Finite Group Databasebeta

← #352849

Subgroups — #352849

Order 512.

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

normal subgroups only

This group has 720 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#33297C211yesmaximal, characteristic, complemented
2#33254C211yesmaximal, characteristic
3#30219C212yesmaximal, complemented
4#30203C212yesmaximal, complemented
5#9838C212yesmaximal, complemented
6#9832C212yesmaximal
7#9828C212yesmaximal, complemented
8#9822C212yesmaximal, complemented
9#7844C211yesmaximal, characteristic, complemented
10C2×#498C2211yescharacteristic
11C2×#493C2211yescharacteristic
12C2×#486C2212yes—
13#2868C2211yescharacteristic
14#2859C2211yescharacteristic
15#2858C2212yes—
16#2071—24no—
17#2042—24no—
18#2042—24no—
19#2023—24no—
20#1993—22no—
21#1992—22no—
22#1990C2212yes—
23#1990C2212yes—
24#1972—22no—
25#1972—22no—
26#1957C2212yes—
27#1957C2212yes—
28#1957C2212yes—
29#1957C2212yes—
30#1956C2212yes—
31#1956C2212yes—
32#1834C2212yes—
33#1834C2212yes—
34#1744C2212yes—
35C4×#97C2211yescharacteristic
36C2×#322C2212yes—
37C2×#322C2211yescharacteristic
38C2×C82C2211yescharacteristic
39C22×D8—22no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8—22no—
42C2×C4⋊Q8C2312yes—
43C2×#127C2312yes—
44C2×#125—21no—
45C2×#124—22no—
46C2×#124—22no—
47#498—24no—
48#493—24no—
49#486—24no—
50#486—24no—
51C2×#106—22no—
52C2×#103—24no—
53C2×#103—24no—
54C2×#102—24no—
55C2×#102—24no—
56C2×#102—24no—
57C2×#102—24no—
58C2×#102—24no—
59C2×#97D412yes—
60C2×#97D412yes—
61C2×C4×C8C2312yes—
62C2×C4×C8C2312yes—
63C2×C4×C8C2311yescharacteristic
64C2×C4×C8C2311yescharacteristic
65#390—22no—
66#390—22no—
67#390—22no—
68#390—22no—
69#389—22no—
70#389—22no—
71#384—22no—
72#384—21no—
73#384C2312yes—
74#384C2312yes—
75#382—21no—
76C43C2311yescharacteristic
77#340—24no—
78#340—24no—
79#340—24no—
80#340—24no—
81#336D414yes—
82#336D414yes—
83#322—22no—
84#322—24no—
85C82—22no—
86C22×Q8—22no—
87C22×Q8—41no—
88C22×D4—22no—
89C22×D4—22no—
90C22×D4—22no—
91C2×D8—44no—
92C2×D8—44no—
93C2×D8—44no—
94C22×C8C2×D414yes—
95C22×C8C2×D414yes—
96C22×C8C2×D412yes—
97C22×C8C2×D412yes—
98C4⋊Q8—22no—
99C4⋊Q8—22no—
100C4⋊Q8—22no—
101C4⋊Q8—22no—
102C4⋊Q8—44no—
103C4⋊Q8—84no—
104#127—22no—
105#127—22no—
106#127—22no—
107#127—22no—
108#125—44no—
109#124—84no—
110#124—84no—
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—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—42no—
122C2×C22⋊C4—42no—
123C2×C22⋊C4—42no—
124C2×C22⋊C4—42no—
125C2×C22⋊C4—42no—
126C2×C42C2×D412yes—
127C2×C42C2×D412yes—
128C2×C42C2×D411yescharacteristic
129C2×C42C2×D411yescharacteristic
130C2×C42C2411yescharacteristic
131#106—44no—
132#103—84no—
133#103—84no—
134#102—48no—
135#102—84no—
136#102—84no—
137#102—84no—
138#102—84no—
139#97—42no—
140#97—42no—
