Finite Group Databasebeta

← #227067

Subgroups — #227067

Order 512.

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

normal subgroups only

This group has 680 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#23063C212yesmaximal
2#22525C212yesmaximal
3C2×#1633C211yesmaximal, characteristic
4#9839C211yesmaximal, characteristic
5#9568C212yesmaximal
6#9224C212yesmaximal
7#9066C211yesmaximal, characteristic
8#8887C212yesmaximal
9#8213C212yesmaximal
10C2×Q82C2211yescharacteristic
11C2×#494C2211yescharacteristic
12C2×#477C2212yes—
13C2×#446C2212yes—
14#2369C2211yescharacteristic
15#2355C2212yes—
16#1993—22no—
17#1990C2212yes—
18#1990C2212yes—
19#1988—22no—
20#1987C2212yes—
21#1947C2212yes—
22#1933—22no—
23#1928—22no—
24#1921—22no—
25#1844C2211yescharacteristic
26#1805—22no—
27#1803C2212yes—
28C4×#107—22no—
29C8×C4⋊C4C2212yes—
30#1763—22no—
31C4×#103C2212yes—
32#1633C418yescomplemented
33C2×#333C2211yescharacteristic
34C2×#333C2211yescharacteristic
35C2×Q8⋊C8C2212yes—
36C2×#322C2211yescharacteristic
37#1296C2212yes—
38#1279C2212yes—
39#1277C2212yes—
40C22×Q16—41no—
41Q82—24no—
42Q82—41no—
43Q82C2×C414yes—
44C2×C4⋊Q8—22no—
45C2×C4⋊Q8—41no—
46C2×C4⋊Q8C2311yescharacteristic
47C2×C4⋊Q8C2311yescharacteristic
48C2×C4×Q8—22no—
49C2×C4×Q8C2312yes—
50#494—41no—
51#494C2×C414yes—
52#477—24no—
53#477C2×C418yes—
54#446—24no—
55#446—42no—
56C2×#107—22no—
57C2×#107—22no—
58C2×#106—41no—
59C2×C4⋊C8—22no—
60C2×C4⋊C8C2312yes—
61C2×C4⋊C8C2312yes—
62C2×#103—22no—
63C2×#103—22no—
64C2×#103D412yes—
65C2×#103D412yes—
66C2×#97—22no—
67C2×C4×C8—22no—
68C2×C4×C8C2312yes—
69C2×C4×C8C2311yescharacteristic
70#391—22no—
71#391—41no—
72#389—22no—
73#387—42no—
74#384C2312yes—
75#382—22no—
76C4×C4⋊C4—22no—
77C4×C4⋊C4—22no—
78C4×C4⋊C4—22no—
79C4×C4⋊C4C2312yes—
80#340—22no—
81#340—22no—
82#340—22no—
83#340—22no—
84#340—22no—
85#340—22no—
86#340—22no—
87#340—22no—
88#340D412yes—
89#340D412yes—
90#336—22no—
91#333—41no—
92#333C2×C414yes—
93Q8⋊C8C2×C418yes—
94#322—41no—
95C22×Q8—21no—
96C22×Q8—21no—
97C22×Q8—41no—
98C22×Q8C2×D412yes—
99C2×Q16—44no—
100C2×Q16—82no—
101C2×Q16—82no—
102C22×C8—21no—
103C22×C8—22no—
104C22×C8—22no—
105C22×C8C2×D412yes—
106C22×C8C2×D412yes—
107C4⋊Q8—28no—
108C4⋊Q8—22no—
109C4⋊Q8—22no—
110C4⋊Q8—21no—
111C4⋊Q8—21no—
112C4⋊Q8—44no—
113C4⋊Q8—44no—
114C4⋊Q8—44no—
115C4⋊Q8C22×C412yes—
116C4⋊Q8C22×C412yes—
117C4×Q8—22no—
118C4×Q8—22no—
119C4×Q8—28no—
120C4×Q8—44no—
121C4×Q8C22×C414yes—
122C4×Q8#4114yes—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4C2×D412yes—
132C2×C4⋊C4C2×D412yes—
133C2×C4⋊C4C2×D412yes—
134C2×C42—22no—
135C2×C42—22no—
136C2×C42—41no—
137C2×C42—42no—
138C2×C42—41no—
139C2×C42C2411yescharacteristic
140#107—28no—
