Finite Group Databasebeta

← #27180926

Subgroups — #27180926

Order 12928.

440 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 143 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#20265730C211yesmaximal, characteristic, complemented
2C2×D4×Dic101C211yesmaximal, characteristic
3#20265479C212yesmaximal, complemented
4#20265475C212yesmaximal, complemented
5#20265465C212yesmaximal, complemented
6#20265455C212yesmaximal, complemented
7#20265220C211yesmaximal, characteristic, complemented
8#20265219C211yesmaximal, characteristic, complemented
9#20265200C211yesmaximal, characteristic, complemented
10#20265182C211yesmaximal, characteristic
11#418×C101C211yesmaximal, characteristic
12C23×Dic101—21no—
13C2×#13108058C2211yescharacteristic
14#15692254C2211yescharacteristic, complemented
15#15692254C2211yescharacteristic, complemented
16#15692246C2211yescharacteristic, complemented
17#15692246C2211yescharacteristic, complemented
18#15692232—21no—
19#15692232—21no—
20C4×Dic202—21no—
21C4×Dic202—21no—
22D4×Dic101—21no—
23D4×Dic101—21no—
24D4×Dic101C2212yes—
25D4×Dic101C2212yes—
26#15692214C2211yescharacteristic
27C2×C4⋊Dic101C2211yescharacteristic
28C2×C22⋊Dic101—21no—
29C2×C4×Dic101C2211yescharacteristic
30#15692189C2212yescomplemented
31#15692188C2212yescomplemented
32#15692187C2212yescomplemented
33#15692186C2212yescomplemented
34#15692184C2212yescomplemented
35#15692184C2212yescomplemented
36#15692176C2212yes—
37#15692176C2212yes—
38C22×D4×C101C2211yescharacteristic
39C2×M16×C101C2211yescharacteristic
40#117×C101C2211yescharacteristic
41#102×C101C2212yes—
42#102×C101C2212yes—
43C2×Dic202—21no—
44C2×Dic202—21no—
45C22×Dic101—21no—
46C22×Dic101—22no—
47C22×Dic101—22no—
48C22×Dic101—44no—
49C22×Dic101D411yescharacteristic
50C22×Dic101D411yescharacteristic
51#13108058—22no—
52C2×#11717579C2312yes—
53C4⋊Dic101—21no—
54C4⋊Dic101—21no—
55C4⋊Dic101—21no—
56C4⋊Dic101C2311yescharacteristic
57C4⋊Dic101C2311yescharacteristic
58C4⋊Dic101C2312yes—
59#13108053—21no—
60#13108053—21no—
61#13108053—21no—
62#13108053—21no—
63C22⋊Dic101—21no—
64C22⋊Dic101—22no—
65C22⋊Dic101—22no—
66C4×Dic101—21no—
67C4×Dic101D411yescharacteristic
68C4×Dic101D411yescharacteristic
69C4×Dic101C2311yescharacteristic
70C4×Dic101C2311yescharacteristic
71C24×C101—21no—
72C2×D4×C101—21no—
73C2×D4×C101—22no—
74C2×D4×C101Q814yescomplemented
75C2×D4×C101C2312yes—
76C22×C4×C101C2311yescharacteristic
77M16×C101—22no—
78C2×C8×C101C2312yes—
79C4⋊C4×C101C2311yescharacteristic
80C4⋊C4×C101C2311yescharacteristic
81C22⋊C4×C101—21no—
82C42×C101—21no—
83C23×C101—24no—
84C23×C101—22no—
85C23×C101—42no—
86C23×C101—42no—
87C23×C101C2×D411yescharacteristic
88D4×C101—24no—
89D4×C101—22no—
90D4×C101C2×Q814yes—
91C2×C4×C101—21no—
92C2×C4×C101—21no—
93C2×C4×C101C2×D411yescharacteristic
94C2×C4×C101C2×Q812yes—
95C2×C4×C101#4112yes—
96C2×C4×C101C2411yescharacteristic
97C2×Dic101—22no—
98C2×Dic101—21no—
99C2×Dic101—21no—
100C2×Dic101—22no—
101C2×Dic101—41no—
102C2×Dic101—44no—
103C2×Dic101—42no—
104C2×Dic101—44no—
105C2×Dic101C2×D411yescharacteristic
106C2×Dic101C2×D411yescharacteristic
107C2×Dic101C2×D411yescharacteristic
108C2×Dic101C2×D411yescharacteristic
109Dic202—42no—
110Dic202—42no—
111C8×C101—22no—
112#11717579—22no—
113C22×C101—21no—
114C22×C101—21no—
115C22×C101—24no—
116C22×C101—48no—
117C22×C101—42no—
118C22×C101—42no—
119C22×C101C22⋊Q812yes—
120C22×C101C22×D411yescharacteristic
121C4×C101—41no—
122C4×C101—41no—
123C4×C101C22⋊Q812yes—
124C4×C101C22×Q811yescharacteristic
125C4×C101C2×#4111yescharacteristic
126Dic101—21no—
127Dic101—21no—
128Dic101—21no—
129Dic101—41no—
130Dic101—42no—
131Dic101—41no—
132Dic101AGL(1,ℤ/8)12yes—
133C2×C101—22no—
134C2×C101—44no—
135C2×C101C2×C22⋊Q811yescharacteristic
