Finite Group Databasebeta

← #6644685

Subgroups — #6644685

Order 512.

1612 conjugacy classes of subgroups.

The subgroup lattice has 563 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35082C211yesmaximal, characteristic
2C2×#2567C211yesmaximal, characteristic, complemented
3#33003C212yesmaximal, complemented
4#32442C214yesmaximal, complemented
5#32348C214yesmaximal, complemented
6#31638C214yesmaximal, complemented
7#31534C214yesmaximal, complemented
8#29638C212yesmaximal, complemented
9#29106C211yesmaximal, characteristic, complemented
10#28314C211yesmaximal, characteristic
11#28118C212yesmaximal, complemented
12#27966C211yesmaximal, characteristic, complemented
13#27909C211yesmaximal, characteristic
14#27644C212yesmaximal, complemented
15#21914C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C2×#480C2212yes—
19C2×#480C2212yes—
20C2×#480C2212yes—
21C2×#448C2212yes—
22#2856C2211yescharacteristic
23#2854C2212yes—
24#2851C2211yescharacteristic
25#2651—22no—
26#2567—22no—
27#2567—22no—
28#2567C2214yes—
29#2567C2214yes—
30#2513C2212yes—
31#2507C2212yes—
32#2454C2212yes—
33#2445C2212yes—
34#2443—22no—
35#2425—22no—
36C2×#395—22no—
37#2335—22no—
38C4×#121C2212yes—
39C2×#391C2211yescharacteristic
40C2×#391C2211yescharacteristic
41C2×#389—22no—
42C2×#384C2211yescharacteristic
43C2×#384C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45#2100C2214yes—
46#2095C2212yes—
47#2093C2214yes—
48#2083C2214yes—
49#2074C2212yes—
50#2071C2214yes—
51#2068C2214yes—
52#2057C2214yes—
53#2057C2214yes—
54#2055C2214yes—
55#2026C2212yes—
56#2023C2214yes—
57#2023C2214yes—
58#2014C2212yes—
59#1992C2212yes—
60#1973C2214yes—
61#1957C2212yes—
62#1947C2214yes—
63#1933C2214yes—
64#1928C2214yes—
65#1917C2214yes—
66#1916C2214yes—
67#1880C2214yes—
68#1880C2214yes—
69#1877C2214yes—
70#1863C2214yes—
71#1863C2214yes—
72#1845C2212yes—
73#1836C2212yes—
74#1823C2212yes—
75#1823C2212yes—
76#1823C2212yes—
77#1817C2212yes—
78#1802C2214yes—
79#1797C2212yes—
80#1787C2212yes—
81#1749C2211yescharacteristic
82C4×C22⋊C8C2211yescharacteristic
83C23×Q8—21no—
84C23×Q8C2311yescharacteristic
85C22×SD16C2314yes—
86C2×C4⋊Q8—22no—
87C2×C4⋊Q8—44no—
88C2×C4⋊Q8C2314yes—
89C2×#127C2311yescharacteristic
90C2×#125C2311yescharacteristic
91C2×#123—22no—
92C2×#123—22no—
93C2×#121—22no—
94C2×#121C2312yes—
95C2×C4×Q8—22no—
96C2×C4×Q8—44no—
97C2×C4×Q8C2314yes—
98C2×C4×D4—22no—
99C2×C4×D4C2312yes—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—22no—
103C22×C4⋊C4—22no—
104C22×C4⋊C4D412yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C42—21no—
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112#480—44no—
113#480—44no—
114#480—44no—
115#448—44no—
116C2×#107C2314yes—
117C2×#107C2314yes—
118C2×#107C2314yes—
119C2×C4⋊C8C2314yes—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×#102C2314yes—
124C2×#102C2314yes—
125C2×#102C2314yes—
126C2×C22⋊C8C2312yes—
127C2×C22⋊C8C2312yes—
128C2×#97C2312yes—
129C2×C4×C8C2312yes—
130#399—22no—
131#398—22no—
132#398—22no—
133#397—22no—
134#397—22no—
