Finite Group Databasebeta

← #6497309

Subgroups — #6497309

Order 512.

1477 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35280C211yesmaximal, characteristic
2#34189C211yesmaximal, characteristic
3#33157C212yesmaximal, complemented
4#33156C212yesmaximal, complemented
5#29946C214yesmaximal, complemented
6#27968C212yesmaximal, complemented
7#27956C212yesmaximal, complemented
8#27883C212yesmaximal, complemented
9#27882C212yesmaximal, complemented
10#27623C214yesmaximal, complemented
11#27609C212yesmaximal
12#27608C212yesmaximal
13#21987C212yesmaximal, complemented
14#21842C212yesmaximal, complemented
15#20719C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C2×C4×#41C2211yescharacteristic
18C2×#432C2212yes—
19#2857C2212yes—
20#2857C2212yes—
21#2629C2212yes—
22#2629C2212yes—
23#2582—22no—
24#2538—22no—
25#2417—22no—
26#2417—22no—
27#2357—22no—
28#2356—22no—
29#2353—22no—
30#2353—22no—
31#2316C2212yes—
32#2314C2212yes—
33#2314C2212yes—
34#2311C2211yescharacteristic
35#2311C2211yescharacteristic
36#2311C2212yes—
37#2285C2211yescharacteristic
38C2×#384C2211yescharacteristic
39#2283C2212yes—
40#2283C2212yes—
41C2×#382C2211yescharacteristic
42#2274C2211yescharacteristic
43#2274C2211yescharacteristic
44#2100C2214yes—
45#2099C2214yes—
46#2085C2214yes—
47#2085C2214yes—
48#2077C2214yes—
49#2077C2214yes—
50#2076C2214yes—
51#2075C2214yes—
52#1993C2212yes—
53#1993C2212yes—
54#1993C2212yes—
55#1993C2212yes—
56#1989C2214yes—
57#1989C2214yes—
58#1948C2214yes—
59#1947C2214yes—
60#1921C2214yes—
61#1920C2214yes—
62#1919C2214yes—
63#1918C2214yes—
64#1853C2214yes—
65#1853C2214yes—
66#1852C2214yes—
67#1852C2214yes—
68#1847C2212yes—
69#1847C2212yes—
70#1836C2212yes—
71#1836C2212yes—
72#1825C2214yes—
73#1825C2214yes—
74#1825C2214yes—
75#1825C2214yes—
76#1818C2212yes—
77#1798C2212yes—
78#1798C2212yes—
79#1780C2212yes—
80#1749C2212yes—
81#1749C2211yescharacteristic
82#1749C2211yescharacteristic
83C22×#41C2311yescharacteristic
84#533—28no—
85C2×C4⋊Q8C2312yes—
86C2×#125C2312yes—
87C2×C22⋊Q8—22no—
88C2×C22⋊Q8—22no—
89C2×C22⋊Q8—22no—
90C4×#41—28no—
91C2×C4×Q8—22no—
92C2×C4×Q8D412yes—
93C2×C4×D4D412yes—
94C2×C4×D4D412yes—
95C2×C4×D4D412yes—
96C2×#117C2312yes—
97C2×#117C2311yescharacteristic
98C2×#117C2311yescharacteristic
99C2×#117C2311yescharacteristic
100C2×#117C2311yescharacteristic
101C2×#117C2311yescharacteristic
102C22×C4⋊C4—22no—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C42C2311yescharacteristic
107C22×C42C2311yescharacteristic
108C22×C42C2311yescharacteristic
109#432—28no—
110C2×#107C2314yes—
111C2×#107C2314yes—
112C2×#106C2314yes—
113C2×#106C2314yes—
114C2×C4⋊C8C2314yes—
115C2×C22⋊C8C2314yes—
116C2×C22⋊C8C2314yes—
117C2×C22⋊C8C2312yes—
118C2×C22⋊C8C2312yes—
119C2×#97C2312yes—
120C2×#97C2312yes—
121#398—24no—
122#398—24no—
123#396—24no—
124#396—24no—
125#391—22no—
126#391—21no—
127#391—21no—
128#391—22no—
129#391—22no—
130#390—21no—
131#390—21no—
132#390—22no—
133#389—24no—
134#389—22no—
135#389—22no—
