Finite Group Databasebeta

← #480039

Subgroups — #480039

Order 512.

1082 conjugacy classes of subgroups.

The subgroup lattice has 385 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34052C211yesmaximal, characteristic
2C2×#1458C212yesmaximal
3#11454C214yesmaximal, complemented
4#11453C214yesmaximal
5#11452C214yesmaximal, complemented
6C22×#117C411yescharacteristic
7C22×#117C411yescharacteristic
8C22×#117C2211yescharacteristic
9C2×#431—22no—
10#2317—22no—
11#2317C412yes—
12#2317C2212yes—
13#2316—22no—
14#2315C412yes—
15#2315C2212yes—
16#2296—22no—
17#2296C412yes—
18#2296C2212yes—
19#2285—22no—
20#1458—22no—
21#1458—22no—
22#1458C2214yes—
23#1458C2214yes—
24C2×#324—22no—
25C2×#323C2214yes—
26#1326C22116yes—
27C24×C4C2×C411yescharacteristic
28C2×#117—21no—
29C2×#117—21no—
30C2×#117—24no—
31C2×#117—21no—
32C2×#117—21no—
33C2×#117—21no—
34C2×#117—21no—
35C2×#117—42no—
36C2×#117—42no—
37C2×#117—42no—
38C2×#117—42no—
39C2×#117C818yescomplemented
40C2×#117C2×C412yes—
41C2×#117C2×C412yes—
42C2×#117C2×C411yescharacteristic
43C2×#117C2×C411yescharacteristic
44C2×#117D412yes—
45C2×#117D412yes—
46C2×#117C2311yescharacteristic
47C2×#117C2311yescharacteristic
48C22×C4⋊C4—21no—
49C22×C4⋊C4C2×C412yes—
50C22×C4⋊C4D412yes—
51C22×C22⋊C4C2×C412yes—
52C22×C22⋊C4C2×C412yes—
53C22×C22⋊C4C2×C411yescharacteristic
54C22×C22⋊C4C2311yescharacteristic
55C22×C42—21no—
56C22×C42C2×C412yes—
57C22×C42D412yes—
58#431—24no—
59#431—24no—
60#431—42no—
61#431—42no—
62C2×C4⋊C8—42no—
63C2×C22⋊C8—24no—
64C2×C4×C8—42no—
65#389—24no—
66#389—24no—
67#389—22no—
68#389—22no—
69#389—22no—
70#389—22no—
71#385—24no—
72#385—24no—
73#385C2×C414yes—
74#385C2×C414yes—
75#385C2×C414yes—
76#385C2314yes—
77#384—22no—
78#384—22no—
79#382—22no—
80#382—22no—
81#380—24no—
82#380—24no—
83C4×C4⋊C4—24no—
84C4×C4⋊C4—24no—
85#324—24no—
86#324—24no—
87#323—24no—
88#323C2318yes—
89C25C22×C411yescharacteristic
90C23×C4—22no—
91C23×C4—22no—
92C23×C4—24no—
93C23×C4—22no—
94C23×C4C22⋊C411yescharacteristic
95C23×C4C22⋊C411yescharacteristic
96C23×C4C22⋊C411yescharacteristic
97C23×C4C22⋊C412yes—
98C23×C4C22⋊C411yescharacteristic
99C23×C4C2×C814yes—
100C23×C4M1614yes—
101C23×C4C22×C411yescharacteristic
102C23×C4C22×C411yescharacteristic
103C23×C4C22×C414yes—
104C23×C4C2×D412yes—
105C22×C8—44no—
106#117—28no—
107#117—28no—
108#117—24no—
109#117—24no—
110#117—41no—
111#117—42no—
112#117—41no—
113#117—42no—
114#117—42no—
115#117—42no—
116#117—42no—
117#117—42no—
118#117—41no—
119#117—41no—
120#117—44no—
121#117—44no—
122#117—42no—
123#117—42no—
124#117—42no—
125#117—42no—
126#117C2×C818yescomplemented
127#117C2×C818yescomplemented
128C2×C4⋊C4—28no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—42no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4—44no—
140C2×C4⋊C4C22⋊C412yes—
141C2×C4⋊C4C22⋊C412yes—
142C2×C4⋊C4C2×D412yes—
143C2×C4⋊C4C2×D412yes—
144C2×C22⋊C4—22no—
145C2×C22⋊C4—24no—
146C2×C22⋊C4—22no—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—42no—
149C2×C22⋊C4—42no—
150C2×C22⋊C4—44no—
151C2×C22⋊C4C22⋊C412yes—
152C2×C22⋊C4C2×C818yes—
153C2×C22⋊C4C22×C414yes—
154C2×C22⋊C4C22×C411yescharacteristic
155C2×C22⋊C4C22×C411yescharacteristic
156C2×C22⋊C4C22×C411yescharacteristic
157C2×C22⋊C4C2×D412yes—
158C2×C22⋊C4C2411yescharacteristic
159C2×C42—28no—
160C2×C42—22no—
161C2×C42—22no—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—42no—
169C2×C42—42no—
170C2×C42C22⋊C412yes—
171C2×C42C22⋊C412yes—
172C2×C42C2×D412yes—
173C2×C42C2×D412yes—
174C4⋊C8—44no—
175C4⋊C8—44no—
176C22⋊C8—28no—
177C22⋊C8—44no—
178C4×C8—44no—
179C4×C8—44no—
180#95—24no—
181#95—24no—
182#95—24no—
