Finite Group Databasebeta

← #472660

Subgroups — #472660

Order 512.

1030 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#15223C211yesmaximal, characteristic
2#15150C211yesmaximal, characteristic
3#11316C212yesmaximal, complemented
4#11292C212yesmaximal, complemented
5#11268C212yesmaximal, complemented
6#11207C212yesmaximal, complemented
7#11200C212yesmaximal
8#10722C212yesmaximal
9#10715C211yesmaximal, characteristic
10C23×M16C2211yescharacteristic
11C22×C22⋊C8C2211yescharacteristic
12C2×#397—22no—
13C2×#389C2212yes—
14C2×#384C2211yescharacteristic
15C2×#382C2211yescharacteristic
16C2×#380C2212yes—
17C2×#380C2212yes—
18C2×C4×C22⋊C4C2211yescharacteristic
19C2×#340C2212yes—
20C2×#340C2212yes—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#336C2212yes—
28C2×#336C2212yes—
29#1445—22no—
30#1445—22no—
31#1439—22no—
32C4⋊#95C2212yes—
33C4⋊#95C2211yescharacteristic
34C4⋊#95C2211yescharacteristic
35C24×C4C2311yescharacteristic
36C22×M16—22no—
37C22×M16—24no—
38C22×M16—42no—
39C22×M16—41no—
40C22×M16—41no—
41C23×C8C2312yes—
42C23×C8C2312yes—
43C22×C4⋊C4—22no—
44C22×C4⋊C4—22no—
45C22×C4⋊C4C2312yes—
46C22×C4⋊C4C2312yes—
47C22×C4⋊C4C2312yes—
48C22×C4⋊C4C2312yes—
49C22×C22⋊C4—22no—
50C22×C22⋊C4—22no—
51C22×C22⋊C4D412yes—
52C22×C42C2311yescharacteristic
53C22×C42C2311yescharacteristic
54C2×C22⋊C8—22no—
55C2×C22⋊C8—41no—
56C2×C22⋊C8—41no—
57#397—48no—
58#389—24no—
59#389—42no—
60#384—41no—
61#384—41no—
62#382—41no—
63#382—41no—
64#380—24no—
65#380—24no—
66#380—24no—
67#380—42no—
68C4×C22⋊C4—41no—
69C4×C22⋊C4—41no—
70C2×#95—22no—
71C2×#95—22no—
72C2×#95—22no—
73C2×#95—22no—
74C2×#95—22no—
75C2×#95—22no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95D412yes—
79#340—24no—
80#340—24no—
81#340—42no—
82#340—42no—
83#340—42no—
84#340—42no—
85#340—42no—
86#340—42no—
87#336—24no—
88#336—42no—
89C25C2×D411yescharacteristic
90C23×C4—22no—
91C23×C4—22no—
92C23×C4—21no—
93C23×C4—22no—
94C23×C4—21no—
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—22no—
98C23×C4—22no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—22no—
102C23×C4C2×D412yes—
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D412yes—
105C23×C4#4112yes—
106C23×C4#4112yes—
107C23×C4C2411yescharacteristic
108C2×M16—44no—
109C2×M16—48no—
110C2×M16—48no—
111C2×M16—44no—
112C2×M16—44no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8#4114yes—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
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—44no—
151C2×C4⋊C4—44no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4—82no—
156C2×C4⋊C4#4114yes—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—42no—
159C2×C22⋊C4—42no—
160C2×C22⋊C4—44no—
161C2×C22⋊C4—44no—
162C2×C22⋊C4—44no—
163C2×C22⋊C4—44no—
164C2×C22⋊C4—44no—
165C2×C22⋊C4—82no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—22no—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42—22no—
174C22⋊C8—44no—
175#95—42no—
176#95—44no—
177#95—44no—
178#95—44no—
179#95—44no—
180#95—44no—
181#95—44no—
182#95—44no—
183#95—82no—
184C24—21no—
185C24—21no—
186C24—22no—
187C24—22no—
188C24—22no—
189C24—22no—
190C24—22no—
191C24—22no—
192C24#12412yes—
193C24C22×D411yescharacteristic
194C22×C4—21no—
195C22×C4—22no—
196C22×C4—21no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—24no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—22no—
213C22×C4—24no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—41no—