136C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
137C2×C101#57511yescharacteristic
138#3077—1011nomaximal, Sylow
139C101#307711yescharacteristic, complemented, Sylow
140#533—1011no—
141C2×C4×D4—1011no—
142#478—1012no—
143#478—1012no—
144#476—1012no—
145#474—1012no—
146#428—1011no—
147#428—1011no—
148#423—1011no—
149#418—1011no—
150#418D10111yescharacteristic
151C22×D4C2×D10111yescharacteristic
152C23×C4—2021no—
153C2×M16—1011no—
154C2×M16C2×D10111yescharacteristic
155C4⋊Q8—1011no—
156C4⋊Q8—1011no—
157#125—1011no—
158#125—1011no—
159C22⋊Q8—2021no—
160C22⋊Q8—2021no—
161C4×Q8—2021no—
162C4×Q8—2021no—
163C4×D4—1012no—
164C4×D4—1012no—
165C4×D4—2021no—
166C4×D4—2021no—
167#117—1011no—
168#117C2×D10111yescharacteristic
169C2×C4⋊C4—1011no—
170C2×C22⋊C4—2021no—
171C2×C42—1011no—
172#107—1012no—
173#107—1012no—
174#106—1012no—
175#106—1012no—
176C4⋊C8—1012no—
177C4⋊C8—1012no—
178#102—1012no—
179#102—1012no—
180#102C2×D10112yes—
181#102C2×D10112yes—
182C24—21no—
183C2×Q8—2021no—
184C2×Q8—2021no—
185C2×D4—21no—
186C2×D4—22no—
187C2×D4Dic20214yescomplemented
188C2×D4C22×D10112yes—
189C22×C4—1011no—
190C22×C4—1011no—
191C22×C4—4044no—
192C22×C4—2022no—
193C22×C4—2022no—
194C22×C4—2021no—
195C22×C4C22×D10111yescharacteristic
196M16—22no—
197M16—2022no—
198C2×C8—1012no—
199C2×C8C22×D10112yes—
200C4⋊C4—1011no—
201C4⋊C4—1011no—
202C4⋊C4—1012no—
203C4⋊C4—2021no—
204C4⋊C4—2021no—
205C4⋊C4—2021no—
206C4⋊C4—2021no—
207C4⋊C4—2021no—
208C4⋊C4—2021no—
209C4⋊C4—2021no—
210C4⋊C4C22×D10111yescharacteristic
211C4⋊C4C22×D10111yescharacteristic
212C22⋊C4—21no—
213C22⋊C4—2022no—
214C22⋊C4—2022no—
215C22⋊C4—2021no—
216C42—21no—
217C42—1011no—
218C42—1011no—
219C42—1011no—
220C42—1011no—
221C42—2021no—
222C23—22no—
223C23—24no—
224C23—42no—
225C23—42no—
226C23D4×D10111yescharacteristic
227Q8—4042no—
228Q8—4042no—
229D4—22no—
230D4—24no—
231D4C2×Dic20214yes—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—1011no—
235C2×C4—1011no—
236C2×C4—1011no—
237C2×C4—1011no—
238C2×C4—4044no—
239C2×C4—4042no—
240C2×C4—4044no—
241C2×C4—4041no—
242C2×C4—2022no—
243C2×C4—2021no—
244C2×C4—2021no—
245C2×C4—2022no—
246C2×C4D4×D10111yescharacteristic
247C2×C4C2×Dic20212yes—
248C2×C4#1310807312yes—
249C2×C4C23×D10111yescharacteristic
250C8—22no—
251C8—2022no—
252C22—21no—
253C22—21no—
254C22—24no—
255C22—48no—
256C22—42no—
257C22—42no—
258C22#1569223212yes—
259C22C2×D4×D10111yescharacteristic
260C4—41no—
261C4—41no—
262C4—1012no—
263C4—4042no—
264C4—4041no—
265C4—4041no—
266C4—2021no—
267C4—2021no—
268C4—2021no—
269C4#1569223212yes—
270C4C22×Dic20211yescharacteristic
271C4C2×#1310807311yescharacteristic
272C2—22no—
273C2—44no—
274C2C2×#1569223211yescharacteristic
275C2AGL(1,ℤ/8)×D10111yescharacteristic
276C2#2026597311yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#418×C101C2
socleC22×C101C22×D4
prime radicalC22×D4×C101C22
hypercenter#418D101
hypocenterC101#3077
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. #20265730C2semidirect
3. #20265479C2semidirect
4. #20265475C2semidirect
5. #20265465C2semidirect
6. #20265455C2semidirect
7. #20265220C2semidirect
8. #20265219C2semidirect
9. #20265200C2semidirect
14. #15692254C22semidirect
15. #15692254C22semidirect
16. #15692246C22semidirect
17. #15692246C22semidirect
30. #15692189C22semidirect
31. #15692188C22semidirect
32. #15692187C22semidirect
33. #15692186C22semidirect
34. #15692184C22semidirect
35. #15692184C22semidirect
74. C2×D4×C101Q8semidirect
139. C101#3077semidirect
187. C2×D4Dic202semidirect

Maximal subgroups

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

subgroupindex
#202657302
C2×D4×Dic1012
#202654792
#202654752
#202654652
#202654552
#202652202
#202652192
#202652002
#202651822
#418×C1012
#3077101