135#396—22no—
136#395—24no—
137#395—24no—
138#395—24no—
139#395—24no—
140#394—22no—
141#394—22no—
142#391—21no—
143#391—22no—
144#391—21no—
145#391C2314yes—
146#391C2312yes—
147#391C2312yes—
148#390—22no—
149#390C2312yes—
150#390C2312yes—
151#390C2312yes—
152#390C2312yes—
153#389—24no—
154#389—24no—
155#389—24no—
156#389—24no—
157#389—22no—
158#389C2312yes—
159#389C2312yes—
160#389C2312yes—
161#389C2312yes—
162#388—22no—
163#388—22no—
164#388—22no—
165#388—22no—
166#388—22no—
167#387—22no—
168#387—22no—
169#386—22no—
170#386—22no—
171#386—22no—
172#385—22no—
173#385—22no—
174#384—21no—
175#384—22no—
176#384—21no—
177#384C2314yes—
178#384C2312yes—
179#384C2312yes—
180#384C2311yescharacteristic
181#382C2311yescharacteristic
182#380—22no—
183#380C2312yes—
184C4×C4⋊C4D412yes—
185C4×C4⋊C4D412yes—
186C4×C4⋊C4C2312yes—
187C4×C4⋊C4C2312yes—
188C4×C4⋊C4C2312yes—
189C4×C22⋊C4—22no—
190C2×#95—22no—
191C2×#95—22no—
192C2×#95D412yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#336C2314yes—
200#336C2314yes—
201#336C2314yes—
202#336C2312yes—
203#336C2312yes—
204C22×Q8—24no—
205C22×Q8—21no—
206C22×Q8—24no—
207C22×Q8—24no—
208C22×Q8—42no—
209C22×Q8—44no—
210C22×Q8—84no—
211C22×Q8C2412yes—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4C2412yes—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—22no—
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D412yes—
223C23×C4C2×D412yes—
224C23×C4C2411yescharacteristic
225C2×SD16—28no—
226C2×SD16—44no—
227C2×SD16—44no—
228C22×C8C2414yes—
229C22×C8C2414yes—
230C4⋊Q8—48no—
231C4⋊Q8—48no—
232#127—42no—
233#125—42no—
234#123—48no—
235#123—84no—
236#121—24no—
237#121—24no—
238#121—84no—
239C4×Q8—48no—
240C4×Q8—48no—
241C4×D4—44no—
242C4×D4—84no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—84no—
265C2×C4⋊C4—84no—
266C2×C4⋊C4C2×D412yes—
267C2×C4⋊C4C2×D412yes—
268C2×C4⋊C4C2×D412yes—
269C2×C4⋊C4C2×D412yes—
270C2×C4⋊C4C2412yes—
271C2×C4⋊C4C2412yes—
272C2×C4⋊C4C2412yes—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2412yes—
275C2×C4⋊C4C2412yes—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—22no—
290C2×C42—42no—
291C2×C42—42no—
292C2×C42—44no—
293C2×C42C2×D412yes—
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D412yes—
296C2×C42C2×D411yescharacteristic
297C2×C42C2412yes—
298C2×C42C2412yes—
299C2×C42C2411yescharacteristic
300C2×C42C2411yescharacteristic
301#107—44no—
302#107—44no—
303#107—44no—
304C4⋊C8—44no—
305#103—28no—
306#103—44no—
307#103—44no—
308#102—44no—
309#102—44no—
310#102—44no—
311C22⋊C8—24no—
312C22⋊C8—24no—
313#97—42no—
314C4×C8—42no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—22no—
320#95—22no—
321#95—22no—
322#95—22no—
323#95—22no—
324#95—22no—
325#95—22no—
326#95C2×D412yes—
327#95C2×D412yes—
328#95C2×D412yes—
329#95C2×D412yes—
330C24—22no—
331C24C22×D411yescharacteristic
332C2×Q8—24no—
333C2×Q8—22no—
334C2×Q8—24no—
335C2×Q8—24no—
336C2×Q8—416no—
337C2×Q8—42no—
338C2×Q8—48no—
339C2×Q8—42no—
340C2×Q8—84no—
341C2×Q8—84no—
342C2×Q8—816no—
