Finite Group Databasebeta

← #266757

Subgroups — #266757

Order 512.

1136 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 1136 conjugacy classes of subgroups — too many to draw. Only its 200 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
1#59817C211yesmaximal, characteristic
2#29287C212yesmaximal, complemented
3C4×#492C211yesmaximal, characteristic
4#10574C212yesmaximal, complemented
5#10349C212yesmaximal, complemented
6#9870C212yesmaximal, complemented
7#9064C212yesmaximal
8#8924C212yesmaximal
9#8585C211yesmaximal, characteristic
10C2×D4×Q8—22no—
11C2×#492C2211yescharacteristic
12C2×C4×SD16—22no—
13#2683—22no—
14#2551C2212yes—
15#2538—22no—
16#2452—24no—
17#2409—22no—
18#2366C2212yes—
19C4×C4⋊Q8C2212yes—
20C4×C4⋊Q8C2211yescharacteristic
21C4×#127C2211yescharacteristic
22#2088—24no—
23#1988—22no—
24#1987C2214yes—
25#1968—22no—
26#1957C2211yescharacteristic
27#1956C2211yescharacteristic
28#1948—22no—
29#1948C2214yes—
30#1844C2211yescharacteristic
31#1842C2212yes—
32C42×C8C2211yescharacteristic
33C2×#331C2212yes—
34#1296C2212yes—
35#1279C2214yes—
36#1279C2214yes—
37C23×Q8—22no—
38C22×SD16—22no—
39D4×Q8—48no—
40D4×Q8—48no—
41C2×C4⋊Q8—24no—
42C2×C4⋊Q8D412yes—
43C2×C4⋊Q8C2311yescharacteristic
44C2×#127C2311yescharacteristic
45C2×C22⋊Q8—24no—
46C2×C22⋊Q8—28no—
47C2×C4×Q8—22no—
48C2×C4×Q8—21no—
49C2×C4×D4—24no—
50C2×C4×D4—22no—
51C2×C4×D4D412yes—
52#492—22no—
53#492Q814yescomplemented
54C4×SD16—48no—
55C2×#106—22no—
56C2×#106—24no—
57C2×C4⋊C8—22no—
58C2×C4⋊C8—22no—
59C2×C4⋊C8C2314yes—
60C2×#103—22no—
61C2×#102—22no—
62C2×C4×C8—21no—
63C2×C4×C8—21no—
64C2×C4×C8C2311yescharacteristic
65C2×C4×C8C2311yescharacteristic
66#396—28no—
67#395—24no—
68#393—24no—
69#391—22no—
70#391—21no—
71#390—24no—
72#390—22no—
73#390D412yes—
74#389—24no—
75#389—22no—
76#389—22no—
77#389D414yes—
78#386—24no—
79#385—24no—
80#384D412yes—
81#384C2311yescharacteristic
82#384C2311yescharacteristic
83#380—22no—
84C4×C4⋊C4—22no—
85C4×C4⋊C4—22no—
86C4×C4⋊C4C2314yes—
87C43C2311yescharacteristic
88#340—24no—
89#340—24no—
90#340—24no—
91#331—24no—
92C22×Q8—24no—
93C22×Q8—22no—
94C22×Q8—21no—
95C22×Q8—416no—
96C22×Q8—84no—
97C22×D4—22no—
98C22×D4C2×D412yes—
99C23×C4—24no—
100C23×C4—22no—
101C2×SD16—28no—
102C2×SD16—44no—
103C2×SD16—44no—
104C22×C8—21no—
105C22×C8—21no—
106C22×C8—24no—
107C4⋊Q8—24no—
108C4⋊Q8—21no—
109C4⋊Q8—22no—
110C4⋊Q8—48no—
111C4⋊Q8—48no—
112C4⋊Q8SD1618yescomplemented
113C4⋊Q8C2×Q812yescomplemented
114#127—22no—
115#127C2×Q812yes—
116#127#4112yes—
117C22⋊Q8—432no—
118C22⋊Q8—416no—
119C4×Q8—44no—
120C4×Q8—44no—
121C4×Q8—44no—
122C4×D4—44no—
123C4×D4—48no—
124C4×D4—48no—
125C4×D4—44no—
126C4×D4—44no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—28no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4C2×D414yes—
138C2×C4⋊C4C2×D414yes—
139C2×C4⋊C4C2×D412yes—
140C2×C22⋊C4—28no—
141C2×C22⋊C4—24no—
142C2×C22⋊C4—24no—
143C2×C22⋊C4—22no—
144C2×C42—24no—
145C2×C42—41no—
146C2×C42C2×D411yescharacteristic
147C2×C42C2×D412yes—
148C2×C42C2×D411yescharacteristic
149C2×C42C2×D412yes—
150C2×C42C2411yescharacteristic
151#106—44no—
152#106—84no—
153C4⋊C8—24no—
154C4⋊C8—24no—
155C4⋊C8—44no—
156C4⋊C8—82no—
157#103—44no—
158#102—44no—
