Finite Group Databasebeta

← #229458

Subgroups — #229458

Order 512.

591 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23162C211yesmaximal, characteristic
2#23069C212yesmaximal
3#22725C211yesmaximal, characteristic
4C2×#1676C211yesmaximal, characteristic
5#9836C211yesmaximal, characteristic
6#9618C211yesmaximal, characteristic
7#9576C212yesmaximal
8#8974C212yesmaximal
9#8927C212yesmaximal
10#8286C211yesmaximal, characteristic
11#8203C211yesmaximal, characteristic
12C2×#500C2211yescharacteristic
13C2×#477C2212yes—
14C2×C8×Q8C2211yescharacteristic
15#2369C2212yes—
16#2355C2212yes—
17#1990C2212yes—
18#1987C2212yes—
19#1956C2212yes—
20#1949—22no—
21#1948C2212yes—
22#1947C2212yes—
23#1943—21no—
24#1935—22no—
25#1927—22no—
26#1918—22no—
27#1918—22no—
28#1846—21no—
29#1844C2211yescharacteristic
30#1776—21no—
31#1776—22no—
32C4×#106—21no—
33C4×#106C2211yescharacteristic
34#1770C2211yescharacteristic
35#1770C2212yes—
36C4×#103C2211yescharacteristic
37#1676C418yescomplemented
38C2×#335C2211yescharacteristic
39C2×#333C2212yes—
40C2×Q8⋊C8C2211yescharacteristic
41C2×#322C2211yescharacteristic
42#1296C2212yes—
43#1279C2211yescharacteristic
44#1279C2212yes—
45#1277C2211yescharacteristic
46C2×C4⋊Q8C2312yes—
47C2×C4×Q8C2311yescharacteristic
48#500—22no—
49#500C2×C414yes—
50#477—24no—
51#477C2×C418yes—
52C8×Q8—22no—
53C8×Q8C2×C414yes—
54C2×#107—21no—
55C2×#107—21no—
56C2×#107—22no—
57C2×#107—22no—
58C2×#106—21no—
59C2×#106—22no—
60C2×#106—21no—
61C2×#106D411yescharacteristic
62C2×#106D411yescharacteristic
63C2×C4⋊C8—21no—
64C2×C4⋊C8C2311yescharacteristic
65C2×C4⋊C8C2312yes—
66C2×C4⋊C8C2311yescharacteristic
67C2×#103—22no—
68C2×#103—42no—
69C2×#103D412yes—
70C2×#97—22no—
71C2×#97—21no—
72C2×C4×C8—21no—
73C2×C4×C8C2311yescharacteristic
74C2×C4×C8C2311yescharacteristic
75C2×C4×C8C2311yescharacteristic
76#391—22no—
77#391—42no—
78#389—22no—
79#389—22no—
80#387—42no—
81#384C2312yes—
82#382—22no—
83C4×C4⋊C4—22no—
84C4×C4⋊C4—22no—
85C4×C4⋊C4—22no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4C2312yes—
88C4×C4⋊C4C2311yescharacteristic
89#340—22no—
90#340—22no—
91#340—22no—
92#340—22no—
93#340—22no—
94#340—22no—
95#340D412yes—
96#340D412yes—
97#336—21no—
98#336—22no—
99#335C2×C414yes—
100#333C2×C418yes—
101Q8⋊C8—22no—
102#322—22no—
103C22×Q8—22no—
104C22×Q8C2×D411yescharacteristic
105C22×C8—22no—
106C22×C8—21no—
107C22×C8—21no—
108C22×C8—21no—
109C22×C8C2×D412yes—
110C22×C8C2×D411yescharacteristic
111C22×C8C2×D411yescharacteristic
112C4⋊Q8—22no—
113C4⋊Q8—22no—
114C4⋊Q8—22no—
115C4⋊Q8C22×C414yes—
116C4×Q8—21no—
117C4×Q8—21no—
118C4×Q8C22×C412yes—
119C4×Q8#4112yes—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4C2×D412yes—
128C2×C4⋊C4C2×D412yes—
129C2×C4⋊C4C2×D412yes—
130C2×C4⋊C4C2×D411yescharacteristic
131C2×C42—21no—
132C2×C42—21no—
133C2×C42—22no—
134C2×C42—42no—
135C2×C42—42no—
136C2×C42C2411yescharacteristic
137#107—24no—
138#107—24no—
139#107—28no—
140#107—82no—
141#106—22no—
142#106—22no—
143#106—44no—
