Finite Group Databasebeta

← #278019

Subgroups — #278019

Order 512.

746 conjugacy classes of subgroups.

The subgroup lattice has 471 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33209C211yesmaximal, characteristic, complemented
2#24651C212yesmaximal
3C8×#121C211yesmaximal, characteristic
4C2×#1673C211yesmaximal, characteristic, complemented
5C2×#1564C211yesmaximal, characteristic, complemented
6#10598C212yesmaximal, complemented
7#10477C212yesmaximal, complemented
8#9731C211yesmaximal, characteristic, complemented
9#9615C211yesmaximal, characteristic, complemented
10#9132C212yesmaximal, complemented
11#8644C211yesmaximal, characteristic, complemented
12C2×#500C2211yescharacteristic
13C2×#461C2212yes—
14C2×C8×D4—21no—
15C2×C8×D4C2211yescharacteristic
16C2×#431C2212yes—
17C22×C4×C8C2211yescharacteristic
18#2857C2212yes—
19C4×#121C2211yescharacteristic
20#2035—22no—
21#2000—22no—
22#1990C2212yes—
23#1981C2212yes—
24#1968—22no—
25#1954C2211yescharacteristic
26#1948C2212yes—
27#1942—21no—
28#1942—21no—
29#1939C2211yescharacteristic
30#1933—22no—
31#1923—21no—
32#1869—21no—
33#1845C2211yescharacteristic
34#1808—21no—
35#1808—21no—
36#1802C2212yes—
37#1797C2211yescharacteristic
38C8×C4⋊C4C2211yescharacteristic
39C8×C22⋊C4—21no—
40#1673—24no—
41#1564—24no—
42C2×#335C2211yescharacteristic
43C2×#335C2211yescharacteristic
44C2×#334C2212yes—
45C2×#332C2212yes—
46C2×#332C2212yes—
47#1296C2212yes—
48#1296C2212yes—
49C22×SD16—22no—
50C23×C8—21no—
51C23×C8D411yescharacteristic
52C23×C8D411yescharacteristic
53C2×C4⋊Q8C2312yes—
54C2×C22⋊Q8—22no—
55C2×#121D411yescharacteristic
56C2×C4×D4—21no—
57C2×C4×D4C2311yescharacteristic
58C22×C42C2311yescharacteristic
59#500—22no—
60#500—22no—
61#461—24no—
62#461—42no—
63C8×D4—22no—
64C8×D4—22no—
65C8×D4—44no—
66#431—24no—
67#431—42no—
68C2×#107—21no—
69C2×#107—21no—
70C2×#107—21no—
71C2×#107—21no—
72C2×#106—22no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8C2311yescharacteristic
75C2×C4⋊C8C2311yescharacteristic
76C2×C4⋊C8C2312yes—
77C2×C4⋊C8C2312yes—
78C2×#103—22no—
79C2×#103—22no—
80C2×#103—22no—
81C2×#102—22no—
82C2×C22⋊C8—21no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—21no—
85C2×C22⋊C8—21no—
86C2×C22⋊C8—21no—
87C2×C4×C8—22no—
88C2×C4×C8—22no—
89C2×C4×C8—21no—
90C2×C4×C8—22no—
91C2×C4×C8D411yescharacteristic
92C2×C4×C8D411yescharacteristic
93C2×C4×C8C2311yescharacteristic
94C2×C4×C8C2311yescharacteristic
95#391—22no—
96#390D411yescharacteristic
97#389—22no—
98#389—22no—
99#389D411yescharacteristic
100#388—21no—
101#386—21no—
102#384C2312yes—
103#380—22no—
104#380—22no—
105#380D411yescharacteristic
106C4×C4⋊C4C2311yescharacteristic
107C4×C22⋊C4—21no—
108#340—22no—
109#340—22no—
110#340—22no—
111#340—22no—
112#336—21no—
113#336—21no—
114#336—21no—
115#335—22no—
116#335D414yescomplemented
117#334—24no—
118#332—24no—
119#332—42no—
120C22×Q8—22no—
121C22×D4—21no—
122C22×D4C2×D411yescharacteristic
123C23×C4—21no—
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4#4111yescharacteristic
127C2×SD16—44no—
128C2×SD16—44no—
129C2×SD16—82no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—22no—
138C22×C8—41no—
139C22×C8—42no—
140C22×C8—42no—
141C22×C8—42no—
142C22×C8SD1614yescomplemented
143C22×C8C2×D411yescharacteristic
144C22×C8C2×D411yescharacteristic
145C22×C8C2×D411yescharacteristic
146C22×C8C2×D411yescharacteristic
147C4⋊Q8—22no—
148C4⋊Q8—22no—
149C4⋊Q8—22no—
150C4⋊Q8—22no—
151C22⋊Q8—82no—
152C22⋊Q8—82no—
153#121—22no—
154#121—22no—
155C4×D4—21no—
156C4×D4—21no—
157C4×D4—21no—
158C4×D4—21no—
159C4×D4—42no—
160C4×D4—42no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4C2×D411yescharacteristic
169C2×C4⋊C4C2×D411yescharacteristic
170C2×C4⋊C4C2×D411yescharacteristic
171C2×C22⋊C4—21no—
172C2×C22⋊C4—21no—
173C2×C22⋊C4—21no—
174C2×C22⋊C4—21no—
175C2×C22⋊C4—21no—
176C2×C22⋊C4—42no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42#4111yescharacteristic
185C2×C42C2411yescharacteristic
186#107—42no—
187#107—42no—
188#107—42no—
189#107—42no—
190#106—82no—
191C4⋊C8—21no—
192C4⋊C8—22no—
193C4⋊C8—21no—
194C4⋊C8—22no—
195C4⋊C8—21no—
196C4⋊C8—22no—
197C4⋊C8—21no—
198C4⋊C8—42no—
199C4⋊C8C2×D414yes—
200#103—44no—
201#103—44no—
202#103—82no—
203#102—82no—
204C22⋊C8—44no—
