Finite Group Databasebeta

← #6435650

Subgroups — #6435650

Order 512.

1762 conjugacy classes of subgroups.

The subgroup lattice has 451 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2851C211yesmaximal, characteristic
2#34012C211yesmaximal, characteristic
3#29944C214yesmaximal, complemented
4#29943C214yesmaximal, complemented
5#29940C218yesmaximal, complemented
6#23747C212yesmaximal, complemented
7#23657C212yesmaximal, complemented
8#23642C214yesmaximal
9#21314C212yesmaximal, complemented
10#21172C212yesmaximal, complemented
11#20626C211yesmaximal, characteristic
12C22×#125C2211yescharacteristic
13C23×C42C2211yescharacteristic
14C2×#428C2214yes—
15C2×#423C2212yes—
16C2×#407C2212yes—
17#2851—22no—
18#2851—22no—
19#2851C2214yes—
20#2851C2214yes—
21C2×#389C2212yes—
22C2×#387—22no—
23C2×#385—22no—
24#2285C2212yes—
25#2285C2212yes—
26#2285C2211yescharacteristic
27#2285C2211yescharacteristic
28#2283—22no—
29#2283—22no—
30#2283C2214yes—
31C2×#382C2211yescharacteristic
32#2280—22no—
33#2280C2212yes—
34C2×#380C2212yes—
35C4×#117—22no—
36C4×#117C2212yes—
37#2101C2218yescomplemented
38#2100C2214yescomplemented
39#2099C2214yescomplemented
40#2085C2218yescomplemented
41#2084C2214yescomplemented
42#2083C2214yescomplemented
43#2077C2218yes—
44#2077C2218yes—
45#2076C2218yes—
46#2075C2218yes—
47#1989C2218yescomplemented
48#1989C2218yescomplemented
49#1949C2218yescomplemented
50#1949C2218yescomplemented
51#1797C2214yes—
52#1797C2214yes—
53#1736C2214yes—
54#1736C2214yes—
55#1736C2214yes—
56#1728C2212yes—
57C24×C4D412yes—
58C24×C4C2311yescharacteristic
59C22×M16C2314yes—
60C2×#125—21no—
61C2×#125—21no—
62C2×#125—28no—
63C2×#125C2314yes—
64C2×#117—22no—
65C2×#117—22no—
66C2×#117C2312yes—
67C2×#117C2312yes—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C22⋊C4—22no—
73C22×C42—28no—
74C22×C42D412yes—
75C22×C42D412yes—
76C22×C42D412yes—
77C22×C42C2312yes—
78C22×C42C2312yes—
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81#428—48no—
82C2×#107C2318yes—
83C2×#106C2318yes—
84#423—28no—
85C2×C4⋊C8C2318yes—
86#407—44no—
87C2×C22⋊C8C2318yes—
88#389—22no—
89#389—22no—
90#389C2314yes—
91#389C2314yes—
92#387—24no—
93#387—24no—
94#387—24no—
95#387—24no—
96#385—24no—
97#385—24no—
98#385—24no—
99#385—24no—
100#384C2312yes—
101#384C2311yescharacteristic
102#384C2311yescharacteristic
103#383—24no—
104#382—21no—
105#382—21no—
106#382—22no—
107#382—22no—
108#382C2314yes—
109#382C2312yes—
110#382C2312yes—
111#381—24no—
112#381—22no—
113#380—22no—
114#380—22no—
115#380—22no—
116#380—22no—
117#380—22no—
118#380—22no—
119#380C2314yes—
120#380C2314yes—
121#380C2314yes—
122#380C2314yes—
123#379—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4C2314yes—
129C4×C4⋊C4C2314yes—
130C4×C22⋊C4—24no—
131C4×C22⋊C4—24no—
132C4×C22⋊C4—24no—
133#376—24no—
134#376—22no—
135C2×#95—22no—
136C2×#95—22no—
137C2×#95—22no—
138C43—22no—
139#340C2318yes—
140#340C2318yes—
141#340C2318yes—
142#340C2318yes—
143#340C2318yes—
144#340C2318yes—
145#336C2314yes—
146#336C2314yes—
147C25C2×D411yescharacteristic
148C23×C4—28no—
149C23×C4—24no—
150C23×C4—22no—
151C23×C4—24no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—44no—
156C23×C4—44no—
157C23×C4C2×D412yes—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D412yes—
160C23×C4C2×D412yes—
161C23×C4C2×D412yes—
162C23×C4C2×D412yes—
163C23×C4C2×D412yes—
164C23×C4C2×D412yes—
165C23×C4C2412yes—
166C23×C4C2411yescharacteristic
167C2×M16—28no—
168C2×M16—28no—
169C2×M16—28no—
170C22×C8C2418yes—
171#125—216no—
172#125—44no—
173#125—44no—
174#117—24no—
175#117—28no—
176#117—24no—
177#117—48no—
178#117—48no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—48no—
193C2×C4⋊C4—48no—
194C2×C4⋊C4C2414yes—
195C2×C4⋊C4C2414yes—
196C2×C4⋊C4C2412yes—
197C2×C4⋊C4C2412yes—
