Finite Group Databasebeta

← #326515

Subgroups — #326515

Order 512.

428 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33290C211yesmaximal, characteristic
2#21732C211yesmaximal, characteristic
3#10365C212yesmaximal
4#10363C212yesmaximal
5#9759C212yesmaximal
6#8794C212yesmaximal
7#8681C212yesmaximal
8#8679C212yesmaximal
9#8267C211yesmaximal, characteristic
10#3160C2211yescharacteristic
11#2960C2212yes—
12#2862C2211yescharacteristic
13#2859C2211yescharacteristic
14#1994C2212yes—
15#1991—21no—
16#1991—21no—
17#1990—21no—
18#1990—21no—
19#1945—21no—
20#1945—21no—
21#1848C2211yescharacteristic
22#1835C2212yes—
23#1835C2212yes—
24#1772—21no—
25#1772—21no—
26C42×C8C2211yescharacteristic
27#1564C2211yescharacteristic
28#1564C2211yescharacteristic
29#1518C2212yes—
30#1518C2212yes—
31#1518C2212yes—
32#1518C2212yes—
33#1452C2212yes—
34#1286C2212yes—
35#1286C2212yes—
36#1286C2212yes—
37#1286C2212yes—
38#1276C2212yes—
39#533C2312yes—
40C2×C4⋊Q8—21no—
41C2×#125—21no—
42#500Q811yescharacteristic, complemented
43#500Q811yescharacteristic, complemented
44#499—21no—
45#499—21no—
46#498Q812yescomplemented
47#446—22no—
48#446—22no—
49C8×Q8—22no—
50C8×Q8—22no—
51#431C2312yes—
52#431C2312yes—
53C2×#108—21no—
54C2×#108—21no—
55#427—21no—
56#427—21no—
57#424—22no—
58#423C2312yes—
59#405—22no—
60C2×C4×C8D411yescharacteristic
61C2×C4×C8D411yescharacteristic
62C2×C4×C8D411yescharacteristic
63C2×C4×C8D411yescharacteristic
64C2×C4×C8C2311yescharacteristic
65C2×C4×C8C2311yescharacteristic
66#391—41no—
67#391—41no—
68#389—41no—
69#389—41no—
70#384—21no—
71#382—21no—
72#382C2312yes—
73#382C2312yes—
74C43C2311yescharacteristic
75#340—22no—
76#340—22no—
77#340—22no—
78#340—22no—
79#339—22no—
80#339—22no—
81#335—21no—
82#335—21no—
83#334—21no—
84#334—21no—
85#328—42no—
86#328—42no—
87#328—42no—
88#328—42no—
89#322—22no—
90#322—22no—
91C22×Q8—41no—
92C2×M16—22no—
93C22×C8C2×D411yescharacteristic
94C22×C8C2×D411yescharacteristic
95C22×C8C2×D411yescharacteristic
96C22×C8C2×D411yescharacteristic
97C4⋊Q8—42no—
98C4⋊Q8—42no—
99C4⋊Q8C2×Q812yes—
100C4⋊Q8C2×Q812yes—
101#125—22no—
102#125—42no—
103#125—42no—
104C22⋊Q8—22no—
105C22⋊Q8—22no—
106C4×Q8—22no—
107C4×Q8—22no—
108#117—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C42—21no—
116C2×C42—21no—
117C2×C42C2×D411yescharacteristic
118C2×C42C2×D411yescharacteristic
119C2×C42C2411yescharacteristic
120#108—42no—
121#108—42no—
122#107—21no—
123#107—21no—
124#107—21no—
125#107—21no—
126#106—22no—
127#106—22no—
128C4⋊C8—22no—
129C4⋊C8—22no—
130C4⋊C8—22no—
131C4⋊C8—22no—
132C4⋊C8—42no—
133C4⋊C8—42no—
134C22⋊C8—22no—
135C22⋊C8—22no—
136#97—42no—
137C4×C8—22no—
138C4×C8—22no—
139C4×C8—22no—
140C4×C8—22no—
141C4×C8—41no—
142C4×C8—41no—
143C4×C8—42no—
144C4×C8C2×Q811yescharacteristic
145C4×C8C2×Q811yescharacteristic
146C4×C8#4111yescharacteristic
147C4×C8#4111yescharacteristic
148C4×C8#4111yescharacteristic
149C4×C8#4111yescharacteristic
150C4×C8#4111yescharacteristic
151C4×C8#4111yescharacteristic
152#95—22no—
153#95—22no—
154#95—22no—
155#95—22no—
156#95—42no—
157C2×Q8—22no—
158C2×Q8—22no—
159C2×Q8—82no—
160C2×Q8—82no—
161C2×Q8—82no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4C22×D411yescharacteristic
167C22×C4H(5,𝔽2)11yescharacteristic
168C22×C4H(5,𝔽2)11yescharacteristic
169M16—42no—
170M16—42no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—42no—
180C2×C8—42no—
181C2×C8—42no—
182C2×C8C22⋊Q811yescharacteristic
183C2×C8C22⋊Q811yescharacteristic
184C2×C8C22⋊Q811yescharacteristic
185C2×C8C22⋊Q811yescharacteristic
186C2×C8#12411yescharacteristic
187C2×C8#12411yescharacteristic
188C2×C8#12411yescharacteristic
189C2×C8#12411yescharacteristic
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—22no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—82no—
201C4⋊C4—82no—
202C4⋊C4—84no—
203C4⋊C4—82no—
204C22⋊C4—42no—
205C42—21no—
206C42—21no—
207C42—42no—
208C42—42no—
209C42—42no—
210C42#13712yes—
211C42#13712yes—
212C42C22×Q811yescharacteristic
213C42C2×#4111yescharacteristic
214C42C2×#4111yescharacteristic
215C42C2×#4111yescharacteristic
216C23#53511yescharacteristic
217Q8—82no—
218Q8—82no—
219Q8—164no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—82no—
231C2×C4—82no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4—82no—
235C2×C4—82no—
236C2×C4C2×C22⋊Q811yescharacteristic
237C2×C4C2×#12411yescharacteristic
238C2×C4#55411yescharacteristic
239C2×C4#55411yescharacteristic
240C2×C4#55511yescharacteristic
241C2×C4#55511yescharacteristic
242C2×C4C2×#13711yescharacteristic
243C2×C4C2×#13711yescharacteristic
244C2×C4#57511yescharacteristic
245C2×C4#57511yescharacteristic
246C8—21no—
247C8—21no—
248C8—21no—
249C8—21no—
250C8—21no—
251C8—21no—
252C8—21no—
253C8—21no—
254C8—82no—
255C8—82no—
256C8—82no—
257C22—21no—
258C22—21no—
259C22#256311yescharacteristic
260C22#321511yescharacteristic
261C22#321511yescharacteristic
262C4—24no—
263C4—21no—
264C4—21no—
265C4—21no—
266C4—21no—
267C4—42no—
268C4—82no—
269C4—82no—
270C4—162no—
271C4—162no—
272C4#307811yescharacteristic
273C4#307811yescharacteristic
274C4#313511yescharacteristic
275C4#313911yescharacteristic
276C2—21no—
277C2—21no—
278C2#2914711yescharacteristic
279C2#2915411yescharacteristic
280C2#2915411yescharacteristic
subgroupG/H
center, socleC22#2563
commutator, Frattini, cosocleC2×C42C24
prime radicalC23#535
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
42. #500Q8semidirect
43. #500Q8semidirect
46. #498Q8semidirect

Maximal subgroups

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

subgroupindex
#332902
#217322
#103652
#103632
#97592
#87942
#86812
#86792
#82672