Finite Group Databasebeta

← #6497310

Subgroups — #6497310

Order 512.

1477 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35280C211yesmaximal, characteristic
2#34189C211yesmaximal, characteristic
3#33158C214yesmaximal, complemented
4#29949C212yesmaximal, complemented
5#29948C212yesmaximal, complemented
6#27969C212yesmaximal, complemented
7#27955C212yesmaximal, complemented
8#27883C212yesmaximal, complemented
9#27882C212yesmaximal, complemented
10#27621C212yesmaximal, complemented
11#27620C212yesmaximal, complemented
12#27606C214yesmaximal
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#2314C2211yescharacteristic
33#2314C2211yescharacteristic
34#2314C2211yescharacteristic
35#2314C2211yescharacteristic
36#2311C2212yes—
37#2311C2212yes—
38#2285C2211yescharacteristic
39C2×#384C2211yescharacteristic
40#2283C2212yes—
41#2283C2212yes—
42C2×#382C2211yescharacteristic
43#2274C2212yes—
44#2101C2214yes—
45#2101C2214yes—
46#2084C2214yes—
47#2083C2214yes—
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#1854C2214yes—
65#1854C2214yes—
66#1853C2214yes—
67#1852C2214yes—
68#1847C2212yes—
69#1847C2212yes—
70#1836C2212yes—
71#1836C2212yes—
72#1825C2214yes—
73#1825C2214yes—
74#1824C2214yes—
75#1823C2214yes—
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×#117C2312yes—
98C2×#117C2311yescharacteristic
99C2×#117C2311yescharacteristic
100C2×#117C2311yescharacteristic
101C22×C4⋊C4—22no—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C42C2311yescharacteristic
105C22×C42C2311yescharacteristic
106C22×C42C2311yescharacteristic
107#432—28no—
108C2×#107C2314yes—
109C2×#107C2314yes—
110C2×#106C2314yes—
111C2×#106C2314yes—
112C2×C4⋊C8C2314yes—
113C2×C22⋊C8C2314yes—
114C2×C22⋊C8C2314yes—
115C2×C22⋊C8C2312yes—
116C2×C22⋊C8C2312yes—
117C2×#97C2312yes—
118C2×#97C2312yes—
119#398—24no—
120#398—24no—
121#396—24no—
122#396—24no—
123#391—22no—
124#391—22no—
125#391—22no—
126#391—22no—
127#390—22no—
128#390—22no—
129#389—24no—
130#389—22no—
131#389—22no—
132#389C2314yes—
133#389C2314yes—
134#388—22no—
135#388—22no—
136#388—22no—
137#388—22no—
138#387—22no—
139#387—22no—
140#387—22no—
141#387—22no—
142#386—22no—
143#386—22no—
144#386—22no—
145#386—22no—
146#384—22no—
147#384C2311yescharacteristic
148#384C2311yescharacteristic
149#384C2312yes—
150#384C2312yes—
151#383—22no—
152#383—22no—
153#382—22no—
154#382C2311yescharacteristic
155#382C2311yescharacteristic
156#382C2312yes—
157#382C2312yes—
158#381—22no—
159#381—22no—
160#380—22no—
161#380—22no—
162#380—22no—
163#380—24no—
164#380C2312yes—
165#380C2312yes—
166#380C2312yes—
167#380C2312yes—
168#380C2314yes—
169#380C2314yes—
170C4×C4⋊C4—22no—
171C4×C4⋊C4—22no—
172C4×C4⋊C4—22no—
173C4×C4⋊C4C2312yes—
174C4×C4⋊C4C2312yes—
175C4×C4⋊C4C2312yes—
176C4×C4⋊C4C2312yes—
177C4×C22⋊C4—22no—
178C4×C22⋊C4—22no—
179C4×C22⋊C4—22no—
180C4×C22⋊C4—22no—
181#376—22no—
182#376—22no—
183C2×#95—22no—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#336C2314yes—
197#336C2312yes—
198#336C2312yes—
199C2×#41—24no—
200C2×#41—24no—
201C2×#41—42no—
202C22×Q8—22no—
203C22×Q8C2×D411yescharacteristic
204C22×D4C2×D411yescharacteristic
205C22×D4C2×D411yescharacteristic
206C22×D4C2×D411yescharacteristic
207C23×C4—22no—
208C23×C4—22no—
209C23×C4C2×D412yes—
210C23×C4C2×D412yes—
211C23×C4C2×D412yes—
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C22×C8C2414yes—
216C22×C8C2414yes—
217C4⋊Q8—28no—
218#125—28no—
219C22⋊Q8—48no—
220C22⋊Q8—84no—
221C22⋊Q8—84no—
222C4×Q8—28no—
223C4×Q8—48no—
224C4×D4—28no—
225C4×D4—28no—
226C4×D4—28no—
227#117—24no—
228#117—24no—
229#117—24no—
230#117—28no—
231#117—44no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—84no—
246C2×C4⋊C4C2×D412yes—
247C2×C4⋊C4C2×D412yes—
248C2×C4⋊C4C2×D412yes—
249C2×C4⋊C4C2412yes—
250C2×C4⋊C4C2412yes—
251C2×C4⋊C4C2414yes—
252C2×C4⋊C4C2414yes—
253C2×C4⋊C4C2412yes—