183#95—24no—
184C24—22no—
185C24—22no—
186C24—82no—
187C24C2×C22⋊C411yescharacteristic
188C24C2×C22⋊C411yescharacteristic
189C24C22×C812yes—
190C24C2×M1612yes—
191C24C23×C411yescharacteristic
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—28no—
195C22×C4—28no—
196C22×C4—28no—
197C22×C4—28no—
198C22×C4—21no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—24no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—28no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—24no—
215C22×C4—44no—
216C22×C4—82no—
217C22×C4—82no—
218C22×C4—84no—
219C22×C4—84no—
220C22×C4C22⋊C814yes—
221C22×C4C22⋊C814yes—
222C22×C4C22⋊C814yes—
223C22×C4C22⋊C814yes—
224C22×C4#9912yes—
225C22×C4#9912yes—
226C22×C4#10114yes—
227C22×C4C2×C22⋊C411yescharacteristic
228C22×C4C2×C22⋊C411yescharacteristic
229C22×C4C2×C22⋊C412yes—
230C22×C4C2×C22⋊C412yes—
231C22×C4C2×C22⋊C412yes—
232C22×C4C2×C22⋊C411yescharacteristic
233C22×C4C2×C22⋊C411yescharacteristic
234C22×C4C22×C812yes—
235C22×C4C22×C812yes—
236C22×C4C2×M1612yes—
237C22×C4C2×M1612yes—
238C22×C4C22×D412yes—
239C2×C8—48no—
240C2×C8—88no—
241C4⋊C4—28no—
242C4⋊C4—28no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—44no—
248C4⋊C4—44no—
249C4⋊C4—84no—
250C4⋊C4—84no—
251C4⋊C4—84no—
252C4⋊C4—84no—
253C22⋊C4—24no—
254C22⋊C4—28no—
255C22⋊C4—42no—
256C22⋊C4—41no—
257C22⋊C4—41no—
258C22⋊C4—41no—
259C22⋊C4—42no—
260C22⋊C4—41no—
261C22⋊C4—44no—
262C22⋊C4—44no—
263C22⋊C4C22×C818yes—
264C42—28no—
265C42—28no—
266C42—22no—
267C42—22no—
268C42—22no—
269C42—22no—
270C42—22no—
271C42—22no—
272C42—22no—
273C42—22no—
274C42—44no—
275C42—42no—
276C42—42no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—24no—
282C23—24no—
283C23—42no—
284C23—42no—
285C23—42no—
286C23—82no—
287C23—82no—
288C23—82no—
289C23—82no—
290C23—84no—
291C23C2×C22⋊C812yes—
292C23C2×C22⋊C812yes—
293C23C2×#9912yes—
294C23C2×#10112yes—
295C23C22×C22⋊C411yescharacteristic
296C23C23×C811yescharacteristic
297C23C22×M1611yescharacteristic
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—28no—
303C2×C4—24no—
304C2×C4—28no—
305C2×C4—24no—
306C2×C4—22no—
307C2×C4—24no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—24no—
311C2×C4—22no—
312C2×C4—28no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—88no—
331C2×C4—88no—
332C2×C4#32418yes—
333C2×C4#32418yes—
334C2×C4C2×C22⋊C812yes—
335C2×C4C2×C22⋊C812yes—
336C2×C4C2×C22⋊C812yes—
337C2×C4C2×C22⋊C812yes—
338C2×C4C2×#9911yescharacteristic
339C2×C4C2×#9912yes—
340C2×C4C2×#9911yescharacteristic
341C2×C4C2×#10112yes—
342C2×C4C2×#10112yes—
343C8—88no—
344C22—22no—
345C22—22no—
346C22—22no—
347C22—21no—
348C22—21no—
349C22—22no—
350C22—42no—
351C22—42no—
352C22—42no—
353C22—41no—
354C22—41no—
355C22—42no—
356C22—42no—
357C22—42no—
358C22—42no—
359C22—84no—
360C22—84no—
361C22C2×#32414yes—
362C22C22×C22⋊C811yescharacteristic
363C22C22×#9911yescharacteristic
364C22C22×#10111yescharacteristic
365C4—22no—
366C4—22no—
367C4—48no—
368C4—42no—
369C4—42no—
370C4—42no—
371C4—42no—
372C4—84no—
373C4C2×#32412yes—
374C4C2×#32414yes—
375C4C2×#32412yes—
376C2—21no—
377C2—21no—
378C2—24no—
379C2—41no—
380C2—41no—
381C2—42no—
382C2C22×#32411yescharacteristic
383C2#3334712yes—
subgroupG/H
centerC2×C4C2×#99
commutatorC23C23×C8
Frattini, cosocleC2×C22⋊C4C24
socleC22C22×#99
prime radicalC25C22×C4
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11454C2semidirect
5. #11452C2semidirect
39. C2×#117C8semidirect
126. #117C2×C8semidirect
127. #117C2×C8semidirect

Maximal subgroups

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

subgroupindex
#340522
C2×#14582
#114542
#114532
#114522