Finite Group Databasebeta

← #506353

Subgroups — #506353

Order 512.

520 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 144 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#33628C212yesmaximal
2#21733C211yesmaximal, characteristic
3#21698C211yesmaximal, characteristic
4#13509C212yesmaximal
5#13415C212yesmaximal
6#13135C214yesmaximal
7#12884C211yesmaximal, characteristic
8#12819C212yesmaximal
9#3160C2211yescharacteristic
10C2×#494C2211yescharacteristic
11#2961—21no—
12C2×#441—21no—
13C2×C4×Q16C2212yes—
14#2327—21no—
15#2229—24no—
16#2227—22no—
17#2225—22no—
18#2159C2214yes—
19#2150C2212yes—
20C2×#362C2214yes—
21#2141C2214yes—
22#1995—21no—
23#1971—21no—
24#1969C2212yes—
25#1960—21no—
26#1946—21no—
27#1945C2212yes—
28#1848C2211yescharacteristic
29#1846C2211yescharacteristic
30#1773—21no—
31#1772C2212yes—
32#1726C2211yescharacteristic
33#1387C2214yes—
34#1376C2212yes—
35#1376C2212yes—
36C22×Q16—21no—
37C22×Q16C2312yes—
38#533—21no—
39C2×C4⋊Q8—21no—
40C2×C4×Q8—22no—
41C2×M32C2314yes—
42#500D412yes—
43#499—21no—
44#499D412yes—
45#498—21no—
46#494—24no—
47#446—44no—
48#441—44no—
49C4×Q16—22no—
50C4×Q16—22no—
51C4×Q16—22no—
52C4×Q16—22no—
53C2×#108—21no—
54C2×#108C2312yes—
55#427—21no—
56#427C2312yes—
57C2×#107C2312yes—
58C2×#106—21no—
59#424—22no—
60#423—21no—
61C2×#103—22no—
62C2×#103—22no—
63#405—22no—
64C2×#97D411yescharacteristic
65C2×#97D411yescharacteristic
66C2×#97D411yescharacteristic
67C2×C4×C8C2311yescharacteristic
68C2×C4×C8C2311yescharacteristic
69C2×C4×C8C2311yescharacteristic
70#391—22no—
71#387—44no—
72#382—21no—
73#376D411yescharacteristic
74#367—24no—
75#366—24no—
76C4⋊C16—22no—
77C4⋊C16—22no—
78#362—28no—
79#358—24no—
80#358—24no—
81#339—22no—
82#339—22no—
83C22×Q8—22no—
84C2×Q16—24no—
85C2×Q16—22no—
86C2×Q16—22no—
87C2×Q16—22no—
88C2×Q16—22no—
89C2×Q16—22no—
90C2×Q16—42no—
91C2×Q16—42no—
92C2×Q16#4114yes—
93C2×M16—22no—
94C22×C8C2×D411yescharacteristic
95C22×C8C2×D411yescharacteristic
96C22×C8C2×D411yescharacteristic
97C22×C8C2411yescharacteristic
98C4⋊Q8—24no—
99C4⋊Q8—22no—
100C4⋊Q8—42no—
101#125—22no—
102C22⋊Q8—42no—
103C4×Q8—42no—
104C4×Q8—42no—
105C4×Q8—42no—
106C4×Q8—42no—
107C4×Q8—42no—
108#117—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4—42no—
112C2×C42—42no—
113C2×C42C2×D411yescharacteristic
114C2×C42C2×D411yescharacteristic
115C2×C42C2×D411yescharacteristic
116M32—24no—
117M32—24no—
118C2×C16—24no—
119#108—24no—
120#108—42no—
121#107—22no—
122#107—22no—
123#107—22no—
124#107C2×D414yes—
125#106—22no—
126#106—22no—
127#106—42no—
128C4⋊C8—44no—
129C4⋊C8—42no—
130#103—42no—
131#103—42no—
132#103—44no—
133#97—22no—
134#97—41no—
135#97—42no—
136#97D814yes—
137C4×C8—22no—
138C4×C8—22no—
139C4×C8—42no—
140C4×C8C2×D411yescharacteristic
141C4×C8C2×D411yescharacteristic
142C4×C8#4111yescharacteristic
143C4×C8#4111yescharacteristic
144#95—22no—
145#95—42no—
146#95—44no—
147C2×Q8—24no—
148C2×Q8—42no—
149C2×Q8—42no—
150C2×Q8—42no—
151C2×Q8—42no—
152C2×Q8—42no—
153C2×Q8—42no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4AGL(1,ℤ/8)12yes—
159C22×C4C22×D411yescharacteristic
160Q16—24no—
161Q16—24no—
162Q16—48no—
163Q16—42no—
164Q16—42no—
165M16—42no—
166M16—42no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—42no—
174C2×C8#12111yescharacteristic
175C2×C8#12111yescharacteristic
176C2×C8C2×D812yes—
177C2×C8C2×D812yes—
178C2×C8C22×D411yescharacteristic
179C2×C8C2×#4112yes—
180C2×C8C2×#4111yescharacteristic
181C2×C82−1+412yes—
182C4⋊C4—24no—
183C4⋊C4—48no—
184C4⋊C4—44no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—82no—
189C4⋊C4—82no—
190C22⋊C4—42no—
191C42—21no—
192C42—21no—
193C42—21no—
194C42—42no—
195C42—82no—
196C42—82no—
197C42#12111yescharacteristic
198C42#12111yescharacteristic
199C42C2×D812yes—
200C42#13712yes—
201C16—44no—
202C23C2×AGL(1,ℤ/8)11yescharacteristic
203Q8—44no—
204Q8—44no—
205Q8—82no—
206Q8—82no—
207Q8—162no—
208C2×C4—22no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—44no—
213C2×C4—41no—
214C2×C4—44no—
215C2×C4—82no—
216C2×C4—82no—
217C2×C4—84no—
218C2×C4—82no—
219C2×C4#45912yes—
220C2×C4#45912yes—
221C2×C4C2×#12111yescharacteristic
222C2×C4#52511yescharacteristic
223C2×C4#53611yescharacteristic
224C2×C4C22×D811yescharacteristic
225C2×C4C2×#13711yescharacteristic
226C2×C4#57511yescharacteristic
227C8—22no—
228C8—22no—
229C8—21no—
230C8—21no—
231C8—82no—
232C8#54812yes—
233C8#55212yes—
234C22—22no—
235C22C2×#45911yescharacteristic
236C22#311011yescharacteristic
237C22#323511yescharacteristic
238C4—22no—
239C4—22no—
240C4—42no—
241C4—81no—
242C4—84no—
243C4—84no—
244C4—162no—
245C4#253611yescharacteristic
246C4#303711yescharacteristic
247C4#310011yescharacteristic
248C4#322611yescharacteristic
249C2—21no—
250C2—21no—
251C2#2820111yescharacteristic
252C2#3359612yes—
subgroupG/H
center, socleC22C2×#459
commutator, Frattini, cosocleC22×C8C24
prime radicalC23C2×AGL(1,ℤ/8)
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1726 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#336282
#217332
#216982
#135092
#134152
#131352
#128842
#128192