Finite Group Databasebeta

← #192153

Subgroups — #192153

Order 512.

1138 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34182C211yesmaximal, characteristic
2#21986C212yesmaximal
3C2×#1467C211yesmaximal, characteristic
4#14346C211yesmaximal, characteristic
5#8857C214yesmaximal
6#8856C214yesmaximal
7#8018C212yesmaximal
8C2×#519—21no—
9C22×C4×Q8C2211yescharacteristic
10C2×#432C2212yes—
11#2351—24no—
12Q8×C4⋊C4—22no—
13Q8×C4⋊C4—22no—
14#2316—21no—
15#2316C2214yes—
16#2285—22no—
17#2285C2212yes—
18C2×#383C2211yescharacteristic
19C2×#382C2211yescharacteristic
20#1949—24no—
21#1853C2214yes—
22#1852C2214yes—
23#1847—22no—
24#1836—22no—
25#1467C418yescomplemented
26C2×#324C2214yes—
27#1446—21no—
28#1445—21no—
29#1441C2212yes—
30#1441C2212yes—
31#1434—22no—
32#1294C2218yes—
33C23×Q8D411yescharacteristic
34#519—48no—
35C2×C4×Q8—24no—
36C2×C4×Q8—24no—
37C2×C4×Q8—22no—
38C2×C4×Q8—22no—
39C2×C4×Q8—24no—
40C2×C4×Q8C2×C414yes—
41C2×C4×Q8Q814yes—
42C2×#117—22no—
43C2×#117—24no—
44C22×C4⋊C4D411yescharacteristic
45C22×C4⋊C4C2312yes—
46C22×C4⋊C4C2311yescharacteristic
47C22×C4⋊C4C2311yescharacteristic
48C22×C42D412yes—
49C22×C42C2311yescharacteristic
50#432—24no—
51#432C2×C418yes—
52C2×#107—24no—
53C2×#106—24no—
54C2×C4⋊C8—22no—
55C2×C22⋊C8C2314yes—
56C2×#97—22no—
57#389—24no—
58#389—24no—
59#389—28no—
60#389—28no—
61#384—22no—
62#384—22no—
63#384D412yes—
64#384D412yes—
65#383—24no—
66#382—24no—
67#382—22no—
68#382—22no—
69#382—22no—
70#380—24no—
71#380—22no—
72#380C2314yes—
73C4×C4⋊C4—24no—
74C4×C4⋊C4—24no—
75C4×C4⋊C4—22no—
76C4×C4⋊C4D414yes—
77C2×#95—21no—
78C2×#95—21no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—21no—
82C2×#95—21no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95C2312yes—
86#340—24no—
87#340—24no—
88#324C2×C4116yes—
89C22×Q8—22no—
90C22×Q8—22no—
91C22×Q8—24no—
92C22×Q8—24no—
93C22×Q8C4⋊C414yes—
94C23×C4—21no—
95C23×C4—21no—
96C23×C4—22no—
97C23×C4C2×D411yescharacteristic
98C23×C4C2×D412yes—
99C23×C4C2×D412yes—
100C23×C4C2×D411yescharacteristic
101C23×C4C2411yescharacteristic
102C22×C8—24no—
103C4×Q8—28no—
104C4×Q8—28no—
105C4×Q8—28no—
106C4×Q8—28no—
107C4×Q8—416no—
108C4×Q8—416no—
109#117—48no—
110#117—48no—
111#117—48no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—81no—
127C2×C4⋊C4—81no—
128C2×C4⋊C4C4⋊C414yes—
129C2×C4⋊C4C22×C414yes—
130C2×C4⋊C4C2×D414yes—
131C2×C4⋊C4C2×D414yes—
132C2×C4⋊C4C2×Q814yes—
133C2×C22⋊C4—24no—
134C2×C42—22no—
135C2×C42—22no—
136C2×C42—24no—
137C2×C42—28no—
138C2×C42—24no—
139C2×C42C4⋊C418yes—
140C2×C42C22×C412yes—
141C2×C42C2×D412yes—
142C2×C42C2×D412yes—
143C2×C42C2×Q812yes—
144C2×C42#4114yes—
145#107—84no—
146#106—84no—
147C4⋊C8—28no—
148C22⋊C8—24no—
149C22⋊C8C22×C418yes—
150#97—28no—
151#95—28no—
152#95—24no—
