Finite Group Databasebeta

← #140909

Subgroups — #140909

Order 512.

541 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 541 conjugacy classes of subgroups — too many to draw. Only its 122 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#33173C211yesmaximal, characteristic
2C4×C22⋊C16C211yesmaximal, characteristic, complemented
3C2×#1348C212yesmaximal, complemented
4C2×#1347C212yesmaximal
5#10861C211yesmaximal, characteristic, complemented
6C2×#495C411yescharacteristic, complemented
7C2×#467C2211yescharacteristic
8C2×#466C2211yescharacteristic
9C2×#465—21no—
10C22×#106—21no—
11C22×C4×C8C2211yescharacteristic
12#2850—21no—
13C2×C22⋊C16C2212yes—
14C2×C4×C16—21no—
15#1970C411yescharacteristic, complemented
16#1969C411yescharacteristic, complemented
17#1968—21no—
18#1958—21no—
19#1846C411yescharacteristic, complemented
20C2×#340C2212yes—
21C4⋊#95—21no—
22#1374—21no—
23#1348—24no—
24#1347—24no—
25C22×Q16C2×C411yescharacteristic
26C22×SD16—21no—
27C22×D8C2×C411yescharacteristic
28C23×C8D411yescharacteristic
29C23×C8C2311yescharacteristic
30C2×#124—21no—
31C2×C22⋊Q8—21no—
32C2×#121—21no—
33C22×C4⋊C4—22no—
34C22×C42D411yescharacteristic
35C22×C16—22no—
36#495—42no—
37#467—22no—
38#467—22no—
39#466—22no—
40#466—22no—
41#465—42no—
42#465—42no—
43C2×#107C2×C411yescharacteristic
44C2×#107C2×C411yescharacteristic
45C2×#106—24no—
46C2×#106—21no—
47C2×#106—42no—
48C2×#106—42no—
49C2×#103—21no—
50C2×#103—21no—
51C2×#102—21no—
52C2×#102—21no—
53C2×C4×C8—42no—
54C2×C4×C8C2×C411yescharacteristic
55C2×C4×C8C2×C411yescharacteristic
56C2×C4×C8D411yescharacteristic
57C2×C4×C8D411yescharacteristic
58#391—21no—
59#390—21no—
60#388—42no—
61#386—42no—
62#382—21no—
63C2×#95—42no—
64C22⋊C16—22no—
65C22⋊C16D414yes—
66C4×C16—42no—
67#340—22no—
68#340—22no—
69C22×Q8—21no—
70C22×D4—21no—
71C22×D4—41no—
72C23×C4—42no—
73C23×C4D812yes—
74C23×C4C2×D411yescharacteristic
75C2×Q16—22no—
76C2×Q16—41no—
77C2×Q16—41no—
78C2×SD16—42no—
79C2×SD16—42no—
80C2×SD16—42no—
81C2×D8—22no—
82C2×D8—41no—
83C2×D8—41no—
84C22×C8—22no—
85C22×C8—22no—
86C22×C8—41no—
87C22×C8—41no—
88C22×C8C22⋊C411yescharacteristic
89C22×C8C22⋊C411yescharacteristic
90C22×C8D812yes—
91C22×C8SD1612yes—
92C22×C8C22×C411yescharacteristic
93C22×C8C2×D411yescharacteristic
94C22×C8C2×D412yes—
95C22×C8#4112yes—
96#124—42no—
97#124—42no—
98C22⋊Q8—41no—
99C22⋊Q8—41no—
100C22⋊Q8—41no—
101C22⋊Q8—41no—
102#121—41no—
103#121—41no—
104#121—41no—
105#121—41no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—41no—
115C2×C22⋊C4—41no—
116C2×C22⋊C4—42no—
117C2×C22⋊C4—42no—
118C2×C22⋊C4—41no—
119C2×C42—41no—
120C2×C42—41no—
121C2×C42C22⋊C411yescharacteristic
122C2×C42C22⋊C411yescharacteristic
123C2×C42SD1612yes—
124C2×C16—24no—
125C2×C16—42no—
126C2×C16—42no—
127#107—22no—
128#107—22no—
129#106—48no—
130#106—42no—
131#103—42no—
132#103—42no—
133#102—42no—
134#102—42no—
135C4×C8—22no—
136C4×C8—22no—
137C4×C8—41no—
138C4×C8—41no—
139#95—42no—
140#95—42no—
141#95—84no—
142C24—41no—
143C24C2×D811yescharacteristic
144C2×Q8—41no—
145C2×Q8—41no—
146C2×Q8—41no—
147C2×Q8—41no—
148C2×Q8—41no—
149C2×Q8—41no—
150C2×D4—41no—
