Finite Group Databasebeta

← #27181522

Subgroups — #27181522

Order 12928.

424 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 125 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#20265725C211yesmaximal, characteristic
2C4⋊Q8×D101C211yesmaximal, characteristic, complemented
3#20265535C214yesmaximal, complemented
4#20265529C211yesmaximal, characteristic, complemented
5#20265528C211yesmaximal, characteristic, complemented
6#20265523C211yesmaximal, characteristic, complemented
7#20265175C214yesmaximal, complemented
8C4×#13108057C211yesmaximal, characteristic, complemented
9#486×C101C211yesmaximal, characteristic, complemented
10C2×#13108073C2212yes—
11C2×Q8×D101—21no—
12C2×#13108057C2212yes—
13#15692255C2211yescharacteristic
14#15692255C2211yescharacteristic
15#15692254—21no—
16#15692254C2212yes—
17#15692253—21no—
18#15692242C2212yes—
19#15692226—21no—
20#15692226—21no—
21D4×Dic101—21no—
22D4×Dic101—21no—
23C4⋊C4×D101—21no—
24C4⋊C4×D101C2212yes—
25C42×D101C2211yescharacteristic
26#15692180C2214yes—
27#15692178C2214yes—
28#15692176C2214yes—
29#15692165C2212yes—
30C4×#11717579C2211yescharacteristic
31C4⋊Q8×C101C2211yescharacteristic
32#127×C101C2211yescharacteristic
33#102×C101C2214yes—
34C4×C8×C101C2211yescharacteristic
35#13108073—28no—
36C2×Dic202—21no—
37C2×Dic202—22no—
38C2×Dic202C2312yes—
39Q8×D101—44no—
40C2×C22⋊D101—22no—
41C22×Dic101—22no—
42C2×C4×D101—22no—
43C2×C4×D101D411yescharacteristic
44C2×C4×D101D411yescharacteristic
45C2×C4×D101C2311yescharacteristic
46#13108057—24no—
47C2×#11717579C2312yes—
48C4⋊Dic101—21no—
49C4⋊Dic101—21no—
50C4⋊Dic101—21no—
51C4⋊Dic101C2312yes—
52#13108053—22no—
53#13108053—22no—
54C22⋊Dic101—22no—
55C22⋊Dic101—22no—
56C4×Dic101D411yescharacteristic
57C4×Dic101D411yescharacteristic
58C4×Dic101C2311yescharacteristic
59C2×Q8×C101—21no—
60C2×D4×C101—21no—
61C2×D4×C101—21no—
62C2×D4×C101C2312yes—
63C2×C8×C101C2312yes—
64C4⋊C4×C101—21no—
65C4⋊C4×C101C2312yes—
66C42×C101C2311yescharacteristic
67C23×C101—22no—
68C22×D101C2×D411yescharacteristic
69Q8×C101—42no—
70D4×C101—24no—
71D4×C101—42no—
72D4×C101—42no—
73C2×C4×C101—22no—
74C2×C4×C101C2×D411yescharacteristic
75C2×C4×C101C2×D411yescharacteristic
76C2×C4×C101C2411yescharacteristic
77C22⋊D101—44no—
78C2×Dic101—22no—
79C2×Dic101—22no—
80C2×Dic101—44no—
81C2×Dic101C2×D411yescharacteristic
82C2×Dic101C2×D411yescharacteristic
83C2×Dic101C2×D411yescharacteristic
84C4×D101—22no—
85C4×D101—22no—
86C4×D101—44no—
87C4×D101#4114yes—
88Dic202—24no—
89Dic202—44no—
90Dic202—42no—
91C8×C101—22no—
92#11717579—22no—
93C22×C101—42no—
94C22×C101—42no—
95C22×C101—42no—
96C22×C101C22×D411yescharacteristic
97C2×D101—21no—
98C2×D101—21no—
99C2×D101#12412yes—
100C4×C101—42no—
101C4×C101AGL(1,ℤ/8)12yes—
102C4×C101#13712yes—
103C4×C101C2×#4112yes—
104Dic101—21no—
105Dic101—21no—
106Dic101—42no—
107Dic101—42no—
108Dic101#12412yes—
109C2×C101—82no—
110C2×C101C2×#12411yescharacteristic