153#95—24no—
154#95—24no—
155#95—41no—
156#95—41no—
157#95—42no—
158#95—42no—
159#95—42no—
160#95—44no—
161#95—44no—
162#95—82no—
163#95—82no—
164C24C22×D411yescharacteristic
165C2×Q8—28no—
166C2×Q8—216no—
167C2×Q8—28no—
168C2×Q8—28no—
169C2×Q8—216no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—24no—
179C22×C4—24no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4C2×C4⋊C412yes—
202C22×C4C2×C4⋊C414yes—
203C22×C4C2×C4⋊C414yes—
204C22×C4C2×C4⋊C412yes—
205C22×C4#12011yescharacteristic
206C22×C4#12011yescharacteristic
207C22×C4#12012yes—
208C22×C4#12312yes—
209C22×C4#12312yes—
210C22×C4AGL(1,ℤ/8)14yes—
211C22×C4#13714yes—
212C22×C4C23×C411yescharacteristic
213C22×C4C22×D412yes—
214C22×C4C22×Q811yescharacteristic
215C22×C4C2×#4112yes—
216C2×C8—28no—
217C2×C8—44no—
218C2×C8—44no—
219C4⋊C4—28no—
220C4⋊C4—28no—
221C4⋊C4—28no—
222C4⋊C4—44no—
223C4⋊C4—44no—
224C4⋊C4—44no—
225C4⋊C4—44no—
226C4⋊C4—44no—
227C4⋊C4—44no—
228C4⋊C4—48no—
229C4⋊C4—48no—
230C4⋊C4—44no—
231C4⋊C4—44no—
232C4⋊C4—84no—
233C4⋊C4—84no—
234C22⋊C4—48no—
235C42—28no—
236C42—28no—
237C42—48no—
238C42—416no—
239C42C4×D414yes—
240C42C4×D414yes—
241C42C22⋊Q814yes—
242C42C22⋊Q814yes—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23C22×C4⋊C411yescharacteristic
248C23C2×#12011yescharacteristic
249C23C2×#12311yescharacteristic
250C23C2×AGL(1,ℤ/8)12yes—
251C23C2×#13712yes—
252Q8—232no—
253C2×C4—24no—
254C2×C4—24no—
255C2×C4—24no—
256C2×C4—24no—
257C2×C4—24no—
258C2×C4—22no—
259C2×C4—24no—
260C2×C4—22no—
261C2×C4—24no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—84no—
292C2×C4—84no—
293C2×C4—82no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4#38512yes—
301C2×C4#38512yes—
302C2×C4#38514yes—
303C2×C4#42818yes—
304C2×C4C2×C4×D412yes—
305C2×C4C2×C22⋊Q812yes—
306C8—48no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—22no—
312C22—22no—
313C22—22no—
314C22—22no—
315C22C2×#38511yescharacteristic
316C22C2×#42812yes—
317C22#300311yescharacteristic
318C22#300411yescharacteristic
319C22#302011yescharacteristic
320C22#302011yescharacteristic
321C22#302711yescharacteristic
322C22#302811yescharacteristic
323C22#309212yes—
324C4—24no—
325C4—28no—
326C4—22no—
327C4—22no—
328C4—22no—
329C4—22no—
330C4—44no—
331C4—84no—
332C4—84no—
333C4—162no—
334C4—161no—
335C4—161no—
336C2—22no—
337C2—22no—
338C2#1836811yescharacteristic
339C2#1836911yescharacteristic
340C2#1837311yescharacteristic
341C2#1837411yescharacteristic
342C2#2084411yescharacteristic
343C2#2085711yescharacteristic
344C2#2089011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
25. #1467C4semidirect

Maximal subgroups

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

subgroupindex
#341822
#219862
C2×#14672
#143462
#88572
#88562
#80182