Finite Group Databasebeta

← #326644

Subgroups — #326644

Order 512.

479 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 479 conjugacy classes of subgroups — too many to draw. Only its 122 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#21736C211yesmaximal, characteristic, complemented
2#21269C211yesmaximal, characteristic
3#10687C211yesmaximal, characteristic, complemented
4#10660C211yesmaximal, characteristic
5#10548C211yesmaximal, characteristic, complemented
6#10036C211yesmaximal, characteristic
7#9580C212yesmaximal, complemented
8#9235C212yesmaximal
9#9050C212yesmaximal
10#8879C212yesmaximal, complemented
11#8800C211yesmaximal, characteristic, complemented
12#3162C2211yescharacteristic
13#3101C2212yes—
14#2961C2212yes—
15C2×#433—21no—
16#2293C2211yescharacteristic
17#1994C2212yes—
18#1955C2211yescharacteristic
19#1955C2211yescharacteristic
20#1946—21no—
21#1945—21no—
22#1940C2211yescharacteristic
23#1940C2211yescharacteristic
24#1923—21no—
25#1923—21no—
26#1849C2211yescharacteristic
27#1773—21no—
28#1772—21no—
29C4×C4⋊C8C2211yescharacteristic
30#1750—21no—
31C4×#97C2211yescharacteristic
32#1571C2212yes—
33#1542C2212yes—
34#1503C2212yes—
35#1486C2212yes—
36#1481C2212yes—
37#1303C2212yes—
38#1301C2212yes—
39#1281C2212yes—
40#1276C2212yes—
41#1275C2211yescharacteristic
42#1275C2211yescharacteristic
43#533C2312yes—
44C2×C4×D4C2311yescharacteristic
45C2×#117—21no—
46#501—22no—
47#501—22no—
48#483—22no—
49#476—22no—
50#461—22no—
51#450—22no—
52#446—42no—
53#446—42no—
54C4×SD16—42no—
55#433—44no—
56#431C2312yes—
57#431C2312yes—
58C2×#108—21no—
59C2×#108—21no—
60#427—21no—
61#427—21no—
62#424—22no—
63#423C2312yes—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8C2311yescharacteristic
67C2×C4⋊C8C2311yescharacteristic
68C2×C22⋊C8—21no—
69C2×C22⋊C8—21no—
70C2×C22⋊C8—21no—
71C2×C22⋊C8—21no—
72#405—22no—
73C2×#97D411yescharacteristic
74C2×#97D411yescharacteristic
75C2×#97C2311yescharacteristic
76C2×#97C2311yescharacteristic
77C2×C4×C8—21no—
78C2×C4×C8D411yescharacteristic
79C2×C4×C8D411yescharacteristic
80#390—21no—
81#389—21no—
82#388—22no—
83C4×C4⋊C4C2311yescharacteristic
84C4×C22⋊C4—21no—
85C43C2311yescharacteristic
86#339—22no—
87#339—22no—
88#336—21no—
89#336—21no—
90#336—21no—
91#335—42no—
92#333—22no—
93#332—22no—
94#328—24no—
95Q8⋊C8—42no—
96D4⋊C8—42no—
97C22×D4#4111yescharacteristic
98C23×C4—21no—
99C2×SD16—42no—
100C2×M16—22no—
101C22×C8—21no—
102C22×C8—21no—
103C22×C8C2×D411yescharacteristic
104C22×C8C2×D411yescharacteristic
105C22×C8C2×D411yescharacteristic
106C22×C8C2×D411yescharacteristic
107C4⋊Q8#4112yes—
108C4⋊Q8#4112yes—
109#125—22no—
110C22⋊Q8—22no—
111C22⋊Q8—22no—
112C4×Q8—22no—
113C4×Q8—22no—
114C4×D4—22no—
115C4×D4—22no—
116#117—22no—
117#117—42no—
118#117—42no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4#4111yescharacteristic
122C2×C22⋊C4—22no—
123C2×C22⋊C4—22no—
124C2×C22⋊C4—21no—
125C2×C42—21no—
126C2×C42—21no—
127C2×C42—21no—
128C2×C42C2×D411yescharacteristic
129C2×C42C2×D411yescharacteristic
130C2×C42C2411yescharacteristic
131#108—42no—
132#108—42no—
133#107—21no—
134#107—21no—
135#107—21no—
136#107—21no—
137#106—22no—
