Finite Group Databasebeta

← #193084

Subgroups — #193084

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#8914C214yesmaximal, complemented
6#8910C214yesmaximal, complemented
7#8033C212yesmaximal, complemented
8C22×#124C2211yescharacteristic
9C2×#432C2212yes—
10C2×#419—21no—
11C2×#418—21no—
12C2×C4×M16C2211yescharacteristic
13#2379—24no—
14#2379—22no—
15#2379C2214yes—
16C2×#391—21no—
17C2×#388—22no—
18C2×#386—21no—
19C2×#385—22no—
20#2285C2211yescharacteristic
21C2×#382C2212yes—
22#1959C2212yes—
23#1959C2212yes—
24#1958—22no—
25#1949—24no—
26#1853—22no—
27#1852—22no—
28#1847C2214yescomplemented
29#1836C2214yescomplemented
30#1533C418yescomplemented
31C2×#329C2214yes—
32#1295C2218yescomplemented
33C23×Q8D411yescharacteristic
34C23×D4D411yescharacteristic
35C24×C4—21no—
36C22×M16—21no—
37C2×#124—22no—
38C2×#124—24no—
39C2×#124C2×C414yes—
40C2×#124D412yes—
41C2×#124Q814yes—
42C2×#124C2312yes—
43C2×#117—21no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4D412yes—
46C22×C4⋊C4D412yes—
47C22×C22⋊C4—22no—
48C22×C22⋊C4—22no—
49C22×C22⋊C4—21no—
50C22×C42—21no—
51C22×C42C2311yescharacteristic
52#432—24no—
53#432C2×C418yescomplemented
54C2×#107—24no—
55C2×#106—24no—
56C2×C4⋊C8—22no—
57#419—44no—
58#418—44no—
59C2×#103—22no—
60C2×#103—22no—
61C2×#102—22no—
62C2×#102—22no—
63C2×C22⋊C8—22no—
64C4×M16—24no—
65C2×#97C2312yes—
66C2×#97C2314yes—
67C2×C4×C8—21no—
68#391—24no—
69#391—24no—
70#388—28no—
71#388—28no—
72#386—24no—
73#386—24no—
74#385—28no—
75#385—28no—
76#384C2311yescharacteristic
77#384C2311yescharacteristic
78#382—22no—
79#382—22no—
80#382—21no—
81#382D414yes—
82#382C2314yes—
83#380—22no—
84C4×C4⋊C4—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88#340—24no—
89#340—24no—
90#329C2×C4116yescomplemented
91C25—21no—
92C22×Q8—21no—
93C22×Q8—22no—
94C22×Q8—42no—
95C22×Q8—42no—
96C22×Q8C4⋊C414yes—
97C22×Q8C2×D412yes—
98C22×D4—21no—
99C22×D4—22no—
100C22×D4—42no—
101C22×D4—42no—
102C22×D4C4⋊C414yes—
103C22×D4C2×D412yes—
104C23×C4—21no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—24no—
108C23×C4—84no—
109C23×C4—82no—
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D412yes—
112C2×M16—24no—
113C2×M16—42no—
114C2×M16—42no—
115C22×C8—22no—
116C22×C8—24no—
117#124—28no—
118#124—28no—
119#124—24no—
120#124—24no—
121#124C4⋊C418yes—
122#124C22×C414yes—
123#124C2×Q814yes—
124#117—42no—
125#117—42no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—82no—
134C2×C4⋊C4—82no—
135C2×C4⋊C4—82no—
136C2×C4⋊C4C2×D414yes—
137C2×C4⋊C4C2×D414yes—
138C2×C22⋊C4—28no—
139C2×C22⋊C4—24no—
140C2×C22⋊C4—28no—
141C2×C22⋊C4—24no—
142C2×C22⋊C4—28no—
143C2×C22⋊C4—24no—
144C2×C22⋊C4—41no—
145C2×C22⋊C4—84no—
146C2×C22⋊C4—82no—
147C2×C42—24no—
148C2×C42—41no—
149C2×C42—82no—
150C2×C42C22×C412yes—
151C2×C42C2×D411yescharacteristic
152C2×C42C2×D412yes—
153C2×C42C2×Q812yes—
154C2×C42#4114yes—
155C2×C42C2411yescharacteristic
156#107—84no—
157#106—84no—
158C4⋊C8—28no—
159#103—44no—
160#103—44no—
161#102—44no—
162#102—44no—
163C22⋊C8—28no—
164#97—24no—
165#97—24no—
166#97C22×C418yescomplemented
167C4×C8—42no—
168#95—28no—
169#95—24no—
170#95—24no—
171#95—24no—
172#95—24no—