159C4×C8—24no—
160C4×C8—21no—
161C4×C8—22no—
162C4×C8—42no—
163C4×C8C2×Q812yescomplemented
164#95—24no—
165#95—24no—
166#95—24no—
167#95—42no—
168C24—22no—
169C2×Q8—28no—
170C2×Q8—22no—
171C2×Q8—416no—
172C2×Q8—42no—
173C2×Q8—42no—
174C2×Q8—41no—
175C2×Q8—44no—
176C2×Q8—44no—
177C2×Q8—84no—
178C2×Q8—832no—
179C2×D4—22no—
180C2×D4—24no—
181C2×D4—24no—
182C2×D4—42no—
183C2×D4—42no—
184C2×D4—42no—
185C2×D4—42no—
186C2×D4—42no—
187C2×D4—42no—
188C2×D4C22⋊Q814yes—
189C22×C4—24no—
190C22×C4—24no—
191C22×C4—22no—
192C22×C4—416no—
193C22×C4—41no—
194C22×C4—48no—
195C22×C4—44no—
196C22×C4—41no—
197C22×C4—84no—
198C22×C4#12011yescharacteristic
199C22×C4#12012yes—
200C22×C4#12011yescharacteristic
201C22×C4C22×D411yescharacteristic
202C22×C4C22×D412yes—
203SD16—416no—
204C2×C8—22no—
205C2×C8—22no—
206C2×C8—22no—
207C2×C8—22no—
208C2×C8—41no—
209C2×C8—41no—
210C2×C8—44no—
211C2×C8—44no—
212C2×C8—44no—
213C4⋊C4—24no—
214C4⋊C4—28no—
215C4⋊C4—24no—
216C4⋊C4—24no—
217C4⋊C4—24no—
218C4⋊C4—42no—
219C4⋊C4—416no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—48no—
225C4⋊C4—44no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—82no—
229C4⋊C4—82no—
230C4⋊C4C2×SD1618yes—
231C22⋊C4—416no—
232C22⋊C4—48no—
233C22⋊C4—82no—
234C22⋊C4—84no—
235C42—22no—
236C42—22no—
237C42—22no—
238C42—41no—
239C42—44no—
240C42—44no—
241C42—82no—
242C42C2×SD1614yes—
243C42AGL(1,ℤ/8)14yes—
244C42C22×Q811yescharacteristic
245C42C2×#4111yescharacteristic
246C422−1+412yes—
247C23—24no—
248C23—44no—
249C23—44no—
250C23—42no—
251C23—42no—
252C23C2×#12011yescharacteristic
253Q8—44no—
254Q8—416no—
255Q8—82no—
256Q8—816no—
257D4—44no—
258D4—44no—
259D4—44no—
260D4—44no—
261D4—44no—
262D4—82no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—28no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—44no—
271C2×C4—44no—
272C2×C4—42no—
273C2×C4—44no—
274C2×C4—44no—
275C2×C4—44no—
276C2×C4—42no—
277C2×C4—44no—
278C2×C4—816no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—88no—
282C2×C4—82no—
283C2×C4—81no—
284C2×C4#45014yes—
285C2×C4#45014yes—
286C2×C4#45312yes—
287C2×C4C2×C22⋊Q811yescharacteristic
288C2×C4#53611yescharacteristic
289C2×C4#54812yes—
290C2×C4D4×Q812yes—
291C2×C4C22×SD1612yes—
292C2×C4C2×AGL(1,ℤ/8)12yes—
293C2×C4#57712yes—
294C2×C4#57812yes—
295C8—24no—
296C8—42no—
297C8—84no—
298C22—48no—
299C22—44no—
300C22—82no—
301C22—82no—
302C22—84no—
303C22#240911yescharacteristic
304C22C2×#45012yes—
305C22#301312yes—
306C22C2×#45311yescharacteristic
307C22#301811yescharacteristic
308C4—24no—
309C4—21no—
310C4—21no—
311C4—41no—
312C4—48no—
313C4—82no—
314C4—84no—
315C4—82no—
316C4—161no—
317C4#162914yes—
318C4#335912yes—
319C4#337312yes—
320C4Q8×SD1612yes—
321C4#338612yes—
322C2—84no—
323C2C2×#162911yescharacteristic
324C2#1867512yes—
325C2#1868811yescharacteristic
326C2#2431211yescharacteristic
327C2#2433012yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #29287C2semidirect
4. #10574C2semidirect
5. #10349C2semidirect
6. #9870C2semidirect
53. #492Q8semidirect
112. C4⋊Q8SD16semidirect
113. C4⋊Q8C2×Q8semidirect
163. C4×C8C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#598172
#292872
C4×#4922
#105742
#103492
#98702
#90642
#89242
#85852