Finite Group Databasebeta

← #201065

Subgroups — #201065

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#60863C211yesmaximal, characteristic
2#34375C211yesmaximal, characteristic, complemented
3#17676C211yesmaximal, characteristic, complemented
4#17591C212yesmaximal
5#9442C214yesmaximal, complemented
6#9440C214yesmaximal, complemented
7#8069C212yesmaximal, complemented
8#3444C2211yescharacteristic
9C2×#533C2211yescharacteristic
10#3237C414yescomplemented
11#3235C414yescomplemented
12#3233—24no—
13#3166C2212yes—
14#3142C2212yes—
15#3093—22no—
16#3079—22no—
17#2920C2211yescharacteristic
18C2×#432C2212yes—
19#2381—22no—
20#2371C2214yes—
21#2355—24no—
22#2332—24no—
23#2314—22no—
24#2283C2212yes—
25#1534C2214yes—
26#1531—24no—
27#1516—24no—
28#1512—24no—
29#1506C2214yes—
30#1505C2214yes—
31#1300C2218yes—
32#533—22no—
33#533—22no—
34#533C2×C412yes—
35#533C2×C414yes—
36#533D414yes—
37#533C2311yescharacteristic
38#532C2×C412yes—
39#532C2311yescharacteristic
40C2×C4⋊Q8D412yes—
41#529—22no—
42#529—22no—
43#529—22no—
44C2×#125C2312yes—
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#501C2×C418yes—
54#499—24no—
55#491C2×C414yes—
56#490C2×C414yes—
57#488—24no—
58#485—28no—
59#482—28no—
60#477—24no—
61#476—24no—
62#475—24no—
63#474—24no—
64#432—22no—
65#432—22no—
66#432—22no—
67#432C2×C414yes—
68#432C2314yes—
69#431—22no—
70#428—24no—
71#423—21no—
72C2×C4⋊C8—22no—
73#417—24no—
74#407—21no—
75C2×C22⋊C8—22no—
76C4×M16C2312yes—
77C2×#97—22no—
78#391—28no—
79#388—28no—
80#387—28no—
81#387—28no—
82#387—28no—
83#384—22no—
84#383—22no—
85#382—22no—
86#382D414yes—
87#381D414yes—
88#380—22no—
89C4×C4⋊C4—24no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4—28no—
92C4×C22⋊C4—22no—
93#376C2314yes—
94#330—28no—
95#328—28no—
96#327—28no—
97C2×#41—22no—
98C2×#41—21no—
99C22×Q8—22no—
100C23×C4C2×D411yescharacteristic
101C23×C4C2×D412yes—
102C2×M16—22no—
103C22×C8—42no—
104C4⋊Q8—22no—
105C4⋊Q8—22no—
106C4⋊Q8C22⋊C414yes—
107C4⋊Q8C22⋊C414yes—
108C4⋊Q8C22×C412yes—
109C4⋊Q8C22×C411yescharacteristic
110#127C22×C411yescharacteristic
111#126—42no—
112#125—22no—
113#125—21no—
114#125—24no—
115#125C22×C414yes—
116#125C2×D414yes—
117#124—22no—
118#124—21no—
119#123—41no—
120#123—42no—
121C22⋊Q8—24no—
122C22⋊Q8—24no—
123C22⋊Q8—24no—
124C22⋊Q8—24no—
125C22⋊Q8—22no—
126C22⋊Q8—22no—
127C22⋊Q8—22no—
128#121—22no—
129#121—22no—
130#121—22no—
131#120—41no—
132C4×Q8—24no—
133C4×Q8—24no—
134C4×Q8—24no—
135C4×Q8—24no—
136C4×Q8—22no—
137C4×Q8—22no—
138C4×Q8—22no—
139C4×D4—22no—
140C4×D4—22no—
141C4×D4—22no—
142C4×D4—42no—
143C4×D4—42no—
144C4×D4—42no—
145#117—24no—
146#117—22no—
147#117—21no—
148#117—42no—
149#117—41no—
150#117C22⋊C418yes—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—28no—
155C2×C4⋊C4C2×D414yes—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—24no—
158C2×C42—22no—
159C2×C42—28no—
160C2×C42—24no—
161C2×C42—41no—
162C2×C42C22×C412yes—
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D412yes—