144#106—42no—
145#106—81no—
146C4⋊C8—24no—
147C4⋊C8—21no—
148C4⋊C8—22no—
149C4⋊C8—21no—
150C4⋊C8—21no—
151C4⋊C8C22×C412yes—
152C4⋊C8C22×C414yes—
153#103—28no—
154#103—42no—
155#103—84no—
156#97—44no—
157#97—42no—
158C4×C8—24no—
159C4×C8—21no—
160C4×C8—21no—
161C4×C8—21no—
162C4×C8—21no—
163C4×C8C22×C412yes—
164C4×C8#4112yes—
165#95—22no—
166#95—21no—
167#95—42no—
168#95—42no—
169#95—42no—
170#95—42no—
171#95—42no—
172C2×Q8—21no—
173C2×Q8—24no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8—42no—
177C2×Q8—82no—
178C2×Q8—82no—
179C2×Q8C4×D412yes—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4C22×D411yescharacteristic
188C22×C4C22×D411yescharacteristic
189C22×C4H(5,𝔽2)11yescharacteristic
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—22no—
193C2×C8—24no—
194C2×C8—22no—
195C2×C8—22no—
196C2×C8—22no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8—42no—
202C2×C8—41no—
203C2×C8—41no—
204C2×C8—41no—
205C2×C8—41no—
206C2×C8C4×D412yes—
207C2×C8C4×D412yes—
208C2×C8#13712yes—
209C2×C8#13712yes—
210C4⋊C4—22no—
211C4⋊C4—21no—
212C4⋊C4—22no—
213C4⋊C4—24no—
214C4⋊C4—24no—
215C4⋊C4—22no—
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—22no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—82no—
228C4⋊C4—162no—
229C4⋊C4C4×D412yes—
230C42—22no—
231C42—22no—
232C42—41no—
233C42—42no—
234C42—42no—
235C42—41no—
236C42—162no—
237C42—162no—
238C42C23×C411yescharacteristic
239C42C2×#4111yescharacteristic
240C42C2×#4111yescharacteristic
241C422−1+411yescharacteristic
242C23D4211yescharacteristic
243Q8—22no—
244Q8—22no—
245Q8—42no—
246Q8—84no—
247Q8—162no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—44no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4C2×C4×D411yescharacteristic
269C2×C4C2×C4×D411yescharacteristic
270C2×C4#52011yescharacteristic
271C2×C4#54711yescharacteristic
272C2×C4#54811yescharacteristic
273C2×C4#55311yescharacteristic
274C2×C4C2×#13711yescharacteristic
275C2×C4C2×#13711yescharacteristic
276C2×C4#57512yes—
277C2×C4#57711yescharacteristic
278C2×C4#57812yes—
279C2×C4#57811yescharacteristic
280C8—22no—
281C8—42no—
282C8—44no—
283C8—42no—
284C8—81no—
285C8—81no—
286C8#44114yes—
287C22#236111yescharacteristic
288C22#326912yes—
289C22#327611yescharacteristic
290C22#327811yescharacteristic
291C22#328812yes—
292C4—21no—
293C4—21no—
294C4—42no—
295C4—42no—
296C4—82no—
297C4—82no—
298C4—84no—
299C4—82no—
300C4—162no—
301C4C2×#44111yescharacteristic
302C4#294311yescharacteristic
303C4#294711yescharacteristic
304C4#295011yescharacteristic
305C4#334912yes—
306C4#336911yescharacteristic
307C4#337411yescharacteristic
308C2#2023112yes—
309C2#2023612yes—
310C2#2300411yescharacteristic
311C2#2303011yescharacteristic
312C2#2304811yescharacteristic
subgroupG/H
center, socle, prime radicalC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
37. #1676C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#231622
#230692
#227252
C2×#16762
#98362
#96182
#95762
#89742
#89272
#82862
#82032