Finite Group Databasebeta

← #492478

Subgroups — #492478

Order 512.

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

normal subgroups only

This group has 391 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#33962C211yesmaximal, characteristic
2#33950C211yesmaximal, characteristic
3#19944C212yesmaximal
4#12504C211yesmaximal, characteristic
5#12490C212yesmaximal
6#12457C211yesmaximal, characteristic
7#12440C211yesmaximal, characteristic
8#12382C212yesmaximal
9#12357C212yesmaximal
10#12083C211yesmaximal, characteristic
11C16×Q16C211yesmaximal, characteristic
12#3395C2212yes—
13#3383C2212yes—
14#3374—22no—
15#2253—21no—
16#2252C2211yescharacteristic
17#2247—21no—
18#2247C2211yescharacteristic
19#2241C2212yes—
20#2229—22no—
21#2225C2212yes—
22#2209—22no—
23#2209C2212yes—
24Q8×C16C2211yescharacteristic
25Q8×C16C2211yescharacteristic
26C4×Q32—21no—
27C4×Q32—21no—
28C4×Q32C2211yescharacteristic
29#1707C2212yes—
30#1700—22no—
31#1674C2212yes—
32#1670C2212yes—
33#1653—22no—
34#1653—22no—
35#1591C2212yes—
36C8×Q16C2211yescharacteristic
37#1371C2211yescharacteristic
38#1370C2211yescharacteristic
39#1364C2212yes—
40#1362C2212yes—
41Q8⋊C16C2211yescharacteristic
42Q8⋊C16C2211yescharacteristic
43#1332C2212yes—
44C8×C16C2211yescharacteristic
45#557—22no—
46#557—22no—
47C2×Q32—22no—
48C2×Q32—21no—
49C2×Q32—21no—
50C2×Q32D411yescharacteristic
51#500C2312yes—
52#499—22no—
53#498—22no—
54#494—22no—
55#494C2312yes—
56#491—42no—
57#487—22no—
58#479—42no—
59#479—42no—
60#475—22no—
61#475—22no—
62#462—22no—
63#462—22no—
64#462—42no—
65#462—42no—
66#446—22no—
67C8×Q8C2311yescharacteristic
68C8×Q8C2311yescharacteristic
69#441—42no—
70C4×Q16—22no—
71C4×Q16—22no—
72C4×Q16D411yescharacteristic
73C4×Q16C2312yes—
74#367—22no—
75#366—21no—
76#366—21no—
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#332—22no—
93#332—22no—
94Q8⋊C8—22no—
95Q8⋊C8D411yescharacteristic
96Q8⋊C8D411yescharacteristic
97C82C2311yescharacteristic
98C2×Q16—22no—
99C2×Q16—22no—
100C2×Q16—42no—
101C2×Q16—42no—
102C2×Q16D811yescharacteristic
103C2×Q16C2×D412yes—
104C4⋊Q8—22no—
105C4⋊Q8—22no—
106C4⋊Q8—42no—
107C4⋊Q8—42no—
108#125—42no—
109#125—42no—
110#125—82no—
111#125—82no—
112C4×Q8—22no—
113C4×Q8—42no—
114C4×Q8—42no—
115C4×Q8—42no—
116C4×Q8—42no—
117C4×Q8C2×D411yescharacteristic
118C4×Q8C2×D411yescharacteristic
119Q32—22no—
120Q32—42no—
121Q32—81no—
122Q32—82no—
123C2×C16—21no—
124C2×C16—21no—
125C2×C16—21no—
126C2×C16C2×D411yescharacteristic
127C2×C16C2×D411yescharacteristic
128#107—22no—
129#107—22no—
130#107D811yescharacteristic
131#107C2×D412yes—
132#106—42no—
133#106—42no—
134#106—82no—
135C4⋊C8—21no—
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#103—42no—
147#103—42no—
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×Q8C2×D811yescharacteristic
160C2×Q8C2×D811yescharacteristic
161Q16—22no—
162Q16—22no—
163Q16—22no—
164Q16—42no—
165Q16—42no—
166Q16—82no—
167Q16—84no—
168Q16—162no—
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⋊C4—82no—
191C4⋊C4C2×D811yescharacteristic
192C4⋊C4C2×D811yescharacteristic
193C42—21no—
194C42—21no—
195C42—42no—
196C42—82no—
197C42—82no—
198C42C22×D411yescharacteristic
199C16—21no—
200C16—41no—
201C16—41no—
202C16—41no—
203C16#13712yes—
204Q8—21no—
205Q8—21no—
206Q8—21no—
207Q8—42no—
208Q8—42no—
209Q8—82no—
210Q8—82no—
211Q8—82no—
212Q8—82no—
213Q8#50812yes—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—82no—
221C2×C4—82no—
222C2×C4D4211yescharacteristic
223C2×C4C22×D811yescharacteristic
224C2×C4#57611yescharacteristic
225C8—22no—
226C8—21no—
227C8—21no—
228C8—21no—
229C8—41no—
230C8—41no—
231C8—82no—
232C8C2×#13711yescharacteristic
233C8#57711yescharacteristic
234C22D4×D811yescharacteristic
235C4—22no—
236C4—21no—
237C4—41no—
238C4—41no—
239C4—41no—
240C4—82no—
241C4—82no—
242C4—162no—
243C4—162no—
244C4#327811yescharacteristic
245C4#329011yescharacteristic
246C4C2×#50811yescharacteristic
247C4#341711yescharacteristic
248C2#1988911yescharacteristic
249C2#3385111yescharacteristic
250C2#3386011yescharacteristic
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

Maximal subgroups

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

subgroupindex
#339622
#339502
#199442
#125042
#124902
#124572
#124402
#123822
#123572
#120832
C16×Q162