Finite Group Databasebeta

← #475603

Subgroups — #475603

Order 512.

905 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23734C212yesmaximal
2C2×#2089C212yesmaximal, complemented
3C2×#2078C212yesmaximal
4#21046C211yesmaximal, characteristic
5C2×#1845C211yesmaximal, characteristic, complemented
6#11354C212yesmaximal, complemented
7#11331C212yesmaximal, complemented
8#11107C212yesmaximal
9#10822C211yesmaximal, characteristic, complemented
10C2×#428C2214yes—
11C22×#107C2212yes—
12C22×#107C2212yes—
13C2×#424C2211yescharacteristic
14C2×#417C2214yes—
15C2×#408C2211yescharacteristic
16C2×#408C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18#2297C2211yescharacteristic
19C2×#384C2212yes—
20#2096—22no—
21#2095—22no—
22#2094—22no—
23#2093—22no—
24#2089—28no—
25#2078—28no—
26#1923—24no—
27#1845—24no—
28C2×#343C2214yes—
29C2×#343C2214yes—
30C2×#341C2212yes—
31C2×#341C2212yes—
32#1729C2211yescharacteristic
33#1728C2211yescharacteristic
34#1728C2212yes—
35C22×#41C2311yescharacteristic
36C22×M16C2311yescharacteristic
37C22×M16C2311yescharacteristic
38C22×M16C2314yes—
39C23×C8C2311yescharacteristic
40C23×C8C2311yescharacteristic
41C2×#117C2311yescharacteristic
42C22×C4⋊C4—24no—
43C22×C4⋊C4C2314yes—
44C22×C42C2311yescharacteristic
45#428—28no—
46#428—44no—
47C2×#107—24no—
48C2×#107—24no—
49C2×#107—24no—
50C2×#107—24no—
51C2×#107—24no—
52C2×#107—44no—
53C2×#106—24no—
54#424—24no—
55C2×C4⋊C8—22no—
56#417—28no—
57#417—44no—
58C2×#103—24no—
59C2×#102—24no—
60#408—24no—
61#408—24no—
62C2×C22⋊C8—22no—
63C2×C22⋊C8—22no—
64C2×C4×C8—24no—
65C2×C4×C8D414yes—
66#391D411yescharacteristic
67#391D411yescharacteristic
68#390D411yescharacteristic
69#390D411yescharacteristic
70#386—22no—
71#384—22no—
72#384—22no—
73#384—24no—
74C4×C22⋊C4—22no—
75#343—28no—
76#343—28no—
77#341—24no—
78#341D418yes—
79#336—22no—
80#336—22no—
81#336—22no—
82#336—22no—
83C2×#41—22no—
84C2×#41—22no—
85C2×#41—22no—
86C2×#41—22no—
87C2×#41—22no—
88C2×#41—22no—
89C22×Q8C2×D411yescharacteristic
90C22×D4—21no—
91C22×D4C2×D411yescharacteristic
92C23×C4—21no—
93C23×C4—24no—
94C23×C4#4112yes—
95C23×C4C2411yescharacteristic
96C2×M16—24no—
97C2×M16—24no—
98C2×M16—24no—
99C2×M16—24no—
100C2×M16—24no—
101C2×M16—22no—
102C2×M16—22no—
103C2×M16—22no—
104C2×M16—22no—
105C2×M16—22no—
106C2×M16—22no—
107C2×M16C2×D418yes—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—24no—
113C22×C8—21no—
114C22×C8—21no—
115C22×C8C2×D412yes—
116C22×C8C2×D414yes—
117C22×C8C2×D412yes—
118C22×C8#4114yes—
119#117—22no—
120#117—22no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—44no—
128C2×C4⋊C4—44no—
129C2×C4⋊C4—88no—
130C2×C4⋊C4C2×D411yescharacteristic
131C2×C4⋊C4C2×D411yescharacteristic
132C2×C22⋊C4—21no—
133C2×C22⋊C4—22no—
134C2×C22⋊C4—22no—
135C2×C42—22no—
136C2×C42—22no—
137C2×C42—21no—
138C2×C42C2×D411yescharacteristic
139C2×C42C2×D411yescharacteristic
140C2×C42C2×D412yes—
141#107—48no—
142#107—48no—
143#107—48no—
144#106—48no—
145C4⋊C8—44no—
146#103—48no—
147#102—48no—
148C22⋊C8—44no—
149C22⋊C8—44no—
150C4×C8—28no—
151#95—22no—
152#95—22no—
153C24—21no—
