Finite Group Databasebeta

← #329821

Subgroups — #329821

Order 512.

713 conjugacy classes of subgroups.

The subgroup lattice has 447 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33109C211yesmaximal, characteristic, complemented
2#33107C211yesmaximal, characteristic
3#33042C211yesmaximal, characteristic
4C2×#1939C211yesmaximal, characteristic, complemented
5#10656C212yesmaximal, complemented
6#10598C212yesmaximal, complemented
7#10587C212yesmaximal, complemented
8#10540C212yesmaximal, complemented
9#9133C211yesmaximal, characteristic
10#9129C211yesmaximal, characteristic, complemented
11#8595C211yesmaximal, characteristic, complemented
12C2×#486C2211yescharacteristic
13C2×#484—21no—
14C2×#481—21no—
15C22×C4⋊C8—21no—
16C22×C4⋊C8C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18#2857C2211yescharacteristic
19#2856C2211yescharacteristic
20C2×C4×C4⋊C4C2211yescharacteristic
21#1990C2211yescharacteristic
22#1982C2212yes—
23#1981C2212yes—
24#1957C2211yescharacteristic
25#1948C2212yes—
26#1947C2212yes—
27#1939—21no—
28#1939—21no—
29#1939C2212yes—
30#1939C2212yes—
31#1934—21no—
32#1934—21no—
33#1928—22no—
34#1926—22no—
35#1893—21no—
36#1893—21no—
37#1874—21no—
38#1872—21no—
39#1834C2211yescharacteristic
40#1797C2211yescharacteristic
41#1797C2211yescharacteristic
42C2×#336—21no—
43C2×#334C2212yes—
44C2×#332C2212yes—
45C2×#331C2212yes—
46#1296C2212yes—
47#1296C2212yes—
48#1277C2212yes—
49C23×C8—21no—
50C23×C8D411yescharacteristic
51C23×C8D411yescharacteristic
52C2×C4⋊Q8C2311yescharacteristic
53C2×#127C2311yescharacteristic
54C2×C22⋊Q8—21no—
55C2×#121—21no—
56C2×#121—21no—
57C2×#121—21no—
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4D411yescharacteristic
60C22×C42—21no—
61C22×C42C2311yescharacteristic
62#486—42no—
63#484—44no—
64#481—44no—
65C2×#107—22no—
66C2×#106—22no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—42no—
71C2×C4⋊C8—44no—
72C2×C4⋊C8C2312yes—
73C2×C4⋊C8C2312yes—
74C2×C4⋊C8C2312yes—
75C2×#103—22no—
76C2×#103—22no—
77C2×#102—21no—
78C2×#102—21no—
79C2×#102—21no—
80C2×#102—21no—
81C2×#102—22no—
82C2×#102—22no—
83C2×C22⋊C8—21no—
84C2×C22⋊C8—21no—
85C2×C4×C8—42no—
86C2×C4×C8D411yescharacteristic
87C2×C4×C8D411yescharacteristic
88C2×C4×C8C2311yescharacteristic
89C2×C4×C8C2311yescharacteristic
90#391—21no—
91#390—21no—
92#390—21no—
93#390—21no—
94#389—21no—
95#389—21no—
96#384C2311yescharacteristic
97#384C2311yescharacteristic
98#380—21no—
99#380—21no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4D412yes—
103C4×C4⋊C4C2312yes—
104C2×#95—21no—
105#340—21no—
106#340—21no—
107#340—21no—
108#340—21no—
109#340—22no—
110#340—22no—
111#336—22no—
112#336—22no—
113#336—21no—
114#336—21no—
115#334—42no—
116#332—42no—
117#331—42no—
118C22×Q8—21no—
119C22×D4—21no—
120C22×D4—21no—
121C22×D4—21no—
122C22×D4—41no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4#4111yescharacteristic
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—41no—
135C22×C8—41no—
136C22×C8—41no—
137C22×C8—42no—
138C22×C8—42no—
139C22×C8—42no—
140C22×C8C2×D411yescharacteristic
141C22×C8C2×D411yescharacteristic
142C22×C8C2×D411yescharacteristic
143C22×C8C2×D411yescharacteristic
144C4⋊Q8—22no—
145C4⋊Q8—22no—
146#127—22no—
147#127—22no—
148C22⋊Q8—42no—
149C22⋊Q8—42no—
150#121—42no—
151#121—42no—
152#121—42no—
153#121—42no—
154#121—82no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4C2×D412yes—
170C2×C4⋊C4C2×D412yes—
171C2×C22⋊C4—41no—
172C2×C22⋊C4—41no—
173C2×C22⋊C4—41no—
174C2×C22⋊C4—41no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—42no—
180C2×C42—42no—
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42#4111yescharacteristic
184C2×C42C2411yescharacteristic
185#107—44no—
186#106—44no—
187C4⋊C8—22no—
188C4⋊C8—22no—
189C4⋊C8—22no—
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—22no—
193C4⋊C8—42no—
194C4⋊C8—44no—
195C4⋊C8—44no—
196#103—44no—
197#103—44no—
198#102—44no—
199#102—42no—
200#102—42no—
201#102—42no—
202#102—42no—
203#102—44no—
204C22⋊C8—42no—
205C22⋊C8—42no—
