Finite Group Databasebeta

← #163626

Subgroups — #163626

Order 512.

1412 conjugacy classes of subgroups.

The subgroup lattice has 409 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2850C211yesmaximal, characteristic
2#20626C212yesmaximal, complemented
3#7710C218yesmaximal, complemented
4#7643C212yesmaximal, complemented
5#7640C212yesmaximal, complemented
6C22×#124C411yescharacteristic
7C22×C22⋊Q8C411yescharacteristic
8C22×#121C411yescharacteristic
9C23×C42C2211yescharacteristic
10C2×#423—22no—
11C2×#407—22no—
12#2850—22no—
13#2850—22no—
14#2850C414yes—
15#2850C2214yes—
16C2×#391C2211yescharacteristic
17C2×#390C2211yescharacteristic
18C2×#388—22no—
19C2×#386—22no—
20C2×#382C411yescharacteristic
21#1797—22no—
22#1797—22no—
23#1797C2214yes—
24#1797C2214yescomplemented
25#1728C2214yescomplemented
26#1272C2218yescomplemented
27#1270C2218yescomplemented
28C23×Q8C2×C411yescharacteristic
29C23×D4—21no—
30C23×D4C2×C411yescharacteristic
31C24×C4C2×C411yescharacteristic
32C24×C4D412yes—
33C22×M16—24no—
34C2×#124—21no—
35C2×#124—21no—
36C2×#124—44no—
37C2×#124C2×C414yes—
38C2×C22⋊Q8—22no—
39C2×C22⋊Q8—44no—
40C2×C22⋊Q8C2×C414yes—
41C2×#121—22no—
42C2×#121—44no—
43C2×#121C2×C414yes—
44C22×C4⋊C4—21no—
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C22⋊C4—21no—
48C22×C22⋊C4—21no—
49C22×C22⋊C4—22no—
50C22×C22⋊C4—22no—
51C22×C42—44no—
52C22×C42C2×C411yescharacteristic
53C22×C42C2×C412yes—
54C22×C42D412yes—
55C22×C42D412yes—
56C22×C42D412yes—
57C22×C42C2311yescharacteristic
58C22×C42C2312yes—
59#423—48no—
60C2×C4⋊C8—24no—
61C2×C4⋊C8—24no—
62#407—48no—
63C2×C22⋊C8—24no—
64C2×C22⋊C8—24no—
65#391—22no—
66#391C2×C412yes—
67#391C2312yes—
68#390—22no—
69#390C2×C412yes—
70#390C2312yes—
71#388—24no—
72#388—24no—
73#388—24no—
74#388—24no—
75#386—24no—
76#386—24no—
77#386—24no—
78#386—24no—
79#382—21no—
80#382—21no—
81#382C2×C414yes—
82C2×#95—22no—
83C2×#95—22no—
84#336—24no—
85#336C2318yes—
86C25—21no—
87C25C22⋊C411yescharacteristic
88C22×Q8—21no—
89C22×Q8—42no—
90C22×Q8—42no—
91C22×Q8—42no—
92C22×Q8C22×C412yes—
93C22×D4—24no—
94C22×D4—21no—
95C22×D4—42no—
96C22×D4—42no—
97C22×D4—42no—
98C22×D4—44no—
99C22×D4—48no—
100C22×D4C22×C412yes—
101C23×C4—24no—
102C23×C4—24no—
103C23×C4—24no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4—42no—
108C23×C4—42no—
109C23×C4—44no—
110C23×C4—44no—
111C23×C4C22⋊C412yes—
112C23×C4C22⋊C412yes—
113C23×C4C22⋊C412yes—
114C23×C4C22⋊C412yes—
115C23×C4C22⋊C411yescharacteristic
116C23×C4C22×C411yescharacteristic
117C23×C4C22×C412yes—
118C23×C4C2×D412yes—
119C23×C4C2×D412yes—
120C23×C4C2×D412yes—
121C2×M16—48no—
122C2×M16—48no—
123C2×M16—48no—
124C22×C8—28no—
125#124—44no—
126#124—48no—
127#124—44no—
128C22⋊Q8—48no—
129C22⋊Q8—48no—
130#121—48no—
131#121—48no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4—44no—
138C2×C4⋊C4C22×C412yes—
139C2×C4⋊C4C22×C412yes—
140C2×C22⋊C4—24no—
141C2×C22⋊C4—24no—
142C2×C22⋊C4—24no—
143C2×C22⋊C4—24no—
144C2×C22⋊C4—24no—
145C2×C22⋊C4—24no—
146C2×C22⋊C4—48no—
147C2×C22⋊C4—44no—
148C2×C22⋊C4—44no—
149C2×C22⋊C4—44no—
150C2×C22⋊C4—44no—
151C2×C42—24no—
152C2×C42—28no—
153C2×C42—44no—
154C2×C42—44no—
155C2×C42—44no—
156C2×C42—42no—
157C2×C42—44no—
158C2×C42C22⋊C411yescharacteristic
159C2×C42C22⋊C412yes—
160C2×C42C22⋊C412yes—
161C2×C42C22⋊C411yescharacteristic
162C2×C42C22×C412yes—
163C2×C42C22×C411yescharacteristic
164C2×C42C2×D412yes—
165C2×C42C2×D412yes—
166C2×C42C2×D412yes—
167C2×C42C2411yescharacteristic
168C4⋊C8—48no—
169C4⋊C8—48no—
170C22⋊C8—48no—
171C22⋊C8—48no—