138#106—22no—
139#106—42no—
140C4⋊C8—22no—
141C4⋊C8—22no—
142C4⋊C8—22no—
143C4⋊C8—22no—
144C4⋊C8—22no—
145C4⋊C8—22no—
146C4⋊C8—42no—
147C4⋊C8—44no—
148C4⋊C8—42no—
149#103—42no—
150#102—42no—
151C22⋊C8—22no—
152C22⋊C8—22no—
153C22⋊C8—42no—
154C22⋊C8—42no—
155C22⋊C8—42no—
156C22⋊C8—42no—
157#97—22no—
158#97—22no—
159#97—22no—
160#97—22no—
161#97—42no—
162C4×C8—22no—
163C4×C8—22no—
164C4×C8—22no—
165C4×C8—22no—
166C4×C8—42no—
167C4×C8—42no—
168#95—22no—
169#95#4112yes—
170C24—21no—
171C2×Q8—22no—
172C2×Q8—22no—
173C2×D4—22no—
174C2×D4—81no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—22no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4C22×D411yescharacteristic
186C22×C4C2×#4111yescharacteristic
187C22×C4C2×#4111yescharacteristic
188SD16—84no—
189M16—42no—
190M16—42no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—41no—
201C2×C8—42no—
202C2×C8—41no—
203C2×C8—42no—
204C2×C8—42no—
205C2×C8—41no—
206C2×C8—41no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8#13512yes—
210C2×C8AGL(1,ℤ/8)12yes—
211C2×C8#13712yes—
212C4⋊C4—22no—
213C4⋊C4—22no—
214C4⋊C4—22no—
215C4⋊C4—22no—
216C4⋊C4—22no—
217C4⋊C4—44no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—81no—
221C22⋊C4—44no—
222C22⋊C4—44no—
223C22⋊C4—42no—
224C22⋊C4—42no—
225C42—21no—
226C42—21no—
227C42—21no—
228C42—42no—
229C42—42no—
230C42—41no—
231C42—41no—
232C42—81no—
233C42#13712yes—
234C42C2×#4112yes—
235C42H(5,𝔽2)12yes—
236C23—42no—
237C23—42no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23#52511yescharacteristic
242Q8—82no—
243Q8—82no—
244D4—42no—
245D4—42no—
246C2×C4—22no—
247C2×C4—22no—
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—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—81no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4#52511yescharacteristic
267C2×C4#53511yescharacteristic
268C2×C4#55112yes—
269C2×C4#55312yes—
270C2×C4C2×#13511yescharacteristic
271C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
272C2×C4C2×#13711yescharacteristic
273C2×C4C2×#13711yescharacteristic
274C8—22no—
275C8—22no—
276C8—22no—
277C8—41no—
278C8—82no—
279C8—82no—
280C8—81no—
281C8—81no—
282C8—82no—
283C22—21no—
284C22—21no—
285C22—82no—
286C22—82no—
287C22—81no—
288C22—81no—
289C22—81no—
290C22#259811yescharacteristic
291C22#310711yescharacteristic
292C22#311011yescharacteristic
293C4—22no—
294C4—22no—
295C4—42no—
296C4—42no—
297C4—82no—
298C4—81no—
299C4—82no—
300C4—82no—
301C4#310811yescharacteristic
302C4#311211yescharacteristic
303C4#320711yescharacteristic
304C4#321811yescharacteristic
305C2—21no—
306C2—21no—
307C2—161no—
308C2#2936911yescharacteristic
309C2#2938511yescharacteristic
310C2#2938611yescharacteristic
subgroupG/H
centerC2×C4#535
commutator, Frattini, cosocleC2×C42C24
socleC22#2598
prime radicalC22×D4#41
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21736C2semidirect
3. #10687C2semidirect
5. #10548C2semidirect
7. #9580C2semidirect
10. #8879C2semidirect
11. #8800C2semidirect

Maximal subgroups

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

subgroupindex
#217362
#212692
#106872
#106602
#105482
#100362
#95802
#92352
#90502
#88792
#88002