Finite Group Databasebeta

← #27170153

Subgroups — #27170153

Order 12928.

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

normal subgroups only

This group has 288 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#20265496C211yesmaximal, characteristic, complemented
3#20265493C211yesmaximal, characteristic, complemented
4#20265114C211yesmaximal, characteristic, complemented
5#20264910C211yesmaximal, characteristic, complemented
6#20264900C211yesmaximal, characteristic, complemented
7#327×C101C211yesmaximal, characteristic, complemented
8C2×D4×D101—21no—
9#15692240—21no—
10#15692237—21no—
11#15692229C2211yescharacteristic
12#15692227—21no—
13#15692225—21no—
14#15692223—21no—
15C2×C4⋊Dic101—21no—
16C2×C4⋊Dic101C2211yescharacteristic, complemented
17C2×#13108051—21no—
18C2×#13108051C2211yescharacteristic
19#15692187—21no—
20#15692186—21no—
21#15692177—21no—
22#15692176—21no—
23C22⋊#11717579C2211yescharacteristic, complemented
24#121×C101C2211yescharacteristic
25C22⋊C8×C101C2211yescharacteristic, complemented
26#95×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—
39C2×#11717579—21no—
40C4⋊Dic101—21no—
41C4⋊Dic101—21no—
42C4⋊Dic101—42no—
43C4⋊Dic101D411yescharacteristic
44#13108053—21no—
45C22⋊Dic101—21no—
46#13108051—21no—
47#13108051—21no—
48#13108051—21no—
49#13108051—21no—
50#13108051—42no—
51C2×D4×C101—21no—
52C2×D4×C101D411yescharacteristic
53C22×C4×C101—21no—
54C22×C4×C101C2311yescharacteristic
55C2×C8×C101—21no—
56C4⋊C4×C101D411yescharacteristic
57C22⋊C4×C101—21no—
58C23×C101—21no—
59C23×C101C2×D411yescharacteristic
60C22×D101—21no—
61C22×D101—21no—
62C22×D101—21no—
63C22×D101—42no—
64C22×D101—41no—
65C22×D101—41no—
66C22×D101—81no—
67C22×D101—81no—
68C22×D101SD1612yes—
69D4×C101—41no—
70D4×C101—41no—
71D4×C101—41no—
72C2×C4×C101—22no—
73C2×C4×C101—21no—
74C2×C4×C101—41no—
75C2×C4×C101—41no—
76C2×C4×C101—41no—
77C2×C4×C101C2×D411yescharacteristic
78C2×C4×C101C2×D411yescharacteristic
79C22⋊D101—42no—
80C22⋊D101—41no—
81C22⋊D101—41no—
82C22⋊D101—81no—
83C2×Dic101—21no—
84C2×Dic101—21no—
85C2×Dic101—41no—
86C2×Dic101—41no—
87C2×Dic101—41no—
88C2×Dic101—41no—
89D404—41no—
90D404—42no—
91C4×D101—42no—
92C8×C101—41no—
93#11717579—41no—
94C22×C101—21no—
95C22×C101—21no—
96C22×C101—41no—
97C22×C101—41no—
98C22×C101—41no—
99C22×C101#12011yescharacteristic
100C22×C101C2×SD1611yescharacteristic
101C22×C101AGL(1,ℤ/8)11yescharacteristic
102C2×D101—22no—
103C2×D101—41no—
104C2×D101—41no—
105C2×D101—42no—
106C2×D101—41no—
107C2×D101—41no—
108C2×D101—41no—
109C2×D101—81no—
110C2×D101—84no—
111C4×C101—21no—
112C4×C101—21no—
113C4×C101—42no—
114C4×C101—41no—
115Dic101—41no—
116Dic101—41no—
117Dic101—81no—
118C2×C101—21no—
119C2×C101—21no—
120C2×C101—81no—
121C2×C101#45011yescharacteristic
122C2×C101#45311yescharacteristic
123C2×C101#45811yescharacteristic
124D101—42no—
125D101—81no—
126#1613—1011nomaximal, Sylow
127C101#161311yescharacteristic, complemented, Sylow
128#534—1011no—
129#481—1011no—
