Finite Group Databasebeta

← #27181304

Subgroups — #27181304

Order 12928.

386 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 121 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#20265713C211yesmaximal, characteristic
2C2×Q8×Dic101C211yesmaximal, characteristic, complemented
3#20265510C211yesmaximal, characteristic, complemented
4#20265496C211yesmaximal, characteristic
5#20265482C212yesmaximal, complemented
6#20265401C212yesmaximal, complemented
7#20265316C211yesmaximal, characteristic
8#20265284C212yesmaximal, complemented
9#20265280C212yesmaximal, complemented
10#20265231C211yesmaximal, characteristic, complemented
11#451×C101C211yesmaximal, characteristic, complemented
12C2×#13108061C2212yes—
13#15692250—21no—
14#15692246C2211yescharacteristic
15#15692244C2211yescharacteristic
16#15692239—21no—
17#15692229C2211yescharacteristic
18Q8×Dic101—21no—
19Q8×Dic101—21no—
20Q8×Dic101C2212yes—
21Q8×Dic101C2212yes—
22Q8×Dic101C2211yescharacteristic
23#15692217C2211yescharacteristic
24#15692215—21no—
25#15692213—21no—
26C2×C4⋊Dic101—21no—
27C2×C4⋊Dic101C2211yescharacteristic
28C2×C4×Dic101—21no—
29C2×C4×Dic101C2211yescharacteristic
30#15692188C2212yes—
31#15692187C2212yes—
32#15692184C2212yes—
33#15692179C2212yes—
34#15692179C2212yes—
35#15692177C2212yes—
36C22⋊#11717579C2211yescharacteristic
37#15692165C2212yes—
38C22×Q8×C101C2211yescharacteristic
39C2×Q16×C101C2212yes—
40C22⋊Q8×C101C2211yescharacteristic
41#103×C101C2212yes—
42C22⋊C8×C101C2211yescharacteristic
43C2×D404C2311yescharacteristic
44C2×C22⋊D101—21no—
45C22×Dic101—21no—
46C22×Dic101D411yescharacteristic
47C22×Dic101D411yescharacteristic
48C2×C4×D101—21no—
49#13108061—24no—
50C2×#11717579C2312yes—
51C4⋊Dic101—21no—
52C4⋊Dic101—22no—
53C4⋊Dic101—22no—
54C4⋊Dic101—21no—
55C4⋊Dic101—21no—
56C4⋊Dic101C2312yes—
57C4⋊Dic101C2311yescharacteristic
58#13108053—21no—
59#13108051—21no—
60#13108051—21no—
61#13108051—21no—
62#13108051—21no—
63C4×Dic101—22no—
64C4×Dic101—22no—
65C4×Dic101—21no—
66C4×Dic101D411yescharacteristic
67C4×Dic101D411yescharacteristic
68C4×Dic101C2311yescharacteristic
69C4×Dic101C2311yescharacteristic
70C2×Q8×C101—24no—
71C2×Q8×C101—21no—
72C2×Q8×C101C2311yescharacteristic
73C2×Q8×C101C2312yes—
74C22×C4×C101—21no—
75C22×C4×C101C2311yescharacteristic
76Q16×C101—24no—
77C2×C8×C101C2312yes—
78C4⋊C4×C101—21no—
79C4⋊C4×C101C2311yescharacteristic
80C22⋊C4×C101—21no—
81C23×C101C2×D411yescharacteristic
82C22×D101—21no—
83Q8×C101—24no—
84Q8×C101—22no—
85Q8×C101—22no—
86Q8×C101#4114yes—
87C2×C4×C101—24no—
88C2×C4×C101—22no—
89C2×C4×C101—22no—
90C2×C4×C101—21no—
91C2×C4×C101—21no—
92C2×C4×C101C2×D411yescharacteristic
93C2×C4×C101C2411yescharacteristic
94C22⋊D101—42no—
95C2×Dic101—22no—
96C2×Dic101—22no—
97C2×Dic101—21no—
98C2×Dic101—41no—
99C2×Dic101—42no—
100C2×Dic101C2×D411yescharacteristic
101C2×Dic101C2×D411yescharacteristic
102C2×Dic101C2×D411yescharacteristic
103C2×Dic101C2×D411yescharacteristic
104D404—22no—
105C4×D101—42no—
106C8×C101—22no—
107#11717579—22no—
108C22×C101—21no—
109C22×C101—21no—
110C22×C101#13712yes—
111C22×C101C22×D411yescharacteristic
112C2×D101—41no—
113C2×D101—41no—
114C2×D101—41no—
115C4×C101—24no—
116C4×C101—21no—
117C4×C101—41no—
118C4×C101—41no—
119C4×C101C2×#4112yes—
120Dic101—21no—
121Dic101—21no—
122Dic101—21no—
123Dic101—21no—
