Finite Group Databasebeta

← #303602

Subgroups — #303602

Order 512.

894 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 894 conjugacy classes of subgroups — too many to draw. Only its 182 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
1Q8×C4⋊Q8C211yesmaximal, characteristic
2#33067C211yesmaximal, characteristic
3#30220C211yesmaximal, characteristic
4#28725C211yesmaximal, characteristic
5#10415C214yesmaximal
6#9702C211yesmaximal, characteristic
7#9641C211yesmaximal, characteristic
8#8451C214yesmaximal
9#8175C211yesmaximal, characteristic
10C2×Q82C2212yes—
11C2×#491—21no—
12#2868—21no—
13#2868C2211yescharacteristic
14#2859C2211yescharacteristic
15#2567—22no—
16#2567C2212yes—
17#2551—22no—
18#2551—24no—
19Q8×C4⋊C4—22no—
20C4×C4⋊Q8—21no—
21C4×C4⋊Q8C2211yescharacteristic
22C42×Q8C2211yescharacteristic
23#2074—22no—
24#2043—24no—
25#1992—21no—
26#1956C2214yes—
27#1948C2212yes—
28#1919—22no—
29#1865—22no—
30#1831C2211yescharacteristic
31#1803C2212yes—
32#1803C2214yes—
33#1767C2211yescharacteristic
34#1744C2211yescharacteristic
35C2×#333C2214yes—
36C2×Q8⋊C8C2212yes—
37C2×Q8⋊C8C2214yes—
38#1279C2212yes—
39Q82—28no—
40Q82—44no—
41C2×C4⋊Q8—24no—
42C2×C4⋊Q8—24no—
43C2×C4⋊Q8—22no—
44C2×C4⋊Q8—22no—
45C2×C4⋊Q8D411yescharacteristic
46C2×C4⋊Q8D414yes—
47C2×C4⋊Q8C2312yes—
48C2×#125—21no—
49C2×C4×Q8—22no—
50C2×C4×Q8—22no—
51C2×C4×Q8D412yes—
52C2×C4×Q8C2312yes—
53C2×C4×Q8C2311yescharacteristic
54#491—44no—
55C2×#106—22no—
56C2×C4⋊C8C2312yes—
57C2×C4⋊C8C2312yes—
58C2×#103—24no—
59C2×#103—22no—
60C2×#103—28no—
61C2×#97—21no—
62C2×C4×C8C2312yes—
63#395—28no—
64#395—24no—
65#391—22no—
66#391D412yes—
67#389—24no—
68#389—28no—
69#389—24no—
70#389—22no—
71#389—22no—
72#389—22no—
73#384—24no—
74#384—22no—
75#384—21no—
76#384—21no—
77#384—22no—
78#384D411yescharacteristic
79#384C2312yes—
80#382—21no—
81C4×C4⋊C4—24no—
82C4×C4⋊C4—22no—
83C4×C4⋊C4—21no—
84C4×C4⋊C4—22no—
85C4×C4⋊C4D412yes—
86C4×C4⋊C4C2311yescharacteristic
87C43—21no—
88C43C2311yescharacteristic
89#340—24no—
90#340—22no—
91#340—22no—
92#336—21no—
93#336—22no—
94#336—22no—
95#333—44no—
96Q8⋊C8—24no—
97Q8⋊C8—44no—
98C22×Q8—24no—
99C22×Q8C2×D412yes—
100C22×Q8C2×D411yescharacteristic
101C22×Q8C2×D414yes—
102C22×C8—22no—
103C22×C8—22no—
104C22×C8—22no—
105C4⋊Q8—22no—
106C4⋊Q8—24no—
107C4⋊Q8—24no—
108C4⋊Q8—48no—
109C4⋊Q8—416no—
110C4⋊Q8—44no—
111C4⋊Q8—44no—
112C4⋊Q8—88no—
113C4⋊Q8—84no—
114C4⋊Q8SD1614yes—
115#125—42no—
116#125—42no—
117C4×Q8—24no—
118C4×Q8—24no—
119C4×Q8—22no—
120C4×Q8—24no—
121C4×Q8—24no—
122C4×Q8—48no—
123C4×Q8—44no—
124C4×Q8—44no—
125C4×Q8#4114yes—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—28no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—42no—
142C2×C4⋊C4C2×D412yes—
143C2×C4⋊C4C2×D411yescharacteristic
144C2×C4⋊C4C2×D412yes—
145C2×C42—22no—
146C2×C42—21no—
147C2×C42—24no—
148C2×C42—22no—
149C2×C42—22no—
150C2×C42—21no—
151C2×C42C2×D412yes—
152C2×C42C2×D411yescharacteristic
153C2×C42C2×D412yes—
154C2×C42C2×D411yescharacteristic
