Finite Group Databasebeta

← #6434925

Subgroups — #6434925

Order 512.

1428 conjugacy classes of subgroups.

The subgroup lattice has 499 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35315C211yesmaximal, characteristic
2C2×#2331C211yesmaximal, characteristic
3#30160C214yesmaximal, complemented
4#30102C212yesmaximal, complemented
5#30070C212yesmaximal, complemented
6#29972C212yesmaximal
7#29913C212yesmaximal
8#29756C212yesmaximal
9#29641C212yesmaximal
10#23845C212yesmaximal, complemented
11#23829C212yesmaximal, complemented
12#23789C214yesmaximal, complemented
13#21579C212yesmaximal
14C4×#483C212yesmaximal
15#20779C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C2×#483C2212yes—
18C2×#483C2212yes—
19#2857C2211yescharacteristic
20#2855C2212yes—
21#2851C2211yescharacteristic
22#2647C2211yescharacteristic
23#2645C2212yes—
24#2642C2211yescharacteristic
25#2533—22no—
26#2528—22no—
27#2505—22no—
28#2493—22no—
29#2331—22no—
30#2331—22no—
31#2331C2212yes—
32#2331C2212yes—
33#2331C2212yes—
34#2331C2212yes—
35#2330C2211yescharacteristic
36#2327C2211yescharacteristic
37#2320C2212yes—
38C4×C22⋊Q8C2212yes—
39C2×#391—22no—
40C2×#389C2212yes—
41C2×#387—22no—
42C2×#382C2211yescharacteristic
43C2×C4×C4⋊C4C2212yes—
44C2×#376C2211yescharacteristic
45#2101C2214yes—
46#2101C2214yes—
47#2097C2214yes—
48#2088C2214yes—
49#2087C2214yes—
50#2085C2214yes—
51#2084C2214yes—
52#2074C2214yes—
53#2066C2214yes—
54#2066C2214yes—
55#2059C2214yes—
56#2058C2214yes—
57#2058C2214yes—
58#2043C2214yes—
59#2024C2214yes—
60#2024C2214yes—
61#1928C2214yes—
62#1927C2214yes—
63#1920C2214yes—
64#1919C2214yes—
65#1872C2212yes—
66#1872C2212yes—
67#1853C2212yes—
68#1852C2212yes—
69#1840C2212yes—
70#1832C2212yes—
71#1823C2212yes—
72#1823C2212yes—
73#1817C2212yes—
74#1797C2212yes—
75#1776C2214yes—
76C4×#106C2214yes—
77#1768C2212yes—
78#1763C2214yes—
79C4×#103C2214yes—
80#1749C2211yescharacteristic
81C4×C22⋊C8C2211yescharacteristic
82#1746C2212yes—
83C23×Q8—21no—
84C2×C4⋊Q8—28no—
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8—21no—
87C2×C4⋊Q8C2311yescharacteristic
88C2×C4⋊Q8C2312yes—
89C2×C4⋊Q8C2312yes—
90C2×#125C2311yescharacteristic
91C2×C22⋊Q8C2312yes—
92C2×C4×Q8C2312yes—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—22no—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C42—22no—
98C22×C42C2311yescharacteristic
99C22×C42C2311yescharacteristic
100C22×C42C2311yescharacteristic
101#483—28no—
102#483—28no—
103C2×#107C2314yes—
104C2×#106C2314yes—
105C2×#106C2314yes—
106C2×#106C2314yes—
107C2×C4⋊C8C2314yes—
108C2×#103C2314yes—
109C2×#103C2314yes—
110C2×#103C2314yes—
111C2×#103C2314yes—
112C2×C22⋊C8C2312yes—
113C2×C22⋊C8C2312yes—
114C2×#97C2312yes—
115C2×C4×C8C2312yes—
116#401—22no—
117#400—22no—
118#400—22no—
119#400—22no—
120#400—22no—
121#400—22no—
122#400—22no—
123#398—22no—
124#391—24no—
125#391—24no—
126#391—24no—
127#391—24no—
128#391C2312yes—
129#391C2312yes—
130#391C2312yes—
131#391C2312yes—
132#389—22no—
133#389—22no—
134#389—22no—
135#389—22no—
136#389C2312yes—
137#389C2312yes—
138#389C2314yes—
139#389C2312yes—
140#389C2314yes—
141#389C2312yes—
142#387—22no—
143#387—22no—
144#387—22no—
145#387—22no—
146#387—22no—
147#387—22no—
148#387—22no—
149#387—22no—
150#387—22no—
151#387—24no—
152#387—24no—
153#387—24no—
154#387—24no—
155#385—22no—
156#385—22no—
157#385—22no—
158#384C2311yescharacteristic
159#383—22no—
160#382—21no—
161#382—21no—
162#382—22no—
163#382C2311yescharacteristic
164#382C2312yes—
165#382C2312yes—
166#381—22no—
167#380—22no—
168#380C2312yes—
169C4×C4⋊C4—22no—
170C4×C4⋊C4—22no—
171C4×C4⋊C4—22no—
172C4×C4⋊C4C2312yes—
173C4×C4⋊C4C2314yes—
174C4×C4⋊C4C2314yes—
175C4×C22⋊C4—22no—
176#376D412yes—
177#376D412yes—
178#376C2311yescharacteristic
179#376C2311yescharacteristic
180#376C2311yescharacteristic
181#376C2311yescharacteristic
182C2×#95—22no—
183C2×#95—21no—
184C2×#95—21no—
185C2×#95—22no—
186C2×#95D412yes—
187C2×#95D412yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#336C2314yes—
197#336C2314yes—
198#336C2314yes—
199#336C2312yes—
200#336C2312yes—
201C22×Q8—28no—
202C22×Q8—22no—
203C22×Q8—22no—
204C22×Q8—84no—
205C22×Q8C2412yes—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4C2×D411yescharacteristic
