Finite Group Databasebeta

← #161207

Subgroups — #161207

Order 512.

1346 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2296C211yesmaximal, characteristic
2#20626C212yesmaximal
3#7641C214yesmaximal, complemented
4#7632C214yesmaximal, complemented
5#7598C214yesmaximal, complemented
6C22×#117C411yescharacteristic
7C22×#117C411yescharacteristic
8C23×C42C2211yescharacteristic
9C2×#423—22no—
10C2×#407—22no—
11#2296—22no—
12#2296—22no—
13#2296C412yes—
14#2296C412yes—
15#2296C2212yes—
16#2296C2212yes—
17C2×#389C412yes—
18C2×#389C2212yes—
19C2×#385—22no—
20C2×C4×C4⋊C4—22no—
21#1797—22no—
22#1797—22no—
23#1797C2214yes—
24#1797C2214yes—
25#1728C2214yes—
26#1271C22116yescomplemented
27C24×C4C2×C411yescharacteristic
28C24×C4D412yes—
29C22×M16—24no—
30C2×#117—21no—
31C2×#117—21no—
32C2×#117—28no—
33C2×#117—21no—
34C2×#117—21no—
35C2×#117—44no—
36C2×#117C2×C412yes—
37C2×#117C2×C412yes—
38C2×#117C2×C412yes—
39C2×#117C2×C412yes—
40C22×C4⋊C4—22no—
41C22×C4⋊C4—22no—
42C22×C4⋊C4C2×C412yes—
43C22×C4⋊C4C2×C412yes—
44C22×C22⋊C4—21no—
45C22×C22⋊C4—21no—
46C22×C42—21no—
47C22×C42—21no—
48C22×C42—44no—
49C22×C42C2×C411yescharacteristic
50C22×C42C2×C411yescharacteristic
51C22×C42C2×C411yescharacteristic
52C22×C42D412yes—
53C22×C42D412yes—
54C22×C42D412yes—
55C22×C42C2311yescharacteristic
56C22×C42C2311yescharacteristic
57C22×C42C2311yescharacteristic
58#423—48no—
59C2×C4⋊C8—24no—
60C2×C4⋊C8—24no—
61#407—48no—
62C2×C22⋊C8—24no—
63C2×C22⋊C8—24no—
64#389—22no—
65#389—22no—
66#389—22no—
67#389—22no—
68#389C2×C414yes—
69#389C2×C414yes—
70#389C2×C414yes—
71#389C2314yes—
72#385—24no—
73#385—24no—
74#385—24no—
75#385—24no—
76C4×C4⋊C4—24no—
77C4×C4⋊C4—24no—
78C4×C4⋊C4—24no—
79C4×C4⋊C4—24no—
80C2×#95—22no—
81C2×#95—22no—
82#336—24no—
83#336C2318yes—
84C25C22⋊C411yescharacteristic
85C23×C4—24no—
86C23×C4—24no—
87C23×C4—24no—
88C23×C4—22no—
89C23×C4—24no—
90C23×C4—24no—
91C23×C4—22no—
92C23×C4—44no—
93C23×C4—44no—
94C23×C4C22⋊C411yescharacteristic
95C23×C4C22⋊C412yes—
96C23×C4C22⋊C412yes—
97C23×C4C22⋊C412yes—
98C23×C4C22⋊C411yescharacteristic
99C23×C4C22⋊C411yescharacteristic
100C23×C4C22×C411yescharacteristic
101C23×C4C22×C411yescharacteristic
102C23×C4C22×C411yescharacteristic
103C23×C4C2×D412yes—
104C23×C4C2×D412yes—
105C23×C4C2×D412yes—
106C2×M16—48no—
107C2×M16—48no—
108C2×M16—48no—
109C22×C8—28no—
110#117—28no—
111#117—28no—
112#117—24no—
113#117—28no—
114#117—44no—
115#117—44no—
116#117—44no—
117#117—44no—
118#117M1618yescomplemented
119C2×C4⋊C4—22no—
120C2×C4⋊C4—28no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—48no—
127C2×C4⋊C4—44no—
128C2×C4⋊C4—44no—
129C2×C4⋊C4—44no—
130C2×C4⋊C4C22×C414yes—
131C2×C4⋊C4C22×C414yes—
132C2×C22⋊C4—28no—
133C2×C22⋊C4—22no—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—22no—
136C2×C22⋊C4—22no—
137C2×C22⋊C4—44no—
138C2×C42—28no—
139C2×C42—28no—
140C2×C42—22no—
141C2×C42—22no—
142C2×C42—22no—
143C2×C42—22no—
144C2×C42—42no—
145C2×C42—42no—
146C2×C42—44no—
147C2×C42—44no—
148C2×C42—42no—
149C2×C42—42no—
150C2×C42—42no—
151C2×C42—42no—
152C2×C42—44no—
153C2×C42C22⋊C412yes—
154C2×C42C22⋊C412yes—
155C2×C42C22⋊C412yes—
156C2×C42C22×C411yescharacteristic
157C2×C42C22×C411yescharacteristic
158C2×C42C22×C411yescharacteristic
159C2×C42C2×D412yes—
160C2×C42C2×D412yes—
161C2×C42C2×D412yes—
162C2×C42C2411yescharacteristic
163C4⋊C8—48no—
164C4⋊C8—48no—
