Finite Group Databasebeta

← #213670

Subgroups — #213670

Order 512.

1334 conjugacy classes of subgroups.

The subgroup lattice has 363 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36624C211yesmaximal, characteristic
2#20714C212yesmaximal, complemented
3#9134C214yesmaximal, complemented
4#9128C214yesmaximal, complemented
5#8729C212yesmaximal, complemented
6#8344C212yesmaximal, complemented
7C2×#532C2211yescharacteristic
8C2×#525C412yes—
9C23×C42C2211yescharacteristic
10C2×#431C2214yes—
11#2857C2212yes—
12#2856C2212yes—
13#2854—22no—
14#2850C414yes—
15#2847—22no—
16C4×#121—22no—
17C4×#121—22no—
18#2297C412yes—
19#2285C2211yescharacteristic
20#1817—22no—
21#1797C2214yes—
22#1797C2214yes—
23#1779—22no—
24C4×C22⋊C8—22no—
25C4×C22⋊C8—22no—
26C2×#333C2214yescomplemented
27C2×#331C2214yescomplemented
28#1296C2218yescomplemented
29C22×#41C2×C412yes—
30C24×C4C2×C411yescharacteristic
31C24×C4D412yes—
32#532—44no—
33C2×C4⋊Q8C2311yescharacteristic
34C2×#127C2311yescharacteristic
35C2×#124C2×C412yes—
36#525—48no—
37C2×#123—22no—
38C2×C22⋊Q8C2×C414yes—
39C2×#121—22no—
40C2×#121—22no—
41C2×#121—22no—
42C2×#121C2×C414yes—
43C2×#120—22no—
44C2×C4×D4—24no—
45C2×C4×D4—22no—
46C2×#117C2×C412yes—
47C22×C42—44no—
48C22×C42C2×C412yes—
49C22×C42C2×C411yescharacteristic
50C22×C42D412yes—
51C22×C42D412yes—
52C22×C42D412yes—
53C22×C42C2311yescharacteristic
54C22×C42C2312yes—
55#431—48no—
56C2×C4⋊C8C2318yes—
57C2×C22⋊C8—24no—
58C2×C22⋊C8—24no—
59C2×C22⋊C8—24no—
60C2×C4×C8—24no—
61#391C2×C414yes—
62#390—22no—
63#390—22no—
64#390C2×C414yes—
65#389—22no—
66#389—22no—
67#388—24no—
68#388—24no—
69#386—24no—
70#386—24no—
71#386—24no—
72#384C2312yes—
73#382C2×C412yes—
74#380—22no—
75#380—22no—
76#380—22no—
77C4×C4⋊C4—22no—
78C4×C22⋊C4—22no—
79C4×C22⋊C4—22no—
80#336—24no—
81#336—24no—
82#333—44no—
83#331—44no—
84C25C22⋊C411yescharacteristic
85C2×#41—44no—
86C2×#41—44no—
87C2×#41—44no—
88C22×Q8C22×C412yes—
89C22×D4—24no—
90C22×D4—22no—
91C22×D4—22no—
92C22×D4C22×C412yes—
93C23×C4—24no—
94C23×C4—28no—
95C23×C4—22no—
96C23×C4—42no—
97C23×C4—42no—
98C23×C4—44no—
99C23×C4—44no—
100C23×C4C22⋊C412yes—
101C23×C4C22⋊C412yes—
102C23×C4C22⋊C412yes—
103C23×C4C22⋊C412yes—
104C23×C4C22⋊C411yescharacteristic
105C23×C4C22×C411yescharacteristic
106C23×C4C22×C412yes—
107C23×C4C2×D412yes—
108C23×C4C2×D412yes—
109C23×C4C2×D412yes—
110C22×C8—28no—
111C4⋊Q8—24no—
112#127—24no—
113#124—44no—
114#123—48no—
115C22⋊Q8—48no—
116#121—48no—
117#121—48no—
118#121—48no—
119#121—48no—
120#120—48no—
121C4×D4—48no—
122C4×D4—416no—
123#117—44no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4C22×C414yes—
129C2×C22⋊C4—22no—
130C2×C22⋊C4—24no—
131C2×C22⋊C4—24no—
132C2×C22⋊C4—24no—
133C2×C22⋊C4—24no—
134C2×C22⋊C4—24no—
135C2×C42—24no—
136C2×C42—28no—
137C2×C42—22no—
138C2×C42—42no—
139C2×C42—44no—
140C2×C42—44no—
141C2×C42—44no—
142C2×C42—44no—
143C2×C42C22⋊C411yescharacteristic
144C2×C42C22⋊C411yescharacteristic
145C2×C42C22⋊C412yes—
146C2×C42C22⋊C412yes—
147C2×C42C22×C411yescharacteristic
148C2×C42C22×C412yes—
149C2×C42C2×D412yes—
150C2×C42C2×D412yes—
151C2×C42C2×D412yes—
152C2×C42C2411yescharacteristic
153C4⋊C8—28no—
154C4⋊C8—28no—
155C22⋊C8—48no—
156C22⋊C8—48no—
157C22⋊C8—48no—
158C4×C8—48no—
159#95—24no—
160#95—24no—
161#95—24no—
162C24—22no—
163C24—24no—
164C24—22no—
165C24—41no—
166C24—41no—
167C24—42no—
168C24C2×C22⋊C411yescharacteristic
