Finite Group Databasebeta

← #496318

Subgroups — #496318

Order 512.

1310 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1310 conjugacy classes of subgroups — too many to draw. Only its 210 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
1C2×#2548C211yesmaximal, characteristic
2#20727C212yesmaximal, complemented
3#13226C218yesmaximal, complemented
4#12699C214yesmaximal, complemented
5C22×#127C411yescharacteristic
6C22×#124C2212yes—
7C22×C4×Q8C2211yescharacteristic
8#2920—22no—
9#2548—22no—
10#2548—22no—
11#2548C414yes—
12#2548C2214yes—
13C2×#394—22no—
14C2×#390—22no—
15C2×#390C412yes—
16C2×C4×C4⋊C4C411yescharacteristic
17C2×#355C22116yescomplemented
18#1821—22no—
19#1801—22no—
20#1801C2214yescomplemented
21#1784C2214yes—
22C4×#101—22no—
23C4×#101C2214yescomplemented
24C23×Q8C2311yescharacteristic
25C23×D4—21no—
26C23×D4—21no—
27C23×D4C2×C412yes—
28C2×#127—22no—
29C2×#127—41no—
30C2×#127—42no—
31C2×#127—41no—
32C2×#127C2×C414yes—
33C2×#124—24no—
34C2×#124—48no—
35C2×#124C2×C414yes—
36C2×#124C2314yes—
37C2×C4×Q8—41no—
38C2×C4×Q8—42no—
39C2×C4×Q8—41no—
40C2×C4×Q8C2×C412yes—
41C2×C4×Q8D412yes—
42C2×C4×Q8D412yes—
43C2×C4×Q8C2312yes—
44C22×C4⋊C4—21no—
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4D412yes—
47C22×C22⋊C4—24no—
48C22×C22⋊C4—24no—
49C22×C42C2×C411yescharacteristic
50C22×C42C2×C412yes—
51C22×C42D412yes—
52#432—42no—
53#432—42no—
54#431—42no—
55#431—42no—
56#423—24no—
57C2×#101—24no—
58C2×#101C2318yes—
59#407—24no—
60C4×M16—24no—
61#394—24no—
62#394—24no—
63#394—24no—
64#394—24no—
65#390—22no—
66#390—22no—
67#390—24no—
68#390—24no—
69#390—24no—
70#390—24no—
71#390C2×C418yes—
72C4×C4⋊C4—22no—
73C4×C4⋊C4C2×C414yes—
74C2×#95—21no—
75#355—416no—
76#343—24no—
77#343—24no—
78C25—22no—
79C22×Q8—24no—
80C22×Q8—41no—
81C22×Q8—41no—
82C22×Q8—42no—
83C22×Q8C22×C411yescharacteristic
84C22×Q8C2×D412yes—
85C22×Q8C2411yescharacteristic
86C22×D4—22no—
87C22×D4—24no—
88C22×D4—24no—
89C22×D4—44no—
90C22×D4—42no—
91C22×D4—42no—
92C22×D4—42no—
93C22×D4—42no—
94C22×D4—48no—
95C22×D4—44no—
96C22×D4—41no—
97C22×D4—42no—
98C22×D4—41no—
99C22×D4—84no—
100C22×D4C22×C414yes—
101C23×C4—22no—
102C23×C4C22⋊C412yes—
103C23×C4C22⋊C411yescharacteristic
104C23×C4C22⋊C411yescharacteristic
105C23×C4C22×C411yescharacteristic
106C23×C4C2×D412yes—
107C2×M16—28no—
108#127—44no—
109#127—44no—
110#127—44no—
111#127—44no—
112#124—216no—
113#124—48no—
114#124—416no—
115C4×Q8—42no—
116C4×Q8—42no—
117C4×Q8—42no—
118C4×Q8—42no—
119C4×Q8—42no—
120C4×Q8—42no—
121C4×Q8—42no—
122C4×Q8—42no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4—44no—
129C2×C4⋊C4C22⋊C411yescharacteristic
130C2×C4⋊C4C22⋊C411yescharacteristic
131C2×C4⋊C4C22⋊C412yes—
132C2×C4⋊C4C22⋊C412yes—
133C2×C4⋊C4C22×C412yes—
134C2×C4⋊C4C2×D412yes—
135C2×C4⋊C4C2×D412yes—
136C2×C22⋊C4—28no—
137C2×C22⋊C4—28no—
138C2×C22⋊C4—28no—
139C2×C22⋊C4—28no—
140C2×C22⋊C4—416no—
141C2×C22⋊C4—88no—
142C2×C42—21no—
143C2×C42—22no—
144C2×C42—21no—
145C2×C42—22no—
146C2×C42—22no—
147C2×C42—22no—
148C2×C42—22no—
149C2×C42—22no—
150C2×C42—44no—
151C2×C42C22⋊C412yes—
152C2×C42C22⋊C412yes—
153C2×C42C22⋊C412yes—
154C2×C42C22×C414yes—
155C2×C42C22×C412yes—
156C2×C42C2×D412yes—
157C2×C42C2×D412yes—
158C4⋊C8—44no—
159C4⋊C8—44no—
160#101—216no—
161#101—48no—
