Finite Group Databasebeta

← #27170170

Subgroups — #27170170

Order 12928.

246 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 51 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#20265754C211yesmaximal, characteristic
2#20265496C211yesmaximal, characteristic, complemented
3#20265493C211yesmaximal, characteristic, complemented
4#20265117C211yesmaximal, characteristic, complemented
5#20264910C211yesmaximal, characteristic, complemented
6#20264902C211yesmaximal, characteristic, complemented
7#327×C101C211yesmaximal, characteristic, complemented
8C2×#13108073—21no—
9#15692233—21no—
10#15692232—21no—
11#15692229C2211yescharacteristic
12#15692228—21no—
13#15692227—21no—
14#15692227—21no—
15C2×C4⋊Dic101C2211yescharacteristic
16C2×#13108053C2211yescharacteristic
17#15692187—21no—
18#15692186—21no—
19#15692177—21no—
20#15692176—21no—
21C22⋊#11717579C2211yescharacteristic
22#15692162—21no—
23#15692162—21no—
24#121×C101C2211yescharacteristic
25C22⋊C8×C101C2211yescharacteristic
26#95×C101C2211yescharacteristic
27#13108073—42no—
28#13108073—42no—
29C2×Dic202—21no—
30C2×D404D411yescharacteristic
31C2×C22⋊D101—21no—
32C2×C22⋊D101—21no—
33C2×C22⋊D101—21no—
34C22×Dic101—21no—
35C22×Dic101—21no—
36C22×Dic101D411yescharacteristic
37C22×Dic101D411yescharacteristic
38C2×C4×D101—21no—
39C2×#11717579—21no—
40C4⋊Dic101—21no—
41C4⋊Dic101—21no—
42C4⋊Dic101D411yescharacteristic
43#13108053—21no—
44#13108053—21no—
45#13108053—21no—
46C22⋊Dic101—21no—
47#13108051—21no—
48#13108051—21no—
49C2×D4×C101—21no—
50C2×D4×C101D411yescharacteristic
51C22×C4×C101—21no—
52C22×C4×C101C2311yescharacteristic
53C2×C8×C101—21no—
54C4⋊C4×C101D411yescharacteristic
55C22⋊C4×C101—21no—
56C23×C101—21no—
57C23×C101C2×D411yescharacteristic
58C22×D101—21no—
59D4×C101—41no—
60D4×C101—41no—
61D4×C101—41no—
62C2×C4×C101—21no—
63C2×C4×C101—41no—
64C2×C4×C101—41no—
65C2×C4×C101—41no—
66C2×C4×C101—41no—
67C2×C4×C101C2×D411yescharacteristic
68C2×C4×C101C2×D411yescharacteristic
69C22⋊D101—42no—
70C22⋊D101—41no—
71C22⋊D101—41no—
72C22⋊D101—81no—
73C2×Dic101—21no—
74C2×Dic101—21no—
75C2×Dic101—21no—
76C2×Dic101—21no—
77C2×Dic101—42no—
78C2×Dic101—41no—
79C2×Dic101—42no—
80C2×Dic101—41no—
81C2×Dic101SD1612yes—
82D404—41no—
83C4×D101—42no—
84Dic202—42no—
85C8×C101—41no—
86#11717579—41no—
87C22×C101—21no—
88C22×C101—21no—
89C22×C101—41no—
90C22×C101—41no—
91C22×C101—41no—
92C22×C101#12011yescharacteristic
93C22×C101C2×SD1611yescharacteristic
94C22×C101AGL(1,ℤ/8)11yescharacteristic
95C2×D101—41no—
96C2×D101—41no—
97C2×D101—41no—
98C4×C101—21no—
99C4×C101—21no—
100C4×C101—41no—
101C4×C101—81no—
102Dic101—22no—
103Dic101—41no—
104Dic101—41no—
105Dic101—81no—
106C2×C101—21no—
107C2×C101—21no—
108C2×C101—81no—
109C2×C101#45011yescharacteristic
110C2×C101#45411yescharacteristic
