Finite Group Databasebeta

← #6484841

Subgroups — #6484841

Order 512.

1352 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35345C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic
3#33157C212yesmaximal
4#33156C212yesmaximal
5#29979C214yesmaximal
6#29295C212yesmaximal, complemented
7#29294C212yesmaximal, complemented
8#29285C214yesmaximal, complemented
9#29223C212yesmaximal, complemented
10#29212C212yesmaximal, complemented
11#29182C212yesmaximal, complemented
12#29181C212yesmaximal, complemented
13#21987C212yesmaximal
14#21842C212yesmaximal
15#20811C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C2×#432C2212yes—
18#2864C2212yes—
19#2859C2211yescharacteristic
20#2859C2211yescharacteristic
21#2629C2212yes—
22#2629C2211yescharacteristic
23#2629C2211yescharacteristic
24#2608—22no—
25#2608—22no—
26#2606—22no—
27#2606—22no—
28#2360—22no—
29#2360—22no—
30#2355—22no—
31Q8×C4⋊C4—22no—
32#2330C2212yes—
33#2330C2211yescharacteristic
34#2330C2211yescharacteristic
35#2330C2212yes—
36#2330C2212yes—
37#2317C2212yes—
38C4×C4⋊Q8C2212yes—
39C4×C4⋊Q8C2212yes—
40C2×#384C2211yescharacteristic
41#2283C2212yes—
42#2274C2212yes—
43C4×#117C2211yescharacteristic
44#2100C2214yes—
45#2099C2214yes—
46#2088C2214yes—
47#2088C2214yes—
48#2087C2214yes—
49#2086C2214yes—
50#2059C2214yes—
51#2059C2214yes—
52#1993C2212yes—
53#1993C2212yes—
54#1993C2212yes—
55#1993C2212yes—
56#1989C2214yes—
57#1989C2214yes—
58#1989C2214yes—
59#1989C2214yes—
60#1948C2214yes—
61#1948C2214yes—
62#1947C2214yes—
63#1947C2214yes—
64#1921C2214yes—
65#1920C2214yes—
66#1919C2214yes—
67#1918C2214yes—
68#1853C2214yes—
69#1852C2214yes—
70#1847C2212yes—
71#1847C2212yes—
72#1843C2212yes—
73#1836C2212yes—
74#1836C2212yes—
75#1832C2212yes—
76#1825C2214yes—
77#1825C2214yes—
78#1798C2212yes—
79#1798C2212yes—
80#1767C2212yes—
81#1767C2211yescharacteristic
82#1767C2211yescharacteristic
83#533—24no—
84#533Q818yes—
85C2×C4⋊Q8C2312yes—
86C2×#125C2311yescharacteristic
87C2×#125C2311yescharacteristic
88C2×#125C2312yes—
89C2×#125C2312yes—
90C2×C22⋊Q8—22no—
91C2×C4×Q8—22no—
92C2×C4×Q8—22no—
93C2×C4×Q8—22no—
94C2×#117C2312yes—
95C2×#117C2311yescharacteristic
96C2×#117C2311yescharacteristic
97C22×C4⋊C4—22no—
98C22×C4⋊C4C2312yes—
99C22×C42C2311yescharacteristic
100#432—28no—
101C2×#107C2314yes—
102C2×#107C2314yes—
103C2×#106C2314yes—
104C2×#106C2314yes—
105C2×C4⋊C8C2314yes—
106C2×C4⋊C8C2314yes—
107C2×C4⋊C8C2312yes—
108C2×C4⋊C8C2312yes—
109C2×C22⋊C8C2314yes—
110C2×#97C2312yes—
111C2×#97C2312yes—
112#398—22no—
113#398—22no—
114#398—22no—
115#398—22no—
116#396—22no—
117#396—22no—
118#396—22no—
119#396—22no—
120#391—22no—
121#391—22no—
122#389—22no—
123#389—22no—
124#389—22no—
125#389—22no—
126#389—22no—
127#389—22no—
128#389—24no—
129#389—24no—
130#389—22no—
