Finite Group Databasebeta

← #480714

Subgroups — #480714

Order 512.

1250 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1250 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
1#35272C211yesmaximal, characteristic
2#17363C212yesmaximal, complemented
3#11557C218yesmaximal, complemented
4#11503C214yesmaximal, complemented
5C2×#535C412yes—
6C22×#127C411yescharacteristic
7C2×C4×#41C2211yescharacteristic
8C2×#432—22no—
9#2856—22no—
10#2619—22no—
11#2548—22no—
12#2541—22no—
13#2408C414yes—
14#2408C2214yes—
15#2309C2212yes—
16#2285C411yescharacteristic
17#1467—22no—
18#1466C2214yes—
19#1464—22no—
20#1464C2214yes—
21#1459—22no—
22#1459C2214yes—
23#1316C22116yes—
24C22×#41C2×C411yescharacteristic
25C23×D4C2×C412yes—
26C23×D4C2×C414yes—
27#535—48no—
28C2×#127—22no—
29C2×#127—21no—
30C2×#127—22no—
31C2×#127—21no—
32C2×#127—41no—
33C2×#127—42no—
34C2×#127—41no—
35C2×#124—22no—
36C2×#121—24no—
37C2×#121—24no—
38C2×#121—24no—
39C2×#120C2×C418yes—
40C4×#41—42no—
41C4×#41—41no—
42C4×#41—41no—
43C2×C4×Q8D412yes—
44C2×C4×D4C2×C412yes—
45C2×C4×D4D412yes—
46C2×C4×D4C2312yes—
47C2×#117C2×C412yes—
48C2×#117D412yes—
49C2×#117C2311yescharacteristic
50C22×C42C2×C411yescharacteristic
51C22×C42D412yes—
52#432—24no—
53#432—24no—
54#432—42no—
55#432—42no—
56C2×C4⋊C8—42no—
57C2×C22⋊C8—24no—
58C2×#97—42no—
59#394—24no—
60#394—24no—
61#392—24no—
62#392—24no—
63#390—24no—
64#390—24no—
65#390—24no—
66#390—24no—
67#384—21no—
68#382—21no—
69#380C2×C414yes—
70#379C2×C414yes—
71#379C2314yes—
72C4×C4⋊C4—22no—
73#324—24no—
74#324—24no—
75#323—24no—
76#323C2318yes—
77C25C22×C414yes—
78C2×#41—24no—
79C2×#41—41no—
80C2×#41—41no—
81C2×#41—42no—
82C22×Q8C22⋊C411yescharacteristic
83C22×D4—28no—
84C22×D4—24no—
85C22×D4—22no—
86C22×D4—22no—
87C22×D4—24no—
88C22×D4—44no—
89C22×D4—42no—
90C22×D4—42no—
91C22×D4—44no—
92C22×D4—44no—
93C22×D4—44no—
94C22×D4—88no—
95C22×D4C22⋊C411yescharacteristic
96C22×D4C22×C412yes—
97C23×C4C22⋊C412yes—
98C23×C4C22⋊C412yes—
99C23×C4C22×C411yescharacteristic
100C23×C4C22×C412yes—
101C23×C4C2×D412yes—
102C22×C8—44no—
103#127—42no—
104#127—42no—
105#127—44no—
106#127—44no—
107#127—48no—
108#127—41no—
109#127—42no—
110#127—41no—
111#124—48no—
112#121—416no—
113#121—88no—
114#121—88no—
115#120—416no—
116C4×Q8—42no—
117C4×Q8—42no—
118C4×D4—42no—
119C4×D4—42no—
120C4×D4—42no—
121C4×D4—42no—
122C4×D4—42no—
123C4×D4—42no—
124#117—42no—
125#117—42no—
126#117—41no—
127#117—41no—
128#117—44no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4C22⋊C412yes—
132C2×C4⋊C4C2×D412yes—
133C2×C4⋊C4C2×D412yes—
134C2×C22⋊C4—28no—
135C2×C22⋊C4—28no—
136C2×C22⋊C4—28no—
137C2×C22⋊C4C22⋊C412yes—
138C2×C22⋊C4C22×C414yes—
139C2×C22⋊C4C22×C411yescharacteristic
140C2×C22⋊C4C2×D412yes—
141C2×C22⋊C4C2411yescharacteristic
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×C42C22⋊C411yescharacteristic
151C2×C42C22⋊C412yes—
152C2×C42C22⋊C412yes—
153C2×C42C22⋊C411yescharacteristic
154C2×C42C2×D412yes—
155C2×C42C2×D412yes—
156C4⋊C8—44no—
157C4⋊C8—44no—
158C22⋊C8—28no—
159C22⋊C8—44no—
160#97—44no—
161#97—44no—
162#95—24no—
163C24—24no—
164C24—28no—
