Finite Group Databasebeta

← #347660

Subgroups — #347660

Order 512.

1040 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C8×Q8C211yesmaximal, characteristic
2#33043C211yesmaximal, characteristic
3#32954C211yesmaximal, characteristic
4#32906C211yesmaximal, characteristic
5C2×#1680C212yesmaximal, complemented
6#9220C212yesmaximal, complemented
7#9218C212yesmaximal, complemented
8#9131C211yesmaximal, characteristic
9#9129C211yesmaximal, characteristic
10#8647C212yesmaximal, complemented
11#8595C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C2×#487C2211yescharacteristic
14C2×#475C2212yes—
15C2×#462C2212yes—
16C2×#451—21no—
17C2×C8×Q8—28no—
18C2×C8×Q8—21no—
19C2×C8×Q8—21no—
20C2×C8×Q8C2212yes—
21C2×C8×Q8C2212yes—
22C22×C4⋊C8—21no—
23C22×C4⋊C8C2211yescharacteristic
24C22×C4×C8—21no—
25C22×C4×C8C2211yescharacteristic
26#2857C2211yescharacteristic
27#2857C2211yescharacteristic
28#1990C2211yescharacteristic
29#1956C2211yescharacteristic
30#1933—21no—
31#1933—21no—
32#1892—21no—
33#1892—21no—
34#1878—22no—
35#1872—21no—
36#1865—22no—
37#1853—21no—
38#1834C2211yescharacteristic
39#1803C2212yes—
40#1803C2212yes—
41#1797C2211yescharacteristic
42#1797C2211yescharacteristic
43#1787—21no—
44#1680—28no—
45C2×#334C2212yes—
46C2×#332C2212yes—
47C2×#332C2212yes—
48#1296C2212yes—
49#1296C2212yes—
50#1277C2212yes—
51C23×Q8D411yescharacteristic
52C22×Q16—22no—
53C23×C8—21no—
54C23×C8D411yescharacteristic
55C23×C8D411yescharacteristic
56C2×C4⋊Q8C2311yescharacteristic
57C2×C4⋊Q8C2311yescharacteristic
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8—21no—
60C2×C4×Q8—24no—
61C2×C4×Q8—24no—
62C2×C4×Q8—21no—
63C2×C4×Q8—21no—
64C2×C4×Q8D412yes—
65C2×C4×Q8C2312yes—
66C22×C4⋊C4—21no—
67C22×C4⋊C4D411yescharacteristic
68C22×C42—21no—
69C22×C42C2311yescharacteristic
70#487—24no—
71#475—24no—
72#475—24no—
73#462—24no—
74#462—24no—
75#451—44no—
76C8×Q8—28no—
77C8×Q8—28no—
78C8×Q8—24no—
79C8×Q8—24no—
80C8×Q8—24no—
81C8×Q8D418yes—
82C2×#106—22no—
83C2×C4⋊C8—28no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—24no—
88C2×C4⋊C8C2312yes—
89C2×C4⋊C8C2312yes—
90C2×C4⋊C8C2312yes—
91C2×#103—22no—
92C2×#103—22no—
93C2×#103—22no—
94C2×#103—21no—
95C2×#103—21no—
96C2×#103—21no—
97C2×#103—21no—
98C2×#103—22no—
99C2×C22⋊C8—21no—
100C2×C22⋊C8—21no—
101C2×C4×C8—28no—
102C2×C4×C8—22no—
103C2×C4×C8—22no—
104C2×C4×C8—24no—
105C2×C4×C8D411yescharacteristic
106C2×C4×C8D411yescharacteristic
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109#391—21no—
110#391—21no—
111#389—21no—
112#389—21no—
113#389—21no—
114#389—21no—
115#384C2311yescharacteristic
116#384C2311yescharacteristic
117#380—21no—
118#380—21no—
119#380—21no—
120#380—21no—
121#340—21no—
122#340—21no—
123#340—21no—
124#340—21no—
125#340—22no—
126#340—22no—
127#336—21no—
128#336—21no—
129#334—24no—
130#332—24no—
131#332—24no—
132C22×Q8—24no—
133C22×Q8—24no—
134C22×Q8—21no—
135C22×Q8—22no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8SD1614yes—
139C22×Q8C2×D412yes—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4#4111yescharacteristic
145C2×Q16—44no—
146C2×Q16—44no—
147C2×Q16—44no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—24no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—41no—
158C22×C8—42no—
159C22×C8—42no—
160C22×C8C2×D411yescharacteristic
161C22×C8C2×D411yescharacteristic
162C22×C8C2×D411yescharacteristic
163C22×C8C2×D411yescharacteristic
164C4⋊Q8—22no—
165C4⋊Q8—22no—
166C4⋊Q8—22no—
167C4⋊Q8—22no—
168C22⋊Q8—42no—
169C22⋊Q8—42no—
170C22⋊Q8—42no—
171C22⋊Q8—42no—
172C4×Q8—28no—
173C4×Q8—24no—
174C4×Q8—24no—
175C4×Q8—24no—
176C4×Q8—24no—
177C4×Q8—22no—
178C4×Q8—22no—
179C4×Q8C2×D414yes—
180C4×Q8#4114yes—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4C2×D412yes—
199C2×C22⋊C4—41no—
200C2×C22⋊C4—41no—
201C2×C42—24no—
202C2×C42—24no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42#4111yescharacteristic
210C2×C42C2411yescharacteristic
211#106—44no—
212C4⋊C8—28no—
213C4⋊C8—24no—
214C4⋊C8—28no—
215C4⋊C8—22no—
216C4⋊C8—22no—
217C4⋊C8—22no—
218C4⋊C8—22no—
