Finite Group Databasebeta

← #494978

Subgroups — #494978

Order 512.

586 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×Q32C211yesmaximal, characteristic
2#21281C211yesmaximal, characteristic
3#20507C211yesmaximal, characteristic
4#13132C212yesmaximal
5#13112C212yesmaximal
6#13015C212yesmaximal
7#12938C212yesmaximal
8C4×#367C211yesmaximal, characteristic
9#12638C211yesmaximal, characteristic
10#12638C211yesmaximal, characteristic
11#12592C211yesmaximal, characteristic
12C22×Q32C2211yescharacteristic
13C2×C4×Q16C2212yes—
14C2×C4×Q16C2211yescharacteristic
15C2×C4×Q16C2211yescharacteristic
16#2320—21no—
17#2276—21no—
18C4×Q32C418yescomplemented
19C2×#367C2212yes—
20C2×#366C2211yescharacteristic
21C2×#358C2212yes—
22C2×#358C2212yes—
23C2×#358C2211yescharacteristic
24C2×#358C2211yescharacteristic
25C2×#346C2211yescharacteristic
26C2×C4×C16C2211yescharacteristic
27#1969C2212yes—
28#1942C2212yes—
29#1935—22no—
30C4×#107C2212yes—
31C4×#107C2211yescharacteristic
32C4×#107C2211yescharacteristic
33#1763—21no—
34#1763—22no—
35#1763—21no—
36#1726C2211yescharacteristic
37#1380C2212yes—
38#1380C2212yes—
39#1380C2212yes—
40#1374C2212yes—
41#1374C2212yes—
42C22×Q16C2311yescharacteristic
43C22×Q16C2311yescharacteristic
44C2×C4×Q8—21no—
45C2×C4×Q8—22no—
46C2×C4×Q8—21no—
47C2×Q32—24no—
48C2×Q32C2×C414yes—
49C22×C16C2312yes—
50C22×C16C2311yescharacteristic
51C22×C16C2311yescharacteristic
52C4×Q16—24no—
53C4×Q16—24no—
54C4×Q16—22no—
55C4×Q16—22no—
56C4×Q16C2×C414yes—
57C4×Q16C2×C414yes—
58C2×#107C2312yes—
59C2×#107C2311yescharacteristic
60C2×#107C2311yescharacteristic
61C2×#107C2312yes—
62C2×#103—21no—
63C2×#103—22no—
64C2×#103—21no—
65C2×#103—22no—
66C2×#97D411yescharacteristic
67C2×#97D412yes—
68C2×C4×C8C2312yes—
69C2×C4×C8C2311yescharacteristic
70#391—22no—
71#389—22no—
72#387—42no—
73#383—41no—
74#382—41no—
75C4×C4⋊C4—21no—
76C4×C4⋊C4—22no—
77C4×C4⋊C4—21no—
78C4×C4⋊C4—41no—
79C4×C4⋊C4—42no—
80#376—41no—
81#376D411yescharacteristic
82#367—24no—
83#366C2×C414yes—
84#358—24no—
85#358—24no—
86#358C2×C414yes—
87#358C2×C414yes—
88#346—22no—
89C4×C16C2×C414yes—
90#340—21no—
91#340—22no—
92#340—22no—
93#340—21no—
94C22×Q8—21no—
95C22×Q8—21no—
96C2×Q16—22no—
97C2×Q16—21no—
98C2×Q16—22no—
99C2×Q16—22no—
100C2×Q16—21no—
101C2×Q16C22×C412yes—
102C2×Q16C22×C412yes—
103C2×Q16#4114yes—
104C22×C8C2×D412yes—
105C22×C8C2×D411yescharacteristic
106C22×C8C2411yescharacteristic
107C4×Q8—28no—
108C4×Q8—24no—
109C4×Q8—24no—
110C4×Q8—44no—
111C4×Q8—42no—
112C4×Q8—42no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—42no—
121C2×C42—42no—
122C2×C42—41no—
123C2×C42—41no—
124C2×C42—42no—
125C2×C42—41no—
126C2×C42—41no—
127C2×C42C2×D412yes—
128C2×C42C2×D411yescharacteristic
129Q32—28no—
130C2×C16—22no—
131C2×C16—22no—
132C2×C16—22no—
133C2×C16—22no—
134C2×C16—22no—
135C2×C16C22×C412yes—
136C2×C16C22×C412yes—
137#107—21no—
