Finite Group Databasebeta

← #278021

Subgroups — #278021

Order 512.

794 conjugacy classes of subgroups.

The subgroup lattice has 475 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33209C211yesmaximal, characteristic, complemented
2#24751C212yesmaximal
3C8×#121C211yesmaximal, characteristic
4C2×#1673C211yesmaximal, characteristic, complemented
5C22×#335C211yesmaximal, characteristic, complemented
6#10602C212yesmaximal, complemented
7#10476C212yesmaximal, complemented
8#9731C211yesmaximal, characteristic, complemented
9#9615C211yesmaximal, characteristic, complemented
10#9131C212yesmaximal, complemented
11#8644C211yesmaximal, characteristic, complemented
12C2×#500C2211yescharacteristic
13C2×#465—22no—
14C2×#461C2212yes—
15C2×C8×D4—21no—
16C2×C8×D4C2211yescharacteristic
17C22×C4⋊C8C2212yes—
18C22×C4×C8C2211yescharacteristic
19#2857C2212yes—
20C4×#121C2211yescharacteristic
21#2066—22no—
22#2055—22no—
23#1990C2212yes—
24#1981C2212yes—
25#1954C2211yescharacteristic
26#1948C2212yes—
27#1942—21no—
28#1942—21no—
29#1939C2211yescharacteristic
30#1923—21no—
31#1892—22no—
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—
41C2×#335—24no—
42C2×#335C2211yescharacteristic
43C2×#335C2212yes—
44C2×#335C2211yescharacteristic
45C2×#332C2212yes—
46C2×#332C2212yes—
47#1296C2212yes—
48#1296C2212yes—
49C22×SD16—22no—
50C23×C8—22no—
51C23×C8—21no—
52C23×C8D411yescharacteristic
53C23×C8D411yescharacteristic
54C2×C4⋊Q8C2312yes—
55C2×C22⋊Q8—22no—
56C2×#121D411yescharacteristic
57C2×C4×D4—21no—
58C2×C4×D4C2311yescharacteristic
59C22×C42C2311yescharacteristic
60#500—22no—
61#500—22no—
62#465—48no—
63#461—24no—
64#461—42no—
65C8×D4—22no—
66C8×D4—22no—
67C8×D4—44no—
68C2×#107—21no—
69C2×#107—21no—
70C2×#107—21no—
71C2×#107—21no—
72C2×#106—22no—
73C2×C4⋊C8—24no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—42no—
77C2×C4⋊C8C2311yescharacteristic
78C2×C4⋊C8C2311yescharacteristic
79C2×C4⋊C8C2312yes—
80C2×C4⋊C8C2312yes—
81C2×#103—22no—
82C2×#103—22no—
83C2×#103—22no—
84C2×#102—22no—
85C2×C22⋊C8—21no—
86C2×C22⋊C8—21no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8—21no—
89C2×C4×C8—22no—
90C2×C4×C8—22no—
91C2×C4×C8—21no—
92C2×C4×C8D411yescharacteristic
93C2×C4×C8D411yescharacteristic
94C2×C4×C8C2311yescharacteristic
95C2×C4×C8C2311yescharacteristic
96#391—22no—
97#390D411yescharacteristic
98#389—22no—
99#389—22no—
100#389D411yescharacteristic
101#388—21no—
102#386—21no—
103#384C2312yes—
104#380—22no—
105#380—22no—
106#380D411yescharacteristic
107C4×C4⋊C4C2311yescharacteristic
108C4×C22⋊C4—21no—
109#340—22no—
110#340—22no—
111#340—22no—
112#340—22no—
113#336—21no—
114#336—21no—
115#336—21no—
116#335—24no—
117#335—22no—
118#335D414yescomplemented
119#332—24no—
120#332—42no—
121C22×Q8—22no—
122C22×D4—21no—
123C22×D4C2×D411yescharacteristic
124C23×C4—21no—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4#4111yescharacteristic
128C2×SD16—44no—
129C2×SD16—44no—
130C2×SD16—82no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—21no—
137C22×C8—22no—
138C22×C8—21no—
139C22×C8—41no—
140C22×C8—44no—
141C22×C8—44no—
142C22×C8—42no—
143C22×C8—42no—
144C22×C8—42no—
145C22×C8—82no—
146C22×C8SD1614yescomplemented
147C22×C8C2×D411yescharacteristic
148C22×C8C2×D411yescharacteristic
149C22×C8C2×D411yescharacteristic
150C22×C8C2×D411yescharacteristic
151C4⋊Q8—22no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C22⋊Q8—44no—
156C22⋊Q8—44no—
157#121—22no—
158#121D814yes—
159C4×D4—21no—
160C4×D4—21no—
161C4×D4—21no—
162C4×D4—21no—
163C4×D4—42no—
164C4×D4—42no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4C2×D411yescharacteristic
173C2×C4⋊C4C2×D411yescharacteristic
174C2×C4⋊C4C2×D411yescharacteristic
175C2×C22⋊C4—21no—
176C2×C22⋊C4—21no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—42no—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42#4111yescharacteristic
189C2×C42C2411yescharacteristic
190#107—42no—
191#107—42no—
192#107—42no—
193#107—42no—
194#106—44no—
195C4⋊C8—21no—
196C4⋊C8—22no—
197C4⋊C8—21no—
198C4⋊C8—22no—
199C4⋊C8—21no—
200C4⋊C8—22no—
201C4⋊C8—21no—
202C4⋊C8—44no—
203C4⋊C8—42no—
204C4⋊C8C2×D414yes—
205#103—44no—
206#103—44no—
207#103—82no—
208#102—44no—
209C22⋊C8—42no—
