Finite Group Databasebeta

← #27171048

Subgroups — #27171048

Order 12928.

612 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 101 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×#13108073C211yesmaximal, characteristic
2C2×#15692170C211yesmaximal, characteristic, complemented
3C2×#15692169C212yesmaximal, complemented
4#20265038C212yesmaximal, complemented
5C2×#99×C101C211yesmaximal, characteristic, complemented
6C2×#13108073—24no—
7C2×#13108073—24no—
8C2×#13108073—22no—
9C2×#13108073—22no—
10C22×Dic202C411yescharacteristic, complemented
11C22×C22⋊D101C411yescharacteristic, complemented
12C22×C22⋊D101C2211yescharacteristic
13C23×Dic101C411yescharacteristic, complemented
14C23×Dic101C2211yescharacteristic
15C22×C4×D101C411yescharacteristic, complemented
16C2×C22⋊Dic101C2212yes—
17#15692170—22no—
18#15692169—24no—
19#15692162—24no—
20C22×D4×C101C2211yescharacteristic
21C2×C22⋊C4×C101C2212yes—
22#99×C101—22no—
23C23×D101C2×C411yescharacteristic
24#13108073—216no—
25#13108073—216no—
26C2×Dic202—22no—
27C2×Dic202—22no—
28C2×Dic202—22no—
29C2×C22⋊D101—22no—
30C2×C22⋊D101—22no—
31C2×C22⋊D101—21no—
32C2×C22⋊D101—21no—
33C2×C22⋊D101—22no—
34C2×C22⋊D101—22no—
35C2×C22⋊D101D414yes—
36C22×Dic101—22no—
37C22×Dic101—22no—
38C22×Dic101—21no—
39C22×Dic101—21no—
40C22×Dic101—22no—
41C22×Dic101—22no—
42C22×Dic101—22no—
43C22×Dic101C2×C411yescharacteristic
44C22×Dic101C2×C411yescharacteristic
45C22×Dic101C2×C411yescharacteristic
46C22×Dic101D414yes—
47C2×C4×D101—22no—
48C2×C4×D101—22no—
49C2×C4×D101—22no—
50C22⋊Dic101—22no—
51C22⋊Dic101—22no—
52C24×C101C2×C411yescharacteristic
53C24×C101C2311yescharacteristic
54C2×D4×C101—22no—
55C2×D4×C101—22no—
56C2×D4×C101D412yes—
57C2×D4×C101D412yes—
58C22×C4×C101—22no—
59C22×C4×C101C2×C411yescharacteristic
60C22⋊C4×C101—22no—
61C22⋊C4×C101—22no—
62C23×C101—21no—
63C23×C101—21no—
64C23×C101—42no—
65C23×C101—42no—
66C23×C101C22⋊C412yes—
67C23×C101C22×C411yescharacteristic
68C23×C101C2×D412yes—
69C23×C101C2×D412yes—
70C23×C101C2×D411yescharacteristic
71C23×C101C2×D411yescharacteristic
72C22×D101—21no—
73C22×D101—21no—
74C22×D101—42no—
75C22×D101C22⋊C412yes—
76D4×C101—24no—
77D4×C101—24no—
78C2×C4×C101—21no—
79C2×C4×C101—21no—
80C2×C4×C101—44no—
81C2×C4×C101—42no—
82C2×C4×C101C22⋊C412yes—
83C22⋊D101—28no—
84C22⋊D101—24no—
85C22⋊D101—24no—
86C2×Dic101—28no—
87C2×Dic101—21no—
88C2×Dic101—21no—
89C2×Dic101—21no—
90C2×Dic101—21no—
91C2×Dic101—21no—
92C2×Dic101—21no—
93C2×Dic101—24no—
94C2×Dic101—24no—
95C2×Dic101—44no—
96C2×Dic101—42no—
97C2×Dic101C22⋊C412yes—
98C2×Dic101C22⋊C412yes—
99C2×Dic101C22⋊C412yes—
100C4×D101—28no—
101Dic202—28no—
102C22×C101—24no—
103C22×C101—22no—
104C22×C101—22no—
105C22×C101—42no—
106C22×C101—41no—
107C22×C101—41no—
108C22×C101—42no—
109C22×C101—42no—
110C22×C101—42no—
111C22×C101C2×C22⋊C411yescharacteristic
112C22×C101C2×C22⋊C411yescharacteristic
113C22×C101C2×C22⋊C411yescharacteristic
114C22×C101#12012yes—
115C22×C101#12012yes—
116C2×D101—22no—
117C2×D101—42no—
118C2×D101—42no—
119C2×D101—42no—
120C4×C101—22no—
121C4×C101—82no—
122Dic101—22no—
123Dic101—22no—
124Dic101—22no—
125Dic101—82no—
126C2×C101—22no—
127C2×C101—22no—
128C2×C101—21no—
129C2×C101—21no—
130C2×C101—42no—
131C2×C101#37911yescharacteristic
132C2×C101#41012yes—