169C24C2×C22⋊C412yes—
170#41—816no—
171C2×Q8—24no—
172C2×Q8—42no—
173C2×Q8—42no—
174C2×D4—24no—
175C2×D4—42no—
176C2×D4—42no—
177C2×D4—48no—
178C2×D4—44no—
179C2×D4—48no—
180C2×D4—44no—
181C2×D4—44no—
182C2×D4—48no—
183C2×D4—44no—
184C2×D4—44no—
185C2×D4—44no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—24no—
189C22×C4—24no—
190C22×C4—22no—
191C22×C4—24no—
192C22×C4—24no—
193C22×C4—24no—
194C22×C4—24no—
195C22×C4—24no—
196C22×C4—24no—
197C22×C4—24no—
198C22×C4—24no—
199C22×C4—24no—
200C22×C4—24no—
201C22×C4—24no—
202C22×C4—24no—
203C22×C4—24no—
204C22×C4—24no—
205C22×C4—22no—
206C22×C4—42no—
207C22×C4—41no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—41no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—416no—
217C22×C4—432no—
218C22×C4—44no—
219C22×C4—44no—
220C22×C4—44no—
221C22×C4#10014yes—
222C22×C4C2×C22⋊C412yes—
223C22×C4C2×C22⋊C412yes—
224C22×C4C2×C22⋊C412yes—
225C22×C4C2×C22⋊C411yescharacteristic
226C22×C4C2×C22⋊C412yes—
227C22×C4#13518yes—
228C22×C4C23×C411yescharacteristic
229C22×C4C22×D412yes—
230C2×C8—48no—
231C2×C8—48no—
232C2×C8—48no—
233C4⋊C4—44no—
234C4⋊C4—44no—
235C4⋊C4—44no—
236C4⋊C4—44no—
237C4⋊C4—48no—
238C22⋊C4—48no—
239C22⋊C4—48no—
240C22⋊C4—48no—
241C22⋊C4—44no—
242C22⋊C4—48no—
243C22⋊C4—48no—
244C42—24no—
245C42—22no—
246C42—24no—
247C42—24no—
248C42—24no—
249C42—41no—
250C42—41no—
251C42—42no—
252C42—42no—
253C42—44no—
254C42H(5,𝔽2)14yes—
255C23—21no—
256C23—22no—
257C23—22no—
258C23—21no—
259C23—22no—
260C23—22no—
261C23—22no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—44no—
266C23—48no—
267C23—48no—
268C23—44no—
269C23—44no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23—44no—
274C23—44no—
275C23C2×#10012yes—
276C23#41614yes—
277C23C22×C22⋊C411yescharacteristic
278Q8—84no—
279D4—84no—
280D4—816no—
281D4—88no—
282D4—88no—
283C2×C4—28no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—216no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—44no—
300C2×C4—48no—
301C2×C4—48no—
302C2×C4—48no—
303C2×C4—48no—
304C2×C4—48no—
305C2×C4—48no—
306C2×C4—48no—
307C2×C4—42no—
308C2×C4—44no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—88no—
314C2×C4C2×#10014yes—
315C2×C4#41312yes—
316C2×C4#41614yes—
317C2×C4#51812yes—
318C2×C4#53414yes—
319C2×C4C2×#13514yes—
320C2×C4#57514yes—
321C8—88no—
322C22—24no—
323C22—24no—
324C22—24no—
325C22—24no—
326C22—22no—
327C22—416no—
328C22—42no—
329C22—42no—
330C22—41no—
331C22—42no—
332C22—41no—
333C22—42no—
334C22—42no—
335C22—84no—
336C22—82no—
337C22—84no—
338C22—82no—
339C22—82no—
340C22#230811yescharacteristic
341C22C22×#10011yescharacteristic
342C22C2×#41311yescharacteristic
343C22C2×#41612yes—
344C22#289612yes—
345C4—24no—
346C4—22no—
347C4—22no—
348C4—24no—
349C4—42no—
350C4—42no—
351C4—84no—
352C4—82no—
353C4#319118yes—
354C2—44no—
355C2—42no—
356C2—162no—
357C2#1770011yescharacteristic
358C2#1771412yes—
359C2#1775711yescharacteristic
360C2#2175211yescharacteristic
361C2#2176012yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36624C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20714C2semidirect
3. #9134C2semidirect
4. #9128C2semidirect
5. #8729C2semidirect
6. #8344C2semidirect
26. C2×#333C22semidirect
27. C2×#331C22semidirect
28. #1296C22semidirect

Maximal subgroups

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

subgroupindex
#366242
#207142
#91342
#91282
#87292
#83442