Finite Group Databasebeta

← #161128

Subgroups — #161128

Order 512.

2718 conjugacy classes of subgroups; only the 266 normal subgroups are shown.

normal subgroups only

This group has 2718 conjugacy classes of subgroups — too many to draw. Only its 266 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#390C211yesmaximal, characteristic
2#20682C212yesmaximal, complemented
3#7606C218yesmaximal, complemented
4#7592C214yesmaximal, complemented
5C24×D4C411yescharacteristic
6C23×C4⋊C4C411yescharacteristic
7C23×C22⋊C4—22no—
8C23×C42C2211yescharacteristic
9C2×#407—22no—
10C2×#390—22no—
11C2×#390—22no—
12C2×#390—22no—
13C2×#390C414yes—
14C2×#390C412yes—
15C2×#390C2212yes—
16C2×#390C2214yes—
17#1817—22no—
18#1817—22no—
19#1817C2214yescomplemented
20#1817C2214yescomplemented
21#1728—22no—
22#1728C2214yescomplemented
23#1270C22116yescomplemented
24C26—21no—
25C23×D4—28no—
26C23×D4—22no—
27C23×D4—21no—
28C23×D4—44no—
29C23×D4—44no—
30C23×D4C2×C414yes—
31C23×D4C2×C412yes—
32C24×C4—21no—
33C24×C4C2×C411yescharacteristic
34C24×C4D412yes—
35C22×M16—24no—
36C22×C4⋊C4—22no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—44no—
39C22×C4⋊C4C2×C412yes—
40C22×C4⋊C4C2×C414yes—
41C22×C22⋊C4—24no—
42C22×C22⋊C4—24no—
43C22×C22⋊C4—24no—
44C22×C22⋊C4—24no—
45C22×C22⋊C4—24no—
46C22×C22⋊C4—24no—
47C22×C22⋊C4—416no—
48C22×C42—44no—
49C22×C42C2×C412yes—
50C22×C42C2×C411yescharacteristic
51C22×C42D412yes—
52C22×C42D412yes—
53C22×C42D412yes—
54C22×C42C2311yescharacteristic
55C22×C42C2312yes—
56#407—48no—
57C2×C22⋊C8—24no—
58C2×C22⋊C8—24no—
59#390—22no—
60#390—24no—
61#390—24no—
62#390—22no—
63#390—24no—
64#390—24no—
65#390—24no—
66#390C2×C414yes—
67#390C2×C418yes—
68#390C2314yes—
69#336—24no—
70#336—24no—
71#336—24no—
72#336C2318yescomplemented
73C25—28no—
74C25—24no—
75C25—22no—
76C25—48no—
77C25—44no—
78C25—48no—
79C25—44no—
80C25C22⋊C411yescharacteristic
81C22×D4—28no—
82C22×D4—216no—
83C22×D4—216no—
84C22×D4—22no—
85C22×D4—24no—
86C22×D4—24no—
87C22×D4—44no—
88C22×D4—48no—
89C22×D4—44no—
90C22×D4—464no—
91C22×D4—48no—
92C22×D4—44no—
93C22×D4—48no—
94C22×D4—48no—
95C22×D4—44no—
96C22×D4M1618yescomplemented
97C22×D4C22×C414yes—
98C23×C4—24no—
99C23×C4—24no—
100C23×C4—24no—
101C23×C4—28no—
102C23×C4—28no—
103C23×C4—22no—
104C23×C4—24no—
105C23×C4—44no—
106C23×C4—48no—
107C23×C4—44no—
108C23×C4C22⋊C412yes—
109C23×C4C22⋊C412yes—
110C23×C4C22⋊C412yes—
111C23×C4C22⋊C412yes—
112C23×C4C22⋊C411yescharacteristic
113C23×C4C22×C412yes—
114C23×C4C22×C411yescharacteristic
115C23×C4C2×D412yes—
116C23×C4C2×D412yes—
117C23×C4C2×D412yes—
118C2×M16—48no—
119C2×M16—48no—
120C2×M16—48no—
121C22×C8—28no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—44no—
125C2×C4⋊C4—48no—
126C2×C4⋊C4—44no—
127C2×C4⋊C4—48no—
128C2×C4⋊C4C22×C414yes—
129C2×C22⋊C4—28no—
130C2×C22⋊C4—28no—
131C2×C22⋊C4—28no—
132C2×C22⋊C4—28no—
133C2×C22⋊C4—416no—
134C2×C22⋊C4—416no—
135C2×C22⋊C4—416no—
136C2×C22⋊C4—416no—
137C2×C22⋊C4—416no—
138C2×C22⋊C4—416no—
139C2×C42—42no—
140C2×C42—44no—
141C2×C42—44no—
142C2×C42—44no—
143C2×C42—44no—
144C2×C42—44no—
145C2×C42—42no—
146C2×C42C22⋊C411yescharacteristic
147C2×C42C22⋊C412yes—
148C2×C42C22⋊C412yes—
149C2×C42C22⋊C411yescharacteristic
150C2×C42C22×C411yescharacteristic
151C2×C42C22×C412yes—
152C2×C42C2×D412yes—
153C2×C42C2×D412yes—
154C2×C42C2×D412yes—
155C2×C42C2411yescharacteristic
156C22⋊C8—48no—
157C22⋊C8—48no—
158C24—22no—
159C24—22no—
160C24—24no—
161C24—24no—
162C24—216no—
163C24—24no—
164C24—28no—
165C24—48no—
166C24—48no—