136#389C2314yes—
137#389C2314yes—
138#388—22no—
139#388—22no—
140#388—22no—
141#388—22no—
142#387—22no—
143#387—22no—
144#387—22no—
145#387—22no—
146#386—21no—
147#386—21no—
148#386—21no—
149#386—21no—
150#386—22no—
151#386—22no—
152#384—22no—
153#384C2312yes—
154#384C2312yes—
155#384C2312yes—
156#383—22no—
157#383—22no—
158#382—22no—
159#382C2312yes—
160#382C2312yes—
161#382C2312yes—
162#381—22no—
163#381—22no—
164#380—22no—
165#380—22no—
166#380—22no—
167#380—24no—
168#380C2312yes—
169#380C2312yes—
170#380C2312yes—
171#380C2312yes—
172#380C2314yes—
173#380C2314yes—
174C4×C4⋊C4—22no—
175C4×C4⋊C4—22no—
176C4×C4⋊C4—22no—
177C4×C4⋊C4C2312yes—
178C4×C4⋊C4C2312yes—
179C4×C4⋊C4C2312yes—
180C4×C4⋊C4C2312yes—
181C4×C22⋊C4—22no—
182C4×C22⋊C4—22no—
183C4×C22⋊C4—22no—
184C4×C22⋊C4—22no—
185#376—22no—
186#376—22no—
187C2×#95—22no—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#336C2314yes—
201#336C2312yes—
202#336C2312yes—
203C2×#41—24no—
204C2×#41—24no—
205C2×#41—42no—
206C22×Q8—22no—
207C22×Q8C2×D411yescharacteristic
208C22×D4C2×D411yescharacteristic
209C22×D4C2×D411yescharacteristic
210C22×D4C2×D411yescharacteristic
211C23×C4—22no—
212C23×C4—21no—
213C23×C4—21no—
214C23×C4C2×D412yes—
215C23×C4C2×D412yes—
216C23×C4C2×D412yes—
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C22×C8C2414yes—
221C22×C8C2414yes—
222C4⋊Q8—28no—
223#125—28no—
224C22⋊Q8—48no—
225C22⋊Q8—84no—
226C22⋊Q8—84no—
227C4×Q8—28no—
228C4×Q8—48no—
229C4×D4—28no—
230C4×D4—28no—
231C4×D4—28no—
232#117—24no—
233#117—24no—
234#117—24no—
235#117—28no—
236#117—24no—
237#117—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—84no—
255C2×C4⋊C4C2×D412yes—
256C2×C4⋊C4C2×D412yes—
257C2×C4⋊C4C2×D412yes—
258C2×C4⋊C4C2412yes—
259C2×C4⋊C4C2412yes—
260C2×C4⋊C4C2414yes—
261C2×C4⋊C4C2414yes—
262C2×C4⋊C4C2412yes—
263C2×C4⋊C4C2412yes—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4C2×D412yes—
274C2×C22⋊C4C2×D412yes—
275C2×C22⋊C4C2×D412yes—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—22no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42C2×D412yes—
286C2×C42C2×D412yes—
287C2×C42C2×D412yes—
288C2×C42#4118yes—
289C2×C42C2411yescharacteristic
290C2×C42C2411yescharacteristic
291C2×C42C2411yescharacteristic
292C2×C42C2411yescharacteristic
293#107—44no—
294#107—44no—
295#106—44no—
296#106—44no—
297C4⋊C8—28no—
298C22⋊C8—28no—
299C22⋊C8—28no—
300C22⋊C8—42no—
301C22⋊C8—42no—
302#97—24no—
303#97—24no—
304#95—22no—
305#95—22no—
306#95—22no—
307#95—22no—
308#95—22no—
309#95—22no—
310#95—22no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95—22no—
315#95—22no—
316#95—22no—
317#95—22no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—24no—
322C24C22×D411yescharacteristic
323C24C22×D411yescharacteristic
324C24C22×D411yescharacteristic
325#41—416no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—41no—
329C2×Q8—44no—
330C2×Q8—44no—
331C2×Q8—44no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4—41no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—44no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—84no—
