Finite Group Databasebeta

← #193085

Subgroups — #193085

Order 512.

1162 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 1162 conjugacy classes of subgroups — too many to draw. Only its 258 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×#2379C211yesmaximal, characteristic
2#21986C212yesmaximal, complemented
3#21204C211yesmaximal, characteristic
4C2×#1533C211yesmaximal, characteristic
5#8911C218yesmaximal, complemented
6#8033C212yesmaximal, complemented
7C22×#124C2211yescharacteristic
8C2×#432C2212yes—
9C2×#419—21no—
10C2×#418—21no—
11C2×C4×M16C2211yescharacteristic
12#2379—24no—
13#2379—22no—
14#2379C2214yes—
15C2×#391—21no—
16C2×#388—22no—
17C2×#386—21no—
18C2×#385—22no—
19#2285C2211yescharacteristic
20C2×#382C2212yes—
21#1959C2214yes—
22#1958—22no—
23#1949—24no—
24#1853—22no—
25#1852—22no—
26#1847C2214yescomplemented
27#1836C2214yescomplemented
28#1533C418yescomplemented
29C2×#329C2214yes—
30#1295C2218yescomplemented
31C23×Q8D411yescharacteristic
32C23×D4D411yescharacteristic
33C24×C4—21no—
34C22×M16—21no—
35C2×#124—22no—
36C2×#124—24no—
37C2×#124C2×C414yes—
38C2×#124D412yes—
39C2×#124Q814yes—
40C2×#124C2312yes—
41C2×#117—21no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4D412yes—
44C22×C4⋊C4D412yes—
45C22×C22⋊C4—22no—
46C22×C22⋊C4—22no—
47C22×C22⋊C4—21no—
48C22×C42—21no—
49C22×C42C2311yescharacteristic
50#432—24no—
51#432C2×C418yescomplemented
52C2×#107—24no—
53C2×#106—24no—
54C2×C4⋊C8—22no—
55#419—44no—
56#418—44no—
57C2×#103—24no—
58C2×#102—24no—
59C2×C22⋊C8—22no—
60C4×M16—24no—
61C2×#97C2312yes—
62C2×#97C2314yes—
63C2×C4×C8—21no—
64#391—24no—
65#391—24no—
66#388—28no—
67#388—28no—
68#386—24no—
69#386—24no—
70#385—28no—
71#385—28no—
72#384C2312yes—
73#382—22no—
74#382—22no—
75#382—21no—
76#382D414yes—
77#382C2314yes—
78#380—22no—
79C4×C4⋊C4—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83#340—24no—
84#340—24no—
85#329C2×C4116yescomplemented
86C25—21no—
87C22×Q8—21no—
88C22×Q8—22no—
89C22×Q8—42no—
90C22×Q8—42no—
91C22×Q8C4⋊C414yes—
92C22×Q8C2×D412yes—
93C22×D4—21no—
94C22×D4—22no—
95C22×D4—42no—
96C22×D4—42no—
97C22×D4C4⋊C414yes—
98C22×D4C2×D412yes—
99C23×C4—21no—
100C23×C4—22no—
101C23×C4—22no—
102C23×C4—24no—
103C23×C4—84no—
104C23×C4—82no—
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D412yes—
107C2×M16—24no—
108C2×M16—42no—
109C2×M16—42no—
110C22×C8—22no—
111C22×C8—24no—
112#124—28no—
113#124—28no—
114#124—24no—
115#124—24no—
116#124C4⋊C418yes—
117#124C22×C414yes—
118#124C2×Q814yes—
119#117—42no—
120#117—42no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—82no—
128C2×C4⋊C4—82no—
129C2×C4⋊C4—82no—
130C2×C4⋊C4C2×D414yes—
131C2×C4⋊C4C2×D414yes—
132C2×C22⋊C4—28no—
133C2×C22⋊C4—24no—
134C2×C22⋊C4—28no—
135C2×C22⋊C4—24no—
136C2×C22⋊C4—28no—
137C2×C22⋊C4—24no—
138C2×C22⋊C4—41no—
139C2×C22⋊C4—84no—
140C2×C22⋊C4—82no—
141C2×C42—24no—
142C2×C42—41no—
143C2×C42—82no—
144C2×C42C22×C412yes—
145C2×C42C2×D411yescharacteristic
146C2×C42C2×D412yes—
147C2×C42C2×Q812yes—
148C2×C42#4114yes—
149C2×C42C2411yescharacteristic
150#107—84no—
151#106—84no—
152C4⋊C8—28no—
153#103—48no—
154#102—48no—
155C22⋊C8—28no—
156#97—24no—
157#97—24no—
158#97C22×C418yescomplemented
