Finite Group Databasebeta

← #231707

Subgroups — #231707

Order 512.

642 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 642 conjugacy classes of subgroups — too many to draw. Only its 198 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#23217C212yesmaximal
2#23163C212yesmaximal
3C2×#1716C211yesmaximal, characteristic
4#9745C212yesmaximal
5#9729C212yesmaximal
6#9066C211yesmaximal, characteristic
7#9066C211yesmaximal, characteristic
8#9059C212yesmaximal
9#8288C212yesmaximal
10C2×#498C2212yes—
11C2×#498C2211yescharacteristic
12C2×#494C2211yescharacteristic
13C2×#487C2212yes—
14#2369C2211yescharacteristic
15#2369C2211yescharacteristic
16#2365C2212yes—
17#1988C2212yes—
18#1987C2212yes—
19#1987C2212yes—
20#1981C2212yes—
21#1944—22no—
22#1933—22no—
23#1928—22no—
24#1920—22no—
25#1919—22no—
26#1863—22no—
27#1776—22no—
28C4×#107C2212yes—
29#1770C2212yes—
30C8×C4⋊C4C2212yes—
31C4×#103—22no—
32#1716C418yescomplemented
33C2×#334C2212yes—
34C2×#333C2211yescharacteristic
35C2×#333C2211yescharacteristic
36C2×#332C2212yes—
37C2×#322C2211yescharacteristic
38#1279C2212yes—
39#1279C2212yes—
40C22×Q16—41no—
41C2×C4⋊Q8C2312yes—
42C2×C4⋊Q8C2311yescharacteristic
43C2×C4⋊Q8C2311yescharacteristic
44C2×C4×Q8—22no—
45#498—24no—
46#498—41no—
47#498C2×C418yes—
48#498C2×C414yes—
49#494—41no—
50#494C2×C414yes—
51#487—24no—
52#487C2×C418yes—
53C2×#107D412yes—
54C2×#107D412yes—
55C2×#106—22no—
56C2×#106—22no—
57C2×#106—41no—
58C2×C4⋊C8C2312yes—
59C2×C4⋊C8C2312yes—
60C2×#103—22no—
61C2×#103—22no—
62C2×#103—22no—
63C2×#103—22no—
64C2×#97—22no—
65C2×C4×C8—22no—
66C2×C4×C8C2311yescharacteristic
67C2×C4×C8C2312yes—
68#391—22no—
69#391—41no—
70#391—41no—
71#389—22no—
72#389—22no—
73#389—22no—
74#387—42no—
75C4×C4⋊C4—22no—
76C4×C4⋊C4—22no—
77C4×C4⋊C4—22no—
78C4×C4⋊C4C2312yes—
79C4×C4⋊C4C2312yes—
80#340—22no—
81#340—22no—
82#340—22no—
83#340—22no—
84#340—22no—
85#340—22no—
86#340D412yes—
87#340D412yes—
88#336—22no—
89#336—22no—
90#334—24no—
91#333—41no—
92#333—41no—
93#332—42no—
94#322C2×C414yes—
95C22×Q8—22no—
96C22×Q8—21no—
97C22×Q8—21no—
98C2×Q16—44no—
99C2×Q16—82no—
100C2×Q16—82no—
101C22×C8—21no—
102C22×C8—22no—
103C22×C8—22no—
104C22×C8C2×D412yes—
105C22×C8C2×D412yes—
106C4⋊Q8—22no—
107C4⋊Q8—22no—
108C4⋊Q8—22no—
109C4⋊Q8—22no—
110C4⋊Q8—22no—
111C4⋊Q8—21no—
112C4⋊Q8—21no—
113C4⋊Q8C22×C412yes—
114C4⋊Q8C22×C414yes—
115C4⋊Q8C22×C412yes—
116C4×Q8—44no—
117C4×Q8—82no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4C2×D412yes—
126C2×C4⋊C4C2×D412yes—
127C2×C4⋊C4C2×D412yes—
128C2×C4⋊C4C2×D412yes—
129C2×C42—22no—
130C2×C42—22no—
131C2×C42—42no—
132C2×C42—41no—
133C2×C42—41no—
134C2×C42C2411yescharacteristic
135#107—24no—
136#107—42no—
137#106—28no—
138#106—28no—
139#106—44no—
140C4⋊C8—22no—
141C4⋊C8—22no—
142C4⋊C8—22no—
143C4⋊C8—22no—
144#103—28no—
145#103—28no—
146#103—44no—
147#103—82no—
148#97—44no—
149C4×C8—21no—
150C4×C8—82no—
151C4×C8C22×C412yes—
152C4×C8C22×C414yes—
153C4×C8#4114yes—
154#95—22no—
155#95—22no—
156#95—42no—
157#95—42no—
158#95—42no—
159#95—42no—
160C2×Q8—24no—
161C2×Q8—22no—
162C2×Q8—22no—
163C2×Q8—42no—
164C2×Q8—41no—
165C2×Q8—41no—
166C2×Q8—42no—
167C2×Q8—42no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×Q8—81no—
171C2×Q8—81no—
172C2×Q8—81no—
173C2×Q8—81no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—42no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4C22×D412yes—
182C22×C4H(5,𝔽2)11yescharacteristic
183Q16—88no—
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—
197C2×C8C4×D414yes—
198C2×C8C4×D414yes—
199C4⋊C4—22no—
200C4⋊C4—24no—
201C4⋊C4—24no—
202C4⋊C4—24no—
203C4⋊C4—24no—
204C4⋊C4—22no—
205C4⋊C4—22no—
206C4⋊C4—22no—
207C4⋊C4—22no—
208C4⋊C4—22no—
209C4⋊C4—22no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—84no—
215C4⋊C4—161no—
216C4⋊C4—161no—
217C4⋊C4#13714yes—
218C42—42no—
219C42—42no—
220C42—42no—
221C42—42no—
222C42—84no—
223C42—161no—
224C42—161no—
225C42C23×C411yescharacteristic
226C42C2×#4112yes—
227C42H(5,𝔽2)11yescharacteristic
228C23D4211yescharacteristic
229Q8—44no—
230Q8—44no—
231Q8—82no—
232Q8—82no—
233Q8—82no—
234Q8—82no—
235Q8—161no—
236Q8—161no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—24no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—44no—
243C2×C4—42no—
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—82no—
255C2×C4—82no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4C2×C4×D412yes—
261C2×C4#51811yescharacteristic
262C2×C4#55012yes—
263C2×C4#55911yescharacteristic
264C2×C4C2×#13712yes—
265C2×C4#57512yes—
266C2×C4#57712yes—
267C2×C4#57812yes—
268C8—24no—
269C8—24no—
270C8—24no—
271C8—82no—
272C8—82no—
273C22#236611yescharacteristic
274C22#326712yes—
275C22#327012yes—
276C22#329111yescharacteristic
277C22#329111yescharacteristic
278C4—21no—
279C4—21no—
280C4—44no—
281C4—82no—
282C4—84no—
283C4—82no—
284C4—82no—
285C4—82no—
286C4—82no—
287C4C4×#13712yes—
288C4#294712yes—
289C4#340012yes—
290C4#340212yes—
291C2#2009212yes—
292C2#2009711yescharacteristic
293C2#2014311yescharacteristic
294C2#2319412yes—
295C2#2319811yescharacteristic
subgroupG/H
center, socle, prime radicalC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
LCSG ▷ 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
32. #1716C4semidirect

Maximal subgroups

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

subgroupindex
#232172
#231632
C2×#17162
#97452
#97292
#90662
#90592
#82882