Finite Group Databasebeta

← #27171434

Subgroups — #27171434

Order 12928.

376 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 97 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×#15692181C212yesmaximal, complemented
3C4×C22⋊Dic101C211yesmaximal, characteristic, complemented
4#20264938C212yesmaximal, complemented
5C22×M16×C101C211yesmaximal, characteristic, complemented
6C2×#13108074C2211yescharacteristic
7#15692238—21no—
8#15692234C411yescharacteristic, complemented
9#15692234C411yescharacteristic, complemented
10#15692229C411yescharacteristic, complemented
11#15692229C411yescharacteristic, complemented
12#15692224—21no—
13C4×C22⋊D101—21no—
14C4×C22⋊D101—21no—
15#15692214C2211yescharacteristic
16C2×C22⋊Dic101—21no—
17C2×C4×Dic101C2212yes—
18#15692181—24no—
19#15692162—21no—
20C23×C4×C101C2211yescharacteristic
21C2×M16×C101—21no—
22C2×M16×C101—24no—
23C2×M16×C101C2212yes—
24C22×C8×C101—21no—
25#13108074—22no—
26#13108074—22no—
27C2×Dic202C2×C411yescharacteristic
28C2×D404C2×C411yescharacteristic
29C2×C22⋊D101—21no—
30C2×C22⋊D101—21no—
31C2×C22⋊D101—21no—
32C22×Dic101—22no—
33C2×C4×D101—21no—
34C4⋊Dic101C2×C411yescharacteristic
35C4⋊Dic101C2×C411yescharacteristic
36#13108053—21no—
37#13108053—21no—
38C22⋊Dic101—24no—
39C22⋊Dic101—21no—
40C22⋊Dic101—21no—
41C22⋊Dic101—21no—
42#13108051—21no—
43#13108051—21no—
44C4×Dic101—21no—
45C4×Dic101—22no—
46C4×Dic101—22no—
47C24×C101D411yescharacteristic
48C22×C4×C101—22no—
49C22×C4×C101—22no—
50C22×C4×C101C2×C411yescharacteristic
51C22×C4×C101C2×C411yescharacteristic
52C22×C4×C101D411yescharacteristic
53C22×C4×C101D411yescharacteristic
54C22×C4×C101D411yescharacteristic
55C22×C4×C101C2311yescharacteristic
56M16×C101—24no—
57M16×C101—22no—
58M16×C101D414yescomplemented
59C2×C8×C101—24no—
60C2×C8×C101—22no—
61C23×C101—22no—
62C23×C101—22no—
63C23×C101—41no—
64C23×C101C22⋊C411yescharacteristic
65C23×C101C22⋊C411yescharacteristic
66C23×C101C2×D411yescharacteristic
67C22×D101—21no—
68C2×C4×C101—22no—
69C2×C4×C101—22no—
70C2×C4×C101—22no—
71C2×C4×C101—21no—
72C2×C4×C101—21no—
73C2×C4×C101—21no—
74C2×C4×C101—21no—
75C2×C4×C101C22⋊C411yescharacteristic
76C2×C4×C101C22⋊C411yescharacteristic
77C2×C4×C101C22×C411yescharacteristic
78C2×C4×C101C2×D412yes—
79C2×C4×C101C2×D411yescharacteristic
80C2×C4×C101#4112yes—
81C22⋊D101—42no—
82C22⋊D101—42no—
83C22⋊D101—42no—
84C2×Dic101—24no—
85C2×Dic101—21no—
86C2×Dic101—21no—
87C2×Dic101—21no—
88C2×Dic101—42no—
89C2×Dic101—42no—
90D404—22no—
91C4×D101—41no—
92C4×D101—41no—
93Dic202—22no—
94C8×C101—24no—
95C22×C101—21no—
96C22×C101—21no—
97C22×C101—21no—
98C22×C101—21no—
99C22×C101—21no—
100C22×C101—21no—
101C22×C101—21no—
102C22×C101—44no—
103C22×C101#10412yescomplemented
104C22×C101C2×C22⋊C411yescharacteristic
105C22×C101#12112yes—
106C2×D101—41no—
107C2×D101—41no—
108C2×D101—41no—
109C4×C101—21no—
110C4×C101—21no—
111C4×C101C4×D412yes—
112C4×C101#12011yescharacteristic
113C4×C101#12311yescharacteristic
114Dic101—44no—
115Dic101—41no—
116Dic101—41no—
117Dic101—41no—
118C2×C101—22no—
119C2×C101—22no—
120C2×C101—41no—
121C2×C101#38611yescharacteristic
122C2×C101C2×#10411yescharacteristic
123C2×C101#42111yescharacteristic
124D101—81no—
125#1871—1011nomaximal, Sylow
