no—
46C42×C8—21no—
47C2×#324C2214yes—
48C2×#323C2214yes—
49#1280C2214yes—
50#1280C2214yes—
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4C2×C411yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C22⋊C4C2×C411yescharacteristic
59C22×C22⋊C4C2×C411yescharacteristic
60C22×C22⋊C4C2×C411yescharacteristic
61C22×C22⋊C4C2×C411yescharacteristic
62C22×C22⋊C4C2×C411yescharacteristic
63C22×C22⋊C4C2×C411yescharacteristic
64C22×C22⋊C4C2311yescharacteristic
65C22×C42C2×C411yescharacteristic
66C22×C42C2×C411yescharacteristic
67C22×C42C2×C411yescharacteristic
68C22×C42C2×C411yescharacteristic
69C22×C42C2311yescharacteristic
70C22×C42C2311yescharacteristic
71C22×C42C2311yescharacteristic
72#431—28no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—22no—
75C2×C22⋊C8C2314yes—
76C2×C22⋊C8C2314yes—
77C2×#97—22no—
78C2×C4×C8—22no—
79C2×C4×C8—22no—
80C2×C4×C8—22no—
81#388—21no—
82#388—21no—
83#388—21no—
84#388—21no—
85#388—21no—
86#388—21no—
87#388—21no—
88#388—21no—
89#388—21no—
90#388—21no—
91#388—21no—
92#388—21no—
93#388—21no—
94#388—21no—
95#388—21no—
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#387—21no—
110#387—21no—
111#387—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#387—21no—
129#387—21no—
130#387—21no—
131#387—21no—
132#387—21no—
133#387—21no—
134#387—21no—
135#387—21no—
136#387—21no—
137C2×#95C2×C411yescharacteristic
138C2×#95C2×C411yescharacteristic
139C2×#95C2×C411yescharacteristic
140C2×#95C2×C411yescharacteristic
141C2×#95C2×C411yescharacteristic
142C2×#95C2×C411yescharacteristic
143C2×#95C2×C411yescharacteristic
144C2×#95C2×C411yescharacteristic
145C2×#95C2×C411yescharacteristic
146C2×#95C2×C411yescharacteristic
147C2×#95C2×C411yescharacteristic
148C2×#95C2×C411yescharacteristic
149C2×#95C2311yescharacteristic
150C2×#95C2311yescharacteristic
151C43D411yescharacteristic
152C43D411yescharacteristic
153C43D411yescharacteristic
154C43D411yescharacteristic
155C43D411yescharacteristic
156C43D411yescharacteristic
157C43D411yescharacteristic
158C43D411yescharacteristic
159#336—22no—
160#336—22no—
161#336—22no—
162#336—22no—
163#336—22no—
164#336—22no—
165#324—28no—
166#323—28no—
167C25C22×C411yescharacteristic
168C23×C4C22×C411yescharacteristic
169C23×C4C22×C411yescharacteristic
170C23×C4C22×C411yescharacteristic
171C23×C4C22×C411yescharacteristic
172C23×C4C22×C411yescharacteristic
173C23×C4C22×C411yescharacteristic
174C23×C4C22×C411yescharacteristic
175C23×C4C22×C411yescharacteristic
176C23×C4C22×C411yescharacteristic
177C23×C4C22×C411yescharacteristic
178C23×C4C22×C411yescharacteristic
179C23×C4C22×C411yescharacteristic
180C23×C4C22×C411yescharacteristic
181C23×C4C2411yescharacteristic
182C22×C8—24no—
183C22×C8—24no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C42—24no—
227C2×C42—24no—
228C2×C42—24no—
229C2×C42—42no—
230C2×C42—42no—
231C2×C42—42no—
232C2×C42—42no—
233C2×C42C22⋊C411yescharacteristic
234C2×C42C22⋊C411yescharacteristic
235C2×C42C22⋊C411yescharacteristic
236C2×C42C22⋊C411yescharacteristic
237C2×C42C22⋊C411yescharacteristic
238C2×C42C22⋊C411yescharacteristic
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×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C4⋊C8—28no—
262C4⋊C8—44no—
263C22⋊C8—24no—
264C22⋊C8—24no—
265C22⋊C8—24no—
266C22⋊C8#4118yes—