141C4×C8—22no—
142C4×C8—22no—
143C4×C8—22no—
144C4×C8—22no—
145C4×C8#4112yes—
146C4×C8#4112yes—
147C4×C8#4114yes—
148#95—41no—
149#95—41no—
150C24—42no—
151C2×Q8—42no—
152C2×Q8—42no—
153C2×Q8—44no—
154C2×Q8—44no—
155C2×Q8—82no—
156C2×Q8—84no—
157C2×D4—42no—
158C2×D4—42no—
159C2×D4—42no—
160C2×D4—42no—
161C2×D4—42no—
162C2×D4—42no—
163C2×D4—44no—
164C2×D4—44no—
165C2×D4—44no—
166C2×D4—44no—
167C2×D4—44no—
168C2×D4—44no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4#12711yescharacteristic
173C22×C4#12711yescharacteristic
174C22×C4#12712yes—
175C22×C4C22×D411yescharacteristic
176C22×C4C22×D412yes—
177D8—88no—
178C2×C8—22no—
179C2×C8—22no—
180C2×C8—22no—
181C2×C8—22no—
182C2×C8—24no—
183C2×C8—24no—
184C2×C8—24no—
185C2×C8—24no—
186C2×C8—24no—
187C2×C8—24no—
188C2×C8#12414yes—
189C2×C8#12414yes—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—44no—
195C4⋊C4—44no—
196C4⋊C4—44no—
197C4⋊C4—44no—
198C4⋊C4—44no—
199C4⋊C4—44no—
200C4⋊C4—84no—
201C4⋊C4—82no—
202C4⋊C4—84no—
203C22⋊C4—84no—
204C22⋊C4—84no—
205C22⋊C4—84no—
206C22⋊C4—84no—
207C42—22no—
208C42—22no—
209C42—24no—
210C42—24no—
211C42C2×#4111yescharacteristic
212C42C2×#4111yescharacteristic
213C42C2×#4112yes—
214C23—82no—
215C23—82no—
216C23—82no—
217C23—82no—
218C23—82no—
219C23—82no—
220C23—82no—
221C23C2×#12711yescharacteristic
222Q8—84no—
223Q8—84no—
224Q8—164no—
225D4—84no—
226D4—84no—
227D4—84no—
228D4—84no—
229D4—84no—
230D4—84no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—82no—
239C2×C4—82no—
240C2×C4—82no—
241C2×C4—82no—
242C2×C4—82no—
243C2×C4—82no—
244C2×C4—82no—
245C2×C4—82no—
246C2×C4—82no—
247C2×C4C2×#12411yescharacteristic
248C2×C4C2×#12411yescharacteristic
249C2×C4#53212yes—
250C2×C4#53212yes—
251C2×C4#57512yes—
252C2×C4#57512yes—
253C2×C4#57614yes—
254C2×C4#57714yes—
255C8—24no—
256C8—24no—
257C8—44no—
258C8—44no—
259C22—162no—
260C22—162no—
261C22—162no—
262C22—162no—
263C22—162no—
264C22—162no—
265C22—162no—
266C22#285811yescharacteristic
267C22#315311yescharacteristic
268C22#315311yescharacteristic
269C22#315412yes—
270C22#315512yes—
271C4—42no—
272C4—41no—
273C4—41no—
274C4—162no—
275C4—162no—
276C4—162no—
277C4#313512yes—
278C4#314012yes—
279C4#317914yes—
280C4#317914yes—
281C2—322no—
282C2#1972212yes—
283C2#1972411yescharacteristic
284C2#1972911yescharacteristic
285C2#3323311yescharacteristic
286C2#3324312yes—
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#493C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33297C2semidirect
3. #30219C2semidirect
4. #30203C2semidirect
5. #9838C2semidirect
7. #9828C2semidirect
8. #9822C2semidirect
9. #7844C2semidirect

Maximal subgroups

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

subgroupindex
#332972
#332542
#302192
#302032
#98382
#98322
#98282
#98222
#78442