Finite Group Databasebeta

← #27180739

Subgroups — #27180739

Order 12928.

500 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 127 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
1C2×#15692274C211yesmaximal, characteristic, complemented
2C2×M16×D101C211yesmaximal, characteristic, complemented
3#20265687C212yesmaximal, complemented
4#20265436C211yesmaximal, characteristic, complemented
5#20265435C211yesmaximal, characteristic, complemented
6#20265427C211yesmaximal, characteristic, complemented
7#20265422C212yesmaximal, complemented
8#20265421C212yesmaximal, complemented
9#20265212C211yesmaximal, characteristic, complemented
10#20265170C212yesmaximal, complemented
11#469×C101C211yesmaximal, characteristic, complemented
12C2×#13108073C2212yes—
13C22×C4×D101C2211yescharacteristic
14#15692274—24no—
15C2×Dic404C2211yescharacteristic, complemented
16C2×#13108062C2212yescomplemented
17C2×#13108058C2211yescharacteristic
18C2×#13108057C2211yescharacteristic
19C2×#13108057C2211yescharacteristic
20M16×D101—24no—
21C2×C8×D101C2211yescharacteristic
22C2×C8×D101C2211yescharacteristic
23#15692237—22no—
24#15692235C2212yes—
25#15692234C2212yes—
26#15692226C2212yes—
27#15692223—22no—
28D4×Dic101—22no—
29#15692215—22no—
30#15692213C2212yes—
31#15692188C2212yescomplemented
32#15692187C2211yescharacteristic, complemented
33#15692180C2212yescomplemented
34#15692178C2212yescomplemented
35#15692176C2212yescomplemented
36C2×D8×C101C2211yescharacteristic, complemented
37C2×M16×C101C2211yescharacteristic
38#121×C101C2212yes—
39#106×C101C2211yescharacteristic, complemented
40#102×C101C2212yescomplemented
41C23×D101D411yescharacteristic
42#13108073—24no—
43#13108073—24no—
44C2×Dic202C2312yes—
45C2×C22⋊D101—22no—
46C2×C22⋊D101—22no—
47C22×Dic101—22no—
48C22×Dic101D411yescharacteristic
49C2×C4×D101—22no—
50C2×C4×D101—22no—
51C2×C4×D101D411yescharacteristic
52C2×C4×D101D411yescharacteristic
53C2×C4×D101C2311yescharacteristic
54C2×C4×D101C2311yescharacteristic
55Dic404—22no—
56#13108062—24no—
57#13108058—22no—
58#13108057—22no—
59#13108057—22no—
60C2×#11717579C2311yescharacteristic
61C2×#11717579C2311yescharacteristic
62C8×D101—22no—
63C8×D101D414yescomplemented
64C4⋊Dic101—22no—
65C4⋊Dic101C2312yes—
66#13108053—22no—
67C22⋊Dic101—22no—
68C22⋊Dic101—22no—
69#13108051—22no—
70#13108051—22no—
71C4×Dic101—22no—
72C2×D4×C101—22no—
73C2×D4×C101C2312yes—
74C22×C4×C101C2311yescharacteristic
75D8×C101—22no—
76M16×C101—22no—
77C2×C8×C101C2311yescharacteristic
78C2×C8×C101C2311yescharacteristic
79C4⋊C4×C101C2312yes—
80C22⋊C4×C101—22no—
81C23×C101—22no—
82C23×C101C2×D411yescharacteristic
83C22×D101—22no—
84C22×D101—42no—
85C22×D101C2×D411yescharacteristic
86C22×D101C2×D411yescharacteristic
87D4×C101—22no—
88D4×C101—22no—
89D4×C101—44no—
90C2×C4×C101—21no—
91C2×C4×C101—21no—
92C2×C4×C101—22no—
93C2×C4×C101C2×D411yescharacteristic
94C2×C4×C101C2411yescharacteristic
95C22⋊D101—44no—
96C22⋊D101—44no—
97C2×Dic101—22no—
98C2×Dic101—22no—
99C2×Dic101—22no—
100C2×Dic101—44no—
101C2×Dic101C2×D411yescharacteristic
102C2×Dic101C2×D411yescharacteristic
103C4×D101—22no—
104C4×D101—21no—
105C4×D101—21no—
106C4×D101—22no—
107C4×D101C2×D412yes—
108C4×D101#4112yes—
109Dic202—22no—
110Dic202—22no—
111C8×C101—21no—
112C8×C101C2×D412yes—
113#11717579—21no—
114#11717579C2×D412yes—
115C22×C101—21no—
116C22×C101—22no—
117C22×C101—42no—
118C22×C101—42no—
119C22×C101—42no—
120C22×C101C22×D411yescharacteristic
121C2×D101—22no—
122C2×D101—21no—
123C2×D101—22no—
124C2×D101—44no—
125C2×D101#12112yes—
126C4×C101—21no—
127C4×C101—42no—
128C4×C101C22×D411yescharacteristic
129C4×C101C2×#4111yescharacteristic
130Dic101—21no—
131Dic101—42no—
132Dic101—42no—
133Dic101#12112yes—
134C2×C101—41no—
135C2×C101—82no—
136C2×C101C2×#12111yescharacteristic
