Finite Group Databasebeta

← #457886

Subgroups — #457886

Order 512.

1138 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2085C212yesmaximal, complemented
2C2×#1809C212yesmaximal
3#15220C212yesmaximal
4#11328C212yesmaximal, complemented
5#11325C212yesmaximal, complemented
6#11219C212yesmaximal, complemented
7#11096C211yesmaximal, characteristic
8#11096C211yesmaximal, characteristic
9#10715C211yesmaximal, characteristic
10C23×M16C2211yescharacteristic
11C22×#107C2212yes—
12C22×#107C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×#106C2212yes—
15C22×C22⋊C8C2211yescharacteristic
16C2×#389C2212yes—
17C2×#389C2212yes—
18C2×#382C2212yes—
19C2×#380C2212yes—
20C2×#380C2212yes—
21C2×C4×C22⋊C4C2211yescharacteristic
22#2085—28no—
23#1809—28no—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2211yescharacteristic
28C2×#340C2211yescharacteristic
29C2×#340C2212yes—
30C2×#336C2212yes—
31C2×#336C2212yes—
32#1443—22no—
33#1441—22no—
34#1439—22no—
35#1439—22no—
36C4⋊#95C2212yes—
37C24×C4C2311yescharacteristic
38C22×M16—24no—
39C22×M16—24no—
40C22×M16—41no—
41C22×M16—41no—
42C22×M16Q814yes—
43C23×C8C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C23×C8C2312yes—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4C2312yes—
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2312yes—
52C22×C22⋊C4—22no—
53C22×C22⋊C4—22no—
54C22×C22⋊C4D412yes—
55C22×C42C2312yes—
56C2×#107—24no—
57C2×#107—24no—
58C2×#107—24no—
59C2×#107—41no—
60C2×#107—42no—
61C2×#107—42no—
62C2×#107—41no—
63C2×#106—24no—
64C2×#106—24no—
65C2×#106—42no—
66C2×C22⋊C8—41no—
67C2×C22⋊C8—41no—
68C2×C22⋊C8Q814yes—
69#389—24no—
70#389—42no—
71#389—42no—
72#389—42no—
73#382—24no—
74#382—42no—
75#380—24no—
76#380—24no—
77#380—42no—
78#380—42no—
79C4×C22⋊C4—41no—
80C4×C22⋊C4—41no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95D411yescharacteristic
90C2×#95D411yescharacteristic
91#340—24no—
92#340—24no—
93#340—41no—
94#340—42no—
95#340—41no—
96#340D418yes—
97#336—24no—
98#336—24no—
99C25C2×D411yescharacteristic
100C23×C4—22no—
101C23×C4—22no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—22no—
106C23×C4—21no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4C2×D412yes—
113C23×C4C2×D412yes—
114C23×C4C2×D411yescharacteristic
115C23×C4C2×Q812yes—
116C23×C4C2×Q812yes—
117C23×C4C2411yescharacteristic
118C2×M16—216no—
119C2×M16—44no—
120C2×M16—44no—
121C2×M16—48no—
122C2×M16—44no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—22no—
132C22×C8—21no—
133C22×C8—21no—
134C22×C8—22no—
135C22×C8C2×D414yes—
136C22×C8C2×Q812yes—
137C22×C8C2×Q812yes—
138C22×C8C2×Q814yes—
139C22×C8#4114yes—
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—22no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—82no—
166C2×C4⋊C4—82no—
167C2×C4⋊C4—82no—
168C2×C4⋊C4C2×D414yes—
169C2×C4⋊C4C2×D414yes—
170C2×C22⋊C4—42no—
171C2×C22⋊C4—42no—
172C2×C22⋊C4—42no—
173C2×C22⋊C4—44no—
174C2×C22⋊C4—44no—
175C2×C22⋊C4—44no—
176C2×C22⋊C4—44no—
177C2×C22⋊C4—82no—
178C2×C22⋊C4—82no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185#107—44no—
186#107—48no—
187#107—44no—
188#106—48no—
189C22⋊C8—44no—
190#95—41no—
191#95—41no—
192#95—44no—
193#95—44no—
194#95—44no—
195#95—44no—
196#95—44no—
197#95—44no—
198#95—44no—
199#95—82no—
200C24—21no—
201C24—21no—
202C24—22no—
203C24—22no—
204C24—22no—
205C24—22no—
206C24—21no—
207C24—21no—
208C24—21no—
209C24—21no—
210C24C4⋊Q812yes—
211C24C22×D411yescharacteristic
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—24no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—24no—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—41no—
245C22×C4—42no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—42no—
249C22×C4—41no—
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×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4C4⋊Q812yes—
