Finite Group Databasebeta

← #480295

Subgroups — #480295

Order 512.

1228 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2296C211yesmaximal, characteristic
2#17357C212yesmaximal
3#11518C214yesmaximal, complemented
4#11517C214yesmaximal
5#11452C214yesmaximal, complemented
6C22×#117C411yescharacteristic
7C22×#117C2211yescharacteristic
8C23×C42C411yescharacteristic
9#2920—22no—
10#2296—22no—
11#2296—22no—
12#2296C412yes—
13#2296C412yes—
14#2296C2212yes—
15#2296C2212yes—
16C2×#389—22no—
17C2×#385C412yes—
18C2×#385C2212yes—
19C2×C4×C4⋊C4—22no—
20#1464—22no—
21#1464C2214yes—
22#1463—22no—
23#1462C2214yes—
24#1458—22no—
25#1458C2214yes—
26#1326C22116yes—
27C24×C4C2×C411yescharacteristic
28C24×C4C2×C412yes—
29C2×#117—21no—
30C2×#117—21no—
31C2×#117—28no—
32C2×#117—42no—
33C2×#117—42no—
34C2×#117C2×C412yes—
35C2×#117C2×C412yes—
36C2×#117C2×C411yescharacteristic
37C2×#117C2×C411yescharacteristic
38C2×#117D412yes—
39C2×#117D412yes—
40C2×#117C2311yescharacteristic
41C2×#117C2311yescharacteristic
42C22×C4⋊C4—21no—
43C22×C4⋊C4—22no—
44C22×C4⋊C4—22no—
45C22×C4⋊C4D412yes—
46C22×C22⋊C4C2×C412yes—
47C22×C22⋊C4C2×C411yescharacteristic
48C22×C22⋊C4C2311yescharacteristic
49C22×C42—21no—
50C22×C42—21no—
51C22×C42—21no—
52C22×C42—21no—
53C22×C42—21no—
54C22×C42—42no—
55C22×C42—42no—
56C22×C42C2×C412yes—
57C22×C42C2×C412yes—
58C22×C42D412yes—
59#432—24no—
60#432—42no—
61#431—24no—
62#431—42no—
63#423—42no—
64#407—24no—
65C4×M16—42no—
66#389—24no—
67#389—24no—
68#389—24no—
69#389—24no—
70#385—22no—
71#385—22no—
72#385—22no—
73#385—22no—
74#385C2×C414yes—
75#385C2×C414yes—
76#385C2×C414yes—
77#385C2314yes—
78C4×C4⋊C4—24no—
79C4×C4⋊C4—24no—
80C4×C4⋊C4—24no—
81C4×C4⋊C4—24no—
82C2×#95—22no—
83C2×#95—22no—
84#324—24no—
85#324—24no—
86#323—24no—
87#323C2318yes—
88C25C22×C411yescharacteristic
89C23×C4—22no—
90C23×C4—22no—
91C23×C4—24no—
92C23×C4—24no—
93C23×C4—22no—
94C23×C4—22no—
95C23×C4—22no—
96C23×C4—48no—
97C23×C4—42no—
98C23×C4—42no—
99C23×C4C22⋊C411yescharacteristic
100C23×C4C22⋊C411yescharacteristic
101C23×C4C22⋊C411yescharacteristic
102C23×C4C22⋊C412yes—
103C23×C4C22⋊C411yescharacteristic
104C23×C4C22×C411yescharacteristic
105C23×C4C22×C411yescharacteristic
106C23×C4C22×C414yes—
107C23×C4C2×D412yes—
108C2×M16—44no—
109#117—28no—
110#117—28no—
111#117—24no—
112#117—28no—
113#117—42no—
114#117—41no—
115#117—41no—
116#117—42no—
117#117—42no—
118#117—42no—
119#117—42no—
120#117—42no—
121#117—41no—
122#117—41no—
123#117M1618yescomplemented
124C2×C4⋊C4—28no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—48no—
134C2×C4⋊C4—48no—
135C2×C4⋊C4C22⋊C412yes—
136C2×C4⋊C4C22⋊C412yes—
137C2×C4⋊C4C2×D412yes—
138C2×C4⋊C4C2×D412yes—
139C2×C22⋊C4—22no—
140C2×C22⋊C4—28no—
141C2×C22⋊C4—42no—
142C2×C22⋊C4—42no—
143C2×C22⋊C4C22⋊C412yes—
144C2×C22⋊C4C22×C414yes—
145C2×C22⋊C4C22×C411yescharacteristic
146C2×C22⋊C4C22×C411yescharacteristic
147C2×C22⋊C4C22×C411yescharacteristic
148C2×C22⋊C4C2×D412yes—
149C2×C22⋊C4C2411yescharacteristic
150C2×C42—28no—
151C2×C42—22no—
152C2×C42—22no—
153C2×C42—22no—
154C2×C42—22no—
155C2×C42—22no—
156C2×C42—22no—
157C2×C42—22no—
158C2×C42—22no—
159C2×C42—22no—
160C2×C42—22no—
161C2×C42—44no—
162C2×C42—44no—
163C2×C42—42no—
164C2×C42—42no—
165C2×C42—42no—
166C2×C42—42no—
167C2×C42—44no—
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—
178#97—44no—
