Finite Group Databasebeta

← #27184183

Subgroups — #27184183

Order 12928.

362 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 117 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#20265912C211yesmaximal, characteristic
2#20265905C211yesmaximal, characteristic, complemented
3#20265570C212yesmaximal, complemented
4#20265550C211yesmaximal, characteristic
5#20265402C212yesmaximal, complemented
6#20265387C211yesmaximal, characteristic, complemented
7#20265381C212yesmaximal, complemented
8#20265377C211yesmaximal, characteristic
9#20265307C211yesmaximal, characteristic, complemented
10#20265276C212yesmaximal, complemented
11C4×Q16×C101C211yesmaximal, characteristic, complemented
12C2×#13108064C2212yes—
13C2×#13108063C2212yes—
14C2×#13108063C2212yes—
15C2×#13108061C2212yes—
16#15692255C2211yescharacteristic
17#15692253C2212yes—
18#15692252C2211yescharacteristic
19#15692247—21no—
20#15692245—22no—
21#15692244C2212yes—
22#15692243—21no—
23#15692240—22no—
24C4×Dic202—21no—
25C4×Dic202C2211yescharacteristic
26#15692220C2212yes—
27C4×D404—21no—
28#15692217C2212yes—
29C4⋊#11717579C2211yescharacteristic
30#15692184C2212yes—
31#15692178C2212yes—
32#15692175C2212yes—
33#15692165C2212yes—
34C2×Q16×C101C2211yescharacteristic
35C4×Q8×C101C2211yescharacteristic
36C4×Q8×C101C2211yescharacteristic
37#107×C101C2211yescharacteristic
38#103×C101C2211yescharacteristic
39#103×C101C2211yescharacteristic
40C4×C8×C101C2211yescharacteristic
41C2×Dic202D411yescharacteristic
42C2×Dic202C2312yes—
43C2×D404—21no—
44C2×D404—21no—
45C2×D404C2312yes—
46C2×C4×D101—22no—
47#13108064—42no—
48#13108063—24no—
49#13108063—42no—
50#13108061—42no—
51C2×#11717579C2312yes—
52C4⋊Dic101—21no—
53C4⋊Dic101—21no—
54C4⋊Dic101D411yescharacteristic
55#13108053—22no—
56#13108053—22no—
57#13108053—22no—
58#13108053D412yes—
59#13108051—22no—
60#13108051—22no—
61#13108051—22no—
62C4×Dic101—22no—
63C4×Dic101C2312yes—
64C2×Q8×C101C2311yescharacteristic
65C2×Q8×C101C2311yescharacteristic
66Q16×C101—22no—
67C2×C8×C101C2311yescharacteristic
68C2×C8×C101C2311yescharacteristic
69C4⋊C4×C101—21no—
70C4⋊C4×C101—21no—
71C4⋊C4×C101C2311yescharacteristic
72C4⋊C4×C101C2311yescharacteristic
73C42×C101—21no—
74C42×C101—21no—
75C42×C101C2311yescharacteristic
76C22×D101—22no—
77Q8×C101—21no—
78Q8×C101—21no—
79Q8×C101—21no—
80Q8×C101—21no—
81C2×C4×C101—21no—
82C2×C4×C101—21no—
83C2×C4×C101—21no—
84C2×C4×C101—21no—
85C2×C4×C101C2×D411yescharacteristic
86C2×C4×C101C2×D411yescharacteristic
87C2×C4×C101C2411yescharacteristic
88C2×Dic101—22no—
89C2×Dic101C2×D412yes—
90C2×Dic101C2×D412yes—
91D404—22no—
92D404—22no—
93D404—42no—
94D404—81no—
95C4×D101—82no—
96Dic202—22no—
97Dic202—22no—
98Dic202—22no—
99C8×C101—21no—
100C8×C101#4112yes—
101#11717579—22no—
102C22×C101C22×D411yescharacteristic
103C2×D101—42no—
104C2×D101—42no—
105C2×D101—42no—
106C4×C101—21no—
107C4×C101—41no—
108C4×C101—41no—
109C4×C101—41no—
110C4×C101—41no—
111C4×C101#13712yes—
112C4×C101C2×#4111yescharacteristic
113C4×C101H(5,𝔽2)11yescharacteristic
114Dic101—22no—
115Dic101—22no—
116Dic101—42no—
117C2×C101#55011yescharacteristic
118C2×C101C2×#13711yescharacteristic
119C2×C101#57711yescharacteristic
120D101—82no—
