Finite Group Databasebeta

← #305863

Subgroups — #305863

Order 512.

565 conjugacy classes of subgroups.

The subgroup lattice has 481 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28790C211yesmaximal, characteristic, complemented
2#26091C211yesmaximal, characteristic
3#26090C211yesmaximal, characteristic
4#22708C211yesmaximal, characteristic
5#10654C211yesmaximal, characteristic, complemented
6#10654C211yesmaximal, characteristic, complemented
7#10579C211yesmaximal, characteristic, complemented
8#10576C211yesmaximal, characteristic, complemented
9#9616C211yesmaximal, characteristic, complemented
10#9594C211yesmaximal, characteristic, complemented
11#8885C211yesmaximal, characteristic
12#8885C211yesmaximal, characteristic
13#8814C211yesmaximal, characteristic, complemented
14#8648C211yesmaximal, characteristic
15C8×#102C211yesmaximal, characteristic
16C2×C8×D4C2211yescharacteristic
17#2571C2211yescharacteristic
18#2552C2211yescharacteristic
19#2354C2211yescharacteristic
20#2354C2211yescharacteristic
21#2094—21no—
22#2092—21no—
23#2085—21no—
24#2065—21no—
25#2065—21no—
26#2059—21no—
27#1991—21no—
28#1949—21no—
29#1948—21no—
30#1948C2211yescharacteristic
31#1948C2211yescharacteristic
32#1948C2211yescharacteristic
33#1948C2211yescharacteristic
34#1948C2211yescharacteristic
35#1947C2211yescharacteristic
36#1941—21no—
37#1939C2211yescharacteristic
38#1939C2211yescharacteristic
39#1920—21no—
40#1919—21no—
41#1854—21no—
42#1854—21no—
43#1845C2211yescharacteristic
44#1845C2211yescharacteristic
45#1844C2211yescharacteristic
46#1844C2211yescharacteristic
47#1834C2211yescharacteristic
48#1834C2211yescharacteristic
49#1807—21no—
50#1802C2211yescharacteristic
51#1802C2211yescharacteristic
52C4×#107C2211yescharacteristic
53C8×C4⋊C4C2211yescharacteristic
54C4×#102C2211yescharacteristic
55#1735—21no—
56C2×#335C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×C82C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1277C2211yescharacteristic
67C23×C8—21no—
68C2×#125—21no—
69C2×C4×D4C2311yescharacteristic
70C2×#117—21no—
71C2×#117—21no—
72C8×D4—22no—
73C8×D4Q814yes—
74C2×#107—21no—
75C2×#107—41no—
76C2×#107—41no—
77C2×#107D411yescharacteristic
78C2×#107D411yescharacteristic
79C2×#106—21no—
80C2×#106—21no—
81C2×#106—21no—
82C2×#106—21no—
83C2×#106—21no—
84C2×#106—21no—
85C2×#106—21no—
86C2×#106—21no—
87C2×#106—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×#102—21no—
95C2×#102—21no—
96C2×#102D411yescharacteristic
97C2×#102D411yescharacteristic
98C2×C22⋊C8—21no—
99C2×C4×C8—21no—
100C2×C4×C8C2311yescharacteristic
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103#400—41no—
104#398—41no—
105#390—21no—
106#390—21no—
107#389—21no—
108#389—21no—
109#389—21no—
110#389—21no—
111#388—41no—
112#388—41no—
113#384—21no—
114#384C2311yescharacteristic
115#384C2311yescharacteristic
116#384C2311yescharacteristic
117#384C2311yescharacteristic
118#382—21no—
119#382—21no—
120#380—21no—
121#380—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4C2311yescharacteristic
125C4×C4⋊C4C2311yescharacteristic
126C4×C4⋊C4C2311yescharacteristic
127#340—21no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340D411yescharacteristic
140#340D411yescharacteristic
141#340D411yescharacteristic
142#340D411yescharacteristic
143#336—21no—
144#336—21no—
145#336—21no—
146#335—22no—
147D4⋊C8Q814yes—
148C82—22no—
149C22×D4C2×D411yescharacteristic
150C23×C4—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—24no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—81no—
158C22×C8—81no—
159C22×C8C2×D411yescharacteristic
160C22×C8C2×D411yescharacteristic
161C22×C8C2×D411yescharacteristic
162C22×C8C2×D411yescharacteristic
163#125—42no—
164#125—81no—
165C4×D4—21no—
166C4×D4—21no—
167C4×D4C2×Q812yes—
168C4×D4C2×Q812yes—
169#117—42no—
170#117—42no—
171#117—81no—
172#117—81no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4C2×D411yescharacteristic
190C2×C4⋊C4C2×D411yescharacteristic
191C2×C4⋊C4C2×D411yescharacteristic
192C2×C4⋊C4C2×D411yescharacteristic