198C2×C4⋊C4C2414yes—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—48no—
206C2×C42—28no—
207C2×C42—24no—
208C2×C42—28no—
209C2×C42—28no—
210C2×C42—28no—
211C2×C42—28no—
212C2×C42—24no—
213C2×C42—24no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42C2×D412yes—
217C2×C42C2×D412yes—
218C2×C42C2×D412yes—
219C2×C42C2×D412yes—
220C2×C42C2×D412yes—
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42#4118yes—
224C2×C42C2411yescharacteristic
225C2×C42C2411yescharacteristic
226C2×C42C2412yes—
227#107—48no—
228#106—48no—
229C4⋊C8—216no—
230C22⋊C8—48no—
231#95—24no—
232#95—24no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95—24no—
237#95—24no—
238#95—24no—
239#95—24no—
240#95—24no—
241#95—24no—
242#95—24no—
243C24—22no—
244C24—24no—
245C24—22no—
246C24—22no—
247C24—42no—
248C24C22×D412yes—
249C24C22×D411yescharacteristic
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—432no—
283C22×C4—48no—
284C22×C4—48no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4#12318yes—
292C22×C4C22×D412yes—
293C22×C4C22×D412yes—
294C22×C4C22×D412yes—
295C22×C4C22×D412yes—
296C22×C4C22×D412yes—
297C22×C4C22×D411yescharacteristic
298C22×C4C2×#4114yes—
299C22×C4H(5,𝔽2)12yes—
300C22×C42−1+414yes—
301C22×C42−1+412yes—
302C22×C4C2511yescharacteristic
303M16—416no—
304C2×C8—28no—
305C2×C8—28no—
306C2×C8—28no—
307C4⋊C4—28no—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—48no—
315C4⋊C4—44no—
316C4⋊C4—44no—
317C4⋊C4—44no—
318C4⋊C4—48no—
319C4⋊C4—48no—
320C4⋊C4—44no—
321C4⋊C4—48no—
322C4⋊C4—44no—
323C4⋊C4—44no—
324C22⋊C4—48no—
325C22⋊C4—48no—
326C22⋊C4—44no—
327C22⋊C4—44no—
328C22⋊C4—48no—
329C22⋊C4—48no—
330C42—24no—
331C42—24no—
332C42—24no—
333C42—24no—
334C42—24no—
335C42—22no—
336C42—22no—
337C42—48no—
338C42—48no—
339C42—44no—
340C42—44no—
341C42#12314yes—
342C42#12314yes—
343C42C2×#4118yes—
344C23—22no—
345C23—22no—
346C23—28no—
347C23—22no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—48no—
354C23—48no—
355C23—44no—
356C23—44no—
357C23C2×#12312yes—
358C23#53612yes—
359C23#53712yes—
360C23#57514yes—
361C23C23×D411yescharacteristic
362C2×C4—216no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—48no—
376C2×C4—48no—
377C2×C4—48no—
378C2×C4—48no—
379C2×C4—48no—
380C2×C4—44no—
381C2×C4—48no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—88no—
398C2×C4—88no—
399C2×C4C2×#12314yes—
400C2×C4C2×#12314yes—
401C2×C4C2×#12312yes—
402C2×C4#53612yes—
403C2×C4#53712yes—
404C2×C4#54118yes—
405C2×C4#54218yes—
406C2×C4#57514yes—
407C2×C4C22×#4112yes—
408C2×C4C2×H(5,𝔽2)11yescharacteristic
409C2×C4C2×2−1+411yescharacteristic
410C2×C4C2×2−1+412yes—
411C8—48no—
412C22—24no—
413C22—22no—
414C22—22no—
415C22—24no—
416C22—24no—
417C22—24no—
418C22—24no—
419C22—416no—
420C22—42no—
421C22—42no—
422C22—42no—
423C22—42no—
424C22#262514yes—
425C22C22×#12311yescharacteristic
426C22C2×#53611yescharacteristic
427C22C2×#53711yescharacteristic
428C22C2×#57512yes—
429C22C2×#57512yes—
430C4—24no—
431C4—24no—
432C4—24no—
433C4—42no—
434C4—42no—
435C4—84no—
436C4—82no—
437C4—82no—
438C4—84no—
439C4—84no—
440C4#262514yes—
441C4C2×#54112yes—
442C4C2×#54212yes—
443C2—24no—
444C2—44no—
445C2C2×#262511yescharacteristic
446C2#6051112yes—
447C2#6086111yescharacteristic
448C2#6086211yescharacteristic
449C2#6091712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29944C2semidirect
4. #29943C2semidirect
5. #29940C2semidirect
6. #23747C2semidirect
7. #23657C2semidirect
9. #21314C2semidirect
10. #21172C2semidirect
37. #2101C22semidirect
38. #2100C22semidirect
39. #2099C22semidirect
40. #2085C22semidirect
41. #2084C22semidirect
42. #2083C22semidirect
47. #1989C22semidirect
48. #1989C22semidirect
49. #1949C22semidirect
50. #1949C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28512
#340122
#299442
#299432
#299402
#237472
#236572
#236422
#213142
#211722
#206262