165C24—44no—
166C24—44no—
167C24—88no—
168C24C2×C22⋊C412yes—
169C24C23×C411yescharacteristic
170#41—48no—
171#41—48no—
172C2×Q8—22no—
173C2×Q8—41no—
174C2×Q8—41no—
175C2×D4—24no—
176C2×D4—24no—
177C2×D4—41no—
178C2×D4—42no—
179C2×D4—41no—
180C2×D4—44no—
181C2×D4—48no—
182C2×D4—48no—
183C2×D4—44no—
184C2×D4—48no—
185C2×D4—44no—
186C2×D4—44no—
187C2×D4—416no—
188C2×D4—44no—
189C2×D4—42no—
190C2×D4—42no—
191C2×D4—44no—
192C2×D4—44no—
193C2×D4—88no—
194C2×D4—88no—
195C2×D4—88no—
196C2×D4—816no—
197C2×D4—816no—
198C2×D4—88no—
199C2×D4—88no—
200C2×D4#10014yes—
201C22×C4—22no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—22no—
205C22×C4—24no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—24no—
215C22×C4—42no—
216C22×C4—44no—
217C22×C4—44no—
218C22×C4—44no—
219C22×C4C2×C22⋊C412yes—
220C22×C4C2×C22⋊C412yes—
221C22×C4C2×C22⋊C411yescharacteristic
222C22×C4C2×C22⋊C411yescharacteristic
223C22×C4C2×C22⋊C412yes—
224C22×C4C2×C22⋊C412yes—
225C22×C4C22×D412yes—
226C2×C8—48no—
227C2×C8—88no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—44no—
232C4⋊C4—48no—
233C22⋊C4—42no—
234C22⋊C4—41no—
235C22⋊C4—42no—
236C22⋊C4—41no—
237C22⋊C4—416no—
238C22⋊C4—44no—
239C22⋊C4—88no—
240C22⋊C4—88no—
241C42—22no—
242C42—22no—
243C42—22no—
244C42—22no—
245C42—21no—
246C42—21no—
247C42—22no—
248C42—22no—
249C42—22no—
250C42—22no—
251C42—44no—
252C42#9911yescharacteristic
253C42#9912yes—
254C42#9911yescharacteristic
255C23—21no—
256C23—22no—
257C23—21no—
258C23—22no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23—48no—
263C23—48no—
264C23—44no—
265C23—44no—
266C23—44no—
267C23—88no—
268C23—816no—
269C23—168no—
270C23—168no—
271C23C2×#10014yes—
272C23C22×C22⋊C411yescharacteristic
273C23#56812yes—
274Q8—44no—
275D4—44no—
276D4—48no—
277D4—88no—
278D4—84no—
279D4—84no—
280D4—88no—
281D4—816no—
282D4—88no—
283D4—1616no—
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—24no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—48no—
307C2×C4—44no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4C2×#9912yes—
311C2×C4C2×#9911yescharacteristic
312C2×C4C2×#9911yescharacteristic
313C2×C4C2×#9912yes—
314C2×C4C2×#10012yes—
315C8—88no—
316C22—22no—
317C22—22no—
318C22—24no—
319C22—44no—
320C22—42no—
321C22—42no—
322C22—42no—
323C22—42no—
324C22—42no—
325C22—82no—
326C22—88no—
327C22—84no—
328C22—84no—
329C22—88no—
330C22—84no—
331C22—1616no—
332C22#287911yescharacteristic
333C22C22×#9911yescharacteristic
334C22C22×#10011yescharacteristic
335C4—22no—
336C4—21no—
337C4—21no—
338C4—22no—
339C4—42no—
340C4—42no—
341C4—42no—
342C4—42no—
343C4—84no—
344C4#212214yes—
345C2—21no—
346C2—21no—
347C2—42no—
348C2—44no—
349C2—164no—
350C2#1739611yescharacteristic
351C2C2×#212212yes—
subgroupG/H
center, socleC22C22×#99
commutatorC24C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radical#35272C2
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17363C2semidirect
3. #11557C2semidirect
4. #11503C2semidirect

Maximal subgroups

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

subgroupindex
#352722
#173632
#115572
#115032