Finite Group Databasebeta

← #224577

Subgroups — #224577

Order 512.

1116 conjugacy classes of subgroups.

The subgroup lattice has 607 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36579C211yesmaximal, characteristic
2Q8×C22⋊C8C211yesmaximal, characteristic, complemented
3#13817C211yesmaximal, characteristic, complemented
4#9713C212yesmaximal
5#9711C211yesmaximal, characteristic, complemented
6#9681C212yesmaximal
7#9679C211yesmaximal, characteristic, complemented
8#8576C212yesmaximal, complemented
9#8571C211yesmaximal, characteristic
10C4×#328C212yesmaximal, complemented
11C4×#327C211yesmaximal, characteristic
12C2×#556C411yescharacteristic, complemented
13C22×C4×Q8C2211yescharacteristic
14C2×C8×Q8—21no—
15#2857C2211yescharacteristic
16#2850C2212yes—
17#2592C412yescomplemented
18#2581C412yescomplemented
19#2568C411yescharacteristic, complemented
20#2538C411yescharacteristic, complemented
21#2481—22no—
22#2469—22no—
23#2421—21no—
24#2409—21no—
25#2362—21no—
26#2355C411yescharacteristic, complemented
27#2352—21no—
28C4×C22⋊Q8C2212yes—
29C4×#121C2211yescharacteristic
30C2×#391—21no—
31C2×C4×C4⋊C4—21no—
32#1984—21no—
33#1984—22no—
34#1797C2211yescharacteristic
35#1797C2212yes—
36C4×C22⋊C8C2211yescharacteristic
37C4×C22⋊C8C2212yes—
38C2×#328C2212yes—
39C2×#328C2212yes—
40C2×#328C2212yes—
41C2×#328C2212yes—
42C2×#327C2211yescharacteristic
43C2×#327C2211yescharacteristic
44C2×#327C2212yes—
45#1436—22no—
46C4⋊#95C2211yescharacteristic
47#1434C2212yes—
48C4×#95C2211yescharacteristic
49C4×#95C2212yes—
50#1294C2211yescharacteristic
51#1294C2211yescharacteristic
52#1294C2212yes—
53C23×Q8D411yescharacteristic
54#556—44no—
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×#126—22no—
57C2×#124C2×C412yes—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8C2312yes—
61C2×#121C2311yescharacteristic
62C2×C4×Q8—44no—
63C2×C4×Q8C2×C412yes—
64C2×C4×Q8C2×C411yescharacteristic
65C2×C4×Q8C2×C411yescharacteristic
66C2×C4×Q8C2×C412yes—
67C2×C4×Q8D411yescharacteristic
68C2×C4×Q8D411yescharacteristic
69C2×C4×Q8D412yes—
70C2×C4×D4—21no—
71C2×C4×D4C2×C411yescharacteristic
72C2×#117—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4D412yes—
76C22×C42—21no—
77C22×C42—22no—
78C22×C42C2312yes—
79C22×C42C2311yescharacteristic
80C8×Q8—44no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—22no—
83C2×C22⋊C8C2311yescharacteristic
84C2×C22⋊C8C2312yes—
85C2×C22⋊C8C2311yescharacteristic
86C2×C4×C8—21no—
87C2×C4×C8—22no—
88#400—22no—
89#400—22no—
90#397—22no—
91#396—22no—
92#395—22no—
93#393—22no—
94#391—28no—
95#391—21no—
96#391—21no—
97#391C2×C412yes—
98#391C2×C412yes—
99#391C2×C412yes—
100#390C2×C411yescharacteristic
101#390C2×C412yes—
102#389—21no—
103#389—21no—
104#389C2×C411yescharacteristic
105#389C2×C412yes—
106#388—22no—
107#388—22no—
108#388—22no—
109#388—22no—
110#388—21no—
111#387—22no—
112#387—22no—
113#387—22no—
114#387—22no—
115#386—22no—
116#386—22no—
117#386—21no—
118#385—22no—
119#384C2×C411yescharacteristic
120#382—21no—
121#382C2×C412yes—
122#380—21no—
123#380—21no—
124#380—21no—
125#380C2311yescharacteristic
126C4×C4⋊C4—21no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—44no—
129C4×C4⋊C4C2×C411yescharacteristic
130C4×C4⋊C4C2×C412yes—
131C4×C22⋊C4—22no—
132C4×C22⋊C4—21no—
133C2×#95—22no—
134C2×#95—22no—
135C2×#95—21no—
136C2×#95—22no—
137C2×#95—21no—
