Finite Group Databasebeta

← #205027

Subgroups — #205027

Order 512.

1330 conjugacy classes of subgroups.

The subgroup lattice has 571 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35019C211yesmaximal, characteristic
2#21934C211yesmaximal, characteristic, complemented
3C2×#1511C211yesmaximal, characteristic, complemented
4C2×#1510C211yesmaximal, characteristic
5C2×#1508C211yesmaximal, characteristic, complemented
6C2×#1507C211yesmaximal, characteristic
7#13807C211yesmaximal, characteristic, complemented
8#9716C214yesmaximal, complemented
9#9713C214yesmaximal, complemented
10C22×C22⋊Q8C2211yescharacteristic
11C22×#121C2211yescharacteristic
12C2×#407C2211yescharacteristic
13C2×#407C2211yescharacteristic
14#2850C2214yes—
15#2639—21no—
16#2639—21no—
17#2591—22no—
18#2561—22no—
19#2504C412yescomplemented
20#2503C412yescomplemented
21#2436C412yescomplemented
22#2435C412yescomplemented
23#2308—22no—
24C2×#381C2211yescharacteristic
25C2×C4×C22⋊C4C2211yescharacteristic
26C2×C4×C22⋊C4C2211yescharacteristic
27#1851—22no—
28#1851—22no—
29#1797C2214yescomplemented
30#1511—24no—
31#1510—24no—
32#1508—24no—
33#1507—24no—
34C2×#328C2212yescomplemented
35C2×#328C2212yescomplemented
36C2×#328C2214yescomplemented
37C2×#327C2212yes—
38C2×#327C2212yes—
39C2×#327C2214yes—
40#1437—22no—
41#1434—21no—
42#1434C2212yescomplemented
43C4×#95—21no—
44C4×#95C2212yescomplemented
45C23×Q8C2×C411yescharacteristic
46C23×D4—21no—
47C23×D4C2×C411yescharacteristic
48C24×C4C2311yescharacteristic
49C22×M16—21no—
50C22×M16—21no—
51C2×#124C2×C414yes—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8—22no—
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8—22no—
57C2×C22⋊Q8—22no—
58C2×C22⋊Q8C2×C412yes—
59C2×C22⋊Q8C2312yes—
60C2×#121—21no—
61C2×#121—21no—
62C2×#121—22no—
63C2×#121—22no—
64C2×#121—22no—
65C2×#121—22no—
66C2×#121C2×C412yes—
67C2×#121C2312yes—
68C22×C4⋊C4—21no—
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4D412yes—
76C22×C42—21no—
77C22×C42—22no—
78C22×C42C2312yes—
79C2×C4⋊C8—24no—
80#407—22no—
81#407—22no—
82#407—22no—
83#407—22no—
84C2×C22⋊C8—21no—
85C2×C22⋊C8—21no—
86C2×C22⋊C8C2312yes—
87C2×C22⋊C8C2312yes—
88#400—22no—
89#400—22no—
90#397—22no—
91#397—22no—
92#394—22no—
93#394—22no—
94#393—22no—
95#393—22no—
96#391C2×C414yes—
97#390C2×C414yes—
98#388—24no—
99#388—24no—
100#386—24no—
101#386—24no—
102#386—24no—
103#386—24no—
104#382C2×C414yes—
105#381—22no—
106#381—24no—
107#381D414yes—
108#379—24no—
109C4×C22⋊C4—22no—
110C4×C22⋊C4C2×C412yes—
111C4×C22⋊C4C2×C412yes—
112C4×C22⋊C4D412yes—
113C4×C22⋊C4D412yes—
114C4×C22⋊C4D414yes—
115C2×#95—21no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95D411yescharacteristic
120C2×#95D411yescharacteristic
121C2×#95C2311yescharacteristic
122C2×#95C2312yes—
123C2×#95C2311yescharacteristic
124#336—24no—
125#328—24no—
126#328—24no—
127#328—28no—
128#327—24no—
129#327—24no—
130#327—28no—
131C25—21no—
132C25C2×D411yescharacteristic
133C22×Q8—21no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—82no—