131#389—22no—
132#389—22no—
133#389—22no—
134#389C2314yes—
135#389C2314yes—
136#389C2312yes—
137#389C2312yes—
138#389C2312yes—
139#389C2312yes—
140#387—22no—
141#387—22no—
142#387—22no—
143#387—22no—
144#387—22no—
145#387—22no—
146#387—22no—
147#387—22no—
148#387—24no—
149#387—24no—
150#385—22no—
151#385—22no—
152#384—22no—
153#384—22no—
154#384C2312yes—
155#384C2312yes—
156#384C2312yes—
157#384C2311yescharacteristic
158#384C2311yescharacteristic
159#383—22no—
160#382—22no—
161#382C2312yes—
162#382C2312yes—
163#381—22no—
164#380C2312yes—
165#380C2312yes—
166C4×C4⋊C4—24no—
167C4×C4⋊C4—22no—
168C4×C4⋊C4—22no—
169C4×C4⋊C4D412yes—
170C4×C4⋊C4D412yes—
171C4×C4⋊C4C2314yes—
172C4×C4⋊C4C2314yes—
173C4×C4⋊C4C2312yes—
174C4×C4⋊C4C2312yes—
175C4×C22⋊C4—22no—
176C4×C22⋊C4—24no—
177C4×C22⋊C4D412yes—
178C4×C22⋊C4D412yes—
179#376—22no—
180#376C2311yescharacteristic
181#376C2311yescharacteristic
182C43C2311yescharacteristic
183C43C2311yescharacteristic
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—
199C22×Q8—22no—
200C23×C4—22no—
201C23×C4C2×D412yes—
202C23×C4C2411yescharacteristic
203C22×C8C2414yes—
204C22×C8C2414yes—
205C4⋊Q8—24no—
206C4⋊Q8C2×Q818yes—
207#125—24no—
208#125—24no—
209#125—24no—
210#125—44no—
211#125C2×Q818yes—
212C22⋊Q8—216no—
213C4×Q8—216no—
214C4×Q8—84no—
215C4×Q8—84no—
216#117—24no—
217#117—24no—
218#117—42no—
219#117C2×Q818yes—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—84no—
242C2×C4⋊C4C2×D412yes—
243C2×C4⋊C4C2×D411yescharacteristic
244C2×C4⋊C4C2×D411yescharacteristic
245C2×C4⋊C4C2414yes—
246C2×C4⋊C4C2412yes—
247C2×C4⋊C4C2412yes—
248C2×C4⋊C4C2412yes—
249C2×C4⋊C4C2412yes—
250C2×C4⋊C4C2414yes—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4C2×D412yes—
255C2×C22⋊C4C2×D411yescharacteristic
256C2×C22⋊C4C2×D411yescharacteristic
257C2×C42—22no—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42C2×D412yes—
263C2×C42C2×D412yes—
264C2×C42C2×D412yes—
265C2×C42C2×D412yes—
266C2×C42C2×D412yes—
267C2×C42C2×Q814yes—
268C2×C42#4114yes—
269C2×C42C2411yescharacteristic
270C2×C42C2411yescharacteristic
271C2×C42C2411yescharacteristic
272C2×C42C2411yescharacteristic
273C2×C42C2411yescharacteristic
274C2×C42C2411yescharacteristic
275#107—44no—
276#107—44no—
277#106—44no—
278#106—44no—
279C4⋊C8—28no—
280C4⋊C8—28no—
281C4⋊C8—42no—
282C4⋊C8—42no—
283C22⋊C8—28no—
284#97—24no—
285#97—24no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—24no—
291#95—24no—
292#95—22no—
293#95—22no—
294#95—24no—
295#95—24no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—22no—
300#95—22no—
301#95—22no—
302#95C2×D412yes—
303#95C2×D412yes—
304#95C2×D412yes—
305#95C2×D412yes—
306C24C22×D411yescharacteristic
307C2×Q8—28no—
308C2×Q8—44no—
309C2×Q8—44no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—44no—