193C2×C4⋊C4C2×D411yescharacteristic
194C2×C4⋊C4C2×D411yescharacteristic
195C2×C4⋊C4C2×D411yescharacteristic
196C2×C22⋊C4—21no—
197C2×C22⋊C4—41no—
198C2×C22⋊C4—41no—
199C2×C22⋊C4—41no—
200C2×C22⋊C4—41no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—41no—
205C2×C42—41no—
206C2×C42C2411yescharacteristic
207#107—22no—
208#107—41no—
209#107—81no—
210#107—82no—
211#107—82no—
212#106—42no—
213#106—42no—
214#106—42no—
215#106—81no—
216#106—81no—
217#106—81no—
218#106—81no—
219#106—81no—
220#106—81no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—42no—
228C4⋊C8—42no—
229C4⋊C8C2×Q812yes—
230C4⋊C8C2×Q812yes—
231#102—22no—
232#102—22no—
233#102—81no—
234#102—81no—
235C22⋊C8—24no—
236C4×C8—21no—
237C4×C8—21no—
238C4×C8—21no—
239C4×C8—81no—
240C4×C8C2×Q812yes—
241C4×C8C2×Q812yes—
242C4×C8#4112yes—
243#95—21no—
244#95—21no—
245#95—21no—
246#95—41no—
247#95—41no—
248#95—41no—
249#95—41no—
250#95—41no—
251#95—41no—
252C24—21no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4C4⋊Q812yes—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4C22×D411yescharacteristic
277C22×C4C22×D411yescharacteristic
278C22×C4H(5,𝔽2)11yescharacteristic
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—22no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—84no—
303C2×C8C22⋊Q812yes—
304C2×C8C22⋊Q812yes—
305C2×C8#13512yes—
306C2×C8#13512yes—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—81no—
336C4⋊C4—81no—
337C4⋊C4—81no—
338C4⋊C4—81no—
339C4⋊C4—82no—
340C4⋊C4—82no—
341C4⋊C4—161no—
342C4⋊C4—161no—
343C4⋊C4C4⋊Q812yes—
344C4⋊C4#13512yes—
345C22⋊C4—22no—
346C22⋊C4—22no—
347C22⋊C4—82no—
348C22⋊C4—82no—
349C22⋊C4—161no—
350C22⋊C4—161no—
351C42—41no—
352C42—41no—
353C42—41no—
354C42—41no—
355C42—41no—
356C42—41no—
357C42—82no—
358C42—82no—
359C42C22×Q811yescharacteristic
360C42C22×Q811yescharacteristic
361C42C2×#4111yescharacteristic
362C422−1+411yescharacteristic
363C23—22no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23D4211yescharacteristic
371D4—22no—
372D4—22no—
373D4—41no—
374D4—41no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4C2×C22⋊Q811yescharacteristic
415C2×C4C2×C4⋊Q811yescharacteristic
416C2×C4#54411yescharacteristic
417C2×C4#54711yescharacteristic
418C2×C4D4×Q811yescharacteristic
419C2×C4#55411yescharacteristic
420C2×C4C2×#13511yescharacteristic
421C2×C4C2×#13511yescharacteristic
422C2×C4C2×#13511yescharacteristic
423C2×C4#57511yescharacteristic
424C2×C4#57611yescharacteristic
425C2×C4#57611yescharacteristic
426C2×C4#57711yescharacteristic
427C2×C4#57811yescharacteristic
428C8—24no—
429C8—22no—
430C8—22no—
431C8—22no—
432C8—22no—
433C8—42no—
434C8—42no—
435C8—81no—
436C8—81no—
437C22—42no—
438C22—42no—
439C22—42no—
440C22—81no—
441C22—81no—
442C22—81no—
443C22—81no—
444C22#244011yescharacteristic
445C22#327311yescharacteristic
446C22#328211yescharacteristic
447C22#328311yescharacteristic
448C22#328711yescharacteristic
449C22#329111yescharacteristic
450C22#329111yescharacteristic
451C4—21no—
452C4—21no—
453C4—42no—
454C4—42no—
455C4—81no—
456C4—81no—
457C4—81no—
458C4—81no—
459C4—81no—
460C4—161no—
461C4—161no—
462C4—161no—
463C4—161no—
464C4#307611yescharacteristic
465C4#308111yescharacteristic
466C4#316011yescharacteristic
467C4#316411yescharacteristic
468C4#333611yescharacteristic
469C4#333911yescharacteristic
470C4#338011yescharacteristic
471C4#338811yescharacteristic
472C2—82no—
473C2#1995411yescharacteristic
474C2#1996111yescharacteristic
475C2#1999711yescharacteristic
476C2#2000411yescharacteristic
477C2#2605111yescharacteristic
478C2#2606611yescharacteristic
479C2#2608111yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #28790C2semidirect
5. #10654C2semidirect
6. #10654C2semidirect
7. #10579C2semidirect
8. #10576C2semidirect
9. #9616C2semidirect
10. #9594C2semidirect
13. #8814C2semidirect

Maximal subgroups

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

subgroupindex
#287902
#260912
#260902
#227082
#106542
#105792
#105762
#96162
#95942
#88852
#88142
#86482
C8×#1022