Finite Group Databasebeta

← #96443

Subgroups — #96443

Order 512.

1446 conjugacy classes of subgroups.

The subgroup lattice has 443 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20715C211yesmaximal, characteristic
2C4×#407C211yesmaximal, characteristic
3C2×C4×#95C211yesmaximal, characteristic
4#7028C214yesmaximal, complemented
5C23×C42—21no—
6C23×C42C2211yescharacteristic
7C2×#432C412yescomplemented
8C2×#407—21no—
9C2×C4×M16—21no—
10C22×#95—21no—
11C22×#95C412yes—
12#1817—21no—
13#1797C412yescomplemented
14#1797C2212yes—
15#1779—21no—
16#1749—21no—
17#1749—21no—
18#1749—21no—
19#1749C412yescomplemented
20C4×C22⋊C8—21no—
21C4×C22⋊C8C2212yes—
22#1728C412yescomplemented
23C4×#95—21no—
24C4×#95—21no—
25C4×#95C412yes—
26C4×#95C2212yes—
27C24×C4—22no—
28C24×C4C2×C411yescharacteristic
29C24×C4D411yescharacteristic
30C24×C4D411yescharacteristic
31C22×M16—22no—
32C22×C42—28no—
33C22×C42—22no—
34C22×C42—22no—
35C22×C42—44no—
36C22×C42—44no—
37C22×C42—44no—
38C22×C42C2×C411yescharacteristic
39C22×C42C2×C411yescharacteristic
40C22×C42C2×C411yescharacteristic
41C22×C42C2×C411yescharacteristic
42C22×C42C2×C411yescharacteristic
43C22×C42D411yescharacteristic
44C22×C42D411yescharacteristic
45C22×C42D411yescharacteristic
46C22×C42D411yescharacteristic
47C22×C42Q811yescharacteristic
48C22×C42Q811yescharacteristic
49C22×C42C2311yescharacteristic
50#432—44no—
51C2×C4⋊C8C2×C414yescomplemented
52#407—28no—
53C2×C22⋊C8—22no—
54C2×C22⋊C8—22no—
55C2×C22⋊C8—22no—
56C2×C22⋊C8—22no—
57C2×C22⋊C8—22no—
58C4×M16—44no—
59C2×#97—22no—
60C2×#97—22no—
61C2×C4×C8—22no—
62C2×#95—28no—
63C2×#95—22no—
64C2×#95—22no—
65C2×#95—22no—
66C2×#95—28no—
67C2×#95C2×C414yes—
68#336—22no—
69#336—22no—
70#336—22no—
71#336C2×C414yescomplemented
72C25C22⋊C411yescharacteristic
73C23×C4—24no—
74C23×C4—24no—
75C23×C4—22no—
76C23×C4—22no—
77C23×C4—28no—
78C23×C4—28no—
79C23×C4—28no—
80C23×C4—24no—
81C23×C4—24no—
82C23×C4—24no—
83C23×C4—24no—
84C23×C4—42no—
85C23×C4—42no—
86C23×C4—42no—
87C23×C4—44no—
88C23×C4—44no—
89C23×C4—44no—
90C23×C4—44no—
91C23×C4C4211yescharacteristic
92C23×C4C4211yescharacteristic
93C23×C4C22⋊C411yescharacteristic
94C23×C4C22⋊C411yescharacteristic
95C23×C4C22⋊C411yescharacteristic
96C23×C4C22⋊C411yescharacteristic
97C23×C4C22⋊C411yescharacteristic
98C23×C4C22⋊C411yescharacteristic
99C23×C4C22⋊C411yescharacteristic
100C23×C4C4⋊C411yescharacteristic
101C23×C4C4⋊C411yescharacteristic
102C23×C4C4⋊C411yescharacteristic
103C23×C4C4⋊C411yescharacteristic
104C23×C4C4⋊C411yescharacteristic
105C23×C4C4⋊C411yescharacteristic
106C23×C4C22×C411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C2×M16—28no—
110C2×M16—44no—
111C2×M16—44no—
112C22×C8—24no—
113C22×C8—24no—
114C2×C42—24no—
115C2×C42—24no—
116C2×C42—28no—
117C2×C42—28no—
118C2×C42—42no—
119C2×C42—42no—
120C2×C42—42no—
121C2×C42—42no—
122C2×C42—42no—
123C2×C42—42no—
124C2×C42—42no—
125C2×C42—42no—
126C2×C42—42no—
127C2×C42—42no—
128C2×C42—432no—
129C2×C42—44no—
130C2×C42—44no—
131C2×C42—44no—
132C2×C42—44no—
133C2×C42C4211yescharacteristic
134C2×C42C4211yescharacteristic
135C2×C42C22⋊C411yescharacteristic
136C2×C42C22⋊C411yescharacteristic
137C2×C42C22⋊C411yescharacteristic
138C2×C42C22⋊C411yescharacteristic
139C2×C42C4⋊C411yescharacteristic
140C2×C42C4⋊C411yescharacteristic
141C2×C42C4⋊C411yescharacteristic
142C2×C42C4⋊C411yescharacteristic
143C2×C42C4⋊C411yescharacteristic
144C2×C42C4⋊C411yescharacteristic
145C2×C42C22×C411yescharacteristic
146C2×C42C22×C411yescharacteristic
147C2×C42C2×D411yescharacteristic
148C2×C42C2×Q811yescharacteristic
149C4⋊C8—44no—
150C22⋊C8—28no—
151C22⋊C8—28no—
152C22⋊C8—44no—
153C22⋊C8—44no—
154C22⋊C8—42no—
155C22⋊C8—42no—
156#97—44no—
157#97—44no—
158C4×C8—44no—
159#95—28no—
160#95—28no—
161#95—28no—
162#95—24no—
163#95M1618yescomplemented
164C24—22no—
165C24—22no—
166C24—22no—
167C24—22no—
168C24—22no—
169C24—21no—
