Finite Group Databasebeta

← #27169686

Subgroups — #27169686

Order 12928.

284 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 73 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#20265688C211yesmaximal, characteristic
2C2×C22⋊#11717579C211yesmaximal, characteristic, complemented
3#20264913C212yesmaximal, complemented
4#20264910C212yesmaximal, complemented
5#381×C101C211yesmaximal, characteristic, complemented
6C2×#13108074C411yescharacteristic, complemented
7C22×#11717579—21no—
8#15692238—21no—
9#15692234C411yescharacteristic, complemented
10#15692234C2211yescharacteristic
11#15692229C411yescharacteristic, complemented
12#15692229C2211yescharacteristic
13#15692224—21no—
14C4×C22⋊D101—21no—
15C4×C22⋊D101—21no—
16#15692214C411yescharacteristic, complemented
17C22⋊#11717579—21no—
18C22⋊#11717579C2212yes—
19C23×C4×C101C2211yescharacteristic
20C2×C22⋊C4×C101—21no—
21#95×C101—21no—
22#95×C101C2212yes—
23#13108074—42no—
24C2×Dic202C2×C411yescharacteristic
25C2×D404C2×C411yescharacteristic
26C2×C22⋊D101—21no—
27C2×C22⋊D101—21no—
28C2×C22⋊D101—21no—
29C2×C4×D101—21no—
30C2×#11717579—22no—
31C2×#11717579—42no—
32C4⋊Dic101C2×C411yescharacteristic
33C4⋊Dic101C2×C411yescharacteristic
34#13108053—21no—
35#13108053—21no—
36C22⋊Dic101—21no—
37C22⋊Dic101—21no—
38C22⋊Dic101—21no—
39#13108051—21no—
40#13108051—21no—
41C4×Dic101—21no—
42C24×C101D411yescharacteristic
43C22×C4×C101—22no—
44C22×C4×C101—42no—
45C22×C4×C101C2×C411yescharacteristic
46C22×C4×C101C2×C411yescharacteristic
47C22×C4×C101D411yescharacteristic
48C22×C4×C101D411yescharacteristic
49C22×C4×C101D411yescharacteristic
50C22×C4×C101C2311yescharacteristic
51C22⋊C4×C101—42no—
52C23×C101—22no—
53C23×C101—22no—
54C23×C101—41no—
55C23×C101C22⋊C411yescharacteristic
56C23×C101C22⋊C411yescharacteristic
57C23×C101C2×D411yescharacteristic
58C22×D101—21no—
59C2×C4×C101—22no—
60C2×C4×C101—22no—
61C2×C4×C101—41no—
62C2×C4×C101—41no—
63C2×C4×C101—41no—
64C2×C4×C101—41no—
65C2×C4×C101—42no—
66C2×C4×C101—42no—
67C2×C4×C101—42no—
68C2×C4×C101C22⋊C411yescharacteristic
69C2×C4×C101C22⋊C411yescharacteristic
70C2×C4×C101C22×C411yescharacteristic
71C2×C4×C101C2×D411yescharacteristic
72C22⋊D101—42no—
73C22⋊D101—42no—
74C22⋊D101—42no—
75C2×Dic101—21no—
76C2×Dic101—21no—
77C2×Dic101—21no—
78D404—41no—
79C4×D101—42no—
80Dic202—41no—
81#11717579—42no—
82C22×C101—21no—
83C22×C101—21no—
84C22×C101—21no—
85C22×C101—21no—
86C22×C101—21no—
87C22×C101—21no—
88C22×C101—21no—
89C22×C101—44no—
90C22×C101#9912yes—
91C22×C101C2×C22⋊C411yescharacteristic
92C22×C101#13512yes—
93C2×D101—41no—
94C2×D101—41no—
95C2×D101—41no—
96C4×C101—21no—
97C4×C101—21no—
98C4×C101—21no—
99C4×C101—21no—
100C4×C101—82no—
101Dic101—41no—
102Dic101—41no—
103Dic101—41no—
104C2×C101—22no—
105C2×C101—22no—
106C2×C101—41no—
107C2×C101C2×#9911yescharacteristic
108C2×C101#41611yescharacteristic
109C2×C101#42111yescharacteristic
110D101—81no—
