Finite Group Databasebeta

← #27170082

Subgroups — #27170082

Order 12928.

308 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 51 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#20265735C211yesmaximal, characteristic
2#20265316C211yesmaximal, characteristic, complemented
3#20265088C211yesmaximal, characteristic, complemented
4#20264912C211yesmaximal, characteristic, complemented
5#20264911C211yesmaximal, characteristic, complemented
6#20264900C211yesmaximal, characteristic, complemented
7#448×C101C211yesmaximal, characteristic, complemented
8C2×D4×D101—21no—
9C2×#13108061—21no—
10#15692240—21no—
11#15692237—21no—
12#15692229C2211yescharacteristic
13#15692227—21no—
14#15692225—21no—
15#15692223—21no—
16C2×#13108051—21no—
17C2×#13108051C2211yescharacteristic
18#15692179—21no—
19C22⋊#11717579C2211yescharacteristic, complemented
20#15692162—21no—
21#15692162C2211yescharacteristic, complemented
22C22×Q8×C101C2211yescharacteristic, complemented
23C2×SD16×C101—21no—
24#121×C101C2211yescharacteristic
25#103×C101—21no—
26C22⋊C8×C101C2211yescharacteristic, complemented
27C23×D101—21no—
28C23×D101D411yescharacteristic
29C2×D404—21no—
30C2×D404D411yescharacteristic
31C2×C22⋊D101—21no—
32C2×C22⋊D101—21no—
33C2×C22⋊D101—21no—
34D4×D101—42no—
35D4×D101—42no—
36C22×Dic101—21no—
37C22×Dic101D411yescharacteristic
38C2×C4×D101—21no—
39#13108061—42no—
40C2×#11717579—21no—
41C4⋊Dic101D411yescharacteristic
42#13108053—21no—
43C22⋊Dic101—21no—
44#13108051—21no—
45#13108051—21no—
46#13108051—21no—
47#13108051—21no—
48#13108051—42no—
49C2×Q8×C101—21no—
50C2×Q8×C101—21no—
51C2×Q8×C101—42no—
52C2×D4×C101—21no—
53C2×D4×C101D411yescharacteristic
54C22×C4×C101—21no—
55C22×C4×C101C2311yescharacteristic
56SD16×C101—42no—
57C2×C8×C101—21no—
58C4⋊C4×C101D411yescharacteristic
59C22⋊C4×C101—21no—
60C23×C101—21no—
61C23×C101C2×D411yescharacteristic
62C22×D101—21no—
63C22×D101—21no—
64C22×D101—21no—
65C22×D101—42no—
66C22×D101—41no—
67C22×D101—41no—
68C22×D101—81no—
69C22×D101—81no—
70C22×D101SD1612yescomplemented
71Q8×C101—22no—
72Q8×C101—22no—
73Q8×C101—41no—
74Q8×C101—41no—
75D4×C101—41no—
76D4×C101—41no—
77D4×C101—41no—
78C2×C4×C101—22no—
79C2×C4×C101—21no—
80C2×C4×C101—41no—
81C2×C4×C101—41no—
82C2×C4×C101—41no—
83C2×C4×C101C2×D411yescharacteristic
84C2×C4×C101C2×D411yescharacteristic
85C22⋊D101—42no—
86C22⋊D101—41no—
87C22⋊D101—41no—
88C22⋊D101—81no—
89C2×Dic101—21no—
90C2×Dic101—21no—
91C2×Dic101—41no—
92C2×Dic101—41no—
93C2×Dic101—41no—
94C2×Dic101—41no—
95D404—41no—
96D404—42no—
97C4×D101—42no—
98C8×C101—41no—
99#11717579—41no—
100C22×C101—21no—
101C22×C101—21no—
102C22×C101—41no—
103C22×C101—41no—
104C22×C101—41no—
105C22×C101#12011yescharacteristic
106C22×C101C2×SD1611yescharacteristic
107C22×C101AGL(1,ℤ/8)11yescharacteristic
108C2×D101—22no—
109C2×D101—41no—
110C2×D101—41no—
111C2×D101—42no—
112C2×D101—41no—
113C2×D101—41no—
114C2×D101—41no—
115C2×D101—81no—
116C2×D101—84no—
117C4×C101—21no—
118C4×C101—21no—
119C4×C101—41no—
120C4×C101—42no—
121Dic101—41no—
122Dic101—41no—
123Dic101—81no—
124C2×C101—21no—
125C2×C101—21no—
126C2×C101—81no—
127C2×C101#45011yescharacteristic
128C2×C101#45511yescharacteristic
129C2×C101B(4,𝔽2)11yescharacteristic
130D101—42no—
