Finite Group Databasebeta

← #6530973

Subgroups — #6530973

Order 512.

2972 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
1C2×#3503C211yesmaximal, characteristic
2C2×#3236C211yesmaximal, characteristic, complemented
3C22×#501C211yesmaximal, characteristic, complemented
4#35138C211yesmaximal, characteristic
5#25847C214yesmaximal, complemented
6#25816C214yesmaximal, complemented
7#25712C214yesmaximal, complemented
8#25677C214yesmaximal, complemented
9#24098C214yesmaximal, complemented
10#24097C214yesmaximal, complemented
11#21841C211yesmaximal, characteristic, complemented
12#21407C211yesmaximal, characteristic, complemented
13#21379C211yesmaximal, characteristic
14#3503—28no—
15#3503—28no—
16C2×Q82C2212yes—
17C2×#557—21no—
18C2×#554C2214yes—
19C2×#554C2212yes—
20C2×D4×Q8—22no—
21C2×D4×Q8—21no—
22C2×#544—22no—
23C2×#544—21no—
24C2×#533C2212yes—
25C2×#533C2211yescharacteristic
26C22×C4⋊Q8C2212yes—
27C22×C4⋊Q8C2211yescharacteristic
28C22×#125C2211yescharacteristic
29C2×#520—21no—
30C2×C4×#41C2211yescharacteristic
31#3236—28no—
32C2×#501—28no—
33C2×#501C2214yes—
34C2×#501C2214yes—
35C2×#501C2212yes—
36C2×#491C2212yes—
37C2×#484C2212yes—
38C2×#483C2212yes—
39C2×#479C2214yes—
40C2×#432C2211yescharacteristic
41C22×#107C2212yes—
42C22×#106C2212yes—
43C22×#97C2211yescharacteristic
44#2851C2214yes—
45#2634—21no—
46#2633—21no—
47#2611—22no—
48#2584C2212yes—
49#2562C2212yes—
50#2487C2212yes—
51#2486C2212yes—
52#2484—21no—
53#2406—21no—
54#2322—22no—
55#2283C2211yescharacteristic
56#2058C2218yes—
57#2057C2218yes—
58#2036C2218yes—
59#2035C2218yes—
60#1993C2214yes—
61#1993C2212yes—
62#1988C2214yes—
63#1985C2212yes—
64#1960C2212yes—
65#1955C2212yes—
66#1892C2214yes—
67#1869C2214yes—
68#1865C2218yes—
69#1863C2218yes—
70#1853C2218yes—
71#1852C2218yes—
72#1824C2214yes—
73#1763C2214yes—
74#1749C2214yes—
75C22×#41D411yescharacteristic
76C23×Q8—22no—
77C23×Q8—21no—
78C23×C8C2312yes—
79Q82—28no—
80Q82—28no—
81#557—216no—
82#554—28no—
83#554—216no—
84#554—28no—
85#554—216no—
86D4×Q8—232no—
87D4×Q8—216no—
88#544—232no—
89#544—216no—
90#533—28no—
91#533—28no—
92#533—24no—
93#533D418yes—
94C2×C4⋊Q8—28no—
95C2×C4⋊Q8—28no—
96C2×C4⋊Q8—24no—
97C2×C4⋊Q8—22no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—24no—
103C2×C4⋊Q8D418yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2314yes—
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2312yes—
109C2×#125—24no—
110C2×#125—22no—
111C2×#125—22no—
112C2×#125—21no—
113C2×#125—21no—
114C2×#125—21no—
115C2×#125—24no—
116C2×#125—24no—
117C2×#125C2312yes—
118C2×#125C2312yes—
119C2×#123—22no—
120C2×C22⋊Q8—22no—
121C2×C22⋊Q8—21no—
122C2×C22⋊Q8—24no—
123C2×C22⋊Q8—24no—
124C2×C22⋊Q8—24no—
125C2×C22⋊Q8C2314yes—
126C2×C22⋊Q8C2312yes—
127#520—216no—
128C4×#41—24no—
129C4×#41—24no—
130C2×C4×Q8—22no—
131C2×C4×Q8—22no—
132C2×C4×Q8—21no—
133C2×C4×Q8D411yescharacteristic
134C2×C4×Q8C2314yes—
135C2×C4×Q8C2312yes—
136C2×C4×Q8C2311yescharacteristic
137C2×C4×D4—22no—
138C2×C4×D4—24no—
139C2×C4×D4D412yes—
140C2×C4×D4D411yescharacteristic
141C2×C4×D4C2311yescharacteristic
142C2×C4×D4C2312yes—
143C2×#117—22no—
144C2×#117—21no—
145C2×#117D412yes—
146C2×#117D411yescharacteristic
147C22×C4⋊C4—22no—
148C22×C4⋊C4—21no—
149C22×C4⋊C4—21no—
150C22×C4⋊C4—21no—
151C22×C4⋊C4C2311yescharacteristic
152C22×C4⋊C4C2311yescharacteristic
153C22×C4⋊C4C2314yes—
154C22×C4⋊C4C2312yes—
155C22×C42C2312yes—
156C22×C42C2311yescharacteristic
157#501—216no—
158#501—216no—
159#501—28no—
160#491—28no—
161#484—28no—
162#483—28no—
