Finite Group Databasebeta

← #96458

Subgroups — #96458

Order 512.

974 conjugacy classes of subgroups.

The subgroup lattice has 419 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20714C211yesmaximal, characteristic
2#20626C211yesmaximal, characteristic
3#13753C211yesmaximal, characteristic
4#7028C214yesmaximal, complemented
5C23×C42C2211yescharacteristic
6C2×#431—21no—
7C2×#423C412yescomplemented
8C2×#407—21no—
9C2×C4×C22⋊C4—21no—
10C2×#376—21no—
11#1817C412yescomplemented
12#1797—21no—
13#1797—21no—
14#1797—21no—
15#1797C412yescomplemented
16#1797C412yescomplemented
17#1797C2212yes—
18#1779—21no—
19C4×C22⋊C8—21no—
20C4×C22⋊C8C2212yes—
21#1728—21no—
22#1434—21no—
23#1434C412yes—
24C22⋊#95—21no—
25C22⋊#95C412yes—
26C4×#95C2212yes—
27C24×C4C2×C411yescharacteristic
28C24×C4D411yescharacteristic
29C24×C4D411yescharacteristic
30C22×M16—22no—
31C22×C22⋊C4—22no—
32C22×C42—22no—
33C22×C42—44no—
34C22×C42C2×C411yescharacteristic
35C22×C42C2×C411yescharacteristic
36C22×C42C2×C411yescharacteristic
37C22×C42C2×C411yescharacteristic
38C22×C42C2×C411yescharacteristic
39C22×C42D411yescharacteristic
40C22×C42D411yescharacteristic
41C22×C42D411yescharacteristic
42C22×C42D411yescharacteristic
43C22×C42Q811yescharacteristic
44C22×C42Q811yescharacteristic
45C22×C42C2311yescharacteristic
46#431—44no—
47#423—44no—
48C2×C4⋊C8—22no—
49C2×C4⋊C8—22no—
50C2×C4⋊C8C2×C414yes—
51#407—44no—
52C2×C22⋊C8—22no—
53C2×C22⋊C8—22no—
54C2×C22⋊C8—22no—
55C2×C22⋊C8—22no—
56C2×C22⋊C8—22no—
57C2×C4×C8—22no—
58C4×C22⋊C4—28no—
59#376—44no—
60C2×#95—22no—
61C2×#95—22no—
62C2×#95—22no—
63C2×#95—22no—
64C2×#95C2×C414yes—
65#336—22no—
66#336—22no—
67#336—22no—
68#336C2×C414yes—
69C25C22⋊C411yescharacteristic
70C23×C4—24no—
71C23×C4—24no—
72C23×C4—22no—
73C23×C4—22no—
74C23×C4—24no—
75C23×C4—42no—
76C23×C4—42no—
77C23×C4—42no—
78C23×C4—42no—
79C23×C4—42no—
80C23×C4—42no—
81C23×C4C4211yescharacteristic
82C23×C4C4211yescharacteristic
83C23×C4C22⋊C411yescharacteristic
84C23×C4C22⋊C411yescharacteristic
85C23×C4C22⋊C411yescharacteristic
86C23×C4C22⋊C411yescharacteristic
87C23×C4C22⋊C411yescharacteristic
88C23×C4C22⋊C411yescharacteristic
89C23×C4C22⋊C411yescharacteristic
90C23×C4C4⋊C411yescharacteristic
91C23×C4C4⋊C411yescharacteristic
92C23×C4C4⋊C411yescharacteristic
93C23×C4C4⋊C411yescharacteristic
94C23×C4C4⋊C411yescharacteristic
95C23×C4C4⋊C411yescharacteristic
96C23×C4C22×C411yescharacteristic
97C23×C4C2×D411yescharacteristic
98C23×C4C2×D411yescharacteristic
99C2×M16—44no—
100C2×M16—44no—
101C2×M16—44no—
102C22×C8—24no—
103C22×C8—24no—
104C2×C22⋊C4—28no—
105C2×C22⋊C4—44no—
106C2×C22⋊C4—44no—
107C2×C42—24no—
108C2×C42—24no—
109C2×C42—24no—
110C2×C42—28no—
111C2×C42—42no—
112C2×C42—42no—
113C2×C42—42no—
114C2×C42—42no—
115C2×C42—42no—
116C2×C42—42no—
117C2×C42—42no—
118C2×C42—42no—
119C2×C42—42no—
120C2×C42—44no—
121C2×C42—42no—
122C2×C42—42no—
123C2×C42C4211yescharacteristic
124C2×C42C4211yescharacteristic
125C2×C42C22⋊C411yescharacteristic
126C2×C42C22⋊C411yescharacteristic
127C2×C42C22⋊C411yescharacteristic
128C2×C42C22⋊C411yescharacteristic
129C2×C42C4⋊C411yescharacteristic
130C2×C42C4⋊C411yescharacteristic
131C2×C42C4⋊C411yescharacteristic
132C2×C42C4⋊C411yescharacteristic
133C2×C42C4⋊C411yescharacteristic
134C2×C42C4⋊C411yescharacteristic
135C2×C42C22×C411yescharacteristic
136C2×C42C22×C411yescharacteristic
137C2×C42C2×D411yescharacteristic
