Finite Group Databasebeta

← #492407

Subgroups — #492407

Order 512.

393 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 393 conjugacy classes of subgroups — too many to draw. Only its 118 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#33950C211yesmaximal, characteristic, complemented
2#33909C211yesmaximal, characteristic
3#19964C212yesmaximal
4#12521C211yesmaximal, characteristic, complemented
5#12498C212yesmaximal, complemented
6#12457C211yesmaximal, characteristic, complemented
7#12451C211yesmaximal, characteristic
8#12405C212yesmaximal, complemented
9#12393C212yesmaximal
10#12083C211yesmaximal, characteristic, complemented
11C16×SD16C211yesmaximal, characteristic
12#3398—22no—
13#3383C2212yes—
14#3342C2212yes—
15#2253—21no—
16#2252C2211yescharacteristic
17#2252C2211yescharacteristic
18#2252C2212yes—
19#2237—22no—
20#2229—22no—
21#2225C2212yes—
22#2225C2212yes—
23#2216—21no—
24Q8×C16C2211yescharacteristic
25C4×Q32—21no—
26C4×Q32C2211yescharacteristic
27D4×C16C2211yescharacteristic
28#2158—21no—
29#1713C2212yes—
30#1690—22no—
31#1674C2212yes—
32#1673C2212yes—
33#1650—22no—
34#1648—22no—
35#1591C2212yes—
36C8×SD16C2211yescharacteristic
37#1371C2211yescharacteristic
38#1367C2211yescharacteristic
39#1364C2212yes—
40#1364C2212yes—
41Q8⋊C16C2211yescharacteristic
42#1332C2212yes—
43D4⋊C16C2211yescharacteristic
44C8×C16C2211yescharacteristic
45#557—22no—
46#556—22no—
47C2×Q32D411yescharacteristic
48C22×C16—21no—
49#500—22no—
50#500C2312yes—
51#500C2312yes—
52#499—22no—
53#497—42no—
54#494—22no—
55#485—42no—
56#479—42no—
57#475—22no—
58#475—22no—
59#467—22no—
60#464—42no—
61#462—22no—
62#462—42no—
63#461—22no—
64#446—22no—
65C8×Q8C2311yescharacteristic
66#441—42no—
67C4×Q16—22no—
68C4×Q16C2312yes—
69C4×SD16D411yescharacteristic
70C8×D4C2311yescharacteristic
71#427—22no—
72#367—22no—
73#367—22no—
74#366—21no—
75#366—21no—
76#366—21no—
77#366—21no—
78#366—42no—
79#366D411yescharacteristic
80C4⋊C16—21no—
81C4⋊C16C2311yescharacteristic
82C4⋊C16C2311yescharacteristic
83#358—22no—
84#358—22no—
85#358D412yes—
86C22⋊C16—21no—
87C4×C16—21no—
88C4×C16C2311yescharacteristic
89C4×C16C2311yescharacteristic
90#335C2312yes—
91#334D411yescharacteristic
92#332—22no—
93#332—22no—
94Q8⋊C8—22no—
95Q8⋊C8D411yescharacteristic
96D4⋊C8D411yescharacteristic
97C82C2311yescharacteristic
98C2×Q16—42no—
99C2×Q16—42no—
100C2×Q16—42no—
101C2×Q16C2×D412yes—
102C2×SD16—42no—
103C2×SD16D811yescharacteristic
104C22×C8—21no—
105C4⋊Q8—22no—
106C4⋊Q8—22no—
107C4⋊Q8—42no—
108#126—82no—
109#125—42no—
110#125—82no—
111#124—42no—
112C22⋊Q8—42no—
113C4×Q8—22no—
114C4×Q8—42no—
115C4×Q8—42no—
116C4×Q8—42no—
117C4×Q8C2×D411yescharacteristic
118C4×D4C2×D411yescharacteristic
119#117—42no—
120Q32—22no—
121C2×C16—21no—
122C2×C16—21no—
123C2×C16—21no—
124C2×C16—42no—
