Finite Group Databasebeta

← #190024

Subgroups — #190024

Order 512.

1198 conjugacy classes of subgroups.

The subgroup lattice has 371 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33996C211yesmaximal, characteristic
2C2×#1464C212yesmaximal
3#8735C214yesmaximal, complemented
4#8734C214yesmaximal
5#7930C214yesmaximal, complemented
6C22×#117C411yescharacteristic
7C22×#117C411yescharacteristic
8C22×#117C2211yescharacteristic
9C2×#432C2214yes—
10#2274—22no—
11#2274—22no—
12#2274C412yes—
13#2274C2212yes—
14#2273C412yes—
15#2273C2212yes—
16C4×#117—22no—
17C4×#117—22no—
18C4×#117C412yes—
19C4×#117C2212yes—
20#1464—22no—
21#1464—22no—
22#1464—22no—
23#1464—22no—
24C2×#324C2214yes—
25C2×#323C2214yes—
26#1282C22116yes—
27C24×C4C2×C411yescharacteristic
28C2×#117—21no—
29C2×#117—21no—
30C2×#117—21no—
31C2×#117—21no—
32C2×#117—28no—
33C2×#117—28no—
34C2×#117—24no—
35C2×#117—24no—
36C2×#117C2×C412yes—
37C2×#117C2×C412yes—
38C2×#117C2×C412yes—
39C2×#117C2×C412yes—
40C2×#117D412yes—
41C2×#117D412yes—
42C2×#117D412yes—
43C2×#117D412yes—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C4⋊C4C2311yescharacteristic
48C22×C22⋊C4C2×C412yes—
49C22×C22⋊C4C2×C411yescharacteristic
50C22×C22⋊C4C2×C412yes—
51C22×C22⋊C4C2311yescharacteristic
52C22×C42—21no—
53C22×C42—21no—
54C22×C42C2×C411yescharacteristic
55C22×C42C2311yescharacteristic
56#432—24no—
57#432—24no—
58#432—24no—
59#432—24no—
60C2×C4⋊C8—24no—
61C2×C22⋊C8C2318yes—
62C2×#97—24no—
63C4×C4⋊C4—24no—
64C4×C4⋊C4—24no—
65C4×C4⋊C4—24no—
66C4×C4⋊C4—24no—
67C4×C4⋊C4—22no—
68C4×C4⋊C4—22no—
69C4×C4⋊C4—22no—
70C4×C4⋊C4—22no—
71C4×C22⋊C4—24no—
72C4×C22⋊C4—24no—
73C4×C22⋊C4—24no—
74C4×C22⋊C4—24no—
75C4×C22⋊C4C2×C414yes—
76C4×C22⋊C4C2×C414yes—
77C4×C22⋊C4C2×C414yes—
78C4×C22⋊C4C2314yes—
79#376—22no—
80#376—22no—
81C43—22no—
82C43—22no—
83#324—24no—
84#324—24no—
85#323—24no—
86#323—24no—
87C25C22×C411yescharacteristic
88C23×C4—24no—
89C23×C4—22no—
90C23×C4—22no—
91C23×C4—24no—
92C23×C4—22no—
93C23×C4—22no—
94C23×C4C22⋊C411yescharacteristic
95C23×C4C22⋊C411yescharacteristic
96C23×C4C22⋊C411yescharacteristic
97C23×C4C22⋊C411yescharacteristic
98C23×C4C22×C411yescharacteristic
99C23×C4C22×C411yescharacteristic
100C23×C4C22×C411yescharacteristic
101C23×C4C22×C411yescharacteristic
102C23×C4C22×C411yescharacteristic
103C23×C4C2411yescharacteristic
104C22×C8—28no—
105#117—28no—
106#117—28no—
107#117—28no—
108#117—28no—
109#117—24no—
110#117—28no—
111#117—28no—
112#117—28no—
113#117—24no—
114#117—24no—
115#117—24no—
116#117—24no—
117#117—24no—
118#117—24no—
119#117—24no—
120#117—24no—
121#117M1618yes—
122C2×C4⋊C4—28no—
123C2×C4⋊C4—28no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4C22⋊C412yes—
131C2×C4⋊C4C22⋊C412yes—
132C2×C4⋊C4C2×D412yes—
133C2×C4⋊C4C2×D412yes—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—22no—
136C2×C22⋊C4—24no—
137C2×C22⋊C4—24no—
138C2×C22⋊C4—24no—
139C2×C22⋊C4—28no—
140C2×C22⋊C4C22⋊C412yes—
141C2×C22⋊C4C22⋊C412yes—
142C2×C22⋊C4M1618yes—
143C2×C22⋊C4C22×C414yes—
144C2×C22⋊C4C22×C414yes—
145C2×C22⋊C4C2×D412yes—
146C2×C22⋊C4C2×D412yes—
147C2×C42—28no—
148C2×C42—28no—
149C2×C42—22no—
150C2×C42—22no—
151C2×C42—22no—
152C2×C42—22no—
153C2×C42—22no—
154C2×C42—22no—
155C2×C42—24no—
156C2×C42—24no—
157C2×C42—24no—
158C2×C42—24no—
159C2×C42C22⋊C412yes—
160C2×C42C22⋊C412yes—
