Finite Group Databasebeta

← #501944

Subgroups — #501944

Order 512.

474 conjugacy classes of subgroups.

The subgroup lattice has 301 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28265C211yesmaximal, characteristic
2Q8×#107C211yesmaximal, characteristic
3#13538C212yesmaximal
4#13481C212yesmaximal
5#13263C211yesmaximal, characteristic
6#13176C211yesmaximal, characteristic
7#13133C212yesmaximal
8#13112C212yesmaximal
9#12888C211yesmaximal, characteristic
10#12812C211yesmaximal, characteristic
11C4×#358C211yesmaximal, characteristic
12C2×C8×Q8C2211yescharacteristic
13C2×C4×Q16C2211yescharacteristic
14#2582—21no—
15Q8×C4⋊C4—21no—
16C2×#366C2212yes—
17C2×C4⋊C16C2211yescharacteristic
18C2×#358C2212yes—
19C2×#358C2211yescharacteristic
20C2×#358C2211yescharacteristic
21C2×C4×C16C2211yescharacteristic
22#2068—22no—
23#1969C2212yes—
24#1947—21no—
25#1947—22no—
26#1942C2212yes—
27#1845C2211yescharacteristic
28#1845C2211yescharacteristic
29#1845C2212yes—
30#1803—21no—
31C4×#107C2212yes—
32C4×#107C2211yescharacteristic
33#1380C2212yes—
34#1380C2211yescharacteristic
35#1380C2212yes—
36#1380C2212yes—
37#1380C2211yescharacteristic
38#1380C2212yes—
39#1374C2212yes—
40#1374C2212yes—
41C22×Q16C2311yescharacteristic
42C2×C4×Q8—21no—
43C2×C4×Q8—41no—
44C2×C4×Q8D411yescharacteristic
45C22×C16C2311yescharacteristic
46C22×C16C2311yescharacteristic
47C22×C16C2312yes—
48C8×Q8—22no—
49C8×Q8—22no—
50C4×Q16—22no—
51C4×Q16—22no—
52C2×#107—21no—
53C2×#107—22no—
54C2×#107—21no—
55C2×#107C2312yes—
56C2×#107C2311yescharacteristic
57C2×#107C2312yes—
58C2×#107C2311yescharacteristic
59C2×#107C2311yescharacteristic
60C2×C4⋊C8D411yescharacteristic
61C2×C4⋊C8D412yes—
62C2×#103—21no—
63C2×#103—22no—
64C2×C4×C8C2312yes—
65C2×C4×C8C2311yescharacteristic
66#398—42no—
67#391—22no—
68#391—41no—
69#389—22no—
70#389—41no—
71#389—41no—
72#389—42no—
73#384—21no—
74#384—22no—
75#384—21no—
76C4×C4⋊C4—22no—
77C4×C4⋊C4—21no—
78#366—24no—
79C4⋊C16—22no—
80#358—24no—
81#358—22no—
82#358—22no—
83C4×C16—22no—
84#340—22no—
85#340—22no—
86#340—21no—
87C22×Q8—21no—
88C22×Q8D811yescharacteristic
89C2×Q16—21no—
90C2×Q16—21no—
91C2×Q16—21no—
92C2×Q16—21no—
93C2×Q16—21no—
94C2×Q16#4112yes—
95C22×C8C2×D412yes—
96C22×C8C2×D411yescharacteristic
97C22×C8C2411yescharacteristic
98C4×Q8—22no—
99C4×Q8—22no—
100C4×Q8—42no—
101C4×Q8—42no—
102C4×Q8—84no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4D811yescharacteristic
117C2×C4⋊C4D812yes—
118C2×C42—41no—
119C2×C42—42no—
120C2×C42—41no—
121C2×C42C2×D412yes—
122C2×C42C2×D411yescharacteristic
123C2×C16—21no—
124C2×C16—21no—
125C2×C16—22no—
126C2×C16—21no—
127C2×C16—21no—
128C2×C16—22no—
129C2×C16—22no—
130C2×C16—21no—
131C2×C16—21no—
132#107—21no—
133#107—22no—
134#107—21no—
135#107—22no—
136#107—21no—
137#107—21no—
138#107—22no—
139#107—22no—
140#107—21no—
141#107—81no—
142#107—81no—
143#107—82no—
144#107#4112yes—
145C4⋊C8—24no—
146C4⋊C8—22no—
147#103—42no—
148#103—82no—
149C4×C8—22no—
150C4×C8—22no—