172#95—24no—
173#95—24no—
174#95—24no—
175#95—24no—
176C24—22no—
177C24—24no—
178C24—24no—
179C24—22no—
180C24—41no—
181C24—41no—
182C24—42no—
183C24—42no—
184C24—44no—
185C24—42no—
186C24—44no—
187C24C2×C22⋊C412yes—
188C24C2×C22⋊C411yescharacteristic
189C2×Q8—24no—
190C2×Q8—42no—
191C2×Q8—42no—
192C2×Q8—42no—
193C2×Q8—42no—
194C2×Q8—42no—
195C2×Q8—88no—
196C2×D4—24no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—42no—
200C2×D4—42no—
201C2×D4—48no—
202C2×D4—48no—
203C2×D4—48no—
204C2×D4—42no—
205C2×D4—88no—
206C2×D4—816no—
207C22×C4—22no—
208C22×C4—24no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—24no—
213C22×C4—22no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—41no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—41no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—416no—
241C22×C4—416no—
242C22×C4—416no—
243C22×C4—42no—
244C22×C4—44no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—44no—
250C22×C4#10014yes—
251C22×C4C2×C22⋊C412yes—
252C22×C4C2×C22⋊C412yes—
253C22×C4C2×C22⋊C411yescharacteristic
254C22×C4C2×C22⋊C412yes—
255C22×C4C2×C22⋊C412yes—
256C22×C4AGL(1,ℤ/8)14yes—
257C22×C4#13714yes—
258C22×C4C23×C411yescharacteristic
259C22×C4C22×D412yes—
260M16—816no—
261C2×C8—48no—
262C2×C8—48no—
263C2×C8—48no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—48no—
269C22⋊C4—48no—
270C22⋊C4—48no—
271C22⋊C4—44no—
272C22⋊C4—44no—
273C22⋊C4—48no—
274C22⋊C4—48no—
275C22⋊C4—48no—
276C42—24no—
277C42—24no—
278C42—24no—
279C42—22no—
280C42—24no—
281C42—42no—
282C42—41no—
283C42—42no—
284C42—41no—
285C42#9914yes—
286C23—21no—
287C23—21no—
288C23—22no—
289C23—28no—
290C23—22no—
291C23—22no—
292C23—22no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—48no—
297C23—48no—
298C23—44no—
299C23—44no—
300C23—42no—
301C23—42no—
302C23—42no—
303C23—44no—
304C23—44no—
305C23—84no—
306C23—84no—
307C23—88no—
308C23—88no—
309C23—84no—
310C23C2×#10012yes—
311C23#41714yes—
312C23#42114yes—
313C23C22×C22⋊C411yescharacteristic
314Q8—84no—
315Q8—84no—
316D4—84no—
317D4—84no—
318D4—816no—
319C2×C4—28no—
320C2×C4—28no—
321C2×C4—28no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—48no—
341C2×C4—48no—
342C2×C4—48no—
343C2×C4—48no—
344C2×C4—48no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—84no—
355C2×C4—84no—
356C2×C4—84no—
357C2×C4C2×#9914yes—
358C2×C4C2×#9912yes—
359C2×C4C2×#10014yes—
360C2×C4#41714yes—
361C2×C4#56812yes—
362C2×C4C2×AGL(1,ℤ/8)12yes—
363C2×C4C2×#13712yes—
364C8—88no—
365C22—22no—
366C22—24no—
367C22—24no—
368C22—22no—
369C22—24no—
370C22—24no—
371C22—24no—
372C22—416no—
373C22—42no—
374C22—42no—
375C22—42no—
376C22—42no—
377C22—82no—
378C22—82no—
379C22—84no—
380C22—84no—
381C22—82no—
382C22—168no—
383C22#156014yes—
384C22#287911yescharacteristic
385C22C22×#9911yescharacteristic
386C22C22×#10011yescharacteristic
387C22C2×#41712yes—
388C22C2×#42112yes—
389C4—22no—
390C4—22no—
391C4—24no—
392C4—24no—
393C4—22no—
394C4—22no—
395C4—24no—
396C4—42no—
397C4—82no—
398C4—82no—
399C4—82no—
400C2—24no—
401C2—44no—
402C2—162no—
403C2#1739611yescharacteristic
404C2#1749212yes—
405C2#1759212yes—
406C2#1766611yescharacteristic
407C2C2×#156011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#121C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20626C2semidirect
3. #7710C2semidirect
4. #7643C2semidirect
5. #7640C2semidirect
24. #1797C22semidirect
25. #1728C22semidirect
26. #1272C22semidirect
27. #1270C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28502
#206262
#77102
#76432
#76402