Finite Group Databasebeta

← #200880

Subgroups — #200880

Order 512.

991 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 991 conjugacy classes of subgroups — too many to draw. Only its 242 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#60862C211yesmaximal, characteristic
2#34348C211yesmaximal, characteristic, complemented
3#17755C211yesmaximal, characteristic, complemented
4#17586C212yesmaximal
5#9430C214yesmaximal, complemented
6#9427C214yesmaximal, complemented
7#7977C212yesmaximal, complemented
8#3444C2211yescharacteristic
9C2×#533C2211yescharacteristic
10#3240—24no—
11#3231C414yescomplemented
12#3228C414yescomplemented
13#3165C2212yes—
14#3141C2212yes—
15#3093—22no—
16#3080—22no—
17#2920C2211yescharacteristic
18C2×#432C2212yes—
19#2382—22no—
20#2381C2214yes—
21#2378—24no—
22#2298—22no—
23#2295—24no—
24#2283C2212yes—
25#1556C2214yes—
26#1552—24no—
27#1516—24no—
28#1515C2214yes—
29#1514C2214yes—
30#1509—24no—
31#1288C2218yes—
32#533—22no—
33#533—22no—
34#533C2×C412yes—
35#533C2×C414yes—
36#533D414yes—
37#533C2311yescharacteristic
38#532C2×C412yes—
39#532C2311yescharacteristic
40C2×C4⋊Q8C2312yes—
41#529—22no—
42#529—22no—
43#529—22no—
44C2×#125D412yes—
45#525—21no—
46#525—22no—
47C2×C22⋊Q8—22no—
48C4×#41—21no—
49C4×#41—22no—
50C2×C4×Q8—22no—
51C2×#117D412yes—
52C22×C42C2311yescharacteristic
53#501—24no—
54#500C2×C414yes—
55#498C2×C414yes—
56#489—24no—
57#487C2×C414yes—
58#486C2×C414yes—
59#485—28no—
60#482—28no—
61#479—28no—
62#478—28no—
63#432—22no—
64#432—22no—
65#432—22no—
66#432—22no—
67#432C2314yes—
68#431C2×C414yes—
69#428—24no—
70#423—21no—
71C2×C4⋊C8—22no—
72#417—24no—
73#407—21no—
74C2×C22⋊C8—22no—
75C4×M16C2312yes—
76C2×#97—22no—
77#391—28no—
78#389—28no—
79#388—28no—
80#387—28no—
81#387—28no—
82#387—28no—
83#383C2314yes—
84#382—22no—
85#381D414yes—
86#380—22no—
87C4×C4⋊C4—24no—
88C4×C4⋊C4D414yes—
89C4×C22⋊C4—22no—
90#376—22no—
91#376—22no—
92C43—22no—
93#332—28no—
94#328—28no—
95#327—28no—
96C2×#41—22no—
97C2×#41—21no—
98C22×Q8—22no—
99C23×C4C2×D411yescharacteristic
100C23×C4C2×D412yes—
101C2×M16—22no—
102C22×C8—42no—
103C4⋊Q8—22no—
104C4⋊Q8—22no—
105C4⋊Q8C22×C412yes—
106C4⋊Q8C22×C411yescharacteristic
107C4⋊Q8C22×C414yes—
108C4⋊Q8C2×D414yes—
109#127C22×C411yescharacteristic
110#126—42no—
111#125—22no—
112#125—22no—
113#125—22no—
114#125—21no—
115#125C22⋊C418yes—
116#124—22no—
117#124—21no—
118#123—41no—
119#123—42no—
120C22⋊Q8—24no—
121C22⋊Q8—24no—
122C22⋊Q8—24no—
123C22⋊Q8—24no—
124C22⋊Q8—22no—
125C22⋊Q8—22no—
126C22⋊Q8—22no—
127#121—22no—
128#121—22no—
129#121—22no—
130#120—41no—
131C4×Q8—24no—
132C4×Q8—24no—
133C4×Q8—24no—
134C4×Q8—24no—
135C4×Q8—22no—
136C4×Q8—22no—
137C4×Q8—22no—
138C4×D4—22no—
139C4×D4—22no—
140C4×D4—22no—
141C4×D4—42no—
142C4×D4—42no—
143C4×D4—42no—
144#117—24no—
145#117—22no—
146#117—21no—
147#117—42no—
148#117—41no—
149#117C22⋊C418yes—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—28no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4C2×D414yes—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—24no—
157C2×C42—22no—
158C2×C42—24no—
159C2×C42—28no—
160C2×C42—41no—