111C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
112C2×C101C2×#13711yescharacteristic
113D101—22no—
114#3141—1011nomaximal, Sylow
115C101#314111yescharacteristic, complemented, Sylow
116#532—1011no—
117C2×C4⋊Q8—1011no—
118#489—1014no—
119#487—1011no—
120#487—1011no—
121#486—1011no—
122#486D10111yescharacteristic, complemented
123#417—1014no—
124C4×M16—1011no—
125C2×#41—1012no—
126C22×Q8—2021no—
127C2×M16—1012no—
128C4⋊Q8—1012no—
129C4⋊Q8—1011no—
130C4⋊Q8—1011no—
131C4⋊Q8—2021no—
132C4⋊Q8—2021no—
133C4⋊Q8C2×D10111yescharacteristic
134#127C2×D10111yescharacteristic
135#124—1012no—
136#121—2021no—
137#121—2021no—
138C4×D4—2021no—
139C4×D4—2021no—
140C2×C4⋊C4—1012no—
141C2×C4⋊C4—2021no—
142C2×C42—1011no—
143#103—1014no—
144#103—1014no—
145#102—1014no—
146#102C2×D10114yes—
147#97—1012no—
148C4×C8—1011no—
149C4×C8C2×D10111yescharacteristic
150#41—2028no—
151C2×Q8—21no—
152C2×Q8—1012no—
153C2×Q8—4044no—
154C2×Q8—2022no—
155C2×Q8—2021no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—2022no—
159C2×D4C22×D10112yes—
160C22×C4—1011no—
161C22×C4—1011no—
162C22×C4—1011no—
163C22×C4—2022no—
164C22×C4—2022no—
165M16—2024no—
166C2×C8—1012no—
167C2×C8C22×D10112yes—
168C4⋊C4—21no—
169C4⋊C4—1012no—
170C4⋊C4—2022no—
171C4⋊C4—2022no—
172C4⋊C4—2021no—
173C4⋊C4—2021no—
174C4⋊C4—2021no—
175C4⋊C4C22×D10112yes—
176C22⋊C4—2022no—
177C22⋊C4—2022no—
178C42—1011no—
179C42—1011no—
180C42—1011no—
181C42C22×D10111yescharacteristic
182C23—22no—
183C23—1011no—
184Q8—42no—
185Q8—4044no—
186Q8—4042no—
187Q8—2024no—
188D4—24no—
189D4—42no—
190D4—42no—
191D4—4044no—
192C2×C4—22no—
193C2×C4—1011no—
194C2×C4—1014no—
195C2×C4—1011no—
196C2×C4—1011no—
197C2×C4—4044no—
198C2×C4—4044no—
199C2×C4—2022no—
200C2×C4—2022no—
201C2×C4—2022no—
202C2×C4—2022no—
203C2×C4D4×D10111yescharacteristic
204C2×C4D4×D10111yescharacteristic
205C2×C4C23×D10111yescharacteristic
206C8—22no—
207C8—2022no—
208C22—42no—
209C22—42no—
210C22—42no—
211C22—1012no—
212C22—2021no—
213C22—2021no—
214C22C2×D4×D10111yescharacteristic
215C4—42no—
216C4—1012no—
217C4—4042no—
218C4—4042no—
219C4—2021no—
220C4—2021no—
221C4#1569227912yes—
222C4#1569228412yes—
223C4#41×D10112yes—
224C2—82no—
225C2—2022no—
226C2#124×D10111yescharacteristic
227C2C2×#1569227911yescharacteristic
228C2C2×#1569228411yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#486×C101C2
socleC22×C101C22×D4
prime radical#20265725C2
hypercenter#486D101
hypocenterC101#3141
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
2. C4⋊Q8×D101C2semidirect
3. #20265535C2semidirect
4. #20265529C2semidirect
5. #20265528C2semidirect
6. #20265523C2semidirect
7. #20265175C2semidirect
8. C4×#13108057C2semidirect
9. #486×C101C2semidirect
115. C101#3141semidirect
122. #486D101semidirect

Maximal subgroups

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

subgroupindex
#202657252
C4⋊Q8×D1012
#202655352
#202655292
#202655282
#202655232
#202651752
C4×#131080572
#486×C1012
#3141101