125C2×C16C2×D411yescharacteristic
126C2×C16C2×D411yescharacteristic
127#107—22no—
128#107—22no—
129#107—22no—
130#107—22no—
131#107—82no—
132#107C2×D412yes—
133#106—42no—
134#106—42no—
135#106D811yescharacteristic
136C4⋊C8—21no—
137C4⋊C8—22no—
138C4⋊C8—42no—
139C4⋊C8C2×D411yescharacteristic
140C4⋊C8C2×D411yescharacteristic
141#103—22no—
142#103—42no—
143#103—42no—
144#103—42no—
145#103—42no—
146#103D811yescharacteristic
147#102D811yescharacteristic
148C22⋊C8—21no—
149#97—42no—
150C4×C8—21no—
151C4×C8C2×D411yescharacteristic
152C4×C8C2×D411yescharacteristic
153C4×C8C2411yescharacteristic
154C2×Q8—22no—
155C2×Q8—42no—
156C2×Q8—42no—
157C2×Q8C2×D811yescharacteristic
158C2×D4C2×D811yescharacteristic
159C22×C4—21no—
160Q16—22no—
161Q16—22no—
162Q16—84no—
163Q16—162no—
164Q16—162no—
165SD16—22no—
166SD16—162no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—41no—
171C2×C8—41no—
172C2×C8—42no—
173C2×C8C2×D811yescharacteristic
174C2×C8C2×D811yescharacteristic
175C2×C8C22×D411yescharacteristic
176C2×C8H(5,𝔽2)11yescharacteristic
177C4⋊C4—21no—
178C4⋊C4—22no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—82no—
185C4⋊C4—82no—
186C4⋊C4—82no—
187C4⋊C4—82no—
188C4⋊C4C2×D811yescharacteristic
189C4⋊C4C2×D811yescharacteristic
190C22⋊C4—21no—
191C22⋊C4—82no—
192C22⋊C4—82no—
193C42—21no—
194C42—42no—
195C42—82no—
196C42—82no—
197C42C22×D411yescharacteristic
198C16—22no—
199C16—21no—
200C16—41no—
201C16—41no—
202C16#13712yes—
203C23—21no—
204Q8—21no—
205Q8—21no—
206Q8—42no—
207Q8—42no—
208Q8—82no—
209Q8—82no—
210Q8—82no—
211Q8—82no—
212D4—21no—
213D4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—82no—
222C2×C4—82no—
223C2×C4D4211yescharacteristic
224C2×C4C22×D811yescharacteristic
225C2×C4#57611yescharacteristic
226C8—22no—
227C8—21no—
228C8—21no—
229C8—41no—
230C8—41no—
231C8—82no—
232C8#50812yes—
233C8C2×#13711yescharacteristic
234C8#57811yescharacteristic
235C22—41no—
236C22—41no—
237C22—41no—
238C22D4×D811yescharacteristic
239C4—21no—
240C4—41no—
241C4—41no—
242C4—41no—
243C4—82no—
244C4—82no—
245C4—162no—
246C4—162no—
247C4#327611yescharacteristic
248C4#329311yescharacteristic
249C4C2×#50811yescharacteristic
250C4#341511yescharacteristic
251C2—81no—
252C2#1988711yescharacteristic
253C2#3385311yescharacteristic
254C2#3386011yescharacteristic
subgroupG/H
center, socleC22D4×D8
commutator, Frattini, cosocleC4×C8C24
prime radicalC2×D4C2×D8
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ C8×SD16 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33950C2semidirect
4. #12521C2semidirect
5. #12498C2semidirect
6. #12457C2semidirect
8. #12405C2semidirect
10. #12083C2semidirect

Maximal subgroups

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

subgroupindex
#339502
#339092
#199642
#125212
#124982
#124572
#124512
#124052
#123932
#120832
C16×SD162