Finite Group Databasebeta

← #315878

Subgroups — #315878

Order 512.

788 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 788 conjugacy classes of subgroups — too many to draw. Only its 170 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#33175C211yesmaximal, characteristic
2#27973C212yesmaximal, complemented
3#27891C212yesmaximal, complemented
4#20764C211yesmaximal, characteristic, complemented
5#10678C212yesmaximal, complemented
6#10671C212yesmaximal, complemented
7#9832C212yesmaximal, complemented
8#8821C212yesmaximal, complemented
9#8147C211yesmaximal, characteristic, complemented
10C2×#493C2211yescharacteristic
11C2×#469—22no—
12C2×C4×M16C2211yescharacteristic
13#2856C2212yes—
14#2285C2211yescharacteristic
15C2×#384C2211yescharacteristic
16#2071—24no—
17#2033—24no—
18#1972—22no—
19#1957C2212yes—
20#1957C2212yes—
21#1957C2212yes—
22#1957C2212yes—
23#1957C2212yes—
24#1925—24no—
25#1874—24no—
26#1845C2212yes—
27#1844C2211yescharacteristic
28#1836—22no—
29#1834C2212yes—
30#1834C2212yes—
31#1834C2212yes—
32#1834C2212yes—
33#1834C2212yes—
34C4×C22⋊C8C2212yes—
35#1736—22no—
36C2×#322C2212yes—
37#1273C2211yescharacteristic
38#1273C2211yescharacteristic
39C22×D8—22no—
40C22×M16D412yes—
41C2×#127C2312yes—
42C2×#121—22no—
43C2×#121—22no—
44C2×#121—22no—
45C2×#117—21no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—22no—
48C22×C42C2311yescharacteristic
49#493—24no—
50#469—48no—
51C2×#107—24no—
52C2×#106—22no—
53C2×#102—24no—
54C2×#102—24no—
55C2×#102—24no—
56C2×#102—24no—
57C2×#102—24no—
58C2×C22⋊C8D414yes—
59C4×M16—24no—
60C2×#97D412yes—
61C2×C4×C8C2311yescharacteristic
62C2×C4×C8C2312yes—
63C2×C4×C8C2312yes—
64C2×C4×C8C2311yescharacteristic
65#390—22no—
66#390—22no—
67#390—22no—
68#384—22no—
69#384—21no—
70#384C2312yes—
71#384C2312yes—
72#384C2312yes—
73#382—21no—
74#380—22no—
75C4×C4⋊C4—22no—
76#340—24no—
77#340—24no—
78#340—24no—
79#340—24no—
80#340—24no—
81#336D414yes—
82#322—24no—
83C22×D4—22no—
84C22×D4—22no—
85C22×D4—22no—
86C22×D4—42no—
87C23×C4—41no—
88C23×C4C2×D412yes—
89C23×C4C2×D411yescharacteristic
90C2×D8—44no—
91C2×D8—44no—
92C2×D8—44no—
93C2×M16—24no—
94C2×M16—24no—
95C2×M16—42no—
96C22×C8C2×D412yes—
97C22×C8C2×D414yes—
98C22×C8C2×D414yes—
99C22×C8C2×D412yes—
100#127—22no—
101#127—22no—
102#127—22no—
103#127—22no—
104#121—48no—
105#121—48no—
106#121—84no—
107#117—44no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—44no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—84no—
121C2×C22⋊C4—41no—
122C2×C22⋊C4—42no—
123C2×C22⋊C4—42no—
124C2×C22⋊C4—42no—
125C2×C42—22no—
126C2×C42—21no—
127C2×C42—21no—
128C2×C42—41no—
129C2×C42C2×D412yes—
130C2×C42C2×D411yescharacteristic
131C2×C42C2411yescharacteristic
132#107—84no—
133#106—44no—
134#102—48no—
135#102—48no—
136#102—84no—
137#102—84no—
138#102—84no—
139C22⋊C8—44no—
140#97—24no—
141C4×C8—22no—
142C4×C8—22no—
143C4×C8—22no—
144C4×C8—22no—
145C4×C8—22no—
146C4×C8#4114yes—
147#95—42no—