167C24—48no—
168C24—432no—
169C24—464no—
170C24—48no—
171C24—48no—
172C24—48no—
173C24—44no—
174C24—44no—
175C24—48no—
176C24—48no—
177C24—816no—
178C24—816no—
179C24—832no—
180C24#9914yes—
181C24C2×C22⋊C411yescharacteristic
182C24C2×C22⋊C412yes—
183C2×D4—216no—
184C2×D4—24no—
185C2×D4—28no—
186C2×D4—416no—
187C2×D4—44no—
188C2×D4—44no—
189C2×D4—464no—
190C2×D4—464no—
191C2×D4—44no—
192C2×D4—48no—
193C2×D4—44no—
194C2×D4—432no—
195C2×D4—416no—
196C2×D4—48no—
197C2×D4C2×M1618yes—
198C22×C4—232no—
199C22×C4—22no—
200C22×C4—24no—
201C22×C4—24no—
202C22×C4—22no—
203C22×C4—24no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—24no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—432no—
228C22×C4—48no—
229C22×C4—48no—
230C22×C4—48no—
231C22×C4—44no—
232C22×C4—48no—
233C22×C4—816no—
234C22×C4#10014yes—
235C22×C4C2×C22⋊C412yes—
236C22×C4C2×C22⋊C412yes—
237C22×C4C2×C22⋊C411yescharacteristic
238C22×C4C2×C22⋊C412yes—
239C22×C4C2×C22⋊C412yes—
240C22×C4C2×M1614yes—
241C22×C4AGL(1,ℤ/8)18yes—
242C22×C4C23×C411yescharacteristic
243C22×C4C22×D412yes—
244M16—816no—
245C2×C8—48no—
246C2×C8—48no—
247C2×C8—48no—
248C4⋊C4—44no—
249C4⋊C4—44no—
250C4⋊C4—48no—
251C22⋊C4—416no—
252C22⋊C4—416no—
253C22⋊C4—416no—
254C22⋊C4—416no—
255C42—41no—
256C42—42no—
257C42—41no—
258C42—42no—
259C42—42no—
260C42—41no—
261C42—42no—
262C42—42no—
263C42—42no—
264C42—41no—
265C23—28no—
266C23—216no—
267C23—28no—
268C23—22no—
269C23—22no—
270C23—22no—
271C23—22no—
272C23—22no—
273C23—22no—
274C23—28no—
275C23—216no—
276C23—28no—
277C23—432no—
278C23—432no—
279C23—432no—
280C23—432no—
281C23—44no—
282C23—44no—
283C23—48no—
284C23—48no—
285C23—416no—
286C23—8128no—
287C23—88no—
288C23—816no—
289C23—88no—
290C23—816no—
291C23—816no—
292C23—816no—
293C23—816no—
294C23#40712yes—
295C23C2×#9912yes—
296C23C2×#9914yes—
297C23C2×#10012yes—
298C23#41814yes—
299C23#42114yes—
300C23C22×C22⋊C411yescharacteristic
301D4—48no—
302D4—48no—
303D4—416no—
304D4—432no—
305C2×C4—216no—
306C2×C4—216no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—28no—
318C2×C4—28no—
319C2×C4—216no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—48no—
327C2×C4—48no—
328C2×C4—48no—
329C2×C4—48no—
330C2×C4—48no—
331C2×C4—44no—
332C2×C4—48no—
333C2×C4—88no—
334C2×C4—816no—
335C2×C4#40712yes—
336C2×C4C2×#10014yes—
337C2×C4#41814yes—
338C2×C4C22×M1612yes—
339C2×C4C2×AGL(1,ℤ/8)14yes—
340C8—88no—
341C22—232no—
342C22—22no—
343C22—24no—
344C22—24no—
345C22—24no—
346C22—24no—
347C22—22no—
348C22—24no—
349C22—24no—
350C22—24no—
351C22—24no—
352C22—416no—
353C22—48no—
354C22—832no—
355C22—864no—
356C22—84no—
357C22—84no—
358C22—88no—
359C22#146214yes—
360C22#151018yes—
361C22C2×#40711yescharacteristic
362C22C22×#9911yescharacteristic
363C22C22×#10011yescharacteristic
364C22C2×#41812yes—
365C22C2×#42112yes—
366C4—24no—
367C4—24no—
368C4—22no—
369C4—22no—
370C4—24no—
371C4—42no—
372C4—42no—
373C4—84no—
374C2—24no—
375C2—28no—
376C2—88no—
377C2C2×#146211yescharacteristic
378C2#1744812yes—
379C2C2×#151012yes—
380C2#1766611yescharacteristic
381C2#1774111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24×D4C4
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. #20682C2semidirect
3. #7606C2semidirect
4. #7592C2semidirect
19. #1817C22semidirect
20. #1817C22semidirect
22. #1728C22semidirect
23. #1270C22semidirect
72. #336C23semidirect
96. C22×D4M16semidirect

Maximal subgroups

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

subgroupindex
C22×#3902
#206822
#76062
#75922