Finite Group Databasebeta

← #314714

Subgroups — #314714

Order 512.

723 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27678C211yesmaximal, characteristic
2#27645C211yesmaximal, characteristic
3#21487C211yesmaximal, characteristic
4C2×#1844C211yesmaximal, characteristic, complemented
5#10654C212yesmaximal, complemented
6#10619C212yesmaximal, complemented
7#9618C212yesmaximal, complemented
8#8725C211yesmaximal, characteristic, complemented
9#8724C211yesmaximal, characteristic, complemented
10#8650C212yesmaximal, complemented
11#8596C211yesmaximal, characteristic, complemented
12C2×#481—21no—
13C2×#436C2211yescharacteristic
14C2×#435—21no—
15C22×#106—21no—
16C22×#106—21no—
17C22×C4×C8C2211yescharacteristic
18C4×#121C2211yescharacteristic
19C2×#384C2211yescharacteristic
20C2×#384C2211yescharacteristic
21#2100—21no—
22#2094—21no—
23#2084—21no—
24#2071—21no—
25#1954C2211yescharacteristic
26#1948C2212yes—
27#1948C2212yes—
28#1939C2211yescharacteristic
29#1923—21no—
30#1874—21no—
31#1869—21no—
32#1853—21no—
33#1844—21no—
34#1844—21no—
35#1844C2212yes—
36#1844C2212yes—
37#1824—21no—
38#1802C2212yes—
39#1802C2212yes—
40#1797C2211yescharacteristic
41#1797C2211yescharacteristic
42#1797C2211yescharacteristic
43#1770C2211yescharacteristic
44#1753—21no—
45C2×#335C2212yes—
46#1296C2212yes—
47#1296C2212yes—
48#1296C2212yes—
49#1296C2212yes—
50#1277C2212yes—
51C22×M16—21no—
52C23×C8D411yescharacteristic
53C23×C8D411yescharacteristic
54C2×#121D411yescharacteristic
55C2×C4×D4—21no—
56C2×C4×D4C2311yescharacteristic
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C42C2311yescharacteristic
64#481—44no—
65#436—24no—
66#435—44no—
67C2×#106—22no—
68C2×#106—22no—
69C2×#106—22no—
70C2×#106—22no—
71C2×#106—22no—
72C2×#106—22no—
73C2×#106—44no—
74C2×#106—44no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8C2312yes—
77C2×C4⋊C8C2312yes—
78C2×C4⋊C8C2311yescharacteristic
79C2×C4⋊C8C2311yescharacteristic
80C2×#102—22no—
81C2×#102—22no—
82C2×C22⋊C8—21no—
83C2×C22⋊C8—21no—
84C2×C22⋊C8—21no—
85C2×C22⋊C8—21no—
86C2×C22⋊C8—21no—
87C2×C22⋊C8—21no—
88C2×#97—21no—
89C2×C4×C8—24no—
90C2×C4×C8D411yescharacteristic
91C2×C4×C8D411yescharacteristic
92C2×C4×C8C2311yescharacteristic
93C2×C4×C8C2311yescharacteristic
94#390D411yescharacteristic
95#389D411yescharacteristic
96#388—21no—
97#386—21no—
98#384—21no—
99#384—21no—
100#384—21no—
101#384—21no—
102#384—21no—
103#384—21no—
104#384C2312yes—
105#384C2312yes—
106#380D411yescharacteristic
107C4×C4⋊C4C2311yescharacteristic
108C4×C22⋊C4—21no—
109#340—22no—
110#340—22no—
111#340—22no—
112#340—22no—
113#336—21no—
114#336—21no—
115#336—21no—
116#336—21no—
117#336—21no—
118#335—24no—
119C22×D4—21no—
120C22×D4C2×D411yescharacteristic
121C23×C4—21no—
122C23×C4—41no—
123C23×C4—41no—
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4#4111yescharacteristic
127C2×M16—42no—
128C2×M16—42no—
129C2×M16—42no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8SD1614yes—
140C22×C8C2×D411yescharacteristic
141C22×C8C2×D411yescharacteristic
142C22×C8C2×D411yescharacteristic
143C22×C8C2×D411yescharacteristic
144#121—22no—
145#121—22no—
146C4×D4—22no—
147C4×D4—22no—
148C4×D4—42no—
149C4×D4—42no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—82no—
170C2×C4⋊C4—82no—
171C2×C4⋊C4—82no—
172C2×C4⋊C4C2×D411yescharacteristic
173C2×C4⋊C4C2×D411yescharacteristic
174C2×C4⋊C4C2×D411yescharacteristic
175C2×C22⋊C4—21no—
176C2×C22⋊C4—21no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—21no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—21no—
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42#4111yescharacteristic
186C2×C42C2411yescharacteristic
187#106—44no—
188#106—44no—
189#106—44no—
190#106—44no—
191#106—44no—
192#106—82no—