137C2×C101#57512yes—
138D101—22no—
139D101—41no—
140#3054—1011nomaximal, Sylow
141C101#305411yescharacteristic, complemented, Sylow
142C2×#135—1011no—
143C22×M16—1011no—
144#525—1012no—
145#470—1011no—
146#470—1011no—
147#469—1011no—
148#469D10111yescharacteristic, complemented
149#468—1012no—
150#468—1012no—
151#427—1011no—
152#416—1012no—
153C2×#41—1012no—
154C23×C4—1011no—
155#135—2024no—
156C2×Q16—1011no—
157C2×SD16—1012no—
158C2×D8C2×D10111yescharacteristic, complemented
159C2×M16—1011no—
160C2×M16—1011no—
161C2×M16—1011no—
162C2×M16—2024no—
163C2×M16C2×D10111yescharacteristic
164C22×C8—1011no—
165C22×C8—1011no—
166#123—2022no—
167C22⋊Q8—1012no—
168C22⋊Q8—1012no—
169#121—1012no—
170#121C2×D10112yes—
171#120—2022no—
172C4×D4—2022no—
173C4×D4—2022no—
174#117—1012no—
175#107—1012no—
176#106—1011no—
177#106C2×D10111yescharacteristic, complemented
178#103—1012no—
179#103—1012no—
180#102—1012no—
181#102C2×D10112yescomplemented
182C24—1011no—
183#41—2024no—
184#41—2024no—
185C2×Q8—1012no—
186C2×D4—22no—
187C2×D4—2022no—
188C2×D4—2022no—
189C2×D4C22×D10112yes—
190C22×C4—1011no—
191C22×C4—1011no—
192C22×C4—1011no—
193C22×C4—1011no—
194C22×C4—1011no—
195C22×C4—2022no—
196C22×C4—2022no—
197C22×C4—2022no—
198C22×C4C22×D10111yescharacteristic
199Q16—2022no—
200SD16—2024no—
201D8—22no—
202M16—22no—
203M16—2022no—
204M16—2022no—
205M16—2022no—
206C2×C8—1011no—
207C2×C8—1014no—
208C2×C8—1011no—
209C2×C8—2022no—
210C2×C8C22×D10111yescharacteristic
211C2×C8C22×D10111yescharacteristic
212C4⋊C4—1012no—
213C4⋊C4—2022no—
214C4⋊C4—2022no—
215C4⋊C4C22×D10112yes—
216C22⋊C4—22no—
217C22⋊C4—2022no—
218C22⋊C4—2022no—
219C22⋊C4—2022no—
220C22⋊C4—2022no—
221C42—2022no—
222C23—22no—
223C23—1011no—
224C23—1011no—
225C23—4042no—
226C23—2022no—
227C23D4×D10111yescharacteristic
228Q8—2022no—
229Q8—2022no—
230D4—22no—
231D4—22no—
232D4—44no—
233D4—4044no—
234D4—4044no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—1011no—
239C2×C4—1012no—
240C2×C4—1012no—
241C2×C4—1011no—
242C2×C4—4044no—
243C2×C4—2022no—
244C2×C4—2022no—
245C2×C4—2022no—
246C2×C4—2022no—
247C2×C4—2022no—
248C2×C4—2021no—
249C2×C4—2021no—
250C2×C4D4×D10111yescharacteristic
251C2×C4C23×D10111yescharacteristic
252C8—21no—
253C8—1012no—
254C8—2021no—
255C8D4×D10112yes—
256C22—21no—
257C22—22no—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—1012no—
262C22—4044no—
263C22—2021no—
264C22—2022no—
265C22—2022no—
266C22C2×D4×D10111yescharacteristic
267C4—21no—
268C4—42no—
269C4—1012no—
270C4—4042no—
271C4—4042no—
272C4—2021no—
273C4C2×D4×D10111yescharacteristic
274C4#41×D10111yescharacteristic
275C2—41no—
276C2—82no—
277C2—4041no—
278C2—2022no—
279C2#121×D10111yescharacteristic
280C2#2026597312yes—
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#469×C101C2
socleC22×C101C22×D4
hypercenter#469D101
hypocenterC101#3054
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×#15692274C2semidirect
2. C2×M16×D101C2semidirect
3. #20265687C2semidirect
4. #20265436C2semidirect
5. #20265435C2semidirect
6. #20265427C2semidirect
7. #20265422C2semidirect
8. #20265421C2semidirect
9. #20265212C2semidirect
10. #20265170C2semidirect
11. #469×C101C2semidirect
15. C2×Dic404C22semidirect
16. C2×#13108062C22semidirect
31. #15692188C22semidirect
32. #15692187C22semidirect
33. #15692180C22semidirect
34. #15692178C22semidirect
35. #15692176C22semidirect
36. C2×D8×C101C22semidirect
39. #106×C101C22semidirect
40. #102×C101C22semidirect
63. C8×D101D4semidirect
141. C101#3054semidirect
148. #469D101semidirect
158. C2×D8C2×D101semidirect
177. #106C2×D101semidirect
181. #102C2×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922742
C2×M16×D1012
#202656872
#202654362
#202654352
#202654272
#202654222
#202654212
#202652122
#202651702
#469×C1012
#3054101