Finite Group Databasebeta

← #140905

Subgroups — #140905

Order 512.

499 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 499 conjugacy classes of subgroups — too many to draw. Only its 114 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×#1350C212yesmaximal, complemented
4C2×#1349C212yesmaximal
5#10844C211yesmaximal, characteristic, complemented
6C2×#495C411yescharacteristic
7C2×#467C2211yescharacteristic
8C2×#466C2211yescharacteristic
9C2×#465—21no—
10C2×#423—21no—
11C22×C4×C8C2211yescharacteristic
12#2850—21no—
13C2×C22⋊C16C2212yes—
14C2×C4×C16—21no—
15#1970C411yescharacteristic
16#1969C411yescharacteristic
17#1968—21no—
18#1958—21no—
19#1846C411yescharacteristic
20C2×#341—21no—
21C2×#341C2212yes—
22#1374—21no—
23#1350—24no—
24#1349—24no—
25C22×Q16C2×C411yescharacteristic
26C22×SD16—21no—
27C22×D8C2×C411yescharacteristic
28C22×M16—22no—
29C23×C8D411yescharacteristic
30C23×C8C2311yescharacteristic
31C2×#124—21no—
32C2×C22⋊Q8—21no—
33C2×#121—21no—
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—21no—
46#423—24no—
47#423—42no—
48C2×C4⋊C8—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—
63C22⋊C16—22no—
64C22⋊C16D414yes—
65C4×C16—42no—
66#341—22no—
67#341—22no—
68#341—42no—
69C22×Q8—21no—
70C22×D4—21no—
71C22×D4—41no—
72C23×C4SD1612yes—
73C23×C4C2×D411yescharacteristic
74C2×Q16—22no—
75C2×Q16—41no—
76C2×Q16—41no—
77C2×SD16—42no—
78C2×SD16—42no—
79C2×SD16—42no—
80C2×D8—22no—
81C2×D8—41no—
82C2×D8—41no—
83C2×M16—24no—
84C2×M16—42no—
85C2×M16—42no—
86C2×M16—42no—
87C2×M16—42no—
88C2×M16—42no—
89C22×C8—41no—
90C22×C8—41no—
91C22×C8—41no—
92C22×C8—41no—
93C22×C8—42no—
94C22×C8C22⋊C411yescharacteristic
95C22×C8C22⋊C411yescharacteristic
96C22×C8D812yes—
97C22×C8SD1612yes—
98C22×C8C22×C411yescharacteristic
99C22×C8C2×D411yescharacteristic
100C22×C8C2×D412yes—
101C22×C8#4112yes—
102#124—42no—
103#124—42no—
104C22⋊Q8—41no—
105C22⋊Q8—41no—
106C22⋊Q8—41no—
107C22⋊Q8—41no—
108#121—41no—
109#121—41no—
110#121—41no—
111#121—41no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—41no—
115C2×C22⋊C4—41no—
116C2×C22⋊C4—42no—
117C2×C22⋊C4—42no—
118C2×C22⋊C4—41no—
119C2×C42—22no—
120C2×C42—22no—
121C2×C42C22⋊C411yescharacteristic
122C2×C42C22⋊C411yescharacteristic
123C2×C42D812yes—
124C2×C16—24no—
125C2×C16—42no—
126C2×C16—42no—
127#107—22no—
128#107—22no—
129#106—42no—
130C4⋊C8—48no—
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—
141C24—41no—
142C24C2×SD1611yescharacteristic
143C2×Q8—41no—
144C2×Q8—41no—
145C2×Q8—41no—
146C2×Q8—41no—
147C2×Q8—41no—
148C2×Q8—41no—
149C2×D4—41no—
150C2×D4—41no—
151C2×D4—41no—
152C2×D4—41no—
153C2×D4—41no—
154C2×D4—41no—
155C2×D4—84no—
156C2×D4—81no—
157C2×D4—81no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—24no—
163C22×C4—24no—
164C22×C4—42no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4#10212yes—
169C22×C4#10212yes—
170C22×C4C2×C22⋊C411yescharacteristic
171C22×C4#12411yescharacteristic
172C22×C4#12711yescharacteristic
173C22×C4C2×D811yescharacteristic
174C22×C4C2×D811yescharacteristic
175C22×C4C2×SD1611yescharacteristic
176Q16—82no—
177SD16—84no—
178D8—82no—
179M16—44no—
180M16—44no—
181M16—82no—
182M16—82no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—42no—
195C2×C8—44no—
196C2×C8—82no—
197C2×C8—82no—
198C2×C8#10212yes—
199C2×C8#10212yes—
200C2×C8C4×D412yes—
201C2×C8#12112yes—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—81no—
207C4⋊C4—81no—
208C22⋊C4—81no—
209C22⋊C4—81no—
210C22⋊C4—84no—
211C22⋊C4—84no—
212C22⋊C4—81no—
213C22⋊C4—81no—
214C42—24no—
215C42—21no—
216C42—21no—
217C42#9912yes—
218C42#10412yes—
219C16—44no—
220C23—22no—
221C23—22no—
222C23—21no—
223C23—21no—
224C23—82no—
225C23—82no—
226C23—81no—
227C23—81no—
228C23—81no—
229C23C2×#10211yescharacteristic
230C23#48611yescharacteristic
231C23#49211yescharacteristic
232Q8—81no—
233Q8—81no—
234Q8—81no—
235Q8—81no—
236D4—81no—
237D4—81no—
238D4—81no—
239D4—81no—
240D4—162no—
241D4—162no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—21no—
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—48no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—81no—
260C2×C4—81no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—81no—
264C2×C4#32714yes—
265C2×C4#35914yes—
266C2×C4#39011yescharacteristic
267C2×C4C2×#9911yescharacteristic
268C2×C4#41011yescharacteristic
269C2×C4C2×#10211yescharacteristic
270C2×C4C2×#10411yescharacteristic
271C2×C4#42111yescharacteristic
272C2×C4#48611yescharacteristic
273C2×C4#49311yescharacteristic
274C8—22no—
275C8—22no—
276C8—22no—
277C8—22no—
278C8—84no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—42no—
286C22—42no—
287C22—162no—
288C22—161no—
289C22—161no—
290C22—162no—
291C22—161no—
292C22C2×#32711yescharacteristic
293C22#151711yescharacteristic
294C22#195711yescharacteristic
295C22C2×#35912yes—
296C22#214812yes—
297C4—24no—
298C4—24no—
299C4—21no—
300C4—21no—
301C4—161no—
302C4—161no—
303C4—161no—
304C4#150211yescharacteristic
305C4#151911yescharacteristic
306C4#196411yescharacteristic
307C4#198011yescharacteristic
308C2—41no—
309C2—41no—
310C2—321no—
311C2#968011yescharacteristic
312C2#1170912yes—
313C2#1186112yes—
314C2#1320811yescharacteristic
315C2#1321411yescharacteristic
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×#1350C2semidirect
5. #10844C2semidirect

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×#13502
C2×#13492
#108442