Finite Group Databasebeta

← #511515

Subgroups — #511515

Order 512.

471 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 471 conjugacy classes of subgroups — too many to draw. Only its 132 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#28737C211yesmaximal, characteristic
2#28675C211yesmaximal, characteristic
3#13475C212yesmaximal
4#13409C212yesmaximal
5#13364C212yesmaximal
6#13264C211yesmaximal, characteristic
7#13150C212yesmaximal
8#13133C212yesmaximal
9#13109C212yesmaximal
10C2×#500C2211yescharacteristic
11C2×#494C2211yescharacteristic
12C2×C8×Q8C2211yescharacteristic
13#2568—21no—
14#2546—21no—
15C2×#366C2212yes—
16C2×C4⋊C16C2211yescharacteristic
17C2×#358C2212yes—
18C2×#358C2212yes—
19C2×#358C2212yes—
20C2×#358C2212yes—
21C2×#346C2211yescharacteristic
22#2086—21no—
23#2086—21no—
24#2056—21no—
25#2056—21no—
26#1988—21no—
27#1981—21no—
28#1969C2212yes—
29#1969C2212yes—
30#1942C2212yes—
31#1942C2212yes—
32C4×#107C2211yescharacteristic
33C4×#107C2211yescharacteristic
34#1380C2212yes—
35#1380C2212yes—
36#1380C2212yes—
37#1374C2212yes—
38#1374C2212yes—
39C22×Q16—21no—
40C22×Q16C2312yes—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—41no—
44C2×C4×Q8D411yescharacteristic
45C22×C16C2312yes—
46C22×C16C2312yes—
47#500—41no—
48#500—41no—
49#494—41no—
50#494—41no—
51C8×Q8—41no—
52C8×Q8—41no—
53C2×#107—21no—
54C2×#107C2312yes—
55C2×#107C2312yes—
56C2×#107C2312yes—
57C2×C4⋊C8D411yescharacteristic
58C2×C4⋊C8D411yescharacteristic
59C2×C4⋊C8D411yescharacteristic
60C2×#103—22no—
61C2×#103—22no—
62C2×C4×C8C2311yescharacteristic
63C2×C4×C8C2311yescharacteristic
64C2×C4×C8C2311yescharacteristic
65#400—41no—
66#400—41no—
67#398—41no—
68#398—41no—
69#391—22no—
70#391—22no—
71#391—41no—
72#391—41no—
73#389—22no—
74#389—22no—
75#382—41no—
76C4×C4⋊C4—21no—
77C4×C4⋊C4—21no—
78#366—24no—
79C4⋊C16—41no—
80#358—24no—
81#358—24no—
82#358—42no—
83#358—42no—
84#346—41no—
85#340—21no—
86#340—21no—
87#340—22no—
88#340—22no—
89C22×Q8—22no—
90C22×Q8—41no—
91C22×Q8D811yescharacteristic
92C2×Q16—22no—
93C2×Q16—22no—
94C2×Q16—22no—
95C2×Q16—22no—
96C2×Q16—22no—
97C2×Q16—22no—
98C2×Q16—42no—
99C2×Q16—82no—
100C22×C8C2×D411yescharacteristic
101C22×C8C2×D411yescharacteristic
102C22×C8C2×D411yescharacteristic
103C22×C8C2411yescharacteristic
104C4⋊Q8—42no—
105C4⋊Q8—42no—
106C4⋊Q8—42no—
107C4⋊Q8—42no—
108C4⋊Q8—84no—
109C4×Q8—41no—
110C4×Q8—41no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4D811yescharacteristic
120C2×C4⋊C4D811yescharacteristic
121C2×C4⋊C4D811yescharacteristic
122C2×C42—41no—
123C2×C42—41no—
124C2×C42C2×D411yescharacteristic
125C2×C42C2×D411yescharacteristic
126C2×C42C2×D411yescharacteristic
127C2×C16—22no—
128C2×C16—22no—
129C2×C16—22no—
130C2×C16—22no—
131C2×C16—22no—
132C2×C16—22no—
133#107—22no—
134#107—22no—
135#107—22no—
136#107—22no—
137#107—22no—
138#107—42no—
139#107#4114yes—
140C4⋊C8—41no—
141C4⋊C8—41no—
142C4⋊C8—41no—
143#103—82no—
144#103—82no—
145C4×C8—21no—
146C4×C8—21no—
147C4×C8—21no—
148C4×C8—21no—
149C4×C8—21no—
150C4×C8—21no—
151#95—41no—
152#95—41no—
153#95—42no—
154#95—42no—
155#95—42no—
156#95—42no—
157#95—42no—
158#95—42no—
159C2×Q8—22no—
160C2×Q8—22no—
161C2×Q8—41no—
162C2×Q8—42no—
163C2×Q8—42no—
164C2×Q8—42no—
165C2×Q8—42no—
166C2×Q8—42no—
167C2×Q8—42no—
168C2×Q8—82no—
169C2×Q8—82no—
170C2×Q8—82no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4C2×D811yescharacteristic
176C22×C4C2×D811yescharacteristic
177C22×C4C2×D811yescharacteristic
178C22×C4C2×D811yescharacteristic
179C22×C4C2×D811yescharacteristic
180C22×C4C2×D811yescharacteristic
181C22×C4C22×D411yescharacteristic
182Q16—42no—
183Q16—42no—
184Q16—42no—
185Q16—42no—
186Q16—84no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—22no—
191C2×C8—22no—
192C2×C8—22no—
193C2×C8C2×#4112yes—
194C2×C8H(5,𝔽2)11yescharacteristic
195C2×C8H(5,𝔽2)11yescharacteristic
196C2×C82−1+412yes—
197C4⋊C4—24no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—41no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—82no—
207C4⋊C4—82no—
208C4⋊C4—82no—
209C4⋊C4—84no—
210C4⋊C4—84no—
211C42—21no—
212C42—21no—
213C42—21no—
214C42—21no—
215C42—21no—
216C42—21no—
217C42—82no—
218C42—82no—
219C16—42no—
220C16—42no—
221C23C22×D811yescharacteristic
222Q8—44no—
223Q8—82no—
224Q8—82no—
225Q8—82no—
226Q8—82no—
227Q8—164no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—44no—
242C2×C4—82no—
243C2×C4—82no—
244C2×C4—82no—
245C2×C4—82no—
246C2×C4—82no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4#50912yes—
254C2×C4#51012yes—
255C2×C4#53411yescharacteristic
256C2×C4#54812yes—
257C2×C4#57511yescharacteristic
258C2×C4#57712yes—
259C8—41no—
260C8—41no—
261C8—41no—
262C8#54212yes—
263C8#55511yescharacteristic
264C8#56311yescharacteristic
265C22#318611yescharacteristic
266C22#335812yes—
267C22C2×#50911yescharacteristic
268C22C2×#51011yescharacteristic
269C22#341712yes—
270C4—22no—
271C4—22no—
272C4—41no—
273C4—41no—
274C4—41no—
275C4—41no—
276C4—84no—
277C4—162no—
278C4—162no—
279C4—162no—
280C4#273111yescharacteristic
281C4#320011yescharacteristic
282C4#337112yes—
283C2#3262811yescharacteristic
284C2#3375911yescharacteristic
285C2#3376811yescharacteristic
286C2#3393412yes—
287C2#3394112yes—
subgroupG/H
center, socle, prime radicalC23C22×D8
commutator, Frattini, cosocleC22×C8C24
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ 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
#287372
#286752
#134752
#134092
#133642
#132642
#131502
#131332
#131092