163#479—216no—
164#432—24no—
165C2×#107—28no—
166C2×#107C2314yes—
167C2×#107C2318yes—
168C2×#106—28no—
169C2×#106C2314yes—
170C2×#106C2318yes—
171C2×C4⋊C8C2312yes—
172C2×#103C2314yes—
173C2×#103C2314yes—
174C2×#103C2318yes—
175C2×C22⋊C8C2312yes—
176C2×C22⋊C8C2318yes—
177C2×#97—24no—
178C2×#97C2312yes—
179C2×#97C2312yes—
180C2×#97C2312yes—
181#400—22no—
182#400—22no—
183#399—22no—
184#398—22no—
185#398—22no—
186#397—22no—
187#395—22no—
188#394—22no—
189#389—24no—
190#389C2314yes—
191#389C2314yes—
192#388—24no—
193#387—24no—
194#387—24no—
195#386—22no—
196#385—24no—
197#385—24no—
198#385—22no—
199#383—21no—
200#382C2312yes—
201#381—21no—
202#381—22no—
203#380C2314yes—
204#380C2314yes—
205#380C2312yes—
206C4×C4⋊C4C2312yes—
207C4×C22⋊C4—21no—
208#376—21no—
209#340C2318yes—
210#340C2318yes—
211#340C2318yes—
212#336C2318yes—
213C2×#41—28no—
214C2×#41—24no—
215C22×Q8—216no—
216C22×Q8—28no—
217C22×Q8—22no—
218C22×Q8—21no—
219C22×Q8—24no—
220C22×Q8—22no—
221C22×Q8—84no—
222C22×Q8—88no—
223C22×Q8C2×D411yescharacteristic
224C22×Q8C2412yes—
225C22×D4C2×D411yescharacteristic
226C22×D4C2×D412yes—
227C23×C4—24no—
228C23×C4—22no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4C2×D412yes—
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D412yes—
234C23×C4C2×D411yescharacteristic
235C23×C4C2412yes—
236C23×C4C2411yescharacteristic
237C22×C8—24no—
238C22×C8—28no—
239C22×C8C2414yes—
240C22×C8C2414yes—
241C4⋊Q8—216no—
242C4⋊Q8—216no—
243C4⋊Q8—232no—
244C4⋊Q8—216no—
245C4⋊Q8—28no—
246C4⋊Q8—28no—
247C4⋊Q8—28no—
248C4⋊Q8—28no—
249C4⋊Q8—22no—
250C4⋊Q8—24no—
251C4⋊Q8—24no—
252C4⋊Q8—22no—
253C4⋊Q8C2×D414yes—
254C4⋊Q8C2×D414yes—
255C4⋊Q8C2×D418yes—
256C4⋊Q8C2×D418yes—
257#125—216no—
258#125—232no—
259#125—216no—
260#125—28no—
261#125—24no—
262#125—24no—
263#125—24no—
264#125—24no—
265#125—44no—
266#125—44no—
267#123—48no—
268C22⋊Q8—216no—
269C22⋊Q8—232no—
270C22⋊Q8—232no—
271C22⋊Q8—232no—
272C22⋊Q8—28no—
273C22⋊Q8—28no—
274C22⋊Q8—24no—
275C22⋊Q8—28no—
276C22⋊Q8C2×D418yes—
277C4×Q8—216no—
278C4×Q8—216no—
279C4×Q8—28no—
280C4×Q8—28no—
281C4×Q8—28no—
282C4×Q8—24no—
283C4×Q8—22no—
284C4×Q8—22no—
285C4×Q8—24no—
286C4×Q8C2×D418yes—
287C4×D4—232no—
288C4×D4—216no—
289C4×D4—28no—
290C4×D4—24no—
291C4×D4—24no—
292C4×D4—24no—
293C4×D4—22no—
294C4×D4—22no—
295#117—216no—
296#117—28no—
297#117—24no—
298#117—28no—
299C2×C4⋊C4—216no—
300C2×C4⋊C4—28no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—28no—
315C2×C4⋊C4—28no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—44no—
329C2×C4⋊C4—44no—
330C2×C4⋊C4C2×D412yes—
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2×D412yes—
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2×D418yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2414yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2414yes—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4C2×D412yes—
346C2×C22⋊C4C2×D412yes—
347C2×C22⋊C4C2×D412yes—
348C2×C42—24no—
349C2×C42—22no—
350C2×C42—24no—
351C2×C42—24no—
352C2×C42—22no—
353C2×C42C2×D412yes—
354C2×C42C2×D412yes—
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D414yes—
358C2×C42C2×D414yes—
359C2×C42C2412yes—
360C2×C42C2411yescharacteristic
361C2×C42C2411yescharacteristic
362#107—216no—
363#107—28no—
364#106—216no—
365#106—28no—
366C4⋊C8—24no—
367#103—28no—
368#103—28no—
369#103—216no—
370C22⋊C8—24no—
371C22⋊C8—216no—
372#97—24no—
373#97—24no—
374#97—24no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—24no—
380#95—24no—
381#95—24no—
382#95—24no—
