Finite Group Databasebeta

← #492451

Subgroups — #492451

Order 512.

406 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 114 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#33942C212yesmaximal, complemented
2#19942C212yesmaximal, complemented
3#12503C211yesmaximal, characteristic
4#12486C212yesmaximal, complemented
5#12480C212yesmaximal, complemented
6#12410C212yesmaximal, complemented
7#12329C212yesmaximal
8#12083C211yesmaximal, characteristic, complemented
9#12054C211yesmaximal, characteristic, complemented
10#3396C2212yes—
11Q8×Q16C2212yes—
12#3375—22no—
13#2252C2211yescharacteristic
14#2246C2211yescharacteristic
15#2243—22no—
16#2240C2212yes—
17#2227C2212yes—
18#2224—22no—
19#2211—22no—
20#2207C2212yes—
21#2183C2212yes—
22#2175—22no—
23C4×Q32C2211yescharacteristic
24#1706C2212yes—
25#1698—22no—
26#1676C2212yes—
27#1668C2212yes—
28#1630—22no—
29#1623—22no—
30#1591C2212yes—
31C8×Q16C2211yescharacteristic
32#1371C2211yescharacteristic
33#1370C2211yescharacteristic
34#1365C2212yes—
35#1361C2212yes—
36Q8⋊C16C2212yes—
37#1332C2212yes—
38C8×C16C2211yescharacteristic
39#559—22no—
40Q82—22no—
41C2×Q32D411yescharacteristic
42C2×SD32—22no—
43C2×D16—41no—
44#500—22no—
45#500C2312yes—
46#495—22no—
47#493C2312yes—
48#492—22no—
49#489—42no—
50#486—22no—
51#477—22no—
52#477—22no—
53#477—42no—
54#474—42no—
55#464—42no—
56#463—42no—
57#460—22no—
58#460—22no—
59#446—22no—
60C8×Q8C2312yes—
61#440—42no—
62C4×Q16—22no—
63C4×Q16D411yescharacteristic
64C4×Q16C2312yes—
65C4×D8—22no—
66#366—22no—
67#366—41no—
68#366D411yescharacteristic
69C4⋊C16—22no—
70C4⋊C16C2312yes—
71#358—22no—
72#358D412yes—
73#357—22no—
74#357—22no—
75#357—22no—
76#346—22no—
77C4×C16C2311yescharacteristic
78C4×C16C2311yescharacteristic
79#335D411yescharacteristic
80#335C2312yes—
81#333—22no—
82#331—22no—
83Q8⋊C8—22no—
84Q8⋊C8D412yes—
85C82C2311yescharacteristic
86C2×Q16D811yescharacteristic
87C2×Q16C2×D412yes—
88C2×SD16—42no—
89C2×SD16—42no—
90C2×SD16—82no—
91C2×D8—22no—
92C2×D8—22no—
93C4⋊Q8—22no—
94C4⋊Q8—42no—
95C4⋊Q8—42no—
96C4⋊Q8—82no—
97#127—22no—
98#124—42no—
99#124—42no—
100#123—82no—
101C4×Q8—22no—
102C4×Q8—42no—
103C4×Q8—42no—
104C4×Q8C2×D412yes—
105C4×D4—42no—
106C4×D4—42no—
107Q32—41no—
108SD32—44no—
109D16—82no—
110C2×C16—21no—
111C2×C16—22no—
112C2×C16C2×D411yescharacteristic
113C2×C16C2×D411yescharacteristic
114#107—22no—
115#107—22no—
116#107—42no—
117#107—42no—
118#107D811yescharacteristic
119#107C2×D412yes—
120C4⋊C8—22no—
121C4⋊C8—22no—
122C4⋊C8—42no—
123C4⋊C8C2×D412yes—
124#103—22no—
125#103—42no—
126#103—42no—
127#103D812yes—
128#102—42no—
129#102—42no—
130#102—42no—
131#102—42no—
132#97—42no—
133C4×C8—22no—
134C4×C8C2×D411yescharacteristic
135C4×C8C2×D411yescharacteristic
136C4×C8C2411yescharacteristic
137C2×Q8—22no—
138C2×Q8—42no—
139C2×Q8—82no—
140C2×Q8C2×D812yes—
141C2×D4—42no—
142C2×D4—42no—
143C2×D4—42no—
144C22×C4—82no—
145Q16—22no—
146Q16—22no—
147Q16—41no—
148SD16—84no—
149SD16—164no—
150SD16—162no—
151D8—42no—
152D8—42no—
153D8—82no—
154C2×C8—21no—
155C2×C8—22no—
156C2×C8—42no—
157C2×C8C2×D811yescharacteristic
158C2×C8C2×D811yescharacteristic
159C2×C8C22×D411yescharacteristic
160C2×C8H(5,𝔽2)11yescharacteristic
161C4⋊C4—22no—
162C4⋊C4—22no—
163C4⋊C4—42no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4—82no—
167C4⋊C4—82no—
168C4⋊C4C2×D812yes—
169C22⋊C4—82no—
170C22⋊C4—82no—
171C22⋊C4—82no—
172C42—22no—
173C42—42no—
174C42—82no—
175C42C22×D411yescharacteristic
176C16—22no—
177C16—21no—
178C16—21no—
179C16—42no—
180C23—82no—
181Q8—22no—
182Q8—22no—
183Q8—42no—
184Q8—42no—
185Q8—162no—
186Q8—322no—
187D4—82no—
188D4—82no—
189D4—82no—
190D4—82no—
191D4—162no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—82no—
197C2×C4—322no—
198C2×C4D4211yescharacteristic
199C2×C4C22×D811yescharacteristic
200C2×C4#57611yescharacteristic
201C8—22no—
202C8—21no—
203C8—21no—
204C8—21no—
205C8—42no—
206C8—82no—
207C8#57511yescharacteristic
208C8#57811yescharacteristic
209C22—162no—
210C22—162no—
211C22—162no—
212C22D4×D811yescharacteristic
213C4—21no—
214C4—42no—
215C4—42no—
216C4—82no—
217C4—82no—
218C4—162no—
219C4#327711yescharacteristic
220C4#329111yescharacteristic
221C4#341411yescharacteristic
222C4#341711yescharacteristic
223C2—322no—
224C2#1988811yescharacteristic
225C2#3384311yescharacteristic
226C2#3385811yescharacteristic
subgroupG/H
center, socleC22D4×D8
commutator, Frattini, cosocleC4×C8C24
prime radical#2246C22
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q16 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33942C2semidirect
2. #19942C2semidirect
4. #12486C2semidirect
5. #12480C2semidirect
6. #12410C2semidirect
8. #12083C2semidirect
9. #12054C2semidirect

Maximal subgroups

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

subgroupindex
#339422
#199422
#125032
#124862
#124802
#124102
#123292
#120832
#120542