137C22×Q8C22×C412yes—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4—21no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4—44no—
144C22×D4—44no—
145C22×D4—42no—
146C22×D4—42no—
147C22×D4—82no—
148C22×D4C22×C412yes—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—22no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—82no—
156C23×C4C22×C411yescharacteristic
157C23×C4C22×C411yescharacteristic
158C23×C4C2×D412yes—
159C23×C4C2×D412yes—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D412yes—
164C23×C4C2×D412yes—
165C23×C4#4112yes—
166C23×C4#4112yes—
167C23×C4C2411yescharacteristic
168C2×M16—42no—
169C2×M16—42no—
170C2×M16—82no—
171C2×M16—82no—
172C22×C8—22no—
173C22×C8—22no—
174#124—84no—
175C22⋊Q8—24no—
176C22⋊Q8—24no—
177C22⋊Q8—22no—
178C22⋊Q8—22no—
179C22⋊Q8—22no—
180C22⋊Q8—22no—
181C22⋊Q8—42no—
182C22⋊Q8—42no—
183C22⋊Q8—42no—
184C22⋊Q8—42no—
185#121—24no—
186#121—24no—
187#121—22no—
188#121—22no—
189#121—22no—
190#121—22no—
191#121—42no—
192#121—42no—
193#121—42no—
194#121—42no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4C22×C412yes—
206C2×C4⋊C4C22×C412yes—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—82no—
226C2×C22⋊C4C22⋊C414yes—
227C2×C22⋊C4C22⋊C414yes—
228C2×C42—24no—
229C2×C42—44no—
230C2×C42—44no—
231C2×C42—82no—
232C2×C42—82no—
233C2×C42—82no—
234C2×C42C22×C414yes—
235C2×C42C2×D414yes—
236C4⋊C8—84no—
237C22⋊C8—22no—
238C22⋊C8—22no—
239C22⋊C8—22no—
240C22⋊C8—22no—
241C22⋊C8—42no—
242C22⋊C8—42no—
243#95—24no—
244#95—24no—
245#95—24no—
246#95—24no—
247#95—24no—
248#95—21no—
249#95—21no—
250#95—22no—
251#95—44no—
252#95—44no—
253#95—82no—
254#95C22⋊C412yes—
255#95C22⋊C412yes—
256#95C22⋊C412yes—
257#95C22⋊C412yes—
258#95C2×D412yes—
259#95C2×D412yes—
260#95C2×D414yes—
261C24—22no—
262C24—22no—
263C24—22no—
264C24—41no—
265C24—41no—
266C24—42no—
267C24—44no—
268C24—42no—
269C24—82no—
270C24—81no—
271C24—81no—
272C24C2×C22⋊C411yescharacteristic
273C24C2×C22⋊C411yescharacteristic
274C24#12112yes—
275C24#12112yes—
276C24C22×D411yescharacteristic
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—44no—
280C2×Q8—42no—
281C2×Q8—82no—
282C2×Q8—84no—
283C2×Q8—84no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—44no—
287C2×D4—48no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—44no—
291C2×D4—44no—
292C2×D4—44no—
293C2×D4—42no—
294C2×D4—82no—
295C2×D4—84no—
296C2×D4—84no—
297C2×D4—84no—
298C2×D4—84no—
299C2×D4—88no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—44no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—82no—
339C22×C4—82no—
340C22×C4—82no—
341C22×C4—82no—
342C22×C4—81no—
343C22×C4—81no—
344C22×C4—82no—
345C22×C4C2×C22⋊C414yes—
346C22×C4C2×C22⋊C414yes—
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4C4×D412yes—
350C22×C4C4×D412yes—
351C22×C4C4×D412yes—
352C22×C4C4×D412yes—
353C22×C4#12112yes—
354C22×C4#12112yes—
355C22×C4#12412yes—
356C22×C4#12412yes—
357C22×C4#12712yes—
358C22×C4#12712yes—