138#107—21no—
139#107—22no—
140#107—22no—
141#107—22no—
142#107C22×C412yes—
143#107C22×C412yes—
144#107#4114yes—
145#103—28no—
146#103—24no—
147#103—24no—
148#103—44no—
149#97—22no—
150#97—42no—
151C4×C8—22no—
152C4×C8—22no—
153C4×C8C22×C412yes—
154C4×C8#4112yes—
155#95—42no—
156#95—41no—
157#95—41no—
158#95—42no—
159#95—42no—
160#95—42no—
161#95D811yescharacteristic
162#95D811yescharacteristic
163#95D812yes—
164C2×Q8—22no—
165C2×Q8—22no—
166C2×Q8—24no—
167C2×Q8—41no—
168C2×Q8—41no—
169C2×Q8—42no—
170C2×Q8—42no—
171C2×Q8—42no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—42no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—42no—
178C22×C4C2×D812yes—
179C22×C4C2×D812yes—
180C22×C4C2×D811yescharacteristic
181C22×C4C2×D811yescharacteristic
182C22×C4C22×D411yescharacteristic
183Q16—22no—
184Q16—22no—
185Q16—22no—
186Q16—24no—
187Q16—42no—
188Q16#11714yes—
189C2×C8—22no—
190C2×C8—22no—
191C2×C8—22no—
192C2×C8—22no—
193C2×C8C4×D412yes—
194C2×C8C23×C411yescharacteristic
195C2×C8C2×#4112yes—
196C2×C8C2×#4112yes—
197C2×C8C2×#4111yescharacteristic
198C4⋊C4—22no—
199C4⋊C4—22no—
200C4⋊C4—24no—
201C4⋊C4—48no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—42no—
205C4⋊C4—44no—
206C4⋊C4—44no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—162no—
210C42—22no—
211C42—22no—
212C42—48no—
213C42—44no—
214C42—44no—
215C42—161no—
216C42—161no—
217C42—162no—
218C42C4×D412yes—
219C42#13512yes—
220C16—22no—
221C16—22no—
222C16—42no—
223C23C22×D811yescharacteristic
224Q8—24no—
225Q8—44no—
226Q8—42no—
227Q8—42no—
228Q8—42no—
229Q8—82no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—44no—
240C2×C4—42no—
241C2×C4—44no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—82no—
246C2×C4—82no—
247C2×C4—81no—
248C2×C4—81no—
249C2×C4—81no—
250C2×C4—81no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4C4×D812yes—
256C2×C4C4×D812yes—
257C2×C4C2×C4×D411yescharacteristic
258C2×C4#52512yes—
259C2×C4C2×#13511yescharacteristic
260C2×C4#57512yes—
261C8—22no—
262C8—42no—
263C8C2×#11711yescharacteristic
264C8C4×#4111yescharacteristic
265C8#52912yes—
266C22C2×C4×D811yescharacteristic
267C22#310012yes—
268C22#341611yescharacteristic
269C22#341611yescharacteristic
270C22#341712yes—
271C4—22no—
272C4—22no—
273C4—44no—
274C4—42no—
275C4—42no—
276C4—82no—
277C4—84no—
278C4—82no—
279C4—82no—
280C4—162no—
281C4#229011yescharacteristic
282C4#294311yescharacteristic
283C4#310212yes—
284C2#2111311yescharacteristic
285C2#3345811yescharacteristic
286C2#3346011yescharacteristic
287C2#3368012yes—
288C2#3368112yes—
subgroupG/H
center, socle, prime radicalC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ #1726 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
18. C4×Q32C4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Q322
#212812
#205072
#131322
#131122
#130152
#129382
C4×#3672
#126382
#125922