205C22⋊C8—42no—
206C22⋊C8—42no—
207C22⋊C8—42no—
208C22⋊C8—42no—
209C4×C8—22no—
210C4×C8—22no—
211C4×C8—21no—
212C4×C8—21no—
213C4×C8—44no—
214C4×C8—42no—
215C4×C8C2×D412yes—
216C4×C8#4112yes—
217#95—21no—
218#95—21no—
219#95—21no—
220#95—42no—
221#95—42no—
222C24—21no—
223C24#12111yescharacteristic
224C2×Q8—42no—
225C2×Q8—42no—
226C2×Q8—42no—
227C2×Q8—42no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×D4—24no—
231C2×D4—21no—
232C2×D4—21no—
233C2×D4—21no—
234C2×D4—21no—
235C2×D4—21no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4#13512yes—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4#12111yescharacteristic
265C22×C4#12111yescharacteristic
266C22×C4#12111yescharacteristic
267C22×C4C2×SD1612yes—
268C22×C4#13712yes—
269C22×C4C22×D411yescharacteristic
270C22×C4C22×D411yescharacteristic
271C22×C4C2×#4111yescharacteristic
272SD16—88no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—28no—
278C2×C8—21no—
279C2×C8—24no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—44no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—42no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—42no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—42no—
294C2×C8—84no—
295C2×C8#12112yes—
296C2×C8#12112yes—
297C2×C8C2×SD1612yes—
298C2×C8C2×SD1612yes—
299C2×C8#13512yes—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—82no—
315C4⋊C4—84no—
316C4⋊C4#13512yes—
317C22⋊C4—24no—
318C22⋊C4—42no—
319C22⋊C4—41no—
320C22⋊C4—41no—
321C22⋊C4—42no—
322C22⋊C4—42no—
323C22⋊C4—84no—
324C42—21no—
325C42—21no—
326C42—21no—
327C42—21no—
328C42—21no—
329C42—21no—
330C42—41no—
331C42—41no—
332C42C22×D411yescharacteristic
333C42C2×#4111yescharacteristic
334C42H(5,𝔽2)11yescharacteristic
335C422−1+411yescharacteristic
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—22no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23#46112yes—
350C23C2×#12111yescharacteristic
351Q8—82no—
352Q8—82no—
353Q8—82no—
354Q8—82no—
355D4—22no—
356D4—22no—
357D4—48no—
358D4—41no—
359D4—41no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—21no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4#46112yes—
399C2×C4#47312yes—
400C2×C4C2×#12111yescharacteristic
401C2×C4#53711yescharacteristic
402C2×C4D4211yescharacteristic
403C2×C4#54711yescharacteristic
404C2×C4#54811yescharacteristic
405C2×C4#55311yescharacteristic
406C2×C4C22×SD1611yescharacteristic
407C2×C4C2×#13511yescharacteristic
408C2×C4C2×#13511yescharacteristic
409C2×C4C2×#13511yescharacteristic
410C2×C4C2×#13711yescharacteristic
411C2×C4#57511yescharacteristic
412C2×C4#57611yescharacteristic
413C2×C4#57611yescharacteristic
414C8—22no—
415C8—22no—
416C8—44no—
417C8—41no—
418C8—81no—
419C8—82no—
420C8—81no—
421C8#46514yes—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—41no—
429C22—42no—
430C22—42no—
431C22—42no—
432C22—81no—
433C22—81no—
434C22—81no—
435C22—81no—
436C22#241611yescharacteristic
437C22C2×#46111yescharacteristic
438C22#303511yescharacteristic
439C22#305011yescharacteristic
440C22#305811yescharacteristic
441C22C2×#47311yescharacteristic
442C22#306711yescharacteristic
443C4—22no—
444C4—21no—
445C4—21no—
446C4—22no—
447C4—42no—
448C4—41no—
449C4—81no—
450C4—81no—
451C4—162no—
452C4—162no—
453C4C2×#46511yescharacteristic
454C4#305611yescharacteristic
455C4#324611yescharacteristic
456C4#324811yescharacteristic
457C4#328011yescharacteristic
458C4#328711yescharacteristic
459C4#333911yescharacteristic
460C4#334111yescharacteristic
461C2—42no—
462C2—82no—
463C2#1910511yescharacteristic
464C2#1912911yescharacteristic
465C2#1919211yescharacteristic
466C2#1921911yescharacteristic
467C2#2475111yescharacteristic
468C2#2477811yescharacteristic
469C2#2480411yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#121D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33209C2semidirect
4. C2×#1673C2semidirect
5. C2×#1564C2semidirect
6. #10598C2semidirect
7. #10477C2semidirect
8. #9731C2semidirect
9. #9615C2semidirect
10. #9132C2semidirect
11. #8644C2semidirect
116. #335D4semidirect
142. C22×C8SD16semidirect

Maximal subgroups

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

subgroupindex
#332092
#246512
C8×#1212
C2×#16732
C2×#15642
#105982
#104772
#97312
#96152
#91322
#86442