Finite Group Databasebeta

← #413441

Subgroups — #413441

Order 512.

1524 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2073C212yesmaximal, complemented
2C2×#1982C212yesmaximal
3#14713C211yesmaximal, characteristic
4C4×#382C211yesmaximal, characteristic, complemented
5#11189C214yesmaximal, complemented
6#11051C212yesmaximal, complemented
7#10849C212yesmaximal, complemented
8#10832C211yesmaximal, characteristic, complemented
9C22×#127C2211yescharacteristic
10C22×C4⋊C8C2212yes—
11C22×#102C2214yes—
12C22×#102C2214yes—
13C2×#390—22no—
14C2×#390—22no—
15C2×#390C2212yes—
16C2×#390C2212yes—
17C2×#384C2211yescharacteristic
18C2×#382C2211yescharacteristic
19C2×#382C2211yescharacteristic
20C2×C4×C4⋊C4C2212yes—
21C2×C43C2211yescharacteristic
22#2073—28no—
23#1982—28no—
24C2×#340C2214yes—
25C2×#340C2214yes—
26C2×#336C2212yes—
27C2×#336C2212yes—
28C4⋊#95C2212yes—
29#1434—22no—
30C4×#95—22no—
31C23×D4—21no—
32C23×D4—21no—
33C23×D4C2312yes—
34C23×C8C2314yes—
35C2×#127—24no—
36C2×#127—24no—
37C2×#127—22no—
38C2×#127D414yes—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—21no—
41C22×C4⋊C4C2312yes—
42C22×C4⋊C4C2312yes—
43C22×C4⋊C4C2312yes—
44C22×C22⋊C4—24no—
45C22×C22⋊C4—24no—
46C22×C42—22no—
47C22×C42D412yes—
48C22×C42D412yes—
49C22×C42C2311yescharacteristic
50C22×C42C2311yescharacteristic
51C22×C42C2311yescharacteristic
52C2×C4⋊C8—24no—
53C2×C4⋊C8—24no—
54C2×C4⋊C8—24no—
55C2×#102—28no—
56C2×#102—28no—
57C2×#102—28no—
58C2×#102—28no—
59C2×#102—44no—
60C2×#102—44no—
61#390—24no—
62#390—24no—
63#390—44no—
64#390—48no—
65#390—48no—
66#384—24no—
67#382—24no—
68#382—22no—
69#382D414yescomplemented
70C4×C4⋊C4—24no—
71C4×C4⋊C4—24no—
72C2×#95—22no—
73C2×#95—22no—
74C2×#95—22no—
75C2×#95—22no—
76C2×#95—22no—
77C43—24no—
78#340—28no—
79#340—44no—
80#336—24no—
81#336—42no—
82C25—22no—
83C22×D4—24no—
84C22×D4—24no—
85C22×D4—22no—
86C22×D4—24no—
87C22×D4—24no—
88C22×D4—22no—
89C22×D4—24no—
90C22×D4—22no—
91C22×D4—24no—
92C22×D4—22no—
93C22×D4—22no—
94C22×D4—44no—
95C22×D4—44no—
96C22×D4—42no—
97C22×D4—42no—
98C22×D4—42no—
99C22×D4—42no—
100C22×D4—44no—
101C22×D4—44no—
102C22×D4C2×D414yes—
103C23×C4—22no—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4C2×D412yes—
107C23×C4C2×D412yes—
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C23×C4C2411yescharacteristic
111C22×C8—24no—
112C22×C8—24no—
113C22×C8—24no—
114C22×C8—24no—
115C22×C8—24no—
116C22×C8—24no—
117C22×C8—24no—
118#127—216no—
119#127—48no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
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—24no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—44no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4—44no—
138C2×C4⋊C4C2×D414yes—
139C2×C4⋊C4#4114yes—
140C2×C22⋊C4—48no—
141C2×C22⋊C4—48no—
142C2×C22⋊C4—48no—
143C2×C22⋊C4—48no—
144C2×C22⋊C4—48no—
145C2×C22⋊C4—48no—
146C2×C42—22no—
147C2×C42—22no—
148C2×C42—24no—
149C2×C42—24no—
150C2×C42—24no—
151C2×C42—24no—
152C2×C42—24no—
153C2×C42—22no—
154C2×C42—21no—
155C2×C42—22no—
156C2×C42—21no—
157C2×C42—42no—
158C2×C42—44no—
159C2×C42—44no—
160C2×C42—44no—
161C2×C42C2×D412yes—
162C2×C42C2×D412yes—
163C2×C42#4112yes—
164C2×C42#4114yes—
165C2×C42#4114yes—
166C2×C42#4112yes—
167C4⋊C8—48no—
168#102—416no—
169#102—416no—
170#95—28no—
171#95—28no—
172#95—44no—
173#95—44no—
174#95—44no—
175C24—24no—
176C24—42no—
177C24—42no—
178C24—42no—
179C24—42no—
180C24—42no—
181C24—42no—
182C24—44no—
183C24—44no—
184C24—44no—
185C24—44no—
186C24C22×D411yescharacteristic
187C2×D4—28no—
188C2×D4—24no—
189C2×D4—28no—
190C2×D4—24no—
191C2×D4—416no—
192C2×D4—44no—
193C2×D4—44no—
194C2×D4—44no—
195C2×D4—44no—
196C2×D4—44no—
197C2×D4—42no—
198C2×D4—44no—
199C2×D4—44no—
200C2×D4—42no—
201C2×D4—44no—
202C2×D4—44no—
203C2×D4—44no—
204C2×D4—48no—
205C2×D4—48no—
206C2×D4—416no—
207C2×D4—88no—
208C2×D4—84no—
209C2×D4—84no—
210C2×D4—88no—
211C2×D4—84no—
212C2×D4—84no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—24no—