383C24C22×D412yes—
384C24C22×D411yescharacteristic
385#41—232no—
386C2×Q8—216no—
387C2×Q8—22no—
388C2×Q8—28no—
389C2×Q8—24no—
390C2×Q8—28no—
391C2×Q8—24no—
392C2×Q8—22no—
393C2×Q8—28no—
394C2×Q8—48no—
395C2×Q8—44no—
396C2×Q8—84no—
397C2×Q8—88no—
398C2×Q8—816no—
399C2×Q8—832no—
400C2×Q8C22×D414yes—
401C2×D4—24no—
402C2×D4—24no—
403C2×D4—24no—
404C2×D4—22no—
405C2×D4#13718yes—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—216no—
415C22×C4—28no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—28no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—42no—
424C22×C4—44no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—44no—
428C22×C4—82no—
429C22×C4—84no—
430C22×C4—88no—
431C22×C4#12014yes—
432C22×C4#12014yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C2511yescharacteristic
444C2×C8—216no—
445C2×C8—24no—
446C2×C8—24no—
447C2×C8—24no—
448C2×C8—28no—
449C2×C82−1+418yes—
450C4⋊C4—216no—
451C4⋊C4—216no—
452C4⋊C4—216no—
453C4⋊C4—216no—
454C4⋊C4—24no—
455C4⋊C4—28no—
456C4⋊C4—28no—
457C4⋊C4—28no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—28no—
461C4⋊C4—28no—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—28no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—48no—
480C4⋊C4—84no—
481C4⋊C4C22×D414yes—
482C4⋊C4C22×D418yes—
483C22⋊C4—216no—
484C22⋊C4—28no—
485C22⋊C4—24no—
486C22⋊C4—24no—
487C22⋊C4—24no—
488C22⋊C4—44no—
489C22⋊C4—48no—
490C22⋊C4—84no—
491C42—216no—
492C42—24no—
493C42—24no—
494C42—28no—
495C42—22no—
496C42—22no—
497C42—82no—
498C42#12014yes—
499C42#12014yes—
500C42C22×D412yes—
501C42C22×D412yes—
502C42C22×D412yes—
503C42C22×D412yes—
504C23—22no—
505C23—22no—
506C23—24no—
507C23—22no—
508C23—24no—
509C23—22no—
510C23C2×#12012yes—
511C23C2×#13714yes—
512C23C2×#13714yes—
513C23C23×D411yescharacteristic
514Q8—28no—
515Q8—28no—
516Q8—416no—
517Q8—48no—
518Q8—44no—
519Q8—88no—
520Q8—816no—
521D4—28no—
522D4—216no—
523C2×C4—216no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—28no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—28no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—44no—
542C2×C4—48no—
543C2×C4—48no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—88no—
557C2×C4—816no—
558C2×C4—84no—
559C2×C4—84no—
560C2×C4C2×#12012yes—
561C2×C4C2×#12012yes—
562C2×C4C2×#12012yes—
563C2×C4C2×#12012yes—
564C2×C4C2×#12012yes—
565C2×C4C2×#13714yes—
566C2×C4C23×D411yescharacteristic
567C2×C4C23×D411yescharacteristic
568C2×C4C2×2−1+414yes—
569C8—28no—
570C8—44no—
571C22—28no—
572C22—22no—
573C22—22no—
574C22—28no—
575C22—24no—
576C22—44no—
577C22—42no—
578C22—42no—
579C22#300414yes—
580C22C22×#12011yescharacteristic
581C22#349912yes—
582C22C22×#13712yes—
583C22#358512yes—
584C4—22no—
585C4—22no—
586C4—22no—
587C4—24no—
588C4—22no—
589C4—22no—
590C4—42no—
591C4—44no—
592C4—42no—
593C4—48no—
594C4—44no—
595C4—84no—
596C4—82no—
597C4—82no—
598C4#348612yes—
599C4#349912yes—
600C2—24no—
601C2—44no—
602C2#3566611yescharacteristic
603C2C2×#300411yescharacteristic
604C2#6036711yescharacteristic
605C2#6250414yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3236C2semidirect
3. C22×#501C2semidirect
5. #25847C2semidirect
6. #25816C2semidirect
7. #25712C2semidirect
8. #25677C2semidirect
9. #24098C2semidirect
10. #24097C2semidirect
11. #21841C2semidirect
12. #21407C2semidirect

Maximal subgroups

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

subgroupindex
C2×#35032
C2×#32362
C22×#5012
#351382
#258472
#258162
#257122
#256772
#240982
#240972
#218412
#214072
#213792