Finite Group Databasebeta

← #7194233

Subgroups — #7194233

Order 512.

1854 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59823C211yesmaximal, characteristic, complemented
2#59817C211yesmaximal, characteristic, complemented
3#32544C214yesmaximal, complemented
4#31454C214yesmaximal, complemented
5#29272C211yesmaximal, characteristic, complemented
6#28156C211yesmaximal, characteristic
7#27813C212yesmaximal, complemented
8#26889C214yesmaximal, complemented
9#26040C212yesmaximal, complemented
10#25318C212yesmaximal, complemented
11#24326C214yesmaximal, complemented
12#23815C211yesmaximal, characteristic, complemented
13#22995C212yesmaximal, complemented
14#22307C211yesmaximal, characteristic, complemented
15#22220C211yesmaximal, characteristic, complemented
16C2×#554—21no—
17C2×D4×Q8—21no—
18C2×D4×Q8C2212yes—
19C2×#486C2211yescharacteristic
20C2×#474C2214yes—
21C2×#459C2212yes—
22C2×C4×D8C2214yes—
23#2732—22no—
24#2683—22no—
25#2552C2212yes—
26#2551C2212yes—
27#2542C2212yes—
28#2538—21no—
29#2538—21no—
30#2538C2214yes—
31#2538C2212yes—
32#2454—22no—
33#2452—22no—
34#2452C2214yes—
35#2417—22no—
36#2409—22no—
37#2366C2212yes—
38#2366C2212yes—
39D4×C4⋊C4C2212yes—
40C4×C4⋊Q8C2211yescharacteristic
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2211yescharacteristic
43C4×#127C2211yescharacteristic
44C4×#127C2211yescharacteristic
45#2099C2212yes—
46#2099C2212yes—
47#2099C2214yes—
48#2086C2214yes—
49#2071C2212yes—
50#2071C2214yes—
51#2071C2212yes—
52#2057C2212yes—
53#2057C2214yes—
54#2054C2214yes—
55#2041C2212yes—
56#2023C2212yes—
57#2023C2214yes—
58#2011C2212yes—
59#1990C2214yes—
60#1990C2211yescharacteristic
61#1988C2212yes—
62#1984C2214yes—
63#1957C2211yescharacteristic
64#1947C2212yes—
65#1939C2212yes—
66#1934C2214yes—
67#1934C2214yes—
68#1925C2214yes—
69#1918C2214yes—
70#1916C2214yes—
71#1879C2214yes—
72#1879C2214yes—
73#1879C2214yes—
74#1862C2214yes—
75#1862C2214yes—
76#1862C2214yes—
77#1845C2211yescharacteristic
78#1831C2211yescharacteristic
79#1802C2214yes—
80#1802C2212yes—
81C8×C4⋊C4C2212yes—
82C4×#102C2212yes—
83C4×#102C2212yes—
84#1744C2211yescharacteristic
85C23×Q8—22no—
86C22×D8C2314yes—
87#554—48no—
88D4×Q8—48no—
89D4×Q8—48no—
90C2×C4⋊Q8—22no—
91C2×C4⋊Q8D411yescharacteristic
92C2×C4⋊Q8C2311yescharacteristic
93C2×C4⋊Q8C2311yescharacteristic
94C2×C4⋊Q8C2314yes—
95C2×#127C2311yescharacteristic
96C2×#127C2311yescharacteristic
97C2×#125—22no—
98C2×C22⋊Q8—22no—
99C2×C22⋊Q8—24no—
100C2×C22⋊Q8—22no—
101C2×C22⋊Q8—22no—
102C2×C22⋊Q8—24no—
103C2×C22⋊Q8—24no—
104C2×C22⋊Q8—22no—
105C2×C4×Q8—21no—
106C2×C4×Q8—21no—
107C2×C4×Q8C2312yes—
108C2×C4×D4—22no—
109C2×C4×D4—22no—
110C2×C4×D4—21no—
111C2×C4×D4—21no—
112C2×C4×D4—21no—
113C2×C4×D4—21no—
114C2×C4×D4C2312yes—
115C2×C4×D4C2314yes—
116C2×C4×D4C2312yes—
117C2×C4×D4C2314yes—
118C22×C4⋊C4—22no—
119#486—24no—
120#474—48no—
121#459—28no—
122C4×D8—48no—
123C2×#107C2314yes—
124C2×#107C2314yes—
125C2×C4⋊C8C2312yes—
126C2×C4⋊C8C2314yes—
127C2×#102C2312yes—
128C2×#102C2314yes—
129C2×#102C2314yes—
130C2×#102C2312yes—
131C2×#102C2312yes—
132C2×#102C2312yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×C4×C8C2311yescharacteristic
136C2×C4×C8C2311yescharacteristic
137C2×C4×C8C2314yes—
138#399—24no—
139#398—24no—
140#396—24no—
141#396—24no—
142#396—24no—
143#396—24no—
144#395—24no—
145#393—24no—
146#391—21no—
147#391—21no—
148#391D412yes—
149#390—22no—
150#390—22no—
151#390—21no—
152#390—21no—
153#390—21no—
154#390—21no—
155#390C2312yes—
