Finite Group Databasebeta

← #345605

Subgroups — #345605

Order 512.

616 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 150 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#26460C212yesmaximal
2#26446C212yesmaximal, complemented
3#10589C212yesmaximal, complemented
4#10587C212yesmaximal
5#10573C212yesmaximal, complemented
6#9842C211yesmaximal, characteristic
7#9599C212yesmaximal, complemented
8#9064C211yesmaximal, characteristic
9#9063C211yesmaximal, characteristic
10C2×#492C2211yescharacteristic
11C2×#474—22no—
12#2551C2212yes—
13#2369C2211yescharacteristic
14#2366C2211yescharacteristic
15#2087—22no—
16#2067—22no—
17#2057—22no—
18#2023—22no—
19#1990C2212yes—
20#1987—22no—
21#1987C2212yes—
22#1982C2212yes—
23#1957C2212yes—
24#1947C2212yes—
25#1947C2212yes—
26#1939C2212yes—
27#1939C2212yes—
28#1921—22no—
29#1862—22no—
30#1845C2211yescharacteristic
31C4×#107C2212yes—
32C2×#334C2212yes—
33C2×#333C2211yescharacteristic
34C2×#331C2211yescharacteristic
35C2×#322C2211yescharacteristic
36#1296C2212yes—
37#1279C2212yes—
38#1279C2212yes—
39C22×SD16—41no—
40C2×C4⋊Q8—22no—
41C2×C4⋊Q8—22no—
42C2×C4⋊Q8C2311yescharacteristic
43C2×#127C2311yescharacteristic
44C2×C4×D4—22no—
45#492—41no—
46#492Q814yescomplemented
47#474—48no—
48C2×#107—22no—
49C2×#107—22no—
50C2×#107—41no—
51C2×#107D412yes—
52C2×#107D412yes—
53C2×C4⋊C8—22no—
54C2×C4⋊C8—22no—
55C2×C4⋊C8C2312yes—
56C2×C4⋊C8C2312yes—
57C2×#102—22no—
58C2×#102—22no—
59C2×#102—22no—
60C2×#102—22no—
61C2×C4×C8C2312yes—
62C2×C4×C8C2311yescharacteristic
63#395—42no—
64#391—41no—
65#390—22no—
66#389—22no—
67#389—22no—
68#389—22no—
69#386—41no—
70#384C2312yes—
71C4×C4⋊C4—22no—
72C4×C4⋊C4C2312yes—
73C4×C4⋊C4C2312yes—
74#340—22no—
75#340—22no—
76#340—22no—
77#340—22no—
78#340—22no—
79#340—22no—
80#340D412yes—
81#340D412yes—
82#336—22no—
83#336—22no—
84#334—42no—
85#333—41no—
86#331—41no—
87#322—41no—
88C22×Q8—21no—
89C22×Q8—42no—
90C22×D4—22no—
91C22×D4—21no—
92C23×C4—41no—
93C2×SD16—44no—
94C2×SD16—82no—
95C2×SD16—82no—
96C22×C8—21no—
97C22×C8—22no—
98C22×C8—22no—
99C22×C8C2×D412yes—
100C22×C8C2×D412yes—
101C4⋊Q8—21no—
102C4⋊Q8—22no—
103C4⋊Q8—44no—
104C4⋊Q8—44no—
105C4⋊Q8—82no—
106C4⋊Q8—82no—
107C4⋊Q8C2×Q812yes—
108#127—21no—
109#127—22no—
110#127C2×Q812yes—
111C4×D4—44no—
112C4×D4—82no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4C2×D412yes—
121C2×C4⋊C4C2×D412yes—
122C2×C4⋊C4C2×D412yes—
123C2×C4⋊C4C2×D412yes—
124C2×C22⋊C4—42no—
125C2×C22⋊C4—41no—
126C2×C42—22no—
127C2×C42—22no—
128C2×C42—41no—
129C2×C42C2411yescharacteristic
130#107—24no—
131#107—24no—
132#107—44no—
133#107—82no—
134#107—82no—
135C4⋊C8—22no—
136C4⋊C8—22no—
137C4⋊C8—22no—
138C4⋊C8—22no—
139C4⋊C8—44no—
140C4⋊C8—82no—
141#102—44no—
142#102—44no—