206C4×C8—22no—
207C4×C8—22no—
208C4×C8—41no—
209C4×C8—41no—
210#95—22no—
211#95—22no—
212#95—41no—
213#95—41no—
214C24—41no—
215C24#12111yescharacteristic
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×D4—41no—
221C2×D4—41no—
222C2×D4—42no—
223C2×D4—42no—
224C2×D4—42no—
225C2×D4—41no—
226C2×D4—41no—
227C2×D4—41no—
228C2×D4—41no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—81no—
232C2×D4—81no—
233C2×D4—82no—
234C2×D4—82no—
235C2×D4—82no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—24no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—41no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4#12111yescharacteristic
257C22×C4#12111yescharacteristic
258C22×C4#12111yescharacteristic
259C22×C4#13512yes—
260C22×C4AGL(1,ℤ/8)12yes—
261C22×C4AGL(1,ℤ/8)12yes—
262C22×C4#13712yes—
263C22×C4C22×D411yescharacteristic
264C22×C4C22×D411yescharacteristic
265C22×C4C2×#4111yescharacteristic
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—44no—
278C2×C8—44no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—84no—
289C2×C8AGL(1,ℤ/8)12yes—
290C4⋊C4—24no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—42no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—42no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—81no—
307C4⋊C4—81no—
308C4⋊C4—82no—
309C4⋊C4#13514yes—
310C22⋊C4—82no—
311C22⋊C4—82no—
312C22⋊C4—82no—
313C22⋊C4—82no—
314C42—21no—
315C42—21no—
316C42—22no—
317C42—22no—
318C42—44no—
319C42—42no—
320C42—42no—
321C42H(5,𝔽2)12yes—
322C422−1+412yes—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—81no—
328C23—81no—
329C23—81no—
330C23—81no—
331C23—81no—
332C23—81no—
333C23—81no—
334C23#46412yes—
335C23#46912yes—
336C23#47112yes—
337C23C2×#12111yescharacteristic
338Q8—82no—
339Q8—82no—
340D4—81no—
341D4—81no—
342D4—81no—
343D4—81no—
344D4—82no—
345D4—82no—
346D4—162no—
347D4—164no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—21no—
361C2×C4—24no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—44no—
365C2×C4—48no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—84no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4#46412yes—
382C2×C4#46912yes—
383C2×C4#53411yescharacteristic
384C2×C4#53611yescharacteristic
385C2×C4#53712yes—
386C2×C4#53912yes—
387C2×C4C2×#13511yescharacteristic
388C2×C4C2×#13512yes—
389C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
390C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
391C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
392C2×C4C2×#13711yescharacteristic
393C2×C4#57511yescharacteristic
394C8—22no—
395C8—41no—
396C8—41no—
397C8—41no—
398C8—82no—
399C8—82no—
400C8—82no—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—161no—
408C22—161no—
409C22—161no—
410C22—161no—
411C22—161no—
412C22—161no—
413C22—161no—
414C22#170214yes—
415C22#261811yescharacteristic
416C22C2×#46411yescharacteristic
417C22#303611yescharacteristic
418C22#305011yescharacteristic
419C22C2×#46911yescharacteristic
420C22C2×#47111yescharacteristic
421C22#306711yescharacteristic
422C4—22no—
423C4—22no—
424C4—22no—
425C4—44no—
426C4—41no—
427C4—82no—
428C4—161no—
429C4—161no—
430C4—161no—
431C4#318811yescharacteristic
432C4#319611yescharacteristic
433C4#323211yescharacteristic
434C4#323911yescharacteristic
435C4#324712yes—
436C4#324712yes—
437C2—24no—
438C2—321no—
439C2#1912311yescharacteristic
440C2C2×#170211yescharacteristic
441C2#1925711yescharacteristic
442C2#1928011yescharacteristic
443C2#2976111yescharacteristic
444C2#2977211yescharacteristic
445C2#2979011yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radical#2856C22
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
1. #33109C2semidirect
4. C2×#1939C2semidirect
5. #10656C2semidirect
6. #10598C2semidirect
7. #10587C2semidirect
8. #10540C2semidirect
10. #9129C2semidirect
11. #8595C2semidirect

Maximal subgroups

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

subgroupindex
#331092
#331072
#330422
C2×#19392
#106562
#105982
#105872
#105402
#91332
#91292
#85952