Finite Group Databasebeta

← #237997

Subgroups — #237997

Order 512.

1087 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 1087 conjugacy classes of subgroups — too many to draw. Only its 190 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×#2399C211yesmaximal, characteristic, complemented
2#21417C211yesmaximal, characteristic
3#17546C212yesmaximal
4#10371C214yesmaximal, complemented
5#10370C214yesmaximal, complemented
6#8098C212yesmaximal, complemented
7#7855C211yesmaximal, characteristic, complemented
8#3444C2211yescharacteristic
9C22×C4×Q8—21no—
10#2920C2212yes—
11#2920C2211yescharacteristic
12#2399—24no—
13#2399—22no—
14#2399C2214yes—
15C2×#398—22no—
16C2×#395—22no—
17C2×C4×C4⋊C4—21no—
18C2×C4×C4⋊C4C2212yes—
19C2×#376C2211yescharacteristic
20#1995C2212yes—
21#1963C2212yes—
22#1896—21no—
23#1896—21no—
24#1759—22no—
25#1509—24no—
26#1504—22no—
27#1504—22no—
28#1503C2214yes—
29#1502C2214yes—
30#1303C2218yes—
31#1274C2214yes—
32C23×Q8—21no—
33#533Q812yescomplemented
34#533C2311yescharacteristic
35#532Q812yescomplemented
36#532C2311yescharacteristic
37#529—22no—
38#529—22no—
39#529—22no—
40#525—21no—
41#525—22no—
42C4×#41—21no—
43C4×#41—22no—
44C2×C4×Q8—24no—
45C2×C4×Q8—24no—
46C2×C4×Q8—48no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—22no—
51C22×C42—22no—
52C22×C42D411yescharacteristic
53C22×C42D411yescharacteristic
54C22×C42C2311yescharacteristic
55#432—22no—
56#432—22no—
57#432—22no—
58#431—22no—
59#431—22no—
60#431C2314yes—
61#423—22no—
62#423C2312yes—
63C2×#104—22no—
64C2×#104—22no—
65#407—22no—
66#407—21no—
67C4×M16—22no—
68C4×M16—21no—
69#398—28no—
70#398—28no—
71#395—28no—
72#395—28no—
73C4×C4⋊C4—24no—
74C4×C4⋊C4—22no—
75C4×C4⋊C4—22no—
76C4×C4⋊C4—24no—
77C4×C4⋊C4D414yes—
78C4×C4⋊C4C2314yes—
79#376—22no—
80#376D412yes—
81#376C2312yes—
82C2×#95—21no—
83C2×#95—21no—
84C2×#95—22no—
85C2×#95D412yes—
86C2×#95D412yes—
87#339—22no—
88#339—22no—
89#328—24no—
90#328—24no—
91#327—24no—
92#327—24no—
93C2×#41—22no—
94C2×#41—21no—
95C22×Q8—24no—
96C22×Q8—44no—
97C22×Q8—48no—
98C23×C4—21no—
99C23×C4—22no—
100C23×C4—21no—
101C23×C4—22no—
102C23×C4C2×D411yescharacteristic
103C23×C4C2×D412yes—
104C2×M16—22no—
105C2×M16—42no—
106C4⋊Q8C2×Q812yescomplemented
107C4⋊Q8C2×Q811yescharacteristic, complemented
108#127C2×Q811yescharacteristic, complemented
109#126—42no—
110#125—22no—
111#125—21no—
112#124—22no—
113#124—21no—
114#123—41no—
115#123—42no—
116C22⋊Q8—22no—
117C22⋊Q8—22no—
118C22⋊Q8—22no—
119#121—22no—
120#121—22no—
121#121—22no—
122#120—41no—
123C4×Q8—22no—
124C4×Q8—22no—
125C4×Q8—22no—
126C4×Q8—416no—
127C4×Q8—48no—
128C4×Q8—48no—
129C4×D4—22no—
130C4×D4—22no—
131C4×D4—22no—
132C4×D4—42no—
133C4×D4—42no—
134C4×D4—42no—
135#117—22no—
136#117—21no—
137#117—41no—
138#117—42no—
139C2×C4⋊C4—28no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—28no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—48no—
145C2×C4⋊C4—48no—
146C2×C4⋊C4—48no—
147C2×C4⋊C4—44no—
148C2×C42—21no—
149C2×C42—21no—
150C2×C42—24no—
151C2×C42—24no—
152C2×C42—24no—
153C2×C42—24no—
154C2×C42—48no—
155C2×C42—41no—
156C2×C42C2×D412yes—
157C2×C42C2×D412yes—
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D412yes—
160C2×C42C2×Q812yes—
161C2×C42#4112yes—
162C2×C42#4112yes—
163C2×C42#4112yes—
164C2×C42C2411yescharacteristic
165C4⋊C8—24no—
166C4⋊C8—24no—
167C4⋊C8—42no—
168#104—44no—
169#104—44no—
170C22⋊C8—24no—
