Finite Group Databasebeta

← #314721

Subgroups — #314721

Order 512.

619 conjugacy classes of subgroups.

The subgroup lattice has 425 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27627C212yesmaximal
2#21487C211yesmaximal, characteristic
3#20817C211yesmaximal, characteristic, complemented
4#10658C212yesmaximal, complemented
5#10625C212yesmaximal, complemented
6#9621C211yesmaximal, characteristic, complemented
7#9620C211yesmaximal, characteristic, complemented
8#8729C212yesmaximal, complemented
9#8652C211yesmaximal, characteristic, complemented
10#8652C211yesmaximal, characteristic, complemented
11#8595C211yesmaximal, characteristic, complemented
12C2×#436C2211yescharacteristic
13C2×#435—21no—
14C22×C4×C8C2211yescharacteristic
15C4×#121C2211yescharacteristic
16#2285C2212yes—
17#2077—22no—
18#2033—22no—
19#1954C2211yescharacteristic
20#1948C2212yes—
21#1947C2212yes—
22#1939C2211yescharacteristic
23#1926—22no—
24#1923—21no—
25#1921—22no—
26#1869—21no—
27#1845C2211yescharacteristic
28#1845C2211yescharacteristic
29#1834C2212yes—
30#1808—21no—
31#1808—21no—
32#1802C2212yes—
33#1802C2212yes—
34#1797C2211yescharacteristic
35#1797C2212yes—
36C4×#107—21no—
37C4×#107—21no—
38#1770C2211yescharacteristic
39#1753—21no—
40C2×#334C2211yescharacteristic
41C2×#334C2211yescharacteristic
42#1296C2212yes—
43#1296C2212yes—
44#1296C2212yes—
45#1296C2212yes—
46#1277C2212yes—
47C22×M16—21no—
48C23×C8D411yescharacteristic
49C23×C8D411yescharacteristic
50C2×#121D411yescharacteristic
51C2×C4×D4—21no—
52C2×C4×D4C2311yescharacteristic
53C2×#117—22no—
54C22×C42C2311yescharacteristic
55#436—22no—
56#436—22no—
57#435—44no—
58C2×#107—21no—
59C2×#107—21no—
60C2×#107—21no—
61C2×#107—21no—
62C2×#107—22no—
63C2×#106—22no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8C2312yes—
66C2×C4⋊C8C2312yes—
67C2×C4⋊C8C2311yescharacteristic
68C2×C4⋊C8C2311yescharacteristic
69C2×#102—22no—
70C2×#102—22no—
71C2×C22⋊C8—21no—
72C2×C22⋊C8—21no—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—21no—
75C2×C22⋊C8—22no—
76C2×#97—21no—
77C2×C4×C8—22no—
78C2×C4×C8—22no—
79C2×C4×C8D411yescharacteristic
80C2×C4×C8D411yescharacteristic
81C2×C4×C8C2311yescharacteristic
82C2×C4×C8C2311yescharacteristic
83#390D411yescharacteristic
84#389D411yescharacteristic
85#388—21no—
86#386—21no—
87#384C2312yes—
88#384C2312yes—
89#382—22no—
90#380—22no—
91#380—22no—
92#380D411yescharacteristic
93C4×C4⋊C4—22no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4C2311yescharacteristic
96C4×C22⋊C4—21no—
97#340—22no—
98#340—22no—
99#340—22no—
100#340—22no—
101#340—22no—
102#340—22no—
103#336—21no—
104#336—21no—
105#336—22no—
106#336—21no—
107#334—22no—
108#334—22no—
109C22×D4—21no—
110C22×D4C2×D411yescharacteristic
111C23×C4—21no—
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C23×C4#4111yescharacteristic
115C2×M16—42no—
116C2×M16—42no—
117C2×M16—42no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—21no—
123C22×C8—21no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—41no—
127C22×C8—41no—
128C22×C8C2×D411yescharacteristic
129C22×C8C2×D411yescharacteristic
130C22×C8C2×D411yescharacteristic
131C22×C8C2×D411yescharacteristic
132#121—42no—
133C4×D4—21no—
134C4×D4—21no—
135C4×D4—21no—
136C4×D4—21no—
137C4×D4—42no—
138C4×D4—42no—
139#117—84no—
140C2×C4⋊C4—21no—
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⋊C4C2×D411yescharacteristic
148C2×C4⋊C4C2×D411yescharacteristic
149C2×C4⋊C4C2×D411yescharacteristic
150C2×C22⋊C4—21no—
151C2×C22⋊C4—21no—
152C2×C22⋊C4—21no—
153C2×C22⋊C4—21no—
154C2×C22⋊C4—21no—
155C2×C22⋊C4—42no—
156C2×C42—21no—
157C2×C42—21no—
158C2×C42—21no—
159C2×C42—21no—
160C2×C42—21no—
161C2×C42—42no—
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42#4111yescharacteristic
165C2×C42C2411yescharacteristic
166#107—44no—
