Finite Group Databasebeta

← #314711

Subgroups — #314711

Order 512.

655 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27624C212yesmaximal
2#21487C211yesmaximal, characteristic
3#20797C211yesmaximal, characteristic, complemented
4#10653C212yesmaximal, complemented
5#10619C212yesmaximal, complemented
6#9620C211yesmaximal, characteristic, complemented
7#9620C211yesmaximal, characteristic, complemented
8#8726C212yesmaximal, complemented
9#8652C211yesmaximal, characteristic, complemented
10#8652C211yesmaximal, characteristic, complemented
11#8595C211yesmaximal, characteristic, complemented
12C2×#436C2211yescharacteristic
13C2×#435—21no—
14C2×#427—21no—
15C2×#427—21no—
16C22×C4×C8C2211yescharacteristic
17C4×#121C2211yescharacteristic
18#2285C2212yes—
19#2075—22no—
20#2033—22no—
21#1954C2211yescharacteristic
22#1947C2212yes—
23#1947C2212yes—
24#1939C2211yescharacteristic
25#1925—22no—
26#1923—21no—
27#1918—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×#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—
58#427—44no—
59#427—44no—
60C2×#107—21no—
61C2×#107—21no—
62C2×#107—21no—
63C2×#107—21no—
64C2×#107—22no—
65C2×#107—22no—
66C2×#106—22no—
67C2×#106—22no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8C2312yes—
70C2×C4⋊C8C2312yes—
71C2×C4⋊C8C2311yescharacteristic
72C2×C4⋊C8C2311yescharacteristic
73C2×#102—22no—
74C2×#102—22no—
75C2×C22⋊C8—21no—
76C2×C22⋊C8—21no—
77C2×C22⋊C8—21no—
78C2×C22⋊C8—21no—
79C2×C22⋊C8—22no—
80C2×#97—21no—
81C2×C4×C8—22no—
82C2×C4×C8—22no—
83C2×C4×C8D411yescharacteristic
84C2×C4×C8D411yescharacteristic
85C2×C4×C8C2311yescharacteristic
86C2×C4×C8C2311yescharacteristic
87#390D411yescharacteristic
88#389D411yescharacteristic
89#388—21no—
90#386—21no—
91#384C2312yes—
92#384C2312yes—
93#382—22no—
94#380—22no—
95#380—22no—
96#380D411yescharacteristic
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4C2311yescharacteristic
100C4×C22⋊C4—21no—
101#340—22no—
102#340—22no—
103#340—22no—
104#340—22no—
105#336—21no—
106#336—21no—
107#336—22no—
108#336—21no—
109#334—22no—
110#334—22no—
111C22×D4—21no—
112C22×D4C2×D411yescharacteristic
113C23×C4—21no—
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C23×C4#4111yescharacteristic
117C2×M16—42no—
118C2×M16—42no—
119C2×M16—42no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—21no—
126C22×C8—21no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8D814yes—
130C22×C8C2×D411yescharacteristic
131C22×C8C2×D411yescharacteristic
132C22×C8C2×D411yescharacteristic
133C22×C8C2×D411yescharacteristic
134#121—42no—
135C4×D4—21no—
136C4×D4—21no—
137C4×D4—21no—
138C4×D4—21no—
139C4×D4—42no—
140C4×D4—42no—
141#117—44no—
142#117—44no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4C2×D411yescharacteristic
151C2×C4⋊C4C2×D411yescharacteristic
152C2×C4⋊C4C2×D411yescharacteristic
153C2×C22⋊C4—21no—
154C2×C22⋊C4—21no—
155C2×C22⋊C4—21no—
156C2×C22⋊C4—21no—
157C2×C22⋊C4—21no—
158C2×C22⋊C4—42no—
159C2×C42—21no—
160C2×C42—21no—
161C2×C42—21no—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—42no—
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42#4111yescharacteristic
168C2×C42C2411yescharacteristic
169#107—44no—
170#107—42no—
171#107—42no—
172#107—42no—
173#107—42no—
174#107—82no—
175#106—44no—
176#106—44no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—21no—
180C4⋊C8—21no—
