Finite Group Databasebeta

← #499632

Subgroups — #499632

Order 512.

443 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 443 conjugacy classes of subgroups — too many to draw. Only its 136 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21737C211yesmaximal, characteristic
2#21289C211yesmaximal, characteristic
3#13541C211yesmaximal, characteristic
4#13536C211yesmaximal, characteristic
5#13521C212yesmaximal
6#13479C212yesmaximal
7#13454C211yesmaximal, characteristic
8#13368C211yesmaximal, characteristic
9#12677C212yesmaximal
10#12660C211yesmaximal, characteristic
11#12648C212yesmaximal
12#3162C2211yescharacteristic
13#2960—21no—
14C2×#441C2211yescharacteristic
15#2320—21no—
16#2159C2212yes—
17C2×#365C2212yes—
18C2×#364C2212yes—
19C2×#362C2212yes—
20#2141C2212yes—
21C2×#347C2212yes—
22#1995—21no—
23#1971C2211yescharacteristic
24#1969C2211yescharacteristic
25#1945—21no—
26#1945C2212yes—
27#1944C2211yescharacteristic
28#1941C2211yescharacteristic
29#1935—21no—
30#1933—21no—
31#1849C2211yescharacteristic
32#1776C2211yescharacteristic
33#1773—21no—
34#1773C2212yes—
35C4×#103—21no—
36#1726C2211yescharacteristic
37#1387C2212yes—
38#1386C2212yes—
39#1386C2212yes—
40#1383C2212yes—
41#1377C2211yescharacteristic
42#1377C2211yescharacteristic
43C22×Q16C2311yescharacteristic
44#533—21no—
45C2×C4×Q8—21no—
46C2×M32C2312yes—
47C2×M32C2312yes—
48#501—22no—
49#501—22no—
50#446—41no—
51#446—41no—
52C8×Q8—41no—
53C8×Q8—41no—
54#441—22no—
55#441Q814yescomplemented
56C2×#108—21no—
57C2×#108C2312yes—
58#427—21no—
59#427C2312yes—
60C2×#107C2311yescharacteristic
61C2×#106C2311yescharacteristic
62C2×#106C2311yescharacteristic
63#424—22no—
64#423—21no—
65C2×#103—21no—
66C2×#103—21no—
67C2×#103—21no—
68#405—22no—
69C2×#97D411yescharacteristic
70C2×#97C2311yescharacteristic
71C2×#97C2311yescharacteristic
72C2×C4×C8D411yescharacteristic
73C2×C4×C8D411yescharacteristic
74C2×C4×C8C2311yescharacteristic
75#391—21no—
76#391—21no—
77#389—21no—
78#389—21no—
79#387—41no—
80#387—41no—
81#383—41no—
82#382—41no—
83C4×C4⋊C4—21no—
84#376D411yescharacteristic
85#367—22no—
86#366—22no—
87#365—24no—
88#364—24no—
89#362—24no—
90#358—22no—
91#358—22no—
92#347—24no—
93#340—21no—
94#340—21no—
95#340—21no—
96#339—22no—
97#339—22no—
98C22×Q8—21no—
99C2×Q16—22no—
100C2×Q16—22no—
101C2×Q16—21no—
102C2×Q16C2×Q812yes—
103C2×M16—22no—
104C22×C8C2×D411yescharacteristic
105C22×C8C2×D411yescharacteristic
106C22×C8C2×D411yescharacteristic
107C22×C8C2411yescharacteristic
108C4⋊Q8—21no—
109C4⋊Q8—21no—
110#125—22no—
111C22⋊Q8—41no—
112C22⋊Q8—41no—
113C4×Q8—24no—
114C4×Q8—42no—
115C4×Q8—41no—
116C4×Q8—41no—
117#117—22no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C42—41no—
125C2×C42—41no—
126C2×C42C2×D411yescharacteristic
127C2×C42C2×D411yescharacteristic
128C2×C42C2×D411yescharacteristic
129M32—22no—
130M32—22no—
131M32—22no—
132M32#4114yes—
133C2×C16—22no—
134C2×C16—22no—
135#108—22no—
136#108—22no—
137#108—42no—
