Finite Group Databasebeta

← #6595137

Subgroups — #6595137

Order 512.

1384 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#36209C211yesmaximal, characteristic
2#36196C211yesmaximal, characteristic
3#33295C211yesmaximal, characteristic, complemented
4#33059C212yesmaximal, complemented
5#29214C211yesmaximal, characteristic, complemented
6#28518C211yesmaximal, characteristic, complemented
7#28076C214yesmaximal, complemented
8#28075C214yesmaximal, complemented
9#27896C212yesmaximal, complemented
10#27881C212yesmaximal, complemented
11#25345C214yesmaximal
12#23064C214yesmaximal, complemented
13C4×#491C211yesmaximal, characteristic, complemented
14#20819C212yesmaximal, complemented
15#20494C211yesmaximal, characteristic
16C2×#491C2211yescharacteristic
17C2×#479C2214yes—
18C2×#436C2212yes—
19C2×C4×M16C2212yes—
20#2859C2211yescharacteristic
21#2859C2211yescharacteristic
22#2857C2212yes—
23#2851C2212yes—
24#2573—22no—
25#2562C2212yes—
26#2536C2212yes—
27#2533—22no—
28#2508—22no—
29#2497C2212yes—
30#2496C2212yes—
31#2495—22no—
32#2489C2214yes—
33#2485—22no—
34#2466—22no—
35#2373—22no—
36#2370C2212yes—
37#2356C2212yes—
38#2346—22no—
39C4×C4⋊Q8C2211yescharacteristic
40C4×#125C2211yescharacteristic
41#2285C2212yes—
42#2283C2212yes—
43C42×D4C2211yescharacteristic
44#2096C2212yes—
45#2095C2212yes—
46#2077C2214yes—
47#2076C2214yes—
48#2059C2214yes—
49#2036C2214yes—
50#2035C2214yes—
51#1992C2212yes—
52#1992C2212yes—
53#1992C2212yes—
54#1992C2211yescharacteristic
55#1991C2214yes—
56#1960C2212yes—
57#1960C2211yescharacteristic
58#1949C2214yes—
59#1935C2214yes—
60#1933C2214yes—
61#1933C2214yes—
62#1928C2214yes—
63#1927C2214yes—
64#1920C2214yes—
65#1918C2214yes—
66#1894C2214yes—
67#1894C2214yes—
68#1847C2212yes—
69#1847C2212yes—
70#1836C2212yes—
71#1836C2212yes—
72#1836C2211yescharacteristic
73#1809C2214yes—
74#1809C2214yes—
75C4×#107C2214yes—
76C4×#106C2214yes—
77C8×C4⋊C4C2212yes—
78#1767C2211yescharacteristic
79C4×#103C2214yes—
80#1753C2214yes—
81#1749C2212yes—
82C4×#97C2211yescharacteristic
83C22×M16C2314yes—
84C2×C4⋊Q8C2311yescharacteristic
85C2×#126—22no—
86C2×#125C2311yescharacteristic
87C2×#125C2312yes—
88C2×#125C2312yes—
89C2×C22⋊Q8C2314yes—
90C2×C4×Q8C2312yes—
91C2×C4×D4D412yes—
92C2×C4×D4C2311yescharacteristic
93C2×#117—22no—
94C22×C42C2312yes—
95#491—24no—
96#479—216no—
97#436—28no—
98C2×#107C2314yes—
99C2×#107C2314yes—
100C2×#106C2314yes—
101C2×#106C2314yes—
102C2×C4⋊C8C2312yes—
103C2×#103C2314yes—
104C2×#103C2314yes—
105C2×#103C2314yes—
106C2×#103C2314yes—
107C2×C22⋊C8C2314yes—
108C4×M16—28no—
109C2×#97C2311yescharacteristic
110C2×#97C2312yes—
111C2×#97C2312yes—
112C2×#97C2311yescharacteristic
113C2×C4×C8C2314yes—
114C2×C4×C8C2312yes—
115#400—22no—
116#400—22no—
117#400—22no—
118#400—24no—
119#398—22no—
120#397—22no—
121#397—24no—
122#396—22no—
123#394—22no—
124#394—22no—
125#391C2312yes—
