Finite Group Databasebeta

← #6434946

Subgroups — #6434946

Order 512.

1376 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35324C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic
3#30162C214yesmaximal, complemented
4#30096C212yesmaximal, complemented
5#30060C212yesmaximal, complemented
6#29977C212yesmaximal
7#29908C212yesmaximal
8#29766C212yesmaximal
9#29645C212yesmaximal
10#23845C212yesmaximal, complemented
11#23831C212yesmaximal, complemented
12#23783C212yesmaximal, complemented
13#23776C212yesmaximal, complemented
14#21591C214yesmaximal
15#20781C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C2×#485C2214yes—
18#2864C2211yescharacteristic
19#2862C2211yescharacteristic
20#2855C2212yes—
21#2645C2212yes—
22#2628C2211yescharacteristic
23#2621C2211yescharacteristic
24#2588—22no—
25#2575—22no—
26#2528—22no—
27#2527—22no—
28#2360—22no—
29#2360—22no—
30#2355—22no—
31Q8×C4⋊C4—22no—
32#2331C2211yescharacteristic
33#2330C2212yes—
34#2330C2212yes—
35#2327C2211yescharacteristic
36#2323C2212yes—
37#2317C2212yes—
38C4×C4⋊Q8C2212yes—
39C4×C4⋊Q8C2212yes—
40#2295C2212yes—
41C2×#384C2211yescharacteristic
42#2274C2212yes—
43C4×#117C2211yescharacteristic
44#2101C2214yes—
45#2099C2212yes—
46#2099C2212yes—
47#2096C2212yes—
48#2096C2212yes—
49#2088C2214yes—
50#2088C2214yes—
51#2085C2214yes—
52#2084C2214yes—
53#2072C2214yes—
54#2068C2214yes—
55#2066C2214yes—
56#2059C2214yes—
57#2058C2214yes—
58#2058C2212yes—
59#2057C2212yes—
60#2043C2214yes—
61#2026C2214yes—
62#2024C2212yes—
63#2024C2212yes—
64#1927C2214yes—
65#1927C2214yes—
66#1921C2214yes—
67#1919C2214yes—
68#1872C2214yes—
69#1853C2214yes—
70#1839C2212yes—
71#1832C2212yes—
72#1825C2214yes—
73#1817C2212yes—
74#1798C2212yes—
75#1776C2214yes—
76C4×#106C2214yes—
77#1767C2211yescharacteristic
78C4×C4⋊C8C2211yescharacteristic
79#1763C2214yes—
80C4×#103C2214yes—
81#1750C2212yes—
82#1746C2212yes—
83#533—28no—
84C2×C4⋊Q8C2311yescharacteristic
85C2×C4⋊Q8C2312yes—
86C2×C4⋊Q8C2311yescharacteristic
87C2×#125C2312yes—
88C2×C22⋊Q8—22no—
89C2×C22⋊Q8C2312yes—
90C2×C4×Q8—22no—
91C2×C4×Q8—22no—
92C2×C4×Q8—22no—
93C2×C4×Q8C2312yes—
94C2×#117C2312yes—
95C2×#117C2311yescharacteristic
96C2×#117C2311yescharacteristic
97C22×C4⋊C4—22no—
98C22×C4⋊C4C2312yes—
99C22×C42C2311yescharacteristic
100#485—216no—
101C2×#107C2314yes—
102C2×#106C2314yes—
103C2×#106C2314yes—
104C2×#106C2314yes—
105C2×C4⋊C8C2312yes—
106C2×C4⋊C8C2312yes—
107C2×#103C2314yes—
108C2×#103C2314yes—
109C2×#103C2314yes—
110C2×#103C2314yes—
111C2×C22⋊C8C2314yes—
112C2×#97C2312yes—
113C2×C4×C8C2312yes—
114#401—22no—
115#400—21no—
116#400—22no—
117#400—22no—
118#400—22no—
119#399—21no—
120#398—21no—
121#398—21no—
122#397—21no—
123#397—21no—
124#395—21no—
125#393—21no—
126#391—22no—
127#391—22no—
128#391—22no—
129#391—22no—
130#391C2312yes—
131#391C2312yes—
132#391C2312yes—
133#391C2312yes—
134#389—22no—