254C2×C4⋊C4C2412yes—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4C2×D412yes—
264C2×C22⋊C4C2×D412yes—
265C2×C22⋊C4C2×D412yes—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42C2×D412yes—
275C2×C42C2×D412yes—
276C2×C42C2×D412yes—
277C2×C42#4118yes—
278C2×C42C2411yescharacteristic
279C2×C42C2411yescharacteristic
280C2×C42C2411yescharacteristic
281C2×C42C2411yescharacteristic
282#107—44no—
283#107—44no—
284#106—44no—
285#106—44no—
286C4⋊C8—28no—
287C22⋊C8—28no—
288C22⋊C8—28no—
289C22⋊C8—42no—
290C22⋊C8—42no—
291#97—24no—
292#97—24no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—22no—
300#95—22no—
301#95—22no—
302#95—22no—
303#95—22no—
304#95—22no—
305#95—22no—
306#95—22no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—24no—
311C24C22×D411yescharacteristic
312C24C22×D411yescharacteristic
313C24C22×D411yescharacteristic
314#41—416no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—41no—
318C2×Q8—44no—
319C2×Q8—44no—
320C2×Q8—44no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—41no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—44no—
352C22×C4—44no—
353C22×C4—84no—
354C22×C4—82no—
355C22×C4—82no—
356C22×C4#12318yes—
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D412yes—
359C22×C4C22×D412yes—
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D412yes—
362C22×C4C22×D412yes—
363C22×C4C22×D411yescharacteristic
364C22×C4C2×#4114yes—
365C22×C42−1+414yes—
366C22×C4C2511yescharacteristic
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—24no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—28no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—44no—
384C4⋊C4—44no—
385C4⋊C4—44no—
386C4⋊C4—48no—
387C4⋊C4—44no—
388C4⋊C4—44no—
389C4⋊C4—82no—
390C4⋊C4—82no—
391C4⋊C4—84no—
392C4⋊C4—84no—
393C4⋊C4—84no—
394C22⋊C4—24no—
395C22⋊C4—24no—
396C22⋊C4—24no—
397C22⋊C4—44no—
398C22⋊C4—44no—
399C22⋊C4—82no—
400C22⋊C4—82no—
401C22⋊C4—82no—
402C22⋊C4—82no—
403C22⋊C4—84no—
404C22⋊C4—84no—
405C42—24no—
406C42—24no—
407C42—22no—
408C42—22no—
409C42—44no—
410C42—44no—
411C42—82no—
412C42—82no—
413C42—82no—
414C42—82no—
415C42#12318yes—
416C42C2×#4114yes—
417C42C2×#4114yes—
418C23—22no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—22no—
427C23—22no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23C2×#12312yes—
432C23#53612yes—
433C23C23×D411yescharacteristic
434Q8—44no—
435Q8—88no—
436D4—44no—
437D4—44no—
438D4—44no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—28no—
448C2×C4—28no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—48no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—84no—
479C2×C4—84no—
480C2×C4—164no—
481C2×C4—164no—
482C2×C4C2×#12314yes—
483C2×C4C2×#12312yes—
484C2×C4C2×#12314yes—
485C2×C4#53612yes—
486C2×C4C22×#4112yes—
487C2×C4C2×2−1+412yes—
488C2×C4#58512yes—
489C8—44no—
490C8—44no—
491C22—24no—
492C22—24no—
493C22—22no—
494C22—22no—
495C22—41no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22—42no—
500C22—42no—
501C22—41no—
502C22—41no—
503C22—42no—
504C22—42no—
505C22—41no—
506C22C22×#12311yescharacteristic
507C22C2×#53611yescharacteristic
508C22#344911yescharacteristic
509C22#358512yes—
510C22#358612yes—
511C4—24no—
512C4—24no—
513C4—22no—
514C4—22no—
515C4—42no—
516C4—42no—
517C4—41no—
518C4—82no—
519C4—84no—
520C4—82no—
521C4—84no—
522C4—82no—
523C4—82no—
524C4#349514yes—
525C2—42no—
526C2—42no—
527C2—81no—
528C2#3531111yescharacteristic
529C2#6052012yes—
530C2#6088512yes—
531C2#6101911yescharacteristic
532C2#6102011yescharacteristic
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. #33158C2semidirect
4. #29949C2semidirect
5. #29948C2semidirect
6. #27969C2semidirect
7. #27955C2semidirect
8. #27883C2semidirect
9. #27882C2semidirect
10. #27621C2semidirect
11. #27620C2semidirect
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
#331582
#299492
#299482
#279692
#279552
#278832
#278822
#276212
#276202
#276062
#219872
#218422
#207192