Finite Group Databasebeta

← #201064

Subgroups — #201064

Order 512.

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

normal subgroups only

This group has 993 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#60861C211yesmaximal, characteristic
2#34372C211yesmaximal, characteristic, complemented
3#17667C211yesmaximal, characteristic, complemented
4#17594C212yesmaximal
5#9442C214yesmaximal, complemented
6#9440C214yesmaximal, complemented
7#8070C212yesmaximal, complemented
8#3444C2211yescharacteristic
9C22×#125C2211yescharacteristic
10#3236C414yescomplemented
11#3234C414yescomplemented
12#3232—24no—
13#3166C2212yes—
14#3142C2212yes—
15#3092—22no—
16#3080—22no—
17#2920C2212yes—
18#2920C2211yescharacteristic
19#2382—22no—
20#2371C2214yes—
21#2345—24no—
22#2336—24no—
23#2313—22no—
24#2283C2212yes—
25#1534C2214yes—
26#1528—24no—
27#1521—24no—
28#1513—24no—
29#1506C2214yes—
30#1505C2214yes—
31#1300C2218yes—
32#533C2×C412yes—
33#533C2311yescharacteristic
34#532C2×C412yes—
35#532C2311yescharacteristic
36#529—22no—
37#529—22no—
38#529—22no—
39C2×#125—22no—
40C2×#125—22no—
41C2×#125—22no—
42C2×#125C2×C414yes—
43C2×#125D414yes—
44C2×#125D412yes—
45C2×#125C2312yes—
46#525—21no—
47#525—22no—
48C4×#41—21no—
49C4×#41—22no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4D412yes—
52C22×C42C2311yescharacteristic
53#501C2×C418yes—
54#499—24no—
55#491C2×C414yes—
56#490C2×C414yes—
57#488—24no—
58#484—24no—
59#483—24no—
60#481—24no—
61#480—24no—
62#479—28no—
63#478—28no—
64#432—22no—
65#432C2×C414yes—
66#432C2314yes—
67#431—22no—
68#431—22no—
69#431—22no—
70#428—24no—
71#423—21no—
72#423—22no—
73#417—24no—
74#407—21no—
75#407—22no—
76C4×M16—22no—
77C4×M16C2312yes—
78#389—28no—
79#387—28no—
80#387—28no—
81#387—28no—
82#387—28no—
83#383D414yes—
84#382—22no—
85#381—22no—
86#380—24no—
87#380D414yes—
88C4×C4⋊C4—22no—
89C4×C4⋊C4—28no—
90C4×C22⋊C4—24no—
91C4×C22⋊C4—22no—
92#376C2314yes—
93#330—28no—
94#328—28no—
95#327—28no—
96C2×#41—22no—
97C2×#41—21no—
98C23×C4—22no—
99C23×C4C2×D411yescharacteristic
100C23×C4C2×D412yes—
101C2×M16—22no—
102C2×M16—42no—
103C4⋊Q8C22×C412yes—
104C4⋊Q8C22×C411yescharacteristic
105#127C22×C411yescharacteristic
106#126—42no—
107#125—24no—
108#125—24no—
109#125—22no—
110#125—24no—
111#125—24no—
112#125—22no—
113#125—22no—
114#125—21no—
115#125—24no—
116#125C22⋊C418yes—
117#125C22×C414yes—
118#125C2×D414yes—
119#124—22no—
120#124—21no—
121#123—41no—
122#123—42no—
123C22⋊Q8—22no—
124C22⋊Q8—22no—
125C22⋊Q8—22no—
126#121—22no—
127#121—22no—
128#121—22no—
129#120—41no—
130C4×Q8—22no—
131C4×Q8—22no—
132C4×Q8—22no—
133C4×D4—22no—
134C4×D4—22no—
135C4×D4—22no—
136C4×D4—42no—
137C4×D4—42no—
138C4×D4—42no—
139#117—22no—
140#117—21no—
141#117—42no—
142#117—41no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—28no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4C22⋊C414yes—
155C2×C4⋊C4C22⋊C414yes—
156C2×C4⋊C4C2×D414yes—
157C2×C22⋊C4—24no—
158C2×C42—28no—
159C2×C42—24no—
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—24no—
179#97C22×C414yes—
180C4×C8—22no—
181C4×C8—42no—
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—21no—
192C2×Q8—22no—
193C2×D4—21no—
194C2×D4—22no—
195C2×D4—41no—
196C2×D4—42no—
197C22×C4—24no—
198C22×C4—24no—
199C22×C4—24no—
200C22×C4—28no—
201C22×C4—42no—
202C22×C4—41no—
203C22×C4—42no—
204C22×C4—82no—
205C22×C4—84no—
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—
217M16—84no—
218C2×C8—24no—
219C2×C8—44no—
220C4⋊C4—28no—
221C4⋊C4—28no—
222C4⋊C4—24no—
223C4⋊C4—28no—
224C4⋊C4—28no—
225C4⋊C4—24no—
226C4⋊C4—24no—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—22no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—84no—
234C4⋊C4—88no—
235C4⋊C4C2×C22⋊C418yes—
236C22⋊C4—48no—
237C22⋊C4—41no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C42—41no—
243C42—42no—
244C42—84no—
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—82no—
263D4—42no—
264D4—82no—
265D4—84no—
266D4—82no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—28no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—48no—
279C2×C4—48no—
280C2×C4—48no—
281C2×C4—48no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—82no—
285C2×C4—84no—
286C2×C4—84no—
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—44no—
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#2100012yes—
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. #34372C2semidirect
3. #17667C2semidirect
5. #9442C2semidirect
6. #9440C2semidirect
7. #8070C2semidirect
10. #3236C4semidirect
11. #3234C4semidirect

Maximal subgroups

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

subgroupindex
#608612
#343722
#176672
#175942
#94422
#94402
#80702