162C22⋊C8—44no—
163C22⋊C8—44no—
164#97—44no—
165C4×C8—44no—
166#95—24no—
167C24—24no—
168C24—44no—
169C24—48no—
170C24—44no—
171C24—48no—
172C24—42no—
173C24—42no—
174C24C2×C22⋊C411yescharacteristic
175C2×Q8—22no—
176C2×Q8—24no—
177C2×Q8—24no—
178C2×Q8—41no—
179C2×Q8—42no—
180C2×Q8—41no—
181C2×Q8—41no—
182C2×Q8—41no—
183C2×Q8—48no—
184C2×Q8—48no—
185C2×Q8H(5,𝔽2)14yes—
186C2×D4—28no—
187C2×D4—44no—
188C2×D4—44no—
189C2×D4—44no—
190C2×D4—44no—
191C2×D4—48no—
192C2×D4—416no—
193C2×D4—44no—
194C2×D4—44no—
195C2×D4—44no—
196C2×D4—88no—
197C2×D4—88no—
198C2×D4—84no—
199C2×D4—88no—
200C2×D4—88no—
201C2×D4—88no—
202C2×D4—88no—
203C2×D4—84no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—24no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—22no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—44no—
220C22×C4—44no—
221C22×C4—44no—
222C22×C4—44no—
223C22×C4#9911yescharacteristic
224C22×C4#9912yes—
225C22×C4#9911yescharacteristic
226C22×C4C2×C22⋊C412yes—
227C22×C4C2×C22⋊C412yes—
228C22×C4C2×C22⋊C411yescharacteristic
229C22×C4C2×C22⋊C412yes—
230C22×C4C2×C22⋊C412yes—
231C22×C4C2×C22⋊C412yes—
232C22×C4C23×C411yescharacteristic
233C22×C4C22×D412yes—
234M16—48no—
235M16—48no—
236C2×C8—48no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—48no—
245C22⋊C4—416no—
246C22⋊C4—416no—
247C22⋊C4—88no—
248C22⋊C4—88no—
249C42—22no—
250C42—22no—
251C42—22no—
252C42—22no—
253C42—22no—
254C42—22no—
255C42—22no—
256C42—22no—
257C42—22no—
258C42—22no—
259C42—22no—
260C42—22no—
261C42—28no—
262C42—44no—
263C23—21no—
264C23—21no—
265C23—22no—
266C23—22no—
267C23—48no—
268C23—48no—
269C23—44no—
270C23—44no—
271C23—44no—
272C23—84no—
273C23—84no—
274C23—84no—
275C23—84no—
276C23—816no—
277C23—816no—
278C23—84no—
279C23—84no—
280C23C2×#9911yescharacteristic
281C23C2×#9911yescharacteristic
282C23C22×C22⋊C411yescharacteristic
283Q8—44no—
284Q8—48no—
285Q8—44no—
286D4—88no—
287D4—88no—
288D4—88no—
289D4—816no—
290D4—88no—
291C2×C4—24no—
292C2×C4—24no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—42no—
311C2×C4—44no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—82no—
320C2×C4—88no—
321C2×C4#35514yescomplemented
322C2×C4C2×#9912yes—
323C2×C4C2×#9912yes—
324C2×C4#41012yes—
325C2×C4#51812yes—
326C2×C4#53514yes—
327C8—88no—
328C22—24no—
329C22—24no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—88no—
337C22—88no—
338C22—84no—
339C22—84no—
340C22—84no—
341C22—168no—
342C22—168no—
343C22C2×#35512yes—
344C22#213412yes—
345C22#230911yescharacteristic
346C22C22×#9911yescharacteristic
347C22C2×#41011yescharacteristic
348C4—24no—
349C4—22no—
350C4—22no—
351C4—22no—
352C4—42no—
353C4—42no—
354C4—42no—
355C4—42no—
356C4—84no—
357C4C2×#35514yes—
358C2—21no—
359C2—21no—
360C2—22no—
361C2—22no—
362C2—164no—
363C2#2179311yescharacteristic
364C2C22×#35511yescharacteristic
365C2C2×#213411yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
prime radicalC22×#127C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20727C2semidirect
3. #13226C2semidirect
4. #12699C2semidirect
17. C2×#355C22semidirect
20. #1801C22semidirect
23. C4×#101C22semidirect
321. C2×C4#355semidirect

Maximal subgroups

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

subgroupindex
C2×#25482
#207272
#132262
#126992