126C101#187111yescharacteristic, complemented, Sylow
127C22×M16D10111yescharacteristic, complemented
128#525—1011no—
129C2×#104—1012no—
130C4×C22⋊C4—1011no—
131#339—1012no—
132C2×#41—1011no—
133C23×C4C2×D10111yescharacteristic
134C2×M16—21no—
135C2×M16—24no—
136C2×M16C2×D10112yes—
137C22×C8—21no—
138#123—2021no—
139C22⋊Q8—1011no—
140C22⋊Q8—1011no—
141#121—1011no—
142#121—1011no—
143#120—2021no—
144C4×D4—2021no—
145C4×D4—2021no—
146#117—1011no—
147C2×C22⋊C4—2021no—
148C2×C42—1012no—
149#104—2024no—
150#95—2021no—
151C24D40411yescharacteristic
152#41—2022no—
153#41—2022no—
154C2×Q8—1011no—
155C2×D4—1011no—
156C2×D4—2021no—
157C2×D4—2021no—
158C2×D4—2021no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—2022no—
162C22×C4—2021no—
163C22×C4C4×D10111yescharacteristic
164C22×C4C4×D10111yescharacteristic
165C22×C4D40411yescharacteristic
166C22×C4C22⋊D10111yescharacteristic
167C22×C4C22⋊D10111yescharacteristic
168C22×C4C22×D10111yescharacteristic
169M16—24no—
170M16—22no—
171M16C22⋊D10114yescomplemented
172C2×C8—24no—
173C2×C8—22no—
174C4⋊C4—1011no—
175C4⋊C4—1011no—
176C4⋊C4—2021no—
177C4⋊C4—2021no—
178C22⋊C4—2024no—
179C22⋊C4—2021no—
180C22⋊C4—2021no—
181C22⋊C4—2021no—
182C22⋊C4—2021no—
183C22⋊C4—2021no—
184C42—2022no—
185C42—2022no—
186C42—2021no—
187C23—22no—
188C23—22no—
189C23—41no—
190C23—2021no—
191C23#1310805111yescharacteristic
192C23#1310805111yescharacteristic
193C23C2×D40411yescharacteristic
194Q8—2022no—
195D4—4042no—
196D4—4042no—
197D4—4042no—
198D4—2022no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—4042no—
207C2×C4—4042no—
208C2×C4—4041no—
209C2×C4—4041no—
210C2×C4—2024no—
211C2×C4—2021no—
212C2×C4—2021no—
213C2×C4—2021no—
214C2×C4#1310805111yescharacteristic
215C2×C4#1310805111yescharacteristic
216C2×C4C2×C4×D10111yescharacteristic
217C2×C4C2×C22⋊D10112yes—
218C2×C4C2×C22⋊D10111yescharacteristic
219C2×C4#1310807412yes—
220C8—24no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—44no—
229C22—4041no—
230C22—4041no—
231C22—4041no—
232C22#1569218112yescomplemented
233C22C2×#1310805111yescharacteristic
234C22#1569222912yes—
235C4—21no—
236C4—21no—
237C4—4044no—
238C4—4041no—
239C4—4041no—
240C4—4041no—
241C4C4×C22⋊D10112yes—
242C4#1569222411yescharacteristic
243C4#1569223811yescharacteristic
244C2—22no—
245C2—22no—
246C2—41no—
247C2—8081no—
248C2#2026505611yescharacteristic
249C2C2×#1569218111yescharacteristic
250C2#2026519311yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D101
commutatorC2×C4×C101C22×C4
FrattiniC22×C4C22×D101
FittingC22×M16×C101C2
socleC22×C101C2×C22⋊C4
prime radical#20265688C2
cosocleC22×C4×C101C23
hypercenterC22×M16D101
hypocenterC101#1871
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
2. C2×#15692181C2semidirect
3. C4×C22⋊Dic101C2semidirect
4. #20264938C2semidirect
5. C22×M16×C101C2semidirect
8. #15692234C4semidirect
9. #15692234C4semidirect
10. #15692229C4semidirect
11. #15692229C4semidirect
58. M16×C101D4semidirect
103. C22×C101#104semidirect
126. C101#1871semidirect
127. C22×M16D101semidirect
171. M16C22⋊D101semidirect
232. C22#15692181semidirect

Maximal subgroups

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

subgroupindex
#202656882
C2×#156921812
C4×C22⋊Dic1012
#202649382
C22×M16×C1012
#1871101