Finite Group Databasebeta

← #508455

Subgroups — #508455

Order 512.

500 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2229C212yesmaximal
2#33311C211yesmaximal, characteristic
3#21276C211yesmaximal, characteristic
4#13507C212yesmaximal
5#13506C211yesmaximal, characteristic
6#13505C211yesmaximal, characteristic
7#13483C212yesmaximal
8#13481C211yesmaximal, characteristic
9#13481C211yesmaximal, characteristic
10#13277C212yesmaximal
11#12879C211yesmaximal, characteristic
12C2×#499—21no—
13C2×#499C2212yes—
14C2×C4×Q16C2211yescharacteristic
15#2859—21no—
16#2295—21no—
17#2229—28no—
18C2×#367C2212yes—
19C2×#366C2212yes—
20C2×C4⋊C16C2212yes—
21C2×#358C2212yes—
22C2×#358C2212yes—
23#1969C2211yescharacteristic
24#1969C2211yescharacteristic
25#1942C2211yescharacteristic
26#1942C2211yescharacteristic
27#1933—21no—
28#1933—21no—
29#1846—21no—
30#1846C2212yes—
31#1845C2211yescharacteristic
32#1845C2211yescharacteristic
33#1844—21no—
34C4×#107C2211yescharacteristic
35C4×#103—21no—
36C42×C8C2211yescharacteristic
37#1380C2212yes—
38#1380C2212yes—
39#1380C2212yes—
40#1380C2212yes—
41#1374C2211yescharacteristic
42#1374C2211yescharacteristic
43#1374C2212yes—
44C22×Q16C2311yescharacteristic
45C2×#125—22no—
46C2×C4×Q8—21no—
47C22×C16C2312yes—
48C22×C16C2312yes—
49#499—24no—
50#499—24no—
51#499—44no—
52C4×Q16—22no—
53C4×Q16D414yes—
54C2×#107—21no—
55C2×#107—21no—
56C2×#107C2312yes—
57C2×#107C2312yes—
58C2×#107C2311yescharacteristic
59C2×#107C2311yescharacteristic
60C2×#107C2311yescharacteristic
61C2×#106—22no—
62C2×#106—22no—
63C2×#103—21no—
64C2×#103—21no—
65C2×#103—21no—
66C2×C4×C8D411yescharacteristic
67C2×C4×C8D411yescharacteristic
68C2×C4×C8D411yescharacteristic
69C2×C4×C8C2311yescharacteristic
70C2×C4×C8C2311yescharacteristic
71C2×C4×C8C2311yescharacteristic
72#391—21no—
73#391—21no—
74#389—21no—
75#389—21no—
76#389—41no—
77#389—41no—
78#389—41no—
79#389—41no—
80#387—41no—
81#387—41no—
82#384—21no—
83#384—21no—
84#382—22no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4—41no—
87#376—41no—
88C43D411yescharacteristic
89#367—24no—
90#366—24no—
91C4⋊C16—24no—
92#358—24no—
93#358—24no—
94#340—21no—
95#340—21no—
96#340—21no—
97C22×Q8—21no—
98C2×Q16—21no—
99C2×Q16—22no—
100C2×Q16—22no—
101C2×Q16C2×D412yes—
102C22×C8C2×D411yescharacteristic
103C22×C8C2×D411yescharacteristic
104C22×C8C2×D411yescharacteristic
105C22×C8C2411yescharacteristic
106#125—44no—
107#125—44no—
108C4×Q8—24no—
109C4×Q8—42no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C42—41no—
125C2×C42—41no—
126C2×C42D811yescharacteristic
127C2×C42D811yescharacteristic
128C2×C42D811yescharacteristic
129C2×C42D811yescharacteristic
130C2×C42C2×D411yescharacteristic
131C2×C42C2×D411yescharacteristic
132C2×C42C2×D411yescharacteristic
133C2×C16—22no—
134C2×C16—22no—
135C2×C16—22no—
136C2×C16—22no—
137C2×C16—22no—
138C2×C16—22no—
139#107—21no—
140#107—22no—
141#107—22no—
142#107—22no—
143#107—22no—
