Finite Group Databasebeta

← #496404

Subgroups — #496404

Order 512.

910 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 910 conjugacy classes of subgroups — too many to draw. Only its 202 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×#2565C211yesmaximal, characteristic
2#20727C212yesmaximal
3#13303C218yesmaximal, complemented
4#12860C212yesmaximal
5#12860C212yesmaximal
6C22×#125C411yescharacteristic
7C22×C4×Q8C2211yescharacteristic
8#2920—22no—
9#2565—22no—
10#2565—22no—
11#2565C414yes—
12#2565C2214yes—
13C2×#398—22no—
14C2×#391C2211yescharacteristic
15C2×#391C2211yescharacteristic
16C2×#387—22no—
17C2×#384C411yescharacteristic
18C2×#382C411yescharacteristic
19C2×C4×C4⋊C4C411yescharacteristic
20#1821—22no—
21#1801—22no—
22#1801C2214yes—
23#1784C2214yes—
24C4×#101—22no—
25C4×#101C2214yes—
26#1394C2218yes—
27#1394C2218yes—
28C23×Q8C2311yescharacteristic
29C2×#125—22no—
30C2×#125—42no—
31C2×#125—41no—
32C2×#125—41no—
33C2×#125C2×C414yes—
34C2×C4×Q8—41no—
35C2×C4×Q8—42no—
36C2×C4×Q8—41no—
37C2×C4×Q8C2×C412yes—
38C2×C4×Q8D412yes—
39C2×C4×Q8D412yes—
40C2×C4×Q8C2312yes—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—22no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C4⋊C4C2×C411yescharacteristic
48C22×C4⋊C4D412yes—
49C22×C42C2×C411yescharacteristic
50C22×C42C2×C411yescharacteristic
51C22×C42C2×C411yescharacteristic
52C22×C42D412yes—
53#432—42no—
54#432—42no—
55#431—42no—
56#431—42no—
57#423—24no—
58C2×#101—24no—
59C2×#101C2318yes—
60#407—24no—
61C4×M16—24no—
62#398—24no—
63#398—24no—
64#398—24no—
65#398—24no—
66#391—22no—
67#391—22no—
68#391C2×C412yes—
69#391C2×C412yes—
70#391C2312yes—
71#391C2312yes—
72#387—24no—
73#387—24no—
74#387—24no—
75#387—24no—
76#384—21no—
77#384—21no—
78#384C2×C414yes—
79#382—21no—
80#382—21no—
81#382C2×C414yes—
82C4×C4⋊C4—22no—
83C4×C4⋊C4C2×C414yes—
84C2×#95—22no—
85C2×#95—21no—
86C2×#95—22no—
87C2×#95—22no—
88#343—24no—
89#343—24no—
90C22×Q8—24no—
91C22×Q8—41no—
92C22×Q8—41no—
93C22×Q8—42no—
94C22×Q8C22×C411yescharacteristic
95C22×Q8C2×D412yes—
96C22×Q8C2411yescharacteristic
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4C22⋊C411yescharacteristic
102C23×C4C22⋊C412yes—
103C23×C4C22⋊C411yescharacteristic
104C23×C4C22×C411yescharacteristic
105C23×C4C2×D412yes—
106C2×M16—28no—
107#125—44no—
108#125—44no—
109#125—44no—
110#125—44no—
111C4×Q8—42no—
112C4×Q8—42no—
113C4×Q8—42no—
114C4×Q8—42no—
115C4×Q8—42no—
116C4×Q8—42no—
117C4×Q8—42no—
118C4×Q8—42no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—44no—
135C2×C4⋊C4—48no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4—42no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4C22⋊C411yescharacteristic
142C2×C4⋊C4C22⋊C411yescharacteristic
143C2×C4⋊C4C22⋊C412yes—
144C2×C4⋊C4C22⋊C412yes—
145C2×C4⋊C4C22×C412yes—
146C2×C4⋊C4C22×C412yes—
147C2×C4⋊C4C22×C412yes—
148C2×C4⋊C4C2×D412yes—
149C2×C4⋊C4C2×D412yes—
150C2×C42—22no—
151C2×C42—21no—
152C2×C42—21no—
153C2×C42—22no—
154C2×C42—22no—
155C2×C42—22no—
156C2×C42—22no—
157C2×C42—21no—
158C2×C42—21no—
159C2×C42—42no—
160C2×C42—42no—
161C2×C42C22⋊C412yes—