135#389—22no—
136#389—22no—
137#389—24no—
138#389—24no—
139#389C2314yes—
140#389C2314yes—
141#389C2312yes—
142#389C2312yes—
143#389C2312yes—
144#389C2312yes—
145#388—22no—
146#388—22no—
147#387—22no—
148#387—22no—
149#387—22no—
150#387—22no—
151#387—22no—
152#387—22no—
153#387—22no—
154#387—22no—
155#387—22no—
156#387—22no—
157#387—24no—
158#387—24no—
159#385—22no—
160#385—22no—
161#385—22no—
162#384—22no—
163#384—22no—
164#384C2312yes—
165#384C2312yes—
166#382—22no—
167#382—22no—
168#382C2311yescharacteristic
169#382C2311yescharacteristic
170#381—22no—
171#380C2312yes—
172C4×C4⋊C4—24no—
173C4×C4⋊C4—22no—
174C4×C4⋊C4—22no—
175C4×C4⋊C4—22no—
176C4×C4⋊C4D412yes—
177C4×C4⋊C4D412yes—
178C4×C4⋊C4C2314yes—
179C4×C4⋊C4C2314yes—
180C4×C4⋊C4C2312yes—
181C4×C22⋊C4—24no—
182C4×C22⋊C4D412yes—
183C4×C22⋊C4D412yes—
184#376—22no—
185#376C2311yescharacteristic
186#376C2311yescharacteristic
187C43C2311yescharacteristic
188C43C2311yescharacteristic
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#336C2314yes—
198#336C2312yes—
199#336C2312yes—
200#336C2314yes—
201#336C2314yes—
202C22×Q8—22no—
203C22×Q8—21no—
204C22×Q8C2412yes—
205C23×C4—22no—
206C23×C4C2×D412yes—
207C23×C4C2411yescharacteristic
208C22×C8C2414yes—
209C22×C8C2414yes—
210C4⋊Q8—28no—
211C4⋊Q8—42no—
212C4⋊Q8—42no—
213#125—28no—
214C22⋊Q8—216no—
215C22⋊Q8—28no—
216C4×Q8—216no—
217C4×Q8—24no—
218C4×Q8—24no—
219C4×Q8—84no—
220C4×Q8—84no—
221#117—24no—
222#117—22no—
223#117—22no—
224#117—42no—
225#117#4118yes—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—84no—
246C2×C4⋊C4C2×D412yes—
247C2×C4⋊C4C2×D411yescharacteristic
248C2×C4⋊C4C2×D411yescharacteristic
249C2×C4⋊C4C2414yes—
250C2×C4⋊C4C2412yes—
251C2×C4⋊C4C2412yes—
252C2×C4⋊C4C2412yes—
253C2×C4⋊C4C2414yes—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4C2×D412yes—
258C2×C22⋊C4C2×D411yescharacteristic
259C2×C22⋊C4C2×D411yescharacteristic
260C2×C42—22no—
261C2×C42—24no—
262C2×C42—22no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—22no—
266C2×C42C2×D412yes—
267C2×C42C2×D412yes—
268C2×C42C2×D412yes—
269C2×C42C2×D412yes—
270C2×C42C2×D412yes—
271C2×C42C2411yescharacteristic
272C2×C42C2411yescharacteristic
273C2×C42C2411yescharacteristic
274C2×C42C2411yescharacteristic
275C2×C42C2411yescharacteristic
276C2×C42C2411yescharacteristic
277#107—28no—
278#106—28no—
279#106—44no—
280#106—44no—
281C4⋊C8—42no—
282C4⋊C8—42no—
283#103—28no—
284#103—28no—
285#103—44no—
286#103—44no—
287C22⋊C8—28no—
288#97—24no—
289C4×C8—24no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—24no—
295#95—24no—
296#95—22no—
297#95—24no—
298#95—24no—
299#95—22no—
300#95—22no—
301#95—22no—
302#95—22no—
303#95—22no—
304#95—22no—
305#95—22no—
306#95—21no—
307#95—21no—
308#95—21no—
309#95—21no—
310#95—22no—
311#95—22no—
312#95C2×D412yes—
313#95C2×D412yes—
314#95C2×D412yes—
315#95C2×D412yes—
