Finite Group Databasebeta

← #320728

Subgroups — #320728

Order 512.

670 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 670 conjugacy classes of subgroups — too many to draw. Only its 170 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#29179C212yesmaximal
3#28717C212yesmaximal
4#21727C211yesmaximal, characteristic
5#10676C211yesmaximal, characteristic
6#10674C211yesmaximal, characteristic
7#10317C211yesmaximal, characteristic
8#10317C211yesmaximal, characteristic
9#9840C212yesmaximal
10#9833C212yesmaximal
11#8263C211yesmaximal, characteristic
12C2×#498C2211yescharacteristic
13C2×#491—22no—
14C2×#487C2212yes—
15C2×#487C2212yes—
16#2868C2211yescharacteristic
17#2859C2211yescharacteristic
18C4×C4⋊Q8C2212yes—
19#2087—24no—
20#2072—24no—
21#1990C2212yes—
22#1990C2212yes—
23#1990C2212yes—
24#1990C2212yes—
25#1990C2212yes—
26#1960—22no—
27#1956C2211yescharacteristic
28#1956C2211yescharacteristic
29#1956C2211yescharacteristic
30#1956C2211yescharacteristic
31#1944—22no—
32#1921—24no—
33#1845C2212yes—
34#1845C2211yescharacteristic
35#1834C2212yes—
36#1803—24no—
37#1776—22no—
38C4×C4⋊C8C2212yes—
39C4×#97C2211yescharacteristic
40C2×#322C2212yes—
41#1273C2211yescharacteristic
42#1273C2211yescharacteristic
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8C2311yescharacteristic
46C2×C4⋊Q8C2311yescharacteristic
47C2×C4⋊Q8C2312yes—
48C2×#125—21no—
49C2×C4×Q8—22no—
50#498—24no—
51#491—48no—
52#487—24no—
53#487—24no—
54#487—42no—
55#487—42no—
56C2×#107—24no—
57C2×#107—22no—
58C2×#106—22no—
59C2×C4⋊C8D414yes—
60C2×#103—24no—
61C2×#103—24no—
62C2×#103—22no—
63C2×#103—22no—
64C2×#103—22no—
65C2×#103—22no—
66C2×#97D412yes—
67C2×#97D412yes—
68C2×C4×C8C2312yes—
69C2×C4×C8C2312yes—
70C2×C4×C8C2311yescharacteristic
71C2×C4×C8C2311yescharacteristic
72#391—22no—
73#389—22no—
74#389—22no—
75#389—22no—
76#389—22no—
77#384—22no—
78#384—22no—
79#384C2312yes—
80#384C2311yescharacteristic
81#384C2311yescharacteristic
82#382—21no—
83C4×C4⋊C4—22no—
84C4×C4⋊C4—22no—
85C43C2311yescharacteristic
86#340—24no—
87#340—24no—
88#340—24no—
89#340—24no—
90#340—24no—
91#340—24no—
92#336D414yes—
93#322—42no—
94C22×Q8—22no—
95C22×Q8—21no—
96C22×Q8—21no—
97C22×C8C2×D414yes—
98C22×C8C2×D414yes—
99C22×C8C2×D412yes—
100C22×C8C2×D412yes—
101C4⋊Q8—22no—
102C4⋊Q8—22no—
103C4⋊Q8—22no—
104C4⋊Q8—22no—
105C4⋊Q8—21no—
106C4⋊Q8—21no—
107C4⋊Q8—22no—
108C4⋊Q8—21no—
109C4⋊Q8—21no—
110C4⋊Q8—44no—
111C4⋊Q8—44no—
112C4⋊Q8#4114yes—
113#125—44no—
114C4×Q8—48no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—42no—
132C2×C42—42no—
133C2×C42C2×D412yes—
134C2×C42C2×D411yescharacteristic
135C2×C42C2×D412yes—
136C2×C42C2×D411yescharacteristic
137C2×C42C2411yescharacteristic
138#107—48no—
139#107—44no—
140#106—28no—
141C4⋊C8—44no—
142#103—48no—
143#103—48no—
144#103—44no—
145#103—44no—
146#103—44no—
147#103—44no—
148#97—24no—
149#97—24no—
150C4×C8—22no—
151C4×C8—22no—
152C4×C8—22no—
153C4×C8—21no—
154C4×C8—22no—
155C4×C8—22no—
156C4×C8—21no—
157#95—42no—
158#95—42no—
159#95—42no—
160C2×Q8—24no—
161C2×Q8—42no—
162C2×Q8—41no—
163C2×Q8—41no—
164C2×Q8—41no—
165C2×Q8—41no—
166C2×Q8—42no—
167C2×Q8—42no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×Q8—44no—
171C2×Q8—44no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4#12711yescharacteristic
179C22×C4#12711yescharacteristic
180C22×C4#12712yes—
181C22×C4C22×D411yescharacteristic
182C22×C4C22×D412yes—
183C2×C8—22no—
184C2×C8—22no—
185C2×C8—22no—
186C2×C8—24no—
187C2×C8—24no—
188C2×C8—22no—
189C2×C8—22no—
190C2×C8—24no—
191C2×C8—24no—
192C2×C8—24no—
193C2×C8—24no—
194C2×C8AGL(1,ℤ/8)14yes—
195C4⋊C4—24no—
196C4⋊C4—24no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—44no—
206C4⋊C4—44no—
207C4⋊C4—44no—
208C4⋊C4—44no—
209C4⋊C4—44no—
210C4⋊C4—84no—
211C4⋊C4—82no—
212C4⋊C4—82no—
213C4⋊C4—82no—
214C4⋊C4—82no—
215C4⋊C4—84no—
216C42—22no—
217C42—24no—
218C42—24no—
219C42—84no—
220C42#13714yes—
221C42C2×#4112yes—
222C42H(5,𝔽2)11yescharacteristic
223C422−1+411yescharacteristic
224C23C2×#12711yescharacteristic
225Q8—48no—
226Q8—84no—
227Q8—82no—
228Q8—82no—
229Q8—82no—
230Q8—82no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—44no—
238C2×C4—82no—
239C2×C4—82no—
240C2×C4—81no—
241C2×C4—81no—
242C2×C4—81no—
243C2×C4—81no—
244C2×C4—82no—
245C2×C4—81no—
246C2×C4—81no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4—81no—
252C2×C4—81no—
253C2×C4#49714yes—
254C2×C4#52511yescharacteristic
255C2×C4#53711yescharacteristic
256C2×C4#54612yes—
257C2×C4#54812yes—
258C2×C4C2×AGL(1,ℤ/8)12yes—
259C2×C4C2×#13712yes—
260C2×C4#57514yes—
261C2×C4#57814yes—
262C8—24no—
263C8—42no—
264C8—44no—
265C8—44no—
266C22#254011yescharacteristic
267C22C2×#49711yescharacteristic
268C22#315111yescharacteristic
269C22#315312yes—
270C22#315612yes—
271C4—24no—
272C4—41no—
273C4—41no—
274C4—84no—
275C4—161no—
276C4—161no—
277C4—162no—
278C4—161no—
279C4—161no—
280C4#311212yes—
281C4#325412yes—
282C4#331514yes—
283C4#337714yes—
284C2#1968112yes—
285C2#1973111yescharacteristic
286C2#1975011yescharacteristic
287C2#2838611yescharacteristic
288C2#2842712yes—
subgroupG/H
center, socle, prime radicalC23C2×#127
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
#291792
#287172
#217272
#106762
#106742
#103172
#98402
#98332
#82632