156#390C2312yes—
157#390C2314yes—
158#390C2314yes—
159#389—22no—
160#389—22no—
161#389—21no—
162#389—21no—
163#389D412yes—
164#389D412yes—
165#389C2312yes—
166#389C2312yes—
167#389C2314yes—
168#389C2314yes—
169#386—24no—
170#386—24no—
171#385—24no—
172#385—24no—
173#384D411yescharacteristic
174#384C2311yescharacteristic
175#384C2311yescharacteristic
176#384C2311yescharacteristic
177#384C2311yescharacteristic
178#384C2314yes—
179#380—22no—
180#380—22no—
181#380—24no—
182C4×C4⋊C4—21no—
183C4×C4⋊C4—21no—
184C4×C4⋊C4C2312yes—
185C4×C4⋊C4C2312yes—
186C4×C4⋊C4C2312yes—
187C4×C4⋊C4C2312yes—
188C43C2311yescharacteristic
189#340C2314yes—
190#340C2312yes—
191#340C2314yes—
192#340C2314yes—
193#340C2312yes—
194#340C2312yes—
195#340C2314yes—
196#340C2312yes—
197#340C2314yes—
198#336C2314yes—
199#336C2312yes—
200#336C2312yes—
201#336C2312yes—
202#336C2314yes—
203#336C2312yes—
204C22×Q8—21no—
205C22×Q8—21no—
206C22×Q8—24no—
207C22×Q8—48no—
208C22×Q8—48no—
209C22×Q8—84no—
210C22×Q8C2×D412yes—
211C22×D4—21no—
212C22×D4—21no—
213C22×D4—21no—
214C22×D4—21no—
215C22×D4C2412yes—
216C22×D4C2412yes—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—24no—
220C23×C4—24no—
221C2×D8—28no—
222C2×D8—28no—
223C2×D8—44no—
224C22×C8C2414yes—
225C22×C8C2412yes—
226C22×C8C2412yes—
227C4⋊Q8—24no—
228C4⋊Q8—28no—
229C4⋊Q8—22no—
230C4⋊Q8—22no—
231C4⋊Q8—44no—
232C4⋊Q8—48no—
233C4⋊Q8D818yescomplemented
234#127—24no—
235#127—22no—
236#127#4114yes—
237#125—48no—
238C22⋊Q8—416no—
239C22⋊Q8—48no—
240C22⋊Q8—48no—
241C22⋊Q8—416no—
242C22⋊Q8—416no—
243C22⋊Q8—48no—
244C22⋊Q8—84no—
245C4×Q8—44no—
246C4×Q8—44no—
247C4×Q8—44no—
248C4×D4—28no—
249C4×D4—24no—
250C4×D4—24no—
251C4×D4—44no—
252C4×D4—44no—
253C4×D4—44no—
254C4×D4—44no—
255C4×D4—48no—
256C4×D4—48no—
257C4×D4—44no—
258C4×D4—44no—
259C4×D4—82no—
260C4×D4—82no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—48no—
279C2×C4⋊C4C2×D412yes—
280C2×C4⋊C4C2×D412yes—
281C2×C4⋊C4C2×D414yes—
282C2×C4⋊C4C2×D412yes—
283C2×C4⋊C4C2×D412yes—
284C2×C4⋊C4C2×D412yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D411yescharacteristic
307C2×C42C2×D411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42C2411yescharacteristic
310C2×C42C2414yes—
311C2×C42C2411yescharacteristic
312C2×C42C2411yescharacteristic
313#107—28no—
314#107—44no—
315C4⋊C8—24no—
316C4⋊C8—44no—
317#102—28no—
318#102—28no—
319#102—24no—
320#102—24no—
321#102—24no—
322#102—24no—
323#102—44no—
324#102—44no—
325C4×C8—22no—
326C4×C8—22no—
327C4×C8—44no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95—22no—
332#95—24no—
333#95—24no—
334#95C2×D414yes—
335#95C2×D414yes—
336C24—22no—
337C24—22no—
338C2×Q8—24no—
339C2×Q8—42no—
340C2×Q8—42no—
341C2×Q8—48no—
342C2×Q8—42no—
343C2×Q8—42no—
344C2×Q8—42no—
345C2×Q8—42no—
346C2×Q8—48no—
347C2×Q8—42no—
348C2×Q8—84no—
349C2×Q8—832no—
350C2×Q8C2×D818yes—
351C2×D4—24no—
352C2×D4—24no—
353C2×D4—24no—
354C2×D4—24no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—24no—
359C2×D4—22no—
360C2×D4—24no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—42no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—81no—
368C2×D4—81no—
369C2×D4—81no—
370C2×D4—81no—
371C2×D4C2×#4114yes—
372C2×D42−1+414yes—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—48no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—48no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—48no—
388C22×C4—84no—
389C22×C4—84no—
390C22×C4—84no—
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D414yes—