143#102—44no—
144#102—82no—
145C4×C8—22no—
146C4×C8—21no—
147C4×C8—22no—
148C4×C8C2×Q812yes—
149#95—22no—
150#95—22no—
151#95—42no—
152#95—42no—
153C24—41no—
154C2×Q8—22no—
155C2×Q8—41no—
156C2×Q8—84no—
157C2×Q8—81no—
158C2×Q8—81no—
159C2×Q8—84no—
160C2×Q8—162no—
161C2×D4—22no—
162C2×D4—24no—
163C2×D4—42no—
164C2×D4—41no—
165C2×D4—42no—
166C2×D4—42no—
167C2×D4—42no—
168C2×D4—42no—
169C2×D4—81no—
170C2×D4—81no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—41no—
176C22×C4—42no—
177C22×C4—84no—
178C22×C4—161no—
179C22×C4C22×D412yes—
180C22×C4H(5,𝔽2)11yescharacteristic
181SD16—88no—
182C2×C8—22no—
183C2×C8—22no—
184C2×C8—22no—
185C2×C8—22no—
186C2×C8—22no—
187C2×C8—22no—
188C2×C8—22no—
189C2×C8—22no—
190C2×C8—22no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—42no—
195C2×C8—42no—
196C2×C8—42no—
197C4⋊C4—22no—
198C4⋊C4—22no—
199C4⋊C4—24no—
200C4⋊C4—22no—
201C4⋊C4—22no—
202C4⋊C4—22no—
203C4⋊C4—22no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—84no—
210C4⋊C4—82no—
211C4⋊C4—84no—
212C4⋊C4—161no—
213C4⋊C4AGL(1,ℤ/8)14yes—
214C4⋊C4#13714yes—
215C22⋊C4—84no—
216C22⋊C4—161no—
217C42—42no—
218C42—42no—
219C42—42no—
220C42—42no—
221C42—161no—
222C42C22×Q811yescharacteristic
223C42H(5,𝔽2)11yescharacteristic
224C422−1+412yes—
225C23—42no—
226C23—82no—
227C23—82no—
228C23—81no—
229C23—81no—
230C23D4211yescharacteristic
231Q8—82no—
232Q8—161no—
233Q8—168no—
234D4—44no—
235D4—44no—
236D4—82no—
237D4—82no—
238D4—82no—
239D4—161no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—24no—
243C2×C4—24no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—81no—
256C2×C4—81no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—164no—
261C2×C4#53712yes—
262C2×C4#54311yescharacteristic
263C2×C4D4×Q812yes—
264C2×C4#56311yescharacteristic
265C2×C4C2×AGL(1,ℤ/8)12yes—
266C2×C4C2×#13712yes—
267C2×C4#57612yes—
268C2×C4#57812yes—
269C8—24no—
270C8—42no—
271C8—42no—
272C8—82no—
273C8—82no—
274C22—84no—
275C22—82no—
276C22—162no—
277C22—161no—
278C22—161no—
279C22#268311yescharacteristic
280C22#326412yes—
281C22#327012yes—
282C22#329111yescharacteristic
283C22#329211yescharacteristic
284C4—21no—
285C4—21no—
286C4—44no—
287C4—44no—
288C4—82no—
289C4—82no—
290C4—82no—
291C4—162no—
292C4#325412yes—
293C4#326212yes—
294C4#338512yes—
295C4#338612yes—
296C2—162no—
297C2#2003212yes—
298C2#2005311yescharacteristic
299C2#2011711yescharacteristic
300C2#3125212yes—
301C2#3125411yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #26446C2semidirect
3. #10589C2semidirect
5. #10573C2semidirect
7. #9599C2semidirect
46. #492Q8semidirect

Maximal subgroups

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

subgroupindex
#264602
#264462
#105892
#105872
#105732
#98422
#95992
#90642
#90632