Finite Group Databasebeta

← #314715

Subgroups — #314715

Order 512.

655 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27625C212yesmaximal
2#21487C211yesmaximal, characteristic
3#20797C211yesmaximal, characteristic, complemented
4#10652C211yesmaximal, characteristic, complemented
5#10651C211yesmaximal, characteristic, complemented
6#10616C211yesmaximal, characteristic, complemented
7#10615C211yesmaximal, characteristic, complemented
8#9618C212yesmaximal, complemented
9#8727C212yesmaximal, complemented
10#8650C212yesmaximal, complemented
11#8596C211yesmaximal, characteristic, complemented
12C2×#436C2211yescharacteristic
13C2×#435—21no—
14C2×#427—21no—
15C2×#427—21no—
16C22×C4×C8C2211yescharacteristic
17C4×#121C2211yescharacteristic
18#2285C2212yes—
19#2077—22no—
20#2034—22no—
21#1954C2211yescharacteristic
22#1948C2212yes—
23#1948C2212yes—
24#1939C2211yescharacteristic
25#1926—22no—
26#1923—21no—
27#1919—22no—
28#1869—21no—
29#1846—21no—
30#1846—21no—
31#1845C2211yescharacteristic
32#1845C2211yescharacteristic
33#1844C2212yes—
34#1802C2212yes—
35#1802C2212yes—
36#1797C2211yescharacteristic
37#1797C2212yes—
38#1770C2211yescharacteristic
39#1753—21no—
40C2×#335C2212yes—
41#1296C2212yes—
42#1296C2212yes—
43#1296C2212yes—
44#1296C2212yes—
45#1277C2211yescharacteristic
46#1277C2211yescharacteristic
47C22×M16—21no—
48C23×C8D411yescharacteristic
49C23×C8D411yescharacteristic
50C2×#121D411yescharacteristic
51C2×C4×D4—21no—
52C2×C4×D4C2311yescharacteristic
53C2×#117—22no—
54C22×C42C2311yescharacteristic
55#436—24no—
56#435—44no—
57#427—44no—
58#427—44no—
59C2×#107—21no—
60C2×#107—21no—
61C2×#107—21no—
62C2×#107—21no—
63C2×#106—22no—
64C2×#106—22no—
65C2×#106—22no—
66C2×#106—22no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8C2312yes—
69C2×C4⋊C8C2312yes—
70C2×C4⋊C8C2311yescharacteristic
71C2×C4⋊C8C2311yescharacteristic
72C2×#102—22no—
73C2×#102—22no—
74C2×C22⋊C8—21no—
75C2×C22⋊C8—21no—
76C2×C22⋊C8—21no—
77C2×C22⋊C8—21no—
78C2×C22⋊C8—22no—
79C2×#97—21no—
80C2×C4×C8—24no—
81C2×C4×C8D411yescharacteristic
82C2×C4×C8D411yescharacteristic
83C2×C4×C8C2311yescharacteristic
84C2×C4×C8C2311yescharacteristic
85#390D411yescharacteristic
86#389D411yescharacteristic
87#388—21no—
88#386—21no—
89#384C2312yes—
90#384C2312yes—
91#382—22no—
92#380—22no—
93#380—22no—
94#380D411yescharacteristic
95C4×C4⋊C4—22no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4C2311yescharacteristic
98C4×C22⋊C4—21no—
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#335—24no—
108C22×D4—21no—
109C22×D4C2×D411yescharacteristic
110C23×C4—21no—
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4#4111yescharacteristic
114C2×M16—42no—
115C2×M16—42no—
116C2×M16—42no—
117C22×C8—22no—
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×C8SD1614yes—
127C22×C8C2×D411yescharacteristic
128C22×C8C2×D411yescharacteristic
129C22×C8C2×D411yescharacteristic
130C22×C8C2×D411yescharacteristic
131#121—42no—
132C4×D4—22no—
133C4×D4—22no—
134C4×D4—42no—
135C4×D4—42no—
136#117—44no—
137#117—44no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4C2×D411yescharacteristic
146C2×C4⋊C4C2×D411yescharacteristic
147C2×C4⋊C4C2×D411yescharacteristic
148C2×C22⋊C4—21no—
149C2×C22⋊C4—21no—
150C2×C22⋊C4—21no—
151C2×C22⋊C4—21no—
152C2×C22⋊C4—21no—
153C2×C22⋊C4—42no—
154C2×C42—22no—
155C2×C42—22no—
156C2×C42—21no—
157C2×C42—42no—
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42#4111yescharacteristic
161C2×C42C2411yescharacteristic
162#107—42no—
163#107—42no—
164#107—42no—
165#107—42no—
166#106—44no—
167#106—44no—