155C2×C42C2411yescharacteristic
156#106—44no—
157C4⋊C8—22no—
158C4⋊C8—24no—
159C4⋊C8—22no—
160#103—48no—
161#103—44no—
162#103—88no—
163#97—42no—
164C4×C8—22no—
165C4×C8—22no—
166#95—28no—
167#95—22no—
168#95—24no—
169#95—21no—
170#95—42no—
171C2×Q8—21no—
172C2×Q8—22no—
173C2×Q8—22no—
174C2×Q8—24no—
175C2×Q8—28no—
176C2×Q8—24no—
177C2×Q8—48no—
178C2×Q8—48no—
179C2×Q8—88no—
180C2×Q8#12412yes—
181C2×Q8C2×SD1614yes—
182C2×Q8#13714yes—
183C22×C4—21no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—21no—
187C22×C4—24no—
188C22×C4—22no—
189C22×C4—42no—
190C22×C4#12012yes—
191C22×C4#12012yes—
192C22×C4C22×D411yescharacteristic
193C22×C4C22×D411yescharacteristic
194C22×C4C22×D411yescharacteristic
195C2×C8—44no—
196C2×C8—42no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—42no—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8—42no—
203C4⋊C4—28no—
204C4⋊C4—21no—
205C4⋊C4—24no—
206C4⋊C4—22no—
207C4⋊C4—28no—
208C4⋊C4—24no—
209C4⋊C4—48no—
210C4⋊C4—48no—
211C4⋊C4—48no—
212C4⋊C4—44no—
213C4⋊C4—44no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—44no—
217C4⋊C4—44no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—84no—
223C4⋊C4—88no—
224C4⋊C4—84no—
225C4⋊C4#12412yes—
226C4⋊C4#13714yes—
227C42—28no—
228C42—24no—
229C42—22no—
230C42—22no—
231C42—22no—
232C42—42no—
233C42—48no—
234C42—44no—
235C42—42no—
236C42—42no—
237C42—42no—
238C42C2×SD1612yes—
239C42#13712yes—
240C42C2×#4112yes—
241C42H(5,𝔽2)12yes—
242C23C2×#12011yescharacteristic
243Q8—22no—
244Q8—24no—
245Q8—28no—
246Q8—44no—
247Q8—88no—
248Q8—168no—
249Q8#49114yescomplemented
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—24no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—24no—
257C2×C4—22no—
258C2×C4—24no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—44no—
268C2×C4—48no—
269C2×C4—44no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4#44814yes—
274C2×C4C2×#12411yescharacteristic
275C2×C4#53511yescharacteristic
276C2×C4#54612yes—
277C2×C4#54612yes—
278C2×C4C22×SD1611yescharacteristic
279C2×C4C2×#13711yescharacteristic
280C2×C4C2×#13712yes—
281C2×C4C2×#13712yes—
282C2×C4#57812yes—
283C8—82no—
284C8—82no—
285C8—82no—
286C22#243511yescharacteristic
287C22C2×#44811yescharacteristic
288C22#300412yes—
289C22#301111yescharacteristic
290C22#301912yes—
291C4—24no—
292C4—22no—
293C4—24no—
294C4—42no—
295C4—44no—
296C4—42no—
297C4—44no—
298C4—84no—
299C4—82no—
300C4—162no—
301C4C2×#49111yescharacteristic
302C4#313911yescharacteristic
303C4#321812yes—
304C4#329812yes—
305C4#332812yes—
306C2#1876812yes—
307C2#1881712yes—
308C2#2586111yescharacteristic
309C2#2587811yescharacteristic
310C2#2588411yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#120
commutator, Frattini, cosocleC2×C42C24
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
249. Q8#491semidirect

Maximal subgroups

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

subgroupindex
Q8×C4⋊Q82
#330672
#302202
#287252
#104152
#97022
#96412
#84512
#81752