138C2×#95C2311yescharacteristic
139C2×#95C2312yes—
140C2×#95C2311yescharacteristic
141#336—22no—
142#336—21no—
143#336—22no—
144#336—21no—
145#328—24no—
146#328—24no—
147#328—42no—
148#328—42no—
149#327—22no—
150#327—22no—
151#327—42no—
152C22×Q8—24no—
153C22×Q8—22no—
154C22×Q8—22no—
155C22×Q8—42no—
156C22×Q8C22⋊C411yescharacteristic
157C22×Q8C22⋊C412yes—
158C22×Q8C22⋊C411yescharacteristic
159C22×Q8C22×C412yes—
160C22×D4—21no—
161C22×D4C22×C411yescharacteristic
162C23×C4—21no—
163C23×C4—22no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4C2×D412yes—
167C23×C4C2×D412yes—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2411yescharacteristic
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—22no—
174C4⋊Q8—42no—
175#126—48no—
176#124—44no—
177C22⋊Q8—22no—
178C22⋊Q8—22no—
179C22⋊Q8—22no—
180C22⋊Q8—22no—
181C22⋊Q8—44no—
182C22⋊Q8—44no—
183#121—21no—
184#121—21no—
185#121—21no—
186#121—21no—
187C4×Q8—24no—
188C4×Q8—24no—
189C4×Q8—44no—
190C4×Q8—44no—
191C4×Q8—42no—
192C4×Q8—42no—
193C4×Q8—44no—
194C4×D4—42no—
195C4×D4—42no—
196C4×D4—42no—
197#117—44no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4C22⋊C412yes—
212C2×C4⋊C4C22⋊C411yescharacteristic
213C2×C4⋊C4C22⋊C412yes—
214C2×C4⋊C4C22⋊C412yes—
215C2×C4⋊C4C22⋊C412yes—
216C2×C4⋊C4C22⋊C412yes—
217C2×C4⋊C4C22⋊C411yescharacteristic
218C2×C4⋊C4C22×C411yescharacteristic
219C2×C4⋊C4C22×C412yes—
220C2×C4⋊C4C22×C411yescharacteristic
221C2×C4⋊C4C22×C411yescharacteristic
222C2×C22⋊C4—21no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—22no—
232C2×C42—24no—
233C2×C42—22no—
234C2×C42—21no—
235C2×C42—22no—
236C2×C42—42no—
237C2×C42—44no—
238C2×C42—42no—
239C2×C42—42no—
240C2×C42—44no—
241C2×C42—44no—
242C2×C42—82no—
243C2×C42C22×C411yescharacteristic
244C2×C42C22×C412yes—
245C2×C42C22×C411yescharacteristic
246C2×C42C22×C412yes—
247C2×C42C2×D411yescharacteristic
248C2×C42C2×D412yes—
249C2×C42C2×D412yes—
250C2×C42C2×D411yescharacteristic
251C4⋊C8—44no—
252C4⋊C8—42no—
253C22⋊C8—21no—
254C22⋊C8—21no—
255C22⋊C8—22no—
256C22⋊C8—22no—
257C22⋊C8—21no—
258C22⋊C8#4112yes—
259C4×C8—44no—
260C4×C8—42no—
261#95—28no—
262#95—28no—
263#95—22no—
264#95—22no—
265#95—21no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—21no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—21no—
276#95—21no—
277#95—21no—
278#95—22no—
279#95—22no—
280#95—21no—
281#95—42no—
282#95—44no—
283#95—81no—
284#95#4112yes—
285C24—21no—
286C24C22×D411yescharacteristic
287C2×Q8—22no—
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8—24no—
291C2×Q8—22no—
292C2×Q8—24no—
293C2×Q8—22no—
294C2×Q8—22no—
295C2×Q8—22no—
296C2×Q8—42no—
297C2×Q8—48no—
298C2×Q8—48no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8#10414yes—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—42no—
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—42no—
344C22×C4—41no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—41no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4C2×C22⋊C412yes—
359C22×C4C2×C22⋊C412yes—
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4C2×C22⋊C412yes—
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4C2×C22⋊C412yes—
364C22×C4C2×C22⋊C411yescharacteristic
365C22×C4#13512yes—
366C22×C4#13512yes—
367C22×C4C23×C411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C2×#4111yescharacteristic
370C22×C4C2×#4111yescharacteristic
371C22×C4H(5,𝔽2)11yescharacteristic