151C4×C8#4112yes—
152C4×C8#4112yes—
153#95—42no—
154#95—41no—
155#95—42no—
156#95—42no—
157#95—42no—
158C2×Q8—22no—
159C2×Q8—22no—
160C2×Q8—22no—
161C2×Q8—41no—
162C2×Q8—41no—
163C2×Q8—41no—
164C2×Q8—41no—
165C2×Q8—41no—
166C2×Q8Q3214yes—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—42no—
170C22×C4—41no—
171C22×C4—42no—
172C22×C4—41no—
173C22×C4C2×D812yes—
174C22×C4C2×D811yescharacteristic
175C22×C4C2×D812yes—
176C22×C4C2×D811yescharacteristic
177C22×C4C22×D411yescharacteristic
178Q16—22no—
179Q16—22no—
180Q16—41no—
181Q16—41no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—22no—
185C2×C8—22no—
186C2×C8—22no—
187C2×C8#12312yes—
188C2×C8C2×#4111yescharacteristic
189C2×C8C2×#4111yescharacteristic
190C2×C8C2×#4111yescharacteristic
191C2×C8C2×#4111yescharacteristic
192C2×C82−1+411yescharacteristic
193C2×C82−1+411yescharacteristic
194C4⋊C4—24no—
195C4⋊C4—22no—
196C4⋊C4—22no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—41no—
202C4⋊C4—41no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—41no—
206C4⋊C4—84no—
207C4⋊C4—82no—
208C4⋊C4—82no—
209C4⋊C4—82no—
210C4⋊C4—161no—
211C4⋊C4—161no—
212C4⋊C4—162no—
213C4⋊C4—162no—
214C42—22no—
215C42—22no—
216C42—84no—
217C42—82no—
218C42—82no—
219C42#12312yes—
220C42#13712yes—
221C16—42no—
222C16—41no—
223C16—41no—
224C23C22×D811yescharacteristic
225Q8—28no—
226Q8—42no—
227Q8—42no—
228Q8—81no—
229Q8—81no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—81no—
240C2×C4—81no—
241C2×C4—81no—
242C2×C4—81no—
243C2×C4—81no—
244C2×C4—82no—
245C2×C4—81no—
246C2×C4—82no—
247C2×C4—82no—
248C2×C4—81no—
249C2×C4—82no—
250C2×C4—81no—
251C2×C4—82no—
252C2×C4—81no—
253C2×C4—82no—
254C2×C4—81no—
255C2×C4#48012yes—
256C2×C4#48012yes—
257C2×C4C2×Q3214yes—
258C2×C4C2×Q3212yes—
259C2×C4#50912yes—
260C2×C4C2×#12311yescharacteristic
261C2×C4#54811yescharacteristic
262C2×C4#54811yescharacteristic
263C2×C4C2×#13711yescharacteristic
264C2×C4#57711yescharacteristic
265C2×C4#57711yescharacteristic
266C8—22no—
267C8—42no—
268C8#52911yescharacteristic
269C8#54111yescharacteristic
270C8#55211yescharacteristic
271C8#55611yescharacteristic
272C22C2×#48011yescharacteristic
273C22#335811yescharacteristic
274C22#335811yescharacteristic
275C22C22×Q3211yescharacteristic
276C22C2×#50911yescharacteristic
277C22#341611yescharacteristic
278C22#341611yescharacteristic
279C4—24no—
280C4—22no—
281C4—42no—
282C4—82no—
283C4—82no—
284C4—161no—
285C4—162no—
286C4—161no—
287C4—162no—
288C4#223614yes—
289C4#242111yescharacteristic
290C4#309911yescharacteristic
291C4#336911yescharacteristic
292C4#337211yescharacteristic
293C2#2499011yescharacteristic
294C2C2×#223611yescharacteristic
295C2#3365311yescharacteristic
296C2#3393011yescharacteristic
297C2#3393011yescharacteristic
298C2#3393511yescharacteristic
299C2#3394011yescharacteristic
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
#282652
Q8×#1072
#135382
#134812
#132632
#131762
#131332
#131122
#128882
#128122
C4×#3582