Finite Group Databasebeta

← #189885

Subgroups — #189885

Order 512.

1096 conjugacy classes of subgroups.

The subgroup lattice has 329 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34182C211yesmaximal, characteristic
2#21822C211yesmaximal, characteristic
3#13961C211yesmaximal, characteristic
4#8717C214yesmaximal
5#8716C214yesmaximal
6#8022C212yesmaximal
7#8018C212yesmaximal
8C2×#519—21no—
9C22×C4×Q8C2211yescharacteristic
10C2×#432C2212yes—
11#2351—24no—
12Q8×C4⋊C4C418yescomplemented
13#2316—21no—
14#2316—21no—
15#2316—21no—
16#2285C2212yes—
17#2285C2212yes—
18#2285C2212yes—
19C2×#383C2211yescharacteristic
20C2×#376C2211yescharacteristic
21#1985—24no—
22#1817C2212yes—
23#1817C2212yes—
24#1446—21no—
25#1445—21no—
26#1436—21no—
27#1436—21no—
28#1434C2212yes—
29#1434C2212yes—
30#1294C2218yes—
31#1294C2218yes—
32C23×Q8D411yescharacteristic
33#519—48no—
34C2×C4×Q8—28no—
35C2×C4×Q8—24no—
36C2×C4×Q8—24no—
37C2×C4×Q8C2×C414yes—
38C2×C4×Q8D414yes—
39C2×#117—22no—
40C2×#117—22no—
41C2×#117—22no—
42C22×C4⋊C4D411yescharacteristic
43C22×C4⋊C4D411yescharacteristic
44C22×C4⋊C4D411yescharacteristic
45C22×C42—21no—
46C22×C42C2311yescharacteristic
47C22×C42C2311yescharacteristic
48C22×C42C2311yescharacteristic
49#432—24no—
50#432—24no—
51C2×C4⋊C8—22no—
52C2×C22⋊C8C2314yes—
53C2×#97—22no—
54#389—28no—
55#389—28no—
56#389—28no—
57#384C2×C414yes—
58#384C2×C414yes—
59#384C2×C414yes—
60#383—24no—
61#382—22no—
62#382—22no—
63#382—22no—
64#380—22no—
65#380—22no—
66#380—22no—
67#380C2314yes—
68C4×C4⋊C4—22no—
69C4×C4⋊C4—22no—
70C4×C4⋊C4—22no—
71C4×C4⋊C4C2×C414yes—
72C4×C4⋊C4C2×C414yes—
73C4×C4⋊C4C2×C414yes—
74#376—24no—
75C2×#95—22no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95—21no—
80C2×#95—21no—
81C2×#95—21no—
82C2×#95C2312yes—
83C2×#95C2312yes—
84#336—24no—
85#336—24no—
86C22×Q8—22no—
87C22×Q8—22no—
88C22×Q8—24no—
89C22×Q8—24no—
90C22×Q8C22⋊C414yes—
91C23×C4—22no—
92C23×C4—22no—
93C23×C4C2×D411yescharacteristic
94C23×C4C2×D411yescharacteristic
95C23×C4C2×D411yescharacteristic
96C23×C4C2×D411yescharacteristic
97C23×C4C2×D411yescharacteristic
98C23×C4C2×D411yescharacteristic
99C23×C4C2411yescharacteristic
100C22×C8—24no—
101C4×Q8—216no—
102C4×Q8—28no—
103C4×Q8—28no—
104C4×Q8—432no—
105#117—48no—
106#117—48no—
107#117—48no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
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—24no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4C22⋊C414yes—
121C2×C4⋊C4C22⋊C414yes—
122C2×C4⋊C4C22⋊C414yes—
123C2×C4⋊C4C22×C418yes—
124C2×C22⋊C4—24no—
125C2×C42—22no—
126C2×C42—22no—
127C2×C42—22no—
128C2×C42—22no—
129C2×C42—24no—
130C2×C42—24no—
131C2×C42—24no—
132C2×C42—44no—
133C2×C42—42no—
134C2×C42C22×C412yes—
135C2×C42C22×C412yes—
136C2×C42C22×C412yes—
137C2×C42C2×D412yes—
138C2×C42C2×D412yes—