126#391C2314yes—
127#389—22no—
128#389—22no—
129#389—22no—
130#389—22no—
131#389—22no—
132#389—22no—
133#389—22no—
134#389—22no—
135#389C2312yes—
136#389C2312yes—
137#389C2312yes—
138#388—22no—
139#388—22no—
140#388—24no—
141#388—24no—
142#388—22no—
143#387—24no—
144#387—22no—
145#387—22no—
146#387—22no—
147#387—22no—
148#387—22no—
149#387—22no—
150#387—22no—
151#385—22no—
152#385—22no—
153#384C2311yescharacteristic
154#384C2312yes—
155#384C2311yescharacteristic
156#384C2311yescharacteristic
157#383—22no—
158#383—22no—
159#382—22no—
160#382—24no—
161#382C2311yescharacteristic
162#382C2312yes—
163#382C2312yes—
164#381—22no—
165#380—22no—
166#380—22no—
167#380—22no—
168#380C2312yes—
169#380C2312yes—
170#380C2312yes—
171#380C2312yes—
172#380C2314yes—
173C4×C4⋊C4—22no—
174C4×C4⋊C4—22no—
175C4×C4⋊C4—22no—
176C4×C4⋊C4—22no—
177C4×C4⋊C4D412yes—
178C4×C4⋊C4C2312yes—
179C4×C4⋊C4C2314yes—
180C4×C4⋊C4C2312yes—
181C4×C4⋊C4C2312yes—
182C4×C4⋊C4C2311yescharacteristic
183C4×C22⋊C4—22no—
184C4×C22⋊C4D412yes—
185C4×C22⋊C4D412yes—
186C4×C22⋊C4C2312yes—
187#376—22no—
188#376—22no—
189C43C2311yescharacteristic
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#336C2312yes—
199#336C2314yes—
200C22×Q8C2412yes—
201C22×D4C2×D411yescharacteristic
202C23×C4C2×D412yes—
203C23×C4C2×D412yes—
204C23×C4C2412yes—
205C2×M16—28no—
206C2×M16—28no—
207C2×M16—44no—
208C22×C8C2412yes—
209C22×C8C2412yes—
210C22×C8C2414yes—
211C4⋊Q8—22no—
212C4⋊Q8—22no—
213#126—48no—
214#125—28no—
215#125—22no—
216#125—22no—
217#125—42no—
218#125—42no—
219C22⋊Q8—44no—
220C22⋊Q8—44no—
221C4×Q8—28no—
222C4×D4—24no—
223C4×D4—44no—
224#117—48no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
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—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4C2×D412yes—
241C2×C4⋊C4C2×D411yescharacteristic
242C2×C4⋊C4C2412yes—
243C2×C4⋊C4C2414yes—
244C2×C4⋊C4C2412yes—
245C2×C4⋊C4C2412yes—
246C2×C4⋊C4C2412yes—
247C2×C4⋊C4C2412yes—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4C2×D412yes—
255C2×C22⋊C4C2×D412yes—
256C2×C22⋊C4C2×D412yes—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42C2×D412yes—
265C2×C42C2×D412yes—
266C2×C42C2×D412yes—
267C2×C42C2×D412yes—
268C2×C42C2411yescharacteristic
269C2×C42C2412yes—
270C2×C42C2411yescharacteristic
271C2×C42C2411yescharacteristic
272#107—28no—
273#107—44no—
274#106—28no—
275#106—44no—
276C4⋊C8#4118yes—
277#103—28no—
278#103—28no—
279#103—28no—
280#103—44no—
281C22⋊C8—28no—
282#97—24no—
283#97—22no—
284#97—42no—
285#97#4114yes—
286C4×C8—28no—
287C4×C8—24no—
288#95—22no—
289#95—24no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—24no—
296#95—24no—
297#95—22no—
298#95—22no—
299#95—24no—
300#95—22no—
301#95—22no—
302#95—22no—
303#95—24no—
304#95—22no—
305#95—22no—
306#95—22no—
307#95—22no—
308#95C2×D411yescharacteristic
309#95C2×D411yescharacteristic