225C22×C4—42no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—42no—
229C22×C4—41no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
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#12412yes—
258C22×C4AGL(1,ℤ/8)14yes—
259C22×C4C2×#4111yescharacteristic
260C22×C4C2×#4112yes—
261C22×C4C2×#4112yes—
262C22×C4C2×#4111yescharacteristic
263C22×C4H(5,𝔽2)11yescharacteristic
264C22×C4H(5,𝔽2)11yescharacteristic
265C22×C4H(5,𝔽2)12yes—
266C22×C4H(5,𝔽2)11yescharacteristic
267C22×C42−1+412yes—
268C22×C42−1+411yescharacteristic
269M16—416no—
270C2×C8—24no—
271C2×C8—24no—
272C2×C8—24no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C4⋊C4—24no—
285C4⋊C4—24no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—88no—
301C4⋊C4—88no—
302C22⋊C4—48no—
303C22⋊C4—88no—
304C22⋊C4—88no—
305C42—42no—
306C42—42no—
307C42—41no—
308C42—41no—
309C42—41no—
310C42—41no—
311C23—24no—
312C23—22no—
313C23—22no—
314C23—22no—
315C23—24no—
316C23—42no—
317C23—42no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23—42no—
322C23—42no—
323C23—41no—
324C23—41no—
325C23—42no—
326C23—41no—
327C23—41no—
328C23—42no—
329C23—42no—
330C23—42no—
331C23—42no—
332C23C2×#12411yescharacteristic
333C23#53411yescharacteristic
334C23#53711yescharacteristic
335C23#54612yes—
336C23#54712yes—
337C23C2×AGL(1,ℤ/8)12yes—
338C23#57512yes—
339C23#57712yes—
340C23#57812yes—
341C2×C4—22no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—44no—
355C2×C4—42no—
356C2×C4—44no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—44no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4#52912yes—
395C2×C4#53811yescharacteristic
396C2×C4#53812yes—
397C2×C4#53811yescharacteristic
398C2×C4#53912yes—
399C2×C4#53911yescharacteristic
400C2×C4#53912yes—
401C2×C4#53911yescharacteristic
402C2×C4#54012yes—
403C2×C4#54212yes—
404C2×C4#55112yes—
405C2×C4#55111yescharacteristic
406C2×C4#55212yes—
407C2×C4#55212yes—
408C2×C4#55911yescharacteristic
409C2×C4#56011yescharacteristic
410C2×C4#56312yes—
411C2×C4#56311yescharacteristic
412C8—44no—
413C8—82no—
414C22—44no—
415C22—44no—
416C22—42no—
417C22—42no—
418C22—42no—
419C22—82no—
420C22—82no—
421C22—82no—
422C22—82no—
423C22#248812yes—
424C22#263911yescharacteristic
425C22#269512yes—
426C22#270911yescharacteristic
427C22#271711yescharacteristic
428C22#313712yes—
429C22#313811yescharacteristic
430C22#314211yescharacteristic
431C22#319412yes—
432C22#320112yes—
433C22#325112yes—
434C22#326011yescharacteristic
435C22#326111yescharacteristic
436C22#330612yes—
437C22#330812yes—
438C22#332312yes—
439C22#332712yes—
440C22#334912yes—
441C22#335012yes—
442C22#335312yes—
443C22#335412yes—
444C4—44no—
445C4—42no—
446C4—82no—
447C4—84no—
448C4—162no—
449C4—162no—
450C4—162no—
451C4#251711yescharacteristic
452C4#264912yes—
453C4#273511yescharacteristic
454C4#273812yes—
455C4#277111yescharacteristic
456C4#283411yescharacteristic
457C2—82no—
458C2#1683011yescharacteristic
459C2#2770812yes—
460C2#2771912yes—
461C2#3019611yescharacteristic
462C2#3020111yescharacteristic
463C2#3162212yes—
464C2#3162412yes—
465C2#3204312yes—
466C2#3227612yes—
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
3. #11316C2semidirect
4. #11292C2semidirect
5. #11268C2semidirect
6. #11207C2semidirect

Maximal subgroups

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

subgroupindex
#152232
#151502
#113162
#112922
#112682
#112072
#112002
#107222
#107152