Finite Group Databasebeta

← #457817

Subgroups — #457817

Order 512.

1162 conjugacy classes of subgroups.

The subgroup lattice has 483 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2059C212yesmaximal, complemented
2C2×#1809C212yesmaximal
3#15223C211yesmaximal, characteristic
4#15216C211yesmaximal, characteristic
5#11325C212yesmaximal, complemented
6#11257C212yesmaximal, complemented
7#11220C212yesmaximal, complemented
8#11095C212yesmaximal
9#10715C211yesmaximal, characteristic
10C23×M16C2211yescharacteristic
11C22×#125C2211yescharacteristic
12C22×#107C2212yes—
13C22×#106C2212yes—
14C22×#106C2212yes—
15C22×C22⋊C8C2211yescharacteristic
16C2×#389C2212yes—
17C2×#382C2211yescharacteristic
18C2×#380C2212yes—
19C2×#380C2212yes—
20C2×C4×C22⋊C4C2211yescharacteristic
21#2059—28no—
22#1809—28no—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#336C2212yes—
29C2×#336C2212yes—
30#1445—22no—
31#1442—22no—
32#1439—22no—
33#1439—22no—
34C4⋊#95C2212yes—
35C4⋊#95C2211yescharacteristic
36C4⋊#95C2211yescharacteristic
37C24×C4C2311yescharacteristic
38C22×M16—24no—
39C22×M16—24no—
40C22×M16—41no—
41C22×M16—41no—
42C22×M16Q814yes—
43C23×C8C2312yes—
44C23×C8C2312yes—
45C2×#125—24no—
46C2×#125—41no—
47C2×#125—41no—
48C2×#125—42no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C22⋊C4—22no—
56C22×C22⋊C4—22no—
57C22×C22⋊C4D412yes—
58C22×C42C2311yescharacteristic
59C22×C42C2311yescharacteristic
60C2×#107—24no—
61C2×#107—24no—
62C2×#107—42no—
63C2×#106—24no—
64C2×#106—24no—
65C2×#106—24no—
66C2×#106—42no—
67C2×#106—42no—
68C2×#106—42no—
69C2×C22⋊C8—41no—
70C2×C22⋊C8—41no—
71C2×C22⋊C8Q814yes—
72#389—24no—
73#389—42no—
74#382—41no—
75#382—41no—
76#380—24no—
77#380—24no—
78#380—42no—
79#380—42no—
80C4×C22⋊C4—41no—
81C4×C22⋊C4—41no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95D412yes—
91#340—24no—
92#340—24no—
93#340—42no—
94#340—42no—
95#340D418yes—
96#336—24no—
97#336—24no—
98C25C2×D411yescharacteristic
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—22no—
105C23×C4—21no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4C2×D412yes—
112C23×C4C2×D412yes—
113C23×C4C2×D411yescharacteristic
114C23×C4C2×Q812yes—
115C23×C4C2×Q812yes—
116C23×C4C2411yescharacteristic
117C2×M16—216no—
118C2×M16—44no—
119C2×M16—44no—
120C2×M16—48no—
121C2×M16—44no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8C2×D414yes—
133C22×C8C2×Q814yes—
134C22×C8C2×Q814yes—
135C22×C8#4114yes—
136#125—44no—
137#125—48no—
138#125—44no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—82no—
167C2×C4⋊C4C2×D414yes—
168C2×C4⋊C4C2×D414yes—
169C2×C22⋊C4—42no—
170C2×C22⋊C4—42no—
171C2×C22⋊C4—42no—
172C2×C22⋊C4—44no—
173C2×C22⋊C4—44no—
174C2×C22⋊C4—44no—
175C2×C22⋊C4—44no—
176C2×C22⋊C4—82no—
177C2×C22⋊C4—82no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—22no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—22no—
186#107—48no—
187#106—48no—
188#106—48no—
189C22⋊C8—44no—
190#95—42no—
191#95—44no—
192#95—44no—
193#95—44no—
194#95—44no—
195#95—44no—
196#95—82no—
197#95—82no—
198#95—82no—
199C24—21no—
200C24—21no—
201C24—22no—
202C24—22no—
203C24—22no—
204C24—22no—
205C24—22no—
206C24—22no—
207C24C4⋊Q812yes—
208C24C22×D411yescharacteristic
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—24no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—41no—
242C22×C4—42no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—42no—