316C24C22×D411yescharacteristic
317C2×Q8—28no—
318C2×Q8—24no—
319C2×Q8—22no—
320C2×Q8—22no—
321C2×Q8—24no—
322C2×Q8—44no—
323C2×Q8—42no—
324C2×Q8—44no—
325C2×Q8—82no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—42no—
338C22×C4—44no—
339C22×C4—84no—
340C22×C4C22×D411yescharacteristic
341C22×C4C22×D411yescharacteristic
342C22×C4C22×D412yes—
343C22×C4C22×D411yescharacteristic
344C22×C4C22×D412yes—
345C22×C4C22×D411yescharacteristic
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D412yes—
348C22×C4C22×D412yes—
349C22×C4C2×#4114yes—
350C22×C4C2511yescharacteristic
351C2×C8—24no—
352C2×C8—24no—
353C2×C8—24no—
354C2×C8—24no—
355C2×C8—24no—
356C2×C8—24no—
357C4⋊C4—216no—
358C4⋊C4—28no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—44no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—44no—
374C4⋊C4—44no—
375C4⋊C4—44no—
376C4⋊C4—44no—
377C4⋊C4—44no—
378C4⋊C4—82no—
379C4⋊C4—82no—
380C4⋊C4—82no—
381C4⋊C4—84no—
382C4⋊C4—84no—
383C4⋊C4—84no—
384C4⋊C4C2×#4118yes—
385C22⋊C4—28no—
386C22⋊C4—24no—
387C22⋊C4—41no—
388C22⋊C4—41no—
389C22⋊C4—44no—
390C22⋊C4—82no—
391C42—24no—
392C42—28no—
393C42—24no—
394C42—24no—
395C42—22no—
396C42—22no—
397C42—21no—
398C42—21no—
399C42—22no—
400C42—22no—
401C42—42no—
402C42—42no—
403C42—44no—
404C42—82no—
405C42—84no—
406C42—84no—
407C422−1+412yes—
408C422−1+412yes—
409C23—22no—
410C23—21no—
411C23—21no—
412C23—22no—
413C23#52512yes—
414C23C23×D411yescharacteristic
415Q8—416no—
416Q8—44no—
417Q8—44no—
418Q8—162no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—28no—
433C2×C4—22no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—48no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—164no—
461C2×C4#52512yes—
462C2×C4#53614yes—
463C2×C4#57714yes—
464C2×C4C22×#4112yes—
465C2×C4C2×2−1+411yescharacteristic
466C2×C4C2×2−1+411yescharacteristic
467C2×C4#58512yes—
468C8—44no—
469C8—44no—
470C22—22no—
471C22—24no—
472C22—41no—
473C22—41no—
474C22—42no—
475C22C2×#52511yescharacteristic
476C22C2×#53611yescharacteristic
477C22#344911yescharacteristic
478C22C2×#57712yes—
479C22#358612yes—
480C4—22no—
481C4—24no—
482C4—24no—
483C4—41no—
484C4—41no—
485C4—41no—
486C4—41no—
487C4—42no—
488C4—48no—
489C4—42no—
490C4—82no—
491C4—84no—
492C4—82no—
493C4—82no—
494C4—82no—
495C4#346112yes—
496C4#346112yes—
497C2—42no—
498C2#3460211yescharacteristic
499C2#6058512yes—
500C2#6088912yes—
501C2#6101211yescharacteristic
502C2#6101811yescharacteristic
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
3. #30162C2semidirect
4. #30096C2semidirect
5. #30060C2semidirect
10. #23845C2semidirect
11. #23831C2semidirect
12. #23783C2semidirect
13. #23776C2semidirect

Maximal subgroups

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

subgroupindex
#353242
#342142
#301622
#300962
#300602
#299772
#299082
#297662
#296452
#238452
#238312
#237832
#237762
#215912
#207812