Finite Group Databasebeta

← #507699

Subgroups — #507699

Order 512.

492 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 150 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#33249C211yesmaximal, characteristic
2#21725C211yesmaximal, characteristic
3#13483C212yesmaximal
4#13364C212yesmaximal
5#13277C212yesmaximal
6#13253C212yesmaximal
7#13150C212yesmaximal
8#13148C212yesmaximal
9#12654C211yesmaximal, characteristic
10C2×#500C2211yescharacteristic
11C2×#499C2211yescharacteristic
12C2×#494C2211yescharacteristic
13C2×#491—21no—
14#2642—21no—
15#2331—21no—
16C2×C4⋊C16C2212yes—
17C2×#358C2212yes—
18C2×#358C2212yes—
19C2×#358C2212yes—
20C2×#358C2212yes—
21C2×#346C2211yescharacteristic
22C2×C4×C16C2211yescharacteristic
23#1993—21no—
24#1971—21no—
25#1971—21no—
26#1969C2212yes—
27#1969C2212yes—
28#1943—21no—
29#1942C2212yes—
30#1846C2211yescharacteristic
31#1776—21no—
32C4×#107C2212yes—
33#1726C2211yescharacteristic
34#1380C2212yes—
35#1380C2212yes—
36#1380C2212yes—
37#1380C2212yes—
38#1374C2212yes—
39C22×Q16—21no—
40C22×Q16C2312yes—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×#125—21no—
44C22×C16C2312yes—
45C22×C16C2312yes—
46#500—24no—
47#499—24no—
48#494—24no—
49#491—44no—
50C2×#107—21no—
51C2×#107C2312yes—
52C2×#107C2312yes—
53C2×#107C2312yes—
54C2×#106—21no—
55C2×#106—21no—
56C2×#103—22no—
57C2×#103—22no—
58C2×#97D411yescharacteristic
59C2×#97D411yescharacteristic
60C2×#97D411yescharacteristic
61C2×C4×C8C2311yescharacteristic
62C2×C4×C8C2311yescharacteristic
63C2×C4×C8C2311yescharacteristic
64#400—42no—
65#400—42no—
66#391—22no—
67#391—22no—
68#391—42no—
69#389—22no—
70#387—42no—
71#382—21no—
72C4×C4⋊C4—22no—
73#376D411yescharacteristic
74C4⋊C16—24no—
75#358—24no—
76#358—24no—
77#358—42no—
78#358—42no—
79#346—41no—
80C4×C16—41no—
81#340—22no—
82#340—22no—
83C22×Q8—22no—
84C22×Q8—41no—
85C2×Q16—22no—
86C2×Q16—22no—
87C2×Q16—22no—
88C2×Q16—22no—
89C2×Q16—22no—
90C2×Q16—22no—
91C2×Q16—42no—
92C2×Q16—42no—
93C2×Q16—42no—
94C22×C8C2×D411yescharacteristic
95C22×C8C2×D411yescharacteristic
96C22×C8C2×D411yescharacteristic
97C22×C8C2411yescharacteristic
98C4⋊Q8—24no—
99C4⋊Q8—42no—
100C4⋊Q8—42no—
101C4⋊Q8—42no—
102#125—42no—
103#125—42no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—42no—
111C2×C42—42no—
112C2×C42C2×D411yescharacteristic
113C2×C42C2×D411yescharacteristic
114C2×C42C2×D411yescharacteristic
115C2×C16—22no—
116C2×C16—22no—
117C2×C16—22no—
118C2×C16—22no—
119C2×C16—22no—
120C2×C16—22no—
121#107—24no—
122#107—22no—
123#107—24no—
124#107—22no—
125#107—22no—
126#107#4114yes—
127#106—42no—
128#106—42no—
129#103—44no—
130#103—44no—
131#97—41no—
132#97—41no—
133#97SD1614yes—
134C4×C8—22no—
135C4×C8—21no—
136C4×C8—21no—
137C4×C8#4114yes—
138#95—42no—
139#95—41no—
140#95—41no—
141#95—42no—
142#95—42no—
143#95—44no—
144#95—42no—
145#95D812yes—
146#95D812yes—
147C2×Q8—44no—
148C2×Q8—42no—
149C2×Q8—42no—
150C2×Q8—42no—
151C2×Q8—42no—
152C2×Q8—42no—
153C2×Q8—42no—
154C2×Q8—162no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4C2×D811yescharacteristic
160C22×C4C2×D812yes—
161C22×C4C2×D811yescharacteristic
162C22×C4C2×D812yes—
163C22×C4C22×D411yescharacteristic
164Q16—42no—
165Q16—42no—
166Q16—42no—
167Q16—42no—
168Q16—84no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8#12412yes—
176C2×C8C2×SD1612yes—
177C2×C8C2×SD1612yes—
178C2×C8C2×#4112yes—
179C2×C8C2×#4112yes—
180C2×C8H(5,𝔽2)11yescharacteristic
181C2×C82−1+411yescharacteristic
182C4⋊C4—24no—
183C4⋊C4—44no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—84no—
190C4⋊C4—84no—
191C4⋊C4—162no—
192C42—21no—
193C42—21no—
194C42—21no—
195C42—21no—
196C42—162no—
197C42#12412yes—
198C42C2×SD1612yes—
199C16—42no—
200C16—42no—
201C23C22×D811yescharacteristic
202Q8—82no—
203Q8—82no—
204Q8—82no—
205Q8—82no—
206Q8—164no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—44no—
217C2×C4—82no—
218C2×C4—82no—
219C2×C4—82no—
220C2×C4—82no—
221C2×C4—82no—
222C2×C4—82no—
223C2×C4—82no—
224C2×C4—82no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4—82no—
228C2×C4#48614yes—
229C2×C4#52511yescharacteristic
230C2×C4C2×#12411yescharacteristic
231C2×C4#53711yescharacteristic
232C2×C4C22×SD1611yescharacteristic
233C2×C4#57511yescharacteristic
234C2×C4#57511yescharacteristic
235C8—22no—
236C8—22no—
237C8—22no—
238C8#55612yes—
239C8#55912yes—
240C22#310011yescharacteristic
241C22C2×#48611yescharacteristic
242C22#324511yescharacteristic
243C22#341612yes—
244C22#341712yes—
245C4—24no—
246C4—42no—
247C4—41no—
248C4—41no—
249C4—84no—
250C4—162no—
251C4—162no—
252C4—162no—
253C4#259211yescharacteristic
254C4#310311yescharacteristic
255C4#314011yescharacteristic
256C4#324611yescharacteristic
257C2#2934611yescharacteristic
258C2#3368311yescharacteristic
259C2#3368711yescharacteristic
260C2#3370312yes—
261C2#3382412yes—
subgroupG/H
center, socle, prime radicalC23C22×D8
commutator, Frattini, cosocleC22×C8C24
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ #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
#332492
#217252
#134832
#133642
#132772
#132532
#131502
#131482
#126542