246C22×C4—41no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4C4⋊Q812yes—
274C22×C4C22×D412yes—
275C22×C4C22×Q811yescharacteristic
276C22×C4C22×Q811yescharacteristic
277C22×C4C2×#4112yes—
278C22×C4H(5,𝔽2)11yescharacteristic
279C22×C4H(5,𝔽2)12yes—
280C22×C4H(5,𝔽2)11yescharacteristic
281C22×C4H(5,𝔽2)11yescharacteristic
282C22×C42−1+411yescharacteristic
283C22×C42−1+412yes—
284M16—416no—
285C2×C8—24no—
286C2×C8—24no—
287C2×C8—24no—
288C2×C8—24no—
289C2×C8—24no—
290C2×C8—24no—
291C2×C8—24no—
292C2×C8—24no—
293C2×C8—24no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8C22⋊Q818yes—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—416no—
304C4⋊C4—42no—
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—42no—
315C4⋊C4—42no—
316C4⋊C4—88no—
317C22⋊C4—48no—
318C22⋊C4—88no—
319C22⋊C4—88no—
320C42—42no—
321C42—42no—
322C42—41no—
323C42—41no—
324C42—41no—
325C42—41no—
326C23—24no—
327C23—22no—
328C23—22no—
329C23—22no—
330C23—24no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—42no—
335C23—41no—
336C23—42no—
337C23—42no—
338C23—42no—
339C23—41no—
340C23—42no—
341C23—41no—
342C23—41no—
343C23—42no—
344C23—42no—
345C23—42no—
346C23—42no—
347C23C2×C4⋊Q811yescharacteristic
348C23#53411yescharacteristic
349C23#53711yescharacteristic
350C23D4212yes—
351C23#54712yes—
352C23#57512yes—
353C23#57512yes—
354C23#57712yes—
355C23#57712yes—
356C2×C4—22no—
357C2×C4—24no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—44no—
363C2×C4—42no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—44no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—42no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4C2×C22⋊Q812yes—
410C2×C4#53512yes—
411C2×C4#53612yes—
412C2×C4#54212yes—
413C2×C4#54212yes—
414C2×C4#54311yescharacteristic
415C2×C4#54311yescharacteristic
416C2×C4#54412yes—
417C2×C4#54411yescharacteristic
418C2×C4#54411yescharacteristic
419C2×C4D4211yescharacteristic
420C2×C4#54612yes—
421C2×C4D4×Q812yes—
422C2×C4#55111yescharacteristic
423C2×C4#55412yes—
424C2×C4#56011yescharacteristic
425C2×C4#56212yes—
426C2×C4#56311yescharacteristic
427C8—28no—
428C8—44no—
429C22—42no—
430C22—42no—
431C22—42no—
432C22—44no—
433C22—44no—
434C22—82no—
435C22—82no—
436C22—82no—
437C22—82no—
438C22#244012yes—
439C22#265911yescharacteristic
440C22#266911yescharacteristic
441C22#267212yes—
442C22#271711yescharacteristic
443C22#316312yes—
444C22#316611yescharacteristic
445C22#316611yescharacteristic
446C22#319712yes—
447C22#320412yes—
448C22#325612yes—
449C22#326011yescharacteristic
450C22#326011yescharacteristic
451C22#326712yes—
452C22#327112yes—
453C22#327512yes—
454C22#327512yes—
455C22#334812yes—
456C22#335012yes—
457C22#335112yes—
458C22#335112yes—
459C4—42no—
460C4—82no—
461C4—82no—
462C4—84no—
463C4—84no—
464C4—162no—
465C4—162no—
466C4#243711yescharacteristic
467C4#262812yes—
468C4#267711yescharacteristic
469C4#269212yes—
470C4#274211yescharacteristic
471C4#284111yescharacteristic
472C2—82no—
473C2#1630311yescharacteristic
474C2#2607512yes—
475C2#2607612yes—
476C2#3029711yescharacteristic
477C2#3030311yescharacteristic
478C2#3051212yes—
479C2#3068312yes—
480C2#3069012yes—
481C2#3227312yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2059C2semidirect
5. #11325C2semidirect
6. #11257C2semidirect
7. #11220C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20592
C2×#18092
#152232
#152162
#113252
#112572
#112202
#110952
#107152