366C22×C4—82no—
367C22×C4—82no—
368C22×C4#12318yes—
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D412yes—
371C22×C4C22×D412yes—
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D412yes—
374C22×C4C22×D412yes—
375C22×C4C22×D411yescharacteristic
376C22×C4C2×#4114yes—
377C22×C42−1+414yes—
378C22×C4C2511yescharacteristic
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—24no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—28no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—44no—
398C4⋊C4—44no—
399C4⋊C4—44no—
400C4⋊C4—48no—
401C4⋊C4—44no—
402C4⋊C4—82no—
403C4⋊C4—82no—
404C4⋊C4—84no—
405C4⋊C4—84no—
406C4⋊C4—84no—
407C22⋊C4—24no—
408C22⋊C4—24no—
409C22⋊C4—24no—
410C22⋊C4—44no—
411C22⋊C4—42no—
412C22⋊C4—42no—
413C22⋊C4—82no—
414C22⋊C4—82no—
415C22⋊C4—82no—
416C22⋊C4—82no—
417C22⋊C4—84no—
418C22⋊C4—84no—
419C42—24no—
420C42—24no—
421C42—22no—
422C42—22no—
423C42—44no—
424C42—42no—
425C42—42no—
426C42—82no—
427C42—82no—
428C42—82no—
429C42—82no—
430C42#12318yes—
431C42C2×#4114yes—
432C42C2×#4114yes—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—22no—
442C23—22no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23C2×#12312yes—
447C23#53612yes—
448C23C23×D411yescharacteristic
449Q8—44no—
450Q8—88no—
451D4—44no—
452D4—44no—
453D4—44no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—28no—
463C2×C4—28no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—48no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—84no—
494C2×C4—84no—
495C2×C4—164no—
496C2×C4—162no—
497C2×C4—162no—
498C2×C4C2×#12314yes—
499C2×C4C2×#12312yes—
500C2×C4C2×#12314yes—
501C2×C4#53612yes—
502C2×C4C22×#4112yes—
503C2×C4C2×2−1+412yes—
504C2×C4#58512yes—
505C8—44no—
506C8—44no—
507C22—24no—
508C22—24no—
509C22—22no—
510C22—22no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22—42no—
516C22—42no—
517C22—41no—
518C22—41no—
519C22—42no—
520C22—42no—
521C22—41no—
522C22C22×#12311yescharacteristic
523C22C2×#53611yescharacteristic
524C22#344911yescharacteristic
525C22#358512yes—
526C22#358612yes—
527C4—24no—
528C4—24no—
529C4—22no—
530C4—22no—
531C4—42no—
532C4—42no—
533C4—41no—
534C4—82no—
535C4—84no—
536C4—82no—
537C4—84no—
538C4—82no—
539C4—82no—
540C4#349514yes—
541C2—42no—
542C2—42no—
543C2—81no—
544C2#3531111yescharacteristic
545C2#6052012yes—
546C2#6088712yes—
547C2#6101711yescharacteristic
548C2#6101811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
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
3. #33157C2semidirect
4. #33156C2semidirect
5. #29946C2semidirect
6. #27968C2semidirect
7. #27956C2semidirect
8. #27883C2semidirect
9. #27882C2semidirect
10. #27623C2semidirect
13. #21987C2semidirect
14. #21842C2semidirect

Maximal subgroups

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

subgroupindex
#352802
#341892
#331572
#331562
#299462
#279682
#279562
#278832
#278822
#276232
#276092
#276082
#219872
#218422
#207192