Finite Group Databasebeta

← #27171572

Subgroups — #27171572

Order 12928.

442 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 87 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#20265684C211yesmaximal, characteristic
2C2×C42×D101C211yesmaximal, characteristic, complemented
3C2×#15692182C211yesmaximal, characteristic, complemented
4#20265140C211yesmaximal, characteristic, complemented
5#20264938C211yesmaximal, characteristic, complemented
6#20264937C211yesmaximal, characteristic, complemented
7C2×#104×C101C211yesmaximal, characteristic, complemented
8C22×C4×D101—21no—
9C22×C4×D101C2211yescharacteristic
10C2×#13108058C2211yescharacteristic
11#15692236—21no—
12#15692231C412yescomplemented
13#15692226C412yescomplemented
14#15692223—21no—
15C4×C22⋊D101—22no—
16#15692214C2211yescharacteristic
17C2×C4×Dic101—21no—
18C2×C4×Dic101C2211yescharacteristic
19C42×D101—22no—
20C42×D101—22no—
21#15692182—22no—
22#15692167—22no—
23C2×#41×C101C2211yescharacteristic
24C2×M16×C101C2211yescharacteristic
25C2×C42×C101C2211yescharacteristic
26#104×C101—22no—
27C23×D101D411yescharacteristic
28C2×C22⋊D101—22no—
29C22×Dic101—21no—
30C22×Dic101D411yescharacteristic
31C2×C4×D101—28no—
32C2×C4×D101—24no—
33C2×C4×D101—21no—
34C2×C4×D101—21no—
35C2×C4×D101—21no—
36C2×C4×D101—21no—
37C2×C4×D101C2×C412yes—
38C2×C4×D101D412yes—
39#13108058—21no—
40#13108058—21no—
41C2×#11717579—21no—
42C4⋊Dic101C2×C412yescomplemented
43#13108053—22no—
44C22⋊Dic101—21no—
45C22⋊Dic101—22no—
46#13108051—22no—
47C4×Dic101—24no—
48C4×Dic101—21no—
49C4×Dic101—21no—
50C4×Dic101D412yescomplemented
51#41×C101—21no—
52#41×C101—21no—
53#41×C101—21no—
54#41×C101—21no—
55C2×Q8×C101C2×C411yescharacteristic, complemented
56C2×D4×C101—21no—
57C2×D4×C101C2×C411yescharacteristic, complemented
58C22×C4×C101—21no—
59C22×C4×C101—21no—
60C22×C4×C101C2311yescharacteristic
61M16×C101—21no—
62M16×C101—21no—
63C2×C8×C101—21no—
64C42×C101—21no—
65C42×C101D412yescomplemented
66C23×C101—21no—
67C23×C101C2×D411yescharacteristic
68C22×D101—21no—
69C22×D101—21no—
70C22×D101—24no—
71C22×D101C22⋊C412yes—
72Q8×C101—21no—
73Q8×C101—21no—
74D4×C101—21no—
75D4×C101—21no—
76D4×C101—41no—
77D4×C101—41no—
78C2×C4×C101—24no—
79C2×C4×C101—22no—
80C2×C4×C101—21no—
81C2×C4×C101—41no—
82C2×C4×C101—41no—
83C2×C4×C101C22×C411yescharacteristic
84C2×C4×C101C2×D411yescharacteristic
85C2×C4×C101C2×D411yescharacteristic
86C2×C4×C101C2×D411yescharacteristic
87C2×C4×C101#4111yescharacteristic
88C2×C4×C101#4111yescharacteristic
89C22⋊D101—44no—
90C2×Dic101—24no—
91C2×Dic101—21no—
92C2×Dic101—22no—
93C2×Dic101—21no—
94C2×Dic101—22no—
95C2×Dic101C22⋊C412yes—
96C4×D101—216no—
97C4×D101—22no—
98C4×D101—22no—
99C4×D101—22no—
100C4×D101—22no—
101C8×C101—41no—
102#11717579—41no—
103C22×C101—22no—
104C22×C101—41no—
105C22×C101—42no—
106C22×C101C2×C22⋊C411yescharacteristic
107C22×C101#12011yescharacteristic
108C22×C101#12311yescharacteristic
109C2×D101—24no—
110C2×D101—24no—
111C2×D101—22no—
112C2×D101—22no—
113C2×D101#10414yescomplemented
114C4×C101—24no—
115C4×C101—41no—
116C4×C101C4×D411yescharacteristic
117C4×C101C4×D411yescharacteristic
118C4×C101#12111yescharacteristic
119C4×C101#12111yescharacteristic
120Dic101—24no—
121Dic101—22no—
122Dic101—42no—
123C2×C101—21no—
124C2×C101—21no—
125C2×C101—81no—
126C2×C101#38611yescharacteristic
127C2×C101C2×#10412yes—
128D101—24no—
129#1896—1011nomaximal, Sylow
130C101#189611yescharacteristic, complemented, Sylow
