Finite Group Databasebeta

← #27181303

Subgroups — #27181303

Order 12928.

386 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 129 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#20265711C211yesmaximal, characteristic
2#20265512C211yesmaximal, characteristic, complemented
3#20265495C211yesmaximal, characteristic
4#20265484C212yesmaximal, complemented
5#20265399C212yesmaximal, complemented
6#20265333C211yesmaximal, characteristic, complemented
7#20265316C211yesmaximal, characteristic
8#20265282C212yesmaximal, complemented
9#20265275C212yesmaximal, complemented
10#20265234C211yesmaximal, characteristic, complemented
11C2×C4×Q8×C101C211yesmaximal, characteristic, complemented
12C2×#13108064C2212yes—
13C2×#13108061C2212yes—
14#15692249—21no—
15#15692247C2211yescharacteristic
16#15692243C2211yescharacteristic
17#15692238—21no—
18#15692234C2211yescharacteristic
19#15692229C2211yescharacteristic
20C4×Dic202C2211yescharacteristic
21C4×D404C2211yescharacteristic
22C4×C22⋊D101—21no—
23#15692212—21no—
24#15692188C2212yes—
25#15692186C2212yes—
26C4⋊#11717579C2212yes—
27#15692179C2212yes—
28#15692179C2212yes—
29#15692178C2212yes—
30#15692175C2212yes—
31C22⋊#11717579C2211yescharacteristic
32C22⋊#11717579C2211yescharacteristic
33#15692166C2212yes—
34C22×Q8×C101C2211yescharacteristic
35C4×Q8×C101—21no—
36C4×Q8×C101—21no—
37C4×Q8×C101C2212yes—
38C4×Q8×C101C2212yes—
39C2×C4⋊C4×C101—21no—
40C2×C4⋊C4×C101C2211yescharacteristic
41C2×C42×C101—21no—
42C2×C42×C101C2211yescharacteristic
43C2×Dic202C2311yescharacteristic
44C2×D404C2311yescharacteristic
45C2×C22⋊D101—21no—
46C2×C4×D101—21no—
47#13108064—24no—
48#13108061—24no—
49C2×#11717579C2312yes—
50C2×#11717579C2312yes—
51C4⋊Dic101C2311yescharacteristic
52C4⋊Dic101C2311yescharacteristic
53#13108053—21no—
54#13108053—21no—
55#13108053—21no—
56#13108053—21no—
57C22⋊Dic101—21no—
58#13108051—21no—
59#13108051—21no—
60#13108051—21no—
61#13108051—21no—
62C4×Dic101—21no—
63C2×Q8×C101—24no—
64C2×Q8×C101—21no—
65C2×Q8×C101C2312yes—
66C22×C4×C101—21no—
67C22×C4×C101—21no—
68C22×C4×C101D411yescharacteristic
69C22×C4×C101D411yescharacteristic
70C22×C4×C101C2311yescharacteristic
71C4⋊C4×C101—22no—
72C4⋊C4×C101—22no—
73C4⋊C4×C101—21no—
74C4⋊C4×C101C2312yes—
75C42×C101—22no—
76C42×C101—22no—
77C42×C101D411yescharacteristic
78C42×C101D411yescharacteristic
79C42×C101C2311yescharacteristic
80C42×C101C2311yescharacteristic
81C23×C101C2×D411yescharacteristic
82C22×D101—21no—
83Q8×C101—24no—
84Q8×C101—22no—
85Q8×C101#4114yes—
86C2×C4×C101—24no—
87C2×C4×C101—22no—
88C2×C4×C101—22no—
89C2×C4×C101—22no—
90C2×C4×C101—22no—
91C2×C4×C101—41no—
92C2×C4×C101—42no—
93C2×C4×C101C2×D411yescharacteristic
94C2×C4×C101C2×D411yescharacteristic
95C2×C4×C101C2×D411yescharacteristic
96C2×C4×C101C2×D411yescharacteristic
97C2×C4×C101C2×D411yescharacteristic
98C2×C4×C101C2411yescharacteristic
99C22⋊D101—42no—
100C2×Dic101—21no—
101C2×Dic101—21no—
102C2×Dic101—21no—
103D404—22no—
104C4×D101—42no—
105Dic202—22no—
106#11717579—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—21no—
118C4×C101—21no—
119C4×C101—21no—
120C4×C101—21no—
121C4×C101—42no—
122C4×C101C2×#4112yes—
123Dic101—41no—
124Dic101—41no—
125Dic101—41no—
