Finite Group Databasebeta

← #6532038

Subgroups — #6532038

Order 512.

1454 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35334C211yesmaximal, characteristic
2#34374C211yesmaximal, characteristic, complemented
3#27621C212yesmaximal, complemented
4#27620C212yesmaximal, complemented
5#27597C212yesmaximal
6#27595C212yesmaximal
7#25070C214yesmaximal, complemented
8#25068C214yesmaximal, complemented
9#25005C214yesmaximal, complemented
10#25003C214yesmaximal, complemented
11#21986C212yesmaximal, complemented
12#21823C211yesmaximal, characteristic, complemented
13#20743C212yesmaximal
14C2×#533C2211yescharacteristic
15C2×C4×#41C2211yescharacteristic
16C2×#432C2212yes—
17#2851—22no—
18#2628—22no—
19#2581C2214yes—
20#2580C2214yes—
21#2564—24no—
22#2382—22no—
23#2365—22no—
24#2355C2214yes—
25Q8×C4⋊C4C2214yes—
26#2323—24no—
27#2316C2211yescharacteristic
28#2316C2211yescharacteristic
29#2314C2212yes—
30#2314C2212yes—
31#2311C2211yescharacteristic
32#2311C2211yescharacteristic
33#2285C2212yes—
34C2×#382C2211yescharacteristic
35#2101C2214yes—
36#2101C2214yes—
37#2096C2212yes—
38#2095C2212yes—
39#2094C2212yes—
40#2093C2212yes—
41#2084C2214yes—
42#2083C2214yes—
43#2074C2212yes—
44#2074C2212yes—
45#2073C2212yes—
46#2073C2212yes—
47#1985C2214yes—
48#1985C2214yes—
49#1949C2214yes—
50#1949C2214yes—
51#1928C2214yes—
52#1927C2214yes—
53#1926C2214yes—
54#1925C2214yes—
55#1921C2214yes—
56#1920C2214yes—
57#1919C2214yes—
58#1918C2214yes—
59#1853C2214yes—
60#1852C2214yes—
61#1847C2214yes—
62#1836C2214yes—
63#1825C2214yes—
64#1818C2212yes—
65#1818C2212yes—
66#1803C2212yes—
67#1803C2212yes—
68#1802C2212yes—
69#1802C2212yes—
70#1788C2214yes—
71#1763C2214yes—
72#1762C2214yes—
73C22×#41C2311yescharacteristic
74#533—28no—
75C2×C4⋊Q8D412yes—
76C2×#125D412yes—
77C2×C22⋊Q8D414yes—
78C4×#41—28no—
79C2×C4×Q8—24no—
80C2×C4×Q8C2312yes—
81C2×C4×Q8C2314yes—
82C2×C4×D4—22no—
83C2×C4×D4C2312yes—
84C2×#117—22no—
85C2×#117—21no—
86C2×#117C2311yescharacteristic
87C2×#117C2311yescharacteristic
88C2×#117C2311yescharacteristic
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2311yescharacteristic
91C22×C42—21no—
92C22×C42C2311yescharacteristic
93#432—28no—
94C2×#107C2314yes—
95C2×#107C2314yes—
96C2×#106C2314yes—
97C2×#106C2314yes—
98C2×C4⋊C8C2312yes—
99C2×C4⋊C8C2312yes—
100C2×#103C2314yes—
101C2×#103C2314yes—
102C2×#102C2314yes—
103C2×#102C2314yes—
104C2×C22⋊C8C2314yes—
105C2×#97C2314yes—
106#400—24no—
107#399—24no—
108#395—24no—
109#393—24no—
110#391—24no—
111#391C2312yes—
112#391C2312yes—
113#390C2312yes—
114#390C2312yes—
115#389—22no—
116#389—22no—
117#389—24no—
118#389—24no—
119#389—24no—
120#389C2314yes—
121#389C2314yes—
122#388—24no—
123#388—24no—
124#387—24no—
125#387—24no—
126#387—24no—
127#387—24no—
128#387—24no—
129#387—24no—
130#386—22no—
131#386—22no—
132#385—24no—
133#385—24no—
134#384C2312yes—
135#384C2312yes—
136#384C2312yes—
137#384C2312yes—
138#382—21no—
139#382—21no—
140#382—22no—
141#382C2312yes—
142#382C2312yes—
143#382C2314yes—
144#380—22no—
145#380—22no—
146#380—22no—
147#380—22no—
148#380—22no—
149#380C2314yes—
150C4×C4⋊C4—24no—
151C4×C4⋊C4—22no—
152C4×C4⋊C4C2314yes—
153C4×C4⋊C4C2314yes—
154C4×C4⋊C4C2314yes—
155C2×#95—22no—
156#340C2314yes—
157#340C2314yes—
158#340C2314yes—
159#340C2314yes—
160#340C2314yes—
161#340C2314yes—
162#340C2314yes—
163#340C2314yes—
164#336C2312yes—
165#336C2312yes—
166#336C2312yes—
167#336C2312yes—
168#336C2314yes—
169#336C2314yes—
170C2×#41—22no—
171C2×#41—22no—
172C2×#41—24no—
173C2×#41—42no—
174C22×Q8C2×D414yes—
175C22×Q8C2411yescharacteristic
176C22×D4—21no—
177C22×D4C2411yescharacteristic
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—21no—
181C23×C4—22no—
182C23×C4C2×D412yes—
183C23×C4C2411yescharacteristic
184C22×C8C2414yes—
185C22×C8C2414yes—
186C4⋊Q8—28no—
187#125—28no—
188C22⋊Q8—216no—
189C4×Q8—28no—
190C4×Q8—28no—
191C4×Q8—24no—
192C4×Q8—24no—
193C4×Q8—88no—
194C4×D4—24no—
195C4×D4—24no—
196C4×D4—48no—
197#117—24no—
198#117—24no—
199#117—24no—
200#117—44no—
201#117—84no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4C2×D412yes—