121#3400—1011nomaximal, Sylow
122C101#340011yescharacteristic, complemented, Sylow
123#559—1011no—
124#557—1011no—
125#497—1012no—
126#492—1011no—
127#464—1012no—
128#462—1011no—
129#461—1012no—
130#460—1011no—
131#446—1011no—
132#440—1012no—
133C4×Q16D10111yescharacteristic, complemented
134C2×Q16—1012no—
135C2×Q16C2×D10111yescharacteristic
136C2×SD16—1012no—
137C2×SD16—1012no—
138C2×SD16—1012no—
139C4⋊Q8—1012no—
140C4⋊Q8—1011no—
141#127—1011no—
142#125—2021no—
143#125—2022no—
144#124—1012no—
145#124—2021no—
146#123—2022no—
147C4×Q8—1011no—
148C4×Q8—1012no—
149C4×Q8—2021no—
150C4×Q8C2×D10111yescharacteristic
151C4×Q8C2×D10111yescharacteristic
152C4×D4—1012no—
153C4×D4—2021no—
154#107C2×D10111yescharacteristic
155C4⋊C8—1012no—
156C4⋊C8—1011no—
157#103—1012no—
158#103C2×D10111yescharacteristic
159#103C2×D10111yescharacteristic
160#102—1012no—
161#97—1012no—
162C4×C8C2×D10111yescharacteristic
163C2×Q8—1011no—
164C2×Q8—1012no—
165C2×Q8C22×D10111yescharacteristic
166C2×Q8C22×D10111yescharacteristic
167C2×D4—1012no—
168C2×D4—2021no—
169C2×D4—2021no—
170C22×C4—2022no—
171Q16—22no—
172Q16—4042no—
173SD16—4042no—
174SD16—4042no—
175SD16—2024no—
176C2×C8—1012no—
177C2×C8C22×D10111yescharacteristic
178C2×C8C22×D10111yescharacteristic
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—1012no—
182C4⋊C4—1011no—
183C4⋊C4—2022no—
184C4⋊C4—2022no—
185C4⋊C4—2022no—
186C4⋊C4—2021no—
187C4⋊C4—2021no—
188C4⋊C4C22×D10111yescharacteristic
189C4⋊C4C22×D10111yescharacteristic
190C22⋊C4—2022no—
191C22⋊C4—2022no—
192C22⋊C4—2022no—
193C42—21no—
194C42—21no—
195C42—1012no—
196C42—2022no—
197C42C22×D10111yescharacteristic
198C23—2022no—
199Q8—21no—
200Q8—21no—
201Q8—21no—
202Q8—21no—
203Q8—2022no—
204Q8—2022no—
205Q8—2022no—
206D4—8081no—
207D4—4042no—
208D4—2022no—
209D4—2022no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—8082no—
215C2×C4—1012no—
216C2×C4—1012no—
217C2×C4—2022no—
218C2×C4D4×D10111yescharacteristic
219C2×C4D4×D10111yescharacteristic
220C2×C4C23×D10111yescharacteristic
221C8—21no—
222C8—2022no—
223C8#1310807412yes—
224C22—4042no—
225C22—4042no—
226C22—4042no—
227C22C2×D4×D10111yescharacteristic
228C4—21no—
229C4—41no—
230C4—41no—
231C4—41no—
232C4—41no—
233C4—4042no—
234C4—2022no—
235C4—2022no—
236C4#1569228712yes—
237C4C2×#1310807411yescharacteristic
238C4#1569230111yescharacteristic
239C2—8082no—
240C2#2026584811yescharacteristic
241C2C2×#1569228711yescharacteristic
242C2#2026598411yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC4×Q16×C101C2
socleC22×C101C22×D4
prime radical#15692252C22
hypercenterC4×Q16D101
hypocenterC101#3400
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q16

Complements

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

NH ≅ G/Ntype
2. #20265905C2semidirect
3. #20265570C2semidirect
5. #20265402C2semidirect
6. #20265387C2semidirect
7. #20265381C2semidirect
9. #20265307C2semidirect
10. #20265276C2semidirect
11. C4×Q16×C101C2semidirect
122. C101#3400semidirect
133. C4×Q16D101semidirect

Maximal subgroups

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

subgroupindex
#202659122
#202659052
#202655702
#202655502
#202654022
#202653872
#202653812
#202653772
#202653072
#202652762
C4×Q16×C1012
#3400101