124Dic101—41no—
125Dic101—42no—
126C2×C101—22no—
127C2×C101#52511yescharacteristic
128C2×C101C2×#13711yescharacteristic
129C2×C101#57511yescharacteristic
130D101—81no—
131#3117—1011nomaximal, Sylow
132C101#311711yescharacteristic, complemented, Sylow
133#529—1011no—
134C2×C4×Q8—1011no—
135#484—1011no—
136#481—1011no—
137#479—1012no—
138#464—1012no—
139#451D10111yescharacteristic, complemented
140#448—1011no—
141#441—1012no—
142#440—1012no—
143#432—1011no—
144C22×Q8C2×D10111yescharacteristic
145C2×Q16C2×D10112yes—
146C2×SD16—1012no—
147#126—2021no—
148#125—1011no—
149#124—1011no—
150#123—2021no—
151C22⋊Q8C2×D10111yescharacteristic
152#121—1011no—
153C4×Q8—1012no—
154C4×Q8—1012no—
155C4×Q8—1011no—
156C4×Q8—2021no—
157C4×Q8—2021no—
158C4×D4—1011no—
159C4×D4—2021no—
160#117—2021no—
161C2×C4⋊C4—1011no—
162C2×C4⋊C4—2021no—
163C2×C42—1011no—
164C2×C42—2021no—
165#107—1012no—
166#106—1012no—
167C4⋊C8—1012no—
168#103—1012no—
169#103—1012no—
170#103C2×D10112yes—
171#102—1012no—
172C22⋊C8—1011no—
173C22⋊C8C2×D10111yescharacteristic
174#97—1012no—
175C2×Q8—24no—
176C2×Q8—21no—
177C2×Q8C22×D10111yescharacteristic
178C2×Q8C22×D10112yes—
179C2×D4—1011no—
180C2×D4—2021no—
181C22×C4—21no—
182C22×C4—1011no—
183C22×C4—1011no—
184C22×C4—2021no—
185C22×C4—2021no—
186C22×C4C22×D10111yescharacteristic
187Q16—24no—
188SD16—2024no—
189C2×C8—1012no—
190C2×C8C22×D10112yes—
191C4⋊C4—21no—
192C4⋊C4—1012no—
193C4⋊C4—1011no—
194C4⋊C4—2021no—
195C4⋊C4—2022no—
196C4⋊C4—2022no—
197C4⋊C4—2021no—
198C4⋊C4—2021no—
199C4⋊C4—2021no—
200C4⋊C4C22×D10111yescharacteristic
201C22⋊C4—21no—
202C22⋊C4—2021no—
203C22⋊C4—2021no—
204C22⋊C4—2021no—
205C22⋊C4—2021no—
206C42—1011no—
207C42—1011no—
208C42—1011no—
209C42—1011no—
210C42—2022no—
211C42—2022no—
212C42—2021no—
213C23—2021no—
214C23D4×D10111yescharacteristic
215Q8—24no—
216Q8—22no—
217Q8—22no—
218Q8#1310807314yes—
219D4—4042no—
220D4—2022no—
221C2×C4—24no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—1011no—
227C2×C4—1011no—
228C2×C4—1011no—
229C2×C4—1011no—
230C2×C4—4042no—
231C2×C4—4042no—
232C2×C4—4041no—
233C2×C4—2022no—
234C2×C4—2021no—
235C2×C4—2022no—
236C2×C4D4×D10111yescharacteristic
237C2×C4C23×D10111yescharacteristic
238C8—22no—
239C8—2022no—
240C22—21no—
241C22—21no—
242C22—4041no—
243C22—4041no—
244C22—4041no—
245C22#1569228712yes—
246C22C2×D4×D10111yescharacteristic
247C4—24no—
248C4—21no—
249C4—41no—
250C4—41no—
251C4—4041no—
252C4—4042no—
253C4—2021no—
254C4—2021no—
255C4—2021no—
256C4—2021no—
257C4C2×#1310807312yes—
258C2—22no—
259C2—8081no—
260C2#2026568311yescharacteristic
261C2C2×#1569228711yescharacteristic
262C2#2026597611yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#451×C101C2
socleC22×C101C22×D4
prime radical#15692229C22
hypercenter#451D101
hypocenterC101#3117
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. C2×Q8×Dic101C2semidirect
3. #20265510C2semidirect
5. #20265482C2semidirect
6. #20265401C2semidirect
8. #20265284C2semidirect
9. #20265280C2semidirect
10. #20265231C2semidirect
11. #451×C101C2semidirect
132. C101#3117semidirect
139. #451D101semidirect

Maximal subgroups

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

subgroupindex
#202657132
C2×Q8×Dic1012
#202655102
#202654962
#202654822
#202654012
#202653162
#202652842
#202652802
#202652312
#451×C1012
#3117101