Finite Group Databasebeta

← #503806

Subgroups — #503806

Order 512.

476 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28828C211yesmaximal, characteristic
2C4⋊C4×Q16C211yesmaximal, characteristic
3#13539C212yesmaximal
4#13516C212yesmaximal
5#13177C211yesmaximal, characteristic
6#13159C211yesmaximal, characteristic
7#12938C212yesmaximal
8#12888C211yesmaximal, characteristic
9#12814C212yesmaximal
10#12812C211yesmaximal, characteristic
11C4×#358C211yesmaximal, characteristic
12C2×C4×Q16C2212yes—
13C2×C4×Q16C2211yescharacteristic
14#2571—21no—
15Q8×C4⋊C4—21no—
16C2×#367C2212yes—
17C2×C4⋊C16C2211yescharacteristic
18C2×C4⋊C16C2212yes—
19C2×#358C2212yes—
20C2×#358C2211yescharacteristic
21C2×#358C2211yescharacteristic
22C2×C4×C16C2211yescharacteristic
23#2085—22no—
24#1948—21no—
25#1942C2212yes—
26#1942C2212yes—
27#1863—22no—
28#1846C2212yes—
29#1845C2211yescharacteristic
30#1845C2211yescharacteristic
31#1803—21no—
32C4×#107C2211yescharacteristic
33C8×C4⋊C4C2211yescharacteristic
34#1380C2212yes—
35#1380C2212yes—
36#1380C2212yes—
37#1380C2212yes—
38#1380C2211yescharacteristic
39#1380C2211yescharacteristic
40#1374C2212yes—
41C22×Q16C2311yescharacteristic
42C2×C4×Q8—21no—
43C2×C4×Q8—22no—
44C22×C16C2311yescharacteristic
45C22×C16C2311yescharacteristic
46C22×C16C2312yes—
47C4×Q16—24no—
48C4×Q16—22no—
49C4×Q16—42no—
50C4×Q16Q814yes—
51C2×#107—21no—
52C2×#107—21no—
53C2×#107C2312yes—
54C2×#107C2312yes—
55C2×#107C2311yescharacteristic
56C2×#107C2311yescharacteristic
57C2×#107C2311yescharacteristic
58C2×#106—22no—
59C2×C4⋊C8D411yescharacteristic
60C2×#103—21no—
61C2×#103—22no—
62C2×C4×C8C2312yes—
63C2×C4×C8C2311yescharacteristic
64#400—42no—
65#389—22no—
66#389—22no—
67#389—41no—
68#389—42no—
69#384—21no—
70#384—21no—
71#384—41no—
72#382—22no—
73#382—41no—
74C4×C4⋊C4—21no—
75C4×C4⋊C4—41no—
76C4×C4⋊C4D411yescharacteristic
77#367—42no—
78C4⋊C16—22no—
79C4⋊C16—42no—
80#358—24no—
81#358—22no—
82#358—22no—
83C4×C16—22no—
84#340—22no—
85#340—21no—
86#340—22no—
87#336D412yes—
88C22×Q8—21no—
89C2×Q16—21no—
90C2×Q16—21no—
91C2×Q16—21no—
92C2×Q16C2×Q812yes—
93C2×Q16#4112yes—
94C2×Q16#4112yes—
95C22×C8C2×D412yes—
96C22×C8C2×D411yescharacteristic
97C22×C8C2411yescharacteristic
98C4×Q8—24no—
99C4×Q8—44no—
100C4×Q8—42no—
101C4×Q8—82no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4D811yescharacteristic
116C2×C4⋊C4D811yescharacteristic
117C2×C42—41no—
118C2×C42—42no—
119C2×C42—41no—
120C2×C42C2×D412yes—
121C2×C42C2×D411yescharacteristic
122C2×C16—21no—
123C2×C16—21no—
124C2×C16—21no—
125C2×C16—21no—
126C2×C16—22no—
127C2×C16—22no—
128C2×C16—21no—
129C2×C16—22no—
130C2×C16—21no—
131#107—21no—
132#107—21no—
133#107—22no—
134#107—21no—
135#107—22no—
136#107—21no—
137#107—22no—
138#107—22no—
139#107—21no—
140#107—81no—
141#107—81no—
142#107C2×Q812yes—
143#106—82no—
144C4⋊C8—22no—
145#103—24no—
146#103—44no—
147C4×C8—22no—
148C4×C8—22no—
149C4×C8C2×Q812yes—
150C4×C8#4112yes—