111C2×C101#45811yescharacteristic
112D101—81no—
113#1616—1011nomaximal, Sylow
114C101#161611yescharacteristic, complemented, Sylow
115#537—1011no—
116#481—1011no—
117#481—1011no—
118#400—1011no—
119#327—1011no—
120#327D10111yescharacteristic, complemented
121#324—1011no—
122C2×#41—2021no—
123C22⋊Q8—2021no—
124C22⋊Q8—2021no—
125#121—1011no—
126#121—2021no—
127#121—2021no—
128#121—2021no—
129#121C2×D10111yescharacteristic
130C2×C4⋊C4—1011no—
131C2×C4⋊C4—1011no—
132#106—2021no—
133#106—2021no—
134#102—2021no—
135#102—2021no—
136C22⋊C8—1011no—
137C22⋊C8C2×D10111yescharacteristic
138#95—2021no—
139#95—2021no—
140#95C2×D10111yescharacteristic
141#41—4042no—
142#41—4042no—
143C2×Q8—2021no—
144C2×D4—21no—
145C2×D4—1011no—
146C2×D4—2021no—
147C2×D4—2021no—
148C2×D4—2021no—
149C2×D4D40411yescharacteristic
150C22×C4—21no—
151C22×C4—1011no—
152C22×C4—1011no—
153C22×C4—2021no—
154C22×C4—2021no—
155C22×C4—2021no—
156C22×C4C22×D10111yescharacteristic
157C2×C8—21no—
158C2×C8—2021no—
159C4⋊C4—1011no—
160C4⋊C4—2021no—
161C4⋊C4—2021no—
162C4⋊C4—2021no—
163C4⋊C4—2021no—
164C4⋊C4—2021no—
165C4⋊C4D40411yescharacteristic
166C22⋊C4—21no—
167C22⋊C4—2021no—
168C22⋊C4—2021no—
169C22⋊C4—2021no—
170C23—21no—
171C23—2021no—
172C23D4×D10111yescharacteristic
173Q8—4042no—
174D4—41no—
175D4—41no—
176D4—41no—
177D4—8081no—
178D4—4041no—
179D4—4041no—
180D4—4041no—
181D4—4042no—
182C2×C4—21no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—1012no—
188C2×C4—4042no—
189C2×C4—4042no—
190C2×C4—4041no—
191C2×C4—4042no—
192C2×C4—4041no—
193C2×C4—2021no—
194C2×C4—2021no—
195C2×C4—2021no—
196C2×C4—2021no—
197C2×C4D4×D10111yescharacteristic
198C2×C4C2×D40411yescharacteristic
199C8—41no—
200C8—4041no—
201C22—21no—
202C22—21no—
203C22—41no—
204C22—41no—
205C22—41no—
206C22—4041no—
207C22—4041no—
208C22—4041no—
209C22#1569222511yescharacteristic
210C22SD16×D10111yescharacteristic
211C22#1569227911yescharacteristic
212C4—21no—
213C4—21no—
214C4—41no—
215C4—81no—
216C4—8081no—
217C4—4041no—
218C4—4041no—
219C4—2022no—
220C2—21no—
221C2—21no—
222C2—81no—
223C2—8081no—
224C2#2026532911yescharacteristic
225C2#2026535111yescharacteristic
226C2#2026536811yescharacteristic
subgroupG/H
centerC22#15692225
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting#327×C101C2
socleC22×C101#120
prime radical#20265754C2
hypercenter#327D101
hypocenterC101#1616
LCSG ▷ C22×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #20265496C2semidirect
3. #20265493C2semidirect
4. #20265117C2semidirect
5. #20264910C2semidirect
6. #20264902C2semidirect
7. #327×C101C2semidirect
114. C101#1616semidirect
120. #327D101semidirect

Maximal subgroups

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

subgroupindex
#202657542
#202654962
#202654932
#202651172
#202649102
#202649022
#327×C1012
#1616101