173#95—42no—
174C24—22no—
175C24—24no—
176C24—44no—
177C24—44no—
178C24—42no—
179C24—42no—
180C24#12011yescharacteristic
181C2×Q8—22no—
182C2×Q8—24no—
183C2×Q8—42no—
184C2×Q8—48no—
185C2×Q8—42no—
186C2×Q8—84no—
187C2×Q8—81no—
188C2×Q8—81no—
189C2×Q8C2×C4⋊C414yes—
190C2×D4—22no—
191C2×D4—24no—
192C2×D4—42no—
193C2×D4—48no—
194C2×D4—42no—
195C2×D4—84no—
196C2×D4—81no—
197C2×D4—81no—
198C2×D4C2×C4⋊C414yes—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—24no—
203C22×C4—24no—
204C22×C4—24no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—44no—
210C22×C4—44no—
211C22×C4—44no—
212C22×C4—44no—
213C22×C4—44no—
214C22×C4—44no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—84no—
218C22×C4—88no—
219C22×C4—816no—
220C22×C4C2×C4⋊C412yes—
221C22×C4C4×D414yes—
222C22×C4#12011yescharacteristic
223C22×C4#12012yes—
224C22×C4C22⋊Q814yes—
225C22×C4#12312yes—
226C22×C4C22×D411yescharacteristic
227C22×C4C22×D412yes—
228M16—48no—
229C2×C8—24no—
230C2×C8—28no—
231C2×C8—44no—
232C2×C8—44no—
233C2×C8—42no—
234C2×C8—42no—
235C4⋊C4—44no—
236C4⋊C4—44no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—44no—
242C4⋊C4—44no—
243C4⋊C4—82no—
244C4⋊C4—82no—
245C4⋊C4—84no—
246C22⋊C4—28no—
247C22⋊C4—28no—
248C22⋊C4—216no—
249C22⋊C4—84no—
250C22⋊C4—88no—
251C22⋊C4—82no—
252C42—82no—
253C42—84no—
254C42C2×C4⋊C412yes—
255C42C4×D414yes—
256C42C22⋊Q814yes—
257C42#12312yes—
258C42C23×C411yescharacteristic
259C42C22×Q811yescharacteristic
260C42C2×#4112yes—
261C23—22no—
262C23—22no—
263C23—24no—
264C23—416no—
265C23—48no—
266C23—42no—
267C23—42no—
268C23—44no—
269C23—44no—
270C23—84no—
271C23—84no—
272C23—88no—
273C23#38512yes—
274C23#45312yes—
275C23#45512yes—
276C23C2×#12011yescharacteristic
277Q8—44no—
278Q8—82no—
279Q8—84no—
280D4—44no—
281D4—82no—
282D4—84no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—24no—
294C2×C4—44no—
295C2×C4—44no—
296C2×C4—48no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—44no—
305C2×C4—88no—
306C2×C4—88no—
307C2×C4—816no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4#38514yes—
313C2×C4#38512yes—
314C2×C4C22×C4⋊C411yescharacteristic
315C2×C4C2×C4×D412yes—
316C2×C4C2×C22⋊Q812yes—
317C2×C4C2×#12311yescharacteristic
318C8—48no—
319C8—44no—
320C22—22no—
321C22—22no—
322C22—24no—
323C22—24no—
324C22—44no—
325C22—48no—
326C22—816no—
327C22—82no—
328C22—82no—
329C22—84no—
330C22#185614yes—
331C22C2×#38511yescharacteristic
332C22C2×#45311yescharacteristic
333C22C2×#45511yescharacteristic
334C22#301912yes—
335C22#302012yes—
336C4—21no—
337C4—21no—
338C4—22no—
339C4—22no—
340C4—22no—
341C4—24no—
342C4—44no—
343C4—84no—
344C4—84no—
345C4—161no—
346C4—161no—
347C2—24no—
348C2—84no—
349C2#1899211yescharacteristic
350C2#1899411yescharacteristic
351C2#1899812yes—
352C2C2×#185611yescharacteristic
353C2#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. #8914C2semidirect
6. #8910C2semidirect
7. #8033C2semidirect
28. #1847C22semidirect
29. #1836C22semidirect
30. #1533C4semidirect
32. #1295C22semidirect
53. #432C2×C4semidirect
90. #329C2×C4semidirect
166. #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
#89142
#89102
#80332