170C24—21no—
171C24#9511yescharacteristic
172C24#9511yescharacteristic
173C24#10414yes—
174C24C2×C22⋊C411yescharacteristic
175C22×C4—216no—
176C22×C4—216no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—28no—
196C22×C4—28no—
197C22×C4—232no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—28no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—416no—
232C22×C4—416no—
233C22×C4—44no—
234C22×C4—416no—
235C22×C4—416no—
236C22×C4—416no—
237C22×C4—416no—
238C22×C4—416no—
239C22×C4—44no—
240C22×C4—416no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4#9511yescharacteristic
244C22×C4#9511yescharacteristic
245C22×C4#9511yescharacteristic
246C22×C4#9511yescharacteristic
247C22×C4#9511yescharacteristic
248C22×C4#9511yescharacteristic
249C22×C4#9914yes—
250C22×C4#10414yes—
251C22×C4#10414yes—
252C22×C4C2×C4211yescharacteristic
253C22×C4C2×C22⋊C411yescharacteristic
254C22×C4C2×C22⋊C411yescharacteristic
255C22×C4C2×C4⋊C411yescharacteristic
256C22×C4C2×C4⋊C411yescharacteristic
257C22×C4C2×C4⋊C411yescharacteristic
258C22×C4C2×M1614yes—
259C22×C4AGL(1,ℤ/8)12yes—
260C22×C4#13712yes—
261M16—416no—
262C2×C8—28no—
263C2×C8—44no—
264C2×C8—44no—
265C2×C8—44no—
266C2×C8—44no—
267C2×C8—44no—
268C42—22no—
269C42—22no—
270C42—22no—
271C42—22no—
272C42—22no—
273C42—22no—
274C42—22no—
275C42—22no—
276C42—41no—
277C42—41no—
278C42—41no—
279C42—41no—
280C42—41no—
281C42—41no—
282C42—41no—
283C42—41no—
284C42—416no—
285C42—416no—
286C23—24no—
287C23—28no—
288C23—24no—
289C23—24no—
290C23—28no—
291C23—24no—
292C23—28no—
293C23—28no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—24no—
307C23—24no—
308C23#33912yes—
309C23#33912yes—
310C23#33912yes—
311C23#33912yes—
312C23#34212yes—
313C23#34412yes—
314C23C2×#9511yescharacteristic
315C23#40712yes—
316C23#41811yescharacteristic
317C23#41911yescharacteristic
318C23C2×#10412yes—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—28no—
322C2×C4—28no—
323C2×C4—28no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—24no—
339C2×C4—28no—
340C2×C4—28no—
341C2×C4—24no—
342C2×C4—216no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—48no—
365C2×C4—464no—
366C2×C4—48no—
367C2×C4—48no—
368C2×C4—48no—
369C2×C4—48no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—48no—
373C2×C4#33912yes—
374C2×C4#33912yes—
375C2×C4#34212yes—
376C2×C4C4×M1612yes—
377C2×C4C2×#9912yes—
378C2×C4#41712yes—
379C2×C4C2×#10412yes—
380C2×C4C2×#10412yes—
381C2×C4#42312yes—
382C2×C4#42812yes—
383C8—48no—
384C8—84no—
385C22—216no—
386C22—216no—
387C22—21no—
388C22—22no—
389C22—22no—
390C22—21no—
391C22—22no—
392C22—22no—
393C22—22no—
394C22—22no—
395C22—21no—
396C22—22no—
397C22—22no—
398C22—21no—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22#129714yes—
408C22#130014yes—
409C22#149012yes—
410C22#149112yes—
411C22#172811yescharacteristic
412C22C2×#33911yescharacteristic
413C22C2×#33911yescharacteristic
414C22C2×#33911yescharacteristic
415C22#173611yescharacteristic
416C22C2×#34211yescharacteristic
417C22C2×#34411yescharacteristic
418C4—22no—
419C4—22no—
420C4—22no—
421C4—22no—
422C4—22no—
423C4—22no—
424C4—22no—
425C4—416no—
426C4—41no—
427C4—41no—
428C4—41no—
429C4—41no—
430C4#150414yes—
431C2—22no—
432C2—24no—
433C2—24no—
434C2—22no—
435C2#786711yescharacteristic
436C2#790411yescharacteristic
437C2#792911yescharacteristic
438C2#802011yescharacteristic
439C2C2×#129711yescharacteristic
440C2C2×#130011yescharacteristic
441C2#809811yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
socleC23C2×#95
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7028C2semidirect
7. C2×#432C4semidirect
13. #1797C4semidirect
19. #1749C4semidirect
22. #1728C4semidirect
51. C2×C4⋊C8C2×C4semidirect
71. #336C2×C4semidirect
163. #95M16semidirect

Maximal subgroups

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

subgroupindex
#207152
C4×#4072
C2×C4×#952
#70282