144#107—22no—
145#107—22no—
146#107—42no—
147#107—42no—
148#107C2×D412yes—
149#106—44no—
150#106—44no—
151#103—24no—
152#103—42no—
153#103—42no—
154C4×C8—22no—
155C4×C8—22no—
156C4×C8—22no—
157C4×C8—22no—
158C4×C8—22no—
159C4×C8C2×D412yes—
160C4×C8#4112yes—
161#95—41no—
162#95—41no—
163#95—41no—
164#95—41no—
165#95—41no—
166#95—41no—
167#95—41no—
168#95—42no—
169#95—42no—
170C2×Q8—22no—
171C2×Q8—41no—
172C2×Q8—41no—
173C2×Q8—41no—
174C2×Q8—41no—
175C2×Q8—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4C2×D811yescharacteristic
183C22×C4C2×D811yescharacteristic
184C22×C4C2×D811yescharacteristic
185C22×C4C2×D811yescharacteristic
186C22×C4C2×D811yescharacteristic
187C22×C4C2×D811yescharacteristic
188C22×C4C22×D411yescharacteristic
189Q16—24no—
190Q16—42no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8#12112yes—
198C2×C8#13712yes—
199C2×C8#13712yes—
200C2×C8C22×D411yescharacteristic
201C2×C8C2×#4111yescharacteristic
202C2×C82−1+412yes—
203C2×C82−1+412yes—
204C4⋊C4—22no—
205C4⋊C4—41no—
206C4⋊C4—44no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—82no—
216C4⋊C4—82no—
217C4⋊C4—82no—
218C4⋊C4—84no—
219C4⋊C4—84no—
220C4⋊C4—82no—
221C4⋊C4—82no—
222C4⋊C4—82no—
223C42—21no—
224C42—21no—
225C42—21no—
226C42—21no—
227C42—22no—
228C42—22no—
229C42—41no—
230C42—41no—
231C42—44no—
232C42—82no—
233C42#12112yes—
234C42AGL(1,ℤ/8)12yes—
235C16—42no—
236C16—42no—
237C23C22×D811yescharacteristic
238Q8—42no—
239Q8—42no—
240Q8—81no—
241Q8—81no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4—81no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4#45912yes—
275C2×C4#45912yes—
276C2×C4#50812yes—
277C2×C4#51012yes—
278C2×C4C2×#12111yescharacteristic
279C2×C4#53611yescharacteristic
280C2×C4#53611yescharacteristic
281C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
282C2×C4C2×#13711yescharacteristic
283C2×C4C2×#13711yescharacteristic
284C8—22no—
285C8—22no—
286C8—22no—
287C8#53612yes—
288C8#54112yes—
289C22C2×#45911yescharacteristic
290C22#322611yescharacteristic
291C22#322611yescharacteristic
292C22C2×#50811yescharacteristic
293C22C2×#51011yescharacteristic
294C22#341411yescharacteristic
295C22#341411yescharacteristic
296C4—22no—
297C4—22no—
298C4—22no—
299C4—22no—
300C4—41no—
301C4—41no—
302C4—82no—
303C4—82no—
304C4—161no—
305C4—161no—
306C4—162no—
307C4—162no—
308C4#262111yescharacteristic
309C4#303911yescharacteristic
310C4#323911yescharacteristic
311C4#323911yescharacteristic
312C2#2986411yescharacteristic
313C2#3357911yescharacteristic
314C2#3358111yescharacteristic
315C2#3379711yescharacteristic
316C2#3379811yescharacteristic
317C2#3380211yescharacteristic
318C2#3380611yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×D8
commutator, Frattini, cosocleC22×C8C24
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ 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
C2×#22292
#333112
#212762
#135072
#135062
#135052
#134832
#134812
#132772
#128792