165C2×C42C2×D412yes—
166C2×C42#4112yes—
167C2×C42#4112yes—
168C2×C42C2411yescharacteristic
169#107—28no—
170#106—28no—
171C4⋊C8—24no—
172C4⋊C8—24no—
173C4⋊C8—42no—
174#103—28no—
175#102—28no—
176C22⋊C8—24no—
177C22⋊C8—24no—
178C22⋊C8—42no—
179#97—24no—
180#97—42no—
181#97C22×C414yes—
182C4×C8—22no—
183#95—28no—
184#95—28no—
185#95—24no—
186#95—24no—
187#95—24no—
188#95C2×D418yes—
189C24#12011yescharacteristic
190#41—44no—
191#41—84no—
192C2×Q8—24no—
193C2×Q8—24no—
194C2×Q8—21no—
195C2×Q8—22no—
196C2×Q8—82no—
197C2×Q8—82no—
198C2×D4—21no—
199C2×D4—22no—
200C2×D4—41no—
201C2×D4—42no—
202C22×C4—24no—
203C22×C4—28no—
204C22×C4—42no—
205C22×C4—41no—
206C22×C4—42no—
207C22×C4—82no—
208C22×C4C2×C22⋊C412yes—
209C22×C4C4×D412yes—
210C22×C4C4×D412yes—
211C22×C4#12011yescharacteristic
212C22×C4#12012yes—
213C22×C4#12112yes—
214C22×C4#12112yes—
215C22×C4#12312yes—
216C22×C4C22×D411yescharacteristic
217C22×C4C22×D412yes—
218M16—44no—
219C2×C8—24no—
220C2×C8—44no—
221C2×C8—84no—
222C4⋊C4—28no—
223C4⋊C4—28no—
224C4⋊C4—24no—
225C4⋊C4—24no—
226C4⋊C4—24no—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—22no—
230C4⋊C4—48no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—88no—
235C4⋊C4C2×C22⋊C418yes—
236C22⋊C4—28no—
237C22⋊C4—41no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—84no—
243C42—28no—
244C42—41no—
245C42—42no—
246C42—84no—
247C42—88no—
248C42C2×C22⋊C412yes—
249C42C4×D412yes—
250C42C4×D412yes—
251C42#12112yes—
252C42#12112yes—
253C42#12311yescharacteristic
254C42#12311yescharacteristic
255C42C23×C411yescharacteristic
256C42C22×D411yescharacteristic
257C42C2×#4112yes—
258C23—22no—
259C23—42no—
260C23—41no—
261C23#38612yes—
262C23C2×#12011yescharacteristic
263Q8—42no—
264Q8—84no—
265Q8—84no—
266Q8—82no—
267D4—42no—
268D4—82no—
269D4—84no—
270D4—82no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—28no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—48no—
285C2×C4—48no—
286C2×C4—48no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—82no—
290C2×C4—84no—
291C2×C4#38612yes—
292C2×C4#38612yes—
293C2×C4#38612yes—
294C2×C4C22×C22⋊C411yescharacteristic
295C2×C4C2×C4×D412yes—
296C2×C4C2×#12112yes—
297C2×C4C2×#12311yescharacteristic
298C8—44no—
299C8—84no—
300C22—22no—
301C22—22no—
302C22—22no—
303C22—22no—
304C22—82no—
305C22—82no—
306C22—81no—
307C22C2×#38611yescharacteristic
308C22#302012yes—
309C4—22no—
310C4—22no—
311C4—22no—
312C4—22no—
313C4—21no—
314C4—21no—
315C4—22no—
316C4—48no—
317C4—88no—
318C4—82no—
319C4—81no—
320C2—22no—
321C2—42no—
322C2—161no—
323C2#2095411yescharacteristic
324C2#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. #34375C2semidirect
3. #17676C2semidirect
5. #9442C2semidirect
6. #9440C2semidirect
7. #8069C2semidirect
10. #3237C4semidirect
11. #3235C4semidirect

Maximal subgroups

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

subgroupindex
#608632
#343752
#176762
#175912
#94422
#94402
#80692