167#107—42no—
168#107—42no—
169#107—42no—
170#107—42no—
171#106—82no—
172C4⋊C8—22no—
173C4⋊C8—22no—
174C4⋊C8—21no—
175C4⋊C8—21no—
176C4⋊C8—21no—
177C4⋊C8—21no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—42no—
181#102—44no—
182#102—82no—
183C22⋊C8—42no—
184C22⋊C8—42no—
185C22⋊C8—42no—
186C22⋊C8—42no—
187C22⋊C8—82no—
188#97—42no—
189C4×C8—22no—
190C4×C8—22no—
191C4×C8—21no—
192C4×C8—21no—
193C4×C8#4112yes—
194C4×C8#4112yes—
195#95—21no—
196#95—21no—
197#95—21no—
198#95—42no—
199#95—42no—
200C24—21no—
201C24#12111yescharacteristic
202C2×D4—21no—
203C2×D4—21no—
204C2×D4—22no—
205C2×D4—22no—
206C2×D4—42no—
207C2×D4—42no—
208C2×D4—42no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—22no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—21no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4#12111yescharacteristic
235C22×C4#12111yescharacteristic
236C22×C4#12111yescharacteristic
237C22×C4C22×D411yescharacteristic
238C22×C4C22×D411yescharacteristic
239C22×C4C2×#4111yescharacteristic
240M16—84no—
241C2×C8—21no—
242C2×C8—21no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—44no—
250C2×C8—44no—
251C2×C8—41no—
252C2×C8—41no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8#12312yes—
264C2×C8#12312yes—
265C2×C8#13512yes—
266C2×C8AGL(1,ℤ/8)12yes—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—82no—
281C4⋊C4—82no—
282C22⋊C4—42no—
283C22⋊C4—42no—
284C22⋊C4—42no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—42no—
288C22⋊C4—84no—
289C42—21no—
290C42—21no—
291C42—21no—
292C42—21no—
293C42—21no—
294C42—21no—
295C42—41no—
296C42—41no—
297C42—84no—
298C42C2×#4111yescharacteristic
299C42C2×#4111yescharacteristic
300C422−1+411yescharacteristic
301C422−1+411yescharacteristic
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23#47212yes—
316C23C2×#12111yescharacteristic
317D4—42no—
318D4—42no—
319D4—84no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4#47312yes—
358C2×C4C2×#12311yescharacteristic
359C2×C4#53611yescharacteristic
360C2×C4#54711yescharacteristic
361C2×C4#54711yescharacteristic
362C2×C4#55211yescharacteristic
363C2×C4#55211yescharacteristic
364C2×C4C2×#13511yescharacteristic
365C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
366C2×C4#57512yes—
367C2×C4#57511yescharacteristic
368C2×C4#57511yescharacteristic
369C2×C4#57712yes—
370C8—22no—
371C8—41no—
372C8—41no—
373C8—81no—
374C8—82no—
375C8—82no—
376C8—81no—
377C8#48214yes—
378C22—22no—
379C22—22no—
380C22—22no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—81no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22—81no—
392C22#248611yescharacteristic
393C22#303511yescharacteristic
394C22#303611yescharacteristic
395C22#305411yescharacteristic
396C22#306011yescharacteristic
397C22C2×#47211yescharacteristic
398C22C2×#47311yescharacteristic
399C4—21no—
400C4—21no—
401C4—22no—
402C4—41no—
403C4—41no—
404C4—81no—
405C4—81no—
406C4—81no—
407C4—162no—
408C4—162no—
409C4C2×#48211yescharacteristic
410C4#309311yescharacteristic
411C4#323311yescharacteristic
412C4#323511yescharacteristic
413C4#334812yes—
414C4#335012yes—
415C2—42no—
416C2—161no—
417C2#1916111yescharacteristic
418C2#1922711yescharacteristic
419C2#1928311yescharacteristic
420C2#1933811yescharacteristic
421C2#2764611yescharacteristic
422C2#2766511yescharacteristic
423C2#2767511yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#121D4
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
3. #20817C2semidirect
4. #10658C2semidirect
5. #10625C2semidirect
6. #9621C2semidirect
7. #9620C2semidirect
8. #8729C2semidirect
9. #8652C2semidirect
10. #8652C2semidirect
11. #8595C2semidirect

Maximal subgroups

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

subgroupindex
#276272
#214872
#208172
#106582
#106252
#96212
#96202
#87292
#86522
#85952