277C22×C4C22×D412yes—
278C22×C4C22×Q811yescharacteristic
279C22×C4C22×Q811yescharacteristic
280C22×C4C2×#4112yes—
281C22×C4H(5,𝔽2)11yescharacteristic
282C22×C4H(5,𝔽2)12yes—
283C22×C4H(5,𝔽2)11yescharacteristic
284C22×C4H(5,𝔽2)11yescharacteristic
285C22×C42−1+411yescharacteristic
286C22×C42−1+412yes—
287M16—416no—
288C2×C8—24no—
289C2×C8—24no—
290C2×C8—24no—
291C2×C8—24no—
292C2×C8—24no—
293C2×C8—24no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—24no—
297C2×C8—24no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8C22⋊Q818yes—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—24no—
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—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—88no—
322C4⋊C4—88no—
323C22⋊C4—48no—
324C22⋊C4—88no—
325C22⋊C4—88no—
326C42—42no—
327C42—42no—
328C42—42no—
329C42—42no—
330C23—22no—
331C23—22no—
332C23—24no—
333C23—22no—
334C23—24no—
335C23—42no—
336C23—42no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23—41no—
341C23—42no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—42no—
350C23—42no—
351C23—42no—
352C23—42no—
353C23C2×C4⋊Q811yescharacteristic
354C23#53411yescharacteristic
355C23#53711yescharacteristic
356C23D4212yes—
357C23#54712yes—
358C23#57512yes—
359C23#57512yes—
360C23#57611yescharacteristic
361C23#57611yescharacteristic
362C23#57811yescharacteristic
363C23#57811yescharacteristic
364C2×C4—22no—
365C2×C4—24no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—44no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—44no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—44no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—44no—
394C2×C4—42no—
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×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4C2×C22⋊Q812yes—
418C2×C4#53712yes—
419C2×C4#53712yes—
420C2×C4#54012yes—
421C2×C4#54112yes—
422C2×C4#54311yescharacteristic
423C2×C4#54311yescharacteristic
424C2×C4#54412yes—
425C2×C4#54411yescharacteristic
426C2×C4#54411yescharacteristic
427C2×C4D4211yescharacteristic
428C2×C4#54612yes—
429C2×C4D4×Q812yes—
430C2×C4#55111yescharacteristic
431C2×C4#55412yes—
432C2×C4#56011yescharacteristic
433C2×C4#56212yes—
434C2×C4#56311yescharacteristic
435C8—28no—
436C8—42no—
437C8—42no—
438C22—42no—
439C22—42no—
440C22—42no—
441C22—44no—
442C22—44no—
443C22—82no—
444C22—82no—
445C22—81no—
446C22—81no—
447C22—81no—
448C22—81no—
449C22#244012yes—
450C22#265911yescharacteristic
451C22#267112yes—
452C22#267511yescharacteristic
453C22#272211yescharacteristic
454C22#316411yescharacteristic
455C22#316411yescharacteristic
456C22#316612yes—
457C22#319712yes—
458C22#319911yescharacteristic
459C22#320211yescharacteristic
460C22#325611yescharacteristic
461C22#325611yescharacteristic
462C22#326211yescharacteristic
463C22#326211yescharacteristic
464C22#326812yes—
465C22#326912yes—
466C22#327312yes—
467C22#327712yes—
468C22#334712yes—
469C22#334912yes—
470C22#335212yes—
471C22#335512yes—
472C4—42no—
473C4—82no—
474C4—84no—
475C4—82no—
476C4—84no—
477C4—162no—
478C4—162no—
479C4#243711yescharacteristic
480C4#262912yes—
481C4#268311yescharacteristic
482C4#270412yes—
483C4#274811yescharacteristic
484C4#284111yescharacteristic
485C2—82no—
486C2#1630511yescharacteristic
487C2#2607312yes—
488C2#2607912yes—
489C2#3029711yescharacteristic
490C2#3030011yescharacteristic
491C2#3057412yes—
492C2#3058012yes—
493C2#3085611yescharacteristic
494C2#3086311yescharacteristic
495C2#3242111yescharacteristic
496C2#3242511yescharacteristic
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×#2085C2semidirect
4. #11328C2semidirect
5. #11325C2semidirect
6. #11219C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20852
C2×#18092
#152202
#113282
#113252
#112192
#110962
#107152