131D101—81no—
132#1601—1011nomaximal, Sylow
133C101#160111yescharacteristic, complemented, Sylow
134#534—1011no—
135#448—1011no—
136#448D10111yescharacteristic, complemented
137#393—1011no—
138#327—1011no—
139#327—1011no—
140#323—1011no—
141C22×Q8C2×D10111yescharacteristic, complemented
142C22×D4—2021no—
143C2×SD16—21no—
144C2×SD16—2021no—
145#123—2021no—
146#123—2021no—
147#121—1011no—
148#121—2021no—
149#121C2×D10111yescharacteristic
150#120—2021no—
151#120—2021no—
152C2×C22⋊C4—1011no—
153C2×C22⋊C4—2021no—
154#103—21no—
155#103—2021no—
156C22⋊C8—1011no—
157C22⋊C8C2×D10111yescharacteristic, complemented
158#95—1011no—
159#95—2021no—
160C24—1011no—
161C24—2021no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8—42no—
165C2×D4—21no—
166C2×D4—1011no—
167C2×D4—4042no—
168C2×D4—4042no—
169C2×D4—2021no—
170C2×D4—2021no—
171C2×D4—2021no—
172C2×D4—2021no—
173C2×D4C22⋊D10111yescharacteristic
174C22×C4—21no—
175C22×C4—1011no—
176C22×C4—2021no—
177C22×C4—2021no—
178C22×C4C22×D10111yescharacteristic
179SD16—42no—
180SD16—4042no—
181C2×C8—21no—
182C2×C8—2021no—
183C4⋊C4—1011no—
184C4⋊C4—2021no—
185C4⋊C4C22⋊D10111yescharacteristic
186C22⋊C4—21no—
187C22⋊C4—4042no—
188C22⋊C4—2021no—
189C22⋊C4—2021no—
190C22⋊C4—2021no—
191C22⋊C4—2021no—
192C22⋊C4—2021no—
193C23—21no—
194C23—8081no—
195C23—8081no—
196C23—1012no—
197C23—4041no—
198C23—4041no—
199C23—4042no—
200C23—2021no—
201C23—2021no—
202C23—2021no—
203C23D4×D10111yescharacteristic
204Q8—22no—
205Q8—22no—
206Q8—41no—
207Q8—41no—
208D4—41no—
209D4—41no—
210D4—41no—
211D4—8081no—
212D4—4041no—
213D4—4042no—
214D4—4042no—
215D4—4041no—
216D4—4041no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—4042no—
223C2×C4—4041no—
224C2×C4—4041no—
225C2×C4—4041no—
226C2×C4—4041no—
227C2×C4—2021no—
228C2×C4—2021no—
229C2×C4D4×D10111yescharacteristic
230C2×C4C2×C22⋊D10111yescharacteristic
231C8—41no—
232C8—4041no—
233C22—21no—
234C22—21no—
235C22—41no—
236C22—41no—
237C22—41no—
238C22—8081no—
239C22—8084no—
240C22—4042no—
241C22—4041no—
242C22—4041no—
243C22—4041no—
244C22—4041no—
245C22—4041no—
246C22—2022no—
247C22#1569222311yescharacteristic
248C22SD16×D10111yescharacteristic
249C22#1569228211yescharacteristic
250C4—21no—
251C4—21no—
252C4—41no—
253C4—42no—
254C4—8081no—
255C4—4041no—
256C4—4041no—
257C2—21no—
258C2—21no—
259C2—81no—
260C2—8081no—
261C2—4042no—
262C2#2026532611yescharacteristic
263C2#2026535311yescharacteristic
264C2#2026536311yescharacteristic
subgroupG/H
centerC22#15692223
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting#448×C101C2
socleC22×C101#120
prime radical#20265735C2
hypercenter#448D101
hypocenterC101#1601
LCSG ▷ C22×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
2. #20265316C2semidirect
3. #20265088C2semidirect
4. #20264912C2semidirect
5. #20264911C2semidirect
6. #20264900C2semidirect
7. #448×C101C2semidirect
19. C22⋊#11717579C22semidirect
21. #15692162C22semidirect
22. C22×Q8×C101C22semidirect
26. C22⋊C8×C101C22semidirect
70. C22×D101SD16semidirect
133. C101#1601semidirect
136. #448D101semidirect
141. C22×Q8C2×D101semidirect
157. C22⋊C8C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202657352
#202653162
#202650882
#202649122
#202649112
#202649002
#448×C1012
#1601101