Finite Group Databasebeta

← #316199

Subgroups — #316199

Order 512.

449 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 449 conjugacy classes of subgroups — too many to draw. Only its 122 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#29812C211yesmaximal, characteristic
2#21995C211yesmaximal, characteristic
3#10635C212yesmaximal, complemented
4#10632C212yesmaximal, complemented
5#9779C211yesmaximal, characteristic, complemented
6#9772C211yesmaximal, characteristic, complemented
7#9196C212yesmaximal
8#9195C212yesmaximal
9#8794C211yesmaximal, characteristic, complemented
10#8792C211yesmaximal, characteristic, complemented
11#8158C211yesmaximal, characteristic
12C2×#494C2211yescharacteristic
13#2921C2211yescharacteristic
14C2×#433C2211yescharacteristic
15#2905C2211yescharacteristic
16#2621C2212yes—
17#2056—22no—
18#2001—22no—
19#1960C2212yes—
20#1950—22no—
21#1855—22no—
22#1849C2211yescharacteristic
23#1848C2211yescharacteristic
24#1838C2211yescharacteristic
25#1837C2211yescharacteristic
26#1835C2211yescharacteristic
27#1546C2212yes—
28#1546C2212yes—
29#1546C2212yes—
30#1546C2212yes—
31#1458C2212yes—
32#1452C2211yescharacteristic
33#1450C2211yescharacteristic
34#1286C2212yes—
35#1286C2212yes—
36#1286C2212yes—
37#1286C2212yes—
38#1281C2212yes—
39C22×Q16—22no—
40C22×M16—21no—
41C2×C4⋊Q8C2312yes—
42C2×C22⋊Q8—22no—
43C2×#117C2311yescharacteristic
44#494—21no—
45#494—21no—
46#494—22no—
47#433—21no—
48#433—21no—
49#433—22no—
50#433—42no—
51#431C2312yes—
52#431C2312yes—
53C2×#108—22no—
54C2×#108—22no—
55#424—42no—
56#423—21no—
57#423C2311yescharacteristic
58#423C2311yescharacteristic
59#423C2311yescharacteristic
60#423C2311yescharacteristic
61C2×C4⋊C8D412yes—
62C2×#103—22no—
63C2×#103—22no—
64#407—21no—
65C2×C22⋊C8D412yes—
66C4×M16—21no—
67C2×#97C2311yescharacteristic
68C2×C4×C8C2311yescharacteristic
69#398—22no—
70#398—22no—
71#397—22no—
72#397—22no—
73#391—22no—
74#382C2312yes—
75#341—22no—
76#341—22no—
77#340—22no—
78#332—42no—
79#332—42no—
80#332—42no—
81#332—42no—
82#324—22no—
83#323—22no—
84#322—21no—
85#322—22no—
86C82—21no—
87C22×Q8—22no—
88C23×C4C2×D411yescharacteristic
89C2×Q16—42no—
90C2×Q16—42no—
91C2×Q16—82no—
92C2×Q16—82no—
93C2×M16—21no—
94C2×M16—21no—
95C2×M16—21no—
96C2×M16—21no—
97C2×M16—41no—
98C2×M16—82no—
99C22×C8—41no—
100C22×C8C2×D412yes—
101C22×C8C2×D412yes—
102C4⋊Q8—22no—
103C4⋊Q8—22no—
104C4⋊Q8—22no—
105C4⋊Q8—22no—
106C22⋊Q8—82no—
107C22⋊Q8—82no—
108#117—21no—
109#117—21no—
110#117—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4#4112yes—
116C2×C22⋊C4—42no—
117C2×C22⋊C4#4112yes—
118C2×C42C2×D411yescharacteristic
119C2×C42C2411yescharacteristic
120#108—82no—
121#108—82no—
122C4⋊C8—22no—
123C4⋊C8—22no—
124C4⋊C8—22no—
125C4⋊C8—22no—
126C4⋊C8—22no—
127C4⋊C8—22no—
128C4⋊C8—22no—
129C4⋊C8—22no—
130C4⋊C8—42no—
131#103—82no—
132#103—82no—
133C22⋊C8—22no—
134C22⋊C8—22no—
135C22⋊C8—22no—
136C22⋊C8—22no—
137C22⋊C8—42no—
138#97—21no—
139#97—21no—
140#97—41no—
141C4×C8—22no—
142C4×C8—22no—
143C4×C8—41no—
144C4×C8#4111yescharacteristic
145C4×C8#4111yescharacteristic
146C4×C8#4112yes—
147#95—22no—
148#95—22no—
149#95—42no—
150#95—42no—
151C24#12411yescharacteristic
152C2×Q8—42no—
153C2×Q8—42no—
154C2×Q8—42no—
155C2×Q8—42no—
156C2×Q8—42no—
157C2×Q8—42no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4#12411yescharacteristic
165C22×C4#12412yes—
166C22×C4C22×D411yescharacteristic
167C22×C4C2×#4112yes—
168Q16—84no—
169Q16—84no—
170M16—42no—
171M16—42no—
172M16—42no—
173M16—42no—
174M16—82no—
175C2×C8—22no—
176C2×C8—22no—
177C2×C8—22no—
178C2×C8—22no—
179C2×C8—22no—
180C2×C8—41no—
181C2×C8—41no—
182C2×C8—41no—
183C2×C8—41no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8—82no—
187C2×C8#12312yes—
188C2×C8#12312yes—
189C4⋊C4—22no—
190C4⋊C4—22no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—84no—
198C22⋊C4—22no—
199C22⋊C4—22no—
200C22⋊C4—84no—
201C42—21no—
202C42—21no—
203C42C2×#4111yescharacteristic
204C42C2×#4111yescharacteristic
205C42H(5,𝔽2)12yes—
206C23—21no—
207C23—22no—
208C23—42no—
209C23C2×#12411yescharacteristic
210Q8—82no—
211Q8—82no—
212Q8—82no—
213Q8—82no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4—84no—
223C2×C4—82no—
224C2×C4—82no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4C2×#12311yescharacteristic
228C2×C4#53511yescharacteristic
229C2×C4#55312yes—
230C2×C4#55312yes—
231C2×C4#57612yes—
232C2×C4#57712yes—
233C8—22no—
234C8—22no—
235C8—42no—
236C8—82no—
237C8—82no—
238C8—81no—
239C8—81no—
240C8—81no—
241C8—81no—
242C22—22no—
243C22—41no—
244C22—42no—
245C22—82no—
246C22#250411yescharacteristic
247C22#313612yes—
248C4—22no—
249C4—22no—
250C4—41no—
251C4—42no—
252C4—162no—
253C4—162no—
254C4#309612yes—
255C4#322011yescharacteristic
256C4#322111yescharacteristic
257C2—22no—
258C2—81no—
259C2#2798611yescharacteristic
260C2#2798812yes—
subgroupG/H
center, socleC22#2504
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#124
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #10635C2semidirect
4. #10632C2semidirect
5. #9779C2semidirect
6. #9772C2semidirect
9. #8794C2semidirect
10. #8792C2semidirect

Maximal subgroups

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

subgroupindex
#298122
#219952
#106352
#106322
#97792
#97722
#91962
#91952
#87942
#87922
#81582