Finite Group Databasebeta

← #324146

Subgroups — #324146

Order 512.

750 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33171C211yesmaximal, characteristic
2#24653C212yesmaximal, complemented
3#21597C211yesmaximal, characteristic, complemented
4#10651C211yesmaximal, characteristic, complemented
5#10617C211yesmaximal, characteristic, complemented
6#10536C211yesmaximal, characteristic, complemented
7#10262C211yesmaximal, characteristic, complemented
8#9572C212yesmaximal, complemented
9#9230C212yesmaximal, complemented
10#9134C212yesmaximal, complemented
11#7872C211yesmaximal, characteristic, complemented
12C2×#493C2211yescharacteristic
13C2×#466—21no—
14C2×#466—21no—
15C2×#460C2212yes—
16C2×#446C2211yescharacteristic
17C2×#431C2212yes—
18C22×C4×C8C2211yescharacteristic
19#2856C2212yes—
20C4×C22⋊Q8C2211yescharacteristic
21#2034—22no—
22#2000—22no—
23#1984C2211yescharacteristic
24#1982C2212yes—
25#1970—21no—
26#1970—21no—
27#1968—22no—
28#1957C2212yes—
29#1948C2212yes—
30#1939C2211yescharacteristic
31#1934—22no—
32#1916—21no—
33#1850—21no—
34#1845C2211yescharacteristic
35#1803C2212yes—
36#1797C2211yescharacteristic
37#1771—21no—
38#1770C2211yescharacteristic
39#1752—21no—
40C2×#331C2212yes—
41C2×#331C2212yes—
42#1296C2212yes—
43#1296C2212yes—
44#1277C2212yes—
45#1277C2211yescharacteristic
46#1277C2211yescharacteristic
47C22×SD16—22no—
48C22×D8—21no—
49C22×D8—21no—
50C23×C8D411yescharacteristic
51C23×C8D411yescharacteristic
52C2×#127C2312yes—
53C2×C22⋊Q8D411yescharacteristic
54C2×#121—22no—
55C2×#121—22no—
56C2×#121—22no—
57C2×C4×Q8C2311yescharacteristic
58C22×C42C2311yescharacteristic
59#493—42no—
60#466—44no—
61#466—44no—
62#460—24no—
63#460—42no—
64#446—42no—
65#431—24no—
66#431—42no—
67C2×#107—21no—
68C2×#107—21no—
69C2×#106—22no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8C2311yescharacteristic
73C2×C4⋊C8C2311yescharacteristic
74C2×C4⋊C8C2312yes—
75C2×C4⋊C8C2312yes—
76C2×#103—22no—
77C2×#102—22no—
78C2×#102—22no—
79C2×#102—22no—
80C2×#102—22no—
81C2×#102—22no—
82C2×C22⋊C8—21no—
83C2×C22⋊C8—22no—
84C2×#97—21no—
85C2×#97—21no—
86C2×C4×C8—22no—
87C2×C4×C8—42no—
88C2×C4×C8D411yescharacteristic
89C2×C4×C8D411yescharacteristic
90C2×C4×C8C2311yescharacteristic
91C2×C4×C8C2311yescharacteristic
92#391D411yescharacteristic
93#390—22no—
94#390—22no—
95#390—22no—
96#389D411yescharacteristic
97#387—21no—
98#385—21no—
99#384C2312yes—
100#380D411yescharacteristic
101C4×C4⋊C4—21no—
102C4×C4⋊C4C2311yescharacteristic
103C4×C22⋊C4—21no—
104#340—22no—
105#340—22no—
106#336—21no—
107#336—21no—
108#336—21no—
109#336—21no—
110#336—21no—
111#331—24no—
112#331—42no—
113C22×Q8C2×D411yescharacteristic
114C22×D4—22no—
115C22×D4—22no—
116C22×D4—22no—
117C22×D4—42no—
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4#4111yescharacteristic
121C2×SD16—44no—
122C2×SD16—44no—
123C2×SD16—82no—
124C2×D8—42no—
125C2×D8—42no—
126C2×D8—42no—
127C2×D8—42no—
128C2×D8—42no—
129C2×D8—42no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—22no—
136C22×C8—41no—
137C22×C8—41no—
138C22×C8—41no—
139C22×C8SD1614yescomplemented
140C22×C8C2×D411yescharacteristic
141C22×C8C2×D411yescharacteristic
142C22×C8C2×D411yescharacteristic
143C22×C8C2×D411yescharacteristic
144#127—22no—
145#127—22no—
146#127—22no—
147#127—22no—
148C22⋊Q8—42no—
149#121—44no—
150#121—44no—
151#121—82no—
152#121—82no—
153#121—84no—
154C4×Q8—22no—
155C4×Q8—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4C2×D411yescharacteristic
163C2×C4⋊C4C2×D411yescharacteristic
164C2×C4⋊C4C2×D411yescharacteristic
165C2×C22⋊C4—21no—
166C2×C22⋊C4—21no—
167C2×C22⋊C4—42no—
168C2×C22⋊C4—42no—
169C2×C22⋊C4—42no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42#4111yescharacteristic
177C2×C42C2411yescharacteristic
178#107—42no—
179#107—42no—
180#106—82no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—42no—
187C4⋊C8—42no—
188C4⋊C8#4114yes—
189#103—82no—
190#102—44no—
191#102—44no—
192#102—44no—
