Finite Group Databasebeta

← #437001

Subgroups — #437001

Order 512.

1044 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#15195C211yesmaximal, characteristic
2#14129C211yesmaximal, characteristic
3#11315C212yesmaximal, complemented
4#11293C212yesmaximal, complemented
5#11267C212yesmaximal, complemented
6#11199C212yesmaximal
7#10921C211yesmaximal, characteristic
8#10880C212yesmaximal, complemented
9#10731C212yesmaximal
10C22×#117C2211yescharacteristic
11C22×C22⋊C8C2212yes—
12C22×C4×C8C2211yescharacteristic
13C2×#389C2212yes—
14C2×#384C2211yescharacteristic
15C2×#380C2211yescharacteristic
16C2×#380C2212yes—
17C2×C4×C22⋊C4C2211yescharacteristic
18C2×#340C2212yes—
19C2×#340C2212yes—
20C2×#340C2212yes—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#336C2212yes—
27C2×#336C2211yescharacteristic
28#1442—22no—
29#1439—22no—
30#1439—22no—
31#1436—22no—
32C4⋊#95C2212yes—
33C4⋊#95C2212yes—
34C4⋊#95C2211yescharacteristic
35C24×C4C2311yescharacteristic
36C23×C8C2312yes—
37C23×C8C2312yes—
38C2×#117—24no—
39C2×#117—42no—
40C2×#117—41no—
41C2×#117—41no—
42C22×C4⋊C4—22no—
43C22×C4⋊C4—22no—
44C22×C4⋊C4C2312yes—
45C22×C4⋊C4C2312yes—
46C22×C4⋊C4C2312yes—
47C22×C4⋊C4C2312yes—
48C22×C22⋊C4—21no—
49C22×C22⋊C4—21no—
50C22×C22⋊C4—22no—
51C22×C22⋊C4D412yes—
52C22×C42—21no—
53C22×C42C2311yescharacteristic
54C22×C42C2311yescharacteristic
55C2×C22⋊C8—24no—
56C2×C22⋊C8—24no—
57C2×C22⋊C8—42no—
58C2×C4×C8—41no—
59C2×C4×C8—42no—
60#389—24no—
61#389—42no—
62#384—41no—
63#384—41no—
64#380—24no—
65#380—24no—
66#380—24no—
67C4×C22⋊C4—41no—
68C4×C22⋊C4—41no—
69C2×#95—21no—
70C2×#95—22no—
71C2×#95—22no—
72C2×#95—22no—
73C2×#95—22no—
74C2×#95—22no—
75C2×#95—22no—
76C2×#95—22no—
77C2×#95D412yes—
78#340—24no—
79#340—24no—
80#340—24no—
81#340—24no—
82#340—42no—
83#340—42no—
84#340—42no—
85#340—42no—
86#336—24no—
87#336—41no—
88C25C2×D411yescharacteristic
89C23×C4—22no—
90C23×C4—22no—
91C23×C4—22no—
92C23×C4—21no—
93C23×C4—21no—
94C23×C4—22no—
95C23×C4—22no—
96C23×C4—22no—
97C23×C4—22no—
98C23×C4—22no—
99C23×C4—22no—
100C23×C4C2×D411yescharacteristic
101C23×C4C2×D412yes—
102C23×C4C2×D412yes—
103C23×C4#4114yes—
104C23×C4C2411yescharacteristic
105C22×C8—22no—
106C22×C8—22no—
107C22×C8—22no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8#4114yes—
119#117—44no—
120#117—48no—
121#117—44no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
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—44no—
145C2×C4⋊C4—44no—
146C2×C4⋊C4—44no—
147C2×C4⋊C4—82no—
148C2×C4⋊C4—82no—
149C2×C4⋊C4—82no—
150C2×C4⋊C4#4114yes—
151C2×C4⋊C4#4114yes—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—24no—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—42no—
156C2×C22⋊C4—44no—
157C2×C22⋊C4—42no—
158C2×C22⋊C4—44no—
159C2×C22⋊C4—82no—
160C2×C22⋊C4—82no—
161C2×C22⋊C4—82no—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42—21no—
165C2×C42—21no—
166C2×C42—21no—
167C2×C42—21no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—44no—
171C2×C42—42no—
172C22⋊C8—48no—
173C4×C8—44no—
174#95—24no—
175#95—42no—
176#95—44no—
177#95—44no—
178#95—44no—
179#95—44no—
180#95—82no—
181#95—82no—
182#95—82no—
183C24—21no—
184C24—21no—
185C24—22no—
186C24—22no—
