Finite Group Databasebeta

← #496311

Subgroups — #496311

Order 512.

1350 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1350 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#20723C212yesmaximal, complemented
3#13223C218yesmaximal, complemented
4#12697C214yesmaximal, complemented
5C2×#535C2212yes—
6C22×#127C411yescharacteristic, complemented
7C2×C4×#41C2211yescharacteristic
8#2856—22no—
9#2619—22no—
10#2548C414yescomplemented
11#2541C2214yes—
12#2408—22no—
13#2408—22no—
14#2309C412yescomplemented
15#2285C411yescharacteristic, complemented
16#2283—22no—
17C2×#353C22116yescomplemented
18#1820—22no—
19#1799—22no—
20#1799C2214yescomplemented
21#1782C2214yes—
22C4×#99—22no—
23C4×#99C2214yescomplemented
24C22×#41C2311yescharacteristic
25C23×D4—22no—
26C23×D4C2×C412yes—
27#535—48no—
28C2×#127—21no—
29C2×#127—21no—
30C2×#127—41no—
31C2×#127—42no—
32C2×#127—41no—
33C2×#127C2×C414yes—
34C2×#127C2314yes—
35C2×#124C2×C414yes—
36C2×#121—24no—
37C2×#121—24no—
38C2×#121—24no—
39C2×#120—24no—
40C4×#41—42no—
41C4×#41—41no—
42C4×#41—41no—
43C2×C4×Q8C2×C412yes—
44C2×C4×D4D412yes—
45C2×C4×D4D412yes—
46C2×C4×D4C2312yes—
47C2×#117C2×C411yescharacteristic
48C2×#117C2×C412yes—
49C2×#117D412yes—
50C22×C42C2×C411yescharacteristic
51C22×C42D412yes—
52C2×#99—24no—
53C2×#99C2318yes—
54#394—24no—
55#394—24no—
56#392—24no—
57#392—24no—
58#390—24no—
59#390—24no—
60#390—24no—
61#390C2×C418yes—
62#384—21no—
63#383—42no—
64#382—21no—
65#382—42no—
66#381—24no—
67#380—22no—
68#380—24no—
69#379—22no—
70#379—22no—
71C4×C4⋊C4—42no—
72C4×C4⋊C4C2×C414yes—
73C4×C22⋊C4—24no—
74#376—42no—
75#353—416no—
76#342—24no—
77#342—24no—
78C25—22no—
79C2×#41—24no—
80C2×#41—41no—
81C2×#41—41no—
82C2×#41—42no—
83C22×Q8C22×C411yescharacteristic
84C22×D4—28no—
85C22×D4—22no—
86C22×D4—24no—
87C22×D4—24no—
88C22×D4—44no—
89C22×D4—44no—
90C22×D4—44no—
91C22×D4—44no—
92C22×D4—42no—
93C22×D4—42no—
94C22×D4—88no—
95C22×D4C22×C414yes—
96C22×D4C2×D412yes—
97C22×D4C2411yescharacteristic
98C23×C4C22⋊C412yes—
99C23×C4C22×C411yescharacteristic
100C23×C4C2×D412yes—
101C23×C4C2×D412yes—
102C23×C4C2×D412yes—
103#127—216no—
104#127—44no—
105#127—44no—
106#127—42no—
107#127—42no—
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—41no—
125#117—41no—
126#117—44no—
127#117—42no—
128#117—42no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—44no—
132C2×C4⋊C4C22⋊C412yes—
133C2×C4⋊C4C22⋊C412yes—
134C2×C4⋊C4C22×C412yes—
135C2×C22⋊C4—22no—
136C2×C22⋊C4—28no—
137C2×C22⋊C4—28no—
138C2×C22⋊C4—28no—
139C2×C22⋊C4—28no—
140C2×C22⋊C4C22⋊C411yescharacteristic
141C2×C22⋊C4C22⋊C411yescharacteristic
142C2×C22⋊C4C2×D412yes—
143C2×C22⋊C4C2×D412yes—
144C2×C42—21no—
145C2×C42—22no—
146C2×C42—21no—
147C2×C42—22no—
148C2×C42—22no—
149C2×C42—22no—
150C2×C42—22no—
151C2×C42—44no—
152C2×C42C22⋊C412yes—
153C2×C42C22⋊C411yescharacteristic
154C2×C42C22⋊C412yes—
155C2×C42C22⋊C412yes—
156C2×C42C22⋊C411yescharacteristic
157C2×C42C22×C414yes—
158C2×C42C22×C412yes—
159#99—216no—
160#99—48no—
161#95—24no—
162#95—44no—
163#95—44no—
164#95—44no—
165#95—44no—
166C24—24no—
167C24—44no—
168C24—44no—
169C24—44no—
170C24—88no—
171C24C2×C22⋊C411yescharacteristic
172C24C22×D412yes—
173#41—48no—
174#41—48no—