133D101—42no—
134#1782—1011nomaximal, Sylow
135C101#178211yescharacteristic, complemented, Sylow
136C22×#41—1011no—
137C2×#99—1011no—
138C2×#99—1012no—
139C2×#99D10111yescharacteristic, complemented
140#381—1012no—
141C2×#41—2024no—
142C2×#41—2024no—
143C2×#41—2022no—
144C2×#41—2022no—
145C22×Q8—1011no—
146C22×D4—1011no—
147C22×D4—1011no—
148C22×D4C2×D10111yescharacteristic
149C23×C4—1011no—
150C23×C4—1011no—
151C23×C4—1011no—
152C2×C22⋊C4—1012no—
153C2×C22⋊C4C2×D10112yes—
154#99—22no—
155#99—2024no—
156#99—2022no—
157#95—2024no—
158C24—1011no—
159C24C4×D10111yescharacteristic
160C24C22×D10111yescharacteristic
161#41—20216no—
162#41—20216no—
163C2×Q8—2022no—
164C2×Q8—2022no—
165C2×Q8—2022no—
166C2×D4—22no—
167C2×D4—22no—
168C2×D4—1014no—
169C2×D4—2022no—
170C2×D4—2022no—
171C2×D4—2022no—
172C2×D4—2022no—
173C2×D4—2021no—
174C2×D4—2021no—
175C2×D4D40412yes—
176C2×D4C22⋊D10112yes—
177C22×C4—22no—
178C22×C4—1011no—
179C22×C4—1011no—
180C22×C4—1011no—
181C22×C4—1014no—
182C22×C4—2022no—
183C22×C4—2022no—
184C22×C4—2022no—
185C22×C4—2022no—
186C22×C4—2022no—
187C22×C4—2022no—
188C22×C4—2022no—
189C22×C4—2021no—
190C22×C4—2021no—
191C22×C4—2022no—
192C22×C4C4×D10111yescharacteristic
193C22⋊C4—22no—
194C22⋊C4—22no—
195C22⋊C4—2022no—
196C22⋊C4—2022no—
197C23—21no—
198C23—21no—
199C23—42no—
200C23—42no—
201C23—1012no—
202C23—4042no—
203C23—2021no—
204C23—2021no—
205C23#1310805112yes—
206C23C2×C4×D10111yescharacteristic
207C23D4×D10112yes—
208C23D4×D10112yes—
209C23C2×C22⋊D10111yescharacteristic
210C23C2×D40411yescharacteristic
211Q8—2028no—
212D4—24no—
213D4—24no—
214D4—2028no—
215D4—2024no—
216D4—2024no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—44no—
220C2×C4—42no—
221C2×C4—1012no—
222C2×C4—1012no—
223C2×C4—1012no—
224C2×C4—4044no—
225C2×C4—4042no—
226C2×C4—2028no—
227C2×C4—2028no—
228C2×C4—2021no—
229C2×C4—2021no—
230C2×C4—2021no—
231C2×C4—2021no—
232C2×C4—2021no—
233C2×C4—2021no—
234C2×C4—2024no—
235C2×C4—2024no—
236C2×C4#1310805112yes—
237C22—24no—
238C22—22no—
239C22—22no—
240C22—42no—
241C22—41no—
242C22—41no—
243C22—42no—
244C22—42no—
245C22—42no—
246C22—4042no—
247C22—4042no—
248C22—4042no—
249C22—2022no—
250C22C22⋊C4×D10111yescharacteristic
251C22C22⋊C4×D10111yescharacteristic
252C22C2×#1310805111yescharacteristic
253C22#1569222312yes—
254C22#1569222512yes—
255C4—22no—
256C4—82no—
257C4—8082no—
258C4—2022no—
259C4—2022no—
260C4—2022no—
261C2—22no—
262C2—22no—
263C2—21no—
264C2—21no—
265C2—42no—
266C2—4042no—
267C2#2026503211yescharacteristic
268C2#2026514912yes—
subgroupG/H
centerC22C22⋊C4×D101
commutatorC23×C101C22×C4
FrattiniC24C22×D101
FittingC2×#99×C101C2
socleC22×C101C2×C22⋊C4
prime radicalC22×#13108073C2
cosocleC24×C101C23
hypercenterC2×#99D101
hypocenterC101#1782
LCSG ▷ C23×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
2. C2×#15692170C2semidirect
3. C2×#15692169C2semidirect
4. #20265038C2semidirect
5. C2×#99×C101C2semidirect
10. C22×Dic202C4semidirect
11. C22×C22⋊D101C4semidirect
13. C23×Dic101C4semidirect
15. C22×C4×D101C4semidirect
135. C101#1782semidirect
139. C2×#99D101semidirect

Maximal subgroups

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

subgroupindex
C22×#131080732
C2×#156921702
C2×#156921692
#202650382
C2×#99×C1012
#1782101