343C2×Q82−1+414yes—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—24no—
347C2×D4—24no—
348C2×D4—44no—
349C2×D4—44no—
350C2×D4—44no—
351C2×D4—44no—
352C2×D4—82no—
353C2×D4—82no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—82no—
384C22×C4—82no—
385C22×C4C22×D412yes—
386C22×C4C22×D412yes—
387C22×C4C22×D412yes—
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D412yes—
390C22×C4C22×D412yes—
391C22×C4C22×D412yes—
392C22×C4H(5,𝔽2)14yes—
393C22×C4C2511yescharacteristic
394SD16—88no—
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—24no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—48no—
412C4⋊C4—48no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—44no—
417C4⋊C4—84no—
418C4⋊C4—84no—
419C4⋊C4—84no—
420C4⋊C4—84no—
421C4⋊C4—82no—
422C4⋊C4—82no—
423C4⋊C4—82no—
424C4⋊C4—82no—
425C4⋊C4—82no—
426C4⋊C4#13718yes—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4—44no—
430C22⋊C4—44no—
431C22⋊C4—44no—
432C22⋊C4—82no—
433C22⋊C4—82no—
434C22⋊C4—82no—
435C22⋊C4—82no—
436C22⋊C4—82no—
437C22⋊C4—82no—
438C42—24no—
439C42—24no—
440C42—22no—
441C42—22no—
442C42—24no—
443C42—41no—
444C42—42no—
445C42—41no—
446C42—48no—
447C42—82no—
448C23—21no—
449C23—21no—
450C23—22no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23#53512yes—
459C23#57614yes—
460C23C23×D411yescharacteristic
461Q8—28no—
462Q8—48no—
463Q8—44no—
464Q8—816no—
465D4—44no—
466D4—44no—
467D4—88no—
468D4—164no—
469C2×C4—28no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—28no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—48no—
490C2×C4—48no—
491C2×C4—48no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—84no—
511C2×C4—84no—
512C2×C4—84no—
513C2×C4—164no—
514C2×C4#53512yes—
515C2×C4C2×#13714yes—
516C2×C4C2×#13714yes—
517C2×C4C2×#13714yes—
518C2×C4C2×H(5,𝔽2)12yes—
519C2×C4C2×2−1+412yes—
520C2×C4#58512yes—
521C8—44no—
522C8—44no—
523C22—24no—
524C22—22no—
525C22—24no—
526C22—22no—
527C22—82no—
528C22—82no—
529C22—82no—
530C22—82no—
531C22—82no—
532C22—82no—
533C22—82no—
534C22#322014yes—
535C22C2×#53511yescharacteristic
536C22#348511yescharacteristic
537C22#348611yescharacteristic
538C22C22×#13712yes—
539C22C2×#57612yes—
540C4—24no—
541C4—24no—
542C4—24no—
543C4—22no—
544C4—42no—
545C4—42no—
546C4—42no—
547C4—44no—
548C4—82no—
549C4—82no—
550C4—82no—
551C4—82no—
552C4—82no—
553C4—82no—
554C4#353514yes—
555C2—24no—
556C2—162no—
557C2#3711811yescharacteristic
558C2C2×#322011yescharacteristic
559C2#6080011yescharacteristic
560C2#6172612yes—
561C2#6184412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2567C2semidirect
3. #33003C2semidirect
4. #32442C2semidirect
5. #32348C2semidirect
6. #31638C2semidirect
7. #31534C2semidirect
8. #29638C2semidirect
9. #29106C2semidirect
11. #28118C2semidirect
12. #27966C2semidirect
14. #27644C2semidirect
15. #21914C2semidirect

Maximal subgroups

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

subgroupindex
#350822
C2×#25672
#330032
#324422
#323482
#316382
#315342
#296382
#291062
#283142
#281182
#279662
#279092
#276442
#219142