210C22⋊C8—42no—
211C22⋊C8—42no—
212C22⋊C8—42no—
213C4×C8—22no—
214C4×C8—22no—
215C4×C8—21no—
216C4×C8—21no—
217C4×C8—42no—
218C4×C8C2×D412yes—
219C4×C8#4112yes—
220#95—21no—
221#95—21no—
222#95—21no—
223#95—42no—
224#95—42no—
225C24—21no—
226C24#12111yescharacteristic
227C2×Q8—42no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—42no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×D4—24no—
234C2×D4—21no—
235C2×D4—21no—
236C2×D4—21no—
237C2×D4—21no—
238C2×D4—21no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4C2×D812yes—
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—22no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4#12111yescharacteristic
266C22×C4#12111yescharacteristic
267C22×C4#12111yescharacteristic
268C22×C4C2×D812yes—
269C22×C4C2×SD1612yes—
270C22×C4#13512yes—
271C22×C4#13712yes—
272C22×C4C22×D411yescharacteristic
273C22×C4C22×D411yescharacteristic
274C22×C4C2×#4111yescharacteristic
275SD16—88no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—28no—
281C2×C8—21no—
282C2×C8—24no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—44no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—42no—
297C2×C8—88no—
298C2×C8—84no—
299C2×C8#12112yes—
300C2×C8#12112yes—
301C2×C8C2×SD1612yes—
302C2×C8C2×SD1612yes—
303C2×C8#13512yes—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—82no—
319C4⋊C4—84no—
320C4⋊C4C2×D812yes—
321C22⋊C4—24no—
322C22⋊C4—42no—
323C22⋊C4—42no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—42no—
327C22⋊C4—84no—
328C42—21no—
329C42—21no—
330C42—21no—
331C42—21no—
332C42—21no—
333C42—21no—
334C42—41no—
335C42—41no—
336C42C22×D411yescharacteristic
337C42C2×#4111yescharacteristic
338C42H(5,𝔽2)11yescharacteristic
339C422−1+411yescharacteristic
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—22no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23#46112yes—
352C23#46612yes—
353C23#47312yes—
354C23C2×#12111yescharacteristic
355Q8—82no—
356Q8—82no—
357Q8—82no—
358Q8—82no—
359D4—22no—
360D4—22no—
361D4—48no—
362D4—41no—
363D4—41no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4#46112yes—
403C2×C4#46612yes—
404C2×C4C2×#12111yescharacteristic
405C2×C4#53711yescharacteristic
406C2×C4D4211yescharacteristic
407C2×C4#54711yescharacteristic
408C2×C4#54811yescharacteristic
409C2×C4#55311yescharacteristic
410C2×C4C22×D811yescharacteristic
411C2×C4C22×SD1611yescharacteristic
412C2×C4C2×#13511yescharacteristic
413C2×C4C2×#13511yescharacteristic
414C2×C4C2×#13711yescharacteristic
415C2×C4#57511yescharacteristic
416C2×C4#57611yescharacteristic
417C2×C4#57811yescharacteristic
418C8—22no—
419C8—22no—
420C8—44no—
421C8—41no—
422C8—81no—
423C8—82no—
424C8—81no—
425C8#46514yescomplemented
426C22—22no—
427C22—22no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—42no—
433C22—42no—
434C22—42no—
435C22—81no—
436C22—81no—
437C22—81no—
438C22—81no—
439C22#167314yescomplemented
440C22#241611yescharacteristic
441C22C2×#46111yescharacteristic
442C22#303511yescharacteristic
443C22C2×#46611yescharacteristic
444C22#306111yescharacteristic
445C22C2×#47311yescharacteristic
446C22#306711yescharacteristic
447C4—22no—
448C4—22no—
449C4—21no—
450C4—21no—
451C4—42no—
452C4—41no—
453C4—81no—
454C4—81no—
455C4—162no—
456C4—162no—
457C4C2×#46511yescharacteristic
458C4#305611yescharacteristic
459C4#324611yescharacteristic
460C4#324811yescharacteristic
461C4D4×D811yescharacteristic
462C4#328511yescharacteristic
463C4#333411yescharacteristic
464C4#333911yescharacteristic
465C2—24no—
466C2—82no—
467C2C2×#167311yescharacteristic
468C2#1912511yescharacteristic
469C2#1919511yescharacteristic
470C2#1922311yescharacteristic
471C2#2475111yescharacteristic
472C2#2478011yescharacteristic
473C2#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. C22×#335C2semidirect
6. #10602C2semidirect
7. #10476C2semidirect
8. #9731C2semidirect
9. #9615C2semidirect
10. #9131C2semidirect
11. #8644C2semidirect
118. #335D4semidirect
146. C22×C8SD16semidirect
425. C8#465semidirect
439. C22#1673semidirect

Maximal subgroups

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

subgroupindex
#332092
#247512
C8×#1212
C2×#16732
C22×#3352
#106022
#104762
#97312
#96152
#91312
#86442