148C24—42no—
149C24#12711yescharacteristic
150C2×D4—42no—
151C2×D4—42no—
152C2×D4—42no—
153C2×D4—42no—
154C2×D4—42no—
155C2×D4—42no—
156C2×D4—44no—
157C2×D4—44no—
158C2×D4—44no—
159C2×D4—44no—
160C2×D4—44no—
161C2×D4—44no—
162C2×D4—84no—
163C2×D4—84no—
164C2×D4—84no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—24no—
171C22×C4—24no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—82no—
176C22×C4—84no—
177C22×C4#12712yes—
178C22×C4#12711yescharacteristic
179C22×C4AGL(1,ℤ/8)14yes—
180C22×C4C22×D411yescharacteristic
181C22×C4C22×D412yes—
182D8—88no—
183M16—48no—
184C2×C8—22no—
185C2×C8—22no—
186C2×C8—22no—
187C2×C8—24no—
188C2×C8—24no—
189C2×C8—22no—
190C2×C8—24no—
191C2×C8—24no—
192C2×C8—22no—
193C2×C8—24no—
194C2×C8—24no—
195C2×C8#12314yes—
196C4⋊C4—44no—
197C4⋊C4—44no—
198C4⋊C4—44no—
199C4⋊C4—44no—
200C4⋊C4—44no—
201C4⋊C4—44no—
202C4⋊C4—44no—
203C4⋊C4—44no—
204C4⋊C4—44no—
205C4⋊C4—82no—
206C4⋊C4—84no—
207C22⋊C4—84no—
208C22⋊C4—84no—
209C22⋊C4—82no—
210C22⋊C4—84no—
211C42—21no—
212C42—21no—
213C42—22no—
214C42—22no—
215C42—82no—
216C42C2×#4112yes—
217C422−1+411yescharacteristic
218C422−1+411yescharacteristic
219C23—21no—
220C23—22no—
221C23—22no—
222C23—21no—
223C23—82no—
224C23—82no—
225C23—82no—
226C23—82no—
227C23—82no—
228C23—82no—
229C23—82no—
230C23#49612yes—
231C23C2×#12711yescharacteristic
232D4—84no—
233D4—84no—
234D4—84no—
235D4—84no—
236D4—84no—
237D4—84no—
238D4—168no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—24no—
242C2×C4—24no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—44no—
247C2×C4—44no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—164no—
258C2×C4#49612yes—
259C2×C4C2×#12311yescharacteristic
260C2×C4#53611yescharacteristic
261C2×C4#54712yes—
262C2×C4#54712yes—
263C2×C4C2×AGL(1,ℤ/8)12yes—
264C2×C4#57514yes—
265C2×C4#57512yes—
266C2×C4#57614yes—
267C8—24no—
268C8—44no—
269C8—42no—
270C8—44no—
271C22—22no—
272C22—24no—
273C22—41no—
274C22—42no—
275C22—41no—
276C22—162no—
277C22—162no—
278C22—162no—
279C22—162no—
280C22—162no—
281C22—162no—
282C22—162no—
283C22#250111yescharacteristic
284C22C2×#49611yescharacteristic
285C22#315311yescharacteristic
286C22#315312yes—
287C22#315512yes—
288C4—22no—
289C4—42no—
290C4—162no—
291C4—162no—
292C4—162no—
293C4#309412yes—
294C4#323412yes—
295C4#334714yes—
296C4#335514yes—
297C2—42no—
298C2—322no—
299C2#1967312yes—
300C2#1972911yescharacteristic
301C2#1975411yescharacteristic
302C2#2791811yescharacteristic
303C2#2794212yes—
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radical#33175C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #27973C2semidirect
3. #27891C2semidirect
4. #20764C2semidirect
5. #10678C2semidirect
6. #10671C2semidirect
7. #9832C2semidirect
8. #8821C2semidirect
9. #8147C2semidirect

Maximal subgroups

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

subgroupindex
#331752
#279732
#278912
#207642
#106782
#106712
#98322
#88212
#81472