Finite Group Databasebeta

← #306125

Subgroups — #306125

Order 512.

576 conjugacy classes of subgroups.

The subgroup lattice has 485 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28835C211yesmaximal, characteristic, complemented
2#26107C211yesmaximal, characteristic
3#26021C211yesmaximal, characteristic
4#22665C211yesmaximal, characteristic
5#10656C211yesmaximal, characteristic, complemented
6#10656C211yesmaximal, characteristic, complemented
7#10582C211yesmaximal, characteristic, complemented
8#10578C211yesmaximal, characteristic, complemented
9#9621C211yesmaximal, characteristic, complemented
10#9593C211yesmaximal, characteristic, complemented
11#8897C211yesmaximal, characteristic
12#8894C211yesmaximal, characteristic
13#8816C211yesmaximal, characteristic, complemented
14#8652C211yesmaximal, characteristic
15#8206C211yesmaximal, characteristic
16C2×#436C2211yescharacteristic
17C2×#428—21no—
18#2571C2211yescharacteristic
19#2552C2211yescharacteristic
20#2354C2211yescharacteristic
21D4×C4⋊C4C2211yescharacteristic
22#2101—21no—
23#2088—21no—
24#2085—21no—
25#2083—21no—
26#2065—21no—
27#2025—21no—
28#1989—21no—
29#1948C2211yescharacteristic
30#1948C2211yescharacteristic
31#1948C2211yescharacteristic
32#1948C2211yescharacteristic
33#1947C2211yescharacteristic
34#1947C2211yescharacteristic
35#1943—21no—
36#1939C2211yescharacteristic
37#1939C2211yescharacteristic
38#1919—21no—
39#1864—21no—
40#1854—21no—
41#1852—21no—
42#1847—21no—
43#1845C2211yescharacteristic
44#1845C2211yescharacteristic
45#1834C2211yescharacteristic
46#1834C2211yescharacteristic
47#1834C2211yescharacteristic
48#1834C2211yescharacteristic
49#1809—21no—
50#1804—21no—
51#1802C2211yescharacteristic
52#1802C2211yescharacteristic
53C4×#106C2211yescharacteristic
54#1770C2211yescharacteristic
55C4×#102C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×C82C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1277C2211yescharacteristic
67C22×M16—21no—
68C2×#125—21no—
69C2×C4×D4—21no—
70C2×C4×D4C2311yescharacteristic
71C2×#117—21no—
72C22×C4⋊C4—21no—
73#436—22no—
74#436Q814yes—
75#428—44no—
76C2×#107—21no—
77C2×#107—21no—
78C2×#107—21no—
79C2×#107—21no—
80C2×#107—21no—
81C2×#107—21no—
82C2×#106—21no—
83C2×#106—21no—
84C2×#106—21no—
85C2×#106—21no—
86C2×#106D411yescharacteristic
87C2×#106D411yescharacteristic
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×#97—21no—
100C2×C4×C8C2311yescharacteristic
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103#400—41no—
104#398—41no—
105#390—21no—
106#389—21no—
107#389—21no—
108#389—21no—
109#389—21no—
110#389—21no—
111#388—41no—
112#385—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⋊C4C2311yescharacteristic
124C4×C4⋊C4C2311yescharacteristic
125C4×C4⋊C4C2311yescharacteristic
126#340—21no—
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—41no—
139#340—41no—
140#340D411yescharacteristic
141#340D411yescharacteristic
142#340D411yescharacteristic
143#340D411yescharacteristic
144#336—21no—
145#336—21no—
146#336—21no—
147#334—22no—
148D4⋊C8Q814yes—
149C82—22no—
150C22×D4C2×D411yescharacteristic
151C23×C4—21no—
152C23×C4—41no—
153C2×M16—24no—
154C2×M16—42no—
155C2×M16—42no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8C2×D411yescharacteristic
162C22×C8C2×D411yescharacteristic
163C22×C8C2×D411yescharacteristic
164C22×C8C2×D411yescharacteristic
165#125—42no—
166#125—81no—
167C4×D4—21no—
168C4×D4—21no—
169C4×D4—42no—
170C4×D4—81no—
171C4×D4C2×Q812yes—
172C4×D4C2×Q812yes—
173#117—42no—
174#117—81no—
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—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—81no—
194C2×C4⋊C4C2×D411yescharacteristic
195C2×C4⋊C4C2×D411yescharacteristic
196C2×C4⋊C4C2×D411yescharacteristic
197C2×C4⋊C4C2×D411yescharacteristic
198C2×C4⋊C4C2×D411yescharacteristic
199C2×C4⋊C4C2×D411yescharacteristic
200C2×C4⋊C4C2×D411yescharacteristic
201C2×C22⋊C4—21no—
202C2×C22⋊C4—41no—
203C2×C22⋊C4—41no—
204C2×C22⋊C4—41no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—41no—
209C2×C42C2411yescharacteristic
210#107—42no—
211#107—42no—
212#107—42no—
213#107—81no—