321C22×C4—84no—
322C22×C4C22⋊Q818yes—
323C22×C4C22×D411yescharacteristic
324C22×C4C22×D412yes—
325C22×C4C22×D411yescharacteristic
326C22×C4C22×D411yescharacteristic
327C22×C4C22×D411yescharacteristic
328C22×C4C22×D411yescharacteristic
329C22×C4C22×D412yes—
330C22×C4C22×D412yes—
331C22×C4C22×D412yes—
332C22×C4C22×Q812yes—
333C22×C4C2×#4112yes—
334C22×C4C2511yescharacteristic
335C2×C8—24no—
336C2×C8—24no—
337C2×C8—24no—
338C2×C8—24no—
339C2×C8—24no—
340C2×C8—24no—
341C4⋊C4—216no—
342C4⋊C4—28no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—24no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—44no—
354C4⋊C4—44no—
355C4⋊C4—44no—
356C4⋊C4—44no—
357C4⋊C4—44no—
358C4⋊C4—44no—
359C4⋊C4—44no—
360C4⋊C4—82no—
361C4⋊C4—82no—
362C4⋊C4—82no—
363C4⋊C4—82no—
364C4⋊C4—82no—
365C4⋊C4—82no—
366C4⋊C4—82no—
367C4⋊C4—82no—
368C4⋊C4—84no—
369C4⋊C4—84no—
370C4⋊C4—84no—
371C4⋊C4C22×Q818yes—
372C22⋊C4—28no—
373C22⋊C4—24no—
374C22⋊C4—41no—
375C22⋊C4—41no—
376C22⋊C4—82no—
377C22⋊C4—82no—
378C42—28no—
379C42—22no—
380C42—22no—
381C42—22no—
382C42—42no—
383C42—42no—
384C42—42no—
385C42—42no—
386C42—82no—
387C42—82no—
388C42—84no—
389C42—84no—
390C42C22⋊Q818yes—
391C42C22×Q812yes—
392C42C22×Q812yes—
393C42C2×#4112yes—
394C42C2×#4112yes—
395C422−1+414yes—
396C23—22no—
397C23—22no—
398C23—21no—
399C23—21no—
400C23C2×C22⋊Q812yes—
401C23C23×D411yescharacteristic
402Q8—416no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—28no—
420C2×C4—22no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—48no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—164no—
444C2×C4C2×C22⋊Q812yes—
445C2×C4C2×C22⋊Q814yes—
446C2×C4C2×C22⋊Q814yes—
447C2×C4#53614yes—
448C2×C4C23×Q811yescharacteristic
449C2×C4C22×#4111yescharacteristic
450C2×C4C2×2−1+412yes—
451C2×C4#58512yes—
452C8—44no—
453C8—44no—
454C22—24no—
455C22—22no—
456C22—41no—
457C22—41no—
458C22—42no—
459C22C22×C22⋊Q811yescharacteristic
460C22C2×#53611yescharacteristic
461C22#344911yescharacteristic
462C22#358512yes—
463C22#358612yes—
464C4—24no—
465C4—24no—
466C4—22no—
467C4—24no—
468C4—48no—
469C4—41no—
470C4—41no—
471C4—41no—
472C4—41no—
473C4—42no—
474C4—82no—
475C4—84no—
476C4—82no—
477C4—82no—
478C4—82no—
479C4#350212yes—
480C4#350712yes—
481C2—42no—
482C2#3517911yescharacteristic
483C2#6049212yes—
484C2#6088712yes—
485C2#6101911yescharacteristic
486C2#6102011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #29295C2semidirect
7. #29294C2semidirect
8. #29285C2semidirect
9. #29223C2semidirect
10. #29212C2semidirect
11. #29182C2semidirect
12. #29181C2semidirect

Maximal subgroups

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

subgroupindex
#353452
#342142
#331572
#331562
#299792
#292952
#292942
#292852
#292232
#292122
#291822
#291812
#219872
#218422
#208112