Finite Group Databasebeta

← #192136

Subgroups — #192136

Order 512.

1222 conjugacy classes of subgroups; only the 266 normal subgroups are shown.

normal subgroups only

This group has 1222 conjugacy classes of subgroups — too many to draw. Only its 266 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2313C211yesmaximal, characteristic
2#21942C212yesmaximal
3C2×#1463C211yesmaximal, characteristic
4#13923C211yesmaximal, characteristic
5#8855C218yesmaximal, complemented
6#8019C212yesmaximal
7C23×C4⋊C4C2211yescharacteristic
8C2×#407C2212yes—
9#2313—24no—
10#2313—22no—
11#2313C2214yes—
12C2×#387—22no—
13C2×#387—22no—
14C2×#381—21no—
15C2×#380C2212yes—
16C2×#380C2211yescharacteristic
17C2×C4×C22⋊C4—21no—
18C2×C4×C22⋊C4C2211yescharacteristic
19#1853—22no—
20#1853C2214yes—
21#1852—22no—
22#1852C2214yes—
23#1809—24no—
24#1463C418yescomplemented
25C2×#324C2214yes—
26#1441C2214yes—
27#1436—22no—
28C22⋊#95—21no—
29C22⋊#95—21no—
30#1294C2218yes—
31C24×C4D411yescharacteristic
32C24×C4D411yescharacteristic
33C24×C4C2311yescharacteristic
34C22×M16—22no—
35C22×C4⋊C4—22no—
36C22×C4⋊C4—22no—
37C22×C4⋊C4—24no—
38C22×C4⋊C4C2×C414yes—
39C22×C4⋊C4D412yes—
40C22×C4⋊C4D412yes—
41C22×C4⋊C4D412yes—
42C22×C4⋊C4Q814yes—
43C22×C4⋊C4C2312yes—
44C22×C4⋊C4C2312yes—
45C22×C22⋊C4—21no—
46C22×C22⋊C4—22no—
47C22×C22⋊C4—21no—
48C22×C42—22no—
49C22×C42—21no—
50C2×#107—24no—
51C2×#106—24no—
52#407—24no—
53#407C2×C418yes—
54C2×C22⋊C8—22no—
55C2×C22⋊C8C2314yes—
56#387—28no—
57#387—28no—
58#387—28no—
59#387—28no—
60#381—24no—
61#381—24no—
62#380—22no—
63#380—22no—
64#380—22no—
65#380—22no—
66#380D414yes—
67#380C2314yes—
68C4×C22⋊C4—24no—
69C4×C22⋊C4—24no—
70C4×C22⋊C4—24no—
71C2×#95—21no—
72C2×#95—21no—
73C2×#95—21no—
74C2×#95—22no—
75C2×#95—21no—
76C2×#95—22no—
77C2×#95—24no—
78C2×#95—21no—
79C2×#95—24no—
80C2×#95C2312yes—
81#340—24no—
82#340—24no—
83#324C2×C4116yescomplemented
84C25C2×D411yescharacteristic
85C23×C4—21no—
86C23×C4—21no—
87C23×C4—21no—
88C23×C4—22no—
89C23×C4—21no—
90C23×C4—22no—
91C23×C4—22no—
92C23×C4—22no—
93C23×C4—22no—
94C23×C4—22no—
95C23×C4—44no—
96C23×C4—82no—
97C23×C4—82no—
98C23×C4C4⋊C414yes—
99C23×C4C4⋊C414yes—
100C23×C4C22×C412yes—
101C23×C4C2×D412yes—
102C23×C4C2×D412yes—
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D412yes—
105C23×C4C2×D412yes—
106C23×C4C2×Q812yes—
107C23×C4#4114yes—
108C23×C4C2411yescharacteristic
109C2×M16—28no—
110C2×M16—82no—
111C2×M16—82no—
112C22×C8—24no—
113C2×C4⋊C4—28no—
114C2×C4⋊C4—28no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—28no—
117C2×C4⋊C4—28no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—82no—
126C2×C4⋊C4—82no—
127C2×C4⋊C4—82no—
128C2×C4⋊C4C4⋊C418yes—
129C2×C4⋊C4C22×C414yes—
130C2×C4⋊C4C2×D414yes—
131C2×C4⋊C4C2×D414yes—
132C2×C4⋊C4C2×Q814yes—
133C2×C22⋊C4—24no—
134C2×C22⋊C4—24no—
135C2×C22⋊C4—24no—
136C2×C22⋊C4—42no—
137C2×C22⋊C4—48no—
138C2×C22⋊C4—42no—
139C2×C22⋊C4—44no—
140C2×C22⋊C4—41no—
141C2×C22⋊C4—41no—
142C2×C42—28no—
143C2×C42—42no—
144C2×C42—84no—
145C2×C42—82no—
146#107—84no—
147#106—84no—
148C22⋊C8—28no—
149C22⋊C8—24no—
150C22⋊C8C22×C418yes—
151#95—24no—
152#95—24no—
153#95—24no—
154#95—28no—
155#95—24no—
156#95—24no—
157#95—24no—
158#95—24no—
159#95—48no—
160#95—42no—
161#95—84no—
162C24—24no—
163C24—22no—
164C24—22no—
165C24—42no—
166C24C2×C4⋊C412yes—