193#102—44no—
194#102—82no—
195C22⋊C8—44no—
196C22⋊C8—42no—
197#97—42no—
198#97—42no—
199C4×C8—22no—
200C4×C8—22no—
201C4×C8—41no—
202C4×C8—41no—
203C4×C8—44no—
204#95—21no—
205#95—21no—
206#95—21no—
207#95—21no—
208#95—21no—
209C24—42no—
210C24#12111yescharacteristic
211C2×Q8—22no—
212C2×Q8—22no—
213C2×Q8—21no—
214C2×Q8—21no—
215C2×D4—42no—
216C2×D4—42no—
217C2×D4—42no—
218C2×D4—42no—
219C2×D4—42no—
220C2×D4—42no—
221C2×D4—42no—
222C2×D4—42no—
223C2×D4—42no—
224C2×D4—42no—
225C2×D4—42no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—82no—
230C2×D4—82no—
231C2×D4—84no—
232C2×D4—84no—
233C2×D4—84no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—41no—
249C22×C4—42no—
250C22×C4#12111yescharacteristic
251C22×C4#12111yescharacteristic
252C22×C4#12111yescharacteristic
253C22×C4C2×SD1612yes—
254C22×C4AGL(1,ℤ/8)12yes—
255C22×C4C22×D411yescharacteristic
256C22×C4C22×D411yescharacteristic
257C22×C4C2×#4111yescharacteristic
258SD16—88no—
259D8—84no—
260D8—84no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—24no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—44no—
272C2×C8—44no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—42no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8C2×SD1612yes—
285C2×C8C2×SD1612yes—
286C2×C8#13512yes—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—82no—
299C22⋊C4—42no—
300C22⋊C4—42no—
301C22⋊C4—84no—
302C22⋊C4—84no—
303C22⋊C4—84no—
304C42—21no—
305C42—21no—
306C42—22no—
307C42—22no—
308C42—41no—
309C42—41no—
310C42—41no—
311C42—41no—
312C42C2×#4112yes—
313C42H(5,𝔽2)12yes—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—82no—
321C23—82no—
322C23—82no—
323C23—82no—
324C23—82no—
325C23—82no—
326C23—82no—
327C23#46012yes—
328C23C2×#12111yescharacteristic
329Q8—42no—
330Q8—42no—
331D4—82no—
332D4—82no—
333D4—82no—
334D4—82no—
335D4—82no—
336D4—82no—
337D4—82no—
338D4—82no—
339D4—162no—
340D4—162no—
341D4—168no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—44no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4#46012yes—
374C2×C4#47212yes—
375C2×C4#53211yescharacteristic
376C2×C4#53411yescharacteristic
377C2×C4#54612yes—
378C2×C4#55312yes—
379C2×C4C22×SD1611yescharacteristic
380C2×C4C2×#13511yescharacteristic
381C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
382C2×C4#57511yescharacteristic
383C2×C4#57512yes—
384C2×C4#57611yescharacteristic
385C2×C4#57611yescharacteristic
386C8—22no—
387C8—22no—
388C8—22no—
389C8—41no—
390C8—44no—
391C8—81no—
392C8—81no—
393C8—82no—
394C22—22no—
395C22—22no—
396C22—22no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—162no—
402C22—162no—
403C22—162no—
404C22—162no—
405C22—162no—
406C22—162no—
407C22—162no—
408C22#254311yescharacteristic
409C22C2×#46011yescharacteristic
410C22#303511yescharacteristic
411C22#305411yescharacteristic
412C22#305811yescharacteristic
413C22C2×#47211yescharacteristic
414C22#306711yescharacteristic
415C4—22no—
416C4—22no—
417C4—22no—
418C4—41no—
419C4—81no—
420C4—81no—
421C4—81no—
422C4—81no—
423C4—162no—
424C4#316711yescharacteristic
425C4#318011yescharacteristic
426C4#318711yescharacteristic
427C4#318811yescharacteristic
428C4#330912yes—
429C4#331912yes—
430C2—42no—
431C2—322no—
432C2#1914611yescharacteristic
433C2#1916011yescharacteristic
434C2#1920311yescharacteristic
435C2#1921911yescharacteristic
436C2#2856211yescharacteristic
437C2#2857411yescharacteristic
438C2#2859211yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radical#33171C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #24653C2semidirect
3. #21597C2semidirect
4. #10651C2semidirect
5. #10617C2semidirect
6. #10536C2semidirect
7. #10262C2semidirect
8. #9572C2semidirect
9. #9230C2semidirect
10. #9134C2semidirect
11. #7872C2semidirect
139. C22×C8SD16semidirect

Maximal subgroups

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

subgroupindex
#331712
#246532
#215972
#106512
#106172
#105362
#102622
#95722
#92302
#91342
#78722