Finite Group Databasebeta

← #329656

Subgroups — #329656

Order 512.

693 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 693 conjugacy classes of subgroups — too many to draw. Only its 138 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#29878C211yesmaximal, characteristic, complemented
2#29690C212yesmaximal
3#21789C211yesmaximal, characteristic
4#10622C212yesmaximal, complemented
5#10480C214yesmaximal, complemented
6#10278C212yesmaximal, complemented
7#9136C212yesmaximal, complemented
8#8598C211yesmaximal, characteristic, complemented
9C2×#532C2211yescharacteristic
10C2×#500C2211yescharacteristic
11C2×#476C2214yes—
12C2×#432C2211yescharacteristic
13C2×#431C2211yescharacteristic
14#2857C2212yes—
15#2087—22no—
16#2076—22no—
17#2058—22no—
18#2041—22no—
19#2034—24no—
20#1990C2212yes—
21#1955—22no—
22#1954C2212yes—
23#1954C2212yes—
24#1834C2211yescharacteristic
25#1802C2214yes—
26#1797C2212yes—
27#1780—22no—
28C2×#334C2212yes—
29C2×#333C2214yes—
30#1296C2214yes—
31#1277C2212yes—
32C22×#41—21no—
33#532—42no—
34#532—42no—
35C2×C4⋊Q8D411yescharacteristic
36C2×C4⋊Q8C2312yes—
37C2×#127D411yescharacteristic
38C2×#124—21no—
39C2×C22⋊Q8—22no—
40C2×#121—21no—
41C2×#121D412yes—
42C2×C4×D4—21no—
43C2×C4×D4C2312yes—
44C22×C42C2311yescharacteristic
45#500—42no—
46#476—44no—
47#476—44no—
48#432—42no—
49#431—42no—
50C2×#107—22no—
51C2×#106—24no—
52C2×C4⋊C8D412yes—
53C2×C4⋊C8C2312yes—
54C2×C4⋊C8C2314yes—
55C2×#102—24no—
56C2×#102—24no—
57C2×C22⋊C8—21no—
58C2×C22⋊C8—22no—
59C2×C22⋊C8—24no—
60C2×C22⋊C8—24no—
61C2×C22⋊C8D412yes—
62C2×#97—21no—
63C2×C4×C8C2311yescharacteristic
64C2×C4×C8C2311yescharacteristic
65#391—22no—
66#389—22no—
67#389—22no—
68#384C2312yes—
69#380—22no—
70#380—22no—
71#340—22no—
72#340—24no—
73#340—24no—
74#336—22no—
75#334—42no—
76#333—44no—
77C2×#41—42no—
78C2×#41—42no—
79C2×#41—84no—
80C22×Q8—21no—
81C22×Q8—22no—
82C22×D4—22no—
83C22×D4—21no—
84C22×D4—21no—
85C22×D4C2×D412yes—
86C23×C4—22no—
87C23×C4C2×D411yescharacteristic
88C23×C4C2×D411yescharacteristic
89C23×C4#4111yescharacteristic
90C22×C8—24no—
91C22×C8—22no—
92C22×C8C2×D412yes—
93C22×C8C2×D412yes—
94C4⋊Q8—24no—
95C4⋊Q8—24no—
96C4⋊Q8—42no—
97#127—42no—
98#124—44no—
99C22⋊Q8—84no—
100#121—44no—
101#121—44no—
102C4×D4—24no—
103C4×D4—24no—
104C4×D4—42no—
105C4×D4—42no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4C2×D412yes—
114C2×C22⋊C4—22no—
115C2×C22⋊C4—22no—
116C2×C22⋊C4—42no—
117C2×C42—22no—
118C2×C42—22no—
119C2×C42C2×D411yescharacteristic
120C2×C42C2×D411yescharacteristic
121C2×C42#4111yescharacteristic
122C2×C42C2411yescharacteristic
123#107—44no—
124#106—48no—
125C4⋊C8—24no—
126C4⋊C8—24no—
127C4⋊C8—24no—
128C4⋊C8—42no—
129#102—48no—
130#102—84no—
131C22⋊C8—42no—
132C22⋊C8—42no—
133C22⋊C8—84no—
134C22⋊C8—84no—
135C22⋊C8—82no—
136#97—42no—
137C4×C8—22no—
138C4×C8—22no—