138#107—21no—
139#107—21no—
140#107—22no—
141#107#4112yes—
142#107#4112yes—
143#106—22no—
144#106—21no—
145#106—22no—
146#106—22no—
147#106C2×Q812yes—
148C4⋊C8—41no—
149C4⋊C8—41no—
150C4⋊C8—42no—
151C4⋊C8—42no—
152#103—24no—
153#103—41no—
154#103—41no—
155#103—42no—
156#97—22no—
157#97—22no—
158#97—42no—
159#97C2×Q812yes—
160#97#4112yes—
161C4×C8—21no—
162C4×C8—21no—
163C4×C8—22no—
164C4×C8—41no—
165C4×C8—42no—
166#95—21no—
167#95—21no—
168#95—41no—
169#95—41no—
170#95—41no—
171#95—41no—
172#95—41no—
173#95—41no—
174#95—41no—
175C2×Q8—22no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8—41no—
181C2×Q8—41no—
182C2×Q8—41no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4AGL(1,ℤ/8)11yescharacteristic
190C22×C4AGL(1,ℤ/8)11yescharacteristic
191C22×C4C22×D411yescharacteristic
192Q16—24no—
193Q16—42no—
194M16—42no—
195M16—42no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—42no—
203C2×C8C22⋊Q812yes—
204C2×C8AGL(1,ℤ/8)11yescharacteristic
205C2×C8AGL(1,ℤ/8)11yescharacteristic
206C2×C8#13712yes—
207C2×C8C22×Q811yescharacteristic
208C2×C8C2×#4111yescharacteristic
209C2×C8C2×#4112yes—
210C2×C8C2×#4112yes—
211C4⋊C4—22no—
212C4⋊C4—22no—
213C4⋊C4—22no—
214C4⋊C4—44no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—41no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—161no—
225C4⋊C4—161no—
226C22⋊C4—42no—
227C42—21no—
228C42—21no—
229C42—21no—
230C42—21no—
231C42—21no—
232C42—44no—
233C42—42no—
234C42—161no—
235C42C22⋊Q812yes—
236C16—24no—
237C16—42no—
238C23C2×AGL(1,ℤ/8)11yescharacteristic
239Q8—42no—
240Q8—42no—
241Q8—82no—
242Q8—82no—
243Q8—81no—
244Q8—81no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—41no—
259C2×C4—81no—
260C2×C4—81no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4C2×C22⋊Q811yescharacteristic
270C2×C4#52511yescharacteristic
271C2×C4#53211yescharacteristic
272C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
273C2×C4C2×#13711yescharacteristic
274C2×C4#57511yescharacteristic
275C8—22no—
276C8—22no—
277C8—21no—
278C8—21no—
279C8—82no—
280C8#52912yes—
281C8#53312yes—
282C22—21no—
283C22—21no—
284C22#307711yescharacteristic
285C22#310911yescharacteristic
286C22#318211yescharacteristic
287C4—42no—
288C4—41no—
289C4—41no—
290C4—81no—
291C4—82no—
292C4—82no—
293C4—82no—
294C4—82no—
295C4—161no—
296C4—161no—
297C4#239611yescharacteristic
298C4#307911yescharacteristic
299C4#311711yescharacteristic
300C4#318011yescharacteristic
301C2—21no—
302C2—21no—
303C2#2378011yescharacteristic
304C2#3369111yescharacteristic
305C2#3369311yescharacteristic
subgroupG/H
center, socleC22#3109
commutator, Frattini, cosocleC22×C8C24
prime radicalC23C2×AGL(1,ℤ/8)
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1726 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
55. #441Q8semidirect

Maximal subgroups

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

subgroupindex
#217372
#212892
#135412
#135362
#135212
#134792
#134542
#133682
#126772
#126602
#126482