139C2×C42C2×D412yes—
140C2×C42#4114yes—
141C2×C42#4114yes—
142C4⋊C8—44no—
143C22⋊C8—24no—
144C22⋊C8—24no—
145#97—44no—
146#95—24no—
147#95—24no—
148#95—24no—
149#95—24no—
150#95—24no—
151#95—24no—
152#95—44no—
153#95—44no—
154#95—42no—
155#95—42no—
156#95—42no—
157#95—44no—
158#95—44no—
159C24C22×D411yescharacteristic
160C2×Q8—216no—
161C2×Q8—216no—
162C2×Q8—28no—
163C2×Q8—28no—
164C2×Q8—28no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—24no—
178C22×C4—28no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
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—42no—
195C22×C4—42no—
196C22×C4C2×C22⋊C412yes—
197C22×C4C2×C22⋊C412yes—
198C22×C4C2×C22⋊C412yes—
199C22×C4C2×C22⋊C412yes—
200C22×C4C2×C22⋊C412yes—
201C22×C4C2×C22⋊C412yes—
202C22×C4#12312yes—
203C22×C4#12312yes—
204C22×C4#12312yes—
205C22×C4#12312yes—
206C22×C4AGL(1,ℤ/8)14yes—
207C22×C4#13714yes—
208C22×C4C23×C411yescharacteristic
209C22×C4C22×D411yescharacteristic
210C22×C4C2×#4112yes—
211C22×C4C2×#4112yes—
212C2×C8—48no—
213C2×C8—44no—
214C4⋊C4—28no—
215C4⋊C4—28no—
216C4⋊C4—28no—
217C4⋊C4—48no—
218C4⋊C4—48no—
219C4⋊C4—48no—
220C4⋊C4—48no—
221C4⋊C4—84no—
222C4⋊C4—84no—
223C4⋊C4—84no—
224C22⋊C4—48no—
225C42—24no—
226C42—24no—
227C42—24no—
228C42—24no—
229C42—48no—
230C42—48no—
231C42—48no—
232C42—48no—
233C42#11718yes—
234C42#12314yes—
235C42#12314yes—
236C23—22no—
237C23—22no—
238C23C22×C22⋊C411yescharacteristic
239C23C2×#12311yescharacteristic
240C23C2×#12311yescharacteristic
241C23C2×AGL(1,ℤ/8)12yes—
242C23C2×#13712yes—
243Q8—232no—
244C2×C4—24no—
245C2×C4—24no—
246C2×C4—24no—
247C2×C4—24no—
248C2×C4—24no—
249C2×C4—24no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—24no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—24no—
260C2×C4—24no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—44no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—48no—
271C2×C4—48no—
272C2×C4—48no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4#38114yes—
289C2×C4#38114yes—
290C2×C4#41814yes—
291C2×C4#41914yes—
292C2×C4C2×#11712yes—
293C2×C4C2×#12312yes—
294C8—84no—
295C22—21no—
296C22—21no—
297C22—22no—
298C22—22no—
299C22—22no—
300C22—22no—
301C22—21no—
302C22—21no—
303C22C2×#38111yescharacteristic
304C22#288412yes—
305C22C2×#41811yescharacteristic
306C22C2×#41911yescharacteristic
307C22#290112yes—
308C22#309212yes—
309C22#309212yes—
310C4—22no—
311C4—22no—
312C4—24no—
313C4—24no—
314C4—24no—
315C4—24no—
316C4—42no—
317C4—42no—
318C4—84no—
319C4—88no—
320C4—162no—
321C2—22no—
322C2—22no—
323C2#1763112yes—
324C2#1763212yes—
325C2#2069111yescharacteristic
326C2#2070011yescharacteristic
327C2#2070611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
12. Q8×C4⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
#341822
#218222
#139612
#87172
#87162
#80222
#80182