Finite Group Databasebeta

← #303582

Subgroups — #303582

Order 512.

938 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 938 conjugacy classes of subgroups — too many to draw. Only its 174 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#36180C211yesmaximal, characteristic
2#33066C211yesmaximal, characteristic
3#30220C211yesmaximal, characteristic, complemented
4#28725C211yesmaximal, characteristic, complemented
5#10414C214yesmaximal, complemented
6#9702C211yesmaximal, characteristic, complemented
7#9641C211yesmaximal, characteristic, complemented
8#8450C214yesmaximal
9#8174C211yesmaximal, characteristic
10C2×#554C2212yes—
11C2×#491—21no—
12#2868C2211yescharacteristic
13#2859C2211yescharacteristic
14#2857—21no—
15#2567C2212yes—
16#2552—22no—
17#2502—22no—
18#2440—24no—
19D4×C4⋊C4—22no—
20C4×C4⋊Q8C2211yescharacteristic
21C2×#384—21no—
22C42×D4C2211yescharacteristic
23#2074—22no—
24#2043—24no—
25#1992—21no—
26#1956C2214yes—
27#1947C2212yes—
28#1918—22no—
29#1862—22no—
30#1831C2211yescharacteristic
31#1803C2214yes—
32#1802C2212yes—
33#1767C2211yescharacteristic
34#1744C2211yescharacteristic
35C2×#333C2214yes—
36C2×Q8⋊C8C2214yes—
37C2×D4⋊C8C2212yes—
38#1279C2212yes—
39#554—28no—
40#554—44no—
41C2×C4⋊Q8D411yescharacteristic
42C2×C4⋊Q8D414yes—
43C2×C4⋊Q8C2312yes—
44C2×#125—22no—
45C2×#125—21no—
46C2×C22⋊Q8—22no—
47C2×C22⋊Q8—24no—
48C2×C4×Q8C2312yes—
49C2×C4×D4—22no—
50C2×C4×D4D412yes—
51C2×C4×D4C2311yescharacteristic
52C22×C4⋊C4—24no—
53C22×C4⋊C4—22no—
54C22×C42—21no—
55#491—44no—
56C2×#107—22no—
57C2×C4⋊C8C2312yes—
58C2×C4⋊C8C2312yes—
59C2×#103—24no—
60C2×#103—28no—
61C2×#102—22no—
62C2×#97—21no—
63C2×C4×C8C2312yes—
64#398—28no—
65#395—24no—
66#391—22no—
67#391D412yes—
68#389—24no—
69#389—24no—
70#389—22no—
71#389—22no—
72#389—22no—
73#385—28no—
74#384—24no—
75#384—22no—
76#384—21no—
77#384—22no—
78#384—21no—
79#384D411yescharacteristic
80#384C2312yes—
81#382—21no—
82#380—24no—
83C4×C4⋊C4—22no—
84C4×C4⋊C4D412yes—
85C4×C4⋊C4C2311yescharacteristic
86C4×C22⋊C4—22no—
87C4×C22⋊C4—21no—
88C43C2311yescharacteristic
89#340—22no—
90#340—24no—
91#340—22no—
92#336—21no—
93#336—22no—
94#336—22no—
95#333—44no—
96Q8⋊C8—44no—
97D4⋊C8—24no—
98C22×Q8C2×D412yes—
99C22×Q8C2×D414yes—
100C22×D4C2×D411yescharacteristic
101C23×C4—22no—
102C23×C4—21no—
103C23×C4—24no—
104C22×C8—22no—
105C22×C8—22no—
106C22×C8—22no—
107C4⋊Q8—22no—
108C4⋊Q8—22no—
109C4⋊Q8—24no—
110C4⋊Q8—24no—
111C4⋊Q8—84no—
112#125—44no—
113#125—44no—
114#125—42no—
115#125—42no—
116C22⋊Q8—416no—
117C22⋊Q8—48no—
118C4×Q8—24no—
119C4×Q8—24no—
120C4×D4—24no—
121C4×D4—24no—
122C4×D4—22no—
123C4×D4—44no—
124C4×D4—44no—
125C4×D4#4114yes—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—28no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—48no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—88no—
142C2×C4⋊C4C2×D412yes—
143C2×C4⋊C4C2×D411yescharacteristic
144C2×C4⋊C4C2×D412yes—
145C2×C22⋊C4—22no—
146C2×C22⋊C4—21no—
147C2×C22⋊C4—24no—
148C2×C42—22no—
149C2×C42—22no—
150C2×C42—21no—
151C2×C42—42no—
152C2×C42—42no—
153C2×C42C2×D412yes—
154C2×C42C2×D411yescharacteristic
155C2×C42C2×D412yes—
156C2×C42C2×D411yescharacteristic
157C2×C42C2411yescharacteristic
158#107—44no—