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D414yes—
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C2511yescharacteristic
400D8—416no—
401C2×C8—22no—
402C2×C8—24no—
403C2×C8—22no—
404C2×C8—24no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—24no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—44no—
424C4⋊C4—48no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—48no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—44no—
431C4⋊C4—42no—
432C4⋊C4—44no—
433C4⋊C4—48no—
434C4⋊C4—48no—
435C4⋊C4—81no—
436C4⋊C4—81no—
437C4⋊C4—81no—
438C4⋊C4—81no—
439C4⋊C4—84no—
440C4⋊C4C2×D818yes—
441C4⋊C4C2×D818yes—
442C4⋊C4H(5,𝔽2)14yes—
443C22⋊C4—48no—
444C22⋊C4—44no—
445C22⋊C4—44no—
446C22⋊C4—48no—
447C22⋊C4—48no—
448C22⋊C4—44no—
449C22⋊C4—44no—
450C22⋊C4—84no—
451C22⋊C4—84no—
452C22⋊C4—82no—
453C22⋊C4—82no—
454C22⋊C4—84no—
455C42—22no—
456C42—24no—
457C42—24no—
458C42—22no—
459C42—22no—
460C42—41no—
461C42—41no—
462C42—44no—
463C42—44no—
464C42C2×D814yes—
465C42C2×#4112yes—
466C422−1+412yes—
467C23—24no—
468C23—24no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23C23×D411yescharacteristic
482Q8—48no—
483Q8—48no—
484Q8—816no—
485Q8—84no—
486D4—24no—
487D4—28no—
488D4—24no—
489D4—48no—
490D4—48no—
491D4—42no—
492D4—42no—
493D4—44no—
494D4—84no—
495D4—84no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—24no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—24no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—24no—
507C2×C4—21no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—24no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—816no—
532C2×C4—88no—
533C2×C4—88no—
534C2×C4—168no—
535C2×C4#54812yes—
536C2×C4#54812yes—
537C2×C4C22×D812yes—
538C2×C4C22×D812yes—
539C2×C4C22×D818yes—
540C2×C4C22×D812yes—
541C2×C4#57712yes—
542C2×C4C22×#4111yescharacteristic
543C2×C4C2×H(5,𝔽2)12yes—
544C2×C4C2×2−1+412yes—
545C2×C4C2×2−1+411yescharacteristic
546C8—42no—
547C8—44no—
548C8—42no—
549C22—44no—
550C22—44no—
551C22—44no—
552C22—44no—
553C22—44no—
554C22—44no—
555C22—82no—
556C22—82no—
557C22—82no—
558C22—82no—
559C22—82no—
560C22—82no—
561C22—82no—
562C22—82no—
563C22C2×#54811yescharacteristic
564C22#348411yescharacteristic
565C22#348611yescharacteristic
566C22C23×D811yescharacteristic
567C22C2×#57711yescharacteristic
568C22#358511yescharacteristic
569C22#358611yescharacteristic
570C4—22no—
571C4—28no—
572C4—22no—
573C4—22no—
574C4—44no—
575C4—41no—
576C4—41no—
577C4—44no—
578C4—82no—
579C4—82no—
580C4—82no—
581C4—82no—
582C4—82no—
583C4#335814yes—
584C4#352012yes—
585C4#355212yes—
586C2—84no—
587C2—84no—
588C2#4533711yescharacteristic
589C2C2×#335811yescharacteristic
590C2#6133311yescharacteristic
591C2#6155911yescharacteristic
592C2#6168611yescharacteristic
593C2#6181711yescharacteristic
594C2#6185011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#28156C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #59823C2semidirect
2. #59817C2semidirect
3. #32544C2semidirect
4. #31454C2semidirect
5. #29272C2semidirect
7. #27813C2semidirect
8. #26889C2semidirect
9. #26040C2semidirect
10. #25318C2semidirect
11. #24326C2semidirect
12. #23815C2semidirect
13. #22995C2semidirect
14. #22307C2semidirect
15. #22220C2semidirect
233. C4⋊Q8D8semidirect

Maximal subgroups

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

subgroupindex
#598232
#598172
#325442
#314542
#292722
#281562
#278132
#268892
#260402
#253182
#243262
#238152
#229952
#223072
#222202