Finite Group Databasebeta

← #330526

Subgroups — #330526

Order 512.

544 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 154 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#33232C211yesmaximal, characteristic
2#29996C211yesmaximal, characteristic
3#29866C211yesmaximal, characteristic
4#21343C211yesmaximal, characteristic
5#10600C212yesmaximal
6#10596C212yesmaximal
7#10487C212yesmaximal
8#10486C212yesmaximal
9#9899C211yesmaximal, characteristic
10#9694C211yesmaximal, characteristic
11#8808C211yesmaximal, characteristic
12C2×#491—21no—
13C2×#491—21no—
14C2×#475C2211yescharacteristic
15C2×#475C2211yescharacteristic
16#2868C2211yescharacteristic
17#2859C2211yescharacteristic
18#2295C2211yescharacteristic
19#2088—22no—
20#2084—22no—
21#2072—22no—
22#2066—22no—
23#1988C2211yescharacteristic
24#1988C2211yescharacteristic
25#1984C2212yes—
26#1960—22no—
27#1956C2212yes—
28#1956C2212yes—
29#1948C2212yes—
30#1939C2212yes—
31#1917—22no—
32#1842C2211yescharacteristic
33#1831C2211yescharacteristic
34#1803C2212yes—
35#1768—21no—
36C4×C4⋊C8C2211yescharacteristic
37#1746—21no—
38C4×#97C2211yescharacteristic
39C2×#335C2212yes—
40C2×#334C2212yes—
41C2×#332C2212yes—
42C2×#332C2212yes—
43#1296C2212yes—
44#1296C2212yes—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8C2311yescharacteristic
48C2×C4⋊Q8C2311yescharacteristic
49C2×#125—21no—
50C2×C4×Q8C2311yescharacteristic
51#491—44no—
52#491—44no—
53#475—42no—
54#475—42no—
55C2×#106—21no—
56C2×#106—21no—
57C2×#106—22no—
58C2×C4⋊C8—22no—
59C2×C4⋊C8—22no—
60C2×C4⋊C8C2311yescharacteristic
61C2×C4⋊C8C2311yescharacteristic
62C2×C4⋊C8C2312yes—
63C2×C4⋊C8C2312yes—
64C2×#103—21no—
65C2×#103—21no—
66C2×#103—22no—
67C2×#103—22no—
68C2×#103—22no—
69C2×#103—22no—
70C2×#103—22no—
71C2×#97D411yescharacteristic
72C2×#97D412yes—
73C2×#97D411yescharacteristic
74C2×C4×C8—21no—
75C2×C4×C8C2312yes—
76#391D412yes—
77#389—22no—
78#389—22no—
79#389D412yes—
80#387—22no—
81#384—22no—
82#384—22no—
83#384C2312yes—
84#382—21no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4C2311yescharacteristic
87#376—21no—
88C43C2311yescharacteristic
89#340—22no—
90#340—22no—
91#340—22no—
92#340—22no—
93#336—22no—
94#336—22no—
95#336—22no—
96#336—22no—
97#336—21no—
98#335—42no—
99#334—42no—
100#332—42no—
101#332—42no—
102C22×Q8—21no—
103C22×Q8—21no—
104C22×Q8C2×D411yescharacteristic
105C22×C8—22no—
106C22×C8—21no—
107C22×C8—21no—
108C22×C8—22no—
109C22×C8C2×D412yes—
110C22×C8C2×D412yes—
111C4⋊Q8—22no—
112C4⋊Q8—22no—
113C4⋊Q8—22no—
114C4⋊Q8—22no—
115C4⋊Q8—44no—
116C4⋊Q8—42no—
117C4⋊Q8—42no—
118#125—42no—
119#125—42no—
120C4×Q8—22no—
121C4×Q8—22no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4C2×D411yescharacteristic
135C2×C4⋊C4C2×D412yes—
136C2×C42—21no—
137C2×C42—21no—
138C2×C42C2×D411yescharacteristic
139C2×C42C2×D412yes—
140C2×C42C2×D411yescharacteristic
141C2×C42#4112yes—
142C2×C42C2411yescharacteristic
143#106—42no—
144#106—42no—
145#106—82no—
146C4⋊C8—22no—
147C4⋊C8—22no—
148C4⋊C8—22no—
149C4⋊C8—22no—
150C4⋊C8—22no—
151C4⋊C8—22no—
152C4⋊C8—44no—
153C4⋊C8—82no—
154#103—44no—
155#103—44no—
156#103—42no—
157#103—42no—
158#103—44no—
159#103—44no—
160#103—82no—
161#97—22no—
162#97—42no—
163#97SD1614yes—
164C4×C8—22no—
165C4×C8—22no—
166C4×C8—42no—
167#95—22no—
168#95—22no—
169#95—22no—
170#95—21no—
171#95—42no—
172C2×Q8—21no—
173C2×Q8—21no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8—42no—
177C2×Q8—42no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8—41no—
181C2×Q8—41no—
182C2×Q8—42no—
183C2×Q8—42no—
184C22×C4—22no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—42no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4#12112yes—
191C22×C4#12112yes—
192C22×C4C22×D411yescharacteristic
193C22×C4C22×D411yescharacteristic
194C22×C4C2×#4111yescharacteristic
195C2×C8—22no—
196C2×C8—22no—
197C2×C8—22no—
198C2×C8—22no—
199C2×C8—22no—
200C2×C8—41no—
201C2×C8—41no—
202C2×C8—42no—
203C2×C8—42no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8C2×SD1614yes—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—22no—
214C4⋊C4—22no—
215C4⋊C4—42no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—82no—
227C4⋊C4—82no—
228C4⋊C4—84no—
229C4⋊C4—84no—
230C4⋊C4—84no—
231C42—22no—
232C42—22no—
233C42—22no—
234C42—42no—
235C42—42no—
236C42—42no—
237C42C2×SD1612yes—
238C42AGL(1,ℤ/8)12yes—
239C42H(5,𝔽2)12yes—
240C422−1+412yes—
241C23C2×#12111yescharacteristic
242Q8—41no—
243Q8—41no—
244Q8—42no—
245Q8—82no—
246Q8—82no—
247Q8—82no—
248Q8—82no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—81no—
263C2×C4—81no—
264C2×C4—82no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4#46014yes—
272C2×C4#53511yescharacteristic
273C2×C4#53611yescharacteristic
274C2×C4#53712yes—
275C2×C4#54212yes—
276C2×C4C22×SD1611yescharacteristic
277C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
278C2×C4#57512yes—
279C2×C4#57712yes—
280C2×C4#57812yes—
281C8—24no—
282C8—42no—
283C8—81no—
284C8—82no—
285C8—81no—
286C8—82no—
287C22#262011yescharacteristic
288C22C2×#46011yescharacteristic
289C22#304011yescharacteristic
290C22#306312yes—
291C22#306912yes—
292C4—24no—
293C4—22no—
294C4—42no—
295C4—81no—
296C4—81no—
297C4—82no—
298C4—162no—
299C4—161no—
300C4—161no—
301C4#322111yescharacteristic
302C4#322311yescharacteristic
303C4#324212yes—
304C4#324612yes—
305C4#325112yes—
306C2#1923912yes—
307C2#1938112yes—
308C2#2982611yescharacteristic
309C2#2985611yescharacteristic
310C2#2986211yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#121
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#332322
#299962
#298662
#213432
#106002
#105962
#104872
#104862
#98992
#96942
#88082