161C2×C42C2×D412yes—
162C2×C42C2×D412yes—
163C2×C42#4114yes—
164C2×C42#4114yes—
165C4⋊C8—44no—
166C4⋊C8—44no—
167C22⋊C8—28no—
168C22⋊C8—28no—
169#97—44no—
170#97—44no—
171#95—24no—
172#95—24no—
173#95—24no—
174#95—24no—
175C24—22no—
176C24—22no—
177C24—81no—
178C24—81no—
179C24C2×C22⋊C411yescharacteristic
180C24C2×C22⋊C411yescharacteristic
181C24C2×M1614yes—
182C24C23×C411yescharacteristic
183C22×C4—216no—
184C22×C4—216no—
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—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—24no—
204C22×C4—24no—
205C22×C4—28no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—28no—
209C22×C4—81no—
210C22×C4—81no—
211C22×C4—82no—
212C22×C4—82no—
213C22×C4—82no—
214C22×C4—82no—
215C22×C4—81no—
216C22×C4—81no—
217C22×C4C2×C22⋊C411yescharacteristic
218C22×C4C2×C22⋊C411yescharacteristic
219C22×C4C2×C22⋊C411yescharacteristic
220C22×C4C2×C22⋊C411yescharacteristic
221C22×C4C2×C22⋊C412yes—
222C22×C4C2×C22⋊C412yes—
223C22×C4C2×C22⋊C412yes—
224C22×C4#11712yes—
225C22×C4#11712yes—
226C22×C4#11714yes—
227C22×C4C2×M1614yes—
228C22×C4C22×D412yes—
229C22×C4C2×#4114yes—
230C2×C8—48no—
231C2×C8—88no—
232C4⋊C4—28no—
233C4⋊C4—28no—
234C4⋊C4—28no—
235C4⋊C4—24no—
236C4⋊C4—24no—
237C4⋊C4—24no—
238C4⋊C4—24no—
239C4⋊C4—28no—
240C22⋊C4—28no—
241C22⋊C4—24no—
242C22⋊C4—24no—
243C22⋊C4—28no—
244C22⋊C4—24no—
245C22⋊C4—24no—
246C22⋊C4—24no—
247C22⋊C4—24no—
248C22⋊C4C2×M1618yes—
249C22⋊C4C2×M1618yes—
250C42—28no—
251C42—28no—
252C42—28no—
253C42—22no—
254C42—22no—
255C42—22no—
256C42—22no—
257C42—28no—
258C42—84no—
259C42—84no—
260C42—84no—
261C42—84no—
262C42#12314yes—
263C42#12314yes—
264C42#12314yes—
265C42#12314yes—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—28no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—82no—
278C23—82no—
279C23—84no—
280C23—84no—
281C23#40712yes—
282C23#40712yes—
283C23C22×C22⋊C411yescharacteristic
284C23C2×#11712yes—
285C23C22×M1612yes—
286C23C22×M1612yes—
287C2×C4—24no—
288C2×C4—24no—
289C2×C4—28no—
290C2×C4—24no—
291C2×C4—24no—
292C2×C4—28no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—28no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—28no—
305C2×C4—28no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—44no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—88no—
321C2×C4—88no—
322C2×C4#38111yescharacteristic
323C2×C4#38112yes—
324C2×C4#38111yescharacteristic
325C2×C4#38114yes—
326C2×C4#40712yes—
327C2×C4#40712yes—
328C2×C4C2×#12314yes—
329C8—88no—
330C22—21no—
331C22—22no—
332C22—21no—
333C22—22no—
334C22—22no—
335C22—22no—
336C22—42no—
337C22—42no—
338C22—42no—
339C22—42no—
340C22—42no—
341C22—42no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—88no—
347C22C2×#38111yescharacteristic
348C22C2×#40711yescharacteristic
349C22C2×#40711yescharacteristic
350C22#292014yes—
351C4—21no—
352C4—21no—
353C4—22no—
354C4—22no—
355C4—22no—
356C4—24no—
357C4—24no—
358C4—48no—
359C4—48no—
360C4—42no—
361C4—42no—
362C2—21no—
363C2—21no—
364C2—24no—
365C2—42no—
366C2—42no—
367C2#2068211yescharacteristic
368C2#2071011yescharacteristic
369C2#2071311yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8735C2semidirect
5. #7930C2semidirect

Maximal subgroups

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

subgroupindex
#339962
C2×#14642
#87352
#87342
#79302