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×Q8D10111yescharacteristic, complemented
135#484—1011no—
136#481—1011no—
137#479—1012no—
138#464—1012no—
139#451—1011no—
140#448—1011no—
141#441—1012no—
142#440—1012no—
143#432—1011no—
144C22×Q8C2×D10111yescharacteristic
145C2×Q16—1012no—
146C2×SD16—1012no—
147#126—2021no—
148#125—1011no—
149#124—1011no—
150#123—2021no—
151C22⋊Q8—1011no—
152#121—1011no—
153C4×Q8—21no—
154C4×Q8—21no—
155C4×Q8—1011no—
156C4×Q8C2×D10112yes—
157C4×Q8C2×D10112yes—
158C4×D4—1011no—
159C4×D4—2021no—
160#117—2021no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4C2×D10111yescharacteristic
163C2×C42—21no—
164C2×C42C2×D10111yescharacteristic
165#107—1012no—
166#106—1012no—
167C4⋊C8—1012no—
168#103—1012no—
169#103—1012no—
170#103—1012no—
171#102—1012no—
172C22⋊C8—1011no—
173C22⋊C8—1011no—
174#97—1012no—
175C2×Q8—24no—
176C2×Q8—21no—
177C2×Q8—1011no—
178C2×Q8C22×D10112yes—
179C2×D4—1011no—
180C2×D4—2021no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—2021no—
184C22×C4C22⋊D10111yescharacteristic
185C22×C4C22⋊D10111yescharacteristic
186C22×C4C22×D10111yescharacteristic
187Q16—2024no—
188SD16—2024no—
189C2×C8—1012no—
190C2×C8—1012no—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—21no—
194C4⋊C4—1011no—
195C4⋊C4—1011no—
196C4⋊C4—2021no—
197C4⋊C4—2021no—
198C4⋊C4—2021no—
199C4⋊C4—2021no—
200C4⋊C4C22×D10112yes—
201C22⋊C4—2021no—
202C22⋊C4—2021no—
203C22⋊C4—2021no—
204C22⋊C4—2021no—
205C22⋊C4—2021no—
206C42—22no—
207C42—22no—
208C42—2021no—
209C42C22⋊D10111yescharacteristic
210C42C22⋊D10111yescharacteristic
211C42C22×D10111yescharacteristic
212C42C22×D10111yescharacteristic
213C23—2021no—
214C23C2×C22⋊D10111yescharacteristic
215Q8—24no—
216Q8—22no—
217Q8—2022no—
218Q8#1310807414yes—
219D4—4042no—
220D4—2022no—
221C2×C4—24no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—4042no—
229C2×C4—2021no—
230C2×C4—2021no—
231C2×C4—2021no—
232C2×C4C2×C22⋊D10111yescharacteristic
233C2×C4C2×C22⋊D10111yescharacteristic
234C2×C4C2×C22⋊D10111yescharacteristic
235C2×C4C2×C22⋊D10111yescharacteristic
236C2×C4C2×C22⋊D10111yescharacteristic
237C2×C4C23×D10111yescharacteristic
238C8—2022no—
239C8—2022no—
240C22—21no—
241C22—21no—
242C22—4041no—
243C22—4041no—
244C22—4041no—
245C22#1569228612yes—
246C22C22×C22⋊D10111yescharacteristic
247C4—24no—
248C4—21no—
249C4—21no—
250C4—21no—
251C4—21no—
252C4—21no—
253C4—42no—
254C4—4041no—
255C4—4041no—
256C4—4041no—
257C4C2×#1310807412yes—
258C2—22no—
259C2—8081no—
260C2#2026568811yescharacteristic
261C2C2×#1569228611yescharacteristic
262C2#2026597511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC2×C4×Q8×C101C2
socleC22×C101C22×D4
prime radical#15692229C22
hypercenterC2×C4×Q8D101
hypocenterC101#3117
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
2. #20265512C2semidirect
4. #20265484C2semidirect
5. #20265399C2semidirect
6. #20265333C2semidirect
8. #20265282C2semidirect
9. #20265275C2semidirect
10. #20265234C2semidirect
11. C2×C4×Q8×C101C2semidirect
132. C101#3117semidirect
134. C2×C4×Q8D101semidirect

Maximal subgroups

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

subgroupindex
#202657112
#202655122
#202654952
#202654842
#202653992
#202653332
#202653162
#202652822
#202652752
#202652342
C2×C4×Q8×C1012
#3117101