159C4⋊C8—22no—
160C4⋊C8—24no—
161C4⋊C8—22no—
162#103—48no—
163#103—88no—
164#102—44no—
165#97—42no—
166C4×C8—22no—
167C4×C8—22no—
168#95—28no—
169#95—22no—
170#95—24no—
171#95—21no—
172#95—42no—
173C24—21no—
174C2×Q8—24no—
175C2×Q8—28no—
176C2×Q8—22no—
177C2×Q8—22no—
178C2×Q8—24no—
179C2×Q8—48no—
180C2×D4—21no—
181C2×D4—22no—
182C2×D4#12412yes—
183C2×D4#13714yes—
184C22×C4—28no—
185C22×C4—22no—
186C22×C4—21no—
187C22×C4—24no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—24no—
191C22×C4—44no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—42no—
195C22×C4—44no—
196C22×C4—48no—
197C22×C4—42no—
198C22×C4—88no—
199C22×C4#12012yes—
200C22×C4#12012yes—
201C22×C4C22×D411yescharacteristic
202C22×C4C22×D411yescharacteristic
203C22×C4C22×D411yescharacteristic
204C2×C8—44no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—42no—
212C4⋊C4—21no—
213C4⋊C4—22no—
214C4⋊C4—24no—
215C4⋊C4—48no—
216C4⋊C4—44no—
217C4⋊C4—48no—
218C4⋊C4—48no—
219C4⋊C4—44no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—44no—
223C4⋊C4—44no—
224C4⋊C4—44no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—84no—
230C4⋊C4—88no—
231C4⋊C4—84no—
232C4⋊C4#12412yes—
233C4⋊C4#13714yes—
234C22⋊C4—28no—
235C22⋊C4—24no—
236C22⋊C4—48no—
237C42—22no—
238C42—22no—
239C42—22no—
240C42—42no—
241C42—42no—
242C42—42no—
243C42—41no—
244C42—41no—
245C42—44no—
246C42#13512yes—
247C42AGL(1,ℤ/8)12yes—
248C42C2×#4112yes—
249C42H(5,𝔽2)12yes—
250C23—22no—
251C23—24no—
252C23—41no—
253C23—41no—
254C23—42no—
255C23C2×#12011yescharacteristic
256Q8—44no—
257Q8—44no—
258Q8—88no—
259D4—22no—
260D4—24no—
261D4#49114yescomplemented
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—24no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—416no—
271C2×C4—41no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—44no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—44no—
283C2×C4—48no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—168no—
288C2×C4#44914yes—
289C2×C4C2×#12411yescharacteristic
290C2×C4#53511yescharacteristic
291C2×C4#54612yes—
292C2×C4#54612yes—
293C2×C4C2×#13511yescharacteristic
294C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
295C2×C4C2×#13712yes—
296C2×C4C2×#13712yes—
297C2×C4#57812yes—
298C8—82no—
299C8—82no—
300C8—82no—
301C22—24no—
302C22—42no—
303C22—42no—
304C22—44no—
305C22—44no—
306C22#243511yescharacteristic
307C22C2×#44911yescharacteristic
308C22#300412yes—
309C22#301111yescharacteristic
310C22#301912yes—
311C4—22no—
312C4—24no—
313C4—42no—
314C4—44no—
315C4—42no—
316C4—82no—
317C4—84no—
318C4—82no—
319C4—162no—
320C4C2×#49111yescharacteristic
321C4#313911yescharacteristic
322C4#321812yes—
323C4#330212yes—
324C4#333012yes—
325C2—44no—
326C2#1875212yes—
327C2#1881712yes—
328C2#2586111yescharacteristic
329C2#2587811yescharacteristic
330C2#2588211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
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
3. #30220C2semidirect
4. #28725C2semidirect
5. #10414C2semidirect
6. #9702C2semidirect
7. #9641C2semidirect
261. D4#491semidirect

Maximal subgroups

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

subgroupindex
#361802
#330662
#302202
#287252
#104142
#97022
#96412
#84502
#81742