Finite Group Databasebeta

← #27171432

Subgroups — #27171432

Order 12928.

376 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 93 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
1C22×#13108058C211yesmaximal, characteristic, complemented
2#20265688C211yesmaximal, characteristic
3C2×#15692183C212yesmaximal, complemented
4#20264937C212yesmaximal, complemented
5C4×C22⋊C4×C101C211yesmaximal, characteristic, complemented
6C2×#13108074C2211yescharacteristic
7C2×#13108058—24no—
8C2×#13108058—21no—
9C2×#13108058C2212yes—
10C22×#11717579—21no—
11#15692238—21no—
12#15692234C411yescharacteristic, complemented
13#15692234C411yescharacteristic, complemented
14#15692229C411yescharacteristic, complemented
15#15692229C411yescharacteristic, complemented
16#15692224—21no—
17C4×C22⋊D101—21no—
18C4×C22⋊D101—21no—
19#15692214C2211yescharacteristic
20#15692183—24no—
21C23×C4×C101C2211yescharacteristic
22C2×C22⋊C4×C101—21no—
23C2×C42×C101C2212yes—
24#95×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—
32C2×C4×D101—21no—
33#13108058—24no—
34#13108058—22no—
35#13108058D414yescomplemented
36C2×#11717579—24no—
37C2×#11717579—22no—
38C4⋊Dic101C2×C411yescharacteristic
39C4⋊Dic101C2×C411yescharacteristic
40#13108053—21no—
41#13108053—21no—
42C22⋊Dic101—21no—
43C22⋊Dic101—21no—
44C22⋊Dic101—21no—
45#13108051—21no—
46#13108051—21no—
47C4×Dic101—21no—
48C24×C101D411yescharacteristic
49C22×C4×C101—22no—
50C22×C4×C101—22no—
51C22×C4×C101—22no—
52C22×C4×C101C2×C411yescharacteristic
53C22×C4×C101C2×C411yescharacteristic
54C22×C4×C101D411yescharacteristic
55C22×C4×C101D411yescharacteristic
56C22×C4×C101D411yescharacteristic
57C22×C4×C101C2311yescharacteristic
58C22⋊C4×C101—24no—
59C42×C101—22no—
60C42×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×C101—24no—
76C2×C4×C101—42no—
77C2×C4×C101—42no—
78C2×C4×C101C22⋊C411yescharacteristic
79C2×C4×C101C22⋊C411yescharacteristic
80C2×C4×C101C22×C411yescharacteristic
81C2×C4×C101C2×D412yes—
82C2×C4×C101C2×D411yescharacteristic
83C2×C4×C101#4112yes—
84C22⋊D101—42no—
85C22⋊D101—42no—
86C22⋊D101—42no—
87C2×Dic101—21no—
88C2×Dic101—21no—
89C2×Dic101—21no—
90D404—22no—
91C4×D101—41no—
92C4×D101—41no—
93Dic202—22no—
94#11717579—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×C101—44no—
112C4×C101C4×D412yes—
113C4×C101#12011yescharacteristic
114C4×C101#12311yescharacteristic
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×M16—1011no—
128#525—1011no—
129C2×#104—1012no—
130C4×C22⋊C4D10111yescharacteristic, complemented
131#339—1012no—
132C2×#41—1011no—
133C23×C4C2×D10111yescharacteristic
134C2×M16—1012no—
135C2×M16—2024no—
136C2×M16—2021no—
137C22×C8—2021no—
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—21no—
148C2×C42C2×D10112yes—
149#104—2024no—
150#95—21no—
151C24C22⋊D10111yescharacteristic
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—22no—
162C22×C4—2021no—
163C22×C4C4×D10111yescharacteristic
164C22×C4C4×D10111yescharacteristic
165C22×C4D40411yescharacteristic
166C22×C4D40411yescharacteristic
167C22×C4C22⋊D10111yescharacteristic
168C22×C4C22×D10111yescharacteristic
169M16—1014no—
170M16—2024no—
171M16—2022no—
172C2×C8—2024no—
173C2×C8—2022no—
174C4⋊C4—1011no—
175C4⋊C4—1011no—
176C4⋊C4—2021no—
177C4⋊C4—2021no—
178C22⋊C4—24no—
179C22⋊C4—2021no—
180C22⋊C4—2021no—
181C22⋊C4—2021no—
182C22⋊C4—2021no—
183C22⋊C4—2021no—
184C42—22no—
185C42—22no—
186C42—2021no—
187C23—22no—
188C23—22no—
189C23—41no—
190C23—2021no—
191C23#1310805111yescharacteristic
192C23#1310805111yescharacteristic
193C23C2×C22⋊D10111yescharacteristic
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—24no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—4041no—
210C2×C4—4041no—
211C2×C4—2021no—
212C2×C4—2021no—
213C2×C4—2021no—
214C2×C4#1310805111yescharacteristic
215C2×C4#1310805111yescharacteristic
216C2×C4C2×C4×D10111yescharacteristic
217C2×C4D4×D10112yes—
218C2×C4C2×D40411yescharacteristic
219C2×C4#1310807312yes—
220C8—2024no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—44no—
229C22—4041no—
230C22—4041no—
231C22—4041no—
232C22#1569218312yescomplemented
233C22C2×#1310805111yescharacteristic
234C22#1569222712yes—
235C4—21no—
236C4—21no—
237C4—44no—
238C4—4041no—
239C4—4041no—
240C4—4041no—
241C4#1569221512yes—
242C4#1569222511yescharacteristic
243C4#1569223911yescharacteristic
244C2—22no—
245C2—22no—
246C2—41no—
247C2—8081no—
248C2#2026505411yescharacteristic
249C2C2×#1569218311yescharacteristic
250C2#2026519511yescharacteristic
subgroupG/H
centerC2×C4C2×D404
commutatorC2×C4×C101C22×C4
FrattiniC22×C4C22×D101
FittingC4×C22⋊C4×C101C2
socleC22×C101C2×C22⋊C4
prime radical#20265688C2
cosocleC22×C4×C101C23
hypercenterC4×C22⋊C4D101
hypocenterC101#1871
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#13108058C2semidirect
3. C2×#15692183C2semidirect
4. #20264937C2semidirect
5. C4×C22⋊C4×C101C2semidirect
12. #15692234C4semidirect
13. #15692234C4semidirect
14. #15692229C4semidirect
15. #15692229C4semidirect
35. #13108058D4semidirect
103. C22×C101#104semidirect
126. C101#1871semidirect
130. C4×C22⋊C4D101semidirect
232. C22#15692183semidirect

Maximal subgroups

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

subgroupindex
C22×#131080582
#202656882
C2×#156921832
#202649372
C4×C22⋊C4×C1012
#1871101