159C4×C8—42no—
160#95—28no—
161#95—24no—
162#95—24no—
163#95—24no—
164#95—24no—
165#95—42no—
166C24—22no—
167C24—24no—
168C24—44no—
169C24—44no—
170C24—42no—
171C24—42no—
172C24#12011yescharacteristic
173C2×Q8—22no—
174C2×Q8—24no—
175C2×Q8—42no—
176C2×Q8—48no—
177C2×Q8—42no—
178C2×Q8—84no—
179C2×Q8—81no—
180C2×Q8—81no—
181C2×Q8C2×C4⋊C414yes—
182C2×D4—22no—
183C2×D4—24no—
184C2×D4—42no—
185C2×D4—48no—
186C2×D4—42no—
187C2×D4—84no—
188C2×D4—81no—
189C2×D4—81no—
190C2×D4C2×C4⋊C414yes—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—24no—
195C22×C4—24no—
196C22×C4—24no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—44no—
202C22×C4—44no—
203C22×C4—44no—
204C22×C4—44no—
205C22×C4—44no—
206C22×C4—44no—
207C22×C4—42no—
208C22×C4—84no—
209C22×C4—88no—
210C22×C4—816no—
211C22×C4C2×C4⋊C412yes—
212C22×C4C4×D414yes—
213C22×C4#12011yescharacteristic
214C22×C4#12012yes—
215C22×C4C22⋊Q814yes—
216C22×C4#12312yes—
217C22×C4C22×D411yescharacteristic
218C22×C4C22×D412yes—
219M16—48no—
220C2×C8—24no—
221C2×C8—28no—
222C2×C8—44no—
223C2×C8—44no—
224C2×C8—42no—
225C2×C8—42no—
226C4⋊C4—44no—
227C4⋊C4—44no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—44no—
231C4⋊C4—44no—
232C4⋊C4—82no—
233C4⋊C4—82no—
234C4⋊C4—84no—
235C22⋊C4—28no—
236C22⋊C4—28no—
237C22⋊C4—216no—
238C22⋊C4—84no—
239C22⋊C4—88no—
240C22⋊C4—82no—
241C42—82no—
242C42—84no—
243C42C2×C4⋊C412yes—
244C42C4×D414yes—
245C42C22⋊Q814yes—
246C42#12312yes—
247C42C23×C411yescharacteristic
248C42C22×Q811yescharacteristic
249C42C2×#4112yes—
250C23—22no—
251C23—22no—
252C23—24no—
253C23—416no—
254C23—48no—
255C23—42no—
256C23—42no—
257C23—44no—
258C23—44no—
259C23—84no—
260C23—88no—
261C23—84no—
262C23#38512yes—
263C23#45312yes—
264C23#45512yes—
265C23C2×#12011yescharacteristic
266Q8—44no—
267Q8—82no—
268Q8—84no—
269D4—44no—
270D4—82no—
271D4—84no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—24no—
283C2×C4—44no—
284C2×C4—44no—
285C2×C4—48no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—88no—
295C2×C4—88no—
296C2×C4—816no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4#38514yes—
301C2×C4#38512yes—
302C2×C4C22×C4⋊C411yescharacteristic
303C2×C4C2×C4×D412yes—
304C2×C4C2×C22⋊Q812yes—
305C2×C4C2×#12311yescharacteristic
306C8—48no—
307C8—44no—
308C22—24no—
309C22—22no—
310C22—22no—
311C22—24no—
312C22—44no—
313C22—48no—
314C22—816no—
315C22—84no—
316C22—82no—
317C22—82no—
318C22#185614yes—
319C22C2×#38511yescharacteristic
320C22C2×#45311yescharacteristic
321C22C2×#45511yescharacteristic
322C22#301912yes—
323C22#302012yes—
324C4—21no—
325C4—21no—
326C4—22no—
327C4—22no—
328C4—22no—
329C4—24no—
330C4—44no—
331C4—84no—
332C4—84no—
333C4—162no—
334C2—24no—
335C2—84no—
336C2#1899312yes—
337C2#1899711yescharacteristic
338C2#1899911yescharacteristic
339C2C2×#185611yescharacteristic
340C2#2086412yes—
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21986C2semidirect
5. #8911C2semidirect
6. #8033C2semidirect
26. #1847C22semidirect
27. #1836C22semidirect
28. #1533C4semidirect
30. #1295C22semidirect
51. #432C2×C4semidirect
85. #329C2×C4semidirect
158. #97C22×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23792
#219862
#212042
C2×#15332
#89112
#80332