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—42no—
200#102—44no—
201#102—82no—
202C22⋊C8—42no—
203C22⋊C8—42no—
204C22⋊C8—42no—
205C22⋊C8—42no—
206C22⋊C8—42no—
207C22⋊C8—42no—
208#97—42no—
209C4×C8—22no—
210C4×C8—22no—
211C4×C8—22no—
212C4×C8#4114yes—
213#95—21no—
214#95—21no—
215#95—21no—
216C24—21no—
217C24#12111yescharacteristic
218C2×D4—22no—
219C2×D4—21no—
220C2×D4—21no—
221C2×D4—22no—
222C2×D4—42no—
223C2×D4—42no—
224C2×D4—42no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—82no—
246C22×C4—82no—
247C22×C4—82no—
248C22×C4—82no—
249C22×C4—82no—
250C22×C4—82no—
251C22×C4#12111yescharacteristic
252C22×C4#12111yescharacteristic
253C22×C4#12111yescharacteristic
254C22×C4C2×SD1612yes—
255C22×C4AGL(1,ℤ/8)12yes—
256C22×C4#13712yes—
257C22×C4#13712yes—
258C22×C4C22×D411yescharacteristic
259C22×C4C22×D411yescharacteristic
260C22×C4C2×#4111yescharacteristic
261M16—84no—
262C2×C8—28no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—28no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8#12312yes—
283C2×C8#12312yes—
284C2×C8C2×SD1612yes—
285C2×C8C2×SD1612yes—
286C2×C8AGL(1,ℤ/8)12yes—
287C2×C8AGL(1,ℤ/8)12yes—
288C4⋊C4—22no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—22no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—82no—
302C4⋊C4—82no—
303C4⋊C4—82no—
304C4⋊C4—82no—
305C4⋊C4—82no—
306C4⋊C4—82no—
307C4⋊C4—82no—
308C4⋊C4—82no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—42no—
312C22⋊C4—42no—
313C22⋊C4—42no—
314C42—21no—
315C42—21no—
316C42—22no—
317C42—22no—
318C42—41no—
319C42—41no—
320C42C2×#4112yes—
321C422−1+412yes—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23#46112yes—
334C23#46812yes—
335C23#47112yes—
336C23C2×#12111yescharacteristic
337D4—42no—
338D4—42no—
339D4—84no—
340C2×C4—28no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—24no—
350C2×C4—22no—
351C2×C4—24no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—164no—
375C2×C4—164no—
376C2×C4#46112yes—
377C2×C4#46812yes—
378C2×C4#47112yes—
379C2×C4C2×#12311yescharacteristic
380C2×C4#53611yescharacteristic
381C2×C4#54712yes—
382C2×C4#55212yes—
383C2×C4C22×SD1611yescharacteristic
384C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
385C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
386C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
387C2×C4C2×#13711yescharacteristic
388C2×C4C2×#13711yescharacteristic
389C2×C4#57712yes—
390C8—22no—
391C8—22no—
392C8—22no—
393C8—81no—
394C8—82no—
395C8—82no—
396C8—81no—
397C8#48114yescomplemented
398C22—22no—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22#169114yes—
412C22#248611yescharacteristic
413C22C2×#46111yescharacteristic
414C22#303811yescharacteristic
415C22C2×#46811yescharacteristic
416C22#305711yescharacteristic
417C22C2×#47111yescharacteristic
418C22C2×#47111yescharacteristic
419C4—22no—
420C4—22no—
421C4—22no—
422C4—22no—
423C4—81no—
424C4—81no—
425C4—81no—
426C4—162no—
427C4—162no—
428C4C2×#48111yescharacteristic
429C4#309211yescharacteristic
430C4#322711yescharacteristic
431C4#323911yescharacteristic
432C4#334412yes—
433C4#334512yes—
434C2—24no—
435C2—161no—
436C2C2×#169111yescharacteristic
437C2#1919111yescharacteristic
438C2#1929111yescharacteristic
439C2#1935211yescharacteristic
440C2#2764511yescharacteristic
441C2#2766311yescharacteristic
442C2#2767211yescharacteristic
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
4. C2×#1844C2semidirect
5. #10654C2semidirect
6. #10619C2semidirect
7. #9618C2semidirect
8. #8725C2semidirect
9. #8724C2semidirect
10. #8650C2semidirect
11. #8596C2semidirect
397. C8#481semidirect

Maximal subgroups

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

subgroupindex
#276782
#276452
#214872
C2×#18442
#106542
#106192
#96182
#87252
#87242
#86502
#85962