151C2×D4—41no—
152C2×D4—41no—
153C2×D4—41no—
154C2×D4—41no—
155C2×D4—41no—
156C2×D4—81no—
157C2×D4—81no—
158C2×D4—84no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—24no—
164C22×C4—24no—
165C22×C4—42no—
166C22×C4—44no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—84no—
171C22×C4—82no—
172C22×C4—82no—
173C22×C4—84no—
174C22×C4#10212yes—
175C22×C4#10212yes—
176C22×C4C2×C22⋊C411yescharacteristic
177C22×C4#12411yescharacteristic
178C22×C4#12711yescharacteristic
179C22×C4C2×D811yescharacteristic
180C22×C4C2×SD1611yescharacteristic
181C22×C4C2×SD1611yescharacteristic
182Q16—82no—
183SD16—84no—
184D8—82no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—24no—
192C2×C8—24no—
193C2×C8—42no—
194C2×C8—42no—
195C2×C8#9912yes—
196C2×C8#10212yes—
197C2×C8#10212yes—
198C2×C8#10412yes—
199C2×C8C4×D412yes—
200C2×C8#12112yes—
201C4⋊C4—44no—
202C4⋊C4—44no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—82no—
208C4⋊C4—82no—
209C4⋊C4—81no—
210C4⋊C4—81no—
211C22⋊C4—84no—
212C22⋊C4—81no—
213C22⋊C4—81no—
214C22⋊C4—84no—
215C22⋊C4—81no—
216C22⋊C4—81no—
217C42—21no—
218C42—21no—
219C42—21no—
220C42—21no—
221C42—42no—
222C16—44no—
223C23—22no—
224C23—21no—
225C23—21no—
226C23—82no—
227C23—82no—
228C23—81no—
229C23—81no—
230C23—81no—
231C23C2×#10211yescharacteristic
232C23#48611yescharacteristic
233C23#49311yescharacteristic
234C23#50814yes—
235Q8—81no—
236Q8—81no—
237Q8—81no—
238Q8—81no—
239D4—81no—
240D4—81no—
241D4—81no—
242D4—81no—
243D4—162no—
244D4—162no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—48no—
261C2×C4—84no—
262C2×C4—88no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—164no—
271C2×C4—162no—
272C2×C4—162no—
273C2×C4#36014yes—
274C2×C4#39011yescharacteristic
275C2×C4C2×#9911yescharacteristic
276C2×C4#41011yescharacteristic
277C2×C4C2×#10211yescharacteristic
278C2×C4C2×#10411yescharacteristic
279C2×C4#42111yescharacteristic
280C2×C4#48611yescharacteristic
281C2×C4#49211yescharacteristic
282C8—22no—
283C8—22no—
284C8—22no—
285C8#32714yes—
286C22—24no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—24no—
292C22—162no—
293C22—161no—
294C22—161no—
295C22—161no—
296C22—162no—
297C22#150211yescharacteristic
298C22#151911yescharacteristic
299C22#195711yescharacteristic
300C22#213812yes—
301C22C2×#36012yes—
302C22#224212yes—
303C22#224912yes—
304C4—24no—
305C4—21no—
306C4—21no—
307C4—42no—
308C4—164no—
309C4—161no—
310C4—161no—
311C4—161no—
312C4C2×#32711yescharacteristic
313C4#151711yescharacteristic
314C4#196411yescharacteristic
315C4#198011yescharacteristic
316C2—22no—
317C2—22no—
318C2—321no—
319C2#968011yescharacteristic
320C2#1174112yes—
321C2#1188512yes—
322C2#1320411yescharacteristic
323C2#1321011yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC2×#466C22
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×C22⋊C16C2semidirect
3. C2×#1348C2semidirect
5. #10861C2semidirect
6. C2×#495C4semidirect
15. #1970C4semidirect
16. #1969C4semidirect
19. #1846C4semidirect

Maximal subgroups

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

subgroupindex
#331732
C4×C22⋊C162
C2×#13482
C2×#13472
#108612