359C22×C4C23×C411yescharacteristic
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C4C2×#4111yescharacteristic
364C22×C4C2×#4111yescharacteristic
365M16—84no—
366M16—84no—
367C2×C8—44no—
368C2×C8—44no—
369C2×C8—42no—
370C2×C8—42no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—44no—
380C22⋊C4—44no—
381C22⋊C4—44no—
382C22⋊C4—48no—
383C22⋊C4—42no—
384C22⋊C4—42no—
385C22⋊C4—44no—
386C22⋊C4—42no—
387C22⋊C4—42no—
388C22⋊C4—44no—
389C22⋊C4—84no—
390C22⋊C4—84no—
391C22⋊C4—84no—
392C22⋊C4—84no—
393C22⋊C4—84no—
394C42—44no—
395C42—44no—
396C42—84no—
397C42—88no—
398C23—22no—
399C23—21no—
400C23—21no—
401C23—22no—
402C23—22no—
403C23—22no—
404C23—22no—
405C23—21no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—22no—
410C23—22no—
411C23—24no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—44no—
417C23—44no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—82no—
422C23—82no—
423C23—82no—
424C23—82no—
425C23—84no—
426C23—84no—
427C23—84no—
428C23—164no—
429C23—164no—
430C23#39012yes—
431C23#39012yes—
432C23C22×C22⋊C411yescharacteristic
433C23C2×#12111yescharacteristic
434C23C2×#12111yescharacteristic
435C23#57512yes—
436C23#57512yes—
437Q8—82no—
438Q8—82no—
439Q8—84no—
440D4—82no—
441D4—82no—
442D4—84no—
443D4—88no—
444D4—82no—
445D4—84no—
446D4—82no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—84no—
500C2×C4—84no—
501C2×C4—84no—
502C2×C4#39012yes—
503C2×C4#39012yes—
504C2×C4C2×C4×D411yescharacteristic
505C2×C4C2×C4×D411yescharacteristic
506C2×C4C2×#12411yescharacteristic
507C2×C4C2×#12711yescharacteristic
508C8—82no—
509C8—82no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—22no—
519C22—22no—
520C22—22no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—82no—
529C22—82no—
530C22—84no—
531C22—82no—
532C22—84no—
533C22—84no—
534C22—82no—
535C22—82no—
536C22—168no—
537C22C2×#39011yescharacteristic
538C22#288312yes—
539C22#289612yes—
540C22#290111yescharacteristic
541C22#290111yescharacteristic
542C22#304011yescharacteristic
543C22#304012yes—
544C22#304011yescharacteristic
545C4—21no—
546C4—21no—
547C4—21no—
548C4—21no—
549C4—22no—
550C4—22no—
551C4—44no—
552C4—44no—
553C4—44no—
554C4—82no—
555C4—84no—
556C4—82no—
557C4—82no—
558C4—82no—
559C2—21no—
560C2—22no—
561C2—21no—
562C2—42no—
563C2—42no—
564C2—162no—
565C2#1762312yes—
566C2#1762512yes—
567C2#2121011yescharacteristic
568C2#2121311yescharacteristic
569C2#2126111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21934C2semidirect
3. C2×#1511C2semidirect
5. C2×#1508C2semidirect
7. #13807C2semidirect
8. #9716C2semidirect
9. #9713C2semidirect
19. #2504C4semidirect
20. #2503C4semidirect
21. #2436C4semidirect
22. #2435C4semidirect
29. #1797C22semidirect
34. C2×#328C22semidirect
35. C2×#328C22semidirect
36. C2×#328C22semidirect
42. #1434C22semidirect
44. C4×#95C22semidirect

Maximal subgroups

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

subgroupindex
#350192
#219342
C2×#15112
C2×#15102
C2×#15082
C2×#15072
#138072
#97162
#97132