228C22×C4—24no—
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—44no—
245C22×C4—44no—
246C22×C4#12114yes—
247C22×C4#12112yes—
248C22×C4#12112yes—
249C22×C4#12312yes—
250C22×C4#12312yes—
251C22×C4#12412yes—
252C22×C4#12412yes—
253C22×C4#13514yes—
254C22×C4AGL(1,ℤ/8)14yes—
255C22×C4AGL(1,ℤ/8)14yes—
256C22×C4C22×D411yescharacteristic
257C22×C4C22×D411yescharacteristic
258C22×C4C2×#4112yes—
259C22×C4C2×#4112yes—
260C22×C4C2×#4111yescharacteristic
261C22×C4C2×#4111yescharacteristic
262C22×C4C2×#4112yes—
263C22×C4H(5,𝔽2)11yescharacteristic
264C22×C4H(5,𝔽2)12yes—
265C22×C42−1+411yescharacteristic
266C2×C8—44no—
267C2×C8—44no—
268C2×C8—44no—
269C2×C8—44no—
270C2×C8—44no—
271C2×C8—44no—
272C2×C8—44no—
273C4⋊C4—24no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—44no—
282C4⋊C4—44no—
283C4⋊C4—84no—
284C4⋊C4—84no—
285C22⋊C4—816no—
286C22⋊C4—816no—
287C42—24no—
288C42—24no—
289C42—48no—
290C42—42no—
291C42—42no—
292C42—48no—
293C42—88no—
294C42#12318yes—
295C42#12618yes—
296C23—44no—
297C23—48no—
298C23—48no—
299C23—44no—
300C23—44no—
301C23—84no—
302C23—84no—
303C23—82no—
304C23—84no—
305C23—82no—
306C23—84no—
307C23—82no—
308C23—82no—
309C23—82no—
310C23—82no—
311C23—82no—
312C23—84no—
313C23—84no—
314C23—84no—
315C23—84no—
316C23—84no—
317C23—84no—
318C23—82no—
319C23C2×#12111yescharacteristic
320C23C2×#12111yescharacteristic
321C23C2×#12311yescharacteristic
322C23#52511yescharacteristic
323C23C2×#12411yescharacteristic
324C23#53511yescharacteristic
325C23#53711yescharacteristic
326C23C2×#13512yes—
327C23C2×AGL(1,ℤ/8)12yes—
328C23C2×AGL(1,ℤ/8)12yes—
329C23#57512yes—
330D4—48no—
331D4—48no—
332D4—816no—
333D4—816no—
334D4—84no—
335D4—84no—
336C2×C4—24no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—84no—
371C2×C4—84no—
372C2×C4—82no—
373C2×C4#39714yes—
374C2×C4#39714yes—
375C2×C4#46318yes—
376C2×C4#48218yes—
377C2×C4#48814yes—
378C2×C4#49014yes—
379C2×C4C2×#12312yes—
380C2×C4C2×#12612yes—
381C2×C4#52912yes—
382C2×C4#53212yes—
383C2×C4#54012yes—
384C2×C4#54212yes—
385C2×C4D4211yescharacteristic
386C2×C4#54612yes—
387C2×C4#54711yescharacteristic
388C2×C4#54711yescharacteristic
389C2×C4#55012yes—
390C2×C4#55312yes—
391C2×C4#55511yescharacteristic
392C2×C4#55612yes—
393C2×C4#55912yes—
394C2×C4#55912yes—
395C8—84no—
396C22—84no—
397C22—84no—
398C22—84no—
399C22—88no—
400C22—88no—
401C22—162no—
402C22—162no—
403C22—164no—
404C22—162no—
405C22—162no—
406C22—164no—
407C22C2×#39711yescharacteristic
408C22#253511yescharacteristic
409C22#253711yescharacteristic
410C22#254811yescharacteristic
411C22#259211yescharacteristic
412C22#262411yescharacteristic
413C22#285411yescharacteristic
414C22C2×#46312yes—
415C22#303512yes—
416C22#303612yes—
417C22#303812yes—
418C22C2×#48212yes—
419C22#309412yes—
420C22#310612yes—
421C22#311512yes—
422C22C2×#48811yescharacteristic
423C22C2×#49011yescharacteristic
424C22#313811yescharacteristic
425C22#314011yescharacteristic
426C22#321212yes—
427C22#321712yes—
428C22#324712yes—
429C22#325112yes—
430C4—24no—
431C4—44no—
432C4—44no—
433C4—84no—
434C4—84no—
435C4—162no—
436C4#207318yescomplemented
437C4#244512yes—
438C4#244712yes—
439C4#251212yes—
440C4#251312yes—
441C2—164no—
442C2#1514711yescharacteristic
443C2C2×#207311yescharacteristic
444C2#2362811yescharacteristic
445C2#2815812yes—
446C2#2825312yes—
447C2#2875211yescharacteristic
448C2#2875611yescharacteristic
449C2#2934811yescharacteristic
450C2#2935611yescharacteristic
451C2#2992512yes—
452C2#3311312yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#127C22
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2073C2semidirect
4. C4×#382C2semidirect
5. #11189C2semidirect
6. #11051C2semidirect
7. #10849C2semidirect
8. #10832C2semidirect
69. #382D4semidirect
436. C4#2073semidirect

Maximal subgroups

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

subgroupindex
C2×#20732
C2×#19822
#147132
C4×#3822
#111892
#110512
#108492
#108322