171C22⋊C8—42no—
172C22⋊C8—42no—
173#97—42no—
174#97—41no—
175C4×C8—24no—
176C4×C8—41no—
177#95—24no—
178#95—24no—
179#95—24no—
180#95—22no—
181#95—22no—
182#95—24no—
183#95C2×D414yes—
184#95C2×D414yes—
185C24#12011yescharacteristic
186#41—42no—
187#41—42no—
188#41—84no—
189C2×Q8—21no—
190C2×Q8—22no—
191C2×Q8—48no—
192C2×Q8—416no—
193C2×Q8—432no—
194C2×D4—21no—
195C2×D4—22no—
196C2×D4—41no—
197C2×D4—42no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—24no—
201C22×C4—24no—
202C22×C4—28no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—44no—
208C22×C4—44no—
209C22×C4—44no—
210C22×C4—44no—
211C22×C4—44no—
212C22×C4—44no—
213C22×C4—44no—
214C22×C4—44no—
215C22×C4—41no—
216C22×C4—42no—
217C22×C4#12011yescharacteristic
218C22×C4#12012yes—
219C22×C4C22⋊Q812yes—
220C22×C4C22⋊Q812yes—
221C22×C4C22⋊Q812yes—
222C22×C4#12412yes—
223C22×C4#12412yes—
224C22×C4#12412yes—
225C22×C4C22×D412yes—
226C22×C4C22×D411yescharacteristic
227M16—42no—
228M16—42no—
229M16—84no—
230C2×C8—44no—
231C2×C8—42no—
232C4⋊C4—22no—
233C4⋊C4—22no—
234C4⋊C4—22no—
235C4⋊C4—48no—
236C4⋊C4—48no—
237C4⋊C4—416no—
238C4⋊C4—416no—
239C4⋊C4—48no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C22⋊C4—41no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C22⋊C4—42no—
247C22⋊C4—42no—
248C42—24no—
249C42—24no—
250C42—22no—
251C42—22no—
252C42—48no—
253C42—41no—
254C42—48no—
255C42—42no—
256C42C22⋊Q812yes—
257C42C22⋊Q812yes—
258C42C22⋊Q812yes—
259C42#12412yes—
260C42#12412yes—
261C42#12411yescharacteristic
262C42#12411yescharacteristic
263C42AGL(1,ℤ/8)14yes—
264C42#13714yes—
265C42C22×Q811yescharacteristic
266C42C2×#4111yescharacteristic
267C42C2×#4112yes—
268C23—21no—
269C23—21no—
270C23—24no—
271C23—41no—
272C23#39312yes—
273C23C2×#12011yescharacteristic
274Q8—432no—
275Q8—42no—
276Q8—82no—
277D4—42no—
278D4—82no—
279D4—81no—
280D4—81no—
281D4—84no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—24no—
288C2×C4—28no—
289C2×C4—24no—
290C2×C4—24no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—44no—
296C2×C4—48no—
297C2×C4—48no—
298C2×C4—416no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—44no—
307C2×C4—84no—
308C2×C4—88no—
309C2×C4—82no—
310C2×C4#39312yes—
311C2×C4#39312yes—
312C2×C4#39312yes—
313C2×C4C2×C22⋊Q812yes—
314C2×C4C2×C22⋊Q811yescharacteristic
315C2×C4C2×#12412yes—
316C2×C4C2×#12411yescharacteristic
317C2×C4C2×AGL(1,ℤ/8)12yes—
318C2×C4C2×#13712yes—
319C8—84no—
320C8—82no—
321C22—24no—
322C22—24no—
323C22—22no—
324C22—22no—
325C22—22no—
326C22—81no—
327C22—82no—
328C22#203114yes—
329C22C2×#39311yescharacteristic
330C22#300311yescharacteristic
331C22#300411yescharacteristic
332C4—22no—
333C4—22no—
334C4—22no—
335C4—22no—
336C4—22no—
337C4—44no—
338C4—48no—
339C4—44no—
340C4—84no—
341C4—81no—
342C4—82no—
343C4#307912yes—
344C4#314112yes—
345C2—21no—
346C2—21no—
347C2—24no—
348C2—161no—
349C2#2345211yescharacteristic
350C2C2×#203112yes—
subgroupG/H
centerC2×C4C2×#124
commutator, Frattini, cosocleC2×C42C24
socleC22C2×#393
prime radical#3444C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2399C2semidirect
4. #10371C2semidirect
5. #10370C2semidirect
6. #8098C2semidirect
7. #7855C2semidirect
33. #533Q8semidirect
35. #532Q8semidirect
106. C4⋊Q8C2×Q8semidirect
107. C4⋊Q8C2×Q8semidirect
108. #127C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#23992
#214172
#175462
#103712
#103702
#80982
#78552