154C24C2×#4111yescharacteristic
155#41—44no—
156#41—44no—
157#41—44no—
158#41—44no—
159C2×Q8—22no—
160C2×Q8—22no—
161C2×Q8—22no—
162C2×D4—22no—
163C2×D4—22no—
164C2×D4—22no—
165C2×D4—42no—
166C2×D4—42no—
167C2×D4—42no—
168C22×C4—24no—
169C22×C4—24no—
170C22×C4—24no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—44no—
181C22×C4—44no—
182C22×C4—44no—
183C22×C4—88no—
184C22×C4#12112yes—
185C22×C4#12112yes—
186C22×C4C22×D411yescharacteristic
187C22×C4C22×D412yes—
188C22×C4C22×D411yescharacteristic
189C22×C4C2×#4112yes—
190C22×C4H(5,𝔽2)12yes—
191C22×C42−1+412yes—
192C22×C42−1+412yes—
193C22×C42−1+412yes—
194M16—28no—
195M16—28no—
196M16—44no—
197M16—44no—
198M16—44no—
199M16—44no—
200C2×C8—22no—
201C2×C8—22no—
202C2×C8—22no—
203C2×C8—22no—
204C2×C8—28no—
205C2×C8—28no—
206C2×C8—28no—
207C2×C8—44no—
208C2×C8—44no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—41no—
212C2×C8—41no—
213C2×C8—41no—
214C2×C8—41no—
215C2×C8#12314yes—
216C2×C8#12714yes—
217C4⋊C4—22no—
218C4⋊C4—22no—
219C4⋊C4—44no—
220C4⋊C4—44no—
221C4⋊C4—44no—
222C4⋊C4—44no—
223C4⋊C4—88no—
224C4⋊C4—88no—
225C22⋊C4—42no—
226C22⋊C4—44no—
227C22⋊C4—44no—
228C42—22no—
229C42—22no—
230C42—22no—
231C42—22no—
232C42—41no—
233C42—41no—
234C23—22no—
235C23—22no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—42no—
240C23—42no—
241C23C2×#12111yescharacteristic
242C23#54112yes—
243C23#55012yes—
244C23#55212yes—
245Q8—44no—
246D4—44no—
247D4—84no—
248C2×C4—216no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—81no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—81no—
264C2×C4—84no—
265C2×C4—88no—
266C2×C4—164no—
267C2×C4—164no—
268C2×C4#47214yes—
269C2×C4#47314yes—
270C2×C4C2×#12311yescharacteristic
271C2×C4C2×#12711yescharacteristic
272C2×C4#53511yescharacteristic
273C2×C4#53611yescharacteristic
274C2×C4#53611yescharacteristic
275C2×C4#53611yescharacteristic
276C2×C4#53614yes—
277C2×C4#53612yes—
278C2×C4#53914yes—
279C2×C4D4212yes—
280C2×C4#54712yes—
281C2×C4#54714yes—
282C2×C4#56314yes—
283C8—24no—
284C8—24no—
285C8—48no—
286C8—81no—
287C8—81no—
288C22—24no—
289C22—22no—
290C22—22no—
291C22—24no—
292C22—82no—
293C22—81no—
294C22—81no—
295C22—81no—
296C22—82no—
297C22#248611yescharacteristic
298C22#254111yescharacteristic
299C22#262111yescharacteristic
300C22#272114yes—
301C22C2×#47212yes—
302C22C2×#47312yes—
303C4—24no—
304C4—24no—
305C4—81no—
306C4—81no—
307C4—81no—
308C4—164no—
309C4#250111yescharacteristic
310C4#262311yescharacteristic
311C4#265811yescharacteristic
312C4#268014yes—
313C4#282111yescharacteristic
314C2—24no—
315C2—161no—
316C2#1706211yescharacteristic
317C2#2767512yes—
318C2#2849711yescharacteristic
319C2#2849811yescharacteristic
320C2#2988612yes—
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#41C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2089C2semidirect
5. C2×#1845C2semidirect
6. #11354C2semidirect
7. #11331C2semidirect
9. #10822C2semidirect

Maximal subgroups

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

subgroupindex
#237342
C2×#20892
C2×#20782
#210462
C2×#18452
#113542
#113312
#111072
#108222