162C2×C42C22⋊C412yes—
163C2×C42C22⋊C412yes—
164C2×C42C22×C412yes—
165C2×C42C22×C412yes—
166C2×C42C22×C412yes—
167C2×C42C2×D412yes—
168C2×C42C2×D412yes—
169C4⋊C8—44no—
170C4⋊C8—44no—
171#101—216no—
172#101—48no—
173C22⋊C8—44no—
174C22⋊C8—44no—
175#97—44no—
176C4×C8—44no—
177#95—24no—
178#95—24no—
179#95—24no—
180#95—24no—
181#95—24no—
182#95—24no—
183#95—24no—
184C24C2×C22⋊C411yescharacteristic
185C2×Q8—22no—
186C2×Q8—24no—
187C2×Q8—24no—
188C2×Q8—41no—
189C2×Q8—42no—
190C2×Q8—41no—
191C2×Q8—41no—
192C2×Q8—41no—
193C2×Q8—48no—
194C2×Q8—48no—
195C2×Q82−1+414yes—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—24no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—24no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—44no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—44no—
221C22×C4—42no—
222C22×C4—44no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4#9912yes—
228C22×C4#9911yescharacteristic
229C22×C4#9911yescharacteristic
230C22×C4C2×C22⋊C412yes—
231C22×C4C2×C22⋊C411yescharacteristic
232C22×C4C2×C22⋊C412yes—
233C22×C4C2×C22⋊C412yes—
234C22×C4C2×C22⋊C412yes—
235C22×C4C2×C22⋊C412yes—
236C22×C4C23×C411yescharacteristic
237C22×C4C22×D412yes—
238M16—48no—
239M16—48no—
240C2×C8—48no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—48no—
251C4⋊C4—48no—
252C4⋊C4—48no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—48no—
256C4⋊C4—44no—
257C4⋊C4—44no—
258C42—22no—
259C42—22no—
260C42—22no—
261C42—22no—
262C42—22no—
263C42—22no—
264C42—22no—
265C42—22no—
266C42—22no—
267C42—22no—
268C42—22no—
269C42—22no—
270C42—42no—
271C42—42no—
272C42—42no—
273C42—42no—
274C23—21no—
275C23—21no—
276C23—22no—
277C23—22no—
278C23C2×#9911yescharacteristic
279C23C2×#9911yescharacteristic
280C23C22×C22⋊C411yescharacteristic
281Q8—44no—
282Q8—48no—
283Q8—44no—
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—24no—
293C2×C4—24no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—44no—
304C2×C4—44no—
305C2×C4—48no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—44no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—44no—
316C2×C4—42no—
317C2×C4—44no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—82no—
322C2×C4—84no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4C2×#9912yes—
328C2×C4C2×#9912yes—
329C2×C4#41012yes—
330C2×C4#51912yes—
331C2×C4#53614yes—
332C8—88no—
333C22—24no—
334C22—24no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22#213212yes—
342C22#213312yes—
343C22#231011yescharacteristic
344C22C22×#9911yescharacteristic
345C22C2×#41011yescharacteristic
346C4—24no—
347C4—22no—
348C4—22no—
349C4—22no—
350C4—22no—
351C4—44no—
352C4—42no—
353C4—42no—
354C4—42no—
355C4—42no—
356C4—82no—
357C4—82no—
358C4—82no—
359C2—21no—
360C2—21no—
361C2—22no—
362C2—22no—
363C2#2182511yescharacteristic
364C2C2×#213211yescharacteristic
365C2C2×#213311yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
prime radicalC24C2×C22⋊C4
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
3. #13303C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25652
#207272
#133032
#128602