310#95C2×D412yes—
311#95C2×D412yes—
312C24C22×D412yes—
313C2×Q8—24no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—24no—
317C2×D4—22no—
318C2×D4—42no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4C22×D412yes—
337C22×C4C22×D412yes—
338C22×C4C22×D412yes—
339C22×C4C22×D411yescharacteristic
340C22×C4C22×D411yescharacteristic
341C22×C4C22×D412yes—
342C22×C4C22×D412yes—
343C22×C4C2511yescharacteristic
344M16—416no—
345C2×C8—24no—
346C2×C8—22no—
347C2×C8—22no—
348C2×C8—24no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8C2×#4114yes—
353C2×C8C2×#4118yes—
354C4⋊C4—22no—
355C4⋊C4—24no—
356C4⋊C4—22no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—22no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—44no—
365C4⋊C4—44no—
366C4⋊C4—44no—
367C4⋊C4—44no—
368C4⋊C4—44no—
369C4⋊C4—44no—
370C4⋊C4—44no—
371C4⋊C4—44no—
372C4⋊C4—44no—
373C4⋊C4—44no—
374C4⋊C4—44no—
375C4⋊C4—82no—
376C4⋊C4—82no—
377C4⋊C4—84no—
378C4⋊C4—84no—
379C22⋊C4—24no—
380C22⋊C4—24no—
381C22⋊C4—42no—
382C22⋊C4—48no—
383C22⋊C4—44no—
384C22⋊C4—44no—
385C22⋊C4—82no—
386C22⋊C4—82no—
387C22⋊C4—84no—
388C42—24no—
389C42—24no—
390C42—22no—
391C42—22no—
392C42—22no—
393C42—42no—
394C42—42no—
395C42—44no—
396C42—48no—
397C42—44no—
398C42—82no—
399C42—82no—
400C42C2×#4114yes—
401C42C2×#4112yes—
402C422−1+412yes—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23—22no—
410C23C23×D411yescharacteristic
411Q8—44no—
412Q8—44no—
413D4—44no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—48no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—84no—
453C2×C4#53212yes—
454C2×C4#53212yes—
455C2×C4#54714yes—
456C2×C4#57514yes—
457C2×C4C22×#4111yescharacteristic
458C2×C4C22×#4112yes—
459C2×C4C2×2−1+411yescharacteristic
460C2×C4#58512yes—
461C8—28no—
462C8—24no—
463C8—42no—
464C22—42no—
465C22—42no—
466C22—42no—
467C22—42no—
468C22—42no—
469C22—42no—
470C22—42no—
471C22C2×#53211yescharacteristic
472C22C2×#54711yescharacteristic
473C22#350011yescharacteristic
474C22C2×#57512yes—
475C22#358612yes—
476C4—22no—
477C4—22no—
478C4—24no—
479C4—24no—
480C4—42no—
481C4—42no—
482C4—42no—
483C4—82no—
484C4—82no—
485C4—82no—
486C4—82no—
487C4—82no—
488C4—82no—
489C4—84no—
490C4#347512yes—
491C4#351812yes—
492C2—82no—
493C2#3618511yescharacteristic
494C2#6067611yescharacteristic
495C2#6072611yescharacteristic
496C2#6125312yes—
497C2#6259112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33295C2semidirect
4. #33059C2semidirect
5. #29214C2semidirect
6. #28518C2semidirect
7. #28076C2semidirect
8. #28075C2semidirect
9. #27896C2semidirect
10. #27881C2semidirect
12. #23064C2semidirect
13. C4×#491C2semidirect
14. #20819C2semidirect

Maximal subgroups

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

subgroupindex
#362092
#361962
#332952
#330592
#292142
#285182
#280762
#280752
#278962
#278812
#253452
#230642
C4×#4912
#208192
#204942