181C4⋊C8—21no—
182C4⋊C8—21no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—42no—
186#102—44no—
187#102—82no—
188C22⋊C8—42no—
189C22⋊C8—42no—
190C22⋊C8—42no—
191C22⋊C8—42no—
192C22⋊C8—82no—
193#97—42no—
194C4×C8—22no—
195C4×C8—22no—
196C4×C8—21no—
197C4×C8—21no—
198C4×C8#4112yes—
199C4×C8#4112yes—
200#95—21no—
201#95—21no—
202#95—21no—
203#95—42no—
204#95—42no—
205C24—21no—
206C24#12111yescharacteristic
207C2×D4—21no—
208C2×D4—21no—
209C2×D4—22no—
210C2×D4—22no—
211C2×D4—42no—
212C2×D4—42no—
213C2×D4—42no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4#12111yescharacteristic
238C22×C4#12111yescharacteristic
239C22×C4#12111yescharacteristic
240C22×C4C2×D812yes—
241C22×C4AGL(1,ℤ/8)12yes—
242C22×C4C22×D411yescharacteristic
243C22×C4C22×D411yescharacteristic
244C22×C4C2×#4111yescharacteristic
245M16—84no—
246C2×C8—28no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—28no—
254C2×C8—41no—
255C2×C8—41no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8#12312yes—
267C2×C8#12312yes—
268C2×C8C2×D812yes—
269C2×C8C2×D812yes—
270C2×C8#13712yes—
271C2×C8#13712yes—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—82no—
286C4⋊C4—82no—
287C22⋊C4—42no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C22⋊C4—42no—
293C22⋊C4—84no—
294C42—21no—
295C42—21no—
296C42—21no—
297C42—21no—
298C42—21no—
299C42—21no—
300C42—41no—
301C42—41no—
302C42—84no—
303C42C2×#4111yescharacteristic
304C42C2×#4111yescharacteristic
305C422−1+411yescharacteristic
306C422−1+411yescharacteristic
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23#45912yes—
321C23C2×#12111yescharacteristic
322D4—42no—
323D4—42no—
324D4—84no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4#45912yes—
363C2×C4C2×#12311yescharacteristic
364C2×C4#53611yescharacteristic
365C2×C4#54711yescharacteristic
366C2×C4#54711yescharacteristic
367C2×C4#55211yescharacteristic
368C2×C4#55211yescharacteristic
369C2×C4C22×D811yescharacteristic
370C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
371C2×C4C2×#13711yescharacteristic
372C2×C4C2×#13711yescharacteristic
373C2×C4#57512yes—
374C2×C4#57712yes—
375C8—22no—
376C8—22no—
377C8—22no—
378C8—81no—
379C8—82no—
380C8—82no—
381C8—81no—
382C8#48014yes—
383C22—22no—
384C22—22no—
385C22—22no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22—81no—
397C22#248611yescharacteristic
398C22C2×#45911yescharacteristic
399C22#303911yescharacteristic
400C22#305411yescharacteristic
401C22#306011yescharacteristic
402C22#306711yescharacteristic
403C22#306711yescharacteristic
404C4—21no—
405C4—21no—
406C4—22no—
407C4—41no—
408C4—41no—
409C4—81no—
410C4—81no—
411C4—81no—
412C4—162no—
413C4—162no—
414C4C2×#48011yescharacteristic
415C4#309211yescharacteristic
416C4#322611yescharacteristic
417C4#323811yescharacteristic
418C4#334812yes—
419C4#335012yes—
420C2—42no—
421C2—161no—
422C2#1913311yescharacteristic
423C2#1918911yescharacteristic
424C2#1929811yescharacteristic
425C2#1936011yescharacteristic
426C2#2764411yescharacteristic
427C2#2766511yescharacteristic
428C2#2767411yescharacteristic
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. #10653C2semidirect
5. #10619C2semidirect
6. #9620C2semidirect
7. #9620C2semidirect
8. #8726C2semidirect
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
#276242
#214872
#207972
#106532
#106192
#96202
#87262
#86522
#85952