Finite Group Databasebeta

← #492472

Subgroups — #492472

Order 512.

391 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 391 conjugacy classes of subgroups — too many to draw. Only its 126 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#33962C211yesmaximal, characteristic
2Q8×Q32C211yesmaximal, characteristic
3#19932C212yesmaximal
4#12504C211yesmaximal, characteristic
5#12488C212yesmaximal
6#12458C211yesmaximal, characteristic
7#12440C211yesmaximal, characteristic
8#12386C212yesmaximal
9#12358C212yesmaximal
10#12083C211yesmaximal, characteristic
11C16×Q16C211yesmaximal, characteristic
12#3395C2212yes—
13#3387—22no—
14Q8×Q16C2212yes—
15#2252—21no—
16#2252C2211yescharacteristic
17#2247—21no—
18#2247C2211yescharacteristic
19#2241C2212yes—
20#2227—22no—
21#2227C2212yes—
22#2209—22no—
23#2209C2212yes—
24Q8×C16C2211yescharacteristic
25Q8×C16C2211yescharacteristic
26C4×Q32—21no—
27C4×Q32—21no—
28C4×Q32C2211yescharacteristic
29#1707C2212yes—
30#1692—22no—
31#1676C2212yes—
32#1670C2212yes—
33#1645—22no—
34#1636—22no—
35#1591C2212yes—
36C8×Q16C2211yescharacteristic
37#1371C2211yescharacteristic
38#1370C2211yescharacteristic
39#1365C2212yes—
40#1362C2212yes—
41Q8⋊C16C2211yescharacteristic
42Q8⋊C16C2211yescharacteristic
43#1332C2212yes—
44C8×C16C2211yescharacteristic
45Q82—22no—
46#557—22no—
47C2×Q32—22no—
48C2×Q32—21no—
49C2×Q32—21no—
50C2×Q32D411yescharacteristic
51#501—42no—
52#500—22no—
53#500C2312yes—
54#498—22no—
55#494—22no—
56#494C2312yes—
57#487—22no—
58#479—42no—
59#477—22no—
60#477—22no—
61#477—42no—
62#475—22no—
63#475—42no—
64#462—22no—
65#462—42no—
66#446—22no—
67C8×Q8C2311yescharacteristic
68C8×Q8C2311yescharacteristic
69#441—42no—
70C4×Q16—22no—
71C4×Q16—22no—
72C4×Q16D411yescharacteristic
73C4×Q16C2312yes—
74#366—21no—
75#366—21no—
76#366—22no—
77#366D411yescharacteristic
78C4⋊C16—21no—
79C4⋊C16—21no—
80C4⋊C16C2311yescharacteristic
81C4⋊C16C2311yescharacteristic
82#358—22no—
83#358—22no—
84#358—42no—
85#358D412yes—
86C4×C16—21no—
87C4×C16—21no—
88C4×C16C2311yescharacteristic
89C4×C16C2311yescharacteristic
90#335D411yescharacteristic
91#335C2312yes—
92#333—22no—
93#332—22no—
94Q8⋊C8—22no—
95Q8⋊C8D411yescharacteristic
96Q8⋊C8D411yescharacteristic
97C82C2311yescharacteristic
98C2×Q16—22no—
99C2×Q16—22no—
100C2×Q16—42no—
101C2×Q16D811yescharacteristic
102C2×Q16C2×D412yes—
103C4⋊Q8—22no—
104C4⋊Q8—22no—
105C4⋊Q8—42no—
106C4⋊Q8—42no—
107C4⋊Q8—42no—
108C4⋊Q8—82no—
109#125—42no—
110#125—82no—
111C4×Q8—22no—
112C4×Q8—42no—
113C4×Q8—42no—
114C4×Q8—42no—
115C4×Q8—42no—
116C4×Q8C2×D411yescharacteristic
117C4×Q8C2×D411yescharacteristic
118Q32—22no—
119Q32—42no—
120Q32—81no—
121Q32—82no—
122C2×C16—21no—
123C2×C16—21no—
124C2×C16—21no—
125C2×C16C2×D411yescharacteristic
126C2×C16C2×D411yescharacteristic
127#107—22no—
128#107—22no—
129#107—42no—
130#107—42no—
131#107D811yescharacteristic
132#107C2×D412yes—
133#106—42no—
134#106—42no—
135#106—42no—
136C4⋊C8—21no—
137C4⋊C8—21no—
138C4⋊C8—22no—
139C4⋊C8—42no—
140C4⋊C8C2×D411yescharacteristic
141C4⋊C8C2×D411yescharacteristic
142#103—22no—
143#103—42no—
144#103—42no—
145#103—42no—
146#103—42no—
147#103—82no—
148#103D811yescharacteristic
149#103D811yescharacteristic
150#97—42no—
151C4×C8—21no—
152C4×C8—21no—
153C4×C8C2×D411yescharacteristic
154C4×C8C2×D411yescharacteristic
155C4×C8C2411yescharacteristic
156C2×Q8—22no—
157C2×Q8—42no—
158C2×Q8—42no—
159C2×Q8—82no—
160C2×Q8C2×D811yescharacteristic
161C2×Q8C2×D811yescharacteristic
162Q16—22no—
163Q16—22no—
164Q16—42no—
165Q16—42no—
166Q16—82no—
167Q16—162no—
168Q16Q3214yescomplemented
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—42no—
173C2×C8C2×D811yescharacteristic
174C2×C8C2×D811yescharacteristic
175C2×C8C22×D411yescharacteristic
176C2×C8H(5,𝔽2)11yescharacteristic
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—22no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—82no—
186C4⋊C4—82no—
187C4⋊C4—82no—
188C4⋊C4—82no—
189C4⋊C4—82no—
190C4⋊C4C2×D811yescharacteristic
191C4⋊C4C2×D811yescharacteristic
192C42—21no—
193C42—21no—
194C42—42no—
195C42—82no—
196C42—82no—
197C42C22×D411yescharacteristic
198C16—21no—
199C16—41no—
200C16—41no—
201C16—41no—
202C16#13712yes—
203Q8—21no—
204Q8—21no—
205Q8—42no—
206Q8—42no—
207Q8—82no—
208Q8—82no—
209Q8—162no—
210Q8—322no—
211Q8C2×Q3212yes—
212Q8C2×Q3212yes—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—82no—
220C2×C4—82no—
221C2×C4D4211yescharacteristic
222C2×C4C22×D811yescharacteristic
223C2×C4#57611yescharacteristic
224C8—21no—
225C8—21no—
226C8—41no—
227C8—41no—
228C8—41no—
229C8—82no—
230C8C2×Q3212yes—
231C8C2×#13711yescharacteristic
232C8#57711yescharacteristic
233C22D4×D811yescharacteristic
234C4—22no—
235C4—22no—
236C4—21no—
237C4—41no—
238C4—41no—
239C4—82no—
240C4—82no—
241C4—162no—
242C4—162no—
243C4#327811yescharacteristic
244C4#329011yescharacteristic
245C4C22×Q3211yescharacteristic
246C4#341711yescharacteristic
247C2#1988911yescharacteristic
248C2D4×Q3211yescharacteristic
249C2#3385911yescharacteristic
subgroupG/H
center, socle, prime radicalC22D4×D8
commutator, Frattini, cosocleC4×C8C24
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
168. Q16Q32semidirect

Maximal subgroups

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

subgroupindex
#339622
Q8×Q322
#199322
#125042
#124882
#124582
#124402
#123862
#123582
#120832
C16×Q162