372C22×C42−1+411yescharacteristic
373C2×C8—22no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—42no—
378C2×C8—41no—
379C2×C8—42no—
380C2×C8—41no—
381C2×C8—42no—
382C4⋊C4—24no—
383C4⋊C4—22no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—21no—
387C4⋊C4—22no—
388C4⋊C4—21no—
389C4⋊C4—22no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—48no—
395C4⋊C4—44no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—44no—
403C4⋊C4—81no—
404C4⋊C4#9914yes—
405C22⋊C4—42no—
406C22⋊C4—44no—
407C22⋊C4—42no—
408C22⋊C4—42no—
409C22⋊C4—41no—
410C22⋊C4—41no—
411C22⋊C4—41no—
412C22⋊C4—41no—
413C22⋊C4—44no—
414C22⋊C4—42no—
415C22⋊C4—44no—
416C22⋊C4—81no—
417C22⋊C4—82no—
418C42—24no—
419C42—22no—
420C42—22no—
421C42—24no—
422C42—42no—
423C42—41no—
424C42—41no—
425C42—42no—
426C42—48no—
427C42—44no—
428C42—42no—
429C42—82no—
430C42—88no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23C22×C22⋊C411yescharacteristic
443C23#54811yescharacteristic
444C23#55011yescharacteristic
445C23C2×#13511yescharacteristic
446C23C2×#13511yescharacteristic
447C23#57711yescharacteristic
448C23#57811yescharacteristic
449Q8—28no—
450Q8—48no—
451Q8—44no—
452Q8—82no—
453Q8—82no—
454D4—81no—
455D4—81no—
456D4—81no—
457D4—81no—
458D4—81no—
459D4—81no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—21no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—21no—
474C2×C4—22no—
475C2×C4—21no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—82no—
523C2×C4—81no—
524C2×C4—82no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—82no—
528C2×C4—81no—
529C2×C4—82no—
530C2×C4—84no—
531C2×C4C2×#9914yes—
532C2×C4C2×#9912yes—
533C2×C4#41612yes—
534C2×C4#41612yes—
535C2×C4C2×#10414yes—
536C2×C4C2×#10412yes—
537C2×C4C4×#4111yescharacteristic
538C2×C4#52011yescharacteristic
539C2×C4#54811yescharacteristic
540C2×C4#55011yescharacteristic
541C8—42no—
542C8—81no—
543C8—82no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22—81no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22#234111yescharacteristic
564C22C22×#9911yescharacteristic
565C22#288411yescharacteristic
566C22C2×#41611yescharacteristic
567C22#289811yescharacteristic
568C22C22×#10411yescharacteristic
569C22#290111yescharacteristic
570C22#336011yescharacteristic
571C22#336411yescharacteristic
572C22#339311yescharacteristic
573C22#339411yescharacteristic
574C4—21no—
575C4—21no—
576C4—21no—
577C4—21no—
578C4—22no—
579C4—24no—
580C4—24no—
581C4—44no—
582C4—42no—
583C4—42no—
584C4—41no—
585C4—41no—
586C4—84no—
587C4—82no—
588C4—82no—
589C4—81no—
590C4—82no—
591C4—81no—
592C4—81no—
593C4#150114yes—
594C2—21no—
595C2—21no—
596C2—21no—
597C2—21no—
598C2—161no—
599C2C2×#150111yescharacteristic
600C2#1754911yescharacteristic
601C2#1760811yescharacteristic
602C2#1761611yescharacteristic
603C2#2253411yescharacteristic
604C2#2254111yescharacteristic
605C2#2255011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. Q8×C22⋊C8C2semidirect
3. #13817C2semidirect
5. #9711C2semidirect
7. #9679C2semidirect
8. #8576C2semidirect
10. C4×#328C2semidirect
12. C2×#556C4semidirect
17. #2592C4semidirect
18. #2581C4semidirect
19. #2568C4semidirect
20. #2538C4semidirect
26. #2355C4semidirect

Maximal subgroups

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

subgroupindex
#365792
Q8×C22⋊C82
#138172
#97132
#97112
#96812
#96792
#85762
#85712
C4×#3282
C4×#3272