209C23×C4C2×D412yes—
210C23×C4C2×D412yes—
211C23×C4C2×D411yescharacteristic
212C23×C4C2411yescharacteristic
213C22×C8C2414yes—
214C22×C8C2414yes—
215C4⋊Q8—28no—
216C4⋊Q8—28no—
217C4⋊Q8—28no—
218C4⋊Q8—28no—
219C4⋊Q8—42no—
220#125—42no—
221C22⋊Q8—24no—
222C22⋊Q8—24no—
223C4×Q8—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—84no—
243C2×C4⋊C4#4118yes—
244C2×C4⋊C4C2414yes—
245C2×C4⋊C4C2412yes—
246C2×C4⋊C4C2412yes—
247C2×C4⋊C4C2412yes—
248C2×C4⋊C4C2414yes—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C42—24no—
252C2×C42—24no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—24no—
256C2×C42—24no—
257C2×C42—42no—
258C2×C42—84no—
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D412yes—
262C2×C42C2×D412yes—
263C2×C42C2411yescharacteristic
264C2×C42C2411yescharacteristic
265C2×C42C2411yescharacteristic
266C2×C42C2411yescharacteristic
267C2×C42C2411yescharacteristic
268C2×C42C2411yescharacteristic
269#107—44no—
270#106—28no—
271#106—28no—
272#106—44no—
273C4⋊C8—44no—
274#103—28no—
275#103—28no—
276#103—44no—
277#103—44no—
278C22⋊C8—24no—
279C22⋊C8—24no—
280#97—24no—
281C4×C8—24no—
282#95—24no—
283#95—24no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—24no—
289#95—24no—
290#95—22no—
291#95—22no—
292#95—22no—
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#95C2×D412yes—
306#95C2×D412yes—
307#95C2×D412yes—
308#95C2×D412yes—
309#95C2×D412yes—
310#95C2×D412yes—
311#95C2×D412yes—
312#95C2×D412yes—
313C24C22×D411yescharacteristic
314C2×Q8—28no—
315C2×Q8—28no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—24no—
319C2×Q8—24no—
320C2×Q8—84no—
321C2×Q8—84no—
322C2×Q8—816no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—28no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4C22×D411yescharacteristic
348C22×C4C22×D411yescharacteristic
349C22×C4C22×D412yes—
350C22×C4C22×D412yes—
351C22×C4C22×D411yescharacteristic
352C22×C4C22×D411yescharacteristic
353C22×C4C22×D412yes—
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D412yes—
356C22×C4C2×#4114yes—
357C22×C4C2511yescharacteristic
358C2×C8—24no—
359C2×C8—24no—
360C2×C8—24no—
361C2×C8—24no—
362C2×C8—24no—
363C2×C8—24no—
364C4⋊C4—28no—
365C4⋊C4—28no—
366C4⋊C4—28no—
367C4⋊C4—24no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—44no—
375C4⋊C4—44no—
376C4⋊C4—44no—
377C4⋊C4—44no—
378C4⋊C4—44no—
379C4⋊C4—82no—
380C4⋊C4—82no—
381C4⋊C4—82no—
382C4⋊C4—82no—
383C4⋊C4—82no—
384C4⋊C4—84no—
385C4⋊C4—84no—
386C4⋊C4C2×#4118yes—
387C22⋊C4—44no—
388C22⋊C4—44no—
389C42—22no—
390C42—22no—
391C42—24no—
392C42—22no—
393C42—22no—
394C42—22no—
395C42—22no—
396C42—22no—
397C42—42no—
398C42—44no—
399C42—82no—
400C42—84no—
401C42—84no—
402C422−1+414yes—
403C23—22no—
404C23—21no—
405C23—21no—
406C23#52512yes—
407C23#57714yes—
408C23C23×D411yescharacteristic
409Q8—44no—
410Q8—44no—
411Q8—88no—
412Q8—88no—
413C2×C4—22no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—22no—
417C2×C4—28no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—28no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—48no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—416no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—84no—
458C2×C4—88no—
459C2×C4#52512yes—
460C2×C4#53614yes—
461C2×C4C22×#4112yes—
462C2×C4C2×2−1+412yes—
463C2×C4#58512yes—
464C8—44no—
465C8—44no—
466C22—22no—
467C22—24no—
468C22—22no—
469C22—24no—
470C22#312314yes—
471C22C2×#52511yescharacteristic
472C22C2×#53611yescharacteristic
473C22#344911yescharacteristic
474C22C2×#57712yes—
475C22#358612yes—
476C4—22no—
477C4—24no—
478C4—42no—
479C4—42no—
480C4—41no—
481C4—41no—
482C4—48no—
483C4—41no—
484C4—41no—
485C4—42no—
486C4—82no—
487C4—82no—
488C4—84no—
489C4—82no—
490C4—82no—
491C4#346114yes—
492C2—24no—
493C2#3460211yescharacteristic
494C2C2×#312311yescharacteristic
495C2#6056611yescharacteristic
496C2#6087212yes—
497C2#6097712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #30160C2semidirect
4. #30102C2semidirect
5. #30070C2semidirect
10. #23845C2semidirect
11. #23829C2semidirect
12. #23789C2semidirect

Maximal subgroups

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

subgroupindex
#353152
C2×#23312
#301602
#301022
#300702
#299722
#299132
#297562
#296412
#238452
#238292
#237892
#215792
C4×#4832
#207792