161C2×C42C22×C412yes—
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D412yes—
164C2×C42C2×D412yes—
165C2×C42#4112yes—
166C2×C42#4112yes—
167C2×C42C2411yescharacteristic
168#107—28no—
169#106—28no—
170C4⋊C8—24no—
171C4⋊C8—24no—
172C4⋊C8—42no—
173#103—28no—
174#102—28no—
175C22⋊C8—24no—
176C22⋊C8—24no—
177C22⋊C8—42no—
178#97—22no—
179#97—24no—
180#97—42no—
181C4×C8C22×C414yes—
182#95—28no—
183#95—28no—
184#95—24no—
185#95—24no—
186#95—24no—
187#95C2×D418yes—
188C24#12011yescharacteristic
189#41—44no—
190#41—84no—
191C2×Q8—24no—
192C2×Q8—24no—
193C2×Q8—21no—
194C2×Q8—22no—
195C2×Q8—84no—
196C2×D4—21no—
197C2×D4—22no—
198C2×D4—41no—
199C2×D4—42no—
200C22×C4—24no—
201C22×C4—28no—
202C22×C4—42no—
203C22×C4—41no—
204C22×C4—42no—
205C22×C4—82no—
206C22×C4C2×C22⋊C412yes—
207C22×C4C4×D412yes—
208C22×C4C4×D412yes—
209C22×C4#12011yescharacteristic
210C22×C4#12012yes—
211C22×C4#12112yes—
212C22×C4#12112yes—
213C22×C4#12312yes—
214C22×C4C22×D411yescharacteristic
215C22×C4C22×D412yes—
216M16—44no—
217C2×C8—24no—
218C2×C8—44no—
219C2×C8—84no—
220C4⋊C4—24no—
221C4⋊C4—24no—
222C4⋊C4—24no—
223C4⋊C4—28no—
224C4⋊C4—28no—
225C4⋊C4—22no—
226C4⋊C4—22no—
227C4⋊C4—22no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—84no—
232C4⋊C4—88no—
233C4⋊C4C2×C22⋊C418yes—
234C22⋊C4—28no—
235C22⋊C4—41no—
236C22⋊C4—42no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—84no—
241C42—28no—
242C42—48no—
243C42—41no—
244C42—42no—
245C42—88no—
246C42C2×C22⋊C412yes—
247C42C4×D412yes—
248C42C4×D412yes—
249C42#12112yes—
250C42#12112yes—
251C42#12311yescharacteristic
252C42#12311yescharacteristic
253C42C23×C411yescharacteristic
254C42C22×D411yescharacteristic
255C42C2×#4112yes—
256C23—22no—
257C23—42no—
258C23—41no—
259C23#38612yes—
260C23C2×#12011yescharacteristic
261Q8—42no—
262Q8—84no—
263Q8—84no—
264Q8—82no—
265D4—42no—
266D4—82no—
267D4—84no—
268D4—82no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—28no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—48no—
281C2×C4—48no—
282C2×C4—48no—
283C2×C4—48no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—82no—
287C2×C4—84no—
288C2×C4#38612yes—
289C2×C4#38612yes—
290C2×C4#38612yes—
291C2×C4C22×C22⋊C411yescharacteristic
292C2×C4C2×C4×D412yes—
293C2×C4C2×#12112yes—
294C2×C4C2×#12311yescharacteristic
295C8—44no—
296C8—84no—
297C22—22no—
298C22—22no—
299C22—22no—
300C22—22no—
301C22—82no—
302C22—82no—
303C22—81no—
304C22C2×#38611yescharacteristic
305C22#302012yes—
306C4—22no—
307C4—22no—
308C4—22no—
309C4—22no—
310C4—21no—
311C4—21no—
312C4—22no—
313C4—48no—
314C4—88no—
315C4—82no—
316C4—81no—
317C2—22no—
318C2—42no—
319C2—161no—
320C2#2095411yescharacteristic
321C2#2099812yes—
subgroupG/H
center, socleC22C2×#386
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radical#3444C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34348C2semidirect
3. #17755C2semidirect
5. #9430C2semidirect
6. #9427C2semidirect
7. #7977C2semidirect
11. #3231C4semidirect
12. #3228C4semidirect

Maximal subgroups

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

subgroupindex
#608622
#343482
#177552
#175862
#94302
#94272
#79772