151#95—41no—
152#95—42no—
153#95—42no—
154#95—42no—
155#95—42no—
156#95D812yes—
157C2×Q8—22no—
158C2×Q8—22no—
159C2×Q8—22no—
160C2×Q8—41no—
161C2×Q8—41no—
162C2×Q8—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4C2×D812yes—
170C22×C4C2×D812yes—
171C22×C4C2×D811yescharacteristic
172C22×C4C2×D811yescharacteristic
173C22×C4C22×D411yescharacteristic
174Q16—22no—
175Q16—22no—
176Q16—22no—
177Q16#12514yes—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—22no—
181C2×C8—22no—
182C2×C8—22no—
183C2×C8C22⋊Q812yes—
184C2×C8C22×Q811yescharacteristic
185C2×C8C2×#4111yescharacteristic
186C2×C8C2×#4111yescharacteristic
187C2×C8C2×#4111yescharacteristic
188C2×C8H(5,𝔽2)11yescharacteristic
189C2×C82−1+411yescharacteristic
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—41no—
194C4⋊C4—44no—
195C4⋊C4—42no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—42no—
199C4⋊C4—41no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—41no—
203C4⋊C4—84no—
204C4⋊C4—82no—
205C4⋊C4—82no—
206C4⋊C4—161no—
207C4⋊C4—161no—
208C4⋊C4—162no—
209C4⋊C4—162no—
210C42—22no—
211C42—22no—
212C42—44no—
213C42—84no—
214C42—82no—
215C42C22⋊Q812yes—
216C42#13712yes—
217C16—41no—
218C16—41no—
219C16—42no—
220C23C22×D811yescharacteristic
221Q8—24no—
222Q8—42no—
223Q8—42no—
224Q8—42no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—81no—
238C2×C4—81no—
239C2×C4—82no—
240C2×C4—81no—
241C2×C4—82no—
242C2×C4—82no—
243C2×C4—81no—
244C2×C4—81no—
245C2×C4—82no—
246C2×C4—81no—
247C2×C4—82no—
248C2×C4—81no—
249C2×C4—82no—
250C2×C4—81no—
251C2×C4#47412yes—
252C2×C4#47412yes—
253C2×C4#50812yes—
254C2×C4#51012yes—
255C2×C4C2×C22⋊Q811yescharacteristic
256C2×C4#54611yescharacteristic
257C2×C4#54811yescharacteristic
258C2×C4C2×#13711yescharacteristic
259C2×C4#57711yescharacteristic
260C2×C4#57711yescharacteristic
261C8—22no—
262C8—42no—
263C8C2×#12511yescharacteristic
264C8#54411yescharacteristic
265C8#55211yescharacteristic
266C8#55311yescharacteristic
267C22C2×#47411yescharacteristic
268C22#329411yescharacteristic
269C22#335811yescharacteristic
270C22C2×#50811yescharacteristic
271C22C2×#51011yescharacteristic
272C22#341611yescharacteristic
273C22#341611yescharacteristic
274C4—22no—
275C4—42no—
276C4—42no—
277C4—44no—
278C4—82no—
279C4—162no—
280C4—161no—
281C4—162no—
282C4—161no—
283C4#222914yes—
284C4#244211yescharacteristic
285C4#308411yescharacteristic
286C4#333111yescharacteristic
287C4#337211yescharacteristic
288C2#2610611yescharacteristic
289C2C2×#222911yescharacteristic
290C2#3363411yescharacteristic
291C2#3387111yescharacteristic
292C2#3389611yescharacteristic
293C2#3393211yescharacteristic
294C2#3393611yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×D8
commutator, Frattini, cosocleC22×C8C24
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ 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
#288282
C4⋊C4×Q162
#135392
#135162
#131772
#131592
#129382
#128882
#128142
#128122
C4×#3582