267#97—44no—
268C4×C8—28no—
269C4×C8—44no—
270C4×C8—44no—
271#95—21no—
272#95—21no—
273#95—21no—
274#95—21no—
275#95—21no—
276#95—21no—
277#95—21no—
278#95—21no—
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—
299C24—21no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—41no—
304C24—41no—
305C24—41no—
306C24—41no—
307C24C23×C411yescharacteristic
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4C2×C22⋊C411yescharacteristic
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4C2×C22⋊C411yescharacteristic
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4#13114yes—
367C22×C4#13114yes—
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C2×#4114yes—
371C2×C8—28no—
372C2×C8—44no—
373C2×C8—44no—
374C2×C8—44no—
375C2×C8—44no—
376C2×C8—44no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C22⋊C4—42no—
392C22⋊C4—42no—
393C22⋊C4—42no—
394C22⋊C4—42no—
395C22⋊C4—42no—
396C22⋊C4—42no—
397C22⋊C4—42no—
398C22⋊C4—42no—
399C22⋊C4—42no—
400C22⋊C4—42no—
401C22⋊C4—42no—
402C22⋊C4—42no—
403C22⋊C4—42no—
404C22⋊C4—42no—
405C42—22no—
406C42—22no—
407C42—22no—
408C42—22no—
409C42—22no—
410C42—22no—
411C42—22no—
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—22no—
425C42—41no—
426C42—41no—
427C42—41no—
428C42—41no—
429C42—41no—
430C42—41no—
431C42—41no—
432C42—41no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—44no—
445C23—82no—
446C23—82no—
447C23—82no—
448C23—82no—
449C23—82no—
450C23—82no—
451C23C22×C22⋊C411yescharacteristic
452C23C4×#4112yes—
453C23C2×#13112yes—
454C23C2×#13112yes—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
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—28no—
473C2×C4—21no—
474C2×C4—21no—
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—21no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
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—44no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4#40812yes—
523C2×C4#40812yes—
524C2×C4#40812yes—
525C2×C4#40812yes—
526C2×C4#41012yes—
527C2×C4#41012yes—
528C2×C4#41312yes—
529C2×C4#41312yes—
530C2×C4#52512yes—
531C2×C4#53212yes—
532C8—48no—
533C8—84no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—81no—
541C22—81no—
542C22—81no—
543C22—81no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22—164no—
548C22#229711yescharacteristic
549C22C2×#40811yescharacteristic
550C22C2×#40811yescharacteristic
551C22C2×#41011yescharacteristic
552C22C2×#41011yescharacteristic
553C22C2×#41311yescharacteristic
554C22C2×#41311yescharacteristic
555C22#296614yes—
556C4—22no—
557C4—22no—
558C4—22no—
559C4—22no—
560C4—22no—
561C4—22no—
562C4—22no—
563C4—22no—
564C4—22no—
565C4—22no—
566C4—41no—
567C4—41no—
568C4—41no—
569C4—41no—
570C4—81no—
571C4—81no—
572C4—81no—
573C4—81no—
574C4—81no—
575C4—81no—
576C4—81no—
577C2—24no—
578C2—161no—
579C2#1737611yescharacteristic
580C2#1738011yescharacteristic
581C2#1738211yescharacteristic
582C2#1738311yescharacteristic
583C2#2137511yescharacteristic
584C2#2142111yescharacteristic
585C2#2142611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20778C2semidirect
3. C4×#431C2semidirect
4. #9880C2semidirect
6. #9082C2semidirect
9. #8154C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28652
#207782
C4×#4312
#98802
#96512
#90822
#90802
#81552
#81542