167C24#12011yescharacteristic
168C24#12011yescharacteristic
169C24#12312yes—
170C24C22×D411yescharacteristic
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—24no—
175C22×C4—24no—
176C22×C4—24no—
177C22×C4—24no—
178C22×C4—24no—
179C22×C4—22no—
180C22×C4—24no—
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—24no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—48no—
191C22×C4—44no—
192C22×C4—48no—
193C22×C4—44no—
194C22×C4—44no—
195C22×C4—44no—
196C22×C4—44no—
197C22×C4—44no—
198C22×C4—44no—
199C22×C4—42no—
200C22×C4—44no—
201C22×C4—44no—
202C22×C4—42no—
203C22×C4—44no—
204C22×C4—44no—
205C22×C4—44no—
206C22×C4—44no—
207C22×C4—44no—
208C22×C4—42no—
209C22×C4—44no—
210C22×C4—82no—
211C22×C4—82no—
212C22×C4—84no—
213C22×C4—84no—
214C22×C4—82no—
215C22×C4—84no—
216C22×C4C2×C4⋊C414yes—
217C22×C4C2×C4⋊C414yes—
218C22×C4C2×C4⋊C412yes—
219C22×C4C4×D414yes—
220C22×C4C4×D414yes—
221C22×C4#12012yes—
222C22×C4C22⋊Q814yes—
223C22×C4C22⋊Q814yes—
224C22×C4#12312yes—
225C22×C4C23×C411yescharacteristic
226C22×C4C22×D412yes—
227C22×C4C22×Q811yescharacteristic
228C22×C4C2×#4112yes—
229M16—88no—
230C2×C8—28no—
231C2×C8—44no—
232C2×C8—44no—
233C4⋊C4—28no—
234C4⋊C4—28no—
235C4⋊C4—216no—
236C4⋊C4—44no—
237C4⋊C4—44no—
238C4⋊C4—44no—
239C4⋊C4—44no—
240C4⋊C4—82no—
241C4⋊C4—82no—
242C22⋊C4—48no—
243C22⋊C4—48no—
244C22⋊C4—48no—
245C22⋊C4—82no—
246C22⋊C4—82no—
247C42—88no—
248C42—84no—
249C23—22no—
250C23—21no—
251C23—21no—
252C23—22no—
253C23—22no—
254C23—24no—
255C23—24no—
256C23—24no—
257C23—24no—
258C23—22no—
259C23—21no—
260C23—21no—
261C23—22no—
262C23—42no—
263C23—42no—
264C23—48no—
265C23—48no—
266C23#38512yes—
267C23#38512yes—
268C23#45412yes—
269C23#45612yes—
270C23C22×C4⋊C411yescharacteristic
271C23C2×#12011yescharacteristic
272C23C2×#12311yescharacteristic
273C23#57514yes—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—24no—
279C2×C4—24no—
280C2×C4—24no—
281C2×C4—24no—
282C2×C4—42no—
283C2×C4—44no—
284C2×C4—42no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—48no—
292C2×C4—48no—
293C2×C4—48no—
294C2×C4—44no—
295C2×C4—44no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—48no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—82no—
307C2×C4—84no—
308C2×C4—82no—
309C2×C4—84no—
310C2×C4—88no—
311C2×C4—88no—
312C2×C4—82no—
313C2×C4—88no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4#38514yes—
318C2×C4C2×C4×D412yes—
319C2×C4C2×C22⋊Q812yes—
320C8—48no—
321C22—22no—
322C22—21no—
323C22—21no—
324C22—24no—
325C22—24no—
326C22—24no—
327C22—24no—
328C22—21no—
329C22—22no—
330C22—21no—
331C22—22no—
332C22—22no—
333C22—22no—
334C22—48no—
335C22—48no—
336C22—42no—
337C22—42no—
338C22#185714yes—
339C22C2×#38511yescharacteristic
340C22#291112yes—
341C22#300512yes—
342C22C2×#45411yescharacteristic
343C22C2×#45611yescharacteristic
344C22#302612yes—
345C22#309312yes—
346C4—22no—
347C4—22no—
348C4—24no—
349C4—44no—
350C4—44no—
351C4—84no—
352C4—84no—
353C4—84no—
354C4—162no—
355C2—22no—
356C2—24no—
357C2—22no—
358C2—42no—
359C2#1832012yes—
360C2#1832112yes—
361C2#2084511yescharacteristic
362C2C2×#185711yescharacteristic
363C2#2088911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #8855C2semidirect
24. #1463C4semidirect
83. #324C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23132
#219422
C2×#14632
#139232
#88552
#80192