139#95—42no—
140#95—42no—
141C24—22no—
142C24#12111yescharacteristic
143#41—88no—
144#41—88no—
145C2×Q8—42no—
146C2×Q8—44no—
147C2×Q8—82no—
148C2×Q8—82no—
149C2×Q8—82no—
150C2×D4—24no—
151C2×D4—24no—
152C2×D4—24no—
153C2×D4—42no—
154C2×D4—42no—
155C2×D4—44no—
156C2×D4—82no—
157C2×D4—82no—
158C2×D4—84no—
159C22×C4—21no—
160C22×C4—22no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—24no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—44no—
171C22×C4—44no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4#12111yescharacteristic
175C22×C4#12111yescharacteristic
176C22×C4#12111yescharacteristic
177C22×C4C22×D411yescharacteristic
178C22×C4C22×D411yescharacteristic
179C22×C4C2×#4111yescharacteristic
180C2×C8—22no—
181C2×C8—22no—
182C2×C8—24no—
183C2×C8—24no—
184C2×C8—44no—
185C2×C8—42no—
186C2×C8—44no—
187C2×C8—44no—
188C2×C8—44no—
189C4⋊C4—24no—
190C4⋊C4—42no—
191C4⋊C4—44no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—44no—
196C4⋊C4—42no—
197C4⋊C4—82no—
198C4⋊C4—84no—
199C22⋊C4—44no—
200C22⋊C4—44no—
201C22⋊C4—84no—
202C42—21no—
203C42—22no—
204C42—21no—
205C42—22no—
206C42H(5,𝔽2)12yes—
207C42H(5,𝔽2)12yes—
208C23—21no—
209C23—22no—
210C23—21no—
211C23—21no—
212C23—22no—
213C23—42no—
214C23—44no—
215C23—42no—
216C23—42no—
217C23—44no—
218C23C2×#12111yescharacteristic
219Q8—84no—
220Q8—164no—
221D4—48no—
222D4—84no—
223D4—84no—
224D4—88no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—42no—
234C2×C4—44no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—44no—
238C2×C4—82no—
239C2×C4—84no—
240C2×C4—82no—
241C2×C4—82no—
242C2×C4—82no—
243C2×C4—82no—
244C2×C4—82no—
245C2×C4—82no—
246C2×C4—82no—
247C2×C4—82no—
248C2×C4#53411yescharacteristic
249C2×C4#53412yes—
250C2×C4#53511yescharacteristic
251C2×C4#53812yes—
252C2×C4#57714yes—
253C2×C4#57712yes—
254C2×C4#57812yes—
255C8—42no—
256C8—42no—
257C8—84no—
258C8—82no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—42no—
263C22—42no—
264C22—82no—
265C22—82no—
266C22—82no—
267C22—84no—
268C22—84no—
269C22#261711yescharacteristic
270C22#304212yes—
271C22#305911yescharacteristic
272C22#306011yescharacteristic
273C22#306812yes—
274C4—22no—
275C4—41no—
276C4—41no—
277C4—41no—
278C4—82no—
279C4—162no—
280C4—162no—
281C4#320012yes—
282C4#320114yes—
283C4#322212yes—
284C2—81no—
285C2—162no—
286C2#1951512yes—
287C2#1951612yes—
288C2#2972311yescharacteristic
289C2#2973711yescharacteristic
290C2#2975111yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
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
1. #29878C2semidirect
4. #10622C2semidirect
5. #10480C2semidirect
6. #10278C2semidirect
7. #9136C2semidirect
8. #8598C2semidirect

Maximal subgroups

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

subgroupindex
#298782
#296902
#217892
#106222
#104802
#102782
#91362
#85982