131#525—1011no—
132C22×C42—1011no—
133C2×#104—1011no—
134C2×#104D10111yescharacteristic, complemented
135#407—1011no—
136#339—1011no—
137#339—1011no—
138C2×#41C2×D10111yescharacteristic
139C23×C4—1011no—
140C23×C4—2021no—
141C2×M16—1011no—
142C2×M16C2×D10111yescharacteristic
143#123—2021no—
144C22⋊Q8—1012no—
145#121—1012no—
146#120—2021no—
147C4×D4—2022no—
148#117—1011no—
149C2×C42—1011no—
150C2×C42—2022no—
151C2×C42—2021no—
152C2×C42—2022no—
153C2×C42C2×D10111yescharacteristic
154#104—22no—
155#104—2022no—
156C22⋊C8—2022no—
157C24—1011no—
158#41—21no—
159#41—21no—
160#41—21no—
161#41—21no—
162C2×Q8C4×D10111yescharacteristic, complemented
163C2×D4—21no—
164C2×D4—2022no—
165C2×D4C4×D10111yescharacteristic, complemented
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—1011no—
169C22×C4—1012no—
170C22×C4—1012no—
171C22×C4—2024no—
172C22×C4—2028no—
173C22×C4—2021no—
174C22×C4—2021no—
175C22×C4—2021no—
176C22×C4—2021no—
177C22×C4—2021no—
178C22×C4C22×D10111yescharacteristic
179M16—21no—
180M16—21no—
181M16—2021no—
182M16—2021no—
183C2×C8—21no—
184C2×C8—2021no—
185C4⋊C4—1012no—
186C4⋊C4—2022no—
187C22⋊C4—2022no—
188C22⋊C4—2022no—
189C22⋊C4—2021no—
190C42—21no—
191C42—1012no—
192C42—2024no—
193C42—2021no—
194C42—2021no—
195C42C22⋊D10112yescomplemented
196C23—21no—
197C23—1012no—
198C23—2024no—
199C23—2021no—
200C23—2021no—
201C23D4×D10111yescharacteristic
202Q8—21no—
203Q8—21no—
204D4—21no—
205D4—21no—
206D4—41no—
207D4—41no—
208D4—4044no—
209C2×C4—24no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—1012no—
215C2×C4—20216no—
216C2×C4—2022no—
217C2×C4—2024no—
218C2×C4—2022no—
219C2×C4—2021no—
220C2×C4—2021no—
221C2×C4—2022no—
222C2×C4—2022no—
223C2×C4—2022no—
224C2×C4—2022no—
225C2×C4C2×C4×D10111yescharacteristic
226C2×C4D4×D10111yescharacteristic
227C2×C4D4×D10111yescharacteristic
228C2×C4C2×C22⋊D10111yescharacteristic
229C2×C4#1310807311yescharacteristic
230C2×C4#1310807411yescharacteristic
231C8—41no—
232C8—4041no—
233C22—22no—
234C22—41no—
235C22—42no—
236C22—1014no—
237C22—2024no—
238C22—2024no—
239C22—2022no—
240C22—2022no—
241C22C22⋊C4×D10111yescharacteristic
242C22#1569222311yescharacteristic
243C22#1569223711yescharacteristic
244C4—24no—
245C4—41no—
246C4—4042no—
247C4—2024no—
248C4—2022no—
249C4#1569221511yescharacteristic
250C4C4×C22⋊D10111yescharacteristic
251C4#1569222611yescharacteristic
252C4#1569222811yescharacteristic
253C2—21no—
254C2—21no—
255C2—81no—
256C2—2024no—
257C2#2026505311yescharacteristic
258C2#104×D10112yes—
subgroupG/H
centerC2×C4D4×D101
commutatorC2×C4×C101C22×C4
FrattiniC22×C4C22×D101
FittingC2×#104×C101C2
socleC22×C101C2×C22⋊C4
prime radical#20265684C2
cosocleC22×C4×C101C23
hypercenterC2×#104D101
hypocenterC101#1896
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. C2×C42×D101C2semidirect
3. C2×#15692182C2semidirect
4. #20265140C2semidirect
5. #20264938C2semidirect
6. #20264937C2semidirect
7. C2×#104×C101C2semidirect
12. #15692231C4semidirect
13. #15692226C4semidirect
42. C4⋊Dic101C2×C4semidirect
50. C4×Dic101D4semidirect
55. C2×Q8×C101C2×C4semidirect
57. C2×D4×C101C2×C4semidirect
65. C42×C101D4semidirect
113. C2×D101#104semidirect
130. C101#1896semidirect
134. C2×#104D101semidirect
162. C2×Q8C4×D101semidirect
165. C2×D4C4×D101semidirect
195. C42C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
#202656842
C2×C42×D1012
C2×#156921822
#202651402
#202649382
#202649372
C2×#104×C1012
#1896101