214#107—81no—
215#107—81no—
216#106—22no—
217#106—22no—
218#106—42no—
219#106—42no—
220#106—81no—
221#106—81no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—81no—
229C4⋊C8—81no—
230C4⋊C8C2×Q812yes—
231C4⋊C8C2×Q812yes—
232#102—41no—
233#102—41no—
234#102—42no—
235#102—81no—
236C22⋊C8—24no—
237#97—42no—
238C4×C8—21no—
239C4×C8—21no—
240C4×C8—21no—
241C4×C8C2×Q812yes—
242C4×C8C2×Q812yes—
243C4×C8#4112yes—
244#95—21no—
245#95—21no—
246#95—21no—
247#95—41no—
248#95—41no—
249#95—41no—
250#95—41no—
251#95—41no—
252#95—41no—
253C24—21no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4C4⋊Q812yes—
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—21no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—82no—
278C22×C4—82no—
279C22×C4—161no—
280C22×C4C22×D411yescharacteristic
281C22×C4C22×D411yescharacteristic
282C22×C4H(5,𝔽2)11yescharacteristic
283M16—48no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—22no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8C22⋊Q812yes—
308C2×C8C22⋊Q812yes—
309C2×C8#13512yes—
310C2×C8AGL(1,ℤ/8)12yes—
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—21no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
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—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—81no—
341C4⋊C4—81no—
342C4⋊C4—81no—
343C4⋊C4—81no—
344C4⋊C4—82no—
345C4⋊C4—82no—
346C4⋊C4—82no—
347C4⋊C4—161no—
348C4⋊C4—161no—
349C4⋊C4C4⋊Q812yes—
350C22⋊C4—22no—
351C22⋊C4—22no—
352C22⋊C4—82no—
353C22⋊C4—82no—
354C22⋊C4—161no—
355C42—41no—
356C42—41no—
357C42—41no—
358C42—41no—
359C42—41no—
360C42—41no—
361C42—82no—
362C42C22×Q811yescharacteristic
363C42C22×Q811yescharacteristic
364C42C2×#4111yescharacteristic
365C422−1+411yescharacteristic
366C23—22no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23D4211yescharacteristic
374D4—22no—
375D4—22no—
376D4—41no—
377D4—41no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—22no—
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—41no—
403C2×C4—41no—
404C2×C4—41no—
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×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—164no—
418C2×C4C2×C22⋊Q811yescharacteristic
419C2×C4C2×C4⋊Q811yescharacteristic
420C2×C4#54411yescharacteristic
421C2×C4#54711yescharacteristic
422C2×C4D4×Q811yescharacteristic
423C2×C4#55411yescharacteristic
424C2×C4C2×#13511yescharacteristic
425C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
426C2×C4#57511yescharacteristic
427C2×C4#57511yescharacteristic
428C2×C4#57611yescharacteristic
429C2×C4#57711yescharacteristic
430C2×C4#57711yescharacteristic
431C2×C4#57811yescharacteristic
432C8—22no—
433C8—42no—
434C8—42no—
435C8—41no—
436C8—41no—
437C8—42no—
438C8—81no—
439C8—81no—
440C8#47814yes—
441C22—42no—
442C22—42no—
443C22—42no—
444C22—81no—
445C22—81no—
446C22—81no—
447C22—81no—
448C22#244011yescharacteristic
449C22#327411yescharacteristic
450C22#327511yescharacteristic
451C22#327511yescharacteristic
452C22#328311yescharacteristic
453C22#329011yescharacteristic
454C22#329011yescharacteristic
455C4—21no—
456C4—21no—
457C4—42no—
458C4—81no—
459C4—81no—
460C4—81no—
461C4—81no—
462C4—81no—
463C4—81no—
464C4—161no—
465C4—161no—
466C4—161no—
467C4—161no—
468C4C2×#47811yescharacteristic
469C4#308311yescharacteristic
470C4#316311yescharacteristic
471C4#316611yescharacteristic
472C4#334811yescharacteristic
473C4#335011yescharacteristic
474C4#338011yescharacteristic
475C4#338611yescharacteristic
476C2—82no—
477C2#1995611yescharacteristic
478C2#1995611yescharacteristic
479C2#2028811yescharacteristic
480C2#2029811yescharacteristic
481C2#2604111yescharacteristic
482C2#2607611yescharacteristic
483C2#2608211yescharacteristic
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. #28835C2semidirect
5. #10656C2semidirect
6. #10656C2semidirect
7. #10582C2semidirect
8. #10578C2semidirect
9. #9621C2semidirect
10. #9593C2semidirect
13. #8816C2semidirect

Maximal subgroups

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

subgroupindex
#288352
#261072
#260212
#226652
#106562
#105822
#105782
#96212
#95932
#88972
#88942
#88162
#86522
#82062