Finite Group Databasebeta

← #511696

Subgroups — #511696

Order 512.

441 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 441 conjugacy classes of subgroups — too many to draw. Only its 128 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#33261C211yesmaximal, characteristic
2#33249C211yesmaximal, characteristic
3#13529C212yesmaximal
4#13481C212yesmaximal
5#13477C212yesmaximal
6#13475C212yesmaximal
7#13453C212yesmaximal
8#13409C212yesmaximal
9#12880C211yesmaximal, characteristic
10C2×#499C2211yescharacteristic
11C2×#494C2211yescharacteristic
12C2×#491—21no—
13#2647—21no—
14#2642—21no—
15C2×#367C2212yes—
16C2×#366C2212yes—
17C2×C4⋊C16C2211yescharacteristic
18C2×C4⋊C16C2211yescharacteristic
19C2×#358C2212yes—
20C2×#358C2212yes—
21C2×#358C2212yes—
22C2×#358C2212yes—
23#1971—21no—
24#1971—21no—
25#1969C2212yes—
26#1969C2212yes—
27#1944—21no—
28#1943—21no—
29#1942C2212yes—
30#1942C2212yes—
31#1846C2211yescharacteristic
32#1845C2211yescharacteristic
33#1836—21no—
34#1726C2211yescharacteristic
35#1380C2212yes—
36#1380C2212yes—
37#1374C2212yes—
38#1374C2212yes—
39C22×Q16—21no—
40C22×Q16C2312yes—
41C2×C4⋊Q8—21no—
42C2×#125—21no—
43C22×C16C2312yes—
44C22×C16C2312yes—
45#499—41no—
46#499—41no—
47#494—41no—
48#494—41no—
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#400—42no—
67#398—42no—
68#391—22no—
69#391—22no—
70#389—22no—
71#389—22no—
72#384—21no—
73#382—21no—
74#376D411yescharacteristic
75#367—42no—
76#366—42no—
77C4⋊C16—41no—
78C4⋊C16—41no—
79#358—42no—
80#358—42no—
81#358—42no—
82#358—42no—
83#340—22no—
84#340—22no—
85C22×Q8—22no—
86C2×Q16—22no—
87C2×Q16—22no—
88C2×Q16—22no—
89C2×Q16—22no—
90C2×Q16—22no—
91C2×Q16—22no—
92C2×Q16—42no—
93C2×Q16—42no—
94C2×Q16—42no—
95C22×C8C2×D411yescharacteristic
96C22×C8C2×D411yescharacteristic
97C22×C8C2×D411yescharacteristic
98C22×C8C2411yescharacteristic
99C4⋊Q8—42no—
100C4⋊Q8—42no—
101#125—42no—
102#125—42no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—42no—
111C2×C42C2×D411yescharacteristic
112C2×C42C2×D411yescharacteristic
113C2×C42C2×D411yescharacteristic
114C2×C16—22no—
115C2×C16—22no—
116C2×C16—22no—
117C2×C16—22no—
118C2×C16—22no—
119C2×C16—22no—
120#107—22no—
121#107—22no—
122#107—22no—
123#107—22no—
124#107—22no—
125#107—22no—
126#107—42no—
127#106—42no—
128#106—42no—
129#103—44no—
130#103—44no—
131#97—22no—
132#97—22no—
133#97—41no—
134C4×C8—21no—
135C4×C8—21no—
136C4×C8—21no—
137C4×C8—21no—
138C4×C8—21no—
139C4×C8—21no—
140#95—42no—
141#95—42no—
142#95—42no—
143#95—42no—
144#95—44no—
145#95—42no—
146#95D812yes—
147#95D812yes—
148C2×Q8—42no—
149C2×Q8—42no—
150C2×Q8—42no—
151C2×Q8—42no—
152C2×Q8—42no—
153C2×Q8—42no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4C2×D811yescharacteristic
159C22×C4C2×D812yes—
160C22×C4C2×D811yescharacteristic
161C22×C4C2×D812yes—
162C22×C4C22×D411yescharacteristic
163Q16—42no—
164Q16—42no—
165Q16—42no—
166Q16—42no—
167Q16—84no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8AGL(1,ℤ/8)12yes—
176C2×C8#13712yes—
177C2×C8H(5,𝔽2)11yescharacteristic
178C2×C8H(5,𝔽2)11yescharacteristic
179C2×C8H(5,𝔽2)11yescharacteristic
180C2×C82−1+411yescharacteristic
181C2×C82−1+411yescharacteristic
182C2×C82−1+411yescharacteristic
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—84no—
190C4⋊C4—84no—
191C4⋊C4—162no—
192C4⋊C4—162no—
193C4⋊C4—162no—
194C42—21no—
195C42—21no—
196C42—21no—
197C42—21no—
198C42—21no—
199C42—21no—
200C16—42no—
201C16—42no—
202C23C22×D811yescharacteristic
203Q8—82no—
204Q8—82no—
205Q8—82no—
206Q8—82no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—21no—
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—82no—
229C2×C4#53511yescharacteristic
230C2×C4#53611yescharacteristic
231C2×C4#53711yescharacteristic
232C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
233C2×C4C2×#13711yescharacteristic
234C2×C4#57511yescharacteristic
235C8—22no—
236C8—22no—
237C8—41no—
238C8#56211yescharacteristic
239C8#56211yescharacteristic
240C8#56311yescharacteristic
241C8#56311yescharacteristic
242C22#320511yescharacteristic
243C22#322611yescharacteristic
244C22#324511yescharacteristic
245C22#341612yes—
246C22#341712yes—
247C4—42no—
248C4—41no—
249C4—41no—
250C4—42no—
251C4—162no—
252C4—162no—
253C4—162no—
254C4—162no—
255C4#282011yescharacteristic
256C4#321511yescharacteristic
257C4#323411yescharacteristic
258C4#325411yescharacteristic
259C2#3281811yescharacteristic
260C2#3379112yes—
261C2#3380812yes—
262C2#3382611yescharacteristic
263C2#3382711yescharacteristic
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
#332612
#332492
#135292
#134812
#134772
#134752
#134532
#134092
#128802