187C24—22no—
188C24—22no—
189C24—22no—
190C24—22no—
191C24#12412yes—
192C24C22×D411yescharacteristic
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—41no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
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#12412yes—
254C22×C4#13514yes—
255C22×C4#13714yes—
256C22×C4C2×#4112yes—
257C22×C4C2×#4112yes—
258C22×C4C2×#4112yes—
259C22×C4C2×#4112yes—
260C22×C4H(5,𝔽2)11yescharacteristic
261C22×C4H(5,𝔽2)12yes—
262C22×C4H(5,𝔽2)11yescharacteristic
263C22×C42−1+412yes—
264C2×C8—24no—
265C2×C8—24no—
266C2×C8—24no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C4⋊C4—24no—
279C4⋊C4—24no—
280C4⋊C4—24no—
281C4⋊C4—24no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
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—88no—
295C4⋊C4—88no—
296C22⋊C4—48no—
297C22⋊C4—48no—
298C22⋊C4—84no—
299C22⋊C4—88no—
300C42—48no—
301C42—42no—
302C42—42no—
303C42—41no—
304C42—41no—
305C42—41no—
306C42—41no—
307C23—22no—
308C23—22no—
309C23—24no—
310C23—22no—
311C23—24no—
312C23—42no—
313C23—42no—
314C23—42no—
315C23—42no—
316C23—42no—
317C23—41no—
318C23—41no—
319C23—42no—
320C23—41no—
321C23—42no—
322C23—42no—
323C23—41no—
324C23—42no—
325C23—42no—
326C23—42no—
327C23—42no—
328C23#52511yescharacteristic
329C23C2×#12411yescharacteristic
330C23#53411yescharacteristic
331C23#54612yes—
332C23#54712yes—
333C23C2×#13512yes—
334C23C2×#13712yes—
335C23#57612yes—
336C23#57712yes—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
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#52912yes—
389C2×C4#52912yes—
390C2×C4#53812yes—
391C2×C4#53912yes—
392C2×C4#53912yes—
393C2×C4#53912yes—
394C2×C4#55111yescharacteristic
395C2×C4#55112yes—
396C2×C4#55212yes—
397C2×C4#55312yes—
398C2×C4#55312yes—
399C2×C4#55312yes—
400C2×C4#55612yes—
401C2×C4#55911yescharacteristic
402C2×C4H(3,𝔽4)11yescharacteristic
403C2×C4#56311yescharacteristic
404C8—44no—
405C8—82no—
406C22—44no—
407C22—44no—
408C22—42no—
409C22—42no—
410C22—42no—
411C22—82no—
412C22—82no—
413C22—82no—
414C22—82no—
415C22#246612yes—
416C22#248812yes—
417C22#259111yescharacteristic
418C22#268411yescharacteristic
419C22#272411yescharacteristic
420C22#310712yes—
421C22#312211yescharacteristic
422C22#312611yescharacteristic
423C22#313111yescharacteristic
424C22#313612yes—
425C22#313911yescharacteristic
426C22#319012yes—
427C22#319812yes—
428C22#330312yes—
429C22#330512yes—
430C22#331412yes—
431C22#331612yes—
432C22#333712yes—
433C22#333912yes—
434C22#335612yes—
435C22#335712yes—
436C4—44no—
437C4—44no—
438C4—42no—
439C4—84no—
440C4—84no—
441C4—162no—
442C4—162no—
443C4#257712yes—
444C4#259512yes—
445C4#278212yes—
446C4#278712yes—
447C2—82no—
448C2#1557911yescharacteristic
449C2#2704112yes—
450C2#2704512yes—
451C2#2770212yes—
452C2#2772112yes—
453C2#2933211yescharacteristic
454C2#2934011yescharacteristic
455C2#3126812yes—
456C2#3246412yes—
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. #11315C2semidirect
4. #11293C2semidirect
5. #11267C2semidirect
8. #10880C2semidirect

Maximal subgroups

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

subgroupindex
#151952
#141292
#113152
#112932
#112672
#111992
#109212
#108802
#107312