138C2×C42C2×Q811yescharacteristic
139C4⋊C8—44no—
140C4⋊C8—44no—
141C4⋊C8—44no—
142C22⋊C8—44no—
143C22⋊C8—44no—
144C22⋊C8—44no—
145C22⋊C8—44no—
146C22⋊C8—44no—
147C4×C8—44no—
148#95—28no—
149#95—28no—
150#95—44no—
151#95—44no—
152#95—44no—
153C24—22no—
154C24—22no—
155C24—22no—
156C24—22no—
157C24—22no—
158C24—41no—
159C24—41no—
160C24#9511yescharacteristic
161C24#9511yescharacteristic
162C24C2×C22⋊C411yescharacteristic
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—28no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—48no—
218C22×C4—48no—
219C22×C4—416no—
220C22×C4—416no—
221C22×C4—44no—
222C22×C4—44no—
223C22×C4—44no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4—44no—
227C22×C4#9511yescharacteristic
228C22×C4#9511yescharacteristic
229C22×C4#9511yescharacteristic
230C22×C4#9511yescharacteristic
231C22×C4#9511yescharacteristic
232C22×C4#9511yescharacteristic
233C22×C4#10414yes—
234C22×C4C2×C4211yescharacteristic
235C22×C4C2×C22⋊C411yescharacteristic
236C22×C4C2×C22⋊C411yescharacteristic
237C22×C4C2×C4⋊C411yescharacteristic
238C22×C4C2×C4⋊C411yescharacteristic
239C22×C4C2×C4⋊C411yescharacteristic
240C22×C4#13114yes—
241C22×C4AGL(1,ℤ/8)12yes—
242C22×C4#13712yes—
243M16—88no—
244C2×C8—44no—
245C2×C8—44no—
246C2×C8—44no—
247C2×C8—44no—
248C2×C8—44no—
249C2×C8—44no—
250C22⋊C4—416no—
251C42—22no—
252C42—22no—
253C42—22no—
254C42—22no—
255C42—22no—
256C42—22no—
257C42—22no—
258C42—22no—
259C42—22no—
260C42—41no—
261C42—41no—
262C42—41no—
263C42—41no—
264C42—41no—
265C42—41no—
266C42—48no—
267C42—48no—
268C42#9914yes—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—28no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—44no—
287C23—44no—
288C23—44no—
289C23—44no—
290C23—44no—
291C23—44no—
292C23#33912yes—
293C23#34412yes—
294C23C2×#9511yescharacteristic
295C23#40812yes—
296C23#41712yes—
297C23#42112yes—
298C2×C4—28no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—48no—
346C2×C4—48no—
347C2×C4—48no—
348C2×C4—84no—
349C2×C4—84no—
350C2×C4—84no—
351C2×C4—84no—
352C2×C4#33912yes—
353C2×C4#34212yes—
354C2×C4#34212yes—
355C2×C4#40512yes—
356C2×C4C2×#9912yes—
357C2×C4#41811yescharacteristic
358C2×C4#41911yescharacteristic
359C2×C4C2×#10412yes—
360C2×C4#42112yes—
361C2×C4#42412yes—
362C2×C4#42812yes—
363C8—84no—
364C8—84no—
365C22—22no—
366C22—22no—
367C22—22no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22—416no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22#149514yes—
387C22#172911yescharacteristic
388C22C2×#33911yescharacteristic
389C22#173311yescharacteristic
390C22#173311yescharacteristic
391C22#173611yescharacteristic
392C22C2×#34211yescharacteristic
393C22C2×#34411yescharacteristic
394C4—22no—
395C4—22no—
396C4—22no—
397C4—22no—
398C4—22no—
399C4—22no—
400C4—22no—
401C4—41no—
402C4—41no—
403C4—41no—
404C4—41no—
405C4—88no—
406C4#150712yes—
407C4#150812yes—
408C2—24no—
409C2—42no—
410C2—42no—
411C2#787011yescharacteristic
412C2#789911yescharacteristic
413C2#792511yescharacteristic
414C2#801911yescharacteristic
415C2#804911yescharacteristic
416C2#807211yescharacteristic
417C2#809311yescharacteristic
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×#423C4semidirect
11. #1817C4semidirect
15. #1797C4semidirect
16. #1797C4semidirect

Maximal subgroups

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

subgroupindex
#207142
#206262
#137532
#70282