179C4×C8—44no—
180#95—24no—
181#95—24no—
182#95—24no—
183#95—24no—
184C24—22no—
185C24—22no—
186C24—22no—
187C24—44no—
188C24C2×C22⋊C411yescharacteristic
189C24C2×C22⋊C411yescharacteristic
190C24C23×C411yescharacteristic
191C22×C4—216no—
192C22×C4—216no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—22no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—28no—
211C22×C4—28no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—48no—
219C22×C4—48no—
220C22×C4—44no—
221C22×C4—44no—
222C22×C4—44no—
223C22×C4—44no—
224C22×C4—416no—
225C22×C4—48no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—44no—
229C22×C4—48no—
230C22×C4—44no—
231C22×C4—44no—
232C22×C4#9912yes—
233C22×C4#9912yes—
234C22×C4#10114yes—
235C22×C4C2×C22⋊C411yescharacteristic
236C22×C4C2×C22⋊C411yescharacteristic
237C22×C4C2×C22⋊C411yescharacteristic
238C22×C4C2×C22⋊C412yes—
239C22×C4C2×C22⋊C412yes—
240C22×C4C2×C22⋊C412yes—
241C22×C4C2×C22⋊C411yescharacteristic
242C22×C4C2×M1614yes—
243C22×C4C22×D412yes—
244M16—88no—
245C2×C8—48no—
246C4⋊C4—28no—
247C4⋊C4—28no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—48no—
253C4⋊C4—48no—
254C4⋊C4—48no—
255C4⋊C4—48no—
256C22⋊C4—28no—
257C22⋊C4—24no—
258C22⋊C4—42no—
259C22⋊C4—41no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—42no—
263C22⋊C4—41no—
264C22⋊C4C2×M1618yes—
265C42—28no—
266C42—28no—
267C42—22no—
268C42—22no—
269C42—22no—
270C42—22no—
271C42—22no—
272C42—22no—
273C42—22no—
274C42—22no—
275C42—44no—
276C42—44no—
277C42—44no—
278C42—42no—
279C42—42no—
280C23—21no—
281C23—21no—
282C23—24no—
283C23—28no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—42no—
289C23—42no—
290C23—44no—
291C23—48no—
292C23—48no—
293C23—44no—
294C23C2×#9912yes—
295C23C2×#10112yes—
296C23C22×C22⋊C411yescharacteristic
297C23C22×M1612yes—
298C2×C4—28no—
299C2×C4—28no—
300C2×C4—28no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—28no—
314C2×C4—28no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—416no—
322C2×C4—44no—
323C2×C4—48no—
324C2×C4—416no—
325C2×C4—48no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—48no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—48no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4#40712yes—
338C2×C4#40712yes—
339C2×C4C2×#9912yes—
340C2×C4C2×#9911yescharacteristic
341C2×C4C2×#9911yescharacteristic
342C2×C4C2×#10112yes—
343C2×C4C2×#10112yes—
344C8—88no—
345C22—24no—
346C22—24no—
347C22—22no—
348C22—22no—
349C22—22no—
350C22—22no—
351C22—22no—
352C22—42no—
353C22—42no—
354C22—42no—
355C22—42no—
356C22—416no—
357C22—42no—
358C22—42no—
359C22#212014yes—
360C22C2×#40711yescharacteristic
361C22C22×#9911yescharacteristic
362C22C22×#10111yescharacteristic
363C4—22no—
364C4—22no—
365C4—22no—
366C4—48no—
367C4—42no—
368C4—48no—
369C4—42no—
370C4—42no—
371C4—42no—
372C4#146312yes—
373C4#146312yes—
374C2—21no—
375C2—21no—
376C2—24no—
377C2—42no—
378C2—42no—
379C2C2×#146311yescharacteristic
380C2C2×#212012yes—
subgroupG/H
centerC2×C4C2×#99
commutatorC24C23×C4
Frattini, cosocleC2×C22⋊C4C24
socleC22C22×#99
prime radicalC25C22×C4
LCSG ▷ C24 ▷ 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. #11518C2semidirect
5. #11452C2semidirect
123. #117M16semidirect

Maximal subgroups

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

subgroupindex
C2×#22962
#173572
#115182
#115172
#114522