175C2×Q8—22no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×D4—24no—
179C2×D4—24no—
180C2×D4—28no—
181C2×D4—42no—
182C2×D4—41no—
183C2×D4—41no—
184C2×D4—44no—
185C2×D4—44no—
186C2×D4—44no—
187C2×D4—48no—
188C2×D4—48no—
189C2×D4—48no—
190C2×D4—44no—
191C2×D4—416no—
192C2×D4—42no—
193C2×D4—42no—
194C2×D4—44no—
195C2×D4—44no—
196C2×D4—88no—
197C2×D4—88no—
198C2×D4—88no—
199C2×D4—816no—
200C2×D4—816no—
201C2×D4—88no—
202C2×D4—88no—
203C2×D4H(5,𝔽2)14yes—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—24no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—24no—
218C22×C4—42no—
219C22×C4—44no—
220C22×C4—44no—
221C22×C4—44no—
222C22×C4—48no—
223C22×C4C2×C22⋊C412yes—
224C22×C4C2×C22⋊C411yescharacteristic
225C22×C4C2×C22⋊C412yes—
226C22×C4C2×C22⋊C412yes—
227C22×C4C2×C22⋊C412yes—
228C22×C4C2×C22⋊C412yes—
229C22×C4C23×C411yescharacteristic
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—44no—
234C4⋊C4—48no—
235C4⋊C4—88no—
236C22⋊C4—41no—
237C22⋊C4—41no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—416no—
241C22⋊C4—44no—
242C22⋊C4—48no—
243C22⋊C4—48no—
244C22⋊C4—88no—
245C22⋊C4—88no—
246C42—22no—
247C42—22no—
248C42—21no—
249C42—21no—
250C42—22no—
251C42—22no—
252C42—22no—
253C42—22no—
254C42—22no—
255C42—22no—
256C42—28no—
257C42—88no—
258C42#9911yescharacteristic
259C42#9912yes—
260C42#9911yescharacteristic
261C23—21no—
262C23—22no—
263C23—21no—
264C23—22no—
265C23—42no—
266C23—42no—
267C23—42no—
268C23—48no—
269C23—48no—
270C23—44no—
271C23—44no—
272C23—44no—
273C23—88no—
274C23—816no—
275C23—168no—
276C23—168no—
277C23#41012yes—
278C23C22×C22⋊C411yescharacteristic
279C23#53514yes—
280Q8—44no—
281D4—44no—
282D4—48no—
283D4—88no—
284D4—84no—
285D4—84no—
286D4—88no—
287D4—816no—
288D4—88no—
289D4—1616no—
290C2×C4—24no—
291C2×C4—24no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—24no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—44no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—48no—
313C2×C4—44no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—88no—
317C2×C4—88no—
318C2×C4—88no—
319C2×C4C2×#9911yescharacteristic
320C2×C4C2×#9911yescharacteristic
321C2×C4C2×#9912yes—
322C2×C4C2×#9912yes—
323C2×C4#51812yes—
324C22—22no—
325C22—22no—
326C22—24no—
327C22—44no—
328C22—42no—
329C22—42no—
330C22—42no—
331C22—42no—
332C22—42no—
333C22—82no—
334C22—88no—
335C22—84no—
336C22—84no—
337C22—84no—
338C22—88no—
339C22—1616no—
340C22#230911yescharacteristic
341C22C22×#9911yescharacteristic
342C22C2×#41011yescharacteristic
343C4—21no—
344C4—21no—
345C4—22no—
346C4—22no—
347C4—42no—
348C4—42no—
349C4—42no—
350C4—42no—
351C4—84no—
352C4—168no—
353C4#212814yes—
354C2—21no—
355C2—21no—
356C2—44no—
357C2—42no—
358C2—164no—
359C2#2179311yescharacteristic
360C2C2×#212812yes—
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×D4C24
prime radical#35272C2
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20723C2semidirect
3. #13223C2semidirect
4. #12697C2semidirect
6. C22×#127C4semidirect
10. #2548C4semidirect
14. #2309C4semidirect
15. #2285C4semidirect
17. C2×#353C22semidirect
20. #1799C22semidirect
23. C4×#99C22semidirect

Maximal subgroups

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

subgroupindex
#352722
#207232
#132232
#126972