130#481—1011no—
131#399—1011no—
132#327—1011no—
133#327D10111yescharacteristic, complemented
134#323—1011no—
135C22×D4—2021no—
136#123—2021no—
137#123—2021no—
138#121—1011no—
139#121—2021no—
140#121C2×D10111yescharacteristic
141#120—2021no—
142#120—2021no—
143C2×C4⋊C4—1011no—
144C2×C4⋊C4—2021no—
145C2×C22⋊C4—1011no—
146C2×C22⋊C4—2021no—
147#106—2021no—
148#106—2021no—
149#102—2021no—
150#102—2021no—
151C22⋊C8—1011no—
152C22⋊C8C2×D10111yescharacteristic, complemented
153#95C2×D10111yescharacteristic, complemented
154C24—1011no—
155C24—2021no—
156C2×D4—21no—
157C2×D4—1011no—
158C2×D4—4042no—
159C2×D4—4042no—
160C2×D4—2021no—
161C2×D4—2021no—
162C2×D4—2021no—
163C2×D4—2021no—
164C2×D4D40411yescharacteristic
165C22×C4—21no—
166C22×C4—1011no—
167C22×C4—2021no—
168C22×C4—2021no—
169C22×C4C22×D10111yescharacteristic
170C2×C8—21no—
171C2×C8—2021no—
172C4⋊C4—1011no—
173C4⋊C4—4042no—
174C4⋊C4—2021no—
175C4⋊C4—2021no—
176C4⋊C4—2021no—
177C4⋊C4D40411yescharacteristic
178C22⋊C4—21no—
179C22⋊C4—4042no—
180C22⋊C4—2021no—
181C22⋊C4—2021no—
182C22⋊C4—2021no—
183C22⋊C4—2021no—
184C22⋊C4—2021no—
185C23—21no—
186C23—8081no—
187C23—8081no—
188C23—1012no—
189C23—4041no—
190C23—4041no—
191C23—4042no—
192C23—2021no—
193C23—2021no—
194C23—2021no—
195C23D4×D10111yescharacteristic
196D4—41no—
197D4—41no—
198D4—41no—
199D4—8081no—
200D4—4041no—
201D4—4042no—
202D4—4042no—
203D4—4041no—
204D4—4041no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—4042no—
211C2×C4—4041no—
212C2×C4—4041no—
213C2×C4—4041no—
214C2×C4—4041no—
215C2×C4—2021no—
216C2×C4—2021no—
217C2×C4D4×D10111yescharacteristic
218C2×C4C2×D40411yescharacteristic
219C8—41no—
220C8—4041no—
221C22—21no—
222C22—21no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—8081no—
227C22—8084no—
228C22—4042no—
229C22—4041no—
230C22—4041no—
231C22—4041no—
232C22—4041no—
233C22—4041no—
234C22—2022no—
235C22#1569222511yescharacteristic
236C22SD16×D10111yescharacteristic
237C22#1569227911yescharacteristic
238C4—21no—
239C4—21no—
240C4—42no—
241C4—41no—
242C4—8081no—
243C4—4041no—
244C4—4041no—
245C2—21no—
246C2—21no—
247C2—81no—
248C2—8081no—
249C2—4042no—
250C2#2026532911yescharacteristic
251C2#2026534611yescharacteristic
252C2#2026536811yescharacteristic
subgroupG/H
centerC22#15692225
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting#327×C101C2
socleC22×C101#120
prime radical#20265735C2
hypercenter#327D101
hypocenterC101#1613
LCSG ▷ C22×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #20265496C2semidirect
3. #20265493C2semidirect
4. #20265114C2semidirect
5. #20264910C2semidirect
6. #20264900C2semidirect
7. #327×C101C2semidirect
16. C2×C4⋊Dic101C22semidirect
23. C22⋊#11717579C22semidirect
25. C22⋊C8×C101C22semidirect
26. #95×C101C22semidirect
127. C101#1613semidirect
133. #327D101semidirect
152. C22⋊C8C2×D101semidirect
153. #95C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202657352
#202654962
#202654932
#202651142
#202649102
#202649002
#327×C1012
#1613101