165C22⋊C8—48no—
166C22⋊C8—48no—
167#95—24no—
168#95—24no—
169#95—24no—
170#95—24no—
171C24—22no—
172C24—22no—
173C24—22no—
174C24—24no—
175C24—42no—
176C24C2×C22⋊C411yescharacteristic
177C24C2×C22⋊C411yescharacteristic
178C24C2×C22⋊C411yescharacteristic
179C22×C4—232no—
180C22×C4—22no—
181C22×C4—24no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
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—24no—
199C22×C4—28no—
200C22×C4—24no—
201C22×C4—24no—
202C22×C4—24no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—416no—
218C22×C4—416no—
219C22×C4—44no—
220C22×C4—44no—
221C22×C4—44no—
222C22×C4—44no—
223C22×C4—44no—
224C22×C4#9914yes—
225C22×C4#10114yes—
226C22×C4C2×C22⋊C412yes—
227C22×C4C2×C22⋊C411yescharacteristic
228C22×C4C2×C22⋊C411yescharacteristic
229C22×C4C2×C22⋊C412yes—
230C22×C4C2×C22⋊C412yes—
231C22×C4C2×C22⋊C411yescharacteristic
232C22×C4C2×M1614yes—
233C22×C4AGL(1,ℤ/8)14yes—
234C22×C4#13714yes—
235C22×C4C23×C411yescharacteristic
236C22×C4C22×D412yes—
237M16—816no—
238C2×C8—48no—
239C2×C8—48no—
240C2×C8—48no—
241C4⋊C4—28no—
242C4⋊C4—24no—
243C4⋊C4—44no—
244C4⋊C4—48no—
245C4⋊C4—44no—
246C4⋊C4—44no—
247C4⋊C4—44no—
248C4⋊C4—48no—
249C4⋊C4—48no—
250C4⋊C4C2×M1618yes—
251C22⋊C4—28no—
252C22⋊C4—28no—
253C22⋊C4—44no—
254C22⋊C4—44no—
255C42—28no—
256C42—28no—
257C42—24no—
258C42—24no—
259C42—24no—
260C42—24no—
261C42—42no—
262C42—41no—
263C42—41no—
264C42—41no—
265C42—42no—
266C42—41no—
267C42—44no—
268C42—44no—
269C23—21no—
270C23—21no—
271C23—22no—
272C23—28no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—22no—
278C23—22no—
279C23—22no—
280C23—22no—
281C23—48no—
282C23—48no—
283C23—44no—
284C23—44no—
285C23#40712yes—
286C23C2×#9912yes—
287C23C2×#10112yes—
288C23#41714yes—
289C23C22×C22⋊C411yescharacteristic
290C2×C4—28no—
291C2×C4—28no—
292C2×C4—28no—
293C2×C4—28no—
294C2×C4—28no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—28no—
308C2×C4—28no—
309C2×C4—28no—
310C2×C4—28no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—44no—
315C2×C4—44no—
316C2×C4—48no—
317C2×C4—48no—
318C2×C4—48no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—416no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—88no—
328C2×C4#40712yes—
329C2×C4C2×#9912yes—
330C2×C4C2×#9912yes—
331C2×C4C2×#10112yes—
332C2×C4C2×#10112yes—
333C2×C4#41714yes—
334C2×C4#42114yes—
335C2×C4C22×M1612yes—
336C2×C4C2×AGL(1,ℤ/8)12yes—
337C2×C4C2×#13712yes—
338C8—88no—
339C22—22no—
340C22—24no—
341C22—24no—
342C22—22no—
343C22—24no—
344C22—24no—
345C22—22no—
346C22—22no—
347C22—416no—
348C22—41no—
349C22—41no—
350C22—42no—
351C22—41no—
352C22—41no—
353C22—42no—
354C22#156114yes—
355C22C2×#40711yescharacteristic
356C22C22×#9911yescharacteristic
357C22C22×#10111yescharacteristic
358C22C2×#41712yes—
359C22C2×#42112yes—
360C4—22no—
361C4—22no—
362C4—24no—
363C4—24no—
364C4—22no—
365C4—22no—
366C4—22no—
367C4—48no—
368C4—42no—
369C4—84no—
370C4#146314yes—
371C2—24no—
372C2—42no—
373C2—42no—
374C2C2×#146311yescharacteristic
375C2#1745112yes—
376C2#1759212yes—
377C2#1766411yescharacteristic
378C2C2×#156111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7641C2semidirect
4. #7632C2semidirect
5. #7598C2semidirect
26. #1271C22semidirect
118. #117M16semidirect

Maximal subgroups

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

subgroupindex
C2×#22962
#206262
#76412
#76322
#75982