216C2×C4⋊C4C2×D412yes—
217C2×C4⋊C4C2×D414yes—
218C2×C4⋊C4C2×D414yes—
219C2×C4⋊C4C2×D414yes—
220C2×C4⋊C4C2412yes—
221C2×C4⋊C4C2412yes—
222C2×C4⋊C4C2412yes—
223C2×C4⋊C4C2414yes—
224C2×C4⋊C4C2414yes—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4C2×D412yes—
231C2×C22⋊C4C2×D412yes—
232C2×C42—21no—
233C2×C42—24no—
234C2×C42—22no—
235C2×C42—24no—
236C2×C42—24no—
237C2×C42—44no—
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42#4118yes—
241C2×C42C2412yes—
242C2×C42C2412yes—
243C2×C42C2412yes—
244#107—44no—
245#107—44no—
246#106—44no—
247#106—44no—
248C4⋊C8—24no—
249C4⋊C8—24no—
250#103—28no—
251#103—44no—
252#102—28no—
253#102—44no—
254C22⋊C8—28no—
255#97—28no—
256#95—24no—
257#95—24no—
258#95—24no—
259#95—24no—
260#95—24no—
261#95—24no—
262#95—24no—
263#95—24no—
264#95—24no—
265#95—24no—
266#95—24no—
267#95—24no—
268C24—21no—
269C24C22×D411yescharacteristic
270#41—48no—
271#41—48no—
272C2×Q8—28no—
273C2×Q8—21no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—44no—
277C2×Q8—44no—
278C2×Q82−1+412yes—
279C2×D4—24no—
280C2×D4—22no—
281C2×D4—21no—
282C2×D4—22no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D42−1+412yes—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—21no—
291C22×C4—24no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—22no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—44no—
303C22×C4—82no—
304C22×C4—82no—
305C22×C4—82no—
306C22×C4—82no—
307C22×C4#12318yes—
308C22×C4C22×D414yes—
309C22×C4C22×D414yes—
310C22×C4C22×D411yescharacteristic
311C22×C4C22×D412yes—
312C22×C4C22×D412yes—
313C22×C4C2×#4114yes—
314C22×C4C2511yescharacteristic
315C2×C8—24no—
316C2×C8—24no—
317C2×C8—24no—
318C2×C8—24no—
319C2×C8—24no—
320C2×C8—24no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—28no—
324C4⋊C4—28no—
325C4⋊C4—28no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—44no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—44no—
338C4⋊C4—44no—
339C4⋊C4—84no—
340C4⋊C4—84no—
341C4⋊C4—82no—
342C4⋊C4—82no—
343C4⋊C4—84no—
344C4⋊C4—84no—
345C22⋊C4—24no—
346C22⋊C4—24no—
347C22⋊C4—42no—
348C22⋊C4—44no—
349C22⋊C4—84no—
350C22⋊C4—84no—
351C22⋊C4—84no—
352C42—24no—
353C42—24no—
354C42—48no—
355C42—42no—
356C42—44no—
357C42—84no—
358C42—84no—
359C42#12314yes—
360C42#12314yes—
361C42C2×#4114yes—
362C42C2×#4114yes—
363C23—22no—
364C23—22no—
365C23—21no—
366C23—21no—
367C23—22no—
368C23—42no—
369C23—42no—
370C23—41no—
371C23—41no—
372C23C2×#12312yes—
373C23C23×D411yescharacteristic
374Q8—24no—
375Q8—416no—
376Q8—42no—
377D4—24no—
378D4—48no—
379D4—42no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—22no—
395C2×C4—48no—
396C2×C4—48no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—42no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—44no—
412C2×C4—88no—
413C2×C4—162no—
414C2×C4—162no—
415C2×C4C2×#12312yes—
416C2×C4C2×#12314yes—
417C2×C4C2×#12314yes—
418C2×C4C22×#4112yes—
419C2×C4C2×2−1+411yescharacteristic
420C2×C4C2×2−1+411yescharacteristic
421C2×C4#58512yes—
422C8—44no—
423C8—44no—
424C22—22no—
425C22—24no—
426C22—42no—
427C22—41no—
428C22—41no—
429C22—42no—
430C22—44no—
431C22—82no—
432C22—81no—
433C22—81no—
434C22C22×#12311yescharacteristic
435C22#348611yescharacteristic
436C22#348711yescharacteristic
437C22#358512yes—
438C22#358612yes—
439C4—22no—
440C4—24no—
441C4—24no—
442C4—42no—
443C4—44no—
444C4—44no—
445C4—82no—
446C4—84no—
447C4—84no—
448C4—82no—
449C4—82no—
450C4#349512yes—
451C4#349512yes—
452C2—42no—
453C2—82no—
454C2#3567211yescharacteristic
455C2#6051712yes—
456C2#6187812yes—
457C2#6194211yescharacteristic
458C2#6194311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#533 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #34374C2semidirect
3. #27621C2semidirect
4. #27620C2semidirect
7. #25070C2semidirect
8. #25068C2semidirect
9. #25005C2semidirect
10. #25003C2semidirect
11. #21986C2semidirect
12. #21823C2semidirect

Maximal subgroups

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

subgroupindex
#353342
#343742
#276212
#276202
#275972
#275952
#250702
#250682
#250052
#250032
#219862
#218232
#207432