141#107—44no—
142#106—82no—
143C4⋊C8—22no—
144C4⋊C8—22no—
145C4⋊C8—22no—
146C4⋊C8—44no—
147C4⋊C8C22×C414yes—
148#103—28no—
149#103—24no—
150#103—42no—
151#103—44no—
152#97—44no—
153C4×C8—22no—
154C4×C8—22no—
155C4×C8—21no—
156C4×C8—44no—
157C4×C8C22×C412yes—
158#95—22no—
159#95—42no—
160#95—42no—
161#95—42no—
162#95—42no—
163C2×Q8—22no—
164C2×Q8—22no—
165C2×Q8—22no—
166C2×Q8—22no—
167C2×Q8—22no—
168C2×Q8—22no—
169C2×Q8—22no—
170C2×Q8—44no—
171C2×Q8—41no—
172C2×Q8—41no—
173C2×Q8—81no—
174C2×Q8—81no—
175C2×Q8—81no—
176C2×Q8—81no—
177C2×Q8—161no—
178C2×Q8—161no—
179C2×Q8C4×D414yes—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—42no—
187C22×C4C22×D412yes—
188C22×C4H(5,𝔽2)11yescharacteristic
189Q16—88no—
190C2×C8—22no—
191C2×C8—22no—
192C2×C8—22no—
193C2×C8—24no—
194C2×C8—22no—
195C2×C8—22no—
196C2×C8—22no—
197C2×C8—22no—
198C2×C8—22no—
199C2×C8—22no—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8—42no—
203C2×C8—42no—
204C2×C8—42no—
205C4⋊C4—24no—
206C4⋊C4—22no—
207C4⋊C4—24no—
208C4⋊C4—22no—
209C4⋊C4—24no—
210C4⋊C4—24no—
211C4⋊C4—22no—
212C4⋊C4—22no—
213C4⋊C4—48no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—82no—
221C4⋊C4—161no—
222C4⋊C4C4×D414yes—
223C4⋊C4#13714yes—
224C42—24no—
225C42—24no—
226C42—44no—
227C42—42no—
228C42—42no—
229C42—161no—
230C42—162no—
231C42C23×C411yescharacteristic
232C42C2×#4112yes—
233C42H(5,𝔽2)11yescharacteristic
234C23D4211yescharacteristic
235Q8—24no—
236Q8—24no—
237Q8—42no—
238Q8—42no—
239Q8—82no—
240Q8—82no—
241Q8—161no—
242Q8—161no—
243Q8—164no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—24no—
247C2×C4—24no—
248C2×C4—24no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—81no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—81no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4C2×C4×D412yes—
268C2×C4#51811yescharacteristic
269C2×C4#54612yes—
270C2×C4#55111yescharacteristic
271C2×C4C2×#13712yes—
272C2×C4#57512yes—
273C2×C4#57812yes—
274C2×C4#57812yes—
275C8—24no—
276C8—42no—
277C8—44no—
278C8—42no—
279C8—82no—
280C22#235811yescharacteristic
281C22#326912yes—
282C22#327012yes—
283C22#329111yescharacteristic
284C22#329211yescharacteristic
285C4—21no—
286C4—21no—
287C4—44no—
288C4—44no—
289C4—44no—
290C4—82no—
291C4—82no—
292C4—82no—
293C4—162no—
294C4C4×#13712yes—
295C4#295012yes—
296C4#332712yes—
297C4#332812yes—
298C2#2009512yes—
299C2#2009711yescharacteristic
300C2#2012511yescharacteristic
301C2#2288112yes—
302C2#2289711yescharacteristic
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. #1633C4semidirect

Maximal subgroups

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

subgroupindex
#230632
#225252
C2×#16332
#98392
#95682
#92242
#90662
#88872
#82132