Finite Group Databasebeta

← #480291

Subgroups — #480291

Order 512.

1140 conjugacy classes of subgroups.

The subgroup lattice has 379 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2317C211yesmaximal, characteristic
2#17357C212yesmaximal
3#11516C214yesmaximal, complemented
4#11512C214yesmaximal, complemented
5#11450C214yesmaximal, complemented
6C22×#117C411yescharacteristic
7C22×#117C2211yescharacteristic
8C23×C4⋊C4C411yescharacteristic
9#2920—22no—
10#2317—22no—
11#2317—22no—
12#2317C412yes—
13#2317C412yes—
14#2317C2212yes—
15#2317C2212yes—
16C2×#389—22no—
17C2×#385—22no—
18C2×#384C2212yes—
19C2×#382C412yes—
20#1464—22no—
21#1464C2214yes—
22#1463C2214yes—
23#1462—22no—
24#1458—22no—
25#1458C2214yes—
26#1325C22116yescomplemented
27C24×C4—21no—
28C24×C4C2×C411yescharacteristic
29C2×#117—21no—
30C2×#117—21no—
31C2×#117—28no—
32C2×#117—42no—
33C2×#117—42no—
34C2×#117C2×C411yescharacteristic
35C2×#117C2×C411yescharacteristic
36C2×#117C2×C412yes—
37C2×#117C2×C412yes—
38C2×#117D412yes—
39C2×#117D412yes—
40C2×#117C2311yescharacteristic
41C2×#117C2311yescharacteristic
42C22×C4⋊C4—21no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—24no—
47C22×C4⋊C4—42no—
48C22×C4⋊C4—42no—
49C22×C4⋊C4C2×C412yes—
50C22×C4⋊C4C2×C412yes—
51C22×C4⋊C4C2×C412yes—
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C22⋊C4—21no—
55C22×C22⋊C4D412yes—
56C22×C42C2×C412yes—
57C22×C42D412yes—
58#432—24no—
59#432—42no—
60#431—24no—
61#431—42no—
62#423—42no—
63#407—24no—
64C4×M16—42no—
65#389—24no—
66#389—24no—
67#389—24no—
68#389—24no—
69#385—24no—
70#385—24no—
71#385—24no—
72#385—24no—
73#384—24no—
74#384C2×C414yes—
75#384C2314yes—
76#382—24no—
77#382C2×C414yes—
78#382C2×C414yes—
79C2×#95—24no—
80#324—24no—
81#324C2318yes—
82#323—24no—
83#323—24no—
84C25C22⋊C411yescharacteristic
85C23×C4—22no—
86C23×C4—22no—
87C23×C4—24no—
88C23×C4—24no—
89C23×C4—22no—
90C23×C4—22no—
91C23×C4—22no—
92C23×C4—22no—
93C23×C4—42no—
94C23×C4—42no—
95C23×C4—84no—
96C23×C4C22⋊C411yescharacteristic
97C23×C4C22⋊C411yescharacteristic
98C23×C4C22⋊C412yes—
99C23×C4C22⋊C411yescharacteristic
100C23×C4C22×C411yescharacteristic
101C23×C4C22×C411yescharacteristic
102C23×C4C22×C411yescharacteristic
103C23×C4C2×D412yes—
104C2×M16—44no—
105#117—28no—
106#117—28no—
107#117—24no—
108#117—28no—
109#117—42no—
110#117—42no—
111#117—42no—
112#117—42no—
113#117—42no—
114#117—42no—
115#117—42no—
116#117—42no—
117#117M1618yescomplemented
118C2×C4⋊C4—28no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—28no—
125C2×C4⋊C4—28no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—44no—
133C2×C4⋊C4—44no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4—44no—
137C2×C4⋊C4—44no—
138C2×C4⋊C4—88no—
139C2×C4⋊C4C22⋊C412yes—
140C2×C4⋊C4C22×C414yes—
141C2×C4⋊C4C22×C411yescharacteristic
142C2×C4⋊C4C22×C411yescharacteristic
143C2×C4⋊C4C22×C411yescharacteristic
144C2×C4⋊C4C2×D412yes—
145C2×C4⋊C4C2411yescharacteristic
146C2×C22⋊C4—28no—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—42no—
150C2×C22⋊C4—42no—
151C2×C22⋊C4C22⋊C412yes—
152C2×C22⋊C4C22⋊C412yes—
153C2×C22⋊C4C2×D412yes—
154C2×C22⋊C4C2×D412yes—
155C2×C42—22no—
156C2×C42—22no—
157C2×C42—22no—
158C2×C42—22no—
159C2×C42—22no—
160C2×C42—28no—
161C2×C42C22⋊C412yes—
162C2×C42C22⋊C412yes—
163C2×C42C22×C414yes—
164C2×C42C2×D412yes—
165C2×C42C2×D412yes—
166C4⋊C8—44no—
167C4⋊C8—44no—
168C22⋊C8—28no—