168#106—44no—
169#106—82no—
170C4⋊C8—22no—
171C4⋊C8—22no—
172C4⋊C8—22no—
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—42no—
177#102—44no—
178#102—82no—
179C22⋊C8—42no—
180C22⋊C8—42no—
181C22⋊C8—42no—
182C22⋊C8—42no—
183C22⋊C8—82no—
184#97—42no—
185C4×C8—22no—
186C4×C8—22no—
187C4×C8—22no—
188C4×C8#4114yes—
189#95—21no—
190#95—21no—
191#95—21no—
192#95—42no—
193#95—42no—
194C24—21no—
195C24#12111yescharacteristic
196C2×D4—22no—
197C2×D4—22no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—42no—
201C2×D4—42no—
202C2×D4—42no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4#12111yescharacteristic
226C22×C4#12111yescharacteristic
227C22×C4#12111yescharacteristic
228C22×C4C2×SD1612yes—
229C22×C4#13712yes—
230C22×C4C22×D411yescharacteristic
231C22×C4C22×D411yescharacteristic
232C22×C4C2×#4111yescharacteristic
233M16—84no—
234C2×C8—28no—
235C2×C8—21no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—21no—
241C2×C8—28no—
242C2×C8—41no—
243C2×C8—41no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—41no—
247C2×C8—41no—
248C2×C8—41no—
249C2×C8—41no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8#12312yes—
255C2×C8#12312yes—
256C2×C8C2×SD1612yes—
257C2×C8C2×SD1612yes—
258C2×C8AGL(1,ℤ/8)12yes—
259C2×C8AGL(1,ℤ/8)12yes—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—82no—
274C4⋊C4—82no—
275C22⋊C4—42no—
276C22⋊C4—42no—
277C22⋊C4—42no—
278C22⋊C4—42no—
279C22⋊C4—42no—
280C22⋊C4—84no—
281C42—21no—
282C42—21no—
283C42—22no—
284C42—22no—
285C42—41no—
286C42—41no—
287C42—84no—
288C42C2×#4112yes—
289C422−1+412yes—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23#46112yes—
304C23C2×#12111yescharacteristic
305D4—42no—
306D4—42no—
307D4—84no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—44no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4#46112yes—
345C2×C4C2×#12311yescharacteristic
346C2×C4#53611yescharacteristic
347C2×C4#54712yes—
348C2×C4#55212yes—
349C2×C4C22×SD1611yescharacteristic
350C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
351C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
352C2×C4C2×#13711yescharacteristic
353C2×C4#57512yes—
354C2×C4#57611yescharacteristic
355C2×C4#57811yescharacteristic
356C8—22no—
357C8—22no—
358C8—22no—
359C8—81no—
360C8—82no—
361C8—82no—
362C8—81no—
363C8#48114yes—
364C22—22no—
365C22—22no—
366C22—22no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22#248611yescharacteristic
379C22C2×#46111yescharacteristic
380C22#303811yescharacteristic
381C22#305411yescharacteristic
382C22#306111yescharacteristic
383C22#306711yescharacteristic
384C22#306711yescharacteristic
385C4—22no—
386C4—22no—
387C4—41no—
388C4—41no—
389C4—81no—
390C4—81no—
391C4—81no—
392C4—162no—
393C4—162no—
394C4C2×#48111yescharacteristic
395C4#309211yescharacteristic
396C4#322711yescharacteristic
397C4#323911yescharacteristic
398C4#334712yes—
399C4#334912yes—
400C2—42no—
401C2—161no—
402C2#1913711yescharacteristic
403C2#1919511yescharacteristic
404C2#1929411yescharacteristic
405C2#1935611yescharacteristic
406C2#2764511yescharacteristic
407C2#2766411yescharacteristic
408C2#2767311yescharacteristic
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. #20797C2semidirect
4. #10652C2semidirect
5. #10651C2semidirect
6. #10616C2semidirect
7. #10615C2semidirect
8. #9618C2semidirect
9. #8727C2semidirect
10. #8650C2semidirect
11. #8596C2semidirect

Maximal subgroups

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

subgroupindex
#276252
#214872
#207972
#106522
#106512
#106162
#106152
#96182
#87272
#86502
#85962