111#1516—1011nomaximal, Sylow
112C101#151611yescharacteristic, complemented, Sylow
113#525—1011no—
114C2×C22⋊C8—1011no—
115#381D10111yescharacteristic, complemented
116#328—1012no—
117#327—1012no—
118C2×#41—1011no—
119C23×C4C2×D10111yescharacteristic
120C22×C8—2021no—
121#123—2021no—
122C22⋊Q8—1011no—
123C22⋊Q8—1011no—
124#121—1011no—
125#121—1011no—
126#120—2021no—
127C4×D4—2021no—
128C4×D4—2021no—
129#117—1011no—
130C2×C22⋊C4—21no—
131C22⋊C8—1012no—
132C22⋊C8—2021no—
133#95—21no—
134#95C2×D10112yes—
135C24C22⋊D10111yescharacteristic
136#41—4042no—
137C2×Q8—1011no—
138C2×D4—1011no—
139C2×D4—2021no—
140C2×D4—2021no—
141C2×D4—2021no—
142C22×C4—22no—
143C22×C4—42no—
144C22×C4—2021no—
145C22×C4C4×D10111yescharacteristic
146C22×C4C4×D10111yescharacteristic
147C22×C4D40411yescharacteristic
148C22×C4D40411yescharacteristic
149C22×C4C22⋊D10111yescharacteristic
150C22×C4C22×D10111yescharacteristic
151C2×C8—4042no—
152C2×C8—2022no—
153C4⋊C4—1011no—
154C4⋊C4—1011no—
155C4⋊C4—2021no—
156C4⋊C4—2021no—
157C22⋊C4—42no—
158C22⋊C4—2021no—
159C22⋊C4—2021no—
160C22⋊C4—2021no—
161C22⋊C4—2021no—
162C22⋊C4—2021no—
163C42—2021no—
164C23—22no—
165C23—22no—
166C23—41no—
167C23—2021no—
168C23#1310805111yescharacteristic
169C23#1310805111yescharacteristic
170C23C2×C22⋊D10111yescharacteristic
171Q8—4041no—
172D4—4041no—
173D4—4042no—
174D4—4042no—
175D4—4042no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—4042no—
186C2×C4—2021no—
187C2×C4—2021no—
188C2×C4—2021no—
189C2×C4#1310805111yescharacteristic
190C2×C4#1310805111yescharacteristic
191C2×C4C2×C4×D10111yescharacteristic
192C2×C4C2×D40411yescharacteristic
193C8—4042no—
194C22—21no—
195C22—21no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—44no—
202C22—4041no—
203C22—4041no—
204C22—4041no—
205C22#1569216912yes—
206C22C2×#1310805111yescharacteristic
207C22#1569227512yes—
208C4—21no—
209C4—21no—
210C4—21no—
211C4—21no—
212C4—82no—
213C4—4041no—
214C4—4041no—
215C4—4041no—
216C2—22no—
217C2—22no—
218C2—41no—
219C2—8081no—
220C2C2×#1569216911yescharacteristic
221C2#2026517111yescharacteristic
222C2#2026519511yescharacteristic
subgroupG/H
centerC22C2×#13108051
commutatorC2×C4×C101C22×C4
FrattiniC22×C4C22×D101
Fitting#381×C101C2
socleC22×C101C2×C22⋊C4
prime radical#20265688C2
cosocleC22×C4×C101C23
hypercenter#381D101
hypocenterC101#1516
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C23×C4 ◁ #381

Complements

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

NH ≅ G/Ntype
2. C2×C22⋊#11717579C2semidirect
3. #20264913C2semidirect
4. #20264910C2semidirect
5. #381×C101C2semidirect
6. C2×#13108074C4semidirect
9. #15692234C4semidirect
11. #15692229C4semidirect
16. #15692214C4semidirect
112. C101#1516semidirect
115. #381D101semidirect

Maximal subgroups

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

subgroupindex
#202656882
C2×C22⋊#117175792
#202649132
#202649102
#381×C1012
#1516101