219C4⋊C8—22no—
220C4⋊C8C2×D414yes—
221#103—42no—
222#103—44no—
223#103—44no—
224#103—44no—
225#103—42no—
226#103—42no—
227#103—42no—
228#103—44no—
229C22⋊C8—42no—
230C22⋊C8—42no—
231C4×C8—28no—
232C4×C8—28no—
233C4×C8—22no—
234C4×C8—22no—
235C4×C8—22no—
236C4×C8C2×D414yes—
237#95—41no—
238#95—41no—
239#95—41no—
240#95—41no—
241C24#12111yescharacteristic
242C2×Q8—24no—
243C2×Q8—216no—
244C2×Q8—24no—
245C2×Q8—216no—
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8—24no—
250C2×Q8—41no—
251C2×Q8—42no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—42no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8#12114yes—
261C2×Q8C2×SD1614yes—
262C2×Q8#13514yes—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—24no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—41no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4#12111yescharacteristic
285C22×C4#12111yescharacteristic
286C22×C4#12111yescharacteristic
287C22×C4C2×SD1612yes—
288C22×C4#13512yes—
289C22×C4#13512yes—
290C22×C4#13712yes—
291C22×C4C22×D411yescharacteristic
292C22×C4C22×D411yescharacteristic
293C22×C4C2×#4111yescharacteristic
294Q16—88no—
295C2×C8—28no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—24no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—44no—
307C2×C8—48no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8#12012yes—
317C2×C8#12012yes—
318C2×C8#13712yes—
319C4⋊C4—28no—
320C4⋊C4—24no—
321C4⋊C4—22no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—41no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—42no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—81no—
337C4⋊C4—81no—
338C4⋊C4—82no—
339C4⋊C4—82no—
340C4⋊C4#12114yes—
341C22⋊C4—82no—
342C22⋊C4—82no—
343C42—28no—
344C42—24no—
345C42—21no—
346C42—22no—
347C42—24no—
348C42—21no—
349C42—22no—
350C42C22×D412yes—
351C42C2×#4112yes—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23#46412yes—
357C23#46512yes—
358C23#47312yes—
359C23C2×#12111yescharacteristic
360Q8—216no—
361Q8—24no—
362Q8—24no—
363Q8—24no—
364Q8—81no—
365Q8—81no—
366Q8—81no—
367Q8—81no—
368Q8—82no—
369Q8—82no—
370Q8#46518yes—
371C2×C4—216no—
372C2×C4—21no—
373C2×C4—24no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—21no—
388C2×C4—24no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—44no—
392C2×C4—48no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4#46412yes—
408C2×C4#46512yes—
409C2×C4C2×#12011yescharacteristic
410C2×C4C2×#12112yes—
411C2×C4#52512yes—
412C2×C4#52511yescharacteristic
413C2×C4C22×SD1611yescharacteristic
414C2×C4C2×#13511yescharacteristic
415C2×C4C2×#13512yes—
416C2×C4C2×#13511yescharacteristic
417C2×C4C2×#13711yescharacteristic
418C2×C4C2×#13711yescharacteristic
419C2×C4#57511yescharacteristic
420C8—22no—
421C8—22no—
422C8—22no—
423C8—41no—
424C8—44no—
425C8—82no—
426C8—82no—
427C22—22no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22#168014yescomplemented
434C22#284711yescharacteristic
435C22C2×#46411yescharacteristic
436C22#303711yescharacteristic
437C22C2×#46511yescharacteristic
438C22#305011yescharacteristic
439C22C2×#47311yescharacteristic
440C22#306711yescharacteristic
441C4—22no—
442C4—28no—
443C4—22no—
444C4—22no—
445C4—41no—
446C4—44no—
447C4—161no—
448C4—161no—
449C4—161no—
450C4—161no—
451C4#300211yescharacteristic
452C4#300411yescharacteristic
453C4C2×#46512yes—
454C4#305012yes—
455C4#310811yescharacteristic
456C4#311711yescharacteristic
457C2—24no—
458C2C2×#168011yescharacteristic
459C2#1912311yescharacteristic
460C2#1926011yescharacteristic
461C2#1926511yescharacteristic
462C2#3290911yescharacteristic
463C2#3292911yescharacteristic
464C2#3294311yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
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
5. C2×#1680C2semidirect
6. #9220C2semidirect
7. #9218C2semidirect
10. #8647C2semidirect
433. C22#1680semidirect

Maximal subgroups

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

subgroupindex
C22×C8×Q82
#330432
#329542
#329062
C2×#16802
#92202
#92182
#91312
#91292
#86472
#85952