169C22⋊C8—44no—
170#97—44no—
171C4×C8—44no—
172#95—28no—
173#95—28no—
174C24—22no—
175C24—21no—
176C24—21no—
177C24—22no—
178C24—42no—
179C24—42no—
180C24C2×C22⋊C411yescharacteristic
181C24C2×C22⋊C411yescharacteristic
182C24C2×C22⋊C411yescharacteristic
183C22×C4—216no—
184C22×C4—216no—
185C22×C4—22no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
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—28no—
202C22×C4—24no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—24no—
208C22×C4—44no—
209C22×C4—44no—
210C22×C4—44no—
211C22×C4—44no—
212C22×C4—44no—
213C22×C4—44no—
214C22×C4—48no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—84no—
218C22×C4—84no—
219C22×C4—88no—
220C22×C4—88no—
221C22×C4#9912yes—
222C22×C4#9912yes—
223C22×C4#10114yes—
224C22×C4C2×C22⋊C411yescharacteristic
225C22×C4C2×C22⋊C411yescharacteristic
226C22×C4C2×C22⋊C412yes—
227C22×C4C2×C22⋊C412yes—
228C22×C4C2×C22⋊C412yes—
229C22×C4C2×C22⋊C411yescharacteristic
230C22×C4C2×M1614yes—
231C22×C4C23×C411yescharacteristic
232C22×C4C22×D412yes—
233M16—88no—
234C2×C8—48no—
235C4⋊C4—28no—
236C4⋊C4—28no—
237C4⋊C4—42no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—42no—
242C4⋊C4—41no—
243C4⋊C4—44no—
244C4⋊C4—44no—
245C4⋊C4—44no—
246C4⋊C4—44no—
247C4⋊C4—88no—
248C4⋊C4—88no—
249C22⋊C4—28no—
250C22⋊C4—28no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C42—28no—
256C42—24no—
257C42—22no—
258C42—22no—
259C42—22no—
260C42—22no—
261C42—22no—
262C42—22no—
263C42—22no—
264C42—22no—
265C42C2×M1618yes—
266C23—21no—
267C23—21no—
268C23—22no—
269C23—22no—
270C23—24no—
271C23—24no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—42no—
281C23—48no—
282C23—48no—
283C23—42no—
284C23—42no—
285C23—42no—
286C23#40712yes—
287C23C2×#9911yescharacteristic
288C23C2×#9911yescharacteristic
289C23C2×#10112yes—
290C23C22×C22⋊C411yescharacteristic
291C2×C4—28no—
292C2×C4—24no—
293C2×C4—24no—
294C2×C4—28no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—24no—
302C2×C4—24no—
303C2×C4—28no—
304C2×C4—28no—
305C2×C4—28no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—44no—
311C2×C4—416no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—48no—
320C2×C4—44no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—84no—
324C2×C4—88no—
325C2×C4—88no—
326C2×C4—84no—
327C2×C4#40712yes—
328C2×C4C2×#9912yes—
329C2×C4C2×#9912yes—
330C2×C4C2×#10112yes—
331C2×C4C2×#10112yes—
332C2×C4C22×M1612yes—
333C8—88no—
334C22—24no—
335C22—24no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—22no—
343C22—22no—
344C22—42no—
345C22—42no—
346C22—42no—
347C22—42no—
348C22—48no—
349C22—48no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22#212812yes—
355C22#213512yes—
356C22C2×#40711yescharacteristic
357C22C22×#9911yescharacteristic
358C22C22×#10111yescharacteristic
359C4—22no—
360C4—22no—
361C4—22no—
362C4—48no—
363C4—42no—
364C4—42no—
365C4—42no—
366C4—42no—
367C4—84no—
368C4#146314yes—
369C2—21no—
370C2—21no—
371C2—22no—
372C2—22no—
373C2—42no—
374C2—42no—
375C2C2×#146311yescharacteristic
376C2C2×#212811yescharacteristic
377C2C2×#213511yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11516C2semidirect
4. #11512C2semidirect
5. #11450C2semidirect
26. #1325C